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Dielectrophoresis of charged colloidal suspensions

J. P. HuangBiophysics and Statistical Mechanics Group, Laboratory of Computational Engineering,

Helsinki University of Technology, P. O. Box 9203, FIN-02015 HUT, Finland, andDepartment of Physics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

Mikko KarttunenBiophysics and Statistical Mechanics Group, Laboratory of Computational Engineering,

Helsinki University of Technology, P. O. Box 9203, FIN-02015 HUT, Finland

K. W. Yu, and L. DongDepartment of Physics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

(Dated: February 2, 2008)

We present a theoretical study of dielectrophoretic (DEP) crossover spectrum of two polarizable particlesunder the action of a nonuniform AC electric field. For two approaching particles, the mutual polarizationinteraction yields a change in their respective dipole moments, and hence, in the DEP crossover spectrum. Theinduced polarization effects are captured by the multiple image method. Using spectral representation theory,an analytic expression for the DEP force is derived. We find that the mutual polarization effects can change thecrossover frequency at which the DEP force changes sign. Theresults are found to be in agreement with recentexperimental observation and as they go beyond the standardtheory, they help to clarify the important questionof the underlying polarization mechanisms.

PACS numbers: 82.70.-y, 77.22.GM, 61.20.Qg, 77.84.Nh

I. INTRODUCTION

When a polarizable particle is subjected to an applied elec-tric field, a dipole moment is induced into it. The movementof colloidal particles in an applied AC electric field is calleddielectrophoresis [1]. It is typically used for micromanipula-tion and separation of biological cellular size particles,andit has recently been successfully applied to submicron sizeparticles as well. Specific applications include diverse prob-lems in medicine, colloidal science and nanotechnology, e.g.separation of nanowires [2], viruses [3], latex spheres [4,5],DNA [6] and leukemic cells [7], as well as lab-on-a-chip de-signs [8].

The dielectrophoretic (DEP) force exerted on a particle canbe either attractive or repulsive depending on the polariz-ability of the particle in comparison to the medium. For anonuniform AC electric field, the magnitude and the direc-tion of the DEP force depends on the frequency, changes insurface charge-density and free charges in the vicinity of theparticle. The frequency at which the DEP force changes itssign is called the crossover frequency (fCF). Analysis of thecrossover frequency as a function of the host medium conduc-tivity can be used to characterize the dielectric properties ofparticles, and is at present the principal method of DEP anal-ysis for submicrometer particles [3, 4].

In the dilute limit, i.e, when a small volume fraction ofcharged particles are suspended in an aqueous electrolyte so-lution, one can focus on the DEP spectrum of an individualparticle ignoring the mutual interactions between the particles.Although the current theory [1] captures some of the essentialphysics in the dilute case, it is not adequate [9, 10, 11, 12, 13].This is due to the fact that even for a single colloidal particlein an electrolyte, it is not established which mechanisms con-

trol its dielectric properties. If the suspension is not dilute,the situation becomes even more complicated due to the mu-tual interactions between the particles. One should also notethat particles may aggregate due to the presence of an exter-nal field, even when the suspension is at the dilute limit underzero field conditions. In this case, the mutual interactionshaveto be included in the description.

In this article, we present a theoretical study of the DEPspectrum of two spherical particles in the presence of anonuniform AC electric field. We use the multiple imagemethod [14], which is able to capture the mutual polarizationeffects. Using the spectral representation theory [15], wede-rive an analytic expression for the DEP force and determinethe crossover frequency. Our theoretical analysis shows thatthe induced mutual polarization interactions plays an impor-tant role in DEP spectrum. In a more general framework, ourresults demonstrate the importance of correlation effects. Thisis analogous to the findings in charged systems where phe-nomena such as overcharging, or charge inversion (see e.g.Refs. [16, 17]), provide spectacular demonstrations of corre-lation effects.

As our starting point, we consider a pair of interactingcharged colloidal particles dispersed in an electrolyte solution.When the two particles approach each other, the mutual po-larization interaction between them leads to changes in theirrespective dipole moments [18], and hence also in the DEPspectrum and crossover frequency. We analyze two cases: 1)longitudinal field (L), in which the field is parallel to the linejoining the centers of particles, and 2) transverse field (T)inwhich the field is perpendicular. The former corresponds topositive dielectrophoresis where a particle is attracted to re-gions of high field and the latter to the opposite case, referredto as negative dielectrophoresis.

2

This paper is organized as follows. In Sec. II we presentthe formalism and derive analytic expressions for the effec-tive dipole factors in spectral representation. In Sec. III, weuse the analytical results to numerically solve the crossoverfrequency, dispersion strength and DEP spectra under differ-ent conditions. This is followed by a discussion of the resultsin Sec. IV.

II. FORMALISM AND ANALYSIS

First, we consider a single charged spherical particle sus-pended in an electrolyte and subjected to a nonuniform ACelectric field. The DEP forceFDEP acting on the particle isthen given by [19]

FDEP =1

4πǫ2D

3Re[b]∇|E|2, (1)

whereD is particle diameter,ǫ2 the real dielectric constant ofhost medium,E the local RMS electric field, andRe[b] thereal part of the dipole factor (also called Clausius-Mossottifactor)

b =ǫ1 − ǫ2ǫ1 + 2ǫ2

. (2)

Here,ǫ1 andǫ2 are the complex dielectric constants of the par-ticle and the host medium, respectively. In order for the twoabove equations to be valid in an AC field, the dielectric con-stant must include dependence on the frequency. The complexfrequency dependent dielectric constant is defined as

ǫ = ǫ +σ

i2πf,

whereǫ is the real dielectric constant,σ denotes conductivity,f the frequency of the external field, andi ≡

√−1.

The conductivity of a particle consists of three components:Its bulk conductivity (σ1bulk), surface effects due to the move-ment of charge in the diffuse double layer (conductancekd),and the Stern layer conductance (ks), i.e.,

σ1 = σ1bulk +4kd

D+

4ks

D. (3)

The diffuse double layer conductancekd can be given as [20]

kd =4F 2

acz2Ξ(1 + 3Λ/z2)

R0T0κ

[

cosh

(

zFaζ

2R0T0

)

− 1

]

, (4)

whereΞ is the ion diffusion coefficient,z the valency of coun-terions,Fa the Faraday constant,R0 the molar gas constant,ζthe electrostatic potential at the boundary of the slip plane andT0 the temperature. The reciprocal Debye lengthκ providinga measure for screening on the system is given by

κ =

√

2czF 2a

ǫ2R0T0, (5)

where c is the electrolyte concentration. ParameterΛ inEq. (4) describes the electro-osmotic contribution tokd, and itis given by

Λ =

(

R0T0

Fa

)22ǫ23ηΞ

, (6)

whereη is the viscosity of medium. In addition, the Sternlayer conductanceks has the form [11]

ks =uµrΣ

2zFa

, (7)

whereu is the surface charge density,Σ molar conductivityfor a given electrolyte, andµr gives the ratio between the ionmobility in the Stern layer to that in the medium.

For a pair of particles at a separationR suspended in anelectrolyte, we have to consider the multiple image effect.Weconsider two spheres in a medium, and apply a uniform elec-tric field E0 = E0z to the suspension. This induces a dipolemoment into each of the particles. The dipole moments of par-ticles 1 and 2 are given byp10 andp20(≡ p10 = ǫ2E0D

3b/8),respectively.

Next, we include the image effects. The dipole momentp10

induces an image dipolep11 into sphere 2, whilep11 inducesanother image dipole in sphere 1. As a result, multiple imagesare formed. Similarly,p20 induces an imagep21 into colloid1. The formation of multiple images leads to an infinite seriesof image dipoles.

In the following, we obtain the sum of dipole momentsinside each particle, and derive the desired expressions fordipole factors. We consider two basic cases: 1) longitudinalfield (L), where the field is parallel to the line joining the cen-ters of particles, and 2) transverse field (T), where the fieldisperpendicular to the line joining the centers of particles.Us-ing the above notation, the effective dipole factors for a pairare given by [14]

bL∗ = b

∞∑

n=0

(2b)n

[

sinhα

sinh(n + 1)α

]3

,

bT∗ = b

∞∑

n=0

(−b)n

[

sinhα

sinh(n + 1)α

]3

, (8)

whereα is defined viacoshα = R/D. The summations inEqs. (8) include the multiple image effects, then = 0 termgiving the dipole factor of an isolated particle.

We have to derive the analytic expressions forRe[b∗L] and

Re[b∗T] to resolve the DEP force in Eq. (1). To do that, we

resort to spectral representation theory. It offers the advan-tage of being able to separate the material parameters (suchasdielectric constant and conductivity) from structural informa-tion [15] in a natural way.

Let us begin by defining a complex material parameters =1/(1− ǫ1/ǫ2). Using this, the dipole factor for a pair takes theform

b∗ =

∞∑

n=1

Fn

s − sn

, (9)

3

wheren is a positive integer, andFn and sn are then−thmicrostructure parameters of the composite material [15].Asan example, the dipole factor of an isolated particle in spectralrepresentation expression becomesb = F1/(s − s1), whereF1 = −1/3 ands1 = 1/3.

In order to obtain expressions for the dipole factorsb∗L

andb∗T

in Eqs. (8), we introduce the following identity

1

sinh3 x=

∞∑

m=1

4m(m + 1) exp[−(1 + 2m)x].

Its application into Eqs. (8) yields the following exact trans-formations:

b∗L

=∞∑

m=1

F(L)m

s − s(L)m

,

b∗T =

∞∑

m=1

F(T )m

s − s(T )m

, (10)

where the m-th components of the microstructure parameterof the composite material are given as

F (L)m

≡ F (T )m

= −4

3m(m + 1) sinh3 α exp[−(2m + 1)α],

s(L)m =

1

3{1 − 2 exp[−(1 + 2m)α]},

s(T )m =

1

3{1 + exp[−(1 + 2m)α]}.

To make this approach more tractable, we introduce di-mensionless dielectric constant and conductivity [21],s =1/(1 − ǫ1/ǫ2) and t = 1/(1 − σ1/σ2), respectively. Now,we are able separate the real and imaginary parts of the ar-guments in the expressions forb∗

Land b∗

Tin Eq. (10). The

argument can be rewritten as

Fm

s − sm

= (Fm

s − sm

+∆ǫm

1 + f2/f2mc

) − i∆ǫmf/fmc

1 + f2/f2mc

(11)

where

∆ǫm = Fm

s − t

(t − sm)(s − sm)(12)

and

fmc =1

2π

σ2s(t − sm)

ǫ2t(s − sm). (13)

The analytic expressions forRe[b∗L] andRe[b∗

T] (Eq. (10)) be-

come

Re[b∗L] =

∞∑

m=1

(F

(L)m

s − s(L)m

+∆ǫ

(L)m

1 + f2/f2(L)mc

),

Re[b∗T] =

∞∑

m=1

(F

(T )m

s − s(T )m

+∆ǫ

(T )m

1 + f2/f2(T )mc

). (14)

Using these, we can obtain the DEP forceFDEP which in-cludes corrections due to the image effects. The DEP spec-trum consists of a series of sub-dispersions with strength∆ǫm

and characteristic frequencyfmc. In particular, the frequencywhich yieldsF = 0, namelyRe[b∗] = 0, is the desiredcrossover frequencyfCF.

−4 −3 −2 −1log10(σ2) (S/m)

6.4

6.8

7.2

log 10

(fC

F) (

Hz)

−4 −3 −2 −1log10(σ2) (S/m)

6.4

6.8

7.2

log 10

(fC

F) (

Hz)

Isolated particleR/D=2.0R/D=1.5R/D=1.03

L

T

FIG. 1: DEP crossover frequency vs. medium conductivity foranisolated particle (solid line) and two particles at different separations.L denotes longitudinal field case and T transverse field case.Param-eters: ζ = 0.12 V, η = 1.0 × 10

−3 Kg/(ms), u = 0.033 C/m2,Σ = 0.014 Sm2/mol, ǫ1 = 2.25ǫ0.

III. NUMERICAL RESULTS

The above formalism enables us to study the effects dueto multiple images under different physical conditions andtocompare the theory to experimental results. In the following,we compare the crossover frequency of an isolated particle tothat of two particles at different separations. We study theef-fects due to multiple images by varying medium conductivity,theζ-potential, medium viscosity, surface charge density, realdielectric constant of the particle and molar conductivity. Fi-nally, we have computed the DEP spectrum and the dispersionstrength.

The common parameters used in all numerical computa-tions are the following: TemperatureT0 = 293 K, dielectricconstant of host mediumǫ2 = 78ǫ0, bulk conductivity of thecolloidal particleσ1bulk = 2.8×10−4 S/m, ion diffusion coef-ficientΞ = 2.5×10−9 m2/s, the ratio between the ion mobilityin the Stern layer to that in the mediumµr = 0.35, particle

4

−4 −3 −2 −1 0log10(σ2) (S/m)

6.0

7.0

8.0

log 10

(fC

F) (

Hz)

−4 −3 −2 −1 0log10(σ2) (S/m)

6.0

7.0

8.0lo

g 10(f

CF)

(H

z)D=90 (nm)D=216 (nm)D=300 (nm)D=550 (nm)

L

T

FIG. 2: DEP crossover frequency vs. medium conductivity when theparticle size is varied. Parameters as in Fig. 1.

diameterD = 2.16 × 10−7m, counterion valencyz = 1. Thedielectric constant of vacuum is denoted byǫ0.

Figure 1 shows the DEP crossover frequency as a func-tion of medium conductivity for an isolated particle and fortwo particles at different separations. In agreement with re-cent experiments [22], we find that a peak in the crossoverfrequency appears at a certain medium conductivity. The ap-pearance of a peak is preceded by an increase offCF uponincreasing medium conductivity and followed by an abruptdrop [11, 22]. Compared to an isolated particle, the multipleimage effect leads to a red-shift (blue-shift) infCF in the lon-gitudinal (transverse) field case. Furthermore, for longitudi-nal (transverse) field, the stronger the polarization interaction,the lower (higher) the crossover frequency. In addition, itisworth noting that the effect of the multiple images is the op-posite in the longitudinal and transverse cases. As the ratioR/D grows, the predicted crossover spectrum approaches tothat of an isolated particle, i.e., at large separations themulti-ple image interaction becomes negligible.

Motivated by a recent experiment [13], we analyzed the ef-fect of particle size on the crossover frequency by keeping the

−4 −3 −2 −1log10(σ2) ((S/m)

6.4

6.8

7.2

log 10

(fC

F) (

Hz)

−4 −3 −2 −1log10(σ2) (S/m)

6.4

6.8

7.2

log 10

(fC

F) (

Hz)

ζ=0.120ζ=0.125ζ=0.130

L

T

FIG. 3: DEP crossover frequency vs. medium conductivity fordif-ferentζ-potentials. Parameters as in Fig. 1.

ratio R/D fixed and varying the particle diameter. In agree-ment with the experiments, we find that the location of thepeak is shifted to higher frequencies and higher conductivitieswhen the diameter of the particle is reduced, see Fig. 2.

Figure 3 displays the effect of theζ-potential. It has beenexperimentally observed by Hughes and Green [11] that de-creasing theζ-potential may red-shift the DEP crossover fre-quency. The system used by them contained many latex beadssuspended in a solution, and hence the multipolar interactionis expected to play a role. Our results are in qualitative agree-ment with the above experimental findings. Furthermore, anincrease in theζ-potential leads to higherfCF in both the lon-gitudinal and transverse field cases. Similarly, increasing thereal part of the dielectric constant leads to an increase infCF,as displayed in Fig. 4. Increasing the viscosity of the medium(figure not shown here), however, has exactly the opposite ef-fect for both the longitudinal and transverse field cases.

Figure 5 shows the effect of molar conductivityΣ oncrossover frequency. For small medium conductivities (here,σ2 < 10−2 S/m), increasingΣ leads to an increase in thecrossover frequency. However, there is a crossover after which

5

−4 −3 −2 −1log10(σ2) (S/m)

6.4

6.8

7.2

log 10

(fC

F) (

Hz)

−4 −3 −2 −1log10(σ2) (S/m)

6.4

6.8

7.2lo

g 10(f

CF)

(H

z)ε1=2.25ε0

ε1=25ε0

ε1=50ε0

L

T

FIG. 4: DEP crossover frequency vs. medium conductivity. The realpart of the dielectric constant is varied. Parameters as in Fig. 1.

lower values ofΣ yield higherfCF. Similar behavior for thelow surface conductivity regime has been observed in exper-iments [11], but the authors are not aware of any systematicstudy of the molar conductivity onfCF. As Fig. 5 shows, theeffect is similar for both longitudinal and transverse fields.

Figure 6 shows the effect of varying the surface chargedensity on the crossover frequency. Variations in the surfacecharge density lead to more pronounced effects in the low fre-quency region, but close to the peak the variations differencesare very small. In addition, the location of the peak is onlyweakly dependent on surface charge density. These resultsare in agreement with the experimental observations of Greenand Morgan [13].

In Fig. 7, we investigate the real part of the dipole factor,and thus the DEP force. The figure shows that the effect dueto multiple image plays an important role at low frequencyregion when the particles separation is not large, whereas itseffect is smaller in the high frequency region. In the low fre-quency region, the DEP force is be enhanced (reduced) due tothe presence of multiple images for longitudinal (transverse)field case. As the particle separation grows, the multiple im-

−4 −3 −2 −1log10(σ2) (S/m)

6.4

6.8

7.2

log 10

(fC

F) (

Hz)

−4 −3 −2 −1 0log10(σ2) (S/m)

6.4

6.8

7.2

log 10

(fC

F) (

Hz)

Σ=0.013Σ=0.014Σ=0.015Σ=0.016

L

T

FIG. 5: The effect of molar conductivity on the DEP crossoverfre-quency. Parameters as in Fig. 1.

age effect becomes negligible as expected. We also studiedthe effect of particle size on the real part of the dipole factorand the effect of multiple images increases as the particle sizedecreases, and the effect is stronger in the longitudinal fieldcase.

Finally, in Fig. 8, we plot the dispersion strengths (∆ǫ(L)m

and ∆ǫ(T )m ) as a function of the characteristic frequencies

(f (L)mc andf

(T )mc ), for m = 1 to 100 with different medium con-

ductivitiesσ2. Here,m is a positive integer, andFm andsm

are the microstructure parameters of the composite material,see Eqs. 11-13. Hence,∆ǫm andfmc are the m-th dispersionstrength and characteristic frequency due to the presence ofmultiple images as discussed in Sec. II.

The advantage of using the spectral representation theoryis shown in Fig. 8. Based on Fig. 7, it may appear thatonly one dispersion exist. Figure 8 shows, however, that sub-dispersions with strength∆ǫm and characteristic frequencyfmc co-exist, and most of them lie close to the main disper-sion. Thus, the spectral representation theory helps us to gainmore detailed information about the system and it provides a

6

−4 −3 −2 −1log10(σ2) (S/m)

6.3

6.8

7.3

log 10

(fC

F) (

Hz)

−4 −3 −2 −1log10(σ2) (S/m)

6.3

6.8

7.3

log 10

(fC

F) (

Hz)

u=0.030u=0.033u=0.036u=0.039

L

T

FIG. 6: The effect of surface charge density on the DEP crossoverfrequency. Parameters as in Fig. 1.

detailed comparison between the longitudinal and transversefield cases.

At a givenσ2, for the longitudinal (transverse) field case,increasingm leads to corresponding sub-dispersions in thecharacteristic frequency due to the presence of multiple im-ages. The crossover frequenciesfc are3.49 × 106 Hz, 1.0 ×107 Hz and1.4×107 Hz (Fig. 8). From Fig. 8 we find that at alower medium conductivity (say,σ2 = 1.0× 10−4 S/m), mul-tiple images have a stronger effect on the DEP spectrum forthe longitudinal field case than for the transverse field. Thisis also apparent in Fig. 7 as well. Moreover, for longitudinalfield case the multiple images play a role in the low frequencyrange (i.e., smaller thanfc). For the transverse field the situa-tion is the opposite. At a largerσ2 (say,σ2 = 5.0× 10−3 S/mor σ2 = 1.0 × 10−2 S/m ), the sub-dispersion strengths forthe two cases have only a minor difference. These observa-tions may partly explain the results of Green and Morgan [13]whose data suggests that there exists a dispersion below thefrequencies predicted by the current theory. The importanceof these observation lies in the fact that they help to clarifythe interesting question of which polarization mechanismsare

4 5 6 7 8 9log10(f) (Hz)

−0.75

−0.25

0.25

0.75

Re[

Dip

ole

fact

or]

4 5 6 7 8 9log10(f) (Hz)

−0.75

−0.25

0.25

0.75

Re[

Dip

ole

fact

or]

Isolated particleR/D=2.0R/D=1.5R/D=1.03 L

T

FIG. 7: DEP spectrum (the real part of the dipole factor). Parametersas in Fig. 1.

present.

IV. DISCUSSION AND CONCLUSION

In this study, we have investigated the crossover spectrumof two approaching polarizable particles in the presence ofa nonuniform AC electric field. When the two particles ap-proach, the mutual polarization interaction between the par-ticles leads to changes in the dipole moments of each of theindividual particles, and hence in the DEP crossover spectrum.This can be interpreted as a correlation effect analogous totheones seen in charged systems [16].

For charged particles, there is a coexistence of an elec-trophoretic and a dielectrophoretic force in the presence ofa nonuniform AC electric field. The DEP force always pointstoward the region of high field gradient. It does not oscil-late with the change of direction of the field. In contrast, theelectrophoretic force points along the direction of field, andhence is oscillatory under the same conditions. How to sepa-rate the DEP force from the electrophoretic force is a question

7

5 6 7log10(fmc) (Hz)

−2.5

−1.5

−0.5

0.5

log 10

(∆ε m

)

L (σ2=1.0X10−4

)T (σ2=1.0X10

−4)

L (σ2=5.0X10−3

)T (σ2=5.0X10

−3)

L (σ2=1.0X10−2

)T (σ2=1.0X10

−2)

FIG. 8: Dispersion strength versus the characteristic frequency fordifferent medium conductivities. Parameters:ζ = 0.12V, η = 1.0×10

−3 Kg/(ms),R/D = 1.03, ǫ1 = 2.25ǫ0, u = 0.033 C/m2, Σ =

0.014 Sm2/mol. The lines are drawn as a guide to the eye.

of interest in many experimental setups [6, 23]. In differentfrequency ranges, either the electrophoretic force or the DEPforce dominates, and the transition from one to the other oc-

curs at a frequencyftr, which has been approximately deter-mined [4]. Here, we have chosen a frequency region whereelectrophoretic effects are negligible and the DEP force dom-inates. In addition, although we are at finite temperature,Brownian motion is not included in our analysis. In experi-ments Brownian motion is always present and has posed diffi-culties in dielectrophoresis of submicrometer particles.How-ever, with current techniques it is possible to access also thisrange [4, 5].

One of the interesting questions is what happens, when thevolume fraction of the suspension becomes large. It turns outthat it is possible to extend our approach by taking into ac-count local field effects which may modify the DEP crossoverspectrum. Work is in progress to address these questions. Inaddition to dielectrophoresis, the extension of the present ap-proach is also of interest from the point of view of electroro-tation.

To summarize, using the multiple image method, we havebeen able to capture mutual polarization effects of two ap-proaching particles in an electrolyte. Using spectral represen-tation theory, we derived an analytic expression for the DEPforce, and using that the crossover frequency was determined.From the theoretical analysis, we find that the mutual polariza-tion effects can change the crossover frequency substantially.

Acknowledgments

This work has been supported by the Research GrantsCouncil of the Hong Kong SAR Government under projectnumber CUHK 4245/01P, and by the Academy of FinlandGrant No. 54113 (M. K.). J.P.H. is grateful to Prof. K. Kaskifor helpful discussions.

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