252 6.6  Electrical Force Tools

However, if the E-​field is uniform, the particle’s dipole moment aligns parallel to the field

lines. If this E-​field vector rotated around the particle, then the finite time taken for the induced

dipole to form (the dipole relaxation time), resulting from charge redistribution in/​on the

particle, lags behind the phase of the E-​field, a phenomenon that becomes increasingly more

significant with increasing rotational frequency of the E-​field. This results in a nonzero angle

between the E-​field vector and the dipole at any given time point, and therefore there is a force

on the induced electrical dipole in a direction of realignment with the rotating E-​field. In other

words, the particle experiences a torque (Figure 6.12c) that causes the particle to rotate out of

phase with the field, either with or against the direction E-​field rotation depending on whether

the phase lag is less or more than half an E-​field period. This effect is called electrorotation.

The speed of electrorotation depends on the particle’s surface density of electrical charge,

its radius a (assuming a spherical particle) and the magnitude E, and frequency ν of the elec­

tric field. The torque G experienced by the bead is given by

(6.35)

G

a E

K v

w

= −

( )





4

3

2

πε

Im

where

εw is the permittivity of the surrounding water-​based pH buffer that embodies the charge

qualities of the solution

Im[K(ν)] is the imaginary component of K, which is the Clausius–​Mossotti factor, which

embodies the charge properties of the bead

The full form of the Clausius–​Mossotti factor is given by ε

ε

ε

ε

b

w

b

w

*

*

*

*

−

(

)

+

(

)

/

2

where εw

* is

the complex permittivity of the surrounding water solution and εb

* is the complex permit­

tivity of the bead. A general complex electrical permittivity ε^* =​ ε − ik/​2πν where ε is the real

part of the complex permittivity and k is the electrical conductivity of the water solution.

Typically, there are four microelectrodes whose driving electric currents are phased in

quadrature, which produces a uniform AC E-​field over an area of a few square microns in

between the microelectrodes (Figure 6.12a). For micron-​sized charged beads, an E-​field rota­

tional frequency of 1–​10 MHz with ~50 V amplitude voltage drop across a microelectrode

gap of a few tens of microns will produce a bead rotation frequency in the range 0.1–​1

kHz. Electrorotation experiments have been applied to studies of single rotary molecular

machines, such as the bacterial flagellar motor (Rowe et al., 2003), to characterize the relation

between the machine’s rotational speed and the level of torque it generates, which is indica­

tive of its mechanism of operation.

6.6.6  ABEL TRAPPING

An arrangement of four microelectrodes, similar to that used for electrorotation described

earlier, can be also used in a DC mode. In doing so, an electrically charged particle in the

center of the electrodes can be controllably moved using dielectrophoresis to compensate

for any random fluctuations in its position due to Brownian diffusion. The potential elec­

trical energy on a particle of net charge q in an electric field of magnitude E moving through

a distance d parallel to the field is qEd, and by the equipartition theorem this indicates that

the mean distance fluctuations at the center of the trap will be ~2kBT/​qE. Also, the size of the

dielectrophoretic force F is given by

(6.36)

F

E

K

w

=

∇(

)

( )





2

3

2

π

ε

ν

a

Re

where Re K ν( )



^{is the real component of the Clausius–​Mossotti factor. If the electric field}

can be adjusted using rapid feedback electronics faster than the time scale of diffusion of the

particle inside a suitable microscope sample flow cell, then the particle’s position in the focal

plane of the microscope can be confined to a region of space covering an area of just a few