A 2D-Planar Dielectrophoretic Model with Electro-Thermally Induced Fluid Motion and the Stability of Trapping Zones

Abstract

Dielectrophoresis (DEP) is an electrokinetics-based phenomenon that involves the motion of a particle due to the interaction between an applied nonuniform electric field and an induced dipole moment. This technique is very effective in particle manipulation and separation. Earlier studies on control-amenable models to describe the motion of a neutrally buoyant, neutrally charged particle in a chamber with a parallel electrode array have restricted the motion of the particle to one dimension. Here, incorporating the electro-thermal fluid motion as well, we present a 2D-planar DEP model and study the effect of electro-thermal fluid motion on particle trapping.

Appendix:  Analysis of the Equilibrium Point/Points on \(\tilde {x}^{*}_{1}=\pm\frac{1}{2}\)

In this case, $\dot{\tilde{x}}_{2}(\tilde{x}_{1} = \pm\frac{1}{2},\tilde {x}_{2})=0$ implies

$$\begin{aligned} \tilde{x}_2^2 \mathrm{e}^{ (\frac{\pi \beta-1}{\beta} ) \tilde{x}_2} = \mp \frac{\bar{v}_{\mathrm{dep}}}{\pi u_0}. \end{aligned}$$

(31)

Note that the existence and uniqueness of a solution for $\tilde{x}_{2}$ in Eq. (31) depends on the signs of the terms $\bar{v}_{\mathrm{dep}}$ and πβ−1. An equilibrium point exists on the $\tilde{x}_{1} =\frac{1}{2}$ line only if $\bar{v}_{\mathrm {dep}}<0$ and similarly on the $\tilde{x}_{1} = -\frac{1}{2}$ line only if $\bar{v}_{\mathrm{dep}}>0$. Assuming that this condition is satisfied, the existence and uniqueness (which depend on the sign of πβ−1) of the solution are studied next.

Case 1 $(\beta\geq\frac{1}{\pi} )$: The LHS of Eq. (31) is such that

$$\begin{aligned} \frac{d}{d \tilde{x}_2} \tilde{x}_2^2 \mathrm{e}^{ (\frac{\pi \beta-1}{\beta} ) \tilde{x}_2} &= \frac {\tilde{x}_2 \mathrm{e}^{\frac{\tilde{x}_2 (\pi \beta-1)}{\beta}} (\tilde{x}_2 (\pi \beta-1)+2 \beta )}{\beta}>0 , \\ \lim_{\tilde{x}_2\rightarrow0} \tilde{x}_2^2 \mathrm{e}^{ (\frac{\pi \beta-1}{\beta} ) \tilde{x}_2} &=0, \\ \lim_{\tilde{x}_2\rightarrow\infty} \tilde{x}_2^2 \mathrm{e}^{ (\frac{\pi \beta-1}{\beta} ) \tilde{x}_2} &=\infty. \end{aligned}$$

Therefore $\tilde{x}_{2}^{2} \mathrm{e}^{ (\frac{\pi \beta-1}{\beta} ) \tilde{x}_{2}}$ is a continuously increasing function of $\tilde{x}_{2}$. Hence (31) has a unique solution. Further, since we have a condition on $|\bar{v}_{\mathrm{dep}}|$ in (28), the unique solution for Eq. (31) is such that $\tilde{x}^{*}_{2}(\bar{v}_{\mathrm{dep}}) < 2 \beta$.

Case 2 $( \beta<\frac{1}{\pi} )$: In this case, we have a solution only if

$$\begin{aligned} |\bar{v}_{\mathrm{dep}}| \leq\frac{4\pi u_0 \beta^2 \mathrm{e}^{-2}}{ (\pi \beta-1)^2}. \end{aligned}$$

(32)

Since $\frac{\mathrm{e}^{-2}}{ (\pi \beta-1)^{2}} > \mathrm{e}^{2( \pi \beta-1) }$, from (28) and (32) we have

$$\begin{aligned} |\bar{v}_{\mathrm{dep}}| < 4 \pi u_0 \beta^2 \mathrm{e}^{2( \pi \beta-1) }<\frac{4\pi u_0 \beta^2 \mathrm {e}^{-2}}{ (\pi \beta-1)^2}. \end{aligned}$$

Therefore, the condition in (28) ensures that the condition in (32) is satisfied. Also, we have two solutions for the equation (31): one solution is such that $\tilde {x}^{*}_{2}(\bar{v}_{\mathrm{dep}})< 2 \beta<\frac{2 \beta}{1-\pi \beta }$ and another is such that $\tilde{x}^{*}_{2}(\bar{v}_{\mathrm{dep}})> \frac{2 \beta}{1-\pi \beta}> 2 \beta$.

The nature of these equilibrium points and their dependence on $\bar {v}_{\mathrm{dep}}$ are studied next.

Nature of the equilibrium points: Let $\tilde{x}^{*}_{2}$ be a solution (in case of $\beta\geq\frac{1}{\pi}$) or one of the solutions (in case of $\beta< \frac{1}{\pi}$) of Eq. (31). Then the Jacobian evaluated at the equilibrium point $(\tilde{x}^{*}_{1} = \pm\frac{1}{2}, \tilde{x}^{*}_{2})$ is given by

$$\begin{aligned} J^* = \begin{bmatrix} \frac{\pm\mathrm{e}^{-\frac{\tilde{x}^*_2}{\beta}} \pi u_0 \tilde {x}^*_2 (\tilde{x}^*_2-2 \beta )}{\beta} & 0 \\ 0 & \frac{\pm\mathrm{e}^{-\frac{\tilde{x}^*_2}{\beta}} \pi u_0 (2 \beta-\tilde{x}^*_2 ) \tilde{x}^*_2-\mathrm{e}^{-\pi \tilde{x}^*_2} \pi\beta\bar{v}_{\mathrm{dep}}}{\beta} \end{bmatrix} \end{aligned}$$

and the corresponding eigenvalues are

$$\begin{aligned} \lambda_1 &= \pm\frac{\pi u_0 \tilde{x}^*_2 \mathrm{e}^{-\frac{\tilde {x}^*_2}{\beta}} (\tilde{x}^*_2-2 \beta )}{\beta}, \\ \lambda_2 &= \frac{\bar{v}_{\mathrm{dep}} \mathrm{e}^{-\pi\tilde {x}^*_2}}{\beta\tilde{x}^*_2} \bigl( \tilde{x}^*_2(1-\pi\beta)-2 \beta\bigr). \end{aligned}$$

The signs of λ ₁ and λ ₂ in the different cases are summarized in Table 1. Note that in either case ($\beta\geq\frac{1}{\pi}$ or $\beta<\frac{1}{\pi}$), we have λ ₁ λ ₂<0 and hence the equilibrium point/points on $\tilde{x}^{*}_{1}=\pm\frac{1}{2}$ is/are always saddle points irrespective of the sign of the parameter $\bar{v}_{\mathrm{dep}}$.

Table 1 Analysis of the equilibrium point/points on $\tilde{x}^{*}_{1}=\pm \frac{1}{2}$