Control-oriented model of dielectrophoresis and electrorotation for arbitrarily shaped objects

The most popular modeling approach for dielectrophoresis (DEP) is the effective multipole (EM) method. It approximates the polarization-induced charge distribution in an object of interest by a set of multipolar moments. The Coulombic interaction of these moments with the external polarizing electric field then gives the DEP force and torque acting on the object. The multipolar moments for objects placed in arbitrary harmonic electric fields are, however, known only for spherical objects. This shape restriction significantly limits the use of the EM method. We present an approach for online (in real time) computation of multipolar moments for objects of arbitrary shapes having even arbitrary internal composition (inhomogeneous objects, more different materials, etc.). We exploit orthonormality of spherical harmonics to extract the multipolar moments from a numerical simulation of the polarized object. This can be done in advance (offline) for a set of external electric fields forming a basis so that the superposition principle can then be used for online operation. DEP force and torque can thus be computed in fractions of a second, which is needed, for example, in model-based control applications. We validate the proposed model against reference numerical solutions obtained using Maxwell stress tensor. We also analyze the importance of the higher-order multipolar moments using a sample case of a Tetris-shaped micro-object placed inside a quadrupolar microelectrode array and exposed to electrorotation. The implementation of the model in Matlab and Comsol is offered for free download.

Figure 1

Illustration of the procedure for extracting the spherical multipolar moments $q_{l,m}$ of arbitrary order for objects having an arbitrary shape. Here a Tetris “$\mathsf{S}$” or “$\mathsf{Z}$”-shaped object placed between two planar electrodes generating the external electric field is used as an example.

Figure 2

Illustration of the superposition principle. Here the three uniform fields are ${\mathbf{E}}_1={\left(1,0,0\right)}^T\mathrm{V}/\mathrm{m}$, ${\mathbf{E}}_2={\left(0,1,0\right)}^T\mathrm{V}/\mathrm{m}$, and $\mathbf{E}={\left(1,1,0\right)}^T\mathrm{V}/\mathrm{m}$. We provide a complete list of all computed spherical multipoles together with their Cartesian equivalents in the Supplemental Material [23].

Figure 3

Geometry used in the FEM solver. The larger sphere $S^{\prime }$ with a radius $R^{\prime }$ is used for setting the potential boundary conditions determining the source polarizing electric field. The smaller sphere $S$ with a radius $R$ defines a virtual surface over which we integrate in Eq. (11).

Figure 4

Generating and denoting the basis elements.

Figure 5

Definition of the angles for rotation of the reference frame.

Figure 6

Dimensions of the Tetris-shaped micro-object used in the example and validation simulations.

Figure 7

Comparison of potentials along the circle laying in the $xy$ plane and enclosing the object as depicted in Fig. 6 for various orders of approximation.

Figure 8

Dimensions of the electrode array and Tetris-shaped micro-object used in the validation simulations.

Figure 9

Comparison of the force prediction obtained by the described model and the reference MST model.

Figure 10

Comparison of the torque prediction obtained by the described model and the reference MST model.