Articles
Electrokinetic Transport in Nanochannels. 1.
Theory
Sumita Pennathur* and Juan G. Santiago
Department of Mechanical Engineering, Stanford University, Stanford, California 94305
Electrokinetic transport in fluidic channels facilitates
control and separation of ionic species. In nanometer-
scale electrokinetic systems, the electric double layer
thickness is comparable to characteristic channel dimen-
sions, and this results in nonuniform velocity profiles and
strong electric fields transverse to the flow. In such
channels, streamwise and transverse electromigration
fluxes contribute to the separation and dispersion of
analyte ions. In this paper, we report on analytical and
numerical models for nanochannel electrophoretic trans-
port and separation of neutral and charged analytes. We
present continuum-based theoretical studies in nanoscale
channels with characteristic depths on the order of the
Debye length. Our model yields analytical expressions for
electroosmotic flow, species transport velocity, stream-
wise-transverse concentration field distribution, and ratio
of apparent electrophoretic mobility for a nanochannel to
(standard) ion mobility. The model demonstrates that the
effective mobility governing electrophoretic transport of
charged species in nanochannels depends not only on
electrolyte mobility values but also on ú potential, ion
valence, and background electrolyte concentration. We
also present a method we term electrokinetic separation
by ion valence (EKSIV) whereby both ion valence and ion
mobility may be determined independently from a com-
parison of micro- and nanoscale transport measurements.
In the second of this two-paper series, we present experi-
mental validation of our models.
The advent of well-defined nanoscale fluidic (nanofluidic)
channel systems has spurred both speculation and experimenta-
tion into their possible applications in the analysis of chemical
and biological species.
1-3
The distinct physical regimes of nano-
fluidic channel systems offer interesting possibilities for new
functionality, including separation and analysis modalities. An
effective technique for pumping liquids in such systems is
electroosmotic flow. Electroosmotic flow is generated by electric
body forces within an electric double layer (EDL) that spontane-
ously forms at solid-liquid interfaces
4
and whose dimensions scale
as the Debye length of the electrolyte.
5
The counterions of the
EDL shield the wall charge within a region that scales with Debye
length. In nanoscale channel systems, channel dimensions are of
the order of the Debye length, transverse electromigration plays
a critical role in determining ion distributions, and a highly
nonuniform velocity profile is established.
6-10
Analytical studies of potential distributions and electrokinetic
transport within long thin channels with significant EDL thick-
nesses date back at least several decades.
4,8-22
Since the EDL
thickness of typical aqueous electrolytes typically ranges from 1
nm to a theoretical maximum of ∼1 µm,
23
fluidic systems with
finite and overlapping double layers can be referred to as
nanoscale electrokinetic systems. Burgeen and Nackache
6
devel-
oped theory for electrokinetic flow in capillary slits with finite
double layers and predicted a high degree of flow retardation in
channel flows having a large Debye length-to-channel height ratio.
Their work examines systems with both low and high nondimen-
sional zeta potentials, ú* ) ú ze/kT, where ú is zeta potential and
kT/ze is the thermal voltage.
23
The work of Burgeen and Nakache
is well complemented by the work of Levine and co-workers.
7-9
* To whom correspondence should be addressed. E-mail: sumita@stanford.edu.
Fax: (650)-723-7657.
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6772 Analytical Chemistry, Vol. 77, No. 21, November 1, 2005 10.1021/ac050835y CCC: $30.25 2005 American Chemical Society
Published on Web 09/28/2005

Collectively these seminal papers are limited to liquid velocity
fields and net current transport and do not address the transport
of individual, charged solute species over long distances (i.e.,
electrophoresis). Other, more recent examples of electrokinetic
flow theory development include work on combined pressure-
driven flow and (streamwise) electrophoretic ion motion,
14
the
effects of adsorption/desorption dynamics on electrophoresis,
15,24
and the dispersion dynamics of neutral solutes in electrokinetic
nanochannels.
17,18
Although a few of these studies investigate the
migration of individual species,
14,15,17,18,24
none of them address
the role of coupling between transverse and streamwise elec-
tromigration fluxes and the effect of this coupling on electro-
phoresis and analyte dispersion. Also, most of the studies are
restricted to the assumption of low ú potential.
14,15,17,24
There are a few recent experimental studies of molecular
transport through planar, shallow nanochannels.
11,25-31
Shu et al.
31
fabricated nanochannels for the application of stretching and
studying the dynamics of 103-kbase T5 phage DNA. Petersen et
al.
28
fabricated nanochannels for DNA separation and demon-
strated a separation between strand lengths of 100 and 1000 base
pairs in a 320-nm-deep channel. Stein et al.
30
performed experi-
ments in nanochannels to quantify the effects of surface charge
on net ion transport. The latter study measured total ion current
as a function of bulk conductivity (i.e., electrolyte conductivity
measured outside of channel) in 70-nm and 1.015-µm channels
with λ
D
ranging from 0.3 to 100 nm. Pu et al.
26
qualitatively
described an ion depletion effect at the interface between nano-
and microchannels. Image intensity data were recorded for 60-
nm-deep channels with electrolyte concentrations of 70 µM, and
ion depletion was characterized with 30 µM fluorescein in
unbuffered solutions. Although a few of these experimental studies
examined the effects of individual species,
26,28,32
none addressed
electrophoretic transport of individual charged species and its
coupling with transverse electromigrative fluxes.
Table 1 summarizes typical examples of theoretical and
experimental studies of electrokinetic flow in systems with a finite
EDL. We use symbols to summarize the content of each paper.
For the theoretical work, they are as follows: u are studies of
liquid velocity fields; ú* > 1 denotes theory with a validity
extending in the high ú potential regime; N > 1 denotes
investigations of distribution of multiple ions in the EDL; κ
S
are
studies of surface conductivity effects;
33
c
i
denotes studies of the
transport of individual neutral species; and the symbol zc
i
is used
to denote transport of charged species. For the experimental work,
they are as follows: I denotes measurements of total ion current;
u are experimental measurements of liquid transport; and c
i
are
measurements of the transport of individual (charged or un-
charged) species.
In this paper, we present theory valid for electrokinetic
transport in nanometer-scale channels. We study electrophoretic
transport of both charged and neutral species in long thin
nanochannels and show that both transverse electromigration and
nonuniform velocities have a significant effect on both net
streamwise transport and dispersion. We present continuum
theory valid for the finite ú* regime and Debye lengths on the
order of the channel height. We also present a method we term
electrokinetic separation by ion valence (EKSIV) that can be used
to determine both ion mobility and valence of analyte ions from
a comparison of micro- and nanoscale transport measurements.
In the second of this two-paper series,
35
we present an experi-
mental validation of the model.
CONTINUUM THEORY MODEL
Liquid Transport Simulations. We first summarize the
governing equations for transport of liquids in channel flows where
λ
D
is on the order of at least one characteristic channel dimension.
We begin with the classical equations describing electrokinetic
flows, as presented by Levich
36
and Probstein.
5
Assuming fully
developed flow and negligible flow to pressure gradients, the
continuum equations are
where ub is the velocity of the liquid, F
E
is the charge density, µ is
the viscosity,  is the permittivity, e is the elementary charge, k is
the Boltzmann constant, T is temperature, z is valence number,
and n
c
is the number density (m
-3
) in the center of the channel
for a channel with nonoverlapped EDLs. Equations 1 are the
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(27) Tas, N. R. J. Micromech. Microeng. 1996, 6, 385.
(28) Petersen, N.; Alarie, J. P.; Ramsey, J. M., Squaw Valley, CA; Northrup, A.
2003; pp 701-703.
(29) Jacobson, S. C.; Moore, A. W.; Ramsey, J. M. Anal. Chem. 1995, 67, 2059-
2063.
(30) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93,1-4.
(31) Shu, D.; Moll, W. D.; Deng, Z.; Mao, C. Nano Lett. 2004,69-73.
(32) Guo, ? Nano Lett. 2004, 4,69-73.
(33) Lyklema, J. J. Phys.: Condens. Matter 2000, 13, 5027-5034.
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Cliffs, NJ, 1962.
Table 1. Theoretical and Experimental Nanochannel
Electrokinetic Studies
reference theory experiments
Burgeen and Nakache
6
ú* > 1, u
Rice and Whitehead
7
u
Levine et al.
8
ú*>1, u
Qiao and Aluru
22
ú* > 1, N > 1, u, κ
S
Griffiths and Nilson
18
ú* > 1, u, c
i
Martin et al.
24
κ
S
, c
i
Datta nd McEldoon
15
u, c
i
Griffiths and Nilson
17
u, c
i
Daiguji et al.
13
ú* > 1, κ
S
,N> 1
Datta and Kotamurthi
14
u, c
i
Stein et al.
30
κ
S
I, u
Guo et al.
32
u, c
i
c
i
Pu et al.
26
N > 1, c
i
I, c
i
Fang et al.
34
c
i
c
i
Peterson et al.
28
u, c
i
u, c
i
current work u,ú* > 1, zc
i
Pennathur and Santiago
35
u, I, c
i
∇‚ub)0; 0 ) µ∇
2
ub+F
e
∇Φ (1)
∇
2
Φ )
-
∑
i)1
N
z
i
en
c
exp(-(z
i
eΦ/kT))

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Analytical Chemistry, Vol. 77, No. 21, November 1, 2005 6773