Hydrodynamic constraints on the energy efficiency of droplet electricity generators

Electric energy generation from falling droplets has seen a hundred-fold rise in efficiency over the past few years. However, even these newest devices can only extract a small portion of the droplet energy. In this paper, we theoretically investigate the contributions of hydrodynamic and electric losses in limiting the efficiency of droplet electricity generators (DEG). We restrict our analysis to cases where the droplet contacts the electrode at maximum spread, which was observed to maximize the DEG efficiency. Herein, the electro-mechanical energy conversion occurs during the recoil that immediately follows droplet impact. We then identify three limits on existing droplet electric generators: (i) the impingement velocity is limited in order to maintain the droplet integrity; (ii) much of droplet mechanical energy is squandered in overcoming viscous shear force with the substrate; (iii) insufficient electrical charge of the substrate. Of all these effects, we found that up to 83% of the total energy available was lost by viscous dissipation during spreading. Minimizing this loss by using cascaded DEG devices to reduce the droplet kinetic energy may increase future devices efficiency beyond 10%.

static energy that was stored previously [2]. Meanwhile, mobile charges accumulate at the water-polymer interface. When the droplet recedes, those charges are detached from the interface and forced to return to the bottom electrode [3]. While this picture predicts the transfer of charges through the DEG with a remarkable accuracy [2], it does not consider the hydrodynamic side of the picture. Yet, the harvested electrical energy accounts at best for 10% of the initial droplet energy, meaning that, in our present understanding, at least 90% of the droplet energy is unaccounted for. A comprehensive DEG model would consider (i) the electrical process at play, (ii) the hydrodynamics (iii) the electrochemical charge stability and (iv) the electrohydrodynamic coupling. Since only 10% of the DEG energy is electrical, we neglect the electrohydrody-namic coupling and focus mainly on the hydrodynamic process. In this simplified view, the DEG hydrodynamics are exactly those of a droplet impacting an inclined plane. Even in this elementary picture, three effects compete to dominate the droplet dynamics: inertia, surface forces, and viscous dissipation. For a spherical droplet of radius a, density ρ, viscosity µ, surface tension γ falling at a velocity U 0 , the ratio of kinetic energy to viscous work is approximately the Reynolds number Re = ρaU 0 µ , while the ratio of kinetic energy to surface energy is connected to the Weber number We = ρU 0 2 a γ . During impacts, the liquid spreads into a thin lamella on the solid where most of the energy conversion occurs [12]. It was noticed quite early that a large fraction of the droplet energy is lost during impact [13].
This loss was attributed to viscous dissipation within the lamella [13][14][15] until experiments of low-viscosity drop impacts on super-hydrophobic and slippery substrates suggested that a large fraction of the energy was actually converted to internal kinetic energy (akin to turbulence) [16][17][18]. Several recent works have since attempted to bridge the gap between these two regimes [12,19,20]. The numerical simulations of Wildeman et al. [12] are of particular interest for this study, as they show the location of viscous losses within the drops during impacts, and quantify the fraction of energy dissipated as viscous work and internal kinetic energy over the entire spreading step with no-slip and free-slip boundary conditions. According to this model, nearly half of the initial kinetic energy of low-viscosity droplets is lost as viscous dissipation in the lamella during spreading, regardless of the slip length, impact velocity and fluid viscosity.
The stability of the lamella formed during the impact is also a key concern for DEG. Intuitively, a splashing droplet releases some surface and kinetic energy in the form of ejected daughter droplets. Lamella breakup depends on a competition between a destabilizing suction and lubrication forces that lift it from the substrate, and a restoring capillary force that pulls the liquid back to the bulk of the droplet [21]. When the restoring force is overwhelmed by the other two, the lamella detaches and breaks into smaller droplets, resulting in splashing [22]. Even if the lamella remains stable and recedes, the liquid film itself may have thinned past its own stability limit and rupture into lower-energy liquid islands [23].
In this paper, we investigate the consequences of the hydrodynamic phenomena on the operating conditions and efficiency of DEG devices. An important hypothesis of our work is that the droplet provides no electrical energy during the spreading phase, and that instead the totality of the energy is obtained during the receding of the lamella. This is supported by a thought experiment where one would drop a neutral conducting disc on the DEG.
The biased capacitor would release the same amount of energy without any work from the disc. However, one would need to overcome the Coulomb force between the mobile charges induced in the disc and the underlying polymer to detach the disc from the DEG, thereby attesting of an electromechanical conversion during recoil. The paper is organized as follows: starting from the total energy available from a falling droplet, we first evaluate the maximum impact velocity allowed for DEG devices depending on the droplet volume.
We then evaluate how this energy is converted into surface energy and turbulent/viscous losses during the spreading phase, followed by the recoil phase. Eventually, we combine our hydrodynamic analysis and the electrical model of Wu et al.
[2] to quantify the energy efficiency of simulated and experimental DEG devices for various operating conditions, and point out to possible improvements for this technology.
Neglecting gravity, the minimum energy of a liquid in contact with a solid surface is obtained when the spherical droplet intersects with the solid surface at the Young contact angle cos θ = γsv−γ sl γ . According to volume conservation, this yields [24]: with A eq,cap = 2πR eq 2 (1 − cos θ), A eq,base = πR eq 2 sin 2 θ, and R eq = 2 2/3 a (2 − 3 cos θ + cos 3 θ) 1/3 , where A eq,cap and A eq,base are the cap and base surface area of the droplet at equilibrium, and R eq is its radius of curvature. Therefore, the total energy available is obtained by subtracting this lowest possible energy from the initial energy:

Maximum impact velocity
The impact velocity is limited by two factors. On the one hand, the droplet may splash, but even without splashing, the film formed by the impacted droplet may still become unstable and rupture. We first recall the splashing conditions according to Riboux and Gordillo [22]. It was experimentally observed that the lamella of impact droplets follow a universal dynamic until an ejection time t e where splashing and non-splashing droplets will diverge [22]. Therefore, the droplet state at t e and particularly the lamella height H t , vertical velocity V v and recoil speed V r determine the impact outcome. √ t e is well-approximated by the real positive root of the fourth-order polynomial: Obtaining t e from Eq. (6) yields the lamella height at the ejection time H t = (a √ 12/π)t e 3/2 , which then yields the recoil V r and vertical V v velocities: with K l − [6/ tan 2 α] (log(19.2λ g /H t ) − log(1 + 19.2λ g /H t )) and K u = 0.3 µ g two hydrodynamic coefficients for the suction and lubrication forces that make up the lift force , and ρ g , µ g and λ g the gas phase density, dynamic viscosity and mean-free path.
Riboux and Gordillo [22] postulate that the droplet will splash if the vertical velocity of the forming lamella exceeds the recoil speed by a factor b max = 0.14, that is: Even in the absence of splashing, the liquid film formed after impact may still rupture.
Surprisingly, even though instability of static liquid films is a well-studied topic [25][26][27], film rupture immediately after droplet impact has received much less attention than the more spectacular splashing. Diman and Chandra [23] have studied the disintegration of liquid films formed after high-speed collision between a droplet and a wall. While their analysis is rather involved, and depends on the liquid-solid contact angle and the size of defects that trigger the liquid film instability, experimental evidence over a range of contact angles, surface roughness and liquids suggest that the liquid film will rupture if the droplet impacts a solid wall above some critical Reynolds number Re c 5000: The available energy and the limiting velocities for splashing (Eq. 10) and film rupturing (Eq. 11) are shown in Fig. 2. Note that smaller droplets tend to splash first while larger ones may not splash but their liquid film will rupture nonetheless. The largest theoretical amount of energy while maintaining droplet integrity is obtained for the largest droplets at velocities close to 1.7 m/s, remarkably close to Xu et al. [1].

Efficiency during spreading and receding
Depending on the impact speed, a sizable fraction of the available energy (Eq. (5)) may be dissipated into turbulent kinetic energy and viscous work during the droplet spreading.
The kinetic-to-surface energy conversion efficiency η s = Vmax Emax , with V max = γ(1 − cos θ)A max the surface energy at maximum spread A max , can be computed directly from experimental (spreading efficiency 30%), respectively. data or numerical simulations, or can be evaluated based on theoretical analyses [14,15,17].
This yields the conversion efficiency η s : A max may be determined from from experimental or simulated droplet impacts, or using simplified models (see [15] and [20] for critical reviews). Among these models, Pasandideh-Fard et al. [14] provide a simple and accurate estimate of the maximum spreading diameter of impinging droplets: In spite of being derived for highly viscous fluids only, this formula was empirically found to work well for low-viscosity fluids as well [15,20]. We refer the reader to Wildeman et al. [12] for a more physically-sound model at low viscosity. The resulting energy efficiency obtained by combining Eq. (12,13) is shown in Fig. 3.
During the recoil phase, the surface energy V max is split into 3 terms: the viscous work during recoil W r , the electrical work during recoil W e and the final mechanical energy remaining in the droplet E ∞ : Experiments with water drops bouncing on super-hydrophobic substrates [18,28] suggest that the viscous dissipation is small during the recoil step, even at high impact velocity, and that most of the surface energy is either restored as external kinetic energy or vibration energy. This is confirmed by the following rough estimate of the droplet viscous dissipation.
We first assume that the recoil speed scales as U r = a max /τ r , with τ r the contact time between the droplet and the substrate. For super-hydrophobic substrates, the contact time, that is a negligible spreading time plus the receding time, is insensitive to the impact velocity [16,29]. Hence, the recoil time scales proportionally to τ r = πτ h / √ 2 with τ h = ρa 3 γ the oscillation period of a levitated drop [29,30]. Using U r as the characteristic recoil velocity, we then compute the volumic shear rate similarly to Pasandideh-Fard [14] and obtain a coarse estimate of the viscous work during recoil: where h bl and h film are the hydrodynamic boundary layer and film thickness during recoil.
with σ the surface charge of the polymer, c p the areal capacitance of the polymer and with: Note that the right-hand side of Eq. (23) cannot be simplified immediately because there is no one-to-one correspondence between A(t) and q 2 (t). Eq. (20) is made of three parts: an efficiency factor η L that accounts for the resistive losses in the liquid, an electrostatic energy E stat and a contribution that depends on the dynamics of the charges and droplet geometry E dyn . We note that E dyn is positive only if q 2 decreases, meaning that this term acts as a generator only when charges are moving out of the droplet.

Available energy
The droplet exchanges electrical energy with the load twice. First during the discharge of the bottom electrode in the liquid, and then when the recoil step. As stated in the introduction, we believe that the discharge process goes without energy exchange between the droplet and the DEG: the liquid merely acts as a conductor to release the stored electrostatic energy. Although the discharge step requires no work from the droplet, the droplet-polymer interface becomes increasingly charged thereafter. During the recoil step, the liquid interface area shrinks such that the liquid-polymer interfacial capacitance decreases, which prompts charges to flow back to the bottom electrode. In order to reduce the contact area between oppositely charged surfaces, some electrowetting work must be provided to overcome the electrostatic energy [31,32]. When charges flow back, they provide resistive electrical work through the load but also through the liquid, thereafter called resistive losses: Upon sufficient recoil, the droplet eventually detaches from the top electrode and the current stops flow through the load. Integrating Eq. (19) shows that the static loss cannot be entirely eliminated unless the droplet area vanishes at t c (see supplementary information).
This can be achieved by using non-uniform surface charges [3]. While there is no constraint on the remaining amount of charges (missing charges can be provided from the electrical ground), a highly charged droplet will need to expend more energy to leave the substrate than a neutral one. Therefore, the least charge remain in the droplet, the most energy is available for the load. We will refer to the additional electrostatic expense at the static loss W stat = q 2 (t∞) 2cpA(t∞) . where t ∞ indicates the time the droplet breaks away from the DEG substrate. By conservation of charge, q(t c ) = q(t ∞ ), and by conservation of energy the work provided to change the droplet surface area must compensate exactly the variation of W stat , therefore: .   energy (83%) is lost as viscous work, in good agreement with Fig. 3 and previous studies [12].
The recoil consumes less than 2% of the total energy E max by itself. The second largest loss contribution ( 10% of the total, 74% of the energy available after recoil) is the difference between the energy after recoil and the electrical energy. This large mismatch is shared across all sizes of droplets regardless of the impact speed, and represents the mechanical energy E ∞ remaining after the droplet detaches from the electrode. By analogy with bouncing droplets [18,28], it is likely that this energy is a combination of internal kinetic energy and surface vibration energy. Next, an optimized load allows extracting almost 75% the electrical energy (resistive losses are negligible and the static losses represent approximately 1% of E max ).
Having identified that most of the energy is dissipated by viscous work and inefficient energy conversion during recoil, we now point to some ways to reduce these losses. The ratio of viscous to capillary work scales as Ca = µU 0 γ and looks insensitive to the main dimensions of the surface, which suggests that micropatterns which were successful at reducing the droplet spreading and contact time [33] may not help minimizing the viscous dissipation.
However, our simulations suggest that decreasing the impact velocity from 1.7 m/s (X5) to U 0 = 0.1 m/s (X5) can cut the viscous dissipation during spreading to only 15% of E max . This is in line with the improvement of restoring coefficient of bouncing Leidenfrost-levitated droplets at lower speed [18]. Therefore, while high-speed impacts generate more energy, lowspeed impacts are considerably more efficient. Nonetheless, low-speed impacts fail to extract the remaining mechanical energy from the droplet, so that only 5.1% of V max is converted to electricity when U 0 = 0.1 m/s (X1) whereas up to 24% of V max becomes electricity when A tentative interpretation is based on the following rough estimate of the energy conversion [2]: with: with E 0 the electrostatic energy stored in the polymer before the droplet impact and ∆A the difference of liquid-solid surface area between the time when the droplet connects to the electrode and when it breaks away from it. The factor 2 ∆A as shown in Fig. 5(b).

IV. CONCLUSION
In this paper, we have used a combination of analytical equations, numerical model and experimental data to map out the energy losses during DEG generation. The energy con-version efficiency of these devices is mainly limited by viscous dissipation and poor capillary to electrical energy conversion. Experimental data and our numerical simulations show that a small fraction of the initial kinetic energy of impinging droplets is converted into surface energy at maximum spread. The remaining energy is lost as shear work and internal kinetic energy. Even though slower impacts provide less peak power, they dissipate much less energy. For applications where the total energy is critical (as opposed to the peak power), slower impact velocities may allow extracting more energy. This may be achieved by cascading generators with small gap height between them, or even harvesting the energy of crawling droplets [34]. Furthermore, increasing the device charge is indeed a key element in the path of improving DEG efficiency. By reducing the impact speed and increasing the surface charge, it makes little doubt that DEG efficiency can be improved beyond the current 10%.