Roughness and cavitations effects on electro-osmotic flows in rough microchannels using the lattice Poisson–Boltzmann methods

Abstract

This paper investigates the effects of roughness and cavitations in microchannels on the electro-osmotic flow behaviors using the Lattice Poisson–Boltzmann methods which combined one lattice evolution method for solving the non-linear Poisson–Boltzmann equation for electric potential distribution with the other lattice evolution method for solving the Navier–Stokes equations for fluid flow. The boundary conditions are correctly treated for consistency between the both. The results show that for the electro-osmotic flows in homogeneously charged rough channels, the flow rate does not vary with the roughness height or the interval space monotonically. The flow rate varies slightly with the roughness height or even increases a little when the roughness is very small, and then decreases when the roughness height is larger than 5% channel width. The flow rate decreases first and then increase with the roughness interval space. An interval space at twice roughness width makes the flow rate minimum. For the heterogeneously charged rough channel, the flow rate increases with the roughness surface potential at a super-linear rate. For the electro-osmotic flows in microchannels with cavitations, the flow rate change little with the cavitations depth when the depth value is very low and decreases sharply when the depth is greater than 3% channel width. The flow rate trends to be a constant when the cavitations are very deep. The flow rate decreases with the cavitations width but increases with the cavitations interval.

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Roughness and cavitations effects on electro-osmotic flows
in rough microchannels using the lattice
Poisson–Boltzmann methods
Moran Wanga,b,*, Jinku Wangc, Shiyi Chena,d
aDepartment of Mechanical Engineering, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, United States
bDepartment of Biological and Agricultural Engineering, University of California, Davis, CA 95616, United States
cSchool of Aerospace, Tsinghua University, Beijing 100084, China
dCollege of Engineering and LTCS, Peking University, Beijing, China
Received 25 August 2006; received in revised form 26 March 2007; accepted 3 May 2007
Available online 22 May 2007
Abstract
This paper investigates the effects of roughness and cavitations in microchannels on the electro-osmotic flow behaviors
using the Lattice Poisson–Boltzmann methods which combined one lattice evolution method for solving the non-linear
Poisson–Boltzmann equation for electric potential distribution with the other lattice evolution method for solving the
Navier–Stokes equations for fluid flow. The boundary conditions are correctly treated for consistency between the both.
The results show that for the electro-osmotic flows in homogeneously charged rough channels, the flow rate does not vary
with the roughness height or the interval space monotonically. The flow rate varies slightly with the roughness height or
even increases a little when the roughness is very small, and then decreases when the roughness height is larger than 5%
channel width. The flow rate decreases first and then increase with the roughness interval space. An interval space at twice
roughness width makes the flow rate minimum. For the heterogeneously charged rough channel, the flow rate increases
with the roughness surface potential at a super-linear rate. For the electro-osmotic flows in microchannels with cavitations,
the flow rate change little with the cavitations depth when the depth value is very low and decreases sharply when the depth
is greater than 3% channel width. The flow rate trends to be a constant when the cavitations are very deep. The flow rate
decreases with the cavitations width but increases with the cavitations interval.
? 2007 Elsevier Inc. All rights reserved.
Keywords: Roughness and cavitation; Electro-osmotic flow; Lattice Poisson–Boltzmann method; Microfluidics
1. Introduction
With growing interests in bio-MEMS and bio-NEMS applications and fuel cell technologies, electrokinetic
flows have become one of the most important non-mechanical techniques in microfluidics and nanofluidics
0021-9991/$ - see front matter ? 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcp.2007.05.001
*Corresponding author. Address: Department of Mechanical Engineering, The Johns Hopkins University, 3400 N. Charles Street,
Baltimore, MD 21218, United States. Tel.: +1 530 754 6770.
E-mail address: moralwang@jhu.edu (M. Wang).
Journal of Computational Physics 226 (2007) 836–851
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[1–6]. Electro-osmotic flows (EOFs) have wide applications for pumping [7,8], separating [3] and mixing [9] in
micro- and nanoscale devices. Due to their many applications, numerical simulations of EOF in micro- and
nanochannels have recently received a great amount of attention [10–29]. The previous studies have mostly
focused on smooth channels which, however, the realistic applications always go beyond. There are many
roughness and cavitations on a real channel wall surface. To the authors’ knowledge, only a little public lit-
erature concerns and presents full analyses of the EOFs in rough channels [16,17].
From the macroscopic point of view, the EOFs are governed by the Poisson–Boltzmann equation for elec-
trical potential distributions and the Navier–Stokes equations for flows [10,13]. Accurate and efficient solution
of the non-linear Poisson–Boltzmann equation is a challenge for both mathematicians and physicists. Up to
now, hundreds of relative research papers appear every year on various scientific journals [30]. Most of
previous researches have employed conventional PDE solvers, such as the finite difference method (FDM)
and the finite element method (FEM) to solve the non-linear Poisson–Boltzmann equation in its linearized
form [11–17]. Several researchers have also gained good predictions by solving the original non-linear
Poisson–Boltzmann equation using FDM [18] or FEM [19], however with suffering from the huge computa-
tional costs from the strong non-linearity. The fast Fourier transform (FFT) [31] and the multi-grid [22,32]
techniques greatly increase the efficiency of the numerical solution of the non-linear Poisson–Boltzmann
equation; however they have seldom been extended for complex geometries.
In recent years, an efficient mesoscopic statistics-based method, the Lattice Boltzmann method (LBM), has
been introduced into modeling the electrical potential distribution in confined domains [23–26,33–38]. Warren
[33] introduced the ‘‘moment propagation’’ method to predict the electrical potential distribution for charged
suspensions. He and Li [34] proposed a different scheme for analyzing the electrochemical processes in an elec-
trolyte by using an independent lattice Boltzmann method to solve the Poisson equation for the ion diffusion.
However, this method was based on a locally electrically neutral assumption so it was not suitable for analyz-
ing the dynamics of charged suspensions [35]. Recently, Wang et al. [38] proposed a lattice evolution method
by tracking the electric potential equation directly on discrete lattices for solving the non-linear Poisson–Boltz-
mann equation accurately and efficiently in confined domains. The method has been combined with the other
lattice Boltzmann method, forming a lattice Poisson–Boltzmann method (LPBM), for applications of model-
ing the EOF in microchannels [24–26].
This paper focuses on the roughness and cavitations effects on EOFs in microchannels. To achieve the
objects, the lattice Poisson–Boltzmann method will be developed for rough or cavitational channels. The con-
sistency of boundary condition implements is correctly considered. The shape and arrangement effects of
roughness or cavitations on the electro-osmotic flow behavior are therefore studied.
2. Numerical method
2.1. Evolution equation for electrical potential distribution
Electric double layer (EDL) theory [10] relates the electrostatic potential and the distribution of counter-
ions and co-ions in the bulk solution by the Poisson equation as follows:
r2w ¼ ?qe
ee0;
ð1Þ
where w is the electrical potential, e the dimensionless dielectric constant of the solution, e0the permittivity of
a vacuum and qethe net charge density. According to classical EDL theory, the equilibrium Boltzmann dis-
tribution equation can be used to describe the ionic number concentration for dilute small-ion solutions.
Therefore, the net charge density distribution can be expressed as the sum of all the ions in the solution
X
where the subscript i represents the ith species, n1is the bulk ionic number concentration, z the valence of the
ions (including the sign), e the absolute value of one proton charge, kbthe Boltzmann constant and T the abso-
lute temperature.
qe¼
i
zieni;1exp
?zie
kbTw
??
;
ð2Þ
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Substituting Eq. (2) into Eq. (1) yields the non-linear Poisson–Boltzmann equation for the electrical poten-
tial in the dilute electrolyte solution
X
Eq. (3) can be solved using 1D or 2D linearized simplifications [20,21], iteration [11,12] or multi-grid methods
[22,32]. Hirabayashi et al. [36,37] ever developed a lattice Bhatnagar-Gross-Krook (BGK) model for the Pois-
son equation where, however, the source term was limited to a linear term or a fluctuation near zero. The mod-
el was hardly used to solve a non-linear Poisson–Boltzmann equation. Here we use the lattice Poisson method
for the non-linear Poisson–Boltzmann equation in confined domains [38] to solve Eq. (3).
The solution of Eq. (3) can be regarded as the steady solution of
r2w ¼ ?1
ee0
i
zieni;1exp
?zie
kbTw
??
:
ð3Þ
ow
ot¼ r2w þ grhsðr;w;tÞ;
where grhs¼
The evolution equation for the electrical potential on the two-dimensional night-directional (D2Q9) dis-
crete square lattices can then be written as [38].
ð4Þ
1
ee0
P
izieni;1expð?zie
kbTwÞ represents the negative term of right hand side (RHS) of Eq. (3).
gaðr þ Dr;t þ dt;gÞ ? gaðr;tÞ ¼ ?1
sg½gaðr;tÞ ? geq
aðr;tÞ? þ
1 ?0:5
sg
??
dt;gxagrhs;
ð5Þ
where r is the position vector, dt,gthe time step, sgthe dimensionless relaxation time, geqthe equilibrium dis-
tribution of evolution variable g:
8
>
and
8
>
The time step
dt;g¼ dx=c0;
where dxis the lattice constant and c0is a pseudo sound speed in the potential field whose value can be artificial
to vary the time step [38]. The dimensionless relaxation time
geq
a¼ -aw
with -a¼
0
1=6
1=12
a ¼ 0
a ¼ 1;2;3;4
a ¼ 5;6;7;8
>
:
<
ð6Þ
xa¼
4=9
1=9
1=36
a ¼ 0
a ¼ 1;2;3;4
a ¼ 5;6;7;8
>
:
<
:
ð7Þ
ð8Þ
sg¼3vdt;g
2d2
x
þ 0:5;
ð9Þ
where v is defined as the potential diffusivity which is equal to unity in the simulations.
The evolution Eqs. (5)–(9) can be proved consistent with the macroscopic non-linear Poisson–Boltzmann
equation, Eq. (4). After evolving on the discrete lattices, the macroscopic electrical potential can be calculated
using
X
Though the electrical potential evolution equations are in an un-steady form, only the steady state result is
realistic, because the electromagnetic susceptibility has not been considered. Although the lattice evolution
method for non-linear Poisson equation is not as efficient as the multi-grid solutions due to its long wavelength
limit, it has the advantages of suitability for complex geometries and parallel computing. Although this paper
only presents 2D cases, the algorithm is easy to extend to a 3D case.
w ¼
a
ðgaþ 0:5dt;ggrhsxaÞ:
ð10Þ
838
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2.2. Evolution equation for fluids with external forces
For the flows with external forces, the continuous Boltzmann–BGK equation with an external force term,
F, is [39–41]
Df
Dt? otf þ ðn ? rÞf ¼ ?f ? feq
s
where f ” f(x,n,t) is the single particle distribution function in the phase space (x,n), n the microscopic veloc-
ity, s the relaxation time and feqthe Maxwell–Boltzmann equilibrium distribution. For a steady fluid im-
mersed in a conservative force field, the equilibrium distribution function is defined by adding a Boltzmann
factor to the Maxwell–Boltzmann distribution
?
where U is the potential energy of the conservative force field, q0the fluid density where U is lowest, R the
ideal gas constant, D the dimension of the calculation space(1D, 2D or 3D) and u the macroscopic velocity.
Here the external force term, F, needs to be chosen carefully. Dimensional analysis led to the following form of
F:
þ F;
ð11Þ
feq¼
q0
ð2pRTÞD=2exp
?U
kbT
?
exp
?ðn ? uÞ2
2RT
!
;
ð12Þ
F ¼G ? ðn ? uÞ
with G being the external force per unit mass [42]. Eq. (3) has a perfect accuracy (relative errors are less than
0.5% when comparing with analytical solutions for a Poiseuille flow), even though ones reported it was an only
first order approximation [43].
The Chapman–Enskog expansion can be used to transform the Boltzmann–BGK equation, Eq. (11), into
the correct continuum Navier–Stokes equations,
RT
feq;
ð13Þ
qou
otþ qu ? ru ¼ ?rP þ lr2u þ FE;
where q is the solution density, P the pressure, l the dynamic fluid viscosity and FEthe electric force density
vector. In general, the electrical body force in electrokinetic fluids can be expressed as
ð14Þ
FE¼ Fextþ qeðEintþ n ? BintÞ þ FV;
where Fextrepresents the external field body forces, including the Lorentz force associated with any externally
applied electric and magnetic field. For only an electrical field, Fext= qeE, where E is the electrical field
strength. Eintand Bintare internally smoothed electrical and magnetic fields due to the motion of the charged
particles inside the fluid. FVis a single equivalent force density due to the intermolecular attraction [20].
The two-dimensional nine-speed (D2Q9) lattice Boltzmann model gives the evolution equation of the dis-
crete density distribution [39]
faðr þ eadt;t þ dtÞ ? faðr;tÞ ¼ ?1
where eais the discrete velocities
8
>
feq
a
the equilibrium distribution
?U
kbT
c2
c4
ð15Þ
sm½faðr;tÞ ? feq
aðr;tÞ? þ dtFa;
ð16Þ
ea¼
ð0;0Þ
ðcosha;sinhaÞc;ha¼ ða ? 1Þp=2
ffiffiffi
?
dtthe time step and smthe dimensionless relaxation time. The external force term is [25]
a ¼ 0
a ¼ 1 ? 4
a ¼ 5 ? 8
2
p
ðcosha;sinhaÞc;ha¼ ða ? 5Þp=2 þ p=4
>
:
<
;
ð17Þ
feq
a¼ xaq0exp
?
1 þ 3ea? u
þ 9ðea? uÞ2
?3u2
2c2
"#
;
ð18Þ
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Fa¼G ? ðea? uÞ
The macroscopic density and velocity can be calculated using
X
qu ¼
a
The dimensionless relaxation time, sm, is a function of the fluid viscosity
sm¼ 3mdt
d2
x
where m is the kinetic viscosity.
For electrokinetic flows in dilute electrolyte solutions, the external electrical force in Eq. (15) can be sim-
plified to:
FE¼ qeE ? qerU;
where U is the stream electrical potential caused by the ion movements in the solution based on the Nernst–
Planck theory. Generally, the stream potential dominates the electro-viscosity effect in pressure driven flows,
but its value is much less than the external potential and can be ignored in electrically driven flows. Therefore,
the external force in the discrete Lattice Boltzmann equation (Eq. 18) should include the pressure and electric
force
RT
feq
aðr;tÞ:
ð19Þ
q ¼
a
X
fa;
ð20Þ
eafa:
ð21Þ
þ 0:5;
ð22Þ
ð23Þ
Fa¼ð?rP þ qeE ? qerUÞ ? ðea? uÞ
qRT
feq
a:
ð24Þ
Eqs. (16)–(24) can then be solved to analyze electrokinetic flows using the LBM as long as the charge density
distribution in the solution is known.
2.3. Boundary conditions
For the evolution equation of the electrical potential, we used a second-order accurate Dirichlet boundary
condition implement on the wall surfaces and the periodic conditions were implemented at both inlet and
outlet.
For the Dirichlet boundary, the unknown distribution functions at the boundary were assumed to be equi-
librium distribution functions with a counter-slip internal energy density with the source, grhs[44]. For exam-
ple, for a straight upper wall, g4, g7and g8are unknown, but can be obtained from the equilibrium distribution
of the local w0:
X
where Spis the sum of known populations coming from the internal nodes and nearest wall nodes
Sp¼ g0þ g1þ g2þ g3þ g5þ g6;
and wsis the boundary value. Thus the unknown distributions are
ga¼ -aw0:
Convex and concave corners exist in roughness and cavitations. They can be treated in a similar way. The
upper-right convex corner, for example, has five unknown populations g1, g2, g5, g6and g8. They also follow
from Eq. (25) with
P
where the known populations g3, g4and g7lead to
w0¼ 3ws? 3Sp? 1:5dt
a
xagrhs;
ð25Þ
ð26Þ
ð27Þ
w0¼12ws? 6dt
axagrhs? 12Sp
7
;
ð28Þ
840
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Sp¼ g0þ g3þ g4þ g7:
Therefore the unknown distributions are determined by Eq. (27).
The lower-left concave corner, for example, has only one unknown population g5which can be calculated
ð29Þ
by
w0¼ 12ws? 6dt
X
a
xagrhs? 12Sp;
ð30Þ
where
Sp¼ g0þ g1þ g2þ g3þ g4þ g6þ g7þ g8;
and therefore the unknown population is
ð31Þ
g5¼ w0=12:
For the evolution for fluid flow, the conventional bounce-back model was mostly used in previous works.
Although this model is easy to deal with the complex geometries, a detail study has shown it has half a lattice
displacement on the boundary [45,46]. To treat the boundary conditions consistent between the electric poten-
tial and fluid flow evolutions, a non-slip model is used here to model the fluid-solid interaction on the wall
surfaces [47].
For still walls, the unknown density distribution functions at the wall are calculated from the local equilib-
rium distribution function with a counter slip velocity. For example, for a straight upper wall, f4, f7and f8are
unknown, and can be obtained from an equilibrium distribution function with a counter slip velocity
ð32Þ
u0¼ 6?ðf1? f3þ f5? f6Þ
q0
c;
ð33Þ
where
q0¼ 6ðf2þ f5þ f6Þ:
Thus the unknown density distributions are
"
ð34Þ
fa¼ xaq01 þ 3u0
cþ 9ðu0Þ2
c2?3ðu0Þ2
2c2
#
:
ð35Þ
For the cases at both convex and concave type corners, it is hard to determine the counter-slip velocities.
Here we simplify the unknown populations determined by the equilibrium distributions at zero counter-slip
velocities. Periodic conditions are implemented at inlet and outlet.
3. Results and discussion
The lattice Poisson–Boltzmann method is applied to simulate the EOF in 2D microchannels by two steps:
firstly the lattice Poisson method solves the non-linear Poisson–Boltzmann equation for a stable electrical
potential distribution; secondly the lattice Boltzmann method is used to simulate the steady electrically-driven
fluid flow in the microchannel. This section shows the effects of boundary implement consistency on the EOF
in straight smooth channels at first, and then analyzes the effects of roughness and cavitations in rough micro-
channels on the EOF characteristics with different shapes and arrangements considered.
The accuracy of the present lattice Poisson–Boltzmann method lies on the lattice size relative to the width
of electric double layer. Previous study showed the lattice size should be smaller than the EDL width for an
accurate numerical simulation [38]. In the simulations of this work, we keep the lattice size much smaller than
the EDL width, i.e. dxis usually 1/10 to 1/3 each EDL width. We initialize a still system and judge the steady
status of electrical potential and velocity distributions by relative errors every 100 steps at the middle point are
smaller than 10?10. The computational efficiency depends also on the chosen dimensionless relaxation times
(s). Well-chosen values of the relaxation times would make each simulation cost minutes to hours on a
2.0G CPU depending on the grid number.
M. Wang et al. / Journal of Computational Physics 226 (2007) 836–851
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3.1. Straight smooth channel cases
The electrically-driven osmotic flows in a straight homogeneously charged channel are simulated using the
consistent lattice evolution methods. We change the external electrical field strength, the wall surface zeta
potential, the bulk ionic molar concentration and the channel width. It is found that the bulk velocity is nearly
proportional to the external electrical field strength, as well as the surface zeta potential, which agree well with
the previous experimental data [48], numerical results [21], and even recent MD simulations in nanochannel
flows [49,50].
Fig. 1 shows the velocity profiles for the various bulk ionic concentrations c1. The channel width is
0.4 lm, the external electric field strength E = 5 · 102V/m and the surface zeta potential ws= ?50 mV
for both walls. The fluid properties are set as those of water at the standard state which are the dielectric
constant ee0= 6.95 · 10?10C2/J m, the density q = 1.0 · 103kg/m3and the viscosity l = 0.89 Pa s. The
results show an optimal ionic concentration that maximizes average velocity. As the ionic molar concentra-
tion decreases from a high value (2 · 10?2), the EDL thickness increases so that although the force is
slightly reduced, the electrical force domain increases and thus the average velocity increases. There exists
a concentration at which the effect of the electrical force can dominate across the entire channel and make
the velocity reach maximum (10?4to 10?3M for current simulations). As the ionic concentration decreasing
(such as from 10?4to 10?6), the force reduction becomes the most important factor and the average velocity
decrease. The lower ionic concentrations also result in a more parabolic-like velocity profile. These results
are similar as those in Ref. [24] qualitatively, however different quantitatively at high ionic concentrations.
The reason lies in the boundary implement inconsistency between the two evolution methods in the previous
research [24].
Fig. 2 shows the velocity profiles for various channel widths for c1= 10?4M, E = 5 · 102V/m and
ws= ?50 mV. The channel width varies from 0.1 lm to 1 lm. Totally different from the previous results
[24], the average velocity shows good monotonicity with the channel width. For channel widths larger
0 0.10.2 0.30.40.5
y/H
0.6 0.70.80.91
0
2
4
6
8
10
12
14
16
18
20
u μm/s
2×10-2M
10-2M
10-3M
10-4M
10-5M
10-6M
Fig. 1. Velocity profiles for various ionic molar concentrations for electrically driven flow. The dotted line: c1= 2 · 10?2M; the solid line:
c1= 10?2M; the dash-dot line: c1= 10?3M; the dashed line: c1= 10?4M; the triangle-line: c1= 10?5M; the square-line: c1= 10?6M.
842
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than double size of the EDL thickness, the maximum velocity seldom changes with the channel width.
However, for channels widths less than double size of the EDL thickness, a smaller channel width leads
to a smaller velocity. Similar results can be found in the previous simulations under the linearization
assumption [21].
3.2. Roughness effects
The roughness is simplified as a group of rectangles on the lower channel wall here, as shown in Fig. 3.
The channel is H in width, with both wall charged with a surface potential ws. The channel is periodic in x
direction. The roughness is w in width and h in height. The roughness is uniformly arranged in the channel
with an interval space D. The three surfaces of each roughness are charged at a surface potential wr. The
A–A section is the middle cross section for each roughness. The electrolyte solution is driven by an external
electrical field E. The fluid parameters are same as used in Section 3.1. Here we consider various cases for
different roughness heights (h), different arrangement interval spaces (D) and different surface potentials
(wr).
Fig. 4 shows the velocity profiles at A–A section for different roughness heights. The ionic concentration of
the electrolyte solution is c1= 10?4M. The external electrical field strength is E = 5 · 102V/m. Both the
channel and roughness surfaces are homogeneously charged at ws= wr= ?50 mV. The channel width is
0.4 lm and divided into 40 lattices. The roughness width is w = H/4 and the interval space D = 3H/4. The
roughness height h changes from 0 (smooth case) to H/4. The results show that the velocity maximum
increases with the roughness height, and the velocity profile becomes asymmetric. The reason is that the
roughness does not only change the channel cross section area, but also change the potential distribution
and therefore the electric driving force on the fluid. Thus there are two factors actually influencing the flow
rate through the channel. Fig. 5 shows the flow rate changing with the roughness height. The results indicate
that the electrically-driven flow rate changes very slowly with the roughness height when the roughness is very
small (h/H < 0.05). A very short roughness (h/H < 0.025) seems even enhancing the flow rate a little bit. The
charged roughness plays two roles: first it changes the electrical field distribution which may enhance the elec-
00.1 0.20.4 0.60.81
0
5
10
15
20
y μm
u μm/s
Fig. 2. Velocity profiles for different channel widths with c1= 10?4M, E = 5 · 102V/m and ws= ?50 mV.
M. Wang et al. / Journal of Computational Physics 226 (2007) 836–851
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tric driving force, and second it also increase the flow resistance. When the former factor dominates, the flow
rate could increase; otherwise the flow rate will decrease. When the roughness height is larger than 1/10 chan-
nel width, the flow rate will decrease sharply with the roughness height. The interesting anomalous variation of
flow rate with the roughness height has not been reported and validated in the previous work. A further anal-
ysis and consideration on the flow mechanism will be performed in the future work using our multi-scale sim-
ulation tools by coupling atomistic and continuum methods.
Fig. 6 shows the flow rates influenced by the roughness interval space when both the roughness width and
height are at H/4. The other parameters are c1= 10?4M, E = 5 · 102V/m, ws= wr= ?50 mV and
H = 0.4 lm. When the roughness interval is smaller than twice the roughness width, the flow rate decrease
with the interval space; however, when the roughness interval is larger than twice the roughness width, a larger
interval space leads to a larger flow rate. It is noticed that even for very sparse roughness cases, such as
D = 15w, the flow rate is lower than 90% that of a smooth channel, which shows that the roughness effect
is not negligible in analysis of electro-osmotic flows in microfluidics.
The heterogeneously charged rough channel is also simulated here. We remain the channel surface potential
ws= ?50 mV and change the roughness surface potential wrfrom ?10 to ?120 mV. The asymmetric potential
Fig. 3. Geometries and boundary conditions for the microchannel with roughness.
05 1015 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y/H
u (μm/s)
smooth
h=0.05 H
h=0.10 H
h=0.15 H
h=0.20 H
h=0.25 H
Fig. 4. Velocity profiles at A-A section for different roughness heights with c1= 10?4M, E = 5 · 102V/m, ws= wr= ?50 mV,
H = 0.4 lm, w = H/4 and D = 3H/4.
844
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boundaries destroy the flow symmetry, as shown in Fig. 7a. Larger values of the surface potential induce lar-
ger velocities near the surface. The flow rate also increases with the roughness surface potential superlinearly,
shown as Fig. 7b. For the current case, when the roughness surface potential is ?120 mV, the flow rate of the
00.050.10.150.20.25
0.8
0.85
0.9
0.95
1
h/H
Qrough/Qsmooth
Fig. 5. Flow rate variation with the roughness height with c1= 10?4M, E = 5 · 102V/m, ws= wr= ?50 mV, H = 0.4 lm, w = H/4 and
D = 3H/4.
0251015
0.66
0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.84
D/w
Qrough/Qsmooth
Fig. 6. Flow rate variation with the roughness interval space with c1= 10?4M, E = 5 · 102V/m, ws= wr= ?50 mV, H = 0.4 lm,
w = H/4 and h = H/4.
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rough channel is even higher than that of the homogeneously charged smooth channel. Such characteristics
could be used to enhance the flow rate in microfluidics by inserting high-surface-potential objects into the
microchannel.
05 10 1520 2530 35
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (μm)
y/H
Ψr=-10 mV
Ψr=-30 mV
Ψr=-50 mV
Ψr=-70 mV
Ψr=-90 mV
Ψr=-120 mV
20406080 100120
0.5
0.6
0.7
0.8
0.9
1
1.1
-Ψr (mV)
Qrough/Qsmooth
Fig. 7. Heterogeneously charged roughness cases where c1= 10?4M, E = 5 · 102V/m, ws= ?50 mV, H = 0.4 lm, w = H/4, D = 3H/4
and h = H/4. (a) Velocity profiles at A–A section (b) flow rate varies with the roughness surface potential.
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3.3. Cavitations effects
Both roughness and cavitations are fundamental elements on a rough channel surface. In macroscopic flow,
no divisions have been made between them. However, in micro channel flow, some difference has been
reported on the flow friction between roughness and cavitations [51,52]. Here we simulate the electro-osmotic
flows in microchannels with cavitations, which has been seldom studied before.
The concerned microchannel with cavitations is shown in Fig. 8. The geometry parameters and the bound-
aries are quite similar as those of roughness. The cavitation is h in depth, w in width and D in interval between
each. Here we only consider the homogeneously charged cases at ws= ?50 mV.
Fig. 9 shows the relationship between the flow rate and the cavitations depth when w equals H/4 and D
equals 7H/4. The other parameters are c1= 10?4M and E = 5 · 102V/m. The relative depth h/H changes
from 0 (smooth channel) to 0.3. When the cavitations are very shallow (h < 3%H), the flow rate almost keep
unchanged with the cavitations depth. With the cavitation depth increasing, the electrically-driven flow rate
decrease sharply. If the cavitations are deep enough (such as h > 10%H), the flow rate decrease becomes slow
Fig. 8. Geometries and boundary conditions for the microchannel with cavitations.
0 0.050.1 0.15
h/H
0.2 0.25
0.985
0.99
0.995
1
Qcavitation/Qsmooth
Fig. 9. Flow rate variation with the cavitations depth at w = H/4 and D = 7H/4 where c1= 10?4M, E = 5 · 102V/m, ws= ?50 mV,
H = 0.4 lm.
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and the flow rate shows to be asymptote-like stable to a constant when the cavitations are very deep
(h ? 20%H). Fig. 9 indicates that the cavitations effect on the flow rate is not as significant as the roughness.
The electrically-driven flow rate difference in between a smooth channel and a cavitational channel is less than
00.10.2 0.30.40.5
0.95
0.96
0.97
0.98
0.99
1
w/H
Qcavitation/Qsmooth
Fig. 10. Flow rate variation with the cavitations width for c1= 10?4M, E = 5 · 102V/m, ws= ?50 mV, H = 0.4 lm.
123456789
0.95
0.955
0.96
0.965
0.97
0.975
0.98
0.985
D/w
Qcavitation/Qsmooth
Fig. 11. Flow rate variation with the cavitations intervals for c1= 10?4M, E = 5 · 102V/m, ws= ?50 mV, H = 0.4 lm.
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3% for the current situations. The cavitations decrease the flow rate in the electro-osmotic channel flows,
which is different from the reported results in previous pressure driven flows [51], however, is consistent with
the atomistic simulation results very recently [53]. The reason may lie in that the cavitations decrease the elec-
tric driving force more than they decrease the surface resistance for electro-osmotic flows.
The cavitations width and interval effects on the flow rate are shown in Figs. 10 and 11. In Fig. 10, the cav-
itations are deep enough, h = 0.3H, to eliminate the depth effect on the flow rate and the cavitations interval
D = 7H/4. The results in Fig. 10 indicate the flow rate decreases with the cavitations width. In Fig. 11, both
the cavitations width and depth are fixed at H/4, and the interval space changes from one width to nine width.
The results indicate that the sparser the cavitations are, the larger the flow rates are and closer the flow rate is
to the value of a smooth channel. Comparing Fig. 11 with Fig. 6 shows the cavitations effect on the flow rate is
much smaller than the roughness effect.
4. Conclusions
The Lattice Poisson–Boltzmann method, which combines two lattice evolution methods solving the non-
linear Poisson equation for electric potential distribution and solving the Navier–Stokes equations for fluid
flow, were used to simulate the electro-osmotic flows in rough microchannels. The boundary conditions were
correctly treated for consistency between both. The roughness and cavitations effects on the flow behaviors
were therefore analyzed with various shapes and arrangements.
For electro-osmotic flows in homogeneously charged rough channels, the flow rate does not vary with the
roughness height or the interval space monotonically. The flow rate varies slightly with the roughness height
when the roughness is very small, and then decreases if the roughness height is greater than 5% of the channel
width. The flow rate decreases first and then increase with the roughness interval space. An interval space at
twice roughness width makes the flow rate minimum. For the heterogeneously charged rough channel, the
flow rate increases with the roughness surface potential at a superlinear rate. For the electro-osmotic flows
in cavitational microchannels, the flow rate almost does not change with the cavitations depth when the depth
value is very low and decreases sharply when the depth is greater than 3% the channel width. The flow rate
trends to be a constant when the cavitations are very deep. The flow rate decreases with the cavitations width
while increases with the cavitations interval. The sparser the cavitations are, the closer the flow rate is to that
for the smooth channel case. Since some interesting and anomalous flow behaviors predicted in this contribu-
tion have not been reported and validated, we will analyze the flow mechanism further using multi-scale sim-
ulation in the future.
Since the electro-osmotic flows in microchannels have many promising applications in microsystems, the
analyzed methods and simulation results presented here provide valuable information for the design and opti-
mization of MEMS/NEMS.
Acknowledgment
The present work is supported by the US NSF-0613085.
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