A numerical investigation on AC electrowetting of a droplet

Abstract

We numerically analyze the AC electric field around a droplet placed on an insulator-covered electrode. The time-averaged effective electrical wetting tension, which is a function of AC frequency, is computed by integrating the Maxwell stress. The computed wetting tension is compared with the experimental result converted from the separately obtained contact-angle data. There is a good agreement between the two results at a low-frequency range and a qualitative agreement at a high-frequency range. Interestingly, the numerical results show that the electric-field strength decreases remarkably in the insulating layer near the TCL as the AC frequency increases. This decrease may account for the delay of the dielectric breakdown of an insulating layer in the AC case, which could be related to the contact-angle saturation phenomenon.

Appendices

Appendix 1: Time average of the Maxwell stress and the RMS electric-field strength

There is the following time-average theorem for the two time-harmonic complex variables of $u = \text{Re} \{ \ifmmode\expandafter\tilde\else\expandafter\sim \fi{u}e{}^{{j\omega t}}\}$ and $v = \text{Re} \{ \ifmmode\expandafter\tilde\else\expandafter\sim \fi{v}e{}^{{j\omega t}}\}$ (Jackson 1999):

$$ {\left\langle {u\,v} \right\rangle } = \frac{1} {2}\text{Re} \{ \ifmmode\expandafter\tilde\else\expandafter\sim \fi{u}\, \ifmmode\expandafter\tilde\else\expandafter\sim \fi{v}^{*} \} . $$

(10)

Here, the bracket ${\left\langle {\text{ }} \right\rangle }$ denotes the time average and the superscript ‘*’ a complex conjugate. The directional vectors are time-independent, so that the time average of the electrical stress in r-direction in Eq. (8) can be rewritten as

$$ {\left\langle {{\left( {{\mathbf{T}} \cdot {\mathbf{n}}} \right)} \cdot {\mathbf{e}}_{{\text{r}}} } \right\rangle } = {\left( {{\left\langle {\mathbf{T}} \right\rangle } \cdot {\mathbf{n}}} \right)} \cdot {\mathbf{e}}_{{\text{r}}} . $$

(11)

The time average of the Maxwell stress tensor becomes from Eq. (10),

$$ {\left\langle {\mathbf{T}} \right\rangle } = \frac{1} {2}\text{Re} {\left\{ { - \frac{1} {2}\varepsilon ({\tilde{\bf{E}}} \cdot {\tilde{\bf{E}}}^{*} ){\mathbf{I}} + \varepsilon {\tilde{\bf{E}}}{\tilde{\bf{E}}}^{*} } \right\}}. $$

(12)

The complex electric field is decomposed as ${{\tilde{\bf{E}}}} = \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{r} {\mathbf{e}}_{r} + \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{z} {\mathbf{e}}_{z}$ which is substituted into Eq. (12) to obtain

$$ {\left\langle {\mathbf{T}} \right\rangle } = \frac{1} {2}\text{Re} {\left\{ { - \frac{1} {2}\varepsilon ( \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{r} \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}^{*}_{r} + \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{z} \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}^{*}_{z} ){\mathbf{I}} + \varepsilon ( \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{r} {\mathbf{e}}_{r} + \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{z} {\mathbf{e}}_{z} )( \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}^{*}_{r} {\mathbf{e}}_{r} + \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}^{*}_{z} {\mathbf{e}}_{z} )} \right\}}. $$

(13)

The time-averaged Maxwell stress in Eq. (13) is substituted to Eq. (11) and normal vector is decomposed as n = n _r e _r+n _z e _z . Then the following result is obtained:

$$ {\left\langle {{\left( {{\mathbf{T}} \cdot {\mathbf{n}}} \right)} \cdot {\mathbf{e}}_{{\text{r}}} } \right\rangle } = \frac{1} {2}\text{Re} {\left\{ {\varepsilon \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}^{*}_{r} {\left( { \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{r} n_{r} + \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{z} n_{z} } \right)} - \frac{1} {2}\varepsilon {\left( { \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{r} \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}^{*}_{r} + \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}_{z} \ifmmode\expandafter\tilde\else\expandafter\sim \fi{E}^{*}_{z} } \right)}n_{r} } \right\}}. $$

(14)

Similarly, the RMS electric-field strength becomes

$$ E_{{rms}} = {\sqrt {{\left\langle {{\mathbf{E}} \cdot {\text{ }}{\mathbf{E}}} \right\rangle }} } = {\sqrt {\tfrac{1} {2}\text{Re} \{ {{\tilde{\bf{E}}}} \cdot {\tilde{\bf{E}}}^{*} \} } } = {\sqrt {E^{2}_{{r,rms}} + E^{2}_{{z,rms}} .} } $$

(15)

Appendix 2: Average value of the Maxwell stress just below the TCL

We introduce the control surface Σ = S∪S ₁∪S ₂···S ₅ as shown in Fig. 13. Here, S denotes the surface used to obtain the average value in the numerical analysis. The conservation of electromagnetic momentum condition for the domain enclosed by Σ is written as follows (Kang et al. 2003):

$$ {\int \limits_\Sigma {{\mathbf{T}} \cdot {\mathbf{n}}\;dS} } = {\int \limits_S {{\mathbf{T}} \cdot {\mathbf{n}}\;dS} } + {\int \limits_{S_{{\text{1}}} \cup S_{2} \cdots S_{5} } {{\mathbf{T}} \cdot {\mathbf{n}}\;dS} } = 0. $$

(16)

Here, n the outward normal vector for each surface. By using Eq. (16), one can calculate the force exerted on the surface S from the sum of the forces acting on other surfaces, that is,

$$ {\int \limits_S {{\mathbf{T}} \cdot {\mathbf{n}}\;dS} } = - {\int \limits_{S_{{\text{1}}} \cup S_{2} \cdots S_{5} } {{\mathbf{T}} \cdot {\mathbf{n}}\;dS}.} $$

(17)

The average force acting on S can be also expressed as follows:

$$ {\left\langle {{\mathbf{T}} \cdot {\mathbf{n}}} \right\rangle }_{S} = \frac{1} {S}{\int \limits_S {{\mathbf{T}} \cdot {\mathbf{n}}\;dS} } = - \frac{1} {S}{\int \limits_{S_{{\text{1}}} \cup S_{2} \cdots S_{5} } {{\mathbf{T}} \cdot {\mathbf{n}}\;dS}.} $$

(18)

It is quite clear the electric fields on the surfaces S ₁, S ₃ and S ₅, which are nonzero contributions, have well-defined finite values. In conclusion, the average stress near the TCL is finite and then the average electric field is also finite.

Fig. 13

Control surface for calculation of average stress just below the TCL