Electrowetting-Controlled Dropwise Condensation with Patterned Electrodes: Physical Principles, Modeling, and Application Perspectives for Fog Harvesting and Enhanced Heat Transfer

Patterning the wettability of solid surfaces is a successful strategy to control the dropwise condensation of vapor onto partially wetting solid surfaces. We followed the condensation of water vapor onto electrowetting-functionalized surfaces with structured co-planar electrodes. A detailed analysis of the experimental distribution of millions of drops reveals that despite the presence of contact angle hysteresis and the occurrence of random drop coalescence events, the preferential drop position closely follows the evolution of the local minima of the numerically calculated drop size-dependent electrostatic energy landscape in two dimensions. Even subtle transitions between competing preferred locations are properly reproduced by the model. Based on this quantitative understanding of the condensation patterns, we discuss a series of important follow-up steps that need to be taken to demonstrate a reliable performance gain in various applications focusing in particular on enhanced heat transfer.


I. INTRODUCTION
The condensation of water vapor onto solid surfaces is integral to many natural processes including dew formation [1] and fog harvesting by animals, like the Namib Desert Beetle [2,3] and Litoria caerulea-, a green tree frog in Australia, [4] and plants, such as the Namib desert plant. [5] Water condensation is also intrinsic to various technological applications like fog harvesting, [6,7] seawater desalination, [8] and heat exchangers for power generation [9] and refrigeration. [10,11] In all cases, efficient condensation and removal (or 'collection') of the condensed liquid is essential. The entire process consists of a series of steps, namely the nucleation of liquid on an initially dry solid surface, the subsequent growth of the liquid phase in form of a film or droplets, and finally the removal of the latter. At first glance, hydrophilic surfaces may seem the most natural choice to promote condensation.
Yet, it has been known for decades that plain hydrophilic surfaces are actually not the best choice because they promote the formation of condensed liquid film (for a review: see [12]).
Compared to films, drops are much easier to manipulate and transport in desired directions by suitable topographical and chemical patterns on the surface. Moreover, particularly in heat transfer, thick films of condensed liquid form a barrier of poor thermal conductivity that prevents direct contact of the vapor with the cooled surface of the condenser and thus reduces the overall heat transfer. Hence, it is usually advantageous to use partially wetting solid surfaces where condensing vapor forms discrete drops that leave parts of the solid surface in direct contact with the to-be-condensed vapor. As these discrete drops are removed, they expose even more bare surface again and thereby free space for a subsequent generation of condensing drops. Like in case of biological or technological fog harvesting surfaces, efficient removal of the condensate drops is therefore essential for the overall performance of the system. Throughout recent years, various efforts have been made to optimize dropwise condensation and the subsequent removal of drops using suitable topographical and chemical surface patterns. [13][14][15][16][17][18][19][20][21][22] Such patterns generate an energy landscape in which condensing drops initially form at either random locations or at preferred hydrophilic nucleation sites. As drops grow with time, they experience the imprinted gradients in wettability, hit geometric boundaries, and coalesce with other drops. In each of these situations, the original configuration of the drop typically becomes unstable and the drop moves towards a location of lower energy.
Examples of such surface patterns include surfaces with alternating hydrophobic and hydrophilic stripes, [13][14][15] surfaces with conical geometries, [3] superhydrophobic surfaces with grooves [16] or nanostructures, [17][18][19] as well as liquid-infused surfaces. [20][21][22] The resulting drop displacements are either driven entirely by capillary and wetting forces or they may be assisted by gravity in case of vertically oriented condenser surfaces. In all cases, drops only move once the driving forces are strong enough to overcome the pinning due to microscopic heterogeneities. [23] The latter are usually quantified by specifying the contact angle hysteresis ∆ cos θ = cos θ r − cos θ a , where, θ r and θ a are the receding and advancing contact angles. This explains the interest in surfaces with low contact angle hysteresis such as superhydrophobic and liquid-infused surfaces for heat transfer applications with dropwise condensation.
The approaches described above all rely on passive wettability patterns imprinted onto the solid surface upon fabrication. In contrast, electrowetting (EW) allows for active tuning of the wettability and controlled transport of drops of conductive liquids such as water on partially wetting hydrophobic surfaces. [24][25][26][27] While generically used in combination with a wire that is immersed directly into the liquid, capacitive coupling between the drop(s) and suitably structured co-planar electrodes on the substrate that are covered by a thin hydrophobic polymer layer allow for similarly efficient control of the wettability locally above the activated electrodes. [27,28] By patterning the electrodes, wettability patterns such as simple traps for drops can be generated and switched on and off at will. [29] Drops that were large compared to the width of a gap between two electrodes, preferentially aligned on the center of the gap. As usual in EW, this minimum of the electrostatic energy E el = −C tot U 2 /2 corresponds to the maximum of the total capacitance between the drop and the electrodes.
In this manner, 't Mannetje et al. [30] demonstrated controlled capture, release, and steering of rolling drops on an inclined plane. Later de Ruiter et al. [31] extended the same principle for drops in microfluidic two phase flow systems for a range of electrode geometries and applied a simple analytical model to calculate the electrical holding force based on the geometric overlap of the trapped drop and the activated electrodes. The idea of manipulating condensing drops by EW was first explored by Kim and Kaviany. [32] Baratian et al. [33] later combined these ideas to study for the first time directly the condensation of water vapor onto EW-functionalized surfaces. For the specific case of parallel interdigitated electrodes aligned along the direction of gravity, they found that the condensation pattern is governed by an electrostatic energy landscape that depends on the size of the condensing drops.
While the initial condensation occurred at random locations, subsequent growth by further condensation and EW-induced coalescence lead to alignment of the drops along the edges of the electrodes. Later, once their diameter became comparable to the width of the electrodes, the drops accumulated at the centers of the gaps between adjacent electrodes. Analyzing the distribution of drop sizes and locations, they showed that the drops decorate the drop sizedependent minima of the (one-dimensional) electrostatic energy landscape perpendicular to the electrodes. EW-induced coalescence events lead to faster drop growth. In combination with the reduced contact angle hysteresis in EW with AC voltage [34] drop shedding occurs on average for smaller drops, as compared to the reference case without EW. [33] According to classical observations in dropwise condensation, such a reduction of the critical shedding radius is accompanied by enhanced heat transfer. [12] A series of follow-up studies confirmed these basic original observations regarding the evolution of the drop distribution for straight interdigitated electrodes. [35][36][37][38][39] Experiments with slightly more complex electrode geometries with zigzag-shaped edges resulted in preferential alignment of the drops not only perpendicular but also along the direction of the electrodes, in qualitative agreement with expectations. [40] That study also indirectly inferred an increased heat transfer from the volume of shedded drops as extracted from video microscopy images. Overall, the experiments suggest that it should be possible to optimize the performance of EW-controlled condensation in heat transfer and other applications by systematically varying electrode geometries and/or excitation patterns. Since experimental brute force optimization of electrode shapes would be very time consuming and costly, it is essential then to extend the existing electrostatic models to arbitrary electrode geometries, and to demonstrate their performance in capturing the complex evolution of drop distribution patterns to enable electrode optimization in silico prior to experimental testing.
The purpose of the present work is therefore twofold: the core of the work consists of a  [29] reproduces the qualitative behavior but underestimates electrostatic energies and forces for small drops. Following the discussion of these results, we evaluate the present status of the field and discuss aspects that we consider essential for the development of EW-controlled condensation from a physical phenomenon towards a technologically relevant application.

A. Experimental Aspects
The present condensation experiments were performed in the same homemade experimental setup ( Figure 1a) that was used in our previous studies. [33,40] The setup consists of a condensation chamber with two inlets at the bottom and an outlet through a fine grid of holes for vapor at the top side. The transparent sample is mounted vertically on one of the side walls and cooled from the back by cooling water (11.5 • C) from a commercial cooler The recorded images are analyzed using a home-built image analysis routine in MATLAB to evaluate the center locations and radii of all the condensing drops (Supporting Information B). The smallest drop size detectable using this method is R min ≈ 4.3 µm.
The interdigitated zigzag electrodes are fabricated using photo-lithography on a glass substrate. The electrodes are subsequently coated with a 2 m thick dielectric layer of Parylene C (PDS2010, SCS Labcoter) using chemical vapour deposition (CVD), and an ultra thin top hydrophobic polymer coating (CytopTM, Asahi Glass Co., Ltd.) using a dip-coating procedure. For the experiments and simulations reported herein, we use interdigitated electrodes with zigzag-shaped edges (Figures 1b-1c). As in ref. [40], the minimum and maximum width of the gap between adjacent electrodes are kept fixed at w g,min = 50µm and w g,max = 250µm, and three different lengths of 500, 1000 and 3000 m are tested.

B. Numerical Aspects
To explain our experimental observations, we developed a numerical model that allows us to calculate the electrostatic energy of a drop as a function of its size and the (x, y) position of its center of mass within the unit cell of the electrode pattern (see zoomed view in Figure   1b). To calculate this energy landscape (E el (x, y; R)), we solve the Poisson equation for a three-dimensional computational domain consisting of the electrodes, the dielectric layer, a water drop, and the surrounding air. Since θ(150 V) ∼ 90 • , we represent the drop by a simple hemisphere with radius R and with a fixed electrical conductivity (10 −5 S/m) that guarantees (for all practical purposes) complete screening of the electric field from the inside of the drop. Note that this hemispherical approximation neglects slight EW-induced distortions of the drop shape (see below). Yet, earlier simulations showed that this merely leads to a minor underestimation of the electrostatic trapping strength for rather weakly deformed drops as in the present experiments. [41] The calculation of E el (x, y; R) starts with the calculation of the distribution of the elec- Here 0 is the permittivity of free space, and is the relative permittivity of the computational domain. ρ e can be related to the current density J using the charge conservation equation where σ is the electrical conductivity of the computational domain. Taking the time derivative of Equation 1, and subsequently substituting Equation 2 in it, we get a second order partial differential equation in φ: Considering a sinusoidal electrical potential φ = φ 0 [e iωt ], and subsequently, considering its Equation 3 can be rewritten as After numerical evaluation of φ(x, y, z) for all allowed drop sizes and (x, y)-location within the unit cell, the total electrostatic energy of the entire system is calculated as where E = −∇φ is the electric field, and the integration represents the volume integral over the entire computational domain.
In the representation of the electrostatic energy landscapes later on ( Figure 5), we make use of symmetries and periodicities to extend the energy landscapes beyond a single unit cell for a more intuitive representation. Finally, note that Equation 4 contains both dielectric and purely conductive contributions. However, for the conductivity of pure water and for the applied (low) frequency, the ionic current dominates the displacement current towards screening the electric field (also see [33]).

A. Evolution of breath figures
As apparent at first glance, the condensate drops form a pattern (breath figure) with well-defined periodicities along both the lateral (x−) and vertical (y−) direction upon condensation onto surfaces with zigzag interdigitated electrodes (Figures 2a-2c). This is in sharp contrast to breath figures with straight interdigitated electrodes under ac-EW, where no periodicity along the y−direction was found. [33] While these observations have qualitatively been reported before, [40] a closer look at the representative Figures 2d-2i reveals a number of additional details: Initially, the small condensate drops are essentially randomly distributed; however, as the drops grow and begin to coalesce, they align parallel to the electrode edges, with a slight preferential displacement towards the gap centers ( Figures   2d-2e). Simultaneously, the drops closer to a gap minimum (i.e. at (x = w/2; y = 0) in Figure 1b) are pulled down towards that minimum and typically grow on their way by coalescence with other drops (Figures 2d-2e). As we will see below, drops at these 'gap minima' are trapped in electrostatic energy minima; as these continue to grow, their lower edge remains close to y = 0, whereas their center gradually moves upwards (Figures 2d-2g). These  (Figure 3b). As the drops coalesce and grow further, they gradually move from the electrode edges towards the gap center (Figures 3b-3d). Drops with a diameter that exceeds the local width w g (y) of the gap, i.e. drop with a critical size R > 0.5w g (y) (red data points) are preferentially found in the center of the gap rather than along the electrode edges (Figure 3c-3d), giving rise to a peculiar bi-modal distribution of the drops (Figure 3c). This bi-modal spatial distribution of drops is unique to the converging electrode geometry. In or along the gap center) [27] . With further increasing drop size (Figure 4d), more drops higher  (Figures 4b-4d) is reminiscent of a 'zipper-like' e ect. It must be noted that the 223 cluster of data points at the gap apex always represent electrically trapped droplets (Figures 4b-4d). of droplets over this region) creating the bigger trapped droplet which continues to grow upward.

227
The bigger drops ' > 0.5F 6,<0G , mostly constituted by the trapped ones, show further interesting 228 lateral migrations which are shown in Figure 5a.   In order to further visualize the spatial evolution of the trapped drops, Figure 4b shows the projected vertical (y-) locations of all drops normalized by the electrode length (y/ ) versus R/ for all three electrode designs ( = 500, 1000, 3000 µm). The tail developing from the gap minimum (y/ = 0) represents the vertical locations of the trapped drops. Initially, the vertical locations of these trapped drops satisfy y ≈ R (solid black line in Figure 4b), as previously observed from Figures 2 and 3. Although for the = 3000 µm electrode (red data points) the trapped drops stay aligned at y ≈ R till gravity-driven shedding (R shed / ≈ 0.3), for the = 1000 µm (blue) and = 500 µm (green) electrodes the drops subsequently deviate from the y = R-trend as these continue to grow by coalescence (Figure 4b). For the = 1000 µm electrode, this deviation begins immediately after the first lateral transition (R 1 / ≈ 0.32), while for the = 500 µm electrode this occurs well before the first lateral transition at R/ ≈ 0.5 ( Figure 4b). Interestingly, once the trapped drops grow bigger, they realign again following y ≈ R for both the = 1000 µm and = 500 µm electrodes above R/ ≈ 0.6 ( Figure 4b). Note that for both = 1000 µm and = 500 µm electrodes, the deviation of the trapped drops from the y = R trend occurs well below the critical shedding radius (R shed ≈ 1 mm); hence, gravity is not the cause of the deviation.

B. Electrostatic energy landscape controls the evolution of breath figures
The details of the drop distributions described above can be understood by considering the 2D electrostatic energy landscape (E el (x, y)) emerging from our numerical calculations. As the drop size approaches 300 µm for = 1000 µm, the electrostatic energy minimum eventually moves from the gap center to the electrode center (Figures 5d-5e). We can predict the drop radius for this and subsequent lateral transitions by calculating the E el for a drop located either at the gap center or at electrode center, for a range of drop radii ( Figure 6). For small drop sizes (R < 320 µm), the total electrostatic energy is smaller when the drop is located at the gap center (black solid line in Figure 6) than when located at the electrode center (red solid line). As the drop size increases further, the location of the lowest electrostatic energy moves alternately between the electrode and the gap centers (compare the relative variations between the black and red solid lines in Figure 6). The characteristic radii at which these transitions occur, R 1 , R 2 , ..., R n in Figure 6 are shown as horizontal dashed lines in Figure 4a, and provide a good description of the transitions observed experimentally. Note that these horizontal transitions of the center of mass as a function of the drop size are well-known for surfaces with parallel stripes of alternating wettability originating from chemical patterning. [42] Tracing the position of the global energy minimum along the y−direction reveals that the drop center indeed moves upward with increasing drop size following slightly below the line y = R, as shown in Figure 4b. The numerical results (squares and triangles in Figure   4b) reproduce the experimental observations with great accuracy. In some cases correlations with lateral transitions of the drop position can be observed.
For the shorter unit cell ( = 500 µm), the evolution of the energy landscape is qualitatively similar. Nevertheless, the two situations cannot be mapped directly onto each other.
For instance, unlike the long electrodes, we find for = 500 µm that the energy minimum in the gap center splits up into two distinct local minima as the drop diameter becomes comparable to between R ∼ 250 . . . ∼ 300 µm (Figures 5g-5i). This leads to a distinct transition of the drop position along the y−direction for R = 280 → 300 µm (Figures 5h-5i Figure 4b is also correctly reproduced for electrodes of both short and intermediate length.
As an alternative to the full numerical calculations, we can also evaluate the energy landscapes by approximating the electrostatic energy using the simple geometric approximation proposed by 't Mannetje et al. [29,30] This analytical calculation involves approximating the condensate drop-dielectric system as an electrical circuit consisting of two parallel plate capacitors in series formed by the overlap between the conducting drop and the electrodes.
The overall capacitance of the system is approximated as C(x, y) = 0 d /d · A cap , where A 2 ), and A 1 and A 2 are the spatially varying overlap areas between the drop footprint and the two electrodes (see inset in Figure 6). The associated electrostatic energy of the system on application of an electrical voltage (U ) can be written as E el,cap = −C(x, y)U 2 /2. [29] The dashed lines in Figure 6 show the electrostatic energy minimum in this approximation for drop centered on the gap (black) and on the electrode (red).
Like in the case of the full numerical model, for small drops (i.e. R <∼ 320 µm) it is more favorable to be centered on the gap, whereas for increasing R there is a succession of transitions between preferred alignment on the electrode center and the gap center. While the energies deviate substantially for the smallest drop sizes (for which the overlap area with one of the electrodes and hence the total energy can vanish), the agreement improves for increasing drop size and the predictions for the various subsequent transitions of the drop positions (R 2 , R 3 , R 4 ) becomes remarkably good.
While some of the aspects described above are very specific to the present electrode configuration, the overall excellent agreement demonstrates the ability of the numerical model to reproduce the experiments, including even subtle aspects such as the transitions between various competing local minima of the overall energy landscape are correctly captured. For not too small droplets, the simple analytical model of geometric overlap also provides reasonable predictions between various competing drop configurations.

IV. DISCUSSION AND PERSPECTIVES
The results presented here clearly demonstrate the flexibility of electric fields in controlling condensation patterns on solid surfaces with submerged co-planar electrodes. While individual drops are obviously subject to their specific local environment, averaging over large ensembles shows that condensed drops decorate the local minima of electrostatic energy landscapes of remarkable complexity. Consequently, the drops undergo gradual translations as well as discrete transitions as local shift or become unstable with increasing drop size.
Apparently, the random character of coalescence events with neighboring drops in combi-

Numerical model
Analytical model the influence of gravity. While some success of this strategy has been demonstrated, [40,43] the overall performance was not impressive. In part, this is probably caused by the fact that the pinning forces increase as the EW-induced reduction of the contact angle hysteresis ceases upon switching off the AC voltage and hence the critical shedding radius increases and the shedding frequency decreases. [44] Alternatively, one could make use of active transport strategies borrowed from EW-based lab-on-a-chip systems, where drops are transported towards activated electrodes. [27] Given the nature of drop condensation, it is obviously not desirable to bring the condensing drops in direct contact with electrodes on top of the functionalized surface. Therefore, structured electrodes, possibly in two layers, should be embedded into the substrate and actuated in such a manner that they lead to a conveyor belt-like directed motion. Such strategies are rather straightforward to implement for surfaces that are flat or covered by some 'moderate' degree of topographic pattern. For intrinsically three-dimensional structures such as meshes that are frequently used for fog harvesting the implementation of any form EW-enhanced condensation and drop removal is much more difficult to realize -notwithstanding initial demonstrations with crossing fibers of switchable wettability. [45] While the effect of EW on the drop distribution patterns is rather striking, the reported consequences for the total condensation rate and the resulting heat transfer are far less impressive. [33,37,40] Applying standard models of dropwise heat transfer, [46] Wikramanayake et al. pointed out that the majority of previous EW experiments were not very enlightening because they were carried out using water vapor in moist air. [37] Under such conditions, it is well-known in the heat transfer community that the overall heat transfer resistance is dominated by the ambient air, which acts as a non-condensable background gas and introduces a diffusive boundary layer at the solid-vapor interface. upon contact with water for several hours. [48] In the presence of electric fields, this effect is even more pronounced and can even be exploited to generate well-controlled charge densities and charge patterns. [49,50] Therefore, the development of reliable hydrophobic fluoropolymer coatings that remain stable throughout the life time of various types of devices has been a long standing challenge in applied EW research. One recommendations from these investigations has been to avoid water as an operating fluid whenever possible. [51] While this is an interesting option for heat transfer devices, it is obviously not possible for fog harvesting applications. In such cases, novel materials with improved stability [52] will be required to achieve the necessary stability of operation. Nevertheless, it is also worth pointing out that extended EW-enhanced drop condensation tests over a period of 40 hours displayedafter some initial degradation within the first 1-2 hours -rather stable operation even for conventional fluoropolymer surfaces (Supporting Information E).

V. CONCLUSIONS
Co-planar electrodes embedded into electrowetting-functionalized surfaces allow to con- Now that the location and center of each drop is known, data can be extracted from the breath figures. This data includes the radius of shedding drops, surface coverage and total volume in the breath figure. The total volume assumes a constant drop contact angle of 115 • to calculate the volume of the spherical caps inside the frame. For drops touching the edge of the frame, the volume inside the frame is estimated assuming hemispherical drops minus (half) spherical caps.

Image lens distortion correction
The recorded images of the breath figure evolution are slightly pincushion distorted.
Projecting the data onto a single small unit cell will result in a spread in the statistical analysis. To correct for a lens distortion correction is applied to analyzed data.
Lens distortion can be expressed as: [53] where u and v are the unobservable distortion-free image coordinates; u d and v d are the corresponding image coordinates with distortion; δ u (u, v) and δ v (u, v) are distortion in u and v direction, respectively.
Distortion can be classified in three categories: radial distortion, decentering distortion and thin prism distortion. Radial distortion commonly exceeds the other distortions by at least an order of magnitude, [54] and since the distortion is our images is minimal (maximum shift of about 15 pixels in a 4000x3000 pixel image) we only radial distortion in our correction.
The first and second terms are predominant, so Equation B3 and B4 can be reduced to We define (x, y) as the coordinate system where (x, y) = (0, 0) is the top-left most corner of the image, as is common in image processing. We define (u, v) as the coordinate system with the origin at the center of the image. w, h are the width and height of the image in pixels, respectively. We assume that our distortion center (principal point) is located at the center of the image, so that To determine the distortion coefficients k 1 and k 2 , we measure the distortion using the  Appendix E: Surface degradation of Teflon surfaces see Figure 11