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Induced dipoles and dielectrophoresis of nanocolloids in electrolytes

Sagnik Basuray and Hsueh-Chia Chang

*

Center for Microﬂuidics and Medical Diagnostics, Department of Chemical and Biomolecular Engineering, University of Notre Dame,

Notre Dame, Indiana 46556, USA

共Received 15 March 2007; published 25 June 2007

兲

Electric induced dipoles of nanocolloids the size of the Debye length are shown to be one order stronger than

predicted by the classical Maxwell-Wagner theory and its extensions. The difference is attributed to normal ion

migration within the diffuse layer, and adsorption onto the Stern layer at the poles. The characteristic relaxation

frequency 共the crossover frequency for dielectrophoresis兲 is shown to be inversely proportional to the RC time

of the diffuse layer capacitance and resistance, and has an anomalous −1 scaling with respect to the product of

the Debye length and the particle size.

DOI: 10.1103/PhysRevE.75.060501 PACS number共s兲: 82.45.⫺h, 47.57.jd, 47.65.⫺d, 66.10.Ed

Dielectrophoresis 共DEP兲 has become an increasingly

popular means of manipulating and identifying immunocol-

loids, bioparticles, and DNA molecules in microﬂuidic de-

vices 关1–3兴. The dielectrophoretic force on the particle re-

sults from induced ac particle dipoles, which can develop by

either conductive or dielectric polarization mechanisms ac-

cording to classical Maxwell-Wagner 共MW兲 theory 关2兴. For

conducting particles with low permittivity, conductive polar-

ization dominates at low frequency and dielectric polariza-

tion at high frequency, with opposite dipole orientations with

respect to the applied ﬁeld and opposite positive 共toward

high ﬁeld兲 and negative 共toward low ﬁeld兲 DEP mobility. As

the length of both the particle capacitor and the resistor is the

particle size a, the dipole orientation and RC relaxation time,

as determined by f

c.m.

=共

˜

P

−

˜

M

兲/共

˜

P

+2

˜

M

兲, the Clausius-

Mossotti factor, is size independent. The complex permittiv-

ity

˜

is related to the real permittivity and conductivity

via the ac ﬁeld frequency

for both particle 共P兲 and medium

共M兲. Hence, there exists a crossover frequency

CO

, deﬁned

by Re关f

c.m.

兴=0, when the induced dipole vanishes, which the

MW theory predicts to be size independent:

MW

=

1

2

冑

共

P

−

M

兲共

P

+2

M

兲

共

M

−

P

兲共

P

+2

M

兲

. 共1兲

This MW theory was recently found to be inaccurate for

nanocolloids 关3–6兴. The

CO

data of Green and Morgan 关6兴

in Fig. 1 for nanosized latex particles clearly show a particle

size dependence. In fact, according to the classical MW

theory Eq. 共1兲, latex particles with permittivity and conduc-

tivity both lower than the medium are not expected to exhibit

any crossover phenomenon. Ermolina and Morgan 关7兴 have

suggested that, for latex and other particles with negligible

native conductivity,

p

should be dominated by Stern and

diffuse layer conduction effects, as described by the classical

theories for electrophoresis, 共see the review in 关8兴兲. With

negligible native conductivity, the corrected particle conduc-

tivity becomes

p

=共2K

s

/a兲+ 共2K

D

/a兲, where K

s

is the Stern

layer conductance and K

D

the diffuse layer conductance,

which includes a dependence on the particle

potential due

to osmotic ﬂow effects. However, that the effective conduc-

tance is the sum of the two conductance contributions from

the Stern and diffuse layers reﬂects the implicit assumption

in the classical theories that the conduction “surface” cur-

rents are tangential. This is true for the Stern layer, which is

always thin compared to the particle, and hence its internal

ﬁeld is always tangential to the latex insulator particle. It is,

however, inaccurate for diffuse layers with Debye length

=共

M

D/

M

兲

1/2

共D is the ion diffusivity兲 comparable to a.

With the normal length scale comparable to the tangential

one in such thick layers, the normal ﬁeld and current become

important in the diffuse layer, and this effect is not captured

by the classical theories. Moreover, modeling just conduction

共electromigration兲 in the diffuse layer is inadequate, as the

neglected diffusive ﬂux can counter electromigration to pro-

duce tangential equilibrium or Poisson-Boltzmann distribu-

*

Author to whom correspondence should be addressed.

hchang@nd.edu

FIG. 1. Green and Morgan’s 关6兴 crossover frequency data for

557 共black triangles兲 and 93 nm 共unﬁlled triangles兲 latex particles.

The classical MW theory with a conducting Stern layer is shown as

a dotted line. The Stern layer conductance is 1.46 nS for 93 nm

particles and 1.92 nS for 557 nm particles, calculated from the data

asymptote at low medium conductivity. The dashed line is from an

extended theory of 关7兴 that includes tangential conduction in the

diffuse layer 共with the same surface conductance as ours and

potential of 67 mV for 93 nm and 45 mV for 557 nm particles from

Fig. 5 of 关7兴兲. The solid line corresponds to the scaling theory of Eq.

共2兲 with a scaling constant of 2.5 and D=5⫻10

−9

m

2

/s.

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tions. We hence expect the improved MW theories to be

valid only at lower conductivity when Ⰷa, or for large

particles, aⰇ.

At the limit of inﬁnitely low medium conductivity, the

Stern layer conduction contribution dominates that from the

diffuse layer even in these theories, and the crossover fre-

quency with negligible native permittivity approaches

⬁

=1/2

共K

s

/

冑

2a

M

兲. Using this limiting value for the particle

data of Green and Morgan 关6兴 to estimate K

s

for different

particle sizes 共see inset of Fig. 2兲, the MW theory with Stern

layer conduction 共dotted curve兲 is shown in Figs. 1 and 2 for

various particle sizes. It is shown to accurately capture the

measured crossover frequencies of large particles, but still

fails to describe the data for nanocolloids with smaller di-

mensions, at medium conductivities corresponding to a ⬃,

as shown in Fig. 1. Although this new crossover frequency is

size dependent, it falls monotonically with respect to the me-

dium conductivity, instead of exhibiting the order-of-

magnitude rise for the smaller particle 共 93 nm兲. The dashed

curve in Fig. 1 corresponds to the extended theory, which

includes diffuse layer conduction and electro-osmotic ﬂow

convection. Using

potential estimates from Fig. 5 of 关7兴 for

all latex particles, and our Stern layer conductance values as

shown in the inset in Fig. 2, the crossover now exhibits a

shallow maximum, but is still lower than the actual data by a

factor of 4. Allowing for a conductivity-dependent

poten-

tial produces similar results.

Considerable empirical evidence 关9–12兴, particularly for

ion-selective conducting particles, suggests that a normal ca-

pacitive current, missing in the classical theories, is respon-

sible for the discrepancy seen in Figs. 1 and 2 at the critical

region of ⬃a. If the migrating ions are allowed to accumu-

late at the opposite poles, an induced dipole that favors posi-

tive dielectrophoresis increases

CO

. If we assume that the

diffuse layer permittivity and conductivity are similar to

those of the bulk medium, the diffuse layer capacitance

would be

M

/ and its resistance along the particle a/

M

.

Consequently, we expect

CO

of this theory to scale as the

inverse RC time of the diffuse layer,

DL

⬃

冉

M

a

M

冊

=

D

a

. 共2兲

This simple expression quantitatively captures

CO

for

smaller particles, as seen in Figs. 1 and 2. Its ratio to

⬁

, the

low-conductivity limiting frequency from the MW theory, is

共1/2

兲共

冑

2

M

/

p

兲共/a兲, and this ratio increases monotoni-

cally with medium conductivity

M

. As is evident from the

93 nm data in Fig. 1, this scaling can be almost a factor of 10

larger than

MW

.

Our theory, which quantitatively conﬁrms the above scal-

ing, uses Stern layer conduction to describe

p

, but describes

both tangential and normal 共conducting and diffusive兲 ﬂuxes

in the diffuse layer explicitly. It follows that of Gonzalez et

al. for electrode double-layer polarization 关12兴 with a Debye-

Hückel linearization for a symmetric monovalent electrolyte,

whose dynamic space charge ﬂuctuation and charge density

are small compared to the dc ion concentrations. However,

Gonzalez et al. consider the thin double-layer limit of

共a / 兲Ⰷ 1, while we include the full Laplacian for the Pois-

son equation and for the charge transport equation for ⬃a.

After a Fourier transform in time, the dimensionless ac com-

ponents of the charge density

共scaled by 2n

0

e兲, the medium

and particle potential 共scaled by k

B

T/e兲, the spatial coordi-

nates scaled by a, and the time scaled by 共

M

/

M

兲, the two

potentials and charge density satisfy

ⵜ

2

M

=−

冉

a

冊

2

, ⵜ

2

P

=0,

ⵜ

2

=

冋

冉

ia

2

D

冊冉

M

M

冊

+

冉

a

冊

2

册

= s

2

冉

a

冊

2

, 共3兲

where

, the dimensionless frequency, has been scaled by

共

M

/

M

兲, and the complex number s is deﬁned by s

2

=1

+i

. The Laplacian on the left of the charge density repre-

sents diffusion, while the two terms on the right represent

accumulation and electromigration, respectively.

With a uniform unit dimensionless far ﬁeld in e

ˆ

Z

, the rel-

evant axisymmetric charge density solution to the Bessel

equation 共3兲 can be expressed as a Bessel function of frac-

tional order, and the particle and medium potentials contain

the related spherical harmonics,

=

BK

3/2

冉

a

sr

冊

r

1/2

cos

,

P

= Dr cos

,

M

=

冢

−

BK

3/2

冉

a

sr

冊

s

2

r

1/2

+

A

r

2

− r

冣

cos

, 共4兲

where the ﬁrst term in the medium potential represents the

polarized space charge distribution in the diffuse layer and

the second term represents the ﬁeld due to the induced par-

ticle dipole. As the space charge contribution to the potential

decays exponentially as exp共−asr/兲, it is clear from Eq. 共4兲

that the potential seen by any second-order applied ﬁeld with

a ﬁnite gradient is predominantly due to the dipole and not

FIG. 2. Scaled data of Green and Morgan for four different latex

particle dimensions in different KCl and NaCl electrolytes with D

=5⫻10

−9

m

2

/s to resolve the data points across a wide range of

/a. The Stern layer conductance using the data asymptote at low

conductivity for different particle sizes is shown as the inset. The

scaling theory of Eq. 共2兲 captures the anomalous rise of the 93 nm

data when ⬃a.

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the space charge. We hence do not need to be concerned with

the Maxwell force on the space charge, nor the resulting

electro-osmotic convection. The space charge is nevertheless

important, as it screens the external ﬁeld such that the par-

ticle dipole is different from the unscreened one as for

共 / a兲Ⰷ 1. With negligible space charge contribution to the

electric force on the particle, the coefﬁcient A for the com-

plex charge of the induced dipole can be used directly

in the classical expression for the dipole force,

4

M

Re共A兲a

3

E

ˆ

·共ⵜ

ˆ

E

ˆ

兲, to obtain the DEP velocity u

DEP

=共

M

a

2

/6

兲Re共A兲ⵜ

ˆ

兩E

ˆ

兩

2

, where E

ˆ

is the dimensional elec-

tric ﬁeld.

The speciﬁc values of A , however, require determination

of the coefﬁcients in Eq. 共4兲. The MW theory uses as surface

boundary conditions the potential continuity 关

兴=0 and the

complex displacement continuity in dimensional form,

冋

˜

n

册

=0 共5兲

where 关·兴 denotes a jump across the particle surface, and the

Stern layer effect is included in the particle conductivity.

Equation 共5兲 is derived by eliminating the surface charge q

s

from the surface charge accumulation due to the conductive

ﬂux imbalance, i

q

s

=关

/

n兴, and the displacement jump

due to the surface charge, −q

s

=关

/

n兴, where the ac sur-

face charge resides at the diffuse/Stern layer interface. How-

ever, an additional boundary condition on this interface is

needed for the space charge density

. For the normal diffuse

layer current to be a charging current, the space charge must

adsorb onto the surface.

Adsorption of counterions at the Stern layer has been well

documented 关13–15兴, and a reversible adsorption isotherm is

usually assumed, as the adsorption and desorption kinetic

times across the Stern layer are expected to be on the order

of nanoseconds or shorter. This corresponds to an ion asso-

ciation reaction, as the counterion is condensing onto a sur-

face charge of opposite sign, and available quasielastic neu-

tron scattering 共QENS兲 and NMR studies of association time

scales 共⬍10

−9

s兲 are indeed much shorter than the inverse

crossover frequencies of nanocolloids, except for very large

hydrated ions 关16兴. We model this Stern layer of thickness

s

to have the same conductivity and permittivity as the me-

dium. The adsorbed Stern layer charge

s

is distinct from the

surface charge q

s

responsible for the complex MW condition

Eq. 共5兲, where q

s

is due to smaller charge carriers than the

adsorbed ions and exists at the inside 共solid兲 boundary of the

Stern layer. We hence exchange the potential continuity

equation at the particle surface with the Stern layer condition

of a potential jump across the Stern layer, with adsorbed

Stern layer charge related to the space charge

by an equi-

librium isotherm

s

=K

eq

,

P

−

M

= K

eq

⬘

冉

S

冊

, 共6兲

where K

eq

⬘

=K

eq

/. Accumulation of

s

in the Stern layer due

to imbalance in particle and medium ﬂuxes yields, using the

same isotherm, the dimensionless condition

冉

a

冊冉

S

冊

i

共

M

−

P

兲 =

r

+

M

r

−

P

M

P

r

. 共7兲

As the ﬁeld is the same in the Stern layer and on the

medium side, we use 共the dimensionless version of兲 Eq. 共5兲

with Eqs. 共6兲 and 共7兲 on the particle surface to solve for the

three complex coefﬁcients in Eq. 共4兲. We use a reasonable

value for the equilibrium constant K

eq

⬘

=K

eq

/=1 and the lit-

erature value 关17兴 for the Stern layer capacitance

m

/

S

=80

Fcm

−2

. The Stern layer RC time is simply the capaci-

tance term in Eq. 共7兲 and yields a characteristic frequency

Stern

=共D / a兲共

S

/兲Ⰶ

DL

. With Stern layer adsorption, the

diffuse layer also becomes a capacitor. However, since both

Stern and diffuse layer capacitors are in series relative to the

normal charging current, it is the diffuse layer with lower

FIG. 3. 共Color online兲 Real part of the dipole strength A for 93

and 557 nm particles for

M

=1 mS m

−1

, =37 nm, K

s

=1.46 nS for

93 nm particles and K

s

=1.92 nS for 557 nm particles, relative per-

mitivities of

M

=78.5 and

P

=2.25, and D =10

−9

m

2

/s. The Stern

layer parameters are K

eq

⬘

=1.0,

m

/

S

=80

Fcm

−2

. Charge density

contours for representative conditions are shown in the insets.

FIG. 4. Comparison of our present theory to Green and Mor-

gan’s data using the parameters in Fig. 3. The Stern layer conduc-

tance for each particle is given in the inset in Fig. 2. The symbols

are the same as in Fig. 2. Dotted curve, MW theory with conducting

Stern layer; solid curve, current theory.

INDUCED DIPOLES AND DIELECTROPHORESIS OF … PHYSICAL REVIEW E 75, 060501共R兲共2007兲

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capacitance and higher inverse RC time that dominates, such

that

DL

of 共2兲 captures the data in Figs. 1 and 2.

The coefﬁcients are determined from a series expansion

of the complex functions. The real part of the charge density

distribution within the diffuse layer is plotted in Fig. 3 for the

two particles of Fig. 1 with roughly the size of the smaller

one. There is negligible space charge in the diffuse layer of

the larger particle for all frequencies, as in the MW theory.

For the smaller particle, the Stern layer capacitor is saturated

at frequencies much lower than

Stern

, and the tangential

conduction in the diffuse layer drains the space charge away.

At higher frequencies, the ac period is too small for charging

current to penetrate the diffuse layer to reach the Stern layer.

These limits, hence, correspond to the conductive and dielec-

tric polarization mechanisms of the MW theory, and there is

again no space charge. At about

Stern

⬃10

5

Hz, however,

the charge density of the smaller particle becomes dramati-

cally higher due to Stern layer charging. Re关A兴 is also shown

in Fig. 3 for both nanocolloids. While the larger colloid

shows a small correction to classical MW dipole intensity

that decreases with increasing frequency, a sharp maximum

is observed for the smaller particle at a frequency near

Stern

.

As shown in Fig. 4, all the crossover frequency data for

various particle sizes by Green and Morgan can be captured

with this model by using the proper a / for each experiment

and using the Stern layer conductance obtained from the

asymptotic crossover of the latex particles at low medium

conductivity. Adjusting the equilibrium constant K

eq

⬘

and the

Stern layer capacitance, the only unknown parameters, over

one order of magnitude does not change the curves signiﬁ-

cantly, provided

Stern

Ⰶ

DL

.

We thank Z. Gagnon, F. Plouraboue, and H. H. Wei for

valuable discussions on this subject and acknowledge ﬁnan-

cial support from the Center of Applied Math and NSF.

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