Breath figures under electrowetting: electrically controlled evolution of drop condensation patterns

Breath figures under electrowetting: electrically controlled evolution of drop condensation patterns

Abstract

Dropwise condensation is a ubiquitous and practically relevant process involving phases of nucleation, growth, coalescence and shedding. Here, we demonstrate that electrowetting (EW) with structured electrodes modifies the condensate droplet pattern on flat hydrophobic surfaces (breath figures) by aligning drops at electrostatic energy minima and enhancing coalescence. Numerical calculations reveal that the unique droplet pattern evolution under EW is governed by the drop size-dependent evolution of the electrostatic energy landscape. We also show using scaling that the evolution of the drop size distribution under EW exhibits significantly different self-similar characteristics over different growth phases. The transition in the droplet size distribution is due to EW-induced coalescence cascades beyond a threshold drop size. The resulting reduced surface coverage, coupled with an earlier drop shedding under EW, significantly enhances the net heat transfer.

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Dropwise condensation of water vapour is intrinsic to natural phenomena like dew formation Beysens (1995), and dew/fog harvesting by animals (e.g. Namib Desert beetle) and plants such as Namib Desert plant Malik et al. (2014). Dropwise condensation of vapour is also utilized in various technologies like water-harvesting systems Milani et al. (2011), heat exchangers for cooling systems Kim and Bullard (2002), and desalination systems Khawaji et al. (2008). The efficacy of these technologies depends on the nucleation, coalescence and growth of the condensate droplets on surfaces, and on their subsequent shedding Beysens (2006); Rose (1967). The pattern formed by condensing droplets is classically referred to as a breath figure due to its similarity with the pattern formed by breathing on a cold surface Beysens and Knobler (1986); Viovy et al. (1988); Fritter et al. (1991). An intriguing feature of breath figures is that the pattern evolution of condensate droplets is self-similar in time, as established by scaling the droplet size distribution Family and Meakin (1988, 1989); Meakin (1992); Narhe et al. (2001); Blaschke et al. (2012). To improve the efficiency of the dropwise condensation process it is essential to control the underlying breath figure characteristics.

Figure 1: (a) Schematic of the substrate used for the condensation experiments. Transparent interdigitated ITO electrodes are patterned on the glass substrate, which is then coated with a hydrophobic dielectric polymer film. Condensation without EW (control) and under EW are performed on identical substrates. A schematic of a conductive droplet under EW is also shown here. (b) Comparison between breath figures without EW (control) (C-i to C-iii) and under EW ( V; kHz) (EW-i to EW-iii) at different time instants. The (e)lectrode-(g)ap geometry underneath the dielectric film is indicated by the solid red and white lines.

In general, dropwise condensation can be controlled by regulating the mobility of the droplets on the surface. Enhanced condensation has been achieved so far by judiciously tailoring the surface characteristics. In this regard, the enhancement of dropwise condensation has been studied on superhydrophobic nanostructured surfaces Boreyko and Chen (2009); Miljkovic et al. (2012a, b, 2013), on superhydrophobic microgrooved and wettability-patterned surfaces Narhe and Beysens (2004); Ghosh et al. (2014), and on liquid impregnated textured surfaces Anand et al. (2012); Tsuchiya et al. (2017). However, in these studies breath figures are altered passively; the droplet mobility on such surfaces is essentially determined by the condensing surface design. In this letter, we describe the use of alternating (AC) electric field in an electrowetting (EW) configuration Mugele and Baret (2005) to actively control the condensate droplet pattern evolution on flat hydrophobic surfaces. EW provides a simple and flexible way of manipulating drops for microfluidics Choi et al. (2012); de Ruiter et al. (2014). Here, we demonstrate how EW actively manipulates the distribution of condensate drops using electrical forces. The evolution of the breath figure under EW is characterized by enhanced coalescence and unique size-dependent alignment of the condensate drops at the minima in the corresponding electrostatic energy landscapes. A scaling analysis shows that the size-distribution of drops is self-similar in time with distinct characteristics in the initial and final growth phases. The transformation in the scaling plot is triggered by EW-induced coalescence cascades beyond a critical drop size. These EW-mediated coalescence cascades increase the average droplet radius, but reduce the surface coverage in contrast.

The condensation experiments are performed on a glass plate coated by a hydrophobic dielectric polymer film (Fig. 1(a)). The glass substrate contains a stripe pattern of transparent interdigitated ITO electrodes (Fig. 1(a)). The electrodes and the intermediate gaps are 200 m in width. For the experimental details see Fig. S1 in the Supplemental Material sup (). For the experiments, a stream of vapour-air mixture at a flow rate of 2 lt/min and a temperature of C is passed through a condensation chamber, in which the substrate is kept at a temperature of C. Condensation experiments are performed both without EW (control) and under EW on identical substrates and under identical experimental conditions. The condensation on the control and EW-functionalized surfaces is monitored using a high resolution camera. We denote the time instant at which reliably detectable condensate drops of radius m are detected for the first time as s (for the image analysis procedure see S8 in sup ()).

At the beginning, small drops appear at random locations on both the control and the EW-functionalized surfaces and grow without coalescence (Fig. 1(b)(C-i), (EW-i)). As the drops grow and coalesce frequently (see Movie S1 in sup ()), the drops on the EW-functionalized surface initially align along the electrode edges, slightly displaced towards the gaps (Fig. 1(b)(EW-ii)). As the droplets grow further and exceed a critical size, sequences of rapid coalescence events (coalescence cascades) create droplets having diameters comparable to the electrode spacing (Fig. S2 in sup ()). Consequently, the resulting breath figure is dominated by approximately monodisperse drops bridging the gap between two adjacent electrodes, and aligned along the gap-centres (Fig. 1(b)(EW-iii)). At the same time, the drops on the control surface remain randomly distributed, and display more polydispersity and on average smaller sizes (Fig. 1(b)(C-iii)). The transformation in the breath figure under EW (from Fig. 1(b)(EW-ii) to (EW-iii)) occurs within a narrow transition period. Eventually, the drops shed earlier under EW compared to the control surface (Fig. S3 in sup ()).

To quantify the unique distribution of droplets under EW, we project the droplets in the breath figure onto unit cells of width equal to the pitch of the electrode pattern, and ranging from one electrode-centre to the adjacent electrode-centre. Within a unit cell, we calculate the variation of the droplet area fraction along the lateral (–) direction. Here, , where is the droplet contact area along a line parallel to the electrode edges (– direction), and is the unit cell area; represents the average over all unit cells in the breath figure, and .

Figure 2: (a) Colour-coded temporal evolution of the average area fraction distribution of drops along the non-dimensionalized lateral co-ordinate (), over one pitch of the electrode pattern under EW ( V). The temporal variations of the peak values in the distribution for different voltages are shown in insets– (I) V, (II) V. The temporal variations of the peak locations in the distribution are shown by star markers in (b). The transition period over which the two side peaks decay and a single peak emerges at the gap-centre is shown in the insets in (a), and in (b), by the grey area. (c) The non-dimensionalized electrostatic energy landscapes corresponding to representative values of the area-weighted average radius of the drops constituting the peak(s) before and after transition. The evolution in the electrostatic energy minima locations with increasing is shown by the solid black line in (b). Inset shows the electric potential distribution for a conductive droplet under EW (schematic in Fig. 1(a)).

The temporal evolution of shows the gradual development of two similar peaks from the gap edges (Fig. 2(a)). The development of these local maxima describes the alignment of the condensate drops on both sides of the gap-centre (Fig. 1(b)(EW-ii)). In contrast, the corresponding distribution on the control surface remains uniform (Fig. S4 in sup ()). At any time instant, the peak value describes the maximum droplet coverage. The peak position is determined by the preferred alignment location of the dominant bigger drops. Over time, as the droplets coalesce and grow (see Movie S1), gradually increases (red symbols in inset (I) in Fig. 2(a)), and concurrently, the locations of the two peaks gradually shift towards the gap-centre (Fig. 2(b)). As the average radius of the drops constituting a peak exceeds a critical value , the coalescence cascades set in. Consequently, a new peak in emerges at the gap-centre (purple symbols in inset (I) in Fig. 2(a)) while the side peaks start to decay; this marks the beginning of the transition period. Within the transition period, the side peaks disappear and a single peak at the gap-centre survives in the end (Fig. 2(a) and inset (I), and Fig. 2(b)). This single peak in reflects the alignment of the bigger condensate drops at the gap-centre (Fig. 1(b)(EW-iii)). Beyond , these drops grow (Fig. 2(a) and inset (I)) while remaining aligned along the gap-centre (Fig. 2(b)). A similar breath figure evolution is observed for other values of the applied voltage, as shown by the evolution of at V (inset (II) in Fig. 2(a)).

To understand the evolution of under EW, we calculate the electrostatic energy profile over the unit cell for a representative condensate droplet of variable size (schematic in Fig. 1(a)). For a conductive droplet having radius equal to , the electric potential () corresponding to a position is calculated by solving using a finite element method (inset in Fig. 2(c)). Then, the total electrostatic energy of the system is calculated as , where is the electric field, is the vacuum permittivity, is the material dielectric constant, and is the computation domain volume. Subsequently, the landscape (non-dimensionalized as ; is the water-air surface tension) is evaluated as a function of the drop position along the unit cell (for the numerical computation methodology see S9 in sup ()). Before the onset of coalescence cascade i.e. for , is symmetric about the gap-centre with minima (electrostatic potential wells) on either side of it (Fig. 2(c)). The locations of these two potential wells gradually shift towards the gap-centre with increasing (dotted line in Fig. 2(c)). The temporal evolution of the peak locations in (symbols) closely follow the electrostatic energy minima locations (solid black line) corresponding to the increase in (Fig. 2(b)). Hence, we conclude that drops formed due to the coalescence of existing droplets under EW migrate to the electrostatic energy minima locations corresponding to their sizes, culminating in the alignment of the drops. As exceeds , droplets under the side peaks with move towards the corresponding electrostatic energy minima even closer to the gap-centre. Motion of these droplets initiates collisions between droplets on both sides of the gap-centre triggering the coalescence cascades (Fig. S5 in sup ()). The resulting drops with diameters comparable to , along with the adjacent electrodes, form two parallel plate capacitors in series, with the dielectric layer as spacers Mannetje et al. (2014). In this case, is symmetric with a single minimum located at the gap-centre (Fig. 2(c)). Consequently, beyond the approximately monodisperse drops remain aligned along the gap-centre (Fig. 2(b)). These EW-induced coalescence cascades result in a sharp increase of the area-weighted average radius of the droplets (Fig. 3). The larger droplets created due to the coalescence cascades give rise to larger value of , as compared to that for the control surface (Fig. 3). Eventually, the drops shed earlier under EW assisted by the AC-EW induced reduction of contact angle hysteresis Li and Mugele (2008) (Fig. 3).

Figure 3: Evolutions of the area-weighted mean radius of the breath figure droplets without EW (control) and under EW ( V) till the first shedding event. The droplet patterns corresponding to different growth regimes of under EW are shown as insets. The similar variation for V is shown in Fig. S6 in sup ().
Figure 4: (a) Self-similar evolution of the droplet size distribution in the breath figure under EW ( V) before and after the transition period. The time evolution is colour coded with times before transition coloured in shades of blue and late times after transition coloured in shades of red. The triangular markers show the estimates for average drop sizes constituting the peaks in (Fig.2(a)). (b) Corresponding scaling plots of obtained using Eq. 1 before and after the transition. The evolution of within the transition period, and the corresponding scaling plots, are shown in the insets in (a) and (b) respectively.
Figure 5: Evolutions of the breath figure surface coverage without EW (control) and under EW ( V). Inset shows the enhanced heat flux associated with the breath figures under EW. Heat transfer data are recorded in a separate setup; see S10 in sup () for a brief overview.

The EW-induced coalescence cascades also significantly alter the evolution of the droplet size distribution from that established for classical breath figures. The droplet size distribution for the breath figures under EW is bimodal exhibiting a power-law decay for smaller droplets and a bell-shaped distribution for larger droplets (Fig. 4(a)). Here, represents the number of droplets of volume per unit droplet volume and surface area. The temporal variation of is self-similar obeying the same scaling relation Family and Meakin (1988, 1989); Meakin (1992)

(1)

that is valid for classical breath figures without EW (Fig. 4(b) and Fig. S7 in sup ()). Here, , and the exponent depends on the dimensionality of the drops and the condensing surface . For , the coalescence-dominated evolution of the bell-shaped distributions for the larger drops is collapsed by the similarity transformation (Eq. 1) on considering the classical value of (grey shaded region of the scaling plots for in Fig. 4(b)). It must be noted that the smaller drops follow a different scaling that is dependent on the spontaneous nucleation and the small scale growth mechanisms Blaschke et al. (2012); Barenblatt (1996). Importantly, the larger droplets, having coalescence-dominated growth, align within the gaps generating the unique breath figure pattern under EW (Fig. 2(a)). This is substantiated by the fact that the estimates corresponding to are close to the peak of the bell-shaped distribution (triangular markers in Fig 4(a)). Hence, for characterizing the breath figure under EW it is sufficient to describe the evolution of the bell-shaped part of the droplet size distribution. Naturally, the EW-induced coalescence cascades break the geometric similarity of the bell-shaped distribution during the transition period (inset in Fig. 4(b)). Consequently, the evolution of under EW loses its self-similarity in the transition period, unlike the size distribution for the control surface which remains self-similar throughout the coalescence-dominated growth regime (Fig. S7 in sup ()). However, for , the evolution of for the larger droplets again exhibits self-similarity, albeit with a significantly different functional form compared to that obtained for (Fig. 4(b)). The almost uniform sizes of the large droplets manifest in the narrower bell-shaped distribution for (compare the blue versus red scaling plots in Fig. 4(b)). Furthermore, due to the large separation of sizes between the dominant monodisperse droplets and the small background droplets in the breath figure (Fig. 1(b)(EW-iii)), the corresponding bell-shaped distributions are no longer peaked at , contrary to that for and for the control surface (Fig. 4(b); Fig. S7 in sup ()). Interestingly, the alignment of the growing monodisperse drops at the electrostatic energy minima, which always remain at the gap-centre, makes the corresponding size distribution analogous to that for condensation on a line Family and Meakin (1989). This evolution of droplet size distribution is unique to the EW-mediated growth of condensate drops. Furthermore, the significant reduction in for the relatively larger droplets, due to the coalescence cascades (Fig. 4(b)), results in enhanced ‘release’ of surface area; this simultaneously reduces the surface coverage of the breath figure (; is the area of the field of view) (Fig. 5). We observe reduction in the steady state value of under EW, with respect to the control surface (Fig. 5). The reduced coupled with the enhanced shedding of condensate drops, due to the EW-induced increased droplet growth and suppression of hysteresis, results in enhanced heat transfer under EW (inset in Fig. 5). Preliminary heat transfer measurements show a striking increase in net heat flux for dropwise condensation under EW (inset in Fig. 5).

In summary, we have shown that AC-EW actively manipulates condensate droplet pattern leading to unique breath figure characteristics. In the breath figure under EW, the coalescing drops align at the corresponding electrostatic energy minima locations, instead of staying restricted to the centre of mass of the parent drops as in classical breath figures. Hence, the breath figure pattern under EW can be further controlled by simply tuning the electrode geometry, and consequently, the electrostatic energy landscape. We have also shown that the breath figure characteristics under EW lead to significantly enhanced heat transfer, as compared to classical scenario. We anticipate that these effects will be very useful for optimizing applications involving dropwise condensation, like heat exchangers and breath figure templated self-assembly Srinivasarao et al. (2012).

Acknowledgements.
We sincerely thank Kripa K. Varanasi from MIT for allowing us to use the heat transfer measurement setup in his lab, and Karim Khalil for his help during the heat transfer measurements. We thank Arjen Pit for his assistance with the numerical calculations, Daniel Wijnperle and B. Robert for their help with the preparation of the condensation substrates. We acknowledge financial support by the Dutch Technology Foundation STW, which is part of the Netherlands Organization for Scientific Research (NWO), and the VICI program (grant 11380).

Footnotes

  1. thanks: These two authors contributed equally.
  2. thanks: These two authors contributed equally.

References

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  27. See Supplemental Material (SM) for video (Movie S1) of dropwise condensation with and without EW, details of the experimental setup/methodology, image analysis procedure, numerical computation methodology, brief overview of the independent heat transfer measurement experiments, example/explanation for EW-induced coalescence cascades, proof of early shedding of drops under EW, evolution of the area-weighted mean radius of the breath figure pattern for different values of the applied voltage, and average droplet area fraction distribution and evolution of droplet size distribution without EW .
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