Breath Figures under Electrowetting: Electrically Controlled Evolution of Drop Condensation Patterns

We show that electrowetting (EW) with structured electrodes significantly modifies the distribution of drops condensing onto flat hydrophobic surfaces by aligning the drops and by enhancing coalescence. Numerical calculations demonstrate that drop alignment and coalescence are governed by the drop-size-dependent electrostatic energy landscape that is imposed by the electrode pattern and the applied voltage. Such EW-controlled migration and coalescence of condensate drops significantly alter the statistical characteristics of the ensemble of droplets. The evolution of the drop size distribution displays self-similar characteristics that significantly deviate from classical breath figures on homogeneous surfaces once the electrically induced coalescence cascades set in beyond a certain critical drop size. The resulting reduced surface coverage, coupled with earlier drop shedding under EW, enhances the net heat transfer.

Figure 1

(a) Schematic of the substrate used for the condensation experiments. Transparent interdigitated ITO electrodes (red) are patterned on the glass substrate (gray), which is then coated with a hydrophobic dielectric polymer film (green). A schematic of a condensate droplet under EW is also shown. (b) Comparison between breath figures without EW [control (C)] [C(i)–C(iii)] and under EW ($U_{\mathrm{rms}}=150\mathrm{V}$; $f=1\mathrm{kHz}$) [EW(i)–EW(iii)] at different time ($t$) instants. The electrode-gap (e, g) geometry underneath the dielectric film is indicated by the solid red and white lines. Gravity points from top to bottom, i.e., along the negative $y$ direction.

Figure 2

(a) Color-coded temporal evolution of the average area fraction distribution of drops $⟨{\overline{S}}_{\overline{x}}⟩$ along the nondimensionalized lateral coordinate ($\overline{x}=x/l$), over one pitch of the electrode pattern under EW ($U_{\mathrm{rms}}=150\mathrm{V}$). The temporal variations of the peak values $⟨{\overline{S}}_{\overline{x}}{⟩}_p$ in $⟨{\overline{S}}_{\overline{x}}⟩$ for different voltages are shown in insets: (I) $U_{\mathrm{rms}}=150\mathrm{V}$, (II) $U_{\mathrm{rms}}=100\mathrm{V}$. The temporal variations of the peak locations in $⟨{\overline{S}}_{\overline{x}}⟩$ are shown by star markers in (b). The transition period is shown by the gray area in the insets in (a) and in (b). (c) The nondimensionalized electrostatic energy ${\overline{E}}_{\mathrm{el}}$ landscapes corresponding to representative values of the area-weighted average radius ($⟨r{⟩}_p=20,40,60,80,100,110\mu \mathrm{m}$) of the drops constituting the peak(s) in $⟨{\overline{S}}_{\overline{x}}⟩$ before (with two minima) and after (with a single minimum) transition. The evolution in the electrostatic energy minima locations with increasing $⟨r{⟩}_p$ is shown by the solid black line in (b). The inset shows the electric potential $\phi$ distribution for a condensate droplet under EW [schematic in Fig. 1].

Figure 3

Evolutions of the area-weighted mean radius $⟨r⟩$ of the breath figure droplets without EW (control) and under EW ($U_{\mathrm{rms}}=150\mathrm{V}$) until the first shedding. The droplet patterns corresponding to different growth regimes of $⟨r⟩$ under EW are shown as insets. The similar $⟨r⟩$ variation for $U_{\mathrm{rms}}=100\mathrm{V}$ is shown in Fig. S6 in [25].

Figure 4

(a) Self-similar evolution of the droplet size ($s$) distribution ($n_s$) in the breath figure under EW ($U_{\mathrm{rms}}=150\mathrm{V}$) before and after the transition period. The time evolution is color coded with times before and after transition, colored in shades of blue and red, respectively. The triangular markers show the $n_s$ estimates for average drop sizes constituting the peaks in $⟨{\overline{S}}_{\overline{x}}⟩$ [Fig. 2]. (b) Corresponding scaling plots of $n_s$ obtained using Eq. (1) before and after the transition. The $n_s$ evolution 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 ($\mathrm{\Gamma }$) without EW (control) and under EW ($U_{\mathrm{rms}}=150\mathrm{V}$). Inset shows the enhanced heat flux under EW. Heat transfer data are recorded in a separate setup; see Sec. S10 in [25] for details.