Review
Sandip Ghosal
Department of Mechanical
Engineering,
Northwestern University,
Evanston, IL, USA
Fluid mechanics of electroosmotic flow and its effect
on band broadening in capillary electrophoresis
Electroosmotic flow (EOF) usually accompanies electrophoretic migration of charged
species in capillary electrophoresis unless special precautions are taken to supress it.
The presence of the EOF provides certain advantages in separations. It is an alterna-
tive to mechanical pumps, which are inefficient and difficult to build at small scales,
for transporting reagents and analytes on microfluidic chips. The downside is that any
imperfection that distorts the EOF profile reduces the separation efficiency. In this
paper, the basic facts about EOF are reviewed from the perspective of fluid mechanics
and its effect on separations in free solution capillary zone electrophoresis is discussed
in the light of recent advances.
Keywords: Band broadening / Capillary electrophoresis / Electroosmosis / Lubrication theory /
Review / Taylor dispersion DOI 10.1002/elps.200305745
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
2 Electroosmotic flow. . . . . . . . . . . . . . . . . . . . . 215
3 The thin EDL limit. . . . . . . . . . . . . . . . . . . . . . . 216
4 EOF in a uniform cylindrical capillary . . . . . . . 216
5 Axially inhomogeneous channels . . . . . . . . . . 217
5.1 Exactly solvable models . . . . . . . . . . . . . . . . . 217
5.2 Potential flow solution . . . . . . . . . . . . . . . . . . . 218
5.3 Lubrication approximation . . . . . . . . . . . . . . . 219
6 Dispersion and EOF . . . . . . . . . . . . . . . . . . . . 220
6.1 Dispersion due to finite size of the EDL . . . . . 220
6.2 Dispersion due to analyte-wall interactions . . 221
6.3 Thermal broadening . . . . . . . . . . . . . . . . . . . . 223
6.4 Dispersion in curved channels . . . . . . . . . . . . 225
7 Summary and conclusions . . . . . . . . . . . . . . . 226
8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
1 Introduction
Electroosmotic flow (EOF) was first reported by Reuss [1]
in 1809 in experiments that demonstrated that water
could be made to percolate through porous clay dia-
phragms through the application of an electric field. The
observed mobility of water was due to the fact that the
clay particles (and many other solid substrates such as
glass, silicon, polymeric materials, minerals of various
kinds, etc.) acquire a surface charge when in contact
with an electrolyte. The immobile surface charge in turn
attracts a cloud of free ions of the opposite sign creating
a thin (,1–10 nm under typical conditions, e.g., univalent
electrolyte at a concentration of 1–100 mol per m3) Debye
layer of mobile charges next to it. The thickness of this
electric double layer (EDL) is determined by a balance be-
tween the intensity of thermal (Brownian) fluctuations and
the strength of the electrostatic attraction to the sub-
strate. In the presence of an external electric field, the
fluid in this charged Debye layer acquires a momentum
which is then transmitted to adjacent layers of fluid
through the effect of viscosity. If the fluid phase is mobile
(such as in a packed bed of particles or in a narrow capil-
lary), it would cause the fluid to flow (electroosmosis). In a
typical separation in capillary electrophoresis (CE)* both
electroosmosis (sometimes also called electroendosmo-
sis) and electrophoresis occur simultaneously. Therefore,
the resultant migration velocity of each species ‘i’ is
uðiÞtotal
ueof þ u
ðiÞ
eph (1)
where the first term is the bulk EOF velocity and the sec-
ond term is the migration velocity relative to still fluid of
species i (generally different for each species ). Due to
Correspondence: Dr. Sandip Ghosal, Department of Mechani-
cal Engineering, Northwestern University, 2145 Sheridan Road,
Evanston, IL 60208, USA
E-mail: s-ghosal@northwestern.edu
Fax: 1847-491-3915
Abbreviations: EDL, electric double layer; HS, Helmholtz-Smo-
luchowski;
-TAS, micro total analysis system
214 Electrophoresis 2004, 25, 214–228
* In this paper CE will always refer to ‘free solution capillary zone
elecrophoresis’, which is the only mode considered here,
though the ideas presented could with appropriate modifica-
tion be useful for other separation modes.
 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Electrophoresis 2004, 25, 214–228 EOF and band broadening 215
the thinness of the EDL (1–10 nm) compared to typical
channel radii (10–100 mm), the electrical driving forces
are localized in a thin sheath at the solid-fluid interface.
EOF can have a number of effects on the efficiency of
separation. First note that if ueofj j4 u ið Þeph







for all i, then all
species move in the same direction enabling single point
detection of charged species of either sign. If the capillary
is uniformly charged and the inlet and outlet are at the
same pressure, it is well-known that the flow is uniform
throughout the capillary cross-section except for a very
thin EDL near the wall where the flow velocity rapidly
decreases from its free stream value to zero at the sub-
strate/fluid interface [2]. This uniform velocity is given by
the Helmholtz-Smoluchowski (HS) formula:
ue ¼ 
eEz
4pm (2)
where e is the dielectric constant of the fluid, E is the
applied electric field, z is the zeta-potential at the electro-
lyte/substrate interface, and m is the fluid viscosity. There-
fore, except in the thin EDL there is no shear in the flow.
Thus, EOF does not add any significant shear induced
axial dispersion (Taylor-Aris dispersion) to the analyte.
Band broadening in this case is purely due to axial molec-
ular diffusion.
Clearly, Eq. (2) cannot be valid if any of the parameters
that enter into the expression vary in the axial direction,
or, if the cross-section varies along the capillary. This is
because any such variation would require the continuity
condition, that the fluid flux through all cross-sections be
the same to be violated. Axial variations usually lead to
induced pressure gradients that drive a ‘Poiseuille’ type
of flow*. Thus, the flat EOF profile becomes distorted
resulting in strong band broadening due to Taylor-Aris dis-
persion [3, 4]. The effect of such axial variability is dis-
cussed in detail in this review.
The rest of this paper is organized in the following way: in
the next section, the basic equations describing EOF in
any conduit are presented. In Section 3, these basic
equations are simplified by introducing the assumption
of thin Debye layers. An exact solution of the EOF prob-
lem in cylindrical capillaries due to Rice and Whitehead is
presented next (Section 4) and compared with the corre-
sponding reduced solution in the case of infinitely thin
Debye layers. In Section 5, the more difficult case of EOF
through capillaries with axial inhomogeneities is consid-
ered and solutions are discussed for two special geome-
tries. In order to handle problems involving inhomoge-
neous channels of arbitrary cross-sectional shapes, the
potential flow and the lubrication approximation are intro-
duced in Sections 5.2 and 5.3. Finally, the dispersive
effects of EOF in homogeneous and inhomogeneous
channels are considered in Section 6. A summary is pre-
sented in Section 7.
2 Electroosmotic flow
The equations describing the velocity field, u, of the fluid
phase are those of momentum conservation:
r0(qtu 1 u ? Hu) ¼ 2 Hp 1 mH2u 2 reHf (3)
and continuity:
H ? u ¼ 0 (4)
where r0 and m are the (constant) density and viscosity of
the fluid, p is the fluid pressure, f is the electric potential,
and the charge density in the EDL, re is related to the
potential by Poisson’s equation
eH2f ¼ 24pre (5)
To close the system, we need an equation for determining
f, which is the Poisson-Boltzmann equation
H2f ¼ 2k2f (6)
where k is a constant determined by the ionic composi-
tion of the electrolyte [5]. The Debye length is defined by
lD ¼ 2p/k. The form (6) incorporates the Debye-Hückel
approximation f kBT/e where kB is the Boltzmann con-
stant, T is the absolute temperature, and e is the electron-
ic charge. At room temperature, kBT/e < 25 mV. The elec-
tric potential at the substrate buffer interface could be as
high as 100 mV . Thus, the Debye-Hückel approximation
is not always satisfied, in which case Eq. (6) should be
replaced by the more accurate but nonlinear Gouy-Chap-
man form [5]. However, Eq. (6) is still useful for the pur-
pose of qualitative understanding even in situations
where it may not be strictly valid over the entire width of
the EDL.
The boundary conditions are those of “no slip” for the fluid
velocity at the solid-fluid interface:
uusolid surface ¼ 0 (7)
and
fusolid surface ¼ z (8)
where the potential at the solid fluid boundary, z, is speci-
fied. Due to the rapid change in the potential at the inter-
face the definition of “at the interface” is somewhat
ambiguous. It is believed that the solid substrate usually
* Sometimes referred to in the literature as ‘laminar flow’. How-
ever, this terminology is inconsistent with usage in fluid
mechanics, since all flows of relevance in CE, including the
‘pure’ EOF are laminar (as opposed to turbulent) due to the
smallness of the Reynolds numbers involved.
 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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