Page 273 - Electrical Properties of Materials

Dielectrophoresis 255

of electric flux imply that D must vary only in the radial direction as 1/r, and y
hence E must also vary in this way.
Shell
Let us now place the dielectric particle between the wire and the shell, on the
x-axis. There may also be a surrounding medium, but the effect will also work E (r)
in a vacuum. Let us assume that the negative charge centre is at x so the field it
Wire EE + dE
experiences is E(x). The force it experiences is therefore –qE in the x-direction.
The field experienced by the positive charge centre has the slightly different V −q +q
x
value E +dE since the positive and negative charge centres are separated by a xx + dx
distance dx. For small distances, dE =(dE/dx)dx, so the force on the positive
charge is +q{E +(dE/dx)dx}. Clearly, the two forces acting on the particle no
longer balance, and there is a net force in the x-direction of

F x = q(dE/dx)dx. (10.77)
Fig. 10.28
Arrangement for generating a
For the geometry here, dE/dx is negative, so the particle experiences an nonuniform electric field, and its
attractive force towards the high-field region at the origin. effect on a dielectric particle.
Now, qdx is simply the polarization of the particle, which as we have already
established depends on the electric field and the polarizability α. Consequently,
we can write the force as
2
F x = αE(dE/dx)=(α/2) d(E )/dx. (10.78)

This fundamental result implies that DEP effects are completely different to
electrostatic forces. There is no dependence on the sign of the electric field, and
2
nothing will happen unless the electric field varies in space, so that d(E )/dx
is non-zero. More generally, the field will vary in more than one direction.
Consequently, there will be forces in other directions. We can guess that F y =
2
(α/2) d(E )/dy, and hence that the total force in a three-dimensional system
can be written as a vector,
2
F =(α/2)∇(E ). (10.79)

Here ∇ is the operator ∂/∂x i+ ∂/∂y j+ ∂/∂z k.
The remaining details are more complicated. Firstly, the polarizability α
depends on any surrounding medium (often a liquid in DEP experiments).
Secondly, the electric field may be time varying rather than static. Clearly the
effect must depend on the properties of both the particle and the medium at the
frequency involved. For a spherical particle of radius R and complex permit-
tivity ε P in a medium of complex permittivity ε M , experiencing a RMS field
E rms at angular frequency ω, the time-averaged DEP force is

3
2
F =2πR ε M Re(f CM ) ∇|E rms | . (10.80)
Here f CM is the Clausius–Mossotti factor,

f CM =(ε P – ε M )/(ε P +2ε M ) (10.81)

that we have already met in Section 10.10, in a simplified form, when we
considered a lossless dielectric in free space.