Discrete Contact Angles and Electric Field Singularity in Electrowetting: A Multi-Scale Complex Potential Analysis

TL;DR

The paper tackles electric-field singularities at the Triple Contact Point in electrowetting by developing a multi-scale framework that couples global and local potentials through conformal mapping and complex analysis. It constructs a far-field solution via a Joukowski transform and extends it to the TCP region, yielding an eigenvalue problem $\tan(\pi\lambda)= k \tan((\pi-\theta)\lambda)$ that binds the local contact angle to the characteristic exponent $\lambda$. Key findings show that admissible contact angles are discrete, with non-singular fields requiring $\text{Re}[\lambda] \ge 1$ and singular behavior confined to $\theta<\pi/2$; high-order modes exhibit degeneracy at special angles and oscillations occur only when both $\theta \to \pi$ and $k\to 1$. The results offer design principles for achieving non-singular electric fields in electrowetting and provide microscopic insight into contact-angle saturation phenomena.

Abstract

This study constructed a multi-scale theoretical framework to resolve the electric field singularity at the Triple Contact Point in electrowetting. Utilizing conformal transformation and complex analysis, we established the structure for both the global potential and local field solutions, complementing the analysis with numerical methods. Our primary finding is that the contact angle $θ$ is not continuously adjustable but is restricted to a discrete set of values, constrained by the characteristic exponent $λ$. Analysis of the complex potential established $\text{Re}[λ] \ge 1$ as the critical condition for a non-singular electric field; conversely, singular solutions ($\text{Re}[λ] < 1$) are localized exclusively in the acute-angle regime ($θ< π/2$). The high-order solution region exhibits a degeneracy phenomenon at specific angles, implying the local field structure is geometrically stable and universally applicable for a wide range of permittivity ratios $k$. Furthermore, we determined that the onset of electric field oscillation requires the simultaneous satisfaction of two critical conditions: the geometry must approach a flat boundary ($θ\to π$) and the dielectric ratio must approach homogeneity ($k \to 1$). These findings provide a solid theoretical basis for designing non-singular electric fields and mitigating the common contact angle saturation phenomenon.

Figures (9)

• Figure 1: The model depicts a thin droplet slit (dashed line) on a dielectric layer (shaded area, permittivity $\epsilon_{\gamma}$) with air above (blank space, permittivity $\epsilon_0$). The slit itself is an equipotential surface (V.cst). The voltage is continuous (V.cts) along the dielectric-air interface (thick black lines). A induced surface charge density $\sigma_b$ resides on the dielectric surface (zero in air). The electrode effect is simplified to a variable point charge (red spot) embedded within the dielectric. V.$\infty$ is the far-field voltage.
• Figure 2: The equipotential lines illustrate the complex potential $w_+(z)$ and $w_-(z)$. The dielectric medium occupies the region $y<0$, while the slit is located on the boundary at $x\in\pm1$. The source charge is positioned at the center of the concentric circles within the dielectric. Note the dense equipotential lines around the two cusps, which resemble flow past a slit. A slight voltage distortion is visible along the interface $y=0$ (the dielectric-air boundary) due to the presence of a non-zero surface charge density.
• Figure 3: The estimated droplet boundary curve for varying charge magnitude $q$. The curves are parametrized by $B$ (representing $q$), where the $B$-axis illustrates the effect of increasing the charge magnitude.
• Figure 4: (a) corresponds to equipotential lines in the air region ($y>0$). This configuration includes the source charge and two induced image charges within the unit disk. (b) corresponds to equipotential lines in the dielectric region ($y<0$). The potential in this region is defined by six point charges, two of which are located at the origin.
• Figure 5: Zoomed-in view near the TCP. The potentials $V_1$ and $V_2$ is induced by some unknown far-field source. The shaded region is the dielectric (potential $V_2$), the unshaded region is the conductive droplet (constant potential $V_3 \equiv 0$), and the upper region is the air (potential $V_1$). $\theta$ is the angle between the dielectric and droplet curve.