Controlling wetting with electrolytic solutions: Phase-field simulations of a droplet-conductor system

The wetting properties of immiscible two-phase systems are crucial in applications ranging from laboratory-on-a-chip devices to field-scale oil recovery. It has long been known that effective wetting properties can be altered by the application of an electric field; a phenomenon coined as electrowetting. Here, we consider theoretically and numerically a single droplet sitting on an (insulated) conductor, i.e., within a capacitor. The droplet consists of a pure phase without solutes, while the surrounding fluid contains a symmetric monovalent electrolyte, and the interface between them is impermeable. Using nonlinear Poisson-Boltzmann theory, we present a theoretical prediction of the dependency of the apparent contact angle on the applied electric potential. We then present well-resolved dynamic simulations of electrowetting using a phase-field model, where the entire two-phase electrokinetic problem, including the electric double layers (EDLs), is resolved. The simulations show that, while the contact angle on scales smaller than the EDL is unaffected by the application of an electric field, an apparent contact angle forms on scales beyond the EDL. This contact angle relaxes in time towards a saturated apparent contact angle. The dependency of the contact angle upon applied electric potential is in good agreement with the theoretical prediction. The only phenomenological parameter in the prediction is shown to depend on the permeability ratio between the two phases. Based on the resulting unified description, we obtain an effective expression of the contact angle which can be used in more macroscopic numerical simulations, i.e. where the electrokinetic problem is not fully resolved.

Figure 1

Schematic setup of the numerical experiment. Here, $\mathbf{d}$ indicates the droplet phase, $\mathbf{s}$ indicates the surrounding phase, and $\mathbf{e}$ indicates the electrode. The figure shows the final state after the application of a potential difference $V_0$ between the two electrodes. Due to the dissolved electrolytes in phase $\mathbf{s}$, an electric double layer, characterized by the Debye length ${\lambda }_{\mathbf{s}}$ is formed near the lower electrode, and an apparent contact angle $\theta$ is formed. Also indicated with a dotted line is the initial state of the droplet (where $V_0=0$), forming the contact angle ${\theta }_0$. Note that the simulations considered herein exploit the indicated axial symmetry of the problem. A close-up view of the contact line shows how the contact angle ${\theta }_0$ persists on small scales, whereas the apparent contact angle $\theta$ is only evident on sufficiently large scales.

Figure 2

Typical mesh used in simulations. The zero-level set of the phase field is shown as a solid yellow line.

Figure 3

Relaxation to the apparent contact angle when an electric field is suddenly applied. The electric potential difference is turned on to $V=2.5$ at time $t=0$. The red color in the surrounding fluid shows the net charge, and thus represents the EDL. (a) to (f) show increasing simulation time.

Figure 4

Comparison of the droplet shape for different droplet size, when the Debye length and other parameters are kept constant. The Debye length is ${\lambda }_{\mathrm{s}}=\sqrt{{\epsilon }_{\mathrm{s}}/\left(2c_0\right)}\simeq 0.071$. The inset shows a close-up of the contact line.

Figure 5

Contact angle in time for a range of potential drops $V_0$. The lines between points are linearly interpolated for visual clarity.

Figure 6

We plot the apparent contact angle as a function of applied potential, for a range of parameters. The simulation sets A–J correspond to the parameter sets reported in Table 2.

Figure 7

We plot the quantities involved in Eq. (22) for a range of parameters.

Figure 8

Computed slopes from the data in Fig. 7 and several other sets of simulations, plotted against permittivity ratio ${\epsilon }_{\mathbf{d}}/{\epsilon }_{\mathbf{s}}$, shown along with a least squares fit.

Figure 9

Collapse of the contact angle data involved in Eq. (40), using the same data as presented in Figs. 6 and 7. The black solid line indicates an exact relationship between the ordinate and the abscissa.

Figure 10

The relaxation times for our simulations obtained by fitting exponential functions to the contact angle in time. The data correspond to what is shown in Fig. 6. Inset: Data collapse obtained by using a dimensionless relaxation time based on surface tension ${\sigma }_{\mathrm{ds}}$, viscosity $\mu$, and the initial wetting length scale ${\ell }_0$. The outliers at low potential/contact angles are due to the poor fit of an exponential function to the data when the contact angle changes only slightly.

Figure 11

Comparison of the evolution of the apparent contact angle as a function of time for the full model, including electrochemistry, and two-phase flow without direct resolution of the electrodynamics but instead using the boundary condition (42) and the relationship (40).