dielectrophoresis and ac electrokinetics: theory

Dielectrophoresis and AC Electrokinetics: Theory

Nicolas G Green

School of Electronics and Computer Science

University of Southampton, Highfield, Southampton, UK

Brief outline of lectures

Quasielectrostatic systems and polarisable particles

• Polarisation and mechanisms

• Polarisation of spherical particles

• Derivation of terms relevant to forces and impedance

• Real particles and other models – shells and surface conductance

• Non-spherical particles – shape factors and dipole moments

• Impedance of suspension – mixture theory

AC electrokinetics: Dielectrophoresis

• Derivation of force expression

• Discussion of scaling versus other physical forces on particles

• Particle-particle interactions – pearl chains

Brief outline of lectures

AC electrokinetics: other forces

• Electrorotation

• Travelling Wave Dielectrophoresis

• Particle-particle interactions – pearl chains

Example calculations of AC electrokinetic forces and torques in real systems

• Analytical models and methods

• Numerical simulation

Other considerations

• Higher order effects and multipoles

• Designing systems for nanoparticles – DEP vs Brownian motion

Quasielectrostatic systems: basic equations

Electrostatics and quasielectrostatics

As discussed in Prof Morgan’s introductory lecture, dielectrophoresis (DEP) is the motion of polarisable particles in non-uniform electric fields. Other AC electrokinetic effects are similar in nature.

In order to discuss the theory behind DEP, we start with a discussion of the basics of polarisation and dielectric materials.

The background equations in electromagnetism are Maxwell’s equations. In the quasi-electrostatic limit, charges are assumed to be stationary or have only small velocities and acceleration. Charge motion does not affect the electric field and magnetic fields are assumed to be negligible.

Basic equations – electrostatics

Coulomb’s Law: force on a charge

The Electric Field

Electrical potential E = .

Q1

Q2 F12

r

1 22

ˆ4 o

Q Q

r12 12F r

2ˆ

4 o

Q

r 12E r

Q

Basic equations

Gauss’s law is the first equation of interest, relating the electric field to the free charge density:

In terms of the electrical potential, this is

If the charge density is zero, this becomes Laplace’s equation

o

Ε

2

o

2 0

Currents

Ohm’s law relates current density, electric field and electrical conductivity

The current density is related to the free charge density by the average drift velocity

Equating these two expressions gives

The last equation is the charge conservation equation:

In the steady state

f J E

cf fJ v

f f cv

E

which relates the electrical conductivity to the mobility of the charge in the electric field.

ff t

J

0f J

The electrical dipole

The simple electrical dipole is formed by the displacement of two equal and opposite charges

d

Q +Q


The dipole moment is defined as the vector

The electrical potential is given by

and the electric field by

Qp d

2 2

cosˆ

4 4o or r

pp r

3ˆˆ(2cos sin )

4 or

pE r θ


The potential energy of the dipole in a uniform field

The force on the dipole is

The torque on the dipole is

cosU Qd E E p

( ) F p E

Γ p E

+Q

Q

EF+

F

Dielectric materials and polarisation

A dielectric material is a material that contains charges which polarise under the influence of an applied electric field.

• The charges are bound within the material and only move short distances

• Negative and positive charges move in opposite directions to form induced dipoles

• Some materials also contain molecules with permanent dipoles – POLAR dielectrics

Average dipole moment of material is proportional to E

The Polarisation (dipole moment per unit volume)

Bound volume charge density

'avp E

'n n avP p E

b P

Polarisation

Polar dielectrics polarise by orientation of fixed dipoles on the molecules in the dielectric.

Non-polar dielectrics polarise by three further mechanisms, with the assumption that total polarisation is the sum of the individual components.

• Electronic polarisation – electron cloud around atoms displaces from nucleus

Positive nucleus

Electron cloud

Unpolarised (no field)

Polarised

E

Centre of negative charge Dipole

Polarisation

• Atomic polarisation – in certain types of solid, charged ions will move in opposite directions in the applied field

• Counterion or interfacial polarisation – long-range charge transport causes polarisation of dielectrics. Charges become trapped at places where the dielectric is inhomogeneous (internal interfaces) or at the surface, causing macroscopic distortion of the field. Similarly, relatively long-range transport of ions occurs along and around the surface of polyelectrolytes. This manifests itself as an increase in the charge storage capacity of the dielectric or an increase in the permittivity of the dielectric.

This last phenomenon is the basis of dielectrophoresis and AC electrokinetics. Similarly, this is important for impedance analysis of non-homogeneous materials such as particle suspensions as will be discussed later.

Equations for dielectric materials

One of the most important terms is the electric flux density, derived from Gauss’Law by expanding the charge term into different types:

i.e. free and bound charge. Using

The vector is referred to as the electric flux density

Gauss’ law can be re-written as

f b

o

E

( )o f E P

o D E P

f D

Other related terms

Other parameters are often defined at this point:

Dielectrics are usually linear and isotropic giving a proportionality between Pand E, referred to as the electrical susceptibility.

Putting this into the expression for D gives

The term in brackets defines the relative permittivity

with the permittivity of the dielectric given as

o ae P E

(1 )o ae o r D E E

1r ae

o r

Real dielectrics and complex permittivity

The equations so far have been for ideal homogeneous dielectrics. A large number of real dielectrics can also be modelled as homogeneous dielectrics but including the conductivity of the material.

A parallel plate capacitor can be used

to illustrate

The capacitance is

and the impedance is

Notes: complex frequency dependent applied potentials:

angular frequency is f and the imaginary unit is defined

Re[ ]i toe

0

V

d

V

0

d

o r

AC

d

1Z

i C

2 1i

Real dielectrics and complex permittivity

Replace the dielectric with a homogeneous material with permittivity and conductivity.

Impedance is

0

V

d

V

0

d

o r

AC

d

1 dR

A

1

1 1

RZ

R i C i RC

1

AC

d Equivalent to with

1

1Z

j C

ε i

with Complex Permittivity

Alternative derivation

A more rigorous definition of complex permittivity can be performed using Gauss’ Law, the charge conservation equation and assuming harmonic fields:

For harmonic fields, the time derivative is

Giving the following equation with

Re[ ]i toe

E E

it

( )fi E( )i

E E

( ) 0 E

fft

J

f D

0i

E

ε i

Complex conductivity

Looking at the currents in the system gives an alternative description:

Displacement current:

Conduction current:

Total current is

giving a complex conductivity

d it

D

J E

f J E

( )f d i J J J E E

i

Summary

Looking again at the polarisation P, for a real dielectric, both conduction currents and displacement currents flow.

This results in a phase shift between the driving field and the induced polarisation, with the polarisability now a complex quantity:

Often the complex permittivity has to include further complex terms arising from relaxation mechanisms and is written as:

1oo

n

P E E

i

Dielectric relaxations

The polarisation of a dielectric involves the movement of charge to create dipoles. The rate of this movement is finite and the time to reach maximum polarisation is characteristic to each polarisation mechanism.

At low frequencies, the dipoles have sufficient time to align with the field. As the frequency increases, the period of time that the dipole moment has to ‘relax’ in and then follow the field decreases. There is a corresponding characteristic frequency for each mechanism:

• at this frequency maximum energy is dissipated

• below this frequency maximum polarisation and energy storage occurs

• well above this frequency, no polarisation occurs due to this mechanism

For example water is a polar molecule and has a relaxation frequency of approximately 21010 Hz. It has a relative permittivity of ~80 below 108 Hz and above 1011 Hz, a relative permittivity of ~2. The failure of the permanent dipoles to orient with the field results in a decrease in energy storage (and permittivity).


Orientational polarisation has the longest relaxation time: atomic and electronic polarisation relaxation frequencies are ~1014 Hz and considered to be constant:

The orientational polarisation is a complex function:

The total polarisation is

ae o ae P E

1o or

orori

P E

Low frequency limiting susceptibility due to orientational mechanism

Orientational relaxation time constant

1or

tot o aeori

P E


Low frequency limit:

High frequency limit:

Therefore:

and

where

1ae or s

1ae

or s

( 1)tot o d P E

Static relative permittivity

“high” frequency relative permittivity

1s

dori


Total complex permittivity is therefore:

Usually written as:

With the Debye relations:

A non-zero ’’ implies a phase angle between D and E and a loss of energy in the system: Joule heating (ohmic loss) from the second term and energy loss due to friction of the rotating molecules from the first term.

1s

oor

ii

i

2 21s

oor

2 2

( )''

1s or

oor

Ideal dielectric relaxation

This figure shows the real and imaginary parts of an ideal Debye relaxation. The behaviour of the real part is sometimes referred to as a dispersion.

Dielectric relaxations: Cole-Cole

The dotted line in the figure represents a spread of relaxation times – typical of some materials, a concept introduced by Cole and Cole:

11 ( )s

o

or

ii

The is a factor which represents the “spread” of the relaxation times. = 0 for the ideal case.

Dielectric relaxations: Cole-Cole

A useful method of analysing this type of data is by using a Cole-Cole plot of imaginary part against real part. For the relaxations shown previously, a Cole-Cole plot shows a circular arc with centre on the axis (single frequency) and below the axis (spread of frequencies)

Also shown is the relationship between the angle and the parameter .

Interfacial polarisation

Real systems often consist of a number of different dielectrics each with different electrical properties. As mentioned previously, charges build up at the discontinuities resulting in what is referred to as interfacial polarisation.

When an electric field is applied to the system, net surface charge accumulates at the interfaces between the different dielectrics due to the differences in electrical properties. Since the polarisabilities of each dielectric are frequency dependent, the magnitude of the surface charge is also frequency dependent and the total complex permittivity of the system exhibits relaxations solely due to the polarisation of the interfaces.

Interfacial polarisation: example

This mechanism can be illustrated using parallel plate capacitors:

With a bit of algebra, this can again be re-written as a single capacitance with an equivalent complex permittivity:

1

1

2

2

0V

d1

d2

d

1 11 2

1 1 2 21 1

R RZ Z Z

i R C i R C

T

AC

d 1

T

Zj C

Interfacial polarisation

This time we use the Debye relations to define

with

equivalent conductivity

and relaxation time

From this example, we can see that measurement of the frequency response in the system can inform us about the properties (including relative sizes) of the dielectrics.

2 2 2 2

( )

1 1lf hf lf hf or

o hf oo

i i

1 2

1 2 2 1hf

d

d d

2 21 1 2 2 2 1

21 2 2 1

( )

( )lf

d d d

d d

1 2 2 1

1 2 2 1o

d d

d d

1 2

1 2 2 1

d

d d

Relaxation frequencies: summary

In general, relaxations due to interfacial polarisation occur at frequencies below those for orientational and electronic/atomic polarisation. The complex permittivity of a real system would exhibit all the different polarisation mechanisms

Polarisable particles and interfacial polarisation

Interfacial polarisation: polarisable particles

In AC electrokinetics and impedance/dielectric spectroscopy, the system is a suspension of dielectric particles in dielectric fluids (generally electrolytes). When an electric field is applied to the system, surface charge accumulates at the interfaces between the particles and the fluid due to the differences in electrical properties.

Points to note:

• both dielectrics polarise

• the charge induced at the interface between the particle and themedium is a net effect of the charges in the two materials

• opposite charge on either side of the particles produces an effective dipole moment on the particle parallel to the field

An electric field is applied to a dielectric particle in a dielectric medium:

• both dielectrics polarise

• charge is induced at the interface between the particle and the medium

• opposite charge on either side of the particles produces an effective dipole moment on the particle parallel to the field

Dielectric particles


m , m

p , p

If the polarisabilities are equal, the charge on either side of the interfaces is balanced.


If the polarisabilities are equal, the charge on either side of the interfaces is balanced.

p

m


If the particle polarisability is greater than the fluid, there is a charge imbalance and the particle has an effective

dipole moment aligned with the field.

Net effective dipole

p

m


If the particle polarisability is less than the fluid, there is again a charge imbalance but the effective dipole moment of the particle is aligned against the field.

Net effective dipole

p

m

Interfacial polarisation: polarisable particles

Further, since the polarisabilities of each dielectric are frequency dependent, the magnitude of the surface charge is also frequency dependent and the total complex permittivity of the system exhibits relaxations solely due to the polarisation of the interfaces.

This is referred to as Maxwell-Wagner interfacial polarisation.

As will be shown, the net effect in this case is that an effective dipole moment is induced in the particle. In impedance spectroscopy, this represents an additional relaxation mechanism by which particle properties can be measured. In AC electrokinetics, this represents a dipole moment on which forces can be generated.

Interfacial polarisation: effective moments

In this section, we will examine the polarisation of particles suspended in a dielectric, which in experiments would be a fluid. The dielectrics will be treated as homogeneous.

The derivation for the electrostatic case is presented as this is simplest way of defining the effective dipole moments of particles are determined.

The modifications for a selection of other simple particle models will then be presented as extensions of this basic derivation.

General assumptions:

• Linear, isotropic (homogeneous) dielectrics

• Electrostatic or quasielectrostatic system

• Single frequency harmonic field

Effective dipole moment of a spherical particle

The simplest case is that of a solid dielectric sphere of radius a suspended in a dielectric medium. The spherical dielectric particle in an applied electric field is a classic problem found in most textbooks.

The applied electric field far from the origin is taken to be uniform and anti-parallel to the z-axis.

Without loss of generality, the sphere can be assumed to have its centre at the origin, making the problem axially symmetric about z and two-dimensional in spherical polar coordinates: r and .

xy

z

a

r (r,,)

ˆE E z


In order to determine the behaviour of this system, we want to solve Laplace’s equation

The general solution of axially symmetric problems is given by a series sum of the Legendre polynomials:

2 0

( 1)

0

( ) (cos )n nn n nA r B r P

2

3

4 2

5 2

(cos )

0 1

1 cos

3cos 12

2

5cos 3cos3

2

35cos 30cos 34

8

63cos 70cos 15cos5

8

nn P


Boundary conditions:

1. The electric field cannot be infinite, so the potential is continuous at r = a.

2. The normal component of D is continuous at r = a. Gauss’s Law gives:

0m pr a r a

ˆ ˆ 0 0pmm m p p m p

r a r ar r

E n E n



3. The electric field E lies along the z-axis implying that the potential with no sphere present must be = Ercos. This must still be true at large r for the case of the potential outside the sphere since the contributions to the electric field from the charges in and around the sphere are inversely proportional to r.

As a consequence, the particular solution for the potential in the medium must contain the term Ercosand no other positive power of r.

( 1)

0

cos ( ) (cos )nm n nEr A r P



4. The electric field must also remain finite inside the particle. As a consequence, must be finite as r 0 implying that the particular solution for the potential inside the sphere must contain no negative powers of r.

The solution for the potential inside the particle is therefore:

0

( ) (cos )np n nB r P


The continuity boundary conditions (1 & 2) for the interface (r = a) give

Expanding:

The Legendre polynomials are orthogonal and the coefficients of each polynomial on either side of the equation must be equal:

20 1 2 20 1 2 22 3

cos (cos )cos ... cos (cos ) ...

A A A PEa B B a B a P

a a a

m p pmm pr r

and

20 1 20 1 22 3, , , ...

A A AB Ea B a B a

a a a


The continuity boundary conditions (1 & 2) for the interface (r = a) give

Expanding:

The Legendre polynomials are orthogonal and the coefficients of each polynomial on either side of the equation must be equal:

0 1 2 21 2 22 3 4

2 cos 3 (cos )cos ... cos 2 (cos ) ...m p

A A A PE B B aP

a a a

m p pmm pr r

and

0 1 21 22 3 4

2 30, , 2 , ...m m m p m p

A A AE B B a

a a a


Combining these equations:

gives

20 1 20 1 22 3, , , ...

A A AB Ea B a B a

a a a

0 1 21 22 3 4

2 30, , 2 , ...m m m p m p

A A AE B B a

a a a

0 0 0A B 0 1n nA B n m and p are both positive numbers & the equivalencies

are opposite in sign

1 113 3

13

2

( 2 ) ( )

m p p p

p m p m

A AE B E

a aA

Ea

31

( )

( 2 )p m

p m

A Ea

3 33 3

1 1 1

3 3 3 31 1

2 2

2 2

p

m

m p m m

a aa B a E A B E

a B a B a E a E

1

3

( 2 )p

p m

B E


Particular solution inside the sphere

and outside the sphere

which can be re-written as

This is the scalar sum of the potential due to the applied field (second term on the right hand side) and the potential from a dipole moment (first term, right hand side).

3

31 cos

2p m

mp m

aEr

r

3cos

2m

pp m

Er

32

coscos

2p m

mp m

Ea Err


Comparison with the equation for the potential from a dipole:

gives by inspection the effective dipole moment of the particle

2

cos

4 mr

p

342

p mm

p m

a

p E

This is the electrostatic model for the particle and would represent the behaviour only of ideal dielectric particles.

A note for later, we can define a relationship between the effective moment and the Legendre polynomial coefficient in the medium A1

31

( )

( 2 )p m

p m

A Ea

14 mp A


In order to determine the behaviour of real dielectrics, the conductivity of the dielectrics must be included. The model is affected by the conductivity through Gauss’s Law and the second boundary condition dealing with the normal component of D.

The continuity equation for transport of charge carriers to and from the surface is

Substituting for the time derivative and equating f

pmm p f

r a r ar r

f pmm p

r a r at r r

0pmm p

r a r ar r


Alternatively, we could have started with the premise that the boundary conditions are determined by the complex equation:

The boundary conditions now including the conductivities of both particle and medium are the same as for the static case, only with the complex permittivity substituted for permittivity in the solution. i.e.

( ) 0 E

3

31 cos

2p m

mp m

aEr

r

3cos

2m

pp m

Er


Again, comparing and

gives for the effective dipole moment

This is sometimes written with an effective polarisability

given by

32

coscos

2p m

mp m

Ea Err

2

cos

4 or

p

342

p mm

p m

a

p E

p E

3 32

p mm m CM

p m

ε εf

ε ε

is the volume

of the sphere

The Clausius-Mossotti Factor

The dipole moment depends on the applied field and the volume of the particle but the frequency dependency is given by the Clausius Mossotti factor:

This is complex, describing a relaxation in the effective polarisability of the particle with a characteristic time of

The characteristic relaxation frequency:

is referred to as the Maxwell-Wagner relaxation frequency

( , )2

p mCM p m

p m

ε εf ε ε

ε ε

2

2p m

MWp m

1

2 2MW

MWMW

f


Plot of the variation of the real and imaginary parts of the Clausius-Mossotti factor with frequency. The high and low frequency limiting values of the real part are shown, as well as the value of the imaginary part at the relaxation frequency.


Attributes:

• At low frequencies, the conductivities of the dielectrics dominate

• At high frequencies, the permittivities dominate

• The relaxation frequency depends on properties of both dielectrics but not on size.

This factor crops up in several different places:

• The impedance of particles

• The frequency variation of the dielectrophoretic force and electroorientational torque depend on the real part

• The frequency variation of the travelling wave dielectrophoretic force and electrorotational torque depend on the imaginary part

More on this later.

Other experimentally relevant models

Other experimentally relevant models

We now extend this simple model to look at experimentally relevant particles by incorporating additional physical parameters of the system:

1.Internal structure and the shell model: biological particles

2.Electrical Double layer and surface conduction: latex spheres

3.Non-spherical particles

We will also examine in brief the permittivity of a mixture of polarisable particles as described by Maxwell’s Mixture Theory

The shell model

Biological particles have a complicated internal structure which is usually modelled using a system of concentric shells. This model determines an equivalent internal permittivity and conductivity for the two internal layers which is then used in the normal expressions.

a1

a2

3 , 3

2 , 2

1 , 1

a1

23 , 23

1 , 1

The shell model

This model can be derived as a boundary value problem by assuming the solutions to Laplace’s equations in the three regions of the system:

11 12

12 1 1 22

3 1 1 2

cos for

cos for

cos for

AEr r a

r

CB r a r a

r

D r a r a

Solution as before outside the sphere

Solution assuming the “applied field” to be the

internal field derived previously plus the dipole

field from the internal sphere

Uniform field inside

The shell model

The boundary conditions at the two interfaces are, as before:

These boundary conditions are combined with the assumed solutions and a great deal of algebra ensues!

1 21 2 1 2 1

322 3 2 3 2

and at

and at

r ar r

r ar r

The shell model

The resulting coefficients are:

the other coefficients are included, we are interested only in the external field

with giving

31 ,23CMA Ea f

31 12

1 323 1 12 ,23

3

( 2 )( )CM

B Ef

3 3 2

123 2

23 23 3 212

3 2

22

2

ε ε

ε ε

ε ε

ε ε

23 1,23

23 12CM

ε εf

ε ε

1 ,2331 3

23 1 12 ,23

3

( 2 )( )CM

CM

fC Ea

f

112

2

a

a

31 ,23 12

1 323 1 12 ,23

3 (1 )

( 2 )( )CM

CM

fD E

f

where

14 mp A Ep 3123,1

~4 afCM

The shell model

Exampleillustration

1.0

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1.0101 103 105 107 109

Frequency (Hz)

61 2.01 10 ma

62 2.0 10 ma

1 78.5 o 2 10 o

3 60 o

4 11 10 Sm

8 12 10 Sm

13 0.5 Sm

The shell model

The shell model can be expanded to multiple shells by extrapolation representing each successive shell using the same equations

a1 a2

3 , 3

2 , 2

1 , 1

a3

4 , 4

a1 a2

34 , 34

2 , 2

1 , 1

a1

23 , 23

1 , 1

The shell model

The effective dipole moment is

and so on through successive levels

3 34 212

34 223 2

3 34 212

34 2

22

2

ε ε

ε ε

ε ε

ε ε

23 1,23

23 12CM

ε εf

ε ε

112

2

a

a

Ep 3123,1

~4 afCM

3 4 323

4 334 3

3 4 323

4 3

22

2

ε ε

ε ε

ε ε

ε ε

223

3

a

a

The shell model

Example illustration

1.0

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1.0101 103 105 107 109

Frequency (Hz)

63 0.5 10 ma

4 30 o 14 0.1 Sm

61 2.01 10 ma

62 2.0 10 ma

1 78.5 o 2 10 o

3 60 o

4 11 10 Sm

8 12 10 Sm

13 0.5 Sm

The shell model

Example illustration

comparison

1.0

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1.0101 103 105 107 109

Frequency (Hz)

63 0.5 10 ma

4 30 o 14 0.1 Sm

61 2.01 10 ma

62 2.0 10 ma

1 78.5 o 2 10 o

3 60 o

4 11 10 Sm

8 12 10 Sm

13 0.5 Sm

Ionic solutions and surface conduction

We now look at another physical phenomena in real systems, the effect of the surface conduction arising from Electrical Double Layer on charged particles. A commonly used type of particles are latex spheres, which are a type of polymer which typically carries a large amount of surface charge when suspended in water.

Aqueous solutions carrying different concentrations of ionic salts are used to control the medium conductivity and the effective polarisability of the particles. The conductivity of an ionic solution is complicated since the current is carried by many different types of ion, each with a different mobility.

The current density for each ion of type j is

j jJ E

j j j j j j jz Fc z n q

valency

the Faraday (9.6487×104 C mol1)

concentrationcharge on the electron

(electrophoretic) mobility (m2V1s1)

Ionic mobility and diffusion

The Einstein relation gives the mobility in terms of the diffusion coefficient

where

or

where aj is the radius of the ion.

or

j j j

qz D

kT

R = NAk(molar) gas constant (8.3143 JK1mol1)

6jj

kTD

a

kTD

f

6j

jj

z q

a

Ions in solution

An ion in aqueous solution is essentially a free charge placed in a dielectric with a high permittivity. The local electric field generated by the ions both orients and binds the water molecules close to the ion: the hydration shell.

The dynamics of the ion are therefore complicated by the formation of this shell. While the electrical force remains the same, the radius of the moving ion has to take into account the bound layer of water: the hydration radius.

Centre of negative charge

Centre of positive charge

Dipole moment

The Electrical Double Layer

Surfaces carry net charges:

• dissociation of the chemical groups on the surface

• adsorption of ions or molecules from the solution onto the surface.

This charge creates a surface electrostatic potential and when immersed in an electrolyte, this electrostatic potential attracts ions of opposite charge (counterions) from the solution and repels ions with like charge (co-ions).

The region of liquid near to the interface has a higher density of counterions and a lower density of co-ions than the bulk. This region is referred to as the diffuse electrical double layer.

When any particle, surface or electrode is immersed in an electrolyte, the surface charge is balanced by an equal (and opposite) amount of excess charge in the double layer. The net result is that the countercharge from the solution effectively screens the surface charge so that electroneutrality is maintained.

Static electrical potentials

The simple model of the diffuse double layer assumes all ions can be represented as point charges.

The electrical potential in the bulk decreases exponentially from the surface, where the characteristic constant is given by:

where is referred to as the reciprocal Debye Length. The Debye length D = −1 is used as the thickness of the Diffuse Double Layer.

y

o

o/e

D=1

diffuse layer

qd

surface

qo

22 2 j o

j

z nq

kT

Surface conduction

We will look at two simple illustrative examples which can easily be related to the theory we have already discussed. The first is due to O’Konski, who presented a theory why insulating particles such as latex spheres were measured as possessing a large conductivity. He postulated that the flux due to the transport of ionic charge carriers associated with the fixed charge on the surface must be considered

En

E

EtJ

dl

En

E

EtJ

Diagram of a surface of specific conductance .

The flux of charge carriers Jthrough line element dl is governed by the component of the electric field E tangential to the surface

dl

Surface conduction

Current density:

Assuming a single counterion:

Extending the dielectric sphere model, we have two free charge components:

Both of which have a continuity equation for the dielectric boundary:

ss t

dqdl

dt J E

s i iz

s f

f pmm p

r a r at r r

2sin

sinps

r at a

Surface conduction

The boundary condition for the surface of the sphere to be included in the derivation of the field is now

Re-solving for the electrical potential around the sphere and extracting the expressions for the effective dipole gives the same solution as before but with the particle complex polarisability replaced with

Note: this is a property of the particle defined by the properties of the medium.

2sin

sinp pm

m pr ar a r a

t r ra

2 p

p p p p s s

iwhere and

a

Typical latex sphere

The Clausius-Mossotti factor for a latex sphere and therefore the polarisation is heavily influenced by this apparent conductivity

1.0

0

-1.0log10(frequency)

1

2

3

a1

a3

a2

Non-spherical particles

We will briefly look at the polarisation of ellipsoidal particles. Details can be found in the textbooks of Stratton and Jones.

An arbitrary ellipsoid is defined with

principle axes 1, 2, 3 and half-lengths

along the axes a1, a2, a3.

3n m nK


The effective polarisability is then different along each of the axes

there are corresponding complex terms equivalent to the Clausius-Mossotti factor:

with depolarising factors

where

3 for 1,2,3n m nK n

3( ( ) )p m

nn p m m

KL

1 2 3 20

1

2 ( )nn

dsL a a a

s a H

2 2 21 2 3( ) ( ) ( )H s a s a s a


The dipole moment therefore depends on the orientation of the particle in the field and the angle between the field and the axes of the particle:

where En gives the electric field in components relative to the axes of the particle. There is also a characteristic relaxation time (frequency) associated with each axis:

We will return to the subject of ellipsoidal particles later.

1 2 34

3 ( )p m

n m np n p m

a a a

L

p E

(1 )

(1 )n p n m

nn p n m

L L

L L

Maxwell’s Mixture Theory

We finish this section with a short aside on the behaviour of a mixture of particles. While not strictly relevant to dielectrophoresis, it is a natural conclusion to the subject of dielectric particles as impedance/dielectric spectroscopy is generally performed on suspensions of many particles.

Dielectric spectroscopy measurements of the variation in the complex permittivity of a suspension of dielectric particles with frequency have been the accepted method for determining the effective polarisability and the induced effective dipole moment of individual particles. Whilst this technique gives the average dielectric properties of the particles, AC electrokinetics and single particle impedance spectropscopy give the unique properties of single particles and the distribution of an ensemble.


Following Jones, we can use a simple method to determine the effective permittivity of a mixture. We consider a spherical volume surrounding our mixture of particles and examine the effective dipole moment of the sphere and the mixture.

Np effp

eff R


3

3

4

42

eff

m CM

p mm

p m

N

f a

N a

p p

E

E

342

eff meff m

eff m

R

p E

The Principle of

Superposition:

This is the mixture formula as given by Maxwell, more often written as:

where is the volume fraction

3

32 2eff m p m

eff m p m

aN

R

2

2

1 21 2

1 1

p m

p m

p m

p m

ε ε

ε εCMmix m m ε ε

CM ε ε

f

f

3

3

aN

R

dielectrophoresis and ac electrokinetics: theory

Documents