Download Study of the Effect of AC Electric Potential Applied on Co

Study of the Effect of AC Electric Potential Applied on Co-planar
Microelectrode Array on Electroosmotic Flow in a Slit Microchannel
by
Naheed Ferdous
Submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
Department of Mechanical Engineering
BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY
Dhaka-1000, Bangladesh
November, 2011
This thesis titled “Study of the Effect of AC Electric Potential Applied on Co-planar
Microelectrode Array on Electroosmotic Flow in a Slit Microchannel”, submitted by
Naheed Ferdous, Student No. 0409102031 Session April 2009 has been accepted as
satisfactory in partial fulfillment of the requirement for the degree of MASTER OF
SCIENCE IN MECHANICAL ENGINEERING on November 20, 2011.
BOARD OF EXAMINERS
Chairman
_______________________
Dr. Noor Al Quddus
Assistant Professor
Department of Mechanical Engineering
Bangladesh University of Engineering and Technology
Dhaka-1000, Bangladesh.
Member
_______________________
Dr. Muhammad Mahbubul Alam
Professor and Head
Department of Mechanical Engineering
Bangladesh University of Engineering and Technology
(Ex-Officio)
Dhaka-1000, Bangladesh.
Member
_______________________
Dr. Mohammad Ali
Professor
Department of Mechanical Engineering
Bangladesh University of Engineering and Technology
Dhaka-1000, Bangladesh.
Member
_______________________
(External)
Dr. Dewan Hasan Ahmed
Assistant Professor
Department of Mechanical and Production Engineering
Ahsanullah University of Science and Technology
Tejgaon, Dhaka, Bangladesh.
ii
CANDIDATE’S DECLARATION
It is hereby declared that this thesis or any part of it has not been submitted elsewhere for the
award of any degree or diploma.
Signature of the Candidate
Naheed Ferdous
iii
CERTIFICATE OF RESEARCH
This is to certify that the work presented in this thesis is carried out by the author under the
supervision of Dr. Noor Al Quddus, Assistant Professor of the Department of Mechanical
Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh.
_______________________
_______________________
Dr. Noor Al Quddus
Naheed Ferdous
iv
Dedicated to My Parents
v
ACKNOWLEDGMENTS
I want to express my gratefulness to my supervisor Dr. Noor Al Quddus who has guided me
through the course of my graduate study and research. Without his guidance, support and
suggestion this thesis was nearly unthinkable to complete. He has drawn my attention to
various aspects of fluid mechanics and motivated me for the fundamental research work. I
have benefitted enormously from our many discussions and his huge engagement with his
students.
I would like to thank my parents and my elder brother for giving me unquestionable support
to continue my graduate study. I also want to express my gratitude to my husband for always
giving me support and inspirations in the course of my graduate study.
I give thanks to the lecturer Mahbubul Islam for his help in proceeding formally with my
thesis.
vi
ABSTRACT
Under the influence of an AC electric field, electrolytes on a planar microelectrodes exhibit
steady fluid flow, termed as AC electroosmosis. A numerical model using finite element
method has been developed to solve the electrokinetic flow parameters of the AC
electroosmosis in a slit microchannel. A thin-layer, low-frequency, nonlinear analysis of the
system is performed including Faradaic currents from electrochemical reactions at the
electrodes. The non equilibrium model consists of Navier-Stokes, continuity, Nernst–Planck
and Poisson equations. In the first step, transient simulations are carried out to obtain a
homogeneous steady state stable periodic regime. The obtained stable period solutions are
then analyzed to compute the time averaged net velocity and other characteristics of the AC
electroosmotic flow. Net flow velocity and flow rate are observed for different geometric
parameters, electric field parameters, electrode position, Debye length and effective
thickness. Dependency of different parameters on the frequency of AC signal is analyzed for
optimal design of AC electroosmotic micropump. Such mechanism can be used to transport,
mix, separate, and manipulate various molecular or colloidal entities e.g. DNA, protein,
polymers etc. in microfluidic chips.
vii
TABLE OF CONTENTS
Item
Page
Title Page
i
Board of Examiners
ii
Candidate’s Declaration
iii
Certificate of Research
iv
Dedication
v
Acknowledgments
vi
Abstract
vii
Table of Contents
viii
Nomenclature
x
List of Table
xii
List of Figures
xiii
Introduction
1
1.1
Background and Motivation
1
1.2
Objective of the Thesis
3
1.3
Outline of the Thesis
4
Chapter 2
Literature Review
5
Chapter 3
Theoretical Description of the Model
12
3.1
Electric Double Layer
12
3.2
Electroosmosis
13
3.3
AC Electroosmosis
14
3.4
Model Geometry
16
3.5
Governing Equations (Equilibrium Model)
17
3.5.1
Electric field and ionic transport
17
3.5.2
Fluid flow
19
Necessity of Non-Equilibrium Model
19
Chapter 1
3.6
viii
3.7
Contribution of Faradaic Current
20
3.8
Non-Dimensionalization of the Equations
23
3.9
Boundary Conditions
24
Numerical Solution Methodology
26
4.1
Finite Element Method
26
4.2
Computer Implementation of the model
26
4.3
Mesh Sensitivity Analysis
27
4.4
Flowchart of the Overall Solution Methodology
28
Results and Discussion
29
5.1
Validation of the Model
29
5.2
Inclusion of Faradaic Current
32
5.3
Asymmetric vs Symmetric Electrodes
36
5.4
Velocity and Electric Field Characteristics
37
5.5
Effect of AC Frequency on Time Averaged Net Velocity and
Flow Rate
46
5.6
Dependence of Time Averaged Net Velocity on Debye Length 48
and Compact Layer Relative Thickness
5.7
Effect of Electrode and Channel Geometry on Time Averaged 51
Net Velocity
Chapter 4
Chapter 5
Conclusion and Recommendations
53
6.1
Concluding Remarks
53
6.2
Recommendation for Future Works
54
Chapter 6
55
References
ix
NOMENCLATURE
Notation
Definition
A
Amplitude [V]
c
Concentration [mol m-3]
D
Diffusivity [m2s-1]
f
Frequency [s-1]
F
The Faraday constant [96,485 Cmol-1]
H
Micro channel height [m]
L
Length of periodic segment [m]
G
Gap size [m]
E
Electrode size [m]
n
Normal unit vector
p
Pressure [Pa]
R
Molar gas constant [ 8.314 J K-1 mol -1]
t
Time [s]
T
Temperature [ 298.15 K]
u
Net velocity [ms-1 ]
û
Time averaged net velocity [ms-1]
x, y
Spatial coordinate [m]

Electrolyte permittivity [Fm-1]

Electric potential [V]

Dynamic viscosity [PaS]
D
The Debye length [m],  D  RT  2 c0 F 2
q
Space charge density [Cm-3 ]
e
Fundamental charge [ 1.6021765×10 −19 C]
s
Compact layer width [m]
x
Superscripts and Subscripts
e
electrode
~
dimensionless
±
Either + or s
OHP
+
0
eq
xi
cation
anion
neutral
Characteristic value
LIST OF TABLES
Table No.
Table I:
Page No.
Dimensionless Parameters …….................................................................. 22
xii
LIST OF FIGURES
Fig no
Fig 3.1
Fig 3.2
Structure of the electric double layer..................................................
Page
12
Geometry of a circular cylindrical microchannel used for modeling
electroosmotic flow
The principle of the AC electro osmotic flow above co-planar
electrodes..........................................................................................
Electrostatic volume force induces the eddy formation (solid line)
and the net flow (dashed line).............................................................
13
Fig 3.5
Structure of the electric double layer (EDL) on an array of
electrodes subjected to an AC potential...........................................
16
Fig 3.6
Scheme of one segment of the AC electroosmotic micropump. The
dashed-dotted lines indicate the periodic boundary conditions.........
17
Fig 4.1
Mesh sensitivity analysis for time averaged net velocity...................
27
Fig 5.1
Model validation with analytical result.............................................
30
Fig 5.2
Schematic Diagram of the AC electroosmotic mircropump. The
dashed dotted line indicates the periodic boundary conditions........
30
Fig 5.3
Comparison of time averaged net velocity between Cervenka model
and present model..............................................................................
31
Fig 5.4
Effect of Faradaic current on net velocity. ......................................
33
Fig 5.5
Net Velocity without Faradaic current.............................................
34
Fig 5.6
Net Velocity with Faradaic current....................................................
35
Fig 5.7
Net velocity comparison for asymmetric and symmetric electrode
arrangements......................................................................................
36
Fig 5.8
Velocity stream lines. .......................................................................
38
Fig 5.9
Velocity contour.................................................................................
38
Fig 5.10
The local orientation and the magnitude of the velocity vector by
arrows...............................................................................................
39
Fig 5.11
Electric field and stream line..........................................................
39
Fig 5.12
Plot of tangential field above electrodes..........................................
40
Fig 3.3
Fig 3.4
xiii
14
15
Fig 5.13
Horizontal component of the velocity vector along a horizontal line
0.001µm above the electrodes..........................................................
41
Fig 5.14
Horizontal component of the velocity vector along a vertical line at
the channel outlet (far from the electrodes) .....................................
42
Fig 5.15
Time averaged net velocity along a vertical line at the channel
outlet (far from the electrodes) ........................................................
43
Fig 5.16
Horizontal component of the velocity vector along a vertical line at
the midpoint between the electrodes...................................................
44
Fig 5.17
Time averaged net velocity along a vertical line at the midpoint
between the electrodes .......................................................................
45
Fig 5.18
Dependence of the time-averaged net velocity on the AC
frequency............................................................................................
46
Fig 5.19
Flow rate at channel outlet and midpoint between electrodes........
47
Fig 5.20
Dependence of the time-averaged net velocity on Debye length ......
48
Fig 5.21
Dependence on the compact layer thickness. ....................................
49
Fig 5.22
Flow reversal.....................................................................................
50
Fig 5.23
Effect of Microchannel height on time averaged net velocity. ........
51
Fig 5.24
Effect of relative size of electrodes...................................................
52
xiv
1. INTRODUCTION
1.1 Background and Motivation
A large family of microfluidic devices employs electrokinetic transport of liquids. The
integration of mechanical components such as high-pressure pump or valve is not required in
electrokinetically driven microfluidic chips. Instead of a pressure gradient, a gradient of
electric potential is imposed over the microfluidic system. It can be provided by either DC or
AC electric field. DC electroosmotic transport in microchannels has been studied intensively
both theoretically [1] and experimentally [2-3]. Origin of the DC electroosmosis (DCEO)
arises from coulombic interaction between an external DC electric field and a mobile electric
charge localized at an electrolyte-dielectric interface. The use of the electroosmosis forced by
a low amplitude AC electric field is a novel area investigated only for several years. The AC
electroosmosis (ACEO) is mostly based on the coulombic interaction between a imposed AC
electric field and a mobile electric charge temporally arising at electrolyte metal electrode
interfaces.
Increasing complexity of microfluidic devices enforces the need for integrated local fluid
control in microchannels. Electric fields applied on microelectrodes can induce
electrokinectic pumping of electrolytes. The electrically induced pumping systems are mostly
based on DC electroosmotic phenomena that usually require high voltages. Integration of
electrodes in microfluidic channels for standard DCEO pumping is seriously limited by
undesirable effects related to high voltages, such as joule heating, significant changes of the
electrolyte composition, and bubble formation. A promising alternative to the classical
electroosmotic pumps relies on microeletrode fluidic systems that exploit AC signals with
very low amplitude.
The developments in microfluidics calls for a means to transport and mix fluid at microscale
with reliability and control. Such microfluidic devices are instrumental to the realization or
improvement of miniature bio- medical-chemical diagnostic kits [4-5], high performance
liquid chromatographs, fuel cells, ion exchange devices [6], chip and micro-circuit cooling,
biochips for drug screening etc. ACEO flows are receiving growing interest in microfluidic
applications because the integration of microelectrodes into microchannels makes possible
the local actuation of electrolytes by means of electric fields [7-8].
Recent developments in micro-fabrication and the technological promise of microfluidic
"labs on a chip" have brought a renewed interest to the study of low Reynolds number flows
[9]. The microscale mixing of miscible fluids must occur without the benefit of turbulence,
by molecular diffusion alone. For extremely small devices, molecular diffusion is relatively
rapid; however, in typical microfluidic devices, the mixing can be prohibitively long. Another
limitation is that the pressure-driven flow rate through small channels decreases with the third
or fourth power of channel size. For pumping, mixing, manipulating and separating on the
micron length scale, many focuses on the use of surface phenomena, owing to the large
1
surface to volume ratios of typical microfluidic devices [10]. ACEO provides one of the most
popular non-mechanical techniques in microfluidics. Electrical forces are used to transport,
mix, separate, and manipulate various molecular or colloidal entities in microfluidic chips.
This principle is utilized to separate proteins of different sizes or charges, DNA, bacteria,
polymers and cells [11].
Under the influence of an AC electric field, electrolytes on a planar microelectrodes exhibit
steady fluid flow, termed as AC electroosmosis. The flow has its origin in the interaction of
the tangential component of the non-uniform field with the induced charge in the electric
double layer on the electrode surfaces. The mechanism of AC electroosmotic driven fluid
flow has recently been shown to be capable of producing unidirectional pumping of liquid on
a microscale, on asymmetric pairs of coplanar microelectrodes as demonstrated
experimentally [12], theoretically [13], and numerically [14]. The measurements have been
analyzed as a function of applied electric field (potential, frequency), electrolyte conductivity,
and position on the electrode. At low potentials, the predicted velocities are in reasonable
agreement with the experiments. One can apply spatially inhomogeneous electric field, vary
the geometry or electrical properties of the polarizable electrode surface etc. The directions
seem promising to investigate numerically and pursue experimentally in real microfluidic
devices.
Coulombic polarization of the microelectrodes is usually considered in the theoretical models
of AC electroosmotic transport. However, Faradaic currents can occur in such systems as a
result of electrochemical reactions. Linear theoretical studies are used to measure fluid flow
of electrolytic solutions induced by a traveling-wave potential applied to an array of coplanar inter digitated microelectrodes [15]. The effect of Faradaic currents on the
electroosmotic slip velocity generated at the electrode/electrolyte interface is taken over the
assumption of perfectly polarizable electrodes.
The combined action of two experimentally relevant effects, i.e Faradaic currents from
electrochemical reactions at the electrodes and differences in ion mobilities of the electrolyte
is taken into account in the analyses of AC electroosmosis [16]. Theoretical results are also
applied for analyzing the influence of Faradaic reactions on AC electroosmotic flows [17].
Electrokinetic flow induced by the pair of electrodes is numerically obtained using the model.
A new prediction of the model is that, for certain values of the parameters, fluid flow can
occur in opposite direction to that obtained in the absence of Faradaic reactions. A linear thindouble-layer analysis is employed where the electrical problem is decoupled from the
mechanical one.
In previous works, the AC electroosmotic motion has been analyzed theoretically under the
assumption that only forces in the diffuse (Debye) layer are relevant. A thin-layer, lowfrequency, linear analysis is performed and the model is applied to the case of an electrolyte
actuated by a traveling-wave signal. A steady liquid motion in opposite direction to the
applied signal is predicted for some ranges of the parameters. This could serve as a partial
explanation for the observed flow reversal in some experiments. Approximations such as
negligible advective currents, linearization of Poisson-Nernst-Planck equations, thin layer
2
approximation etc are made for this solution. However, the need for a complete analysis
which must be fully nonlinear is felt for explanation of the underlying mechanism of AC
electroosmosis considering Faradaic currents.
Motivated by this, a complete nonlinear model for low-frequency ACEO flow including
Faradaic currents from electrochemical reactions at the electrodes has been developed in the
present study. A two dimensional geometric model is utilized to obtain the solution for
velocity distribution in a slit microchannel. Faradaic current is incorporated in the nonlinear
(non equilibrium) analysis by using electrochemical reaction at electrode surface. The
electric potential is solved assigning known potentials to the electrodes with sinusoidal
function along with solving the double layer charge density using charge conservation
equations. The motion of the bulk fluid is numerically calculated, using no slip condition at
the electrode surface as the boundary condition. A commercial finite element package is used
to obtain the solution. Horizontal component of velocity is calculated for one cycle when the
system goes to a steady state from transient phase. Total velocity for different fraction of time
in one steady state cycle is determined and time averaged net velocity is calculated for one
complete cycle. Volumetric flow rate per unit width of microchannel is computed by
integrating velocity. Responsive behavior of fluid velocity and flow rate in symmetric
microelectrode array with the change of applied AC frequency, concentration of electrolyte
i.e Debye length and geometric parameters of microchannel are investigated.
1.2 Objective of the Thesis
The specific objectives of the research work are as follows:
(a) To develop a 2-D finite element model employing nonlinear non equilibrium
approach.
(b) To study the electroosmotic effect when asymmetric AC electric potential is applied
on symmetric electrodes
(c) To investigate the effect of geometry, applied electric field, electrolyte concentration
on the time averaged net velocity of electrolyte
(d) To study the effect of Faradaic current on net flow for coplanar asymmetric and
symmetric microelectrodes
The interesting consequences of shape and field asymmetries, which generally lead to
electroosmotic pumping or electrophoretic motion in AC fields will give some basic issues
for the microfluidic devices.
3
1.3 Outline of the Thesis
A detail literature review is provided in Chapter 2. The theoretical description of the model is
described in Chapter 3. The slit microchannel geometric configuration, governing equations
for fluid flow, ion transport, electric field, boundary conditions, non-dimensionalization of
the model are described in this chapter. The origin of the electrokinetic transport and the
principle of AC electroosmotic micropumps are also shortly described. The multiphysical
model of electro-microfluidic systems based on the Poisson-Nernst-Planck-Navier-Stokes
approach and its possible simplifications is briefly described in this section. In Chapter 4,
numerical solution methodology of the model is described. A brief description of the
procedure involved in finite element analysis is provided. A flowchart depicts the overall
solution methodology. Chapter 5 presents the simulation results. First the model is validated
by comparing the calculated electroosmotic velocity of fluid flowing through a channel,
against analytical and numerical results found in the literature. In the later sections, the values
of the time averaged net velocity and flow rate are presented for different AC frequency
values, electrolyte concentration, geometric parameters of microchannel. In Chapter 6, the
conclusion and recommendations for future works are presented.
4
2. LITERATUE REVIEW
Under the influence of an AC electric field, electrolytes on planar microelectrodes exhibit
fluid flow. The nonuniform electric field generated by the electrodes interacts with the
suspending fluid through a number of mechanisms giving rise to body forces and fluid flow.
The mechanism responsible for this motion has been termed AC electroosmosis (ACEO). In
other models, it is a continuous flow driven by the interaction of the oscillating electric field
and the charge at the diffuse double layer on the electrodes.
Experiments were performed to find optimal conditions for obtaining particle and bacterial
assembly lines on electrodes by ACEO and preliminary results showed good resolution at a
concentration of 104 bacteria/ml, indicating that combining ACEO with impedance
measurement can improve the sensitivity of particle electrical detection [4]. Due to different
bacterial impedance signatures in tap water and PBS buffers, experiments were performed to
find the optimal voltage and frequency ranges such that a trapping converging flow exists on
the electrodes and the assembled cells exhibit sensitive impedance spectrum signatures [5].
Two types of ACEO devices, in the configurations of planar interdigitated electrodes and
parallel plate electrodes, and a biased ACEO technique were studied, which provided added
flexibility in particle manipulation and line assembly [6]. A real-time particle concentration
technique using a novel electrokinetic method was used to bridge the gap between the
detectable level and infectious level of bacterial solutions [7].
The dynamics of a particle with both polarizability and net charges in a non-uniform AC
electric trapping field was investigated through the study of AC electrokinetics and AC
electrophoretic (ACEP) phenomena [8]. Velocity was measured for a new type of on-chip
micro-pump that exploits the AC electro-kinetic forces acting in the volume of a fluid in the
presence of a temperature gradient [9]. Covalent surface modification techniques, in
particular surface oxidation procedures, was employed as a mean to modify polymer
microfluidic channels for the purpose of modulating microflow [10]. Due to the fact that skin
electrical resistance can be controlled by an alternating current (AC) electric field, a human
epidermal membrane was modeled to assess the effects of AC voltage and frequency and
direct current (DC) offset on the flux of neutral and ionic model permeants [11].
The velocity of fluid flow on microelectrodes at frequencies below the charge relaxation
frequency of the electrolyte was shown by detailed experimental measurements [12]. The
velocity of latex tracer particles was measured as a function of applied signal frequency and
potential, electrolyte conductivity, and position on the electrode surface. The data were
discussed in terms of a linear model of AC electroosmosis: the interaction of the nonuniform
AC field and the induced electrical double layer.
The fluid flow was predominant at frequencies of the order of the relaxation frequency of the
electrode electrolyte system. A theoretical approach was developed to this problem using a
5
linear double layer analysis [13]. The theoretical results were compared with the experiments,
and a good correlation was found.
ACEO was studied experimentally and theoretically using linear analysis: Potential drop
across the double layer at the surface of the electrodes was calculated numerically using a
linear double layer model [14]. Experimental observations of the fluid flow profile were pre
obtained by superimposing images of particle movement in a plane normal to the electrode
surface. These experimental streamlines demonstrated that the fluid flow was driven at the
surface of the electrodes. Experimental measurements of the impedance of the electrical
double layer on the electrodes were also reported and was used in numerical analysis. The
AC electroosmotic flow at the surface of the electrodes was calculated using the HelmholtzSmoluchowski formula. The bulk fluid flow driven by this surface velocity was numerically
calculated as a function of frequency and good agreement was found between the numerical
and experimental streamlines.
Net fluid flow of electrolytic solutions induced by a traveling-wave potential applied to an
array of co-planar microelectrodes was reported in literature [15]. At low applied voltages the
flow was driven in the direction of the traveling-wave potential, as expected by linear and
weakly nonlinear theoretical studies. The flow was driven at the surfaces of the electrodes by
electrical forces acting in the diffuse electrical double layer. The pumping mechanism has
been analyzed theoretically under the assumption of perfectly polarizable electrodes. Here,
the study was extended to include the effect of Faradaic currents on the electroosmotic slip
velocity generated at the electrode/electrolyte interface. The electrokinetic equations was
integrated under the thin-double-layer and low-potential approximations. Finally, the
pumping of electrolyte induced by a traveling-wave signal applied to a microelectrode array
was analyzed using this linear model.
Previous analyses of AC electroosmosis was extended [16] to account for the combined
action of two experimentally relevant effects: (i) Faradaic currents from electrochemical
reactions at the electrodes, and (ii) differences in ion mobilities of the electrolyte. In previous
works, the AC electroosmotic motion has been analyzed theoretically under the assumption
that only forces in the diffuse (Debye) layer are relevant. Here, it was shown that different ion
mobilities of a 1-1 aqueous solution made the charged zone to expand from the Debye layer
to include the diffusion layer. The Faradaic currents were included later and, as an attempt to
explore both factors simultaneously, a thin-layer, low-frequency, linear analysis of the
system was performed. Finally, the model was also applied to the case of an electrolyte
actuated by a traveling-wave signal. A steady liquid motion in opposite direction to the
applied signal was predicted for some ranges of the parameters.
The simplest microelectrode structure was chosen to analyze the electrokinetic flow induced
by a pair of coplanar symmetric microelectrodes[17]. The theoretical results were applied for
analyzing the influence of Faradaic reactions on the flows. The liquid was assumed to be a
1:1 electrolyte with ions of different diffusivities. For simplicity, in the model only the
cations were responsible for the Faradaic reactions while the anions did not react at the
6
electrodes. The prediction of the model was that, for certain values of the parameters, fluid
flow can occur in opposite direction to that obtained in the absence of Faradaic reactions.
The general phenomenon of ‘induced-charge electroosmosis’ (ICEO) was described as the
nonlinear electroosmotic slip that occurs when an applied field acts on the ionic charge it
induces around a polarizable surface. Motivated by a simple physical picture, ICEO flows
were calculated around conducting cylinders in steady (DC), oscillatory (AC), and suddenly
applied electric fields [18]. The system represented perhaps the clearest example of nonlinear
electrokinetic phenomena. This physically motivated approach was complemented and
verified using a matched asymptotic expansion to the electrokinetic equations in the thindouble-layer and low-potential limits. ICEO slip velocities vary as u s ∝E20L, where E0 is the
field strength and L is a geometric length scale, and are set up on a time scale τc =λDL/D,
where λD is the screening length and D is the ionic diffusion constant. ICEO microfluidic
pumps and mixers were proposed and analyzed under low applied potentials. Similar flows
around metallic colloids with fixed total charge have been described in the Russian literature.
ICEO flows around conductors with fixed potential, on the other hand, have no colloidal
analogue and offer further possibilities for microfluidic applications.
A physical description of ICEO and the nonlinear electrokinetic slip at a polarizable surface
were given in the context of some new techniques for microfluidic pumping and mixing [19].
ICEO generalizes ‘‘AC electroosmosis’’ at microelectrode arrays to various dielectric and
conducting structures in weak DC or AC electric fields. The basic effect produces micro
vortices to enhance mixing in microfluidic devices, while various broken symmetries—
controlled potential, irregular shape, nonuniform surface properties, and field gradients—can
be exploited to produce streaming flows. Although the qualitative picture of ICEO was
emphasized, the mathematical theory (for thin double layers and weak fields) was also briefly
described and applied to a metal cylinder with a dielectric coating in a suddenly applied DC
field.
Experimental and numerical investigations of ICEO on a planar electrode surface directly in
contact with a high conductivity electrolytic solution was reported [20]. Symmetric rolls of
ICEO flow was produced on the electrode by placing it in an AC electric field. The slip
velocity was measured for a range of AC voltages and frequencies using micro particle image
velocimetry (µPIV). The slip velocity was also calculated by finite element simulations based
on a linear and a nonlinear model of electrical double layer, respectively. The µPIV
measurements were found to be much lower (two and half orders of magnitude) than the
velocities predicted by the linear model. The linear model is valid only under Debye Huckel
approximation which does not hold true for practical situations. The nonlinear model, on the
other hand, predicts velocities which are lower than the linear model and closer to the
experimental values. The nonlinearity reduces discrepancy between experimental and
numerical results by approximately an order of magnitude. The nonlinear model accounts for
nonlinear capacitance of the double layer and lateral conduction of charge in the double layer.
7
Numerical and experimental studies were also used to investigate the increase in efficiency of
microfluidic AC electroosmotic pumps by introducing nonplanar geometries with raised steps
on the electrodes [21]. The effect of the step height on AC electroosmotic pump performance
was analyzed . AC electroosmotic pumps with three-dimensional electroplated steps were
fabricated on glass substrates and pumping velocities of low ionic strength electrolyte
solutions were measured systematically using a custom microfluidic device.
The response of a model micro electrochemical cell to a large AC voltage of frequency was
investigated [22]. To bring out the basic physics, the simplest possible model of a symmetric
binary electrolyte was considered which was confined between parallel-plate blocking
electrodes, ignoring any transverse instability or fluid flow. The resulting one-dimensional
problem was analyzed by matched asymptotic expansions in the limit of thin double layers
and two features are considered in the strongly nonlinear regime—significant salt depletion in
the electrolyte near the electrodes and, at very large voltage, the breakdown of the quasi
equilibrium structure of the double layers. The former leads to the prediction of “AC
capacitive desalination” since there was a time-averaged transfer of salt from the bulk to the
double layers, via oscillating diffusion layers. The latter was associated with transient
diffusion limitation, which drives the formation and collapse of space-charge layers, even in
the absence of any net Faradaic current through the cell. It was also predicted that steric
effects of finite ion sizes (going beyond dilute-solution theory) act to suppress the strongly
nonlinear regime in the limit of concentrated electrolytes, ionic liquids, and molten salts.
Beyond the model problem, the reduced equations for thin double layers, based on uniformly
valid matched asymptotic expansions, provided a useful mathematical framework to describe
additional nonlinear responses to large AC voltages, such as Faradaic reactions, electroosmotic instabilities, and induced-charge electrokinetic phenomena.
Theoretical models were presented for the time-dependent voltage of an electrochemical cell
in response to a current step, including effects of diffuse charge (or “space charge”) near the
electrodes on Faradaic reaction kinetics [23]. The full model was based on the classical
Poisson-Nernst-Planck equations with generalized Frumkin- Butler-Volmer boundary
conditions to describe electron-transfer reactions across the Stern layer at the electrode
surface. In practical situations, diffuse charge is confined to thin diffuse layers (DLs) which
poses numerical difficulties for the full model but allows simplification by asymptotic
analysis. For a thin quasi equilibrium DL, effective boundary conditions were derived on the
quasi-neutral bulk electrolyte at the diffusion time scale, valid up to the transition time, where
the bulk concentration vanishes due to diffusion limitation. The thin-DL problem was
integrated analytically to obtain a set of algebraic equations, whose (numerical) solution
compares favorably to the full model. In the Gouy-Chapman and Helmholtz limits, where the
Stern layer is thin or thick compared to the DL, respectively, simple analytical formulas were
derived for the cell voltage versus time. The full model also described the fast initial
capacitive charging of the DLs and super limiting currents beyond the transition time, where
the DL expands to a transient non-equilibrium structure. The well-known Sand equation was
extended for the transition time to include all values of the super limiting current beyond the
diffusion-limiting current.
8
Two mathematical models of the electrokinetic flow were presented based on: (i) momentum
balance of the electrolyte, (b) continuity equation, (c) molar balances of the components of
the uni-univalent electrolyte, and (d) Poisson equation of electrostatics [24]. The
electroosmotic flow was induced by the interaction of a surface electric charge with a
perpendicularly imposed electric field. Both the models were characterized by the formation
of extremely thin surface layers with gradients of electric potential, pressure, concentrations
and velocity. In order to solve such problems, an anisotropic mesh of rectangular finite
elements was developed. The first model described a classical electroosmotic problem – the
electrolyte dosing in a microfluidic channel with an axially imposed DC electric field .
Stationary distributions of the model variables were computed for various sets of model
parameters: applied voltage, density of the surface electric charge, microchannel diameter,
electrolyte concentration etc. The interaction of a low-concentrated biological analyte with a
receptor bound on the microchannel walls was studied by means of dynamical simulations.
The second model dealt with AC electroosmosis that was based on the application of AC
voltage on two electrodes of different size. The formed asymmetric electric field pattern
caused the zigzag motion of electrolyte in microcapillary with one dominant direction. The
dynamical simulations were aimed at the effects of voltage frequency, electrode size and
electrolyte concentration on velocity of the electrolyte flow.
A full dynamic description was used, instead of the linearized model to show the results of
the mathematical modeling of AC electroosmotic micropump [25]. Skewed hybrid
discretization meshes were employed in order to accurately capture the main features of the
studied system. Also, zig-zag electrode arrangements were introduced for traveling-wave
electroosmotic micropumps. The detailed analysis of the system behavior was presented by
means of the examination of the model properties.
AC electroosmotic micropumps were suggested to be powerful tools for electrolyte dosing in
various micro and nanofluidic systems. Two modeling approaches were compared for
studying the AC electroosmosis in the following micro and nanochannel systems: (i) a
traveling- wave AC pump with a spatially continuous wave of electric potential applied on a
planar boundary, (ii) a traveling- wave AC pump with a wave of electric potential applied on
a set of discrete planar electrodes, and (iii) an AC pump with a set of non-planar electrodes
[26]. The equilibrium approach was based on the use of capacitor–resistor boundary
conditions for electric potential and the slip boundary conditions for velocity at electrode
surfaces. The non-equilibrium approach used the mathematical model based on the Poisson
equation and the non-slip boundary conditions. Discrepancies have been observed between
the predictions given by the both models and then their possible reasons were identified. The
comparison of the equilibrium and non-equilibrium results further showed three important
actualities: (a) how the equilibrium model overestimates or underestimates the net velocity,
(b) how the velocity maxima in the frequency characteristics can be shifted, if the equilibrium
model assumptions are not satisfied, (c) the parametric region where the equilibrium model is
applicable. The limitations of the equilibrium and non-equilibrium models were discussed
and selected predictions with available experimental data were compared.
9
The coplanar asymmetric arrangements of the forcing electrodes were analyzed by means of
the following two mathematical models: 1) the classical slip model, which is based on a
capacitor–resistor representation of the spatial domain, and 2) the nonslip model, which is
based on the Poisson–Navier–Stokes–Nernst–Planck approach to the entire domain, including
the electric double layers [27] . Both the models predicted similar results in many lowamplitude regimes. However, the nonslip model gave a much better insight on the highamplitude (nonlinear) behavior of the micropumps. Most important findings obtained by the
nonslip model were summarized as follows: 1) There are optimal values of the electrode and
gap size ratios that are generally different from those obtained by the slip model; 2) the
micropump performance is relatively insensitive with respect to the electrode size ratio; 3)
there is an optimal vertical confinement that enables to attain high net velocities; 4) flow
reversals on frequency, amplitude, and certain geometry characteristics are observed; 5) the
energy efficiency of these pumps is very low; and 6) the Joule heating effect is negligible.
The nonslip model characteristics were also discussed to explain the observed differences
between predictions of the models. Convergence analysis dealing with the precision of
numerical results obtained by the nonslip model was presented.
AC electroosmotic pumping of fluids in a microchannel was studied with two thin coplanar
electrodes, deposited on one dielectric wall and placed in an aqueous solution [28]. Two
different numerical formulations were compared. In the first one, the transport equations
(mass, molar, and momentum balances) for the ion concentration and the Poisson equation
for the electric field were solved simultaneously in the time domain without any
simplification of the computational domain except that the dielectric walls are neglected. In
the second one, a time average model was assumed, and the Debye layer was represented by
an approximate 1-D model. In both cases, the finite-element method implemented in the
COMSOL commercial software was used as a numerical tool. The results for the electric
field, space charge, and velocity distributions were compared with a qualitative agreement
between both models.
Micropumps are able to manipulate small volumes of liquid samples and involve no moving
parts. The pumps find their use in a variety of bio applications or clinical diagnostics. The
micropump was realized by an array of microelectrodes coated along the microchannel, on
which an AC electric field was applied. The AC voltage was typically in the range of few
volts. Theoretical models describing the AC electroosmotic transport usually consider the
Coulombic polarization of the microelectrodes. However, Faradaic currents can occur in
such systems as a result of electrochemical reactions. A mathematical model of an AC
electroosmotic system was presented with electrochemical reactions [29]. The model was
based on the balances of mass and electric charge and the kinetic equations for the electrode
reaction. The theory of the electrolyte dynamics at polarized surfaces for larger applied
voltages taking into account the steric effects (formation of condensed ionic layers) has been
also published by other authors.
10
The analytical validation of Poisson-Boltzmann (PB) equation was computed with Comsol
Multiphysics, in the case of a polarized surface in contact with the electrolyte [30]. Comsol
Multiphysics algorithms easily handled the highly nonlinear aspect of the PB equation. The
limitations of the PB model, that considers ions as point like charges, were outlined. To
account for the steric effects of the ion crowding at the charged surface, the Modified
Poisson-Boltzmann model was analyzed for symmetric electrolytes. The MPB equation was
then coupled to the complex AC electrokinetic and the Navier-Stokes equations to simulate
the AC electroosmosis flow observed inside an interdigitated electrodes microsystem.
11
3. THEORETICAL DESCRIPTION OF THE MODEL
3.1 Electric Double Layer
The electroosmotic transport can be induced by an external electric field in fluidic systems
where the electric polarization (charging) of solid surfaces occurs. Some organic
(polystyrene, plexiglass) or inorganic polymers (glass) gain a surface electric charge if
immersed in an electrolyte. The fixed charge can arise, e.g., from dissociation of surface
chemical groups of either the polymer substrate itself or of adsorbed additives. Counter ions
present in the electrolyte are attracted to the charged solid phase via the coulombic force and
the electric double layer(EDL) is formed. In the immediate proximity, the counter ions are
tightly bound to the charged surface and thus become immobile. This thin part of EDL is
called the Stern layer.
OHP(Outer Helmholtz plane)
Stern layer
Diffuse Layer
Distribution of Electric
field potential
Bulk solution
Counter ion distribution
Co ion distribution
Electric double layer
 
  0
  Vs
Figure 3.1: Structure of the electric double layer
Further away from the solid phase in the diffusive part of EDL, the attraction columbic force
is relatively weak and the counter ions remain mobile. The diffusive layer is usually much
wider than the Stern layer. The imaginary surface between the diffusive and the Stern layers
is called the outer Helmholtz plane – OHP, as shown in Fig. 3.1. The electric potential
12
localized on this surface and related to the reference potential value in the bulk solution ( 0 )
is an important characteristic of EDL, so-called zeta-potential (  ).
Let us note that the EDL structure, depicted in Fig. 3.1 represents a typical steady state that is
established when: (i) the dielectric or the electrode surface is in a contact with the electrolyte
for a sufficient time, (ii) any temperature changes and bulk concentration changes do not
occur in space and time, and (iii) the electrolyte does not move. These conditions are often
not satisfied in real applications and thus the EDL structure can be more complex. The EDL
width is approximately equal to the Debye length λ D. For a symmetric uni-univalent
electrolyte, the Debye length can be estimated as
3.1
D  RT /(2c0 F 2 )
where the symbols R, T, c0, and F denote the molar gas constant, temperature, the electrolyte
concentration in the bulk, and Faraday’s constant, respectively. The permittivity of the
environment ε is equal to the product of the vacuum permittivity ε0 and the dielectric constant
(relative permittivity) of the environment r.
The fixed electric charge on the solid phase is usually negative (depends on the substrate and
the electrolyte pH value). Then, the electric potential decreases from the electrolyte bulk to
the solid surface. In the same direction, the anion concentration decreases and the cation
concentration increases with respect to the bulk concentration. Hence, the electrolyte in the
EDL region does not satisfy the electro neutrality condition, i.e., a nonzero concentration of a
mobile electric charge exists in EDL.
3.2 Electroosmosis
Channel Wall
EDL
r
a
x
EDL
Channel Wall
Electric Field, Ex
Fig 3.2: Geometry of a microchannel used for modeling electroosmotic flow
13
Electrokinetic techniques provide some of the most popular small-scale non mechanical
strategies for manipulating particles and fluids. A surface with charge density q in an aqueous
solutions attracts a screening cloud of oppositely charged counter-ions to form the
electrochemical ‘double layer’, which is effectively a surface capacitor. The excess diffuse
ionic charge exponentially screens the electric field set up by the surface charge (Fig 3.2). An
externally applied electric field exerts a body force on the electrically charged fluid in this
screening cloud, driving the ions and the fluid into motion. The resulting electroosmotic fluid
flow appears to ‘slip’ just outside the screening layer of width λD.
This basic electrokinetic phenomenon gives rise to electroosmosis which find wide
application in analytical chemistry, microfluidics, colloidal self-assembly, and other
emerging technologies. Electroosmotic flow occurs when an electric field is applied along a
channel with charged walls, wherein the electroosmotic slip at the channel walls gives rise to
plug flow in the channel. Because the electroosmotic flow velocity is independent of channel
size, (in contrast to pressure-driven flow, which depends strongly upon channel size),
electroosmotic pumping presents a natural and popular technique for fluid manipulation in
small channels.
3.3 AC Electroosmosis
Alternating electric fields can generate a net steady motion of aqueous saline solutions over
microelectrode structures. The term AC electroosmosis (ACEO) refers to the fluid motion
generated on top of electrodes by the interaction between an AC electric field and the
electrical charge that this field induces at the electrode/electrolyte interface— i.e., the charge
induced in the electrical double layer.
E
E
Ey
Ey
Ex
Ex
Δ
Fig 3.3: The principle of the AC electroosmotic flow above co-planar electrodes.
ACEO flows are receiving growing interest in microfluidic applications because the
integration of microelectrodes into microchannels makes possible the local actuation of
electrolytes by means of electric fields.
14
In the most frequent co-planar arrangement of the electrodes, the AC electric field has the
tangential and the normal components, as shown in Fig. 3.3. The normal component induces
electrode polarization via coulombic force (capacitive charging). Then, the tangential
component of the electric field forces the accumulated electric charge to move along the
electrodes. The highest tangential coulombic force was predicted and observed at the
electrode edges.
f
Δ
Fig 3.4: Electrostatic volume force induces the eddy formation (solid line) and the net
flow (dashed line)
As the electric charge is formed by ions of finite size, the moving ionic particles pull the
surrounding liquid via viscous forces, as shown in Fig. 3.4. The combination of coulombic,
pressure and viscous forces in the liquid result in the formation of eddies above the
electrodes. The velocity of the net flow strongly depends on several parameters: the
frequency and the amplitude of the imposed electric field, the concentration of the used
electrolyte and the geometric properties of the microfluidic system.
The induced charge in the double layer lags behind the applied signal due to the finite
charging time of the double layer. The ions are, therefore, subjected to a tangential electrical
force in the direction of the traveling wave that, by viscous friction, is transmitted to the fluid.
The characteristic charging time of the double layer is given by the product of the typical
resistance of the bulk L /σ S and the typical capacitance of the double layer  S / λD, tc = (
/σ) (L/ λD ). Here,  and σ are the permittivity and conductivity of the liquid, respectively, L
and S denote the typical length and area of the system, and λD is the Debye length. This
charging time typically lies in the range of milliseconds.
The frequency dependence is given by characteristic rates of two transport processes: (i) the
rate of the electrode polarization (the formation of the EDL) and (ii) the rate/frequency of an
15
AC electric field. There are three qualitatively different regimes that can arise above the
electrodes.
If the period of the applied signal, T, is very large compared to tc , counterions have time to
accumulate in the double layer. The electrical force is, then, mainly normal to the surface and
negligible flow occurs. If the period of the applied signal T is much shorter than tc , the
accumulation of induced charge in the double layer becomes negligible and, again, no lateral
motion is generated. Maximum lateral force and fluid flow occur for T~ tc, which typically
corresponds to an applied frequency in the range of kilohertz.
A significant feature of ACEO flow is its dependence on the square of the electric field
amplitude. This has important consequences for AC fields: if the direction of the electric field
in the above picture (Fig 3.3) is reversed, so are the signs of the induced surface charge and
screening cloud. The resultant ACEO flow, however, remains unchanged: the net flow
generically occurs away from the poles, and towards the equator.
3.4 Model Geometry
A 1:1 aqueous solution X+Y− e.g., H+Cl− placed on top of an array of electrodes subjected
to an AC signal Fig. 3.5. The behavior of the system close to the plane of the electrodes is the
area of interest. Above this plane, three distinct layers are distinguished: compact, diffuse,
and diffusion. AC electroosmosis is usually explained in terms of the phenomena that
happen in the combination of compact and diffuse layers known as electric double layer,
abbreviated as EDL. In this paper, the effect of the charges situated in the diffusion layer is
also examined. The outer Helmholtz plane (OHP) is localized between the compact and
diffuse layers.
.
Bulk
Diffusion
Diffuse
OHP
1

s
Compact
Vs
Fig 3.5: Structure of the electric double layer (EDL) on an array of electrodes subjected
to an AC potential.
16
The coplanar arrangement of AC electroosmotic pumps is considered (Fig.3.6). The studied
microfluidic pump can be represented by a single segment of a long microfluidic channel. We
assume that a microchannel consists of an infinitely large series of segments with periodic
boundary conditions.
L
H
y
x
G2/2
E1
G1
E2
G2/2
Fig 3.6: Scheme of one segment of the AC electroosmotic micropump. The dasheddotted lines indicate the periodic boundary conditions.
The microchannel width is much larger than the other characteristic dimensions of the
segments, so the AC electroosmotic pump can be described as a 2-D object with length L and
height H. The electrodes have lengths E1 and E2, and they are separated by gap G1 . Both
electrodes are G2/2 gap away from each periodic boundary.
A symmetric mono-monovalent water electrolyte (e.g., Hydrogen chloride) is considered as
the fluid in microchannels. No changes of density, viscosity, and temperature are expected.
Faradic reactions are assumed on the electrodes. The formation of the condensed EDL is not
considered in this paper.
3.5 Governing Equations (Equilibrium Model)
At the electrode and electrolyte interfaces, double layers are formed with a typical thickness
given by the Debye length. The Debye length is negligibly small compared to the typical size
the system. Since the electrodes are long, both compared to gap width and the size of the
periodic segment observed in the fluid, the problem can be considered to be two dimensional.
When an AC voltage is applied to the electrodes, an electrical current is established in the
solution. The following analysis assumes that electrolysis does not occur at the electrode
surfaces, i.e., the electrodes are considered to be perfectly polarizable, and the double layer
behaves in a linear manner.
3.5.1 The electrical field and ionic transport
When the double layer is in quasi equilibrium, the bulk electrolyte behaves in a resistive
manner and the double layer in a capacitive manner.
17
As a result, the potential in the bulk electrolyte satisfies Laplace’s equation
3.5.1(a)
2  0
with the boundary condition just outside the double layer on the electrode surface given by

 q EDL

y
t
3.5.1(b)
where q EDL is the charge per unit area in the double layer. In this equation we assume that
lateral currents along the double layer are negligible. The relationship between the charge
and the potential drop across the double layer depends on the model used. If the voltage drop
kT


 0.025V  there is a linear
across the diffuse double layer is sufficiently small  d 
e


relationship between the charge and the voltage, i.e., q EDL  C EDL (  V ) , and the equation
can be written with complex amplitudes as


= iq DL  iC EDL (  V j )
y
3.5.1(c)
where C EDL is the capacitance per unit of area of the total double layer (diffuse plus compact
layers), and Vj is the potential applied to the electrode j. This capacitance C EDL is The
compact layer capacitance is approximately independent of the electrolyte concentration. The
properties of the diffuse part of the double layer depend on electrolyte concentration as given
by the Gouy-Chapman theory [31]. Although experimentally the potential drop across the
diffuse double layer can exceed 0.025 V, nevertheless the linear analysis can give useful
information into the mechanism governing fluid flow.
Since the conductivity of the solid surface is negligible, the boundary condition at the solid
interface in the fluid is


0
3.5.1(d)
y
In deriving these boundary conditions, the presence of lateral currents along the double layer
has been neglected. In this theoretical analysis, the applied and natural surface potentials are
assumed to be small. In this case, the intrinsic mean ion density of the diffuse double layer is
small and the lateral currents are negligible. For high ion concentrations in the double layer,
surface currents might be comparable to the currents in the normal direction. Owing to
electrode polarization, the electric field in the bulk electrolyte is frequency dependent. When
the frequency is low, most of the applied voltage is dropped across the double layer but when
the frequency is high most of the applied voltage is dropped across the bulk electrolyte.
18
3.5.2 Fluid Flow
Once the potential is solved, the electroosmotic velocity just outside the double layer at the
surface of the electrodes can be calculated from this solution.
a. The solution in the diffuse layer
The fluid motion is caused by electrical stresses that are nonzero only in the diffuse double
layer, since the charge density in the bulk is zero. These stresses result in a rapidly varying
velocity profile in the diffuse double layer, changing from zero at the wall to a finite value
just outside the double layer. This velocity can be imagined as slip velocity at electrode
surface to calculate the bulk motion. In the thin double layer approximation, for diffuse layers
in quasi equilibrium and on a perfectly polarizable metal surface, the slip electroosmotic
velocity is given by the Helmholtz-Smoluchowski formula

 
u    d
 d Ex

X 
3.5.2(a)
In this expression, Ex is the tangential field just outside the diffuse layer,  is the fluid
viscosity, and d = (   ) represents the difference between the potential  on the outer
side of the diffuse layer and the potential  on the inner side of this layer, at the nonslip
plane. This equation gives the tangential velocity on the electrodes; the normal velocity is
zero.
b. The solution in the bulk
To obtain the velocity in the bulk, the Navier-Stokes equations must be solved. Since, for
microelectrodes the Reynolds number is usually very small,  muL /   102 , we neglect the
inertial terms in the Navier-Stokes equations. In the absence of externally applied body
forces, these equations reduce to
 2 u  p  0
and
u  0
3.5.2(b)
The boundary conditions are (i) the tangential AC electroosmotic velocity on the electrodes,
Eqn.3.5.2(a); (ii) zero tangential velocity on the glass; and (iii) zero normal velocity on every
boundary.
3.6 Necessity of Non-Equilibrium Model
In Equilibrium model of AC electroosmosis, the computational domain is usually divided
into the capacitor domains (vicinities of the polarized surfaces, where electric double layers
(EDL) are formed) and the resistor domain (the electrolyte bulk). It is assumed that electric
potential in an arbitrary point of the system can be represented by a product of a complex
time-independent function and a time-dependent function (e.g., harmonic). Equilibrium
19
model is valid only for systems in the thermodynamic equilibrium. The AC electroosmotic
systems are rarely close to the equilibrium because of the intensive convection transport at
the electrode surfaces.
Because of the mentioned limitation, more complex Poisson–Boltzmann models and non
equilibrium models have been developed. The non-equilibrium models describe the electric
potential distribution with the use of the Poisson equation and zero velocity is applied on
solid surfaces. The models based on the Poisson equation should satisfactorily describe the
behavior of the AC electroosmotic systems with non-equilibrated EDLs.
The electric potential field satisfies the Poisson equation
 2  q,
3.6(a)
q  F (c   c  )
In order to evaluate the field of electric charge density, two molar balances for the anion (-)
and the cation (+) have to be used
c 
   J 
t
3.6(b)
The total flux density of ions is given by the sum of the convective and electro migrationdiffusion contributions (the Nernst–Planck equation)
J   uc   D(c   c 
F
 )
RT
3.6(c)
The velocity and pressure fields in the electrolyte are described by the Navier–Stokes
equation and the continuity equation for incompressible Newtonian fluids
(
u
 u  u)   2 u  p  q ,
t
.u  0
3.6(d)
The non-slip boundary conditions are used on the electrolyte-solid interfaces. The Dirichlet
boundary conditions are used for electric potential on the discrete electrodes. No Faradaic
current through the solid-electrolyte interfaces is considered. The insulating boundary
conditions are used for electric potential on the non-electrode solid boundaries. Periodical
boundary conditions applied in all other boundaries.
3.7 Contribution of Faradaic Current
The simplest form of AC electroosmosis theory is a linear theory that assumes that only
forces in the diffuse layer are present and that the electrodes are perfectly polarizable, i.e.,
there are no Faradaic currents. At certain range of frequencies the direction of net fluid flow
changes. The possible reason for this flow reversal is said to be Faradaic current. Faradaic
20
currents depolarize the electrode-electrolyte interface leading to lower electroosmotic
velocities. Oscillations of total ionic concentration close to the electrodes is originated by
Faradaic currents at low frequencies.
In our model we only include two charged species and consider the pair of ions to be
asymmetric in the sense that their mobilities are different. For simplicity, we assume that the
cations react reversibly at the electrodes to produce neutral molecules according to a simple
one-step one-electron redox process of the form X++e−↔X and that the electrodes are
blocking for the anions. The sign of the reacting species, however, is not essential to the
model. The introduction of the chemical reaction requires the inclusion of one neutral
species, X, in the mathematical analysis.
In our system, there is an AC applied voltage on the electrodes and this signal induces
charges and motion in the liquid. The behavior of our system can be modeled in terms of four
functions: the electric potential,  and the concentrations of positive, negative, and neutral
species, c+, c−, and c0, respectively. Our model describes a two-dimensional system, which
approximates the behavior of a solution on top of an array of electrodes in the form of long
strips. We denote as x the coordinate tangential to the plane of the electrodes and as y the
coordinate normal to this plane. The different quantities verify the Poisson-Nernst-Planck
(PNP) equations in the domain.
The electric potential obeys Poisson’s equation,
2  q,
q  F (c   c  )
3.7.1
The concentration of ions satisfy Nernst-Planck equations,
c 
   J  ,
t
J   uc   D  (c   c 
F
 )
RT
3.7.2(a)
c 
   J  ,
t
J   uc   D  (c   c 
F
 )
RT
3.7.2(b)
c 0
   J 0 ,
t
J 0  uc 0  D 0 c 0
The diffusion coefficients
D   9.312e 9 m2 s-1 (Diffusion Coefficient of H+)
D   2.032e 9 m2 s-1 (Diffusion Coefficient of Cl-)
D 0  2.2e 9 m2 s-1 (Diffusion Coefficient of Water)[31]
To these equations we must add the Navier-Stokes equation for the liquid motion,
21
3.7.2(c)
3.7.3(a)
u  0
(
u
 u  u)   2 u  p  q
t
TABLE I
DIMENSIONLESS PARAMETERS
Symbol
Parameter
~
xxL
~
yy L
Dimensionless x-coordinate
Dimensionless y-coordinate
Dimensionless delta operator
Dimensionless Debye length
~
  L
~
D  D L
~
h H L
~
re  E1 E 2
~
rg  G 2 G1
~
le  E1  E 2  L
~
t  t t 0 , t 0  D L D
~
f  ft0
~
c   c c
eq
~
c  c c eq
~
c 0  c 0 ceq
~ 
RT
  , 0 
0
F
~
A  A 0
~ u
u
D
L
p
~
p 0
2c RT
2
  RT 
Ra 


D  F 

Sc 
D
( /  D )
~
s 
( s / s )


Height to length ratio
Electrode length ratio
Gap size ratio
Equivalent length
Dimensionless time
Dimensionless frequency
Dimensionless concentration of cation
Dimensionless concentration of anion
Dimensionless concentration of water
Dimensionless potential
Dimensionless amplitude
Dimensionless velocity
Dimensionless pressure
Rayleigh number (Ra= 0.372)
Schmidt number (Sc=348)
Compact layer relative thickness
22
3.7.3(b)
3.8 Non-Dimensionalization of the Equations
The model equations are transformed into a dimensionless form. The meaning of all the used
symbols is summarized in Table I. The spatial coordinates and the segment dimensions are
scaled by the factor L. Combination of Eqs. 3.7.1-3.7.3(b) and Table I gives the
dimensionless form of the non-equilibrium mathematical model.
The electric potential field satisfies the Poisson equation
3.8.1
1
~ ~
 2   ~2 q~,
D

~
~
q  0.5(c  ~
c)
The concentrations of ions satisfy Nernst-Planck equations,
c~  ~ ~ ~~  ~ ~  ~  ~ ~
  D   uc  c  c   0
~t
3.8.2(a)
c~  ~ ~ ~~  ~ ~  ~  ~ ~
~   D   u c   c  c   0
t
3.8.2(b)
c~ 0 ~ ~ ~~ 0 ~ ~ 0
~   D   u c  c  0
t
3.8.2(c)






The velocity and pressure fields in the electrolyte are described by the Navier-Stokes
Equation and the continuity equation for an incompressible Newtonian fluid
1  ~
u ~ ~ ~ ~  ~ ~ 2 ~ Ra ~ ~ ~ ~ ~
 u    D  u  ~  p  q 
 ~  D u
Sc   t
D



~ ~
u
0
3.8.3(a)
3.8.3(b)
where Sc is the Schmidt number and Ra is the Rayleigh number defined by
Ra 
  RT 


D  F 
2
Sc 

D
Hence in this section, the physical meaning of the dimensionless criteria is shortly discussed.
Because the Schmidt number is high (particularly Sc=500>>1; the concentration and electric
charge boundary layers will be much thinner than the momentum boundary layers. It means
that the velocity field must be affected on a much higher distance from the electrodes than the
concentration fields. Different behavior, not typical for microfluidic applications, can be
expected for electrolytes with extremely low viscosities and with high diffusivities of ions.
23
From the definition, the Rayleigh number expresses a ratio between the forces that
~
destabilizes and stabilizes a static fluid. In our modification, the dimensionless ratio Ra /  D2
is a relative ratio between the destabilizing electric body force and the stabilizing viscous
force. This ratio is very high in the presented parametrical studies as well as in all AC
microfluidic pumps. A low value of the ratio necessarily leads to zero net velocity.
3.9 Boundary Conditions
The boundary conditions at the OHP (y=0) for these functions are the continuity of the
displacement vector, assuming a linear compact layer of width λs and permittivity s
3.9(a)
s

 s  
s
y
where  s denotes the voltage drop across the compact layer (Fig 3.5)
s  Vs (t )   ( y  0),
Vs ,1  A sin 2ft,
Vs , 2  0
3.9(b)
We assume that the electrodes are blocking to the anions,
D c 
F 
c 
 D
0
RT y
y
3.9(c)
whereas the fluxes of positive and neutral species are related through Faradaic currents,
 Dc
F 
c  J F
 D

RT y
y
e
 D0
J
c 0
 F
y
e
3.9(d)
3.9(e)
To model this current we use the Butler-Volmer equation
JF
 F s 
 (1   ) F s 

 K 0 c 0 exp
  K  c exp 

e
RT
 RT 


with  as the transfer coefficient, where the reaction constants
K0 = 1e-10 m3mol-1s-1 (Reaction rate constant for water)
K+ = 1e-7 m3mol-1s-1 (Reaction rate constant for H+)
24
3.9(f)
The boundary condition for the liquid velocity at the OHP is the no-slip condition
u0
3.9(g)
The equations can be made dimensionless using adequate scales. Equation 3.9(a) becomes
~
~
~ ~ 
 s    s  D ~
y
~ ~
~
s  Vs (t )   ( y  0),
3.9(h)
~
~
~
Vs ,1  A sin 2f ~t ,
Vs , 2  0
3.9(i)
~ ( /  D )
where  s 
which is the compact layer thickness. It gives the ratio between the
( s / s )
capacitance of the diffuse layer to the compact layer. This can be interpreted as a ratio
between effective thicknesses:
~
 s  ( s /  s ) /  D
Equation 3.9(d)-3.9(g) becomes
~
 ~
c
~
c ~  ~ 0
y
y
3.9(j)
~
JFL
~
c
 
~
c ~  ~ 
y
y
eD  c eq
3.9(k)
J L
c~ 0
 ~   F0
y
eD c eq
3.9(l)
and the Butler-Volmer equation becomes
JFL
L
~
~
 [K0~
c 0 exp  s  K  ~
c  exp  (1   ) s ]
eDc eq D
 


3.9(m)
Equation 3.9(h) becomes
~
u0
3.9(n)
25
4. NUMERICAL SOLUTION METHODOLOGY
4.1 Finite Element Method
Finite element method (FEM) is employed for solving the governing equations of AC
electroosmosis. The method is well described in literature [32] and is widely used for solving
differential equations in many areas of engineering and science. In finite element method, a
geometrically complex domain of the problem is discussed with simple elements. The
elements are connected to each other at nodal points. The responses of the dependent
variables are assumed a priori. State variables are approximated by basis functions, which are
formulated from polynomials. Approximating piecewise function for state variables are
known as test functions. Replacing the shape function in the discretized weak equations by
the shape functions associated to each computational node gives a single algebraic equation
which satisfies the discretized form of the governing equations. In this way a system of
algebraic equations are formed. These equations describe the relationship between the
coefficients of the test functions. Solving these algebraic equations for these coefficients the
behavior of the dependent variable is obtained.
4.2 Computer Implementation of the Model
Poisson, Nernst Planck, Navier-Stokes and continuity equations are solved in a time
dependent domain. In present analysis a two dimensional geometry is discretized by
triangular mesh. Higher number of mesh elements is used on the electrode surface. This
facilitates capturing the higher electric field and/or ion concentration gradient near the
electrode surface .
A general feature of all types of the AC electrokinetic systems is that the systems do not
work in steady condition. After the initial transient, they can attain a stable periodic regime.
This aspect of the AC systems substantially increases the demands on numerical analysis of
model equations. From technical point of view, the most interesting velocity characteristic of
the flow is the net velocity which is obtained as the tangential velocity averaged over one
period of the stable periodic regime.
Intel Core i3 processor and 4GB RAM are employed to obtain the solutions presented here.
26
4.3 Mesh Sensitivity Analysis
The mesh sensitivity analysis is performed by observing the improvement in the calculated
values of time averaged net velocity with increasing number of mesh elements. A slit
~
~
~ H
microchannel having h  0.333, ~
re  1, ~
rg  9 , le  0.333 is considered. Here, h  , is
L
E
the dimensionless height of the channel, ~
re  1 , is the ratio of the electrode dimensions,
E2
G
~ E  E2
~
rg  2 , is the ratio of the gap sizes, le  1
, is the relative size of the electrode
G1
L
domains. Tests are performed from nearly 2000 elements to 6000 elements. The problem is
time dependent, memory requirement for computation is higher than the stationary problem.
Figure 4.1: Mesh sensitivity analysis for time averaged net velocity
Moreover, Poisson and Nernst-Planck equations are coupled with the Navier-Stokes and
continuity equations, which increases the number of degrees of freedom for each mesh node.
This means, at each node dependent variables c+, c-, c0 and V are solved with the dependent
variables of Navier-Stokes equation (u, v, p). For this reason, number of mesh elements is
taken adjustable to minimize the memory and time requirement.
27
Time averaged net velocity for different number of mesh elements is shown in Fig. 4.1. It is
well noticeable that, the value cannot be improved significantly by taking higher number of
elements above 6000.
4.4 Flowchart of the Overall Solution Methodology
Geometry modeling
(Finite channel length approximation)
Defining governing equations
Non-dimensionalization of the governing equations
Defining boundary conditions
Non-dimensionalization of boundary conditions
Discretizing the geometry into finite number of mesh elements
Applying boundary conditions
Solving Poisson, Nernst Planck, Navier-Stokes and continuity equations
simultaneously with time dependent solver
Calculating horizontal component of velocity for one cycle when the system goes to a
steady state from transient phase
Calculating Time averaged net velocity for one complete cycle
28
5. RESULTS AND DISCUSSION
This chapter shows the dependence of time averaged net velocity and volumetric flow rate on
different parameters. The developed model utilized in present analysis is validated by
comparing the calculated values of time averaged net velocity with those available in the
literature. Later, time averaged net velocity is presented for different values of frequency,
Debye length, compact layer thickness, microchannel height and electrodes position.
5.1 Validation of the Model
Electroosmotic flow in a straight tube is presented having a low surface potential [31]. The
flow is considered as fully developed. A microchannel of radius a with a charged surface
bearing a surface potential of  is considered as illustrated in Fig 3.2. The coordinate system
used is cylindrical with r representing the radial direction and x representing the axial
direction. A symmetric ( z : z) electrolyte is considered flowing in the channel. The
electroosmotic velocity in x-direction is given by
u x (r )  u x 
a 2 px
4
  r  2  
1     
  a   
 I 0 ( r / D ) 
1  I (a /  )  E x

0
D 
5.1
dp
,  is electrolyte viscosity,  is electrolyte permittivity,  D is Debye
dx
length and I 0 is the zeroth-order modified Bessel function of the first kind.
where, p x  
29
Fig 5.1: Model validation with analytical result
A finite element solution is obtained for  =1,  = 1e-3 Pa s,  =8.8541878e-12*78.5 C2 N-1
m-2 and  D =10-7m. From Fig 5.1, it is seen that the result obtained from present model is in
good agreement with the analytical results.
Values of time averaged net velocity are calculated utilizing the developed non-equilibrium
model.
L
H
y
x
G2 /2
E1
G1
E2
G2/2
Fig 5.2: Schematic diagram of the AC electroosmotic mircropump. The dashed dotted
line indicates the periodic boundary conditions.
30
These calculated values are compared with the published numerically computed results [27].
Fig 5.2 indicates the Cervenka model of an AC electroosmotic micropump. The coplanar
arrangement of AC electroosmotic pump is considered here. A periodic segment of a long
microfluidic channel is represented by this model. Length of this segment is L and height is
H. The length of the electrodes are E1 and E2, and the gap between them are G1 and gap from
~
each periodic boundary is G /2. For this model, the values are h  0.333, ~
r  1.667,
2
e
~
~ H
E
~
rg  10, le  0.2667 where h  , is the dimensionless height of the channel, ~
re  1 ,
L
E2
G
~ E  E2
is the ratio of the electrode dimensions, ~
,
rg  2 , is the ratio of the gap sizes, le  1
L
G1
is the relative size of the electrode domains. That means, E1 is 1.667 times larger than E2 i.e.
the electrodes are arranged in an asymmetric way.
Fig 5.3: Comparison of time averaged net velocity between Cervenka model and
present model
Time averaged net velocity for different dimensionless frequency is shown in Fig 5.3 for
Cervenka model and present model. Only magnitude of the time averaged net velocity is
considered here. Dimensionless frequency are taken in the range of 0.1 to 50. The velocity of
any point varies with the varying electric field caused by AC cycle. The net velocity is
defined as the average of the velocity over a cross section of the microchannel. For a certain
31
dimensionless frequency, the net velocity goes to a steady state from transient states. For
example, it takes 100 cycles for the system to go in a steady state when the dimensionless
~
frequency is, f = 0.4. The average value of the velocities at 100th cycle is taken as the time
averaged net velocity. In Fig. 5.3, the net velocities are taken at the middle point of the two
electrodes.
v when compared to the
The percentage of error for the value of calculated net velocity, ~
~
~
values of [27] for dimensionless frequency, f  1 are within 0.949 % . Above f  1 , the
maximum error is 5.3%. So, values of dimensionless frequency is taken in the range of 0.01
to 1 in the present study. Parametric dependencies computed by present model are plotted by
empty markers. The Cervenka results are represented by filled markers (Fig 5.3).
5.2 Inclusion of Faradaic Current
The above models consider that the electrodes are perfectly polarizable i.e., there is no
Faradaic current. For certain ranges of frequency, the direction of predicted flow may change
while considering Faradaic current. Assuming a linear compact layer width s and Debye
length D , a dimensionless parameter can be defined as Compact layer relative thickness,
( /  D )
~
, where  s is the permittivity of electrolyte at the compact layer and  is the
s 
( s /  s )
permittivity of electrolyte at other regions. It gives the ratio between the capacitance of the
diffuse layer to the compact layer. This can be interpreted as a ratio between effective
thicknesses i.e., ratio between compact layer and diffuse (Debye) layer thickness.
~
s  (s /  s ) / D
32
Fig 5.4: Effect of Faradaic current on net velocity.
~
~
~
~
~
re  1.667, ~
rg  10 , le  0.2667 , A = 0.75, s =4 ,  D =3.3e-3
h  0.333, ~
The magnitude of time averaged net velocity is shown for asymmetric arrangements of
electrodes taking Faradaic currents and no Faradaic currents in consideration. No Faradaic
currents means considering the electrodes as perfectly polarizable. Faradaic currents
depolarize the electrode-electrolyte interface i.e charge leaks from the double layer leading to
lower electroosmotic velocity. This phenomenon is seen in Fig 5.4. Net velocity decreases by
an order because of the effect of Faradaic currents. Net velocity is taken at logarithmic scale
to show the difference of order in two cases. The shape of the net velocity profile with
increasing dimensionless frequency is somewhat similar. Sharp changes are noticed for net
velocity when considering Faradaic current. The slope of the curve is relatively small without
Faradaic current.
33
Fig 5.5: Net velocity without Faradaic current.
~
~
~
~
~
~
h  0.333, re  1.667, ~
rg  10 , le  0.2667 , A = 0.75, s =4 ,  D =3.3e-3
No flow reversal is seen for this system when taking no Faradaic currents. Spatial direction x
is taken positive from left to right side. Negative sign of velocity indicated the net flow is in
the opposite direction of positive x. In the range of dimensionless frequency mentioned above
in Fig 5.5, velocities are negative. That means fluid flows in the same direction for
~
dimensionless frequency, f  0.01  1 while considering no Faradaic current.
34
Fig 5.6: Net velocity with Faradaic current.
~
~
~
~
~
re  1.667, ~
rg  10 , le  0.2667 , A = 0.75, s =4 ,  D =3.3e-3
h  0.333, ~
~
Flow reversal occurs after dimensionless frequency, f ≈ 0.2 in this system while taking
Faradaic current into consideration. Fluid velocity goes positive from negative direction.
That means the direction of fluid flow changes in the above range of dimensionless
~
frequency, f  0.01  1 while considering Faradaic current.
In both cases, the change in the magnitude of net velocity is low up to frequency 0.2. After
this frequency, net velocity significantly increases. For no Faradaic current, net velocity
dramatically increases and for Faradaic current inclusion, flow reversal occurs and net
velocity also increases in that opposite direction. Again it is seen in more detail that the
changes are sharp in the net velocity profile while considering Faradaic current. The changes
in net velocity is very low up to frequency 0.2 for no Faradaic current. It remains almost the
same.
Faradaic current predicts (i) flow reversal and (ii) lower electroosmotic velocity. So, the
calculation of time averaged net velocity and its dependences on selected parameters should
be done taking Faradaic current into consideration. The electrode arrangement of above
35
models are asymmetric. In the next section, time averaged net velocity is explored for two
different arrangements of electrodes i.e., symmetric and asymmetric. The following sections
show the characteristics of velocity and electric field using Faradaic current at electrode
surface.
5.3 Asymmetric vs Symmetric Electrodes
~
The spatial parameters for asymmetric electrodes are ~
re  1.667, ~
rg  10 and le  0.2667,
which means one electrode is 1.667 times larger than the other one and both of them are at
equal distance away from periodic boundaries.
~
Fig 5.7: Net velocity comparison for asymmetric h  0.333, ~
re  1.667, ~
rg  10 ,
~
~
~
le  0.2667 , A = 0.75, s =4 ,  D =3.3e-3 and symmetric electrode arrangement
~
~
~
~
r  9 , l  0.333 , A = 0.75,  =4,  =3.3e-3
h  0.333, ~
r  1, ~
e
g
e
s
36
D
~
Symmetric electrode arrangement ~
re  1, ~
rg  9 , le  0.333 shows that two electrodes are
equal in size and again they are equal distance away from periodic boundaries. Time
averaged net velocity is calculated at the middle plane between two electrodes for the two
~
~
cases. Dimensionless amplitude is taken as, A = 0.75, compact layer relative thickness is, s
=4, which means compact layer is four times larger than diffuse (Debye) layer and
~
dimensionless Debye length is taken as,  D =3.3e-3.
Figure 5.7 shows the comparison of net velocity for asymmetric and symmetric electrode
arrangements. Negative sign of velocity shows the direction of fluid flow as it is in the
opposite direction of positive x-direction considered in the model geometry. Fluid usually
flows in the opposite direction of produced electric field. In comparison of magnitude, it is
seen that for lower frequency symmetric electrode arrangement exerts lower net velocity than
asymmetric arrangements. For higher frequencies, symmetric electrodes exert higher net
velocities than asymmetric electrodes. For symmetric electrodes, flow reversal occurs at
dimensionless frequency ≈ 0.6 and for asymmetric electrodes, it happens after the value of
dimensionless frequency ≈ 0.2. That means flow reversal occurs at a higher frequency when
electrodes are symmetric. So, symmetric electrode arrangement is a better design considering
flow reversal as for a certain range of frequency, desired net flow can be obtained.
~
In the range of dimensionless frequency, f  0.01  1 , the value of time averaged net velocity
is higher for symmetric arrangement than for asymmetric arrangement. The further
calculation of the velocity and Electric field characteristics, behavior of time averaged net
velocity and flow rate within a range of dimensionless frequency are discussed below while
considering Faradaic current in symmetric arrangement of electrodes. Effect of Debye length,
microchannel height, compact layer relative thickness, the ratio of the electrode dimensions,
the ratio of the gap sizes and the relative size of the electrode domains are calculated using
Faradaic currents in the following sections.
5.4 Velocity and Electric Field Characteristics
Electroosmosis depends on surface charges at the electrode/electrolyte interface. Most
materials will acquire fixed surface charges when coming into contact with a fluid that
contains ions (either an electrolyte or a dielectric liquid with ionic impurities or generated
locally via reactions). The surface charges attract counterions from the solution and repel coions from the surface to maintain local charge neutrality. Consequently, an excess of charges
is built up near the electrode surface, thus forming an electrical double layer. The ions in the
electrical double layer are mobile and will migrate under the influence of an electric field
tangential to the electrode surface. Due to fluid viscosity, fluid surrounding the ions will
move along, inducing so-called electroosmosis.
37
Electrodes
Fig 5.8: Velocity stream lines for ACEO in a slit microchannel
~
~
~
~
~
~
re  1, ~
h  0.333, ~
rg  9 , le  0.333  D =3.3e-3 , A = 0.75, s =4, f =0.3
,
Figure 5.8 shows the stream lines within slit microchannel. Two vortices are created at the
electrode edges. One central flow is noticed in the middle of the electrodes.
Electrodes
Fig 5.9: Velocity contour for ACEO in a slit microchannel
~
~
~
~
~
~
re  1, ~
rg  9 , le  0.333  D =3.3e-3 , A = 0.75, s =4, f =0.3
h  0.333, ~
,
38
Electrodes
Fig 5.10: The local orientation and the magnitude of the velocity vector by arrows
for ACEO in a slit microchannel
~
~
~
~
~
~
~
~
h  0.333, re  1, rg  9 , le  0.333  D =3.3e-3 , A = 0.75, s =4, f =0.3
,
The interaction of the electric field and the space charge in the EDL generates the electric
force, which moves the fluid. The calculated flow velocity distribution is very non uniform
(Fig. 5.9 ). At points closer to the channel center and ends, the velocity rapidly decreases, but
at all points, the fluid uniformly flows in the horizontal direction (Fig 5.10).
Electrodes
Fig 5.11: Electric field and stream line for ACEO in a slit microchannel
~
~
~
~
~
~
h  0.333, ~
re  1, ~
rg  9 , le  0.333  D =3.3e-3 , A = 0.75, s =4, f =0.3
,
39
As the electric force is a product of the space charge density and the electric field, the
strongest force occurs close to the electrode edges ( Fig 5.11).
~
~
~~
When positive potential Vs  A sin(2f t ) and 0 potential are applied on left and right
electrodes respectively, electric field is created as shown in Fig 5.12.
Fig 5.12: Plot of tangential field above electrodes.
~
~
~
~
~
~
h  0.333, ~
re  1, ~
rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4, f =0.3
Tangential component of electric field created above electrodes are shown in Fig 5.12. It is
seen that x component of electric field is maximized at two edges of electrodes.
40
Fig 5.13: Horizontal component of the velocity vector along a horizontal line 0.001µm
~
~
~
~
~
above the electrodes. h  0.333, ~
r  1, ~
r  9 , l  0.333,  =3.3e-3 , A = 0.75,  =4,
e
e
g
D
s
~
f =0.3
A detailed information about the flow characteristics is provided in Fig. 5.13. The local
velocity vector along a horizontal line just above the electrodes (0.001 µm above electrodes)
varies rather dramatically; the maximum value is more than 0.19 µm/s at a point between
both electrodes. Flow direction changes at both edges of the electrodes. Velocity reaches 0.5
µm/s for outer edge of positive electrode but changes direction and goes to -0.13 µm/s near
other end. At the inner edge of the negative electrode, the flow reaches maximum velocity
and then reverses its direction and reaches a value of about 0.08 µm/s.
41
Fig 5.14: Horizontal component of the velocity vector along a vertical line at the channel
outlet (far from the electrodes)
~
~
~
~
~
~
~
~
h  0.333, re  1, rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4, f =0.3
The net velocity along a vertical direction at the channel outlet is shown in Fig 5.14. At
frequency 0.3, the system goes at steady state for 100 cycle. Net velocity is calculated for
every 1/10th time of 100th cycle. 0t means the starting of the cycle and 1t means the end of the
cycle. It is observed that the velocity changes direction in the middle of the cycle that is at the
time 0.5t. The value of net velocity goes to a high from 0t to 0.1t but then decreases with
time. At 0.5t, flow is reversed. After that, net velocity increases then decreases and finally at
the end of the cycle the flow is again at the previous direction that is opposite to created
tangential electric field. The magnitude is also very close for the starting and ending of the
cycle. The net velocity at channel outlet is parabolic in nature. It goes to a maximum at the
middle point of the microchannel and velocity is 0 at walls. At time 0.1t, the velocity reaches
maximum at the middle point of channel radius, the value is -0.1 µm/s. After flow reversal, at
time 0.7t, it goes to maximum at opposite direction, the value is 0.09 µm/s.
42
Fig 5.15: Time averaged net velocity along a vertical line at the channel outlet (far from
the electrodes)
~
~
~
~
~
~
~
~
h  0.333, re  1, rg  9 , le  1  D =3.3e-3 , A = 0.75, s =4, f =0.3
,
Fig 5.15 shows the average values of Fig 5.14. In the vertical direction at the channel outlet,
the flow profile is parabolic with the maximum value of about 0.0048 µm/s (see Fig. 5.15).
43
Fig 5.16: Horizontal component of the velocity vector along a vertical line at the
midpoint between the electrodes
~
~
~
~
~
~
~
~
h  0.333, re  1, rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4, f =0.3
The net velocity along a vertical direction at the middle point between electrodes is shown in
Fig 5.16. It is observed that the velocity changes direction in the middle of the cycle that is at
the time 0.5t, at the same time as the velocity at discharge. At 0.14 µm , net velocity changes
direction for every fraction of time of the cycle. The value of net velocity goes to a high from
0t to 0.1t but then decreases with time, just like above case. At 0.5t, flow is reversed. After
that, net velocity increases then decreases and finally at the end of the cycle the flow is again
at the previous direction that is opposite to created tangential electric field. The magnitude is
also very close for the starting and ending of the cycle. The net velocity changes sharply up to
0.14 µm above electrodes. After that point, velocity is parabolic in nature. It goes to a
maximum value of 0.05 µm/s at 0.24 µm from and velocity is 0 at walls. At time 0.1t, the
velocity reaches maxim at 0.3 µm from electrodes, the value is -0.45 µm/s.
44
Fig 5.17: Time averaged net velocity vector along a vertical line at the midpoint between
the electrodes
~
~
~
~
~
~
~
~
h  0.333, re  1, rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4, f =0.3
Fig 5.17, shows the time averaged net velocity in the middle plane between two electrodes.
Near the electrodes the velocity increases with the increasing distance from electrode, goes to
a maximum at 0.025 µm above electrodes. The maximum velocity is 0.02 µm/s for the above
~
case of dimensionless frequency f =0.3. From 0.025 µm above the electrodes, the velocity
decreases and goes to zero at 0.149 µm above electrodes. Then the velocity changes direction
i.e. flow reversal occurs. Dramatic change in net velocity is observed in the range of 0 µm to
0.025 µm range area above the electrodes. Rate of change of the magnitude of net velocity is
comparatively lower at 0.025 µm to 0.1 µm region. After the flow reversal, the region of 0.2
to 0.3 µm above electrodes shows that the magnitude of velocity is nearly equal. The velocity
is parabolic in shape from the point 0.0125 µm to the wall (0.33 µm). The velocity is 0 at
both wall. Parabolic nature velocity is in the reversed direction. It reaches maximum of
magnitude 0.025 µm/s at the region of 0.225 µm to 0.275 µm above electrodes.
45
5.5 Effect of AC Frequency on Time Averaged Net Velocity and Flow Rate
Fig 5.18: Dependence of the time-averaged net velocity on the AC frequency.
~
~
~
~
~
h  0.333, ~
re  1, ~
rg  9 , le  1,  D =3.3e-3 , A = 0.75, s =4.
The dependences of the dimensionless time-averaged net velocity on the dimensionless
frequency of the electric field are shown in Fig 5.18. Velocity decreases in the domain of
lower frequency, increases in relatively higher frequency and reaches maximum. At higher
frequency, flow reversal occurs and again it goes to another maximum in the opposite
~
direction of fluid flow. Velocity maxima (in the absolute value) is located around f ≈ 0.3. It
should be noted that the precise location of the maximum depends on the particular choice of
the characteristic length. In this paper, the length of the periodic domain is chosen to be the
~
characteristic length. Flow reversal occurs at f ≈ 0.6.
The volumetric flow per width of microchannel shows the same nature as the net velocity
while changing with frequency. It goes to a maximum at frequency 0.3. The maximum flow
rate at channel outlet is 0.005 µm3/s.
46
Fig 5.19: Flow rate at channel outlet and midpoint between electrodes
~
~
~
~
~
re  1, ~
rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4.
h  0.333, ~
Comparative flow rate between channel outlet and middle point (Fig 5.19) shows that flow
rate at channel outlet is more or less greater than that of middle point for every frequency.
Both the curves show same nature. It goes to a maximum at frequency 0.3. The value of
maximum flow rate is 0.008 µm3/s.
47
5.6 Dependence of Time Averaged Net Velocity on Debye length and Compact Layer
Relative Thickness
Fig 5.20: Dependence of the time-averaged net velocity on Debye length
~
~
~
~
re  1, ~
rg  9 , le  0.333, f = 0.3 , A = 0.75
h  0.333, ~
Fig 5.20 shows that the effect of reducing the equilibrium concentration of neutral species
( HCl Solution) on net velocity. The velocity of the net flow dramatically increases with an
increasing Debye length. As Debye length is proportional to ceq -0.5 according to Eq. (3.1), it
can be said that net flow increases with decreasing electrolyte concentration of neutral
species.
Compact layer relative thickness can be interpreted as a ration between effective thickness of
compact layer and diffuse (Debye) layer. Increasing Compact layer relative thickness means
~
decreasing length of Debye length. Here, s represents Compact layer relative thickness.
~
Velocity decreases with increasing s . Negative sign shows the direction of fluid flow. Flow
~
reversal is noticed for Dimensionless frequency 1 in case of s = 4. That is, Fluid flow
~
changes direction at this point. But no flow reversal occurs in the same frequency if s is =
48
~
~
0.5 or 0.2 ( when s is less than 1). s value is greater than or equal to 1 means, compact
~
layer is of the same order or larger than the diffuse layer. Increasing value of s causes
decreasing electroosmotic velocity. From equation 3.5.2(a) it can be seen that electroosmotic
velocity is proportional to difference between the potential  on the outer side of the diffuse
layer and the potential  on the inner side of this layer, at the nonslip plane.
Fig: 5.21: Dependence on the compact layer thickness.
~
~
~
r  1, ~
r  9 , l  0.333 , A = 0.75
h  0.333, ~
e
e
g
The presence of a thick compact layer reduces the voltage drop across the diffuse layer,
lowering the electroosmotic velocity. In all the three conditions of relative compact layer
frequency, the time averaged net velocity increases with increasing dimensionless AC
frequency.
~
Increasing value of s means decreasing value of Debye length thickness. It can be said that
~
with the decreasing value of s , electroosmotic velocity increases. That means with the
increasing Debye length thickness, velocity increases ( Fig 5.21).
49
Diffusion
Diffuse
Compact layer
~
~
~
~
~
Fig 5.22: Flow reversal. h  0.333, ~
re  1, ~
rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4,
~
f =1
If the compact layer is very thin there is no reverse flow, but if it is of same order or larger
than the diffuse layer, there could be flow reversal since the voltage drop in the diffuse
(Debye) layer is greatly reduced and can be overcome by the opposite voltage drop across the
diffusion layer.
50
5.7 Effect of Electrode and Channel Geometry on Time Averaged Net Velocity
Fig 5.23: Effect of Microchannel height on time averaged net velocity.
~
~
~
~
~
~
re  1, ~
rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4, f =0.3
In this section, the effects of the geometric parameters of the system on the micropump
performance is discussed. The dependences of the dimensionless time averaged net velocity
on the microchannel height are shown in Fig 5.23. There is clear maxima of the time
averaged net velocity in the computed dependences. It means that a certain confinement leads
to a higher net velocities than those in the unconfined channels. Micropumps with a larger
vertical dimension are not affected by the top solid boundary.
E
The relative size of electrodes is defined by ~
re  1 which means the ration of the lengths of
E2
electrodes. ~
r =1 means the electrodes are of same size. Here we see the effect of increasing
e
~
re on time averaged net velocity. Four values of ~
re are presented here. Value of ~
re is
51
Fig 5.24: Effect of relative size of electrodes
~
~
~
~
~
~
rg  9 , le  0.333,  D =3.3e-3 , A = 0.75, s =4, f =0.3
increasing from 1 to 3. ~
re =3 means left electrode is 3 times greater than right electrode. Here,
left electrode is positive and right electrode is negative. It seems that with the increasing
length of positive electrodes, time average net velocity increases. Maximum value is 0.0024
µm/s for the given system which is for ~
re =1 i.e. symmetric electrodes. For ~
re =2, flow
reversal occurs and then velocity increases in that direction with increasing relative size of
electrode lengths ( ~
re ).
52
6. CONCLUSION AND RECOMMENDATIONS
6.1 Concluding Remarks
A fully nonlinear analysis is performed to understand the mechanism of AC electroosmosis
with Faradaic currents. The mobilities of the species are taken different and the compact layer
is assumed to be not very thin compared to the diffuse layer. ACEO with Faradaic current for
symmetric electrodes is considered in this analysis.
In this chapter, conclusions are drawn based on the analysis performed in the previous
chapter. The analyses are performed first through investigating the effect of dimensionless
AC frequency on the time averaged net velocity of fluid moving along a slit microchannel by
varying the frequency from 0.01 to 1. Later the analyses are performed for different Debye
length, compact layer relative thickness, microchannel height and electrodes position. From
the results of time averaged net velocity and direction of flow of fluid flowing through silt
micro channel for different parameters; the following inferences can be drawn:
-
At very low frequency, net velocity is higher when considering Faradaic current.
After frequency 0.02, net velocity with Faradaic current is lower than net velocity
without Faradaic current. Changes in velocity profile is sharper with Faradaic current
than without Faradaic current.
-
In the range of frequency from 0.01 to 1, there is no flow reversal while taking no
Faradaic current. But in the same region of frequency, flow reversal occurs after
frequency ≈0.2 while taking Faradaic current into consideration.
-
Taking absolute value of net velocity, symmetric electrode arrangement exerts lower
net velocity than asymmetric arrangements at lower frequency. And for higher
frequencies, symmetric electrodes exert higher net velocities than asymmetric
electrodes. For symmetric electrodes, flow reversal occurs at dimensionless
frequency ≈ 0.6 and for asymmetric electrodes, it happens after the value of
dimensionless frequency ≈ 0.2.
-
For the dimensionless amplitude of applied sinusoidal signal at the positive electrode
being 0.75, time averaged net velocity decreases at lower frequency region and
increases at higher frequency region. Maximum velocity attained at dimensionless
frequency 0.3. Flow reversal occurs at dimensionless frequency 0.6 when relative
compact layer thickness is 4. Volumetric flow rate is more or less same for the middle
point between electrodes and at the channel outlet. The maximum flow rate at channel
outlet is 0.005 µm3/s.
-
The velocity of the net flow dramatically increases with an decreasing electrolyte
concentration and increasing Debye length up to frequency 0.02; then it increases
53
slowly at higher frequency region. Electroosmotic velocity increases with decreasing
compact layer relative thickness.
-
Up to frequency 0.1, the time averaged net velocity drastically changes from going to
maximum and then decreases sharply with the increase of height to length ration of
slit microchannel. After that point, net velocity decreases slowly. Top solid boundary
of microchannel is not affected when the vertical dimension is large. Maximum
velocity is attained at electrode length ratio 1. When positive electrode is double than
negative electrode, flow reversal occurs.
Optimal set of model parameters for AC pump can be recommended based of the present
numerical analysis of the dimensionless model. Real parameters for any particular case can
be computed from all the dimensionless parameters. Geometrical parameters can be set to
attain particular net velocity for different electrolyte concentration. These results can be a
guideline for the selection of an experimental geometry.
6.2 Recommendations for Future Work
Some possible directions of the future works are as follows:
-
Frequency values are taken from 0.01 to 1. Investigation can be carried out in a much
broader parameter range to precisely examine the qualitative and quantitative
behavior of AC electroosmosis in slit microchannel with symmetric electrode.
-
Zero potential is taken for the negative electrode in the present study. Negative AC
signal can be incorporated at the bottom electrode. A DC offset can be added to the
AC signal so that one electrode potential remains greater than zero and the other
lower than zero.
-
Geometrically more complex electrodes should be taken in consideration. Since it is
predicted that the counter rotating regions of fluid observed above the electrode arrays
inhibit net flow, a three-dimensional electrode design should be incorporated where
the inhibition should be taken care of.
-
A simple model for Faradaic currents is used here to investigate its effect on the
electrokinetic flows generated on top of coplanar symmetric electrodes. The liquid is
taken as symmetric uni-univalent electrolyte where only one ionic species react at the
electrodes. More reacting ionic species should be included that is different
metal/electrolyte systems should be investigated for a better understanding of the
electrochemical process in a more realistic way.
AC electroosmotic pumps can be a promising tool for various microfluidic applications.
Further analysis can be carried out to predict the effect of Faradaic current on the pumping
behavior of symmetric and asymmetric electrodes subjected to AC signal.
54
7. REFERENCES
[1]. Přibyl, M., Šnita, D. and and Marek, M., Multiphysical Modeling of DC and AC
Electroosmosis in Micro and Nanosystems, Institute of Chemical Technology, Prague, Czech
Republic, Recent Advances in Modeling and Simulation, pp. 501-522.
[2]. Becker, H. and Locascio, L.E. Polymer microfluidic devices. Talanta, 56, 2002, 2, 267287, 0039-9140
[3]. Beebe, D.J., Mensing, G.A. and Walker, G.M., Physics and applications of microfluidics
in biology, Annual Review Of Biomedical Engineering, 4, 2002, 261-268, 1523-9829
[4]. Wu, J., Ben, Y. and Chang, H.,"Particle detection by electrical impedance spectroscopy
with asymmetric-polarization AC electroosmotic trapping", Microfluid Nanofluid, Vol. 1,
2005, pp. 161–167.
[5]. Wu, J., Ben, Y., Battigelli, D. and Chang, H.," Long-Range AC Electroosmotic Trapping
and Detection of Bioparticles", American Chemical Society, Vol. 44, 2005, pp. 2815-2822
[6]. Lian, M., Islam, N. and and Wu, J., " Particle Line Assembly/Patterning by Microfluidic
AC Electroosmosis", Institute of Physics Publishing Journal of Physics: Conference Series
34, 2006, pp. 589–594.
[7]. Wu, J., " Biased AC Electro-Osmosis for On-Chip Bioparticle Processing", IEEE
Transactions on nanotechnology, Vol. 5, March 2006, No. 2.
[8]. Guan, W., Park, J., Krsti, P. S. and Reed, M. A., " Non-vanishing ponderomotive AC
electrophoretic effect for particle trapping", IOP Publishing, Nanotechnology 22, 2011,
245103, pp. 8.
[9]. Stubbe, M., Holtapples, M. and Gimsa, J., “A new working principle for ac electrohydrodynamic on-chip micro-pumps,” Journal of Physics. D: Appl Phys. 40, 2007, pp. 68506856.
[10]. Krishnamoorthy, S., Feng, J., Henry, A. C., Locascio, L. E., Hickman, J. J. and
Sundaram, S.,"Simulation and experimental characterization of electroosmotic flow in
surface modified channels", Microfluid Nanofluid, Vol. 2, 2006, pp. 345–355.
[11]. Yan, G., Peck, K. D., Zhu, H., HiguchiI, W. I. and S. Kevin, S., " Effects of
Electrophoresis and Electroosmosis during Alternating Current Iontophoresis across Human
Epidermal Membrane", Journal of Pharmaceutical Sciences, Vol. 94, 2005, pp. 547–558.
[12]. Green, N.G., Ramos, A., Morgan, H., and Castellanos A., “ Fluid flow induced by
nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental
measurements,” Physical Review E, vol. 61, 2000, Number 4.
55
[13]. Gonzalez, A., Ramos, A., Green, N. G., Castellanos, A. and Morgan, H., “Fluid flow
induced by nonuniform ac electric fields in electrolytes on microelectrodes.II. A linear
double-layer analysis,” Physical Review E, Vol. 61, 2000, Number 4.
[14]. Green, N.G., Ramos, A., Gonzalez, A., Morgan, H. and Cstellanos, A., “ Fluid flow
induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Observations
of streamlines and numerical simulation,” The American Physical Society, Physical Review
E 66,026305, 2002, Number 4.
[15]. Ramos, A., González, A., García-Sánchez, P. and Castellanos, A., "A linear analysis of
the effect of Faradaic currents on traveling-wave electroosmosis", Journal of Colloid and
Interface Science, Vol. 309, 2007, pp. 323–331.
[16]. Gonzalez, A., Ramos, A., Sanchez, P. G. and Castellanos, A., "Effect of the combinded
action of Faradaic currents and mobility differences in ac electroosmosis, " The American
Physical society, Physical Review E 81, 016320, 2010.
[17]. García-Sánchez, P., Ramos, A., González, A., "Effects of Faradaic currents on AC
electroosmotic flows with coplanar symmetric electrodes", Colloids and Surfaces A:
Physicochem. Eng. Aspects 376, 2011, pp. 47–52.
[18]. Squires, T. M. and Bazant, M. Z., " Induced charge electro-osmosis," Journal of Fluid
Mechanics, Vol. 509, 2004, pp. 217-252.
[19]. Bazant, M. Z., Squires, T. M., “ Induced-Charge electrokinetic phenomena: theory
and microfluidin applications,” Physical Review Letters, Vol. 92, 2004, Number 6.
[20]. Soni, G., Squires, T. M. and Meinhart, C. D., “Study of Non Linear Effects in
Electrokinetics,” Excerpt from the Proceedings of the COMSOL Conference 2007, Boston.
[21]. Urbanski, J. P., Levitan, J. A., Burch, D. N., Thorsen, T. and Bazant, M. Z.," The effect
of step height on the performance of three-dimensional ac electro-osmotic microfluidic
pumps", Journal of Colloid and Interface Science, Vol. 309, 2007, pp. 332–341.
[22]. Olesen, L. H., Bazant, M. Z. and Bruus, H.," Strongly nonlinear dynamics of
electrolytes in large ac voltages", Physical Review E 82, 011501, 2010.
[23]. Soestbergen, M. V., Biesheuvel, P. M. and Bazant, M. Z.," Diffuse-charge effects on the
transient response of electrochemical cells", Physical Review E 81, 021503, 2010.
[24]. M. Přibyl, M. and Šnita, D. ,"Poisson based modeling of DC and AC electroosmosis in
microfluidic channels", Excerpt from the Proceedings of the COMSOL Users Conference,
2006, Prague.
[25]. Hrdlicka, J., Cervenka, P., Pribyl, M. and Snita. D.," Mathematical Modeling of ZigZag Traveling-Wave Electro-Osmotic Micropumps", Excerpt from the Proceedings of the
COMSOL Conference 2009 Milan
56
[26]. Hrdlicka, J., Cervenka, P., Pribyl, M. and Snita, D., “ Mathematical modeling of AC
electroosmosis in microfluidic and nanofluidic chips using equilibrium and non-equilibrium
approaches,” Journal of Appl Electrochem, Vol. 40, 2010, pp. 967-980.
[27]. Cervenka, P., Hrdlicka, J., Pribyl, M. and Snita, D., “ Toward high net velocities in AC
electroosmosis micropumps based on asymmetric coplanar electrodes,” IEEE transactions on
industry applications, Vol. 46, 2010, No. 5.
[28]. Pribyl, M.
and
Adamiak, K., "Numerical Models for AC Electroosmotic
Micropumps," IEEE Transactions on Industry Applications, Vol. 46, 2010, Number6
[29]. Cervenka, P., Hrdlicka, J., Pribyl, M. and Snita, D., "Study on Faradaic and
Coulombic interactions at microelectrodes utilized in AC electroosmotic
micropumps",
Research work of Institute of Chemical Technology, Prague,
Department of Chemical
Engineering, Technick´a 5, 166 28 Praha 6, Czech Republic, 2010, MSMT no. 21.
[30]. Pham, P., Howorth, M., Chrétien, A. P. and Tardu, S., “ Numerical Simulation of the
Electric Double Layer based on Poisson-Boltzman Models for AC Electroosmosis Flows,”
Excerpt from the Proceedings of the COMSOL Users Conference 2007.
[31]. Masliyah, J. H. and Bhattacharjee, S., Electokinetic and Colloid Transport Phenomena,
May 11, 2005.
[32]. Reddy, J. N., Finite Element Method, third edition, Tata McGraw-Hill, 2008.
57