The impact of raindrops on Salvinia molesta leaves: effects of trichomes and elasticity

1. The trichome cover of Salvinia

Leaves of Salvinia species, a genus of floating water ferns, are superhydrophobic due to large and highly water-repellent leaf hairs (=trichomes) covering the upper leaf surface (figure 1, [1,2]). When pushed under water, a persistent air layer develops between the trichomes and is held for many days [1]. This can be explained by a combination of factors, particularly the flexibility of the trichomes, their geometry and their superhydrophobicity. Trichome complexity differs between Salvinia species. In some species, such as S. molesta, the trichomes are carried as groups of four by emergences and are arranged similar to an ‘egg-beater’: the four trichomes are fused at their tips which are topped by pads of dead cells (figure 1) [3]. Whereas all cell walls of the trichome are covered with water-repellent wax nano crystals, the tip pad shows no wax and is hydrophilic. This ‘egg-beater’ trichome is particularly in the focus of research, because the interface between air and water was shown to be pinned by the hydrophilic tip [1] and the enclosed air acts as ‘pneumatic spring’ when subjected to pressure fluctuations [4].

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Whereas the technical potential of these fascinating surface effects is substantial, including drag reduction for ships or antifouling technology, it is not clear whether and how a selective advantage is provided to the plant. It is quite reasonable that superhydrophobicity of the upper leaf surface of Salvinia benefits photosynthesis, because this side of the Salvinia leaf is equipped with stomata (figure 1). A water film covering the leaf surface would considerably hamper diffusion of CO₂, thereby reducing photosynthesis. The presence, however, of such a large and complex trichome layer represents a high investment in biomass and metabolic energy. The question thus arises about the benefit of such a costly structure which maintains an air layer for days when immersed. It has been suggested that the entrapped air volume could serve as a buoyancy aide, assisting immersed plants to swiftly return back to the water surface (LM Harper 1986, unpublished report) [5]. Other floating plants, however, manage without such a structure, and, moreover, the leaves of Salvinia show—as common among aquatic plant species—extensive intercellular spaces aiding buoyancy [6].

Salvinia molesta has its original habitat in Brazil [7], within a humid tropical region which is subjected to frequent and heavy rainfall [8]. Rain hitting the surface of S. molesta is relevant for leaf function, because stomata are exposed to the drop impact and leaf gas exchange is reduced if water covers the stomatal pores. Collisions with rain drops may also lead to wax erosion and therefore to loss of superhydrophobicity [9]. It can be expected that this process will be promoted by the high frequency of rainfall events. Rain drops can attain diameters of several millimetres and reach maximum velocities of about 9–10 m s⁻¹ [10,11]. A cover of pillars within the submillimetre range, as is the case for the S. molesta trichomes, has a marked influence on surface wettability [12] and drop impact [13]. For instance, splashing and/or drop spreading may be prevented or substantially reduced on a hairy surface [13]. Furthermore, the trichome structure is complex, not only with respect to the eggbeater but also because the entire surface is covered by superhydrophobic wax crystals [1].

Numerous scenarios are possible when a drop collides with a solid, depending on the characteristics of the liquid (density, viscosity), the drop (size and speed) and the surface (structure, wettability, structure and elasticity), which is a topic of intense research [14–20] including drop impact on leaf surfaces [21–23]. In the present study, the collision of water drops with the surface of S. molesta is investigated, with special attention to the effect of the trichomes. In order to characterize the process of drop impact on S. molesta, studies with a high-speed camera were conducted. Leaves of S. molesta were exposed to drop impact events of different kinetic energy, by increasing the height of the drop fall, and also the size of the drop. These experimental settings include conditions similar to natural rain to be analysed.

2. Material and methods

2.2. Experimental set-up

Drop impact from different heights on the upper leaf surface of S. molesta was recorded with a high-speed camera (figure 2). Two different camera types were used: (1) FASTCAM SA3 Model 120 K-M2 and (2) FASTCAM Mini AX 100 type 540K-C-4GB. The FASTCAM SA3 Model 120 K-M2 was used with a LEICA optical microscope MZ7.5, whereas a Nikon camera objective, an AF-S Micro NIKKOR 40 mm, was installed with the FASTCAM Mini AX 100 type 540K-C-4. These different optical utensils differed with respect to zoom capabilities, levels of required illumination and working distance. Furthermore, the FASTCAM SA3 Model 120 K-M2 provides greyscale videos. The experiments were carried out at the Department of Geosciences of the University of Tübingen.

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To produce falling drops, a pipette attached to a laboratory stand was used (figure 2). Considered falling heights H were: H = 0.1 m, 0.3 m, 0.5 m and 4.3 m. Since the present study focuses on the impact of rain drops, drop sizes representing larger rain drops were used. Rain drops of more than 2.5 mm radius are not uncommon [8], and even much larger rain drops with more than 4 mm radius occur in tropical and subtropical regions [10]. For the heights of H = 0.1 m … 0.5 m, a pipette was used which produced drops with an average radius of R = 2.2 mm ± 0.11 mm. To account for extreme raindrop impact events, meaning drops larger than with a 2.5 mm radius and reaching the maximum velocity of rain drops (about 9 m s⁻¹), an experimental set-up allowing for a falling height of H = 4.3 m and a drop radius of about 4.5 mm was devised. The height of 4.3 m was achieved by fastening the laboratory set-up to the railing of a staircase at the Geoscience department at the second floor level (figure 2). The leaf samples were placed on a table in the central staircase below the lab stand at the level of the first floor. To obtain larger drops with 4.5 mm radius, a pipette with a cut tip was used.

Five different combinations of falling height and drop size were considered, termed as ‘set-up 1’ to ‘set-up 5’ in the following. All experimental set-ups are summarized in table 1, together with their characteristic parameters. In all set-ups, tap water was used and the experiments were performed under room temperature (about 20°C). Except for set-up 4, for the drop impact experiments, the leaves were placed in a Petri dish on a thin layer of water. The leaf arrangements in the Petri dish were varied, similar to natural arrangements of floating S. molesta leaves in their habitat. In set-up 4, the leaf was fixed to an inflexible base plate.

Table 1. Quantities and numerical values characterizing set-up 1 to 5.

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	symbol (unit)	set-up 1	set-up 2	set-up 3, 4a	set-up 5
drop data
falling height	H (m)	0.1	0.3	0.5	4.30
drop radius	R (mm)	2.2	2.2	2.2	4.5
terminal velocity	v (m s−1)	1.40	2.42	3.13	9.19
pressures
dynamic pressure	pdyn (kPa)	0.98	2.9	4.9	42.2
water hammer pressureb	pwh (kPa)	415	718	927	2720
atmospheric pressure	patm (kPa)	100
time scales
crashing time	tcr (ms)	3.14	1.82	1.40	0.98
contact timec	tco (ms)	12	9	14 (52)	24
energies
kinetic energy of drop at impact	Ekin (μJ)	44	130	218	16 119
elastic energy of hairs hit by dropd	Ehair (μJ)	224	224	224	936
elastic energy of leaf base hit by drope	Eleaf (μJ)	0.41	3.6	10.1	12 789
wetting energyf	Ewet (μJ)	2.23	2.23	2.23	14.22
viscous energy loss	Evisc (μJ)	0.014	0.024	0.031	0.039
formation energy of a secondary dropletg	Esurf (μJ	3.6
kinetic energy of a secondary dropleth	Ejump (μJ)	3.3
rotational energy of a secondary dropleti	Erot (μJ)	33
energy of a hovering secondary dropletj	Ehov (μJ)	5.26

^aIn set-up 3, the leaf was placed on a thin layer of water, whereas in set-up 4 it was fixed to an inflexible base plate.

^bCalculated by using f = 0.2 and speed of sound in water c = 1480 m s⁻¹.

^cThe value in parentheses is related to set-up 4 where the leaf was fixed to the base plate, in contrast to set-up 3 where it floated on a thin layer of water.

^dAccording to equation (3.5) and assuming k = 4.6 N m⁻¹, $h=(\frac{2}{3})L=2000\hspace{0.17em}\mu \mathrm{m}$, α = 1.6 mm⁻².

^eOther leaf properties: Young’s modulus E = 10 MPa, Poisson’s number η = 0.27, thickness d = 1 mm, radius a = 16 mm. Notice that the structures of equations (3.3), (3.6) and (A.7) imply that $E_{\mathrm{el}}^{\mathrm{base}}$ is proportional to v⁴.

^fAccording to (A.15) of the electronic supplementary material, r_h = 35.5 μm, θ = 161°, L = 3 mm.

^gAccording to equation (3.7), assuming a droplet radius r = 2 mm.

^hAccording to equation (3.8), assuming a droplet radius r = 2 mm and a vertical ‘jumping height’ of h = 10 mm.

ⁱAssuming a droplet radius r = 1 mm and a time t_rot = 1 ms for one revolution.

^jAccording to equation (3.13), R_d = 2 mm.

For each set-up, a number of trials recorded by the described high-speed equipment were conducted. To allow for observing different details and situations, magnification (and therefore optical equipment) and working distance was varied, as well as time resolution (in frames per second = fps). The various parameters were partially interdependent from each other, or made certain compromises necessary. For example, higher fps required a stronger illumination, and particularly so in case of the stereo microscope. Therefore—because extreme illumination damaged the leaf samples—recordings in which the stereo microscope was used were limited with respect to fps. In the case of set-up 5, with 4.3 m falling height, the drop splash was so extreme that a higher working distance was necessary. For the series of experiments recorded with the FASTCAM SA3 Model 120 K-M2 and the optical microscope, the leaves were fixed to the base plate to allow for observing the drop impact with high magnification. This arrangement had the additional effect of reducing the elastic response of the leaves upon impact.

3. Results: drop impact behaviour and analysis of energy conversion

3.1. Observations for the different set-ups

3.1.1. Set-up 1

After impact, the drop sinks into the trichome stand (figure 3, electronic supplementary material, videos S1–S2). In most cases, a few satellite droplets are separated at the periphery of the flattening drop, although in some cases, there are no satellite droplets at all. Elastic deformation particularly of the trichomes can be observed, and also moderate elastic deformation of the leaf (see electronic supplementary material, videos S1–S2). After sinking into the trichome cover, the drop bounces off with a pancake-like shape (‘pancake-lifting’, [19]), and rolls off the leaf surface. During impact, kinetic energy is converted into surface energy (the drop starts to flatten) and elastic energy (the leaf is deformed). During the subsequent swinging back of the deformed leaf portion elastic energy is converted into surface energy (the drop adopts pancake shape) and kinetic energy (the ‘pancake’ bounces up from the elastic ‘springboard’) [19]. The flexible trichomes promote the elastic processes. As is consistent with the general descriptions of pancake-lifting, the peripheral part of the ‘pancake’ is separated from the surface by an air layer (figure 3). There is no residual water left on the surface. For video material, see electronic supplementary material, videos S1–S2.

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3.1.2. Set-up 2

The process of drop impact and bouncing-off is similar to set-up 1 (figure 4, electronic supplementary material, video S3–S4). Also, the lift-off of the water still largely occurs as ‘pancake-bouncing’. According to the videos, residual water remained in 25% of the cases left within the tricome cover, obviously trapped by the ‘egg-beaters’.

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3.1.3. Set-up 3

The elastic deformation of the leaf is much more pronounced when compared to set-up 1 and set-up 2 (figure 5; electronic supplementary material, videos S5–S6). The lifting ‘pancake’ is largely fragmented into satellite droplets, and the bouncing-off process may be described as a mixture between ‘splashing’ and ‘pancake-lifting’ (figure 5). Elastic deformation is distinctly visible. According to the videos, residual water remained in 42.6% of the cases within the trichome layer. For video material, see electronic supplementary material, videos S5–S6.

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3.1.4. Set-up 4

This set-up is nearly identical to Set-up 3, the only difference is the fixation of the leaf to the base plate. Consequently, elastic deformation of the leaf base, and therefore ‘spring-boarding’ is minimal. Therefore, the impact energy is rather converted into surface forming and into kinetic energy of secondary droplets (visible as ‘splashing’) than into elastic energy. Since ‘spring-boarding’ is virtually absent in set-up 4, pancake-lifting was not observed and the largest part of the drop appears to be ‘swallowed’ by the trichome cover upon impact (figure 7; electronic supplementary material, videos S7–S8). Also, the absence of the spring-boarding effect has a considerable effect on the time required for the water to recede from the surface. The parameter t_co describes the ‘contact time’ of a bouncing drop, meaning the time span between first contact with the surface (the trichomes in this case) and leaving the surface. If residual water remains on the surface, the end of the contact time was defined as the time when the ‘bouncing process’ is finished and the water movements have come to an end. As shown by figure 6, t_co is much higher for set-up 4 when compared to set-up 1–3.

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The most distinct feature, however, of set-up 4 is the quite slow rising of water from below the egg-beaters (figure 7; electronic supplementary material, videos S7–S8). This slow process, termed ‘emerging process’ (EP) throughout the rest of the text, is characteristic for set-up 4, as it was observed in almost all cases. Often, a portion of a water body already residing above the egg-beaters was connected to the rest of the water patch and appeared to ‘suck’ the residual water upwards through a more or less thin ‘water bridge’ (figure 8). This can be explained by the energetically favourable state of a water drop above the trichome cover as compared to a water patch surrounded by hydrophobic trichomes: Once some of the water has reached the top, the residual water is driven upwards by surface forces. A prerequisite for this mechanism is a connection between both water portions. The details showed a high variation, such as number and position of water portions being attached to the upper part of the egg-beaters, size of the residual water body being trapped within the trichome stand, the number of connections between residual water to drops on the trichome surface, and the density and distribution of trichomes.

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In addition, the draining of the residual water by the EP may be incomplete, leaving residual water trapped between the trichomes. Obviously, some water patches did not succeed in bringing a portion of water above the trichomes to drive the rest upwards. In fact, this happens quite often, in about 20% of the observed cases. Drops rolling over the trichome surface can, however, make contact with residual water. Then, the subsequent coalescence process drives water upwards into the ‘upper’ drop (figure 8). For video material, see electronic supplementary material, videos S7–S8.

3.1.5. Set-up 5

The drop hits the leaves with extreme splashing and the drop bursts into a large number of small satellite droplets bouncing off the surface with high velocity (figure 9; electronic supplementary material, video S9). This is consistent with observations of a spreading water film fragmenting strongly when originating from a drop impact with high speed [22]. Despite this strong impact event, however, residual water remained within the trichome stand in all cases. Also, the EP still occurs after the impact event as shown by figure 9 and electronic supplementary material, video S9. A number of drops attached to the egg-beaters grow by drawing residual water from the trichome cover during the EP. In contrast to set-up 4, however, t_co is much lower and the EP occurs faster (figure 6), and the emerged drops are occasionally able to bounce off with low speed. For video material, see electronic supplementary material, video S9.

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3.2. Where does the kinetic energy of the impact go?

The kinetic energy of the drop impact calculates from the drop’s mass m and impact velocity v as

$$E_{\mathrm{kin}}=\frac{1}{2}m{v}^2=\frac{2\pi }{3}\rho R^3{v}^2.$$3.1

In the second version, mass has been replaced by mass density ρ and drop radius R. The energy transfer of E_kin to the plant tissue during impact reflects the process of reducing the drop velocity v to zero: the lowermost molecules, having already made contact with the leaf, have lost their momentum completely, whereas the water molecules located at the upper end of the drop are still moving. The interface separating the two regions travels with velocity v through the impacting drop, implying that for the drops in set-ups 1–5 the ‘crashing time’

$$t_{\mathrm{cr}}\approx \frac{2R}{v},$$3.2

is in the milliseconds range [24,25]. The impact area is during this time exposed to the pressure

$$p_{\mathrm{dyn}}=\frac{\rho v^2}{2}.$$3.3

Mitigation of the impact pressure can be attained either by prolonging the duration of the energy transfer or by expanding the area of interaction between impacting drop and leaf. The situation is similar to the effect of an airbag during a car crash: the bag protects the passengers from severe injuries by providing a large contact area with the persons and by extending the deceleration period via a slow deflation of the bag. Both effects reduce the kinetic energy input per body volume and time which is the crucial quantity for structural damage. The videos of the set-ups described above indicate that upon drop impact, a catena of processes is initiated which dissipates the impact energy both spatially and temporally.

Experimental quantification of all relevant parameters and processes is, however, extremely difficult, if not impossible, due to the rapid sequence of events and the complexity of the involved biological structures. Instead, the question will be answered to where the impact energy goes by estimating the energy conversions involved in the successive phases identified from the videos. In this way, it will be attempted to provide indirect evidence for the dissipation catena.

There are two relevant time scales on which the impact interactions occur. The first, the ‘crashing time’ t_cr (cf. equation (3.2)), is defined in terms of the life span of the drop after impact. Once this period is over and the original form of the drop has been transformed into other shapes by the impact forces, the water originating from the original drop is still in a state of agitation as can be seen in the videos of set-up 1–5. Various numbers of secondary droplets of different sizes form and are, for instance, being ejected from the water mass on the leaf, some do fall back onto the leaf, while others hover for some time above the leaf hairs. These movements persist for tens of milliseconds, and t_co as derived from the video observations are in the range of t_co ≈ 9 ms … 52 ms (figure 6).

Figure 6 shows that set-up 1–3 have very similar t_co, with set-up 2 showing the lowest value. Maximum t_co is reached with set-up 4. Comparison of the videos led to grouping the energy conversion taking place during impact into three consecutive phases: firstly, buffering of impact energy, secondly, removal of impact energy from the leaf, and, thirdly, final processes which include the EP. These three phases will be described in the following.

3.2.1. First phase: buffering of impact energy

3.2.1.1. Elasticity of the hair cover

The hair cover of the Salvinia species features a collective elasticity whenever an object (such as a drop) which is large enough to cover several hairs is pressed towards the leaf base. Ditsche et al. [5] found that the hair cover behaves according to Hooke’s Law: if it is pushed towards the leaf base by the distance h, the elasticity energy stored in one hair amounts to

$$E=\frac{1}{2}k{h}^2,$$3.4

where the constant k characterises the elasticity of the hair cover as a whole ([5] term k the stiffness). A drop of radius R hits απR² hairs (α denotes the density of the hairs per unit surface area). The elastic energy that is stored in these απR² hairs when they have been pushed towards the leaf base by the distance h amounts to

$$E_{\mathrm{hair}}=\frac{1}{2}k{h}^2\alpha \pi R^2.$$3.5

Figure 10 illustrates this expression for set-up 1–5 (cf. table 1) and compares it with the impact energies in these set-ups. It suggests that for set-ups 1–4 the trichomes are able to absorb the impact energy completely, provided they can be pushed downwards by an impacting drop by the distances h = 884 μm for set-up 1, h = 1528 μm for set-up 2, and h = 1976 μm for set-up 3/4. In the case of set-up 5, however, the affected trichome group can absorb at most 13% of the impact energy.

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3.2.1.2. Elasticity of the leaf base

After penetrating the trichome cover of Salvinia, the drop hits and deforms the leaf base. Hereby, the drop loses kinetic energy that is converted into elastic energy and stored for a small time span in the deformed leaf. To roughly estimate the amount of converted energy, the Salvinia leaf is modelled in a simplified way as a circular, isotropic and elastic plate of radius a and thickness d ≪ a with elastic constants E (Young’s modulus) and η (Poisson’s number) (details are described in the electronic supplementary material). Standard procedures [26,27] lead to a lengthy expression for the deformation w(r) (electronic supplementary material, equation (A.7)) and to the following equation for the elastic energy content of the leaf at maximum deformation:

$$E_{\mathrm{leaf}}={\int }_V({\sigma }_{rr}{ϵ}_{rr}+{\sigma }_{\phi \phi }{ϵ}_{\phi \phi })\hspace{0.17em}\mathrm{d}V=\frac{\pi E{d}^3}{6(1-{\eta }^2)}{\int }_0^a[{(w^{″})}^2+2\eta \frac{w^{″}{w}^{\prime }}{r}+\frac{{(w^{\prime })}^2}{r^2}]r\hspace{0.17em}\mathrm{d}r$$3.6

The σ_ik and ε_ik denote stresses and strains, respectively.

The expression for E_leaf (equation (3.6), and similarly equation (A.7) of the electronic supplementary material for w(r)) depends on Young’s modulus E. Since specific values for S. molesta are not available in the literature, calculations were performed within the value range E = 5 MPa … 50 MPa as reported by Onoda et al. [28] and figure 11 displays the dependency of E_leaf on E for this interval. It should be noted that leaves represent composite constructions, consisting of different layers (cuticle, epidermis, mesophyll) and internal air spaces (intercellular air spaces). In fact, leaves of S. molesta show extensive intercellular air spaces which are expected to enhance elasticity.

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Figure 11 shows that for setup 1 to 4, the leaf can absorb only part of the impact energy (the related solid curves remain below the dashed lines), and this part increases with decreasing Young’s modulus. In the case of set-up 5, however, the leaf can absorb considerable amounts of the impact energy, and even completely, provided the leaf’s Young modulus is smaller than about 10 MPa (in view of the provisional character of expression (3.6) this value should be taken as a rough approximation). Interestingly, this behaviour is complementary to the energy absorption capacity of the leaf’s hair cover which can absorb the complete impact energy for set-up 1–4 but only part of this energy for set-up 5 (cf. figure 10). In any case, if $E_{\mathrm{leaf}}⪅10\hspace{0.17em}\mathrm{MPa}$ the elasticity of both the leaf and its trichome cover are able to absorb the impact energy for all set-ups discussed.

There are also some more possibilities of how the leaf surface of Salvinia can absorb energy: wetting of the hydrophobic leaf, energy dissipation by viscosity or entrainment of air bubbles. These various processes can, however, consume only small amounts of the impact energy, and are partially complex and/or speculative, and are therefore described in the electronic supplementary material.

3.2.2. Second phase: getting rid of impact energy

3.2.2.1. Springboard effect

The most striking second phase event is the springboard effect: the leaf, having been deflected from its equilibrium position by the impact, snaps back and accelerates upwards the water mass of the impacting drop. By this, energy is converted from elastic to kinetic and finally taken away from the leaf.

The oscillations of the leaf in the aftermath of the impact (not clearly visible in the attached videos) abate after some time, indicating that the remaining elastic energy has been converted into heat.

3.2.2.2. Secondary droplets

With increasing impact velocity, the original drop splits up into secondary droplets whose number and size may vary considerably. The formation of these secondary droplets also consumes energy:

The formation of the air/water interface of a secondary droplet of radius r requires the energy

$$E_{\mathrm{surf}}=4\pi \gamma r^2.$$3.7

If the droplet is subsequently accelerated to the velocity v_jump or if it jumps up to the (energetically equivalent vertical) height h_jump the additional energy

$$E_{\mathrm{jump}}=\frac{2\pi }{3}\rho r^3{v}_{\mathrm{jump}}^2=\frac{4\pi }{3}{r}^3\rho g{h}_{\mathrm{jump}}$$3.8

is required.

Figure 12 illustrates the sum E_surf + E_jump.

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For example, in set-up 5, numerous secondary droplets are ejected and reach a vertex height h_jump of their trajectory. If the original drop is split up into, say, 100 secondary droplets (with a correspondingly small size of about 1 mm) that reach a height of 2 cm, the energy demand is E_surf + E_jump ≈ 173 μJ (see equations (3.7) and (3.8)). This is, according to table 1, only about 1% of the kinetic impact energy of set-up 5. (Notice, however, that E_surf + E_jump ≈ 173 μJ is approximately the magnitude of the impact energy of set-ups 1–4.)

3.2.2.3. Other possible effects

There are other effects conceivable which may contribute to the final dissipation of energy, such as rotation of secondary droplets, leaf oscillations or entrainment of air. These effects could, however, not be clearly observed from the videos. Their potential capability for energy conversion is described in the electronic supplementary material.

3.2.3. Third phase: the emerging process

Towards the end of the drop/leaf interactions characterized by the time scale t_co (table 1), it can be observed for set-ups 4 and 5, that water rises quite slowly from the leaf surface, ‘climbs up’ between the trichomes and forms secondary droplets above the tips of the trichomes. This process is much slower than the ‘immediate’ impact events. In the following, it will be attempted to explain the EP, its time delay and the energy source of this phenomenon.

As noted above, covering the hydrophobic leaf surface of S. molesta with a water layer by drop spreading consumes a portion of the impact energy. As long as the thickness of the water film on the leaf surface exceeds the capillary length λ_c (equation (A.14) of the electronic supplementary material) the wetting energy remains stored in the leaf/water-film system. But as soon as the water film on the leaves has sufficiently thinned out during spreading, the surface forces acting between water and leaf surface exceed the weight of the water film and contract parts of the water layer. Owing to the presence of the hydrophobic trichomes, it is to be expected that the thickness of the water film will be irregular and/or the water film will be disintegrated, particularly in set-up 5 because of the high kinetic energy. Therefore, there will be separate ‘contraction islands’. The content of these receding patches may climb upwards between neighbouring trichomes, guided by the hydrophobicity of the trichomes and the leaf surface. This process is complex and it is not attempted—within the scope of this contribution—to describe the dynamics of this scenario in detail. To provide a rough explanation of the EP, two balance equations are evaluated in the following describing the initial point of this process, a water patch of thickness h_d and radius r_d, and its end point, a drop of radius R_d formed at a height H_d above the leaf surface:

$$\pi r_d^2{h}_d=\frac{4\pi }{3}{R}_d^3$$3.9

and

$$-\pi \gamma r_d^2\cos \theta =4\pi \gamma R_d^2+\pi r_d^2{h}_d\rho g{H}_d.$$3.10

The first equation equals the amount of water of the contracted patch to the water mass of the newly formed drop on the trichomes; the second equation states that the wetting energy that was stored in the patch equals the surface energy of the sphere plus the energy that is necessary to lift it to the height H_d. Solving equations (3.9) and (3.10) for h_d and r_d yields

$$h_d=-\frac{\gamma R_d\cos \theta }{H_d{R}_d\rho g+3\gamma }$$3.11

and

$$r_d=2R_d\sqrt{-\frac{H_d{R}_d\rho g+3\gamma }{3\gamma \cos \theta }}.$$3.12

The energy that is required to form a spherical drop of radius R_d and to lift it against gravity by the vertical distance H_d is given by the right-hand side of equation (3.10)

$$E_{\mathrm{hov}}=4\pi \gamma R_d^2+\frac{4\pi }{3}{R}_d^3\rho g{H}_d.$$3.13

To lift a drop from the original water patch to the top of the trichomes, a vertical distance has to be covered which corresponds to the length of the trichomes (L ≈ 3 mm, [5]) plus the radius R_d of the final drop: H_d = L + R_d. Figure 13 illustrates equations (3.11–3.13) with this vertical distance H_d = L + R_d. Notice that the thickness of the original water patch, h_d is, according to figure 13c, well below ${\lambda }_{c,{\mathrm{H}}_2\mathrm{O}}=2.7\hspace{0.17em}\mathrm{mm}$: the condition $h_d<{\lambda }_c$ guarantees the availability of the energy stored in the water film/leaf system. For $h_d>{\lambda }_c$ the weight of the water layer would prevent the dewetting of the leaf, and ultimately the ‘climbing process’. The results as shown in figure 13 therefore support the suggested explanation for the EP and how water is able to climb against gravity from below the trichomes after all energy conversions originating in the impact are seemingly finished. Probably, the quite slow EP is additionally driven by coalescence: once a part of a water patch succeeds in rising to the top of the trichome cover, the rest of the water which is still trapped between hydrophobic trichomes will gain an energetically much more favourable state when coalescing with the ‘surface drop’. This requires a continuous connection between both water bodies as is usually observed for ‘successful’ EPs (figure 8).

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3.2.4. Summary of energy conversions

Comparing the magnitudes of the energy conversions described in this section, it is obvious that the impact energy is dissipated by two stages related to the elasticity of trichomes and leaf:

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In a first phase, the buffer phase, the elastic structures absorb and buffer the impact energy obviously without any leaf damage.

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In a second phase, the dissipation phase, a large portion of the buffered energy is transferred via the springboard effect to secondary droplets that carry the energy away from the leaf. Additional processes such as leaf oscillations can contribute to the final dissipation of the stored energy. These contribute, however, only minor amounts of dissipation and/or cannot be reliably assessed from the videos (e.g. the rotation energy of secondary droplets) (see electronic supplementary material, Supplementary Information).

4. Discussion and conclusions

Given the size of the trichomes (lengths of about 3 mm, with the entire egg-beater structure showing a diameter of almost half a millimetre) and their areal density (about 1.5 mm⁻² to 3 mm⁻²) [5,29], falling drops easily penetrate the trichome cover. Owing to their elastic qualities, the trichomes are then able to absorb a certain part of the kinetic energy, which is—in the case of larger rain drops—only a minor contribution to the entire energy conversion, which takes largely place with the elastic deformation of the leaves. In their natural habitat, S. molesta will rapidly form dense and thick mats on the water surface in which new leaves are partially covering older leaves [2]. This arrangement, including additional factors such as the extensive internal air spaces of leaves of S. molesta, will greatly enhance the ability for buffering drop impacts by elastic responses (which will be complex in detail). The role of elastic responses of the leaves in swift shedding of water after drop impact was demonstrated by set-up 4 in which the leaves were fixed: here, it took a much longer period of time to remove water from the surface.

Swift removal of water after drop impact is, in fact, usually described as an important ability of biological surfaces [19,21,24,30]. From the results of wetting experiments, it would be expected that the superhydrophobic surface of S. molesta is able to vigourously repell water [1]. In the case of drop impact, however, the trichomes of S. molesta leaves appear to delay the shedding process, exactly because of their water-repellency: apparently, the quite narrow and also constricted spaces between the egg-beaters hamper the ascent of water. Therefore, water can be trapped beneath the egg-beaters, as is visible on various video recordings. This was particularly observed when the water film was very thin due to strong spreading driven by high kinetic energy (in set-up 5) or when spring-boarding was suppressed (in set-up 4). Obviously, there was not enough energy available for all of the contracting water patches to pass the narrow passages between the egg-beaters. In fact, in set-up 5, with the highest drop velocity and largest drop size, some residual water remained under the egg-beaters in all experiments. As a conclusion, it appears as if the trichome cover of S. molesta is not an asset for the reliable and swift shedding of rain water. Rather, the egg-beater trichomes appear to be quite often ‘in the way’ when water attempts to leave the surface after drop impact. Probably, the slow EP is a consequence of the eggbeaters hampering water repulsion: sufficient energy is available for a part of the water patch to reach the trichome tops, and the rest of the water is then driven upwards by surface forces.

Comparison of the contact times of set-up 3 (t_co = 14 ms, table 1) and set-up 4 (t_co = 52 ms) suggests a relation between the elasticity of the leaf (in set-up 4 it was fixed to an inflexible base plate, in contrast to set-up 3 where it floated on a thin layer of water) and its ability to avoid being wetted. Due to the different elasticities of the substrates, a drop in set-up 4 experiences a higher deceleration and distortion compression than in set-up 3. Since the surface forces—they tend to keep a drop together—are identical in both set-ups, a break-up of the drop (and the loitering of its remnants close to the leaf) is more likely in set-up 4, whereas in set-up 3 the deformed drop (pancake) performs rather a quick rebound. Vasileiou et al. [31] reported similar results.

Most of the water, however, manages to bounce off the leaves of S. molesta after drop impact. Also, during a natural rain event, water drops continuously hit the leaf. A dynamic process will take place for a while, and during that time, drops will be bouncing off, satellite droplets will be formed and falling back to the surface, and also residual water will be hit, both by satellite droplets and rain drops. Additionally, water drops driven above the trichome tops can roll along the trichome cover, together with satellite droplets which have fallen back to the leaf. Numerous drops will coalesce, including drops rolling above the trichome cover and residual water patches. In these events, the residual water will be ‘sucked up’ by the drops at the trichome tops (figure 8 and electronic supplementary material video S10).

At the end of such a rain event pouring down on a dense and extensive mat of S. molesta, it is to be expected that some water patches will cover the leaf ground at least partially. At this stage, unhindered gas exchange through the stomata is probably ensured by the network of microgrooves [32] that remain filled with air even if overlain by a thick water layer: These microgrooves are distinct indentations formed by the convex shape of adjacent epidermis cells. Stomatal cells, in contrast, have a flat or slightly concave surface and are located at recessed positions, at vertices of the network of grooves. As discussed in §A.7 of the electronic supplementary material, the geometry of the grooves and the superhydrophobicity of Salvinia’s epidermal cells imply that it is energetically favourable to keep the network of microgrooves and stomatal cells dry. It should also be noted, that the leaves of S. molesta are not planar, but often cup-shaped or—particularly in older mats—almost ‘folded up’ (meaning that both halves of a leaf are quite close together). The trichome cover will prevent water from penetrating nooks and folds in such dense mats. Additionally, the observed coalescence processes between drops rolling along the trichome cover and residual water are possibly a quite effective way to ‘scrub’ the leaf surface from remaining water patches.

Salvinia molesta is ‘one of the world’s worst aquatic weeds’ [2], mainly due to its extraordinarily high productivity which enables this species to double its biomass within a few days. For this, a unhindered and continuous gas exchange is necessary to supply the assimilation system with CO₂. The unique and dense trichome cover of S. molesta is very likely an important asset for preventing blockage of gas exchange in various ways and therefore assists the photosynthesis machinery in gaining high productivity.

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Competing interests

We declare we have no competing interests.

Funding

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