AC Electrokinetics forces and torques. AC electrokinetics. There are several forces and torques which fall under this category.

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1 AC lectrokinetics forces and torques AC electrokinetics There are several forces and torques which fall under this category. Dielectrophoretic force force on a polarisable particle in a non-uniform field Travelling Wave Dielectrophoretic force force in a field with spatially varying phase lectrorotational torque produces rotation of polarisable particles in rotating electric fields lectroorientational torque aligns non-spherical particles in uniform or nonuniform fields We will discuss these effect but devote most time to the first and most common effect dielectrophoresis. We are also only interested in AC potentials and fields.

2 An aside - electrophoresis The term dielectrophoresis is related to lectrophoresis, which is the movement of a particle with a non-zero net charge produced by the Coulomb force. Biological particles generally have a finite surface charge density (usually negative, due to the presence of acid groups on the surface) and observation of the movement of these particles in a uniform electric field is used both to characterise and also to separate particles. The static force in this case is F Q ds P S q In AC fields, this produces an oscillatory motion which reduces in magnitude with frequency. At frequencies where dielectrophoretic experiments are typically performed, this motion is negligible. Dielectrophoresis of polarisable particles The term Dielectrophoresis was coined by Herbert Pohl in 1951 and is the movement of a particle arising from the interaction of a non-uniform electric field and the effective dipole moment induced in a particle. As we discussed previously, the effective dipole moment is frequency dependent, changing in magnitude and direction. How does this relate to the dielectrophoretic movement of the particles?

3 Dielectrophoresis Simple non-uniform field Dielectrophoresis p m Net effective dipole m p Dielectrophoretic movement The non-uniform field results in a different electrical force on the two poles of the induced dipole. The net effect is to produce a movement of the particle

4 Dielectrophoresis p m Net effective dipole m p Dielectrophoretic movement When the field is reversed, the dipole moment is also reversed. The net force points in the same direction and the net time averaged dielectrophoretic motion for this particle is non-zero. Dielectrophoresis p m Net effective dipole m Dielectrophoretic movement p Again there is a force imbalance. However as the dipole now opposes the applied field, the force points in the opposite direction.

5 Force on a infinitesimal dipole A dipole p = Qd in a non-uniform electric field experiences a net force since the electric field and hence the Coulomb force on the two charges is different. +Q d (r + d) F Q( rd) Q( r) Assuming the length of d is small compared with a typical dimension of the non-uniformity of the electric field, can be expanded around r using the vector Taylor series: x z y r Q (r) A( x dx) A( x) dxa higher order terms Force on a infinitesimal dipole The force is then F Q( r) Q( d. ) higher order terms Q( r) which, neglecting higher order terms becomes F ( p ) DP This force is only valid if the length of the dipole is smaller than a typical dimension of the field non-uniformity. In other words, if the magnitude of the electric field does not vary significantly across the dimensions of the dipole, this expression for the force is correct. This assumption referred to as the dipole approximation. If this assumption does not hold, then multipole force terms must be considered. An alternative and more exact method is to use the Maxwell stress tensor. These will be discussed later.

6 AC dielectrophoresis Dielectrophoresis will of course occur in DC fields and for fixed dipoles. Most of the applications, however, are for polarisable particles in AC harmonic fields. We begin by defining the basic terms as phasors. Arbitrary harmonic potential: ( x, t) Re[ ( x) e it ] i R I The electric field i t x (, t) Re[ x ( ) e ] i R I If there is no spatial variation in phase, then without loss of generality, the field and potential phasors can be assumed to be real. R AC dielectrophoresis In a harmonic AC field, the time averaged force is From previous lectures, the dipole moment is 1 Re[( ) * ] FDP p Re[] real part of p i t e Complex conjugate of the field For real phasors (no phase variation) 1 FDP Re[ ]( ) Using the vector identity: ( AB ) ( A) B( B) AB( A) A( B) ( ) ( ) ( ) and the irrotationality of the electric field: 0

7 AC dielectrophoresis the force expression can be written as which is generally written as 1 FDP Re[ ] ( ) 4 F 1 DP Re 4 or using RMS values for the field F DP 1 Re rms This last equation is also the equation for a DC field. AC dielectrophoresis Inspection of this equation shows the following dependencies: the volume of the particle an important factor as the range of particle radii of interest covers 10m down to 10nm (9 orders of magnitude) the gradient of the field magnitude squared (proportional to the energy density in the field). the real part of the effective polarisability and therefore, on the permittivity and conductivity of both the particle and the suspending medium, as well as the frequency of the applied electric field.

Dielectrophoresis of a spherical particle Substituting for the effective polarisability of a homogeneous spherical dielectric particle, we obtain F DP 3 p m ma Re p m This is the dipole approximation

8 Dielectrophoresis of a spherical particle Substituting for the effective polarisability of a homogeneous spherical dielectric particle, we obtain F DP 3 p m ma Re p m This is the dipole approximation for spatially non-uniform fields with constant phase. If the electric field is highly non-uniform for example, then the dipole approximation doesn t hold and higher order terms must be considered. This will be looked at briefly later. Dielectrophoresis of a spherical particle There are two regimes for DP related to the value of the Clausius-Mossotti factor, which itself depends on particle and medium properties but most importantly on the frequency of the applied field. If the real part is Positive: DP force acts towards regions of high field strength (positive dielectrophoresis) 1 0 Negative: DP force acts away from regions of high field strength (negative dielectrophoresis) log 10 (frequency)

9 An aside on force direction For design purposes or to understand an experiment, it is important to understand exactly what it is about the system that makes it work. In this case, let us think about how DP works and how it relates to the dipole approximation. An aside on force direction In this simple arrangement of electrodes, we generate a non-uniform electric field which has two interesting regions of symmetry. Let s examine a highly polarisable particle in these two locations.

10 Dielectrophoresis and phase We will now examine the dielectophoretic force for the more general case of both non-uniform magnitude and phase in the applied field. The electric field is with complex phasor x (, t) Re[ x ( ) e it ] i R I The correct form for the time averaged force on the dipole is now: 1 1 FDP p * * Re[( ) ] Re[ ( ) ] Complex conjugate Dielectrophoresis and phase The derivation requires two vector identities, assumes the irrotationality of the field and that the field has zero divergence in a homogeneous dielectric. ( AB) ( A) B( B) AB( A) A( B) * * * ( ) ( ) ( ) * * * * * ( ) ( ) ( ) ( ) ( ) 0 and ( AB) ( B) A( A) B( B) A( A) B ( ) ( ) ( ) ( ) ( ) * * * ( ) ( ) ( ) * * * * * 0 giving ( ) ( ) ( ) * * *

11 Dielectrophoresis and phase The force then becomes 1 1 FDP 4 4 * * Re[ ( )] Re[ ( )] i i i * ( ) (Re[ ] Re[ ]) (Im[ ] Im[ ]) (Re[ ] Im[ ]) (Im[ ] Re[ ]) (Re[ ] Re[ ]) (Im[ ] Im[ ]) Re[ ] (Re[ ] Re[ ]) (Im[ ] Im[ ]) Im[ ] (Re[ ] Re[ ]) (Im[ ] Im[ ]) * Re[ ( )] Re[ ] Re[ ] Im[ ] Dielectrophoresis and phase The force then becomes 1 1 FDP 4 4 * * Re[ ( )] Re[ ( )] * ( ) (Re[ ] Re[ ]) i (Im[ ] Re[ ]) i (Re[ ] Im[ ]) (Im[ ] Im[ ]) i(im[ ] Re[ ]) i(re[ ] Im[ ]) i (Re[ ] Im[ ]) Im[ ] (Re[ ] Im[ ]) i Re[ ] (Re[ ] Im[ ]) * Re[ ( )] Im[ ] (Re[ ] Im[ ])

12 Dielectrophoresis and phase The force can be re-written as 1 1 FDP Re[ ] Im[ ]( (Re[ ] Im[ ])) 4 where Re[ ] Im[ ] The first term in the force expression depends on the frequency in the same manner the dielectrophoretic force. The second term in the force equation depends, however, on the imaginary part of the effective polarisability, or rather the imaginary part of the Clausius-Mossotti factor. This force is zero at high and low frequencies, rising to a maximum value at the Maxwell-Wagner interfacial relaxation frequency. Travelling Wave Dielectrophoresis The phase dependent component is generally referred to as the travelling wave dielectrophoretic force component, related to the design of electrode array shown below consecutive phase shifted signals generate a travelling potential and a travelling electric field above the array Direction of travel of field p F V sin( t) V cos( t) V sin( t) V cos( t) o o o o

Travelling Wave Dielectrophoresis This is a good example to consider since, as we will discuss later in the modelling section, the DP force acts vertically and the TWD force acts along the array.

13 Travelling Wave Dielectrophoresis This is a good example to consider since, as we will discuss later in the modelling section, the DP force acts vertically and the TWD force acts along the array. Direction of travel of field p F V sin( t) V cos( t) V sin( t) V cos( t) o o o o The real and imaginary parts of the Clausius-Mossotti factor for a solid homogeneous particle are shown as an example. TWD only really works in the frequency range shown by the grey area, since a negative DP repulsion is required to levitate, and an imaginary part is required to produce a finite translational force along the array. Dielectrophoresis summary The dielectrophoretic force, whether simply the classical force expression or including the travelling wave component of the force is related to the spatial non-uniformities of the electric field. The exact shape of the field is required to understand the dielectrophoretic force in practical devices this will be discussed in more detail later. Measurement of the dielectrophoretic force or more correctly the dielectrophoretic velocity provides information on the dielectric spectrum of the particle i.e. a measurement of the Clausius-Mossotti factor which then allows you to determine the polarisability and the electrical properties of the particle. The interaction of a particle and the fluid medium and the effect on moving particles for DP will be summarised at the end of this section.

14 Torques: electrorotation and electroorientation lectrorotation lectrorotation is a phenomena which is typically observed in rotating electric fields. The delay in the response of the polarisation to the applied field results in the dipole moment lagging the field. y p x A schematic electrorotation setup. Four signals, successively 90 o out of phase are applied to four electrodes encircling the particle.

15 lectrorotation The derivation we will start from is to look at an infinitesimal dipole. This model is at the root of both electrorotation and electroorientiation. The equation for the instantaneous torque on the dipole is found from d d ΓROT Q ( Q) Qd p F Q +Q F + Again, for the time averaged torque in a harmonic AC field 1 Re[ * ] ΓROT p lectrorotation Using the same approach as before and the induced dipole formula: 1 Re[ ( * )] ΓROT * ( ) (Re[ ] Re[ ]) i(im[ ] Re[ ]) i(re[ ] Im[ ]) (Im[ ] Im[ ]) i(im[ ] Re[ ]) i(re[ ] Im[ ]) i (Re[ ] Im[ ]) Im[ ](Re[ ] Im[ ]) i Re[ ](Re[ ] Im[ ]) * Re[ ( )] Im[ ](Re[ ] Im[ ])

16 lectrorotation The equation for the electrorotational torque is then given by or for a spherical particle giving ΓROT Im[ ](Re[ ] Im[ ]) ΓROT 3 p m 4 ma Im (Re[ ] Im[ ]) p m Aside: rotating fields ( xˆ iyˆ) Re[ ] xˆ Im[ ] iyˆ Re[ ] Im[ ] xˆ yˆ zˆ det Γ ROT 4 ma Im p m 3 p m lectrorotation summary As for the travelling wave force, imaginary part of the Clausius-Mossotti factor peaks at the Maxwell-Wagner interfacial frequency and so does the electrorotational torque. One electrorotational peak is found for each interface as in the case of biological particles, although they are not always well separated. 1 0 There is also clearly a relationship between lectrorotation and Travelling Wave dielectrophoresis log 10 (frequency) In fact, as will be shown in the section on modelling, travelling wave dielectrophoresis and electrorotation always occur together. In addition, dielectrophoresis will also occur, making electrorotation observations across a broad range of frequencies challenging.

17 lectro-orientation lectroorientation is the orientation of a non-spherical particle with the electric field. Without going into much detail, we have the dipole moment for each axis: L a1aa3 0 p 4 aaa 1 3mK K p m 3( L ( p m) m) 1 ( s ds a ) H H ( sa ) ( sa ) ( sa ) 1 3 are the cartesian coordinates arranged x y z x There are three torque components: 1 Re[ * * ] Γ p p giving Γ aaa 1 3m ( L L) Re[ K K ] 3 lectrorotation & lectro-orientation For illustration: lectrorotation lectroorientation

Particle-particle interactions A quick look at particle-particle interactions without the aid of equations p p F q F q Schematic diagram of the attractive force

18 Particle-particle interactions A quick look at particle-particle interactions without the aid of equations p p F q F q Schematic diagram of the attractive force between two fixed dipoles aligned by an applied uniform electric field. Particle-particle interactions The complexity of these interactions for induced dipoles can be shown:

19 Particle-particle interactions The complexity of these interactions for induced dipoles can be shown: More polarisable particles Less polarisable particles Particle-particle interactions The complexity of these interactions for induced dipoles can be shown: More polarisable particles Less polarisable particles

phenomenogical models would look something like: chains of like particles parallel to

20 Particle-particle interactions The stability for a mixture is more different Mixture of two particles Particle-particle interactions The behaviour of a mixture based on these phenomenogical models would look something like: chains of like particles parallel to applied field unlike particles align alternately and perpendicular to the applied field positive DP negative DP

21 Note: ffect of the fluid Fluid exerts viscous drag on particles as they move or rotate: F fv For a sphere, friction factor is f 6a The velocity is F v f Arb ( f / m) t (1 e ) containing the momentum relaxation time m a / f The terminal velocity is v T F f Arb Note: ffect of the fluid Steady state dielectrophoresis: For a sphere v v DP DP Re[ ] f 3 a m Re[ f CM] 6a Define DP mobility Such that v DP DP DP a m Re[ f CM] 6 Notes: velocity is proportional to diameter squared implying that smaller particles do not require such a large degree of increase in voltage to observe motion.

22 Note: ffect of the fluid Steady state electrorotation: Rotational friction factor f 8 a 3 Steady state angular velocity R( ) Γ ROT f For a sphere m R( ) Im f CM Notes: rotation rate is now independent of particle size, depending only on the differences in the dielectric properties

NANOPARTICLE MANIPULATION BY DIELECTROPHORESIS

Physica Macedonica 61, (1) p. 39-44 ISSN 149-7168 NANOPARTICLE MANIPULATION BY DIELECTROPHORESIS A. Neculae, R. Giugiulan and M. Lungu West University of Timisoara, Faculty of Physics, Blv. V. Parvan no.4,