J. Fluid Mech. (2003), vol. 491, pp. 285–300.
c© 2003 Cambridge University Press
DOI: 10.1017/S0022112003005330 Printed in the United Kingdom
285
The effect of wall interactions in
capillary-zone electrophoresis
By SANDIP GHOSAL
Department of Mechanical Engineering, Northwestern University,
2145 Sheridan Road, Evanston, IL 60208, USA
(Received 11 September 2002 and in revised form 23 April 2003)
Capillary-zone electrophoresis (CZE) is an efficient separation method in analytical
chemistry. It exploits the difference in electrophoretic migration speeds between
charged molecular species in aqueous solution when an external electric field is
applied to achieve separation. In most cases the electrophoretic migration of species
is also accompanied by a bulk electro-osmotic flow in the capillary due to the presence
of a zeta-potential at the capillary wall. Adsorption of charged species at the wall
could modify this zeta-potential in a non-uniform manner. This induces axial pressure
gradients, so that the flow is no longer uniform over the capillary cross-section. The
resulting shear-induced dispersion of the sample is a serious cause of band broadening
in CZE particularly for species such as proteins and peptides which adsorb strongly
on capillary walls. The problem of the spatio-temporal evolution of the sample
concentration is studied in the presence of such wall interactions. An asymptotic
theory is developed that is valid provided axial variations have characteristic length
scales that are much larger than the capillary radius and temporal variations have
acharacteristic time scale much larger than the characteristic diffusion time over
a capillary radius. These conditions are normally satisfied in CZE, except when
the sample is close to the inlet, on account of the capillary length being very
much larger than its radius. It is shown that the cross-sectionally averaged sample
concentration obeys a one-dimensional partial differential equation. Further, the full
three-dimensional concentration field may be calculated once the cross-sectionally
averaged concentration field is known. The reduced system is integrated numerically
and is shown to lead to predictions consistent with known observations on CZE in
the presence of wall interactions.
1. Introduction
Capillary zone electrophoresis (CZE) is one of the methods employed in analytical
chemistry for the separation of mixtures of chemical species by exploiting their dif-
ferent electrophoretic mobilities in aqueous solution (see Weinberger 2000; Jorgenson
1987 for a basic introduction). Since its discovery in the early 1970s, CZE has gained
in popularity over more classical gel electrophoresis methods due to its superior
resolution, short analysis times and small sample size requirement. In recent years,
it has been the object of renewed interest due to the possibility of miniaturizing the
device and integrating it in the ‘Lab on a chip’. Such a device could in principle
accomplish biochemical experiments at speeds that are orders of magnitude faster
than are possible with existing technology while requiring extremely small quantities
of samples and reagents (see e.g. Jakeway, de Mello & Russell 2000).

286 S. Ghosal
In its simplest form, a benchtop CZE device consists of a micro-capillary
(characteristic diameter 10− 100 µmand10− 100 cm long) connecting two reservoirs.
The micro-capillary and both the reservoirs are filled with an ionic aqueous solution
(the buffer) of known pH. (We only discuss ‘free solution’ CZE as opposed to
separation modes where the capillary is filled with a sieving gel.) An electric potential
difference (∼ 30 kV) is applied between the inlet and outlet reservoirs by means of
electrodes. The sample (analyte) is introduced as a plug near the inlet, and allowed
to migrate towards the outlet due to the electro-osmotic flow induced by the electric
field. Any particular species of molecule migrates with a velocity u
e
+ v
e
where u
e
is
the electro-osmotic velocity of the fluid in the capillary and v
e
is the electrophoretic
migration velocity specific to the chemical species. The electro-osmotic velocity may
be calculated from the well known (Probstein 1994) formula:
u
e
=−


d
ζ
∗
E
4piµ
(1.1)
(in CGS units) where

d
is the dielectric constant of the liquid, ζ
∗
is the wall zeta-
potential, E is the applied electric field and µ is the dynamic viscosity of the liquid.
This formula is valid provided the zeta-potential is constant over the capillary wall
and the Debye layer thickness is very much smaller than the capillary radius. The
latter condition is usually satisfied in the applications we are concerned with. Since
v
e
is different for different species of molecules, the analyte separates into zones
of homogeneous composition each with a characteristic electrophoretic mobility. A
ultra-violet absorbance detector near the outlet end detects the arrival of each zone by
monitoring the attenuation of ultra-violet light, which alters the electrical signal from
aphotodetector. The spectrum consist of a series of peaks in the electrical output.
Each peak corresponds to the arrival at the detector of a zone which determines the
electrophoretic mobility of the respective species.
Under ideal circumstances, the electro-osmotic flow profile is uniform (except in a
very thin, ∼ 1–10 nm, boundary layer – the Debye layer – in which the flow speed
decreases rapidly in order to satisfy the ‘no-slip’ boundary condition at the wall)
over the capillary cross-section. Thus, there is negligible shear-induced dispersion
so that ‘diffusion limited’ separation, where resolution is limited only by molecular
diffusion, is potentially realizable. In practice, there are multiple known mechanisms
of band broadening that prevent the diffusion-limited theoretical resolution from
being realized (see the recent review by Gaˇs&Kendler 2000). In this paper we
address the problem of dispersion caused by variability in the wall zeta-potential
due to adsorption of analytes from the fluid stream. Perturbations due to such
inhomogeneity not only alter the flow profile, but also the bulk flow rate, as shown by
Anderson & Idol (1985) and Ghosal (2002c). Thus, in addition to band broadening,
the elution times are altered making it difficult to calculate mobilities with accuracy.
Adsorption also causes loss of some of the sample.
Analytical results on dispersion in the presence of wall adsorption have been
presented by Gaˇs et al. (1995) and
ˇ
Stˇedr ´y, Gaˇs&Kendler (1995). Though their
analyses take account of the fact that analyte is lost to the wall, the consequent
modification of the zeta-potential and therefore the hydrodynamic flow field have
been neglected. Indeed, the hydrodynamics is restricted to the trivial case of uniform
flow at constant velocity independent of the adsorption process. However, recent
theoretical as well as experimental work indicates that the modification of the the
zeta-potential by adsorption, and the consequent perturbation of the hydrodynamic
field, is an important, if not the principal cause of dispersion (Ghosal 2002a,b,c;