# Electrostatics - Dielectrophoresis

## Part I: About charge and polarization

Our demonstration toward dielectrophoresis begins whith a single charge generating an electric field around it, and the principle of electrical polarization. All is detailed including analytical modeling.

### Charges and electrostatic dipoles

An electric charge at a point is a microscopic particle with an electric charge $Q$ not equal to $0$. The surrounding electrical field $\overrightarrow{E}$ is given by:

$$\overrightarrow{E\left(\overrightarrow{r}\right)}=\frac{Q}{4\pi {\u03f5}_{0}{r}^{2}}\overrightarrow{{u}_{r}}$$

where

- $\overrightarrow{r}$ is the position vector precising where the field is calculated
- ${\u03f5}_{0}$ is the void electrical permittivity
- $\stackrel{}{{u}_{r}}$ is the unit vector in the direction defined by $\overrightarrow{r}$

If now we consider a second charge $-Q$ at a distance $a$ from $+Q$, along ${O}_{z}$ axis so that the distance between $O$ and any of the two charges is equal.

Schematics of an electrostatic dipole with 2 punctual charges

The total contribution of the two charges generates an electric field at any point $M$, located by the $\overrightarrow{r}$ vector in the orthonormal referential exposed before. If $r$ is large enough before $a$ then the electric field is given by:

$$\overrightarrow{E\left(\overrightarrow{r}\right)}=\frac{\overrightarrow{p}.\overrightarrow{r}}{4\pi {\u03f5}_{0}{r}^{3}}\overrightarrow{{u}_{r}}$$

$$\overrightarrow{p}=Q.a.\overrightarrow{{u}_{z}}$$

$\overrightarrow{p}$ is known as *dipolar moment*.

### Polarizability

Some materials, at a certain scale, made of linked charges. These kind of materials are called *dielectric*.
Their charge is neutral from a macroscopic point of view, but they locally show positive and negative charges.

A permanent external electric field *polarization*.

There are 3 kinds of polarizations:

*the orientational polarization*, that concerns materials containing natural dipoles at steady state ; they juste reorganize themselves following the electric field*the atomic polarization*: some molecules are made of ions assembly ; under and electric field these ions will move in response to the electric excitation*the electronic polarization*, this one can be done in any kind of material: electrons gaz around atoms will have a tendancy to move conforming to the electric field.

For any of the considered object, we will define a local dipole:

$$\overrightarrow{p}=\alpha .\overrightarrow{E}$$

where $\alpha $ is a scalar coefficient.

What we call *polarizability* is the capacity to create oriented dipoles or to reorganize existing dipoles conforming to an external electric fied applied to a material.

The material polarizability is obtained by a contribution of all local polarizable entities. So we can define a *polarization vector* $\overrightarrow{P}$ so that:

$$\overrightarrow{P}=\frac{\overrightarrow{{\mathrm{dp}}_{\mathrm{or}}}+\overrightarrow{{\mathrm{dp}}_{\mathrm{at}}}+\overrightarrow{{\mathrm{dp}}_{\mathrm{\xe9l}}}}{\mathrm{d\tau}}$$

in which:

- $\overrightarrow{d{p}_{\mathrm{i}}}$ are the dipolar moments included in a volume $d\tau $
- $d\tau $ is a volume element.

If we appply now an electric field $\overrightarrow{E}$ with an alternative amplitude, we will see a similar microscopic changing of the dipoles, so so will change the polarization vector. But due to physical rules, dipoles cannot change instantly (they depend on charges carriers movement, these ones being more or less mobile). So, above a certain frequency, some of the dipoles won't be able any more to follow electric field variations. This means that the polarization vector will submit a changing according to the dipoles state, and will take other characteristics.

If we go on increasing the frenquency, we will see the same changing as one goes along with the different polarizations mechanisms falling down because of the high speed changing in electric field direction.

Let's note ${f}_{c}$ the first transition frequency from which the polarization does not correspond anymore to the electric field amplitude in static mode. We will talk about *depolarization*. This is directly linked to an internal electric field and become more and more important with the different cut frequency (orientational, atomic and electronic). We will come back on this phenomenon later.

### Polarization effect on the electric field

The polarization vector $\overrightarrow{P}$ in a dielectric material is directly linked to the external electric field $\overrightarrow{{E}_{0}}$. This link can be established, and for a linear homogeneous isotropic material, we have:

$$\overrightarrow{P}=\chi .\overrightarrow{{E}_{0}}={\u03f5}_{0}.{\chi}_{0}.{E}_{0}$$

with $\chi ={\u03f5}_{0}.{\chi}_{r}$.

What we call *electrical susceptibility* for a linear homogeneous isotropic material, and noted $\chi $, is the amplitude factor between the induced polarization vector in the material and the surrounding electrical field. We also note ${\chi}_{r}$ the factor so that:

$$\chi ={\u03f5}_{0}.{\chi}_{r}$$

${\chi}_{r}$ is called *relative electrical susceptibility*. The electrical susceptibility can be a tensor in the case of non-linear and/or non-isotropic material.

So, in the case of a linear, homogeneous isotropic material, we have:

$$\u03f5={\u03f5}_{0}(1+{\chi}_{r})$$

The induced electric field $\overrightarrow{{E}_{\mathrm{int}}}$ in a dielectric material has an external component, that is the external field $\overrightarrow{{E}_{0}}$ itself, plus the induced polarization vector $\overrightarrow{{E}_{p}}$.

$$\overrightarrow{{E}_{\mathrm{int}}}=\overrightarrow{{E}_{0}}+\overrightarrow{{E}_{\mathrm{p}}}$$

We can show that the existence of a polarization effect in a material can be represented as a combination of a surface and volume distributions of charges in the material. So we note:

$$\{\begin{array}{ccc}{\rho}_{\mathrm{p}}& =& -{\mathrm{div}}_{M}\left(\overrightarrow{P}\right)\\ {\sigma}_{\mathrm{p}}& =& \overrightarrow{P}.\overrightarrow{n}\end{array}$$

where ${\rho}_{\mathrm{p}}$ is the virtual volume charge density, and ${\sigma}_{\mathrm{p}}$ is the virtual surface charge density. The divegence calculations being made at the point $M$ in the material, we can calculate the induced electrical field $\overrightarrow{{E}_{\mathrm{p}}}$ in a geometrically limited defined material.