17.1 Dielectrophoresis [DEP top]

Dielectrophoresis (DEP) isoften used in microsystems as a mechanism for manipulating particles. It is appealing because the dielectrophoretic force on a particle scales with the characteristic length scale of the system to the -3 power, and dielectrophoretic forces are quite large when small devices are employed. Further, particle response varies based on the frequency and phase of the applied field. Because the user can change particle response by changing a setting on a function generator, DEP measurements afford great flexibility to the user.

The term dielectrophoresis refers to the Coulomb response of an electrically polarized object in a nonuniform electric field. In contrast to linear electrophoresis, it (a) does not require that the object have a net charge and (b) has a nonzero time-averaged effect even if AC electric fields are used.

Consider, as an example, a spherical, uncharged, uniform, ideal dielectric particle with a finite polarizability, expressed using its electrical permittivity ε_p, suspended in empty space. If a uniform electrical field is applied to this system, the sphere polarizes (Figure 17.2), and there is a net positive charge at one end of the sphere and a net negative charge on the other end of the sphere. Given that the electric field is uniform, the Coulomb forces on either end of the sphere are equal and opposite, and the net Coulomb force is zero. If the electric field is nonuniform, however, the side of the sphere with the larger electric field feels a larger attractive force, and the net force moves the particle toward the region of high electric field. Motion toward high electric field regions istermed positive dielectrophoresis.

Microfluidic applications involve particles suspended in a medium (usually an aqueous solution) with electrical permittivity ε_m. For the moment, we assume that the medium is a perfect dielectric as well. In this case, the arguments are similar to above, except that the particle and the medium both polarize. For a particle suspended in a medium, the net force on the particle is dependent on the difference between the polarization of the particle and the polarization of the medium. If the medium polarizes less than the particle, the particle experiences positive dielectrophoresis and moves toward the high electric field region. If the medium polarizes more than the particle, the particle experiencesnegative dielectrophoresis and moves toward the low electric field region. In both cases, the direction of motion of the particle is a function of the electric field magnitude, but not its polarity. Thus the dielectrophoretic response of an uncharged, uniform, ideal dielectric object in an ideal dielectric medium is independent of whether a DC or AC field is used, or even the frequency of the field, as long as the permittivity is independent of frequency.

The above description highlights the basic physics of dielectrophoresis—by controlling the polarization of a particle with respect to the suspending medium, charge is generated at the interface between the particle and the medium. This charge is called Maxwell-Wagner interfacial charge. The motion of the particle is dictated by the sign and magnitude of this charge. In the upcoming sections, we quantify the response of a uniform, uncharged sphere and extend the analysis to include media and particles with finite conductivity and permittivity; describe Maxwell equivalent body techniques for describing nonuniform isotropic particles; and extend the analysis to account for surface charge and the attendant electrical double layer. We also describe the response of nonspherical particles, and describe electrorotation and traveling-wave DEP, both phenomena based on applied electric fields with spatially varying phase.

While the analysis in this section is mostly focused on predicting _DEP, the velocity of a particle in a nonuniform field is the end result. In a manner analogous with electrophoretic mobility or electroosmotic mobility, the dielectrophoretic velocity can be described using adielectrophoretic mobility, μ_DEP:

which is written in terms of the magnitude of the applied electric field, i.e., the field that would have existed if the particle were absent. We derive expressions for the dielectrophoretic mobility in the sections to follow.

17.1.1 Inferring the Coulomb force on an enclosed volume from the electric field outside the volume

All dielectrophoretic analysis techniques are enabled by the fundamental concept that the only information required to determine the Coulomb force on an enclosed volume is the material properties and electric field on the outer surface of that enclosed volume. This property follows mathematically from Maxwell’s equations– for example, in the absence of magnetic fields, we can show that the force on an enclosed volume is given by

where S is the surface of the volume and  is a unit outward normal.

This leads to an enormous simplification when we use electrodynamic solutions to predict the force on a particle. Equation 17.2 illustrates that the force on the particle can be determined from only the electric field and permittivity at the surface. Thus two different particles, if they result in the same solution for the electric field in the suspending medium, feel the same electrostatic force, even if they lead to a different solution for the electric field inside the particle. Further, given two alternate systems suspended in a uniform medium and an arbitrary volume large enough to enclose either one, if the two systems lead to the same electric field on the outside of the volume, the force on each system is the same.

This result enables multiple analytical tools. For example, if we can replace a particle with a simpler structure that creates the same electric field at the surface of the particle, then the force on the simple structure predicts the force on the physical particle. This is the basis for the dipole approximation for the dielectrophoretic force on a sphere, or the Maxwellian equivalent body for particles of regular shape but nonuniform material properties. Even if such a simple analytical approach is not possible (for example, for irregularly shaped objects), we need only find the solution by integrating Equation 17.2, the so-called Maxwell stress tensor approach. The electric field inside the particle need not be known.

17.1.2 The force on an uncharged, uniform, isotropic sphere in a linearly varying electric field with uniform, isotropic phase

We solve for the force on a particle in terms of material properties and the applied field _ext( ). By applied field, we refer to the electric field that would exist if the particle were absent. The electric field and potential in the presence of the particle are written as ( ) and ϕ( ).

The dielectrophoretic force on an uncharged, uniform, isotropic, dielectric sphere (Figure 17.3) of radius a located at the origin in a steady linearly varying electric field with uniform, isotropic phase can be derived by first solving for the electrical potential ϕ around a sphere in a uniform applied field (here we take _ext = E₀ẑ) to evaluate its effective dipole and then calculating the force on that effective dipole owing to a linear variation in electric field. Thus we solve Gauss’s law for ( ) using the Laplace equation:

(17.3)

with boundary conditions at r = ∞:

(17.4)

and at r = a:

(17.5)

where subscripts m and p refer to the medium and particle, respectively, and the n-direction is normal to the surface and directed outward, toward the suspending medium (cf. Equation 5.23).

Rather than solving this steady problem, however, we consider AC electric fields, which is analogous and more general. For AC fields, the general approach is the same, but we combine the charge conservation and Poisson equations to create a complex version of the Laplace equation, which we can then solve in terms of the complex permittivity.

To determine the effective dipole induced in a sphere by a uniform applied electric field, we consider a sphere with no surface charge and uniform isotropic properties ε_p and σ_p embedded in a medium with uniform, isotropic properties ε_m and σ_m. We assume a uniform applied sinusoidal electric field

(17.6)

where E₀ is the peak magnitude of the applied electric field. We write the analytical representation of the applied field as

(17.7)

As described in Section 5.3, the charge conservation equation (without convection)

(17.8)

and the Poisson equation

(17.9)

can be combined and written as an equation for the complex electric displacement:

(17.10)

or, in terms of the complex permittivity = ε+σ∕jω, as an equation for the electrical potential:

(17.11)

with boundary conditions at r →∞:

(17.12)

and at r = a:

(17.13)

Here subscripts m and p indicate medium (fluid) and particle, respectively. The electrostatic condition (Gauss’s law) and the electrodynamic condition (charge conservation) are simultaneously specified with this complex equation, and the solution provides the response at all frequencies. The solution is written in terms of several complex quantities, which can be directly related to the dynamics of system response. This system has a solution in term of Legendre polynomials, as described in the appendix in Section F.1.1. Inside the sphere r < a, the solution is

(17.14)

while the solution outside the sphere r > a is

(17.15)

The solution outside the sphere consists of a term corresponding to the electric field ( = -E₀rcosϑexpjωt = E₀zexpjωt) and a term corresponding to a point dipole (Section F.1.1) at the origin. In terms of the Legendre polynomials bounded at infinity, i.e., the multipolar expansion, this dipole has coefficient B₁ = E₀a³expjωt, corresponding to an electrical dipole with dipole moment phasor

The quantity

(17.17)

is called the Clausius-Mossotti factor; the Clausius-Mossotti factor is equal to the first-order (dipolar) polarization coefficient, often denoted by ⁽¹⁾.

The electrical potential outside r = a is therefore identical to the electrical potential that would exist if the sphere were replaced with an oscillating point dipole oscillating at frequency ω with dipole moment phasor

(17.18)

Returning to real quantities, this effective dipole moment is given by

(17.19)

or, equivalently,

(17.20)

where

(17.21)

and

(17.22)

The magnitude of the dipole moment is proportional to , the magnitude of , and is proportional to the magnitude of the net charge buildup at either end of the sphere; the phase lag between the applied electric field and the dipole moment is , the angle of . Equivalently, Re is the portion of the dipole moment in phase with the applied electric field, and Im is the portion of the dipole moment that is 90 degrees out of phase with the applied electric field.

Since a point dipole at the origin can be used to create the same electric field outside the particle, and given our earlier observations in Section 17.1.1, the force on the volume enclosing the particle is the same as the force on any volume enclosing a point dipole at the origin with dipole moment specified by Equation 17.16. Since the force on a dipole is given by = ⋅∇_ext, the force on a particle is given by the dot product of the applied electric field gradient and the effective dipole moment. This relation is valid if the applied electric field varies linearly, and this linear variation is small enough over the length of the sphere that the effective induced dipole calculated for a uniform field is still accurate. The force on the sphere is thus given by the force on a dipole in a linearly varying electric field ( = ⋅∇_ext for the steady-state case).

Since we are working with the analytic representations and wish to take the product of  and , we must note that ≠ and therefore use Equation G.36. The analytical representation of the force on the particle can therefore be written as:

(17.23)

returning to real quantities, the resulting instantaneous force is given by

(17.24)

or, applying the inverse chain rule,

(17.25)

Thus, the force on the particle has a DC component proportional to Re, and a component at 2ω with magnitude proportional to . While the force at 2ω is significant, the observed velocity at 2ω is damped by the viscous drag of the fluid surrounding the particle, and the velocity magnitude at 2ω is smaller than the DC motion by the factor , where τ_p = 2a²ρ_p∕9η is the characteristic time of particle equilibration. Thus at high frequencies, the oscillatory particle motion is damped. In addition to this, even undamped oscillatory particle motion is difficult to observe with standard microscopy setups. Because of this, we focus on the time-averaged force

where the brackets denote time-averaging. For a DC applied field _ext, the result has a similar form, but has double the magnitude:

since the average value of cos² is (1∕2).

Since the effective dipole  and the applied electric field _ext are collinear, the torque ×_ext on a uniform sphere is zero. From the above results, we can immediately write the dielectrophoretic mobility of a sphere: For a spherical particle of radius a, for which the drag force is = 6πηa, thedielectrophoretic mobility μ_DEP is given by

The real part of the Clausius-Mossotti factor, Re, ranges from 1 (for _p ≫_m) to -0.5 (for _m ≫_p). The sign of Re determines whether particles move up (Re > 0) or down (Re < 0) gradients in the squared electric field magnitude. Stated in different terms, the sign of Re determines whether particles are attracted to, or repelled from, regions of high electric field magnitude.

The gradient indicates that the dielectrophoretic force is related to the spatial variation of electric field magnitude squared. This means that F_DEP is independent of the polarity of electric field vectors, so particles experience a time-averaged force even with AC fields, whether harmonic or non-harmonic.

The force on a sphere is proportional to the sphere volume. DEP is a second-order effect that scales with applied voltage squared and with system characteristic length to the -3 power. In the high-frequency limit,  can be replaced by ε in Equation 17.17; DEP effects are strictly due to the polarization of the medium and particle as expressed by their permittivities. In the low-frequency limit,  can be replaced by σ in Equation 17.17; DEP effects are strictly a function of the conductivities of the medium and particle.

17.1.3 Maxwellian equivalent body for inhomogeneous, spherically symmetric particles

While the result for an uncharged homogeneous sphere (Equation 17.27) is straightforward to evaluate, many objects are poorly approximated as homogeneous. In particular, biological cells have a complicated internal structure, a gross model of which might include a lipid bilayer encapsulating a conductive cytosol with a defined nucleus. If the body can be approximated as spherically symmetric with several discrete layers, however, an effective dipole can again be defined.

Consider an uncharged sphere with a radius a₂ with a core of radius a₁ suspended in a uniform medium. Let the properties of the core be ε₁ and σ₁, the properties of the shell be ε₂ and σ₂, and the properties of the medium be ε_m and σ_m. Given a uniform harmonic applied field, this system consists of the complex equation

(17.29)

with boundary conditions at r →∞:

(17.30)

at r = a₁:

(17.31)

and at r = a₂:

(17.32)

This system, while algebraically more tedious than the one for a homogeneous sphere, can also be solved using Legendre polynomials, and the solution outside the particle can again be written in terms of the uniform applied field plus the response of a dipole at the origin. The solution for the effective dipole can be written in a form identical to Equation 17.16:

(17.33)

if we set the radius in Equation 17.33 equal to the outer radius of the composite particle (a = a₂) and write the effective complex permittivity _p of the particle as a function of the properties and thicknesses of the two particle components:

As was the case for a homogeneous uncharged sphere, we can determine the force on the uncharged core-shell sphere because it induces an electric field outside the particle that is mathematically equivalent to that generated by the effective dipole using the definitions in Equations 17.33 and 17.34. For particles with more than one shell, the effective permittivity of the particle is determined by repeated application of Equation 17.34, starting from the core, calculating the effective permittivity of the core plus one shell, then calculating the effective permittivity of that plus the second shell, and so on working out through all of the shells.

Maxwellian equivalent body for thin outer shells

When the outer shell is vanishingly thin as compared to the core, Equation 17.34 takes on a simplified form. Consider the solution from Equation 17.34, but let Δa = a₂ -a₁ with Δa ≪ a₂. We can write a linear expansion (a₂³∕a₁³ = 1+3Δa∕a₂) and show that the effective permittivity is given by

If σ₂ ≫ σ₁ and ε₂ ≫ ε₁, as might be the case for an electrolyte film at the surface of a polystyrene bead, this simplifies to

which is reminiscent of the parallel relation for capacitors, because in this limit the core is the primary resistance to current, current goes primarily through the shell layer, and thus the field lines are almost parallel to the shell surface. If σ₂ ≪ σ₁ and ε₂ ≪ ε₁, as might be observed in a biological cell with a low-conductivity, low-permittivity lipid bilayer at its surface, this simplifies to

which is reminiscent of the series relation for capacitors, because in this limit the shell is the primary resistance to current, and thus the field lines are normal to the shell surface.

17.1.4 Dielectrophoresis of charged spheres

The analysis for spheres with no surface charge often gives a good estimate for the performance of relatively large spheres in the thin double limit, especially when the particle itself is relatively conductive. However, if the particle is not conductive or if the double layer is not thin, the presence of charge on the surface, and particularly the double layer, effects the net dipole on the particle. In fact, the response of the double layer routinely dominates the dielectrophoretic response of nanoparticles.

Response owing to fixed surface charge In a system with no free charge (e.g., a solid particle suspended in space), the presence of uniform fixed surface charge on the surface of a particle does not directly lead to a change in the dielectrophoretic response of the particle. For example, consider a particle with radius a and a surface charge density q′′. If this system is exposed to a uniform DC or AC electric field, the solution for the electric field outside the particle is unchanged except for an additional steady term given by

(17.38)

which is just the electric potential induced by a charge 4πa²q′′, the total charge on the particle, at the origin. Thus the presence of fixed charge leads to a net Coulomb force on the particle, but it does not change the induced dipole caused by the electric field, or the dielectrophoretic response.

Response owing to property variation in the electrical double layer While a fixed charge density on the surface of an object does not lead to a change in the induced dipole, in aqueous systems, the surface charge density is balanced by the charge in the electrical double layer, and the ion distribution in the electrical double layer does change the dielectrophoretic response. If the double layer is spherically symmetric and the electromigration of ions can be approximated with an effective ohmic conductivity, the effect of the ion concentration in the double layer can be incorporated into a general multishell model for the effective dipole response of the particle, or, if the double layer is thin, the effect of the increased conductivity and reduced permittivity can be handled with a thin shell model as discussed in Section 17.1.3. This approach neglects the fluid convection, and accounts for all ion electromigration (including convective transport) through the use of an effective ohmic double layer conductivity. If convective processes distort the double layer and cause double layer fluid properties to vary from spherical symmetry (cf. Figure 13.4), then spherically-symmetric shell models cannot predict the effective dipole of the particle—no simple model exists for describing dielectrophoretic response when this occurs. Double layer asymmetry occurs at low frequencies (the so-called alpha relaxation), when the cycle is long enough that an appreciable amount of charge can build up in the double layer on either end of a particle and convection can carry this charge beyond the boundaries of the static double layer. We also discuss double layer asymmetry in Section 13.2.3 as a reason for nonlinearity in the electrophoretic response of particles—there, electrophoretic response is nonlinear when the double layer becomes distorted and generates an adverse electric field.

17.1.5 Dielectrophoresis of non-spherical objects

Dielectrophoresis of spheres is analytically convenient because the effective dipole and, in turn, the dielectrophoretic force can be calculated straightforwardly. For non-spherical objects, the calculations are more complex.

Maxwell stress tensor approach

The dielectrophoretic force on a particle with irregular shape can be determined by solving the Laplace equation for the potential distribution inside and outside the particle, and integrating the Maxwell stress tensor  is given in the absence of magnetic fields by

to give the force:

where S denotes the surface of the particle and  is an outward-pointing unit normal.

For a harmonic electric field, the time-averaged form of the stress tensor is

(17.41)

where is the unit or identity tensor, denotes time-averaging, and the asterisk denotes complex conjugation.

The Maxwell stress tensor technique is the most general, but because of its computational demands, it is used only when analytical techniques are inapplicable to the particle under study.

Multipole expansion

The multipole expansion is a compromise between the effective dipole techniques and the Maxwell stress tensor techniques. A multipolar expansion, in general, is an expansion of the electric field in powers of 1∕r corresponding to solutions of the Laplace equation obtained by separation of variables (Appendix F). For some simple geometries, an exact result can be obtained with a small number of terms. For example, for a sphere in a weakly varying electric field, the exact result requires only a uniform field term and an effective dipole term. Multipole expansions are always implemented with a finite number of terms, and in that sense are always approximations of the (exact) Maxwell stress tensor description. This approach is useful in cases where the particle geometry is simple, but either (a) variations in the electric field are on the same length scale as the particle, or (b) the dipole moment (the n = 1 term of the general multipole expansion) is zero (such as at a local field null point).

While a general multipole yields the most general result possible from a series expansion, linear multipoles describe most systems, especially for fields that are linearly polarized. The linear multipole solution assumes that the particle and domain are axisymmetric, and is generated using separation of variables applied to the axisymmetric Laplace equation (see Section F.1.1).

The force on a linear multipole is

(17.42)

and the accuracy of this approximation is good if the electric field is close to axisymmetric on the length scale of the particle.

DEP of ellipsoids

In general, the shape of a polarized particle influences the electric field it creates. As particle shape deviates from spherical, the applicability of the equivalent dipole representation based on a spherical particle rapidly decreases. Consider a prolate ellipsoidal particle with a long axis, a₁, and equivalent minor axes, a₂ = a₃. The effective dipole moment has three vector components:

(17.43)

where each axis has its own Clausius-Mossotti factor:

(17.44)

defined using depolarization factors L_i:

(17.45)

This analysis leads to three dipole moments, one along each axis of the ellipsoid. If the field gradient is not aligned with the principal axis, the effective dipole will lead to a torque that rotates the particle’s principal axis to be in line with the applied field. If the electric field gradient is aligned with the principal axis, there is no net torque on the particle, and the dielectrophoretic force can be evaluated in terms of the dipole moment along the principal axis, using the depolarization factor

(17.46)

where the eccentricity e is given by

(17.47)

The dielectrophoretic mobility for a prolate ellipsoid with its long axis parallel to the direction of the applied electric field is found by combining equations 8.40, 17.43, and 5.33:

(17.48)

17.1.6 Nonuniform and anisotropic phase effects

A harmonic electric field with uniform and isotropic phase (as assumed in Section 17.1.2) leads to a particle force proportional to Re, and the spatial dependence comes from the ∇(₀ ⋅₀) term. The torque on a particle with an applied field with uniform, isotropic phase is zero.

A particle feels a torque if the phase is anisotropic. This is normally referred to as electrorotation (ROT). When the phase is anisotropic, i.e., when the phases of the different vector components of the electric field are unequal, we refer to the field as a circularly polarized or rotating electric field. Figure 17.4 shows an electrode setup used for electrorotation.

If the phase is nonuniform, the particle experiences an additional force and torque. The force resulting from this nonuniform phase is the basis of traveling-wave DEP techniques. Figure 17.5 illustrates an interdigitated electrode array used for traveling-wave DEP.

Both electrorotation and traveling-wave DEP are analyzed by allowing the phase of the applied signal to vary as a function of both the position and the electric field component. Thus each component i = 1,2,3 of the applied electric field is written as E_0,i( ) = E_0,i( )cos. The analytic representation of each component is _0,i = E_0,iexpj. Writing each component _0,i = E_0,iexpjα_i defines a complex vector quantity  that captures the nonuniformity and anisotropy. The applied electric field can then be written as

(17.49)

Comparing this to Equation 17.7, the only difference is that the real vector ₀ has been replaced with the phasor . The effects of this are described in detail in the following sections.

Electrorotation

The instantaneous torque on an induced dipole is given by = ×_ext, and the time-averaged torque is given by or ×^*. The effective dipole is

(17.50)

Evaluating the time-averaged cross product using the guidelines in Section G.3, we find

(17.51)

The torque is proportional to Re[]×Im[] and to Im. The vector product Re[]×Im[] has three components that quantify how much the applied electric field is rotating around the three coordinate axes. If the phase is isotropic, then Re and Im point in the same direction, and the torque is zero. The imaginary component Im is a measure of the dipole component that is 90 degrees out of phase with the applied field. The dipole moment in phase with the field induces no torque, since it points in the same direction as the electric field, but the dipole moment 90 degrees out of phase with the field is oriented normal to the electric field and induces a torque. Thus the cross-product quantifies how much the field is rotating, and the imaginary part of the Clausius-Mossotti factor quantifies what fraction of the effective dipole lags behind to induce the torque.

Electrorotation is often quantified by measuring particle rotation rates as a function of electric field rotation frequency—this measurement is often termed an ROT spectrum and the process electrorotation spectroscopy. Because the real and imaginary parts of the Clausius-Mossotti factor are related by the Kramers-Krönig relation, ROT spectra inform dielectrophoretic experiments and vice versa, and electrorotation is maximum when DEP is at its minimum.

Rotating electric fields can be created by forming a quadrupolar configuration and driving each electrode at a different phase (again, φ = 0, π∕2, π, and 3π∕2 form the components of the ABCD configuration)or by using an octode cage consisting of two planar, quadrupolar electrode arrays assembled facing one another. One quadrupolar array is offset by a few degrees (by adding a phase increment between the signals driving the quadrupolar arrays).

Traveling-wave DEP

We can repeat the analysis on a sphere with nonuniform electric field phase. In this case, the time-averaged force

The first term in this equation is equal to the time-averaged dielectrophoretic force with uniform phase. The second term arises only in the presence of a spatially non-uniform electric field phase. The second term illustrates the physical behavior of particles under the influence of a spatially-varying phase. Unlike the uniform-phase effects, which are maximized when the dipole moment and field are in phase and the gradient of the field squared is high, the spatially-varying-phase effects are maximized when the dipole moment and field are π∕2 out of phase and when the curl of the real and imaginary parts of the electric field are large.

Traveling-wave DEP uses an array of electrodes patterned in a microfluidic channel. The electrode array is composed of alternating, independently driven electrodes with different phase (e.g., α = 0, π∕4, π∕2, 3π∕4, 0, ...). The electrode array is aligned at an angle to the direction of flow. These signals — irrespective of phase — levitate particles against gravity within the flow field owing to irrotational, negative dielectrophoresis, and the varying phase drives particles transverse to the direction of flow according to the imaginary component of the Clausius-Mossotti factor.

Creating spatially varying phase signals in microfluidic systems is most often accomplished using arrays of interdigitated electrodes. By applying a different phase to various electrodes in an array, a spatially varying field is produced. The electrode array itself is often fabricated at an angle to the microfluidic channel, because the twDEP forces that are created drive particles perpendicular to electrode orientation. This configuration leads to sorting as a function of the imaginary component of the Clausius-Mossotti factor transverse to the direction of flow:

Electrode configurations

Micro-fabricated electrodes are generally the most practical and straightforward method for creating non-uniform electric fields. They benefit from a long history of fabrication techniques and technologies, and are quite flexible in terms of implementation. In practical applications, their potential major limitations are fouling and electrolysis at low electric field frequencies. Some device designs may also require complex multi-level microfabrication increasing the cost of devices as well as the time required to create them.

As previously discussed, changing the shape and orientation of electrodes, in addition to modulating the frequency and phase applied, can give rise to dielectrophoretic particle trapping, dielectrophoretic sorting, electrorotation, and traveling wave dielectrophoresis effects. We now explore specific geometries and their key experimental parameters.

Interdigitated Electrode Array The interdigitated electrode array is a common electrode configuration used in DEP studies. The electrode array consists of two sets of electrodes, grounded and energized, that alternate spatially. This creates a non-uniform field in the region of the electrode array that traps particles against a flow (Figure 17.6).

Castellanos, et al. examined the case of two adjacent electrodes in detail in [187]. Their results detail the relative effects of DEP and hydrodynamics on the motion of particles adjacent to the electrode array. Castellanos and co-workers analyzed an interdigitated, two-electrode configuration. The electrodes were energized with a 5V potential and separated by a distance of 25 μm. In general, motion due to gravity and dielectrophoresis vary with particle volume, and dielectrophoresis dominates provided an appropriate electric potential is chosen (frequency and magnitude). Brownian motion decreases with increasing particle size, depending inversely on . For a particle size in the micron regime, this can be overcome by dielectrophoresis with a relatively low electric field. A similar argument can be made for the effects of buoyancy.

Interdigitated electrode arrays have been used extensively to retain particles of interest from a microchannel flow, or to filter out unwanted particles from an analyte stream. Electrodes are typically gold and patterned on a glass substrate using a lift-off procedure. Only a few relevant design parameters need be optimized: electrode width, inter-electrode distance, electrode length, and fluidic channel depth. With these parameters set, variations in applied electric field magnitude and frequency are left to vary in the experiment.

Electric fields and particle motion are easily modeled (and have been calculated analytically in [188]) in interdigitated electrode systems and can lead to consistent results when accurate particle models are used [189, 187]. The advantages of planar, interdigitated electrode arrays are simplicity of fabrication and analysis. Disadvantages of the interdigitated electrode array are the potential for permanent particle adhesion during positive dielectrophoresis and the inherent “binary” separation achieved under a particular set of experimental conditions.

Castellated Electrodes Castellated electrodes are similar to interdigitated electrodes. Rather than straight electrodes, however, the castellated electrode array consists of square-wave shaped electrodes. These patterns are usually placed parallel to each other and consist of alternating low- and high-separation regions, creating alternating low and high electric field regions, respectively. Two different configurations of castellated electrodes are used, pictured here: symmetric and offset.

Castellated electrode arrays have been used to focus particles from a well-mixed solution into a line of particles for subsequent analysis. This technique has been used in DEP sorting devices and micro-scale dielectric spectroscopy experiments, where the electrical response of individual particles is examined [190, 191]. Both straight and castellated interdigitated electrode arrays can sustain a fluid flow and potentially trap particles against fluid drag forces using DEP. This characteristic has been used to measure not only the sign, but also the magnitude, of DEP forces as a function of frequency. This is accomplished by measuring the number of particles collected by the array at varying electric field frequencies. Castellated electrode arrays, however, are most typically used to concentrate samples (in the high electric field regions, under positive DEP) or to pattern particles at a specific location. The advantage of castellated interdigitated electrodes is the localization of high electric field regions.

Angled Electrodes Angled electrodes can be used to separate particles based on DEP response or as a pre-concentration system to create a localized stream of particles. Castellated electrodes have been used as “concentrators” as well, but the mechanism is slightly different. Castellated electrodes act to focus particles using negative dielectrophoresis (referring to the sign of , meaning particles are directed away from regions of high electric field) and focused by the cumulative action of a series of high field regions generated between two parallel castellated electrodes. Angled electrodes, in contrast, rely on negative dielectrophoresis to trap particles against fluidic drag forces. Dielectrophoretic forces and fluidic drag forces parallel to the direction of flow balance, and a net force parallel to the electrode results (Figure 17.9).

Angled electrodes have a few advantages that have led researchers to utilize them to preferentially guide particles or trap them. By angling the electrodes, with reference to the channel, particles trapped against a channel wall by nDEP (see Section 17.1.6) can be displaced transverse to the direction of flow. In this manner, they have been used to preferentially direct particles to different outlets or focus particles [192, 193, 194].

Traps Electrode-based DEP traps generally consist of geometries that tend to trap single particles. The goal is usually to create a system of addressable particle traps to observe individual particle responses to a stimulus or to study biological particle interactions as a function of distance. Methods used to achieve addressable trapping include point-lid, quadrupole cages, ring-dot, and “DEP microwell” geometries (Figure 17.10) [195].

Related work on dielectrophoresis from our research group can be found here.

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