Hsueh-Chia Chang,
University of Notre Dame, Notre Dame, IN 46556
January 29, 2001
Abstract
Displacement of air bubbles in a circular capillary by electrokinetic ow is shown to be
possible when the lm ow around the bubble is less than the bulk ow behind it. In our
experiments, lm ow reduction is achieved by a surfactant-endowed interfacial double layer
with an opposite charge from the wall double layer. Increase in the lm conductivity relative
to the bulk due to expansion of the double layers at low electrolyte concentrations decreases
the eld strength in the lm and further reduces lm ow. Within a large window in the total
ionic concentration C
t
, these mechanisms conspire to induce fast bubble motion. The speed of
short bubbles (about the same length as the capillary diameter) can exceed the electroosmotic
velocity of liquid without bubble and can be achieved with a low voltage drop. Both mechanisms
disappear at high C
t
with thin double layers and very low values of zeta-potentials. Since the
capillary and interfacial zeta-potentials at low concentrations scale as log C
1
t
and log C
1/3
t
,
respectively, lm ow resumes and bubble velocity vanishes in that limit despite a higher relative
lm conductivity. The bubble velocity within the above concentration window is captured with
a matched asymptotic Bretherton analysis which yields the proper scaling with respect to a
large number of experimental parameters.
1
Introduction
There is considerable interest in using electrokinetic ow for drug delivery through tissues and driv-
ing liquids through micron-level channels in micro-laboratories and micro-reactors. Electrokinetic
ow occurs when the dielectric channel wall induces a charge separation near its surface such that
the counter ion concentration decreases away from the wall while the coion concentration increases
[1]. Both approach the same value far into the electroneutral bulk electrolyte and there is hence a
net charge near the wall. This net charge is conned to a thin double layer of thickness , the
Debye length specied by a balance between diusion and potential gradient, and also introduces
a normal potential variation within the layer that can be obtained by a simple integration of the
Poisson equation. The potential dierence across the double layer (zeta-potential) is a function of
the wall material and the total ionic concentration concentration C
t
. In the presence of a tangential
external electric eld E, this charge separation near the wall introduces a net tangential body force,
which is proportional to E.
Since the charge is proportional to
d
2 dn
2
from the Poisson equation and since the tangential force
E is balanced by the viscous dissipation � 2
u
n
2
, where n
is the normal derivative, the tangential
velocity u within the double layer scales as E. As a result, the velocity rapidly approaches a
constant electrokinetic velocity beyond the thin double layer. Also, since u scales linearly with
1 respect to and assuming zero slip at the reference point for the zeta potential, one obtains the
classical at electrokinetic velocity prole
u
c
= 0 c
E
�
(1)
where the proportionality constant
is the dielectric permittivity of the medium and subscript c
refers to a cylindrical capillary (our channels of choice) with a diameter d. Hence, as long as the
channel size d is much larger than the double layer thickness, this electrokinetic phenomenon acts
as a surface force to the bulk uid that imposes the surface slip velocity (1) at the wall. As a result,
the liquid ow rate u
c
A is proportional to the channel cross-sectional area A
d
2
, contrary to
d
4
scaling of the pressure-driven ow, and this is a great advantage for small channels.
In the above cited applications of electrokinetic ow, parallel transport of long bubbles and
organic liquid drops with the electrokinetically driven electrolyte is often desired. The drops can
be drugs or blood capsules and the air bubbles can be used to separate samples along channels
of microlaboratories. Unfortunately, the at electrokinetic velocity prole, which allows eective
uid pumping through small channels, now can become an obstacle. The requirement that the
electric current is constant through the capillary and through the lm around the bubble (or drop)
results in
1/A intensication of the electric eld E in the lm, where A is the cross-section area
of the liquid lm around the bubble or drop. If the bubble (drop) interface is mobile (without
viscous traction) or the capillary and interfacial zeta potentials are identical, the at velocity
prole (1) of the electroosmotic ow extends from the capillary double layer across the entire lm,
as in the capillary behind the bubble. The respective total ow rates, the product of velocity
and liquid area, are then identical within the capillary and around the bubble. As a result, the
bubble remains stationary while the electrokinetically driven liquid ows around it. Hence, one
should somehow reduce the lm ow in order to accumulate liquid behind the bubble to build
up a pressure gradient, which then displaces the bubble to accommodate the accumulated liquid;
and/or introduce interfacial traction such that the electrokinetically driven liquid will drag the
bubble along.
The addition of surfactants endows traction on the interface [2]. Ionic surfactants will, however,
also introduce a double layer to the interface [3] with a bubble zeta potential
b
of the interface
dierent from the capillary zeta potential
c
, in general. Depending on the relative values and
signs of the zeta potentials, these two asymmetric double layers across the lm will produce a
normal electrokinetic velocity gradient across the lm and, hence, can reduce the ow rate below
that of a at velocity prole if the corresponding bubble electrokinetic velocity u
b
is lower than its
counterpart u
c
on the capillary (see Figure 1 b). The largest bubble speed would occur when
b
and c
are large but of dierent signs the capillary and bubble double layers are oppositely charged.
In the present work, we study experimentally and theoretically, following the classical Bretherton
problem of pressure-driven bubble transport [4], the possibility of displacing air bubbles by an
electrokinetically driven electrolyte solution in a cylindrical capillary (see Figure 1 a). We have
carried out experiments with air bubbles in KCl/H
2
SO
4
solutions in a capillary (3 cm length,
radius R = 0.25 mm, the measured
c
is positive, indicating a negatively charged double layer)
over ranges of KCl concentrations C
0
(10
6
10
2
mol/liter), air bubble lengths l
b
(1.620 R) and
applied voltages (10120 V). We observe bubble motion only after 10
5
mol/liter of SDS (anionic
surfactant which induces a positively charged interfacial double layer with
b
< 0 ) is added to
the solution and only within specic windows of applied voltage (2080 V) and KCl concentration
2 (10
5
10
3
mol/liter), with an optimal concentration of 10
4
mol/liter where the bubble velocity is
at its maximum. Bubbles can move with wide-ranging speeds over 4 orders of magnitude, including
an astonishing maximum of 3 mm/sec for the shortest bubbles with length l
b
2R. This high end
is comparable to the electroosmotic velocity of KCl solution without the air bubble. In contrast,
the introduction of a single bubble increases the required pumping pressure by orders of magnitude
in pressuredriven ow [11]. This suggests bubble transport in microchannels is only feasible with
properly designed electrokinetic ow.
Using a modied version of Bretherton analysis which includes transport within the double
layers, we obtain satisfactory prediction of the bubble speed as a function of l
b
, zeta potentials,
voltage applied and the total ionic concentration C
t
which is the sum of the surfactant and elec-
trolyte concentrations. We show that, while the presence of interfacial traction is necessary for
bubble transport, the window in electrolyte concentration where bubble motion is possible and the
bubble speed are mostly determined by the eective drag from asymmetric double layers. The
increase in the lm conductivity relative to the bulk caused by double layers expansion at low
electrolyte concentrations (C
0
< 10
3
mol/liter) and the resulting decrease in lm electric eld are
shown to be responsible for the observed fast bubble motion. Correspondingly, the vanishing double
layer thickness and the decrease in the absolute values of the zeta-potentials with increasing con-
centration dene the upper concentration bound for bubble motion. At very low concentrations,
the positive electrokinetic ow at the capillary exceeds the negative ow at the interface, again
permitting positive lm ow and slow bubble speed. At these low concentrations, however, the
lm around the bubble cannot be sustained because of unscreened electrostatic attraction between
oppositely charged bubble interface and capillary wall. This results in breaking the electric current
through the lm and produces the lower concentration bound for bubble motion.
2
Experiments
Two open acrylic electrode chambers are connected to both ends of a horizontal glass capillary
tube with diameter d = 0.5 mm and length L = 3 cm. The chambers house two platinum foil
electrodes. The chambers and connecting capillary are lled with a working electrolyte solution,
KCl/SDS with a small addition of sulfuric acid. The H
2
SO
4
concentration C
a
remains constant for
all experiments at C
a
= 3.62
� 10
5
mol/liter. The KCl concentration C
0
, however, ranges from
10
6
to 10
2
moles/liter.
Before each new series of experiments, the capillary is carefully cleaned with distilled water and
ethyl alcohol and then thoroughly rinsed rst with distilled wa