Electrothermally driven flows in ac electrowetting

Mixing within sessile drops can be enhanced by generating internal flow patterns using ac electrowetting. While for low ac frequencies, the flow patterns have been attributed to oscillations of the drop surface, we provide here the driving mechanism of the hitherto unexplained high-frequency flows. We show that: (1) the electric field in the liquid bulk becomes important, leading to energy dissipation due to Joule heating and a temperature increase of several degrees Celsius, and (2) the fluid flow at these frequencies is generated by electrothermal effect, i.e., gradients in temperature give rise to gradients in conductivity and permittivity, the electric field acting on these inhomogeneities induces an electrical body force that generates the flow. We solved numerically the equations for the electric, temperature and flow fields. The temperature is obtained from a convection-diffusion equation where Joule heating is introduced as a source term. From the solution of the electric field and the temperature, we compute the electrical force that acts as a body force in Stokes equations. Our numerical results agree with previous experimental observations.

Figure 1

(Color online) Hydrodynamic flows observed in electrowetting experiments (reproduced with permission from Ref. [5]). An aqueous droplet with conductivity 1.3 mS/m and volume $5\mu \text{l}$ is subjected to an ac signal of amplitude $80{\text{V}}_{\text{rms}}$. (a) At low frequencies the motion is generated by the oscillation of the drop. (b) The amplitude of the oscillations decreases as the frequency increases. (c) Another type of flow appears at higher frequencies, and its origin remains unexplained.

Figure 2

Problem domain. The problem has axial symmetry and a semispherical cap is assumed for the drop shape. A domain much larger than the drop is considered so at the outer boundaries we impose room temperature and zero electrical current.

Figure 3

(Color online) Solution for the electric potential (contour lines), temperature (surface plot) and velocity field (arrows) for a conductivity of $1.3\times {10}^{-2}\text{S}/\text{m}$ and a signal of 100 V and two different frequencies [(a)1 kHz and (b) 100 kHz]. Typical velocities in (b) are around 1 mm/s.

Figure 4

(Color online) Electrothermal force for the case of Fig. 3b, ($1.3\times {10}^{-2}\text{S}/\text{m}$ 100 V, 100 kHz). Normalized arrows indicate the force direction. The force is only intense near the tip of the needle and with a maximum magnitude of nearly $4\times {10}^6\text{N}/{\text{m}}^3$.

Figure 5

(Color online) Fluid velocity generated by electrothermal effect versus frequency of the applied signal. The amplitude of the signal is 50 V. The inset shows the maximum velocity obtained for each conductivity.

Figure 6

(Color online) Fluid velocity generated by electrothermal effect versus amplitude of the applied signal. The frequency of the signal is 50 kHz.