Adopting the Navier slip conditions, we analyze the fully developed electroosmotic flow in hydrophobic microducts of general cross section under the Debye–Hückel approximation. The method of analysis includes series solutions which their coefficients are obtained by applying the wall boundary conditions using the least-squares matching method. Although the procedure is general enough to be applied to almost any arbitrary cross section, eight microgeometries including trapezoidal, double-trapezoidal, isosceles triangular, rhombic, elliptical, semi-elliptical, rectangular, and isotropically etched profiles are selected for presentation. We find that the flow rate is a linear increasing function of the slip length with thinner electric double layers (EDLs) providing higher slip effects. We also discover that, unlike the no-slip conditions, there is not a limit for the electroosmotic velocity when EDL extent is reduced. In fact, utilizing an analysis valid for very thin EDLs, it is shown that the maximum electroosmotic velocity in the presence of surface hydrophobicity is by a factor of slip length to Debye length higher than the Helmholtz–Smoluchowski velocity. This approximate procedure also provides an expression for the flow rate which is almost exact when the ratio of the channel hydraulic diameter to the Debye length is equal to or higher than 50.

References

References
1.
Kirby
,
B. J.
,
2010
,
Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices
,
Cambridge University
,
New York
.
2.
Reuss
,
F. F.
,
1809
, “
Charge-Induced Flow
,”
Imperial Society of Naturalists of Moscow
, Vol.
3
, pp.
327
344
.
3.
Sadeghi
,
A.
,
Kazemi
,
Y.
, and
Saidi
,
M. H.
,
2013
, “
Joule Heating Effects in Electrokinetically Driven Flow Through Rectangular Microchannels: An Analytical Approach
,”
Nanoscale Microscale Thermophys. Eng.
,
17
(
3
), pp.
173
193
.
4.
Smoluchowski
,
M.
,
1903
, “
Contribution à la théorie l′endosmose électrique et de quelques phénomènes corrélatifs
,”
Bull. Acad. Sci. Cracovie
,
8
, pp.
182
199
.
5.
Helmholtz
,
H.
,
1879
, “
Studien über electrische Grenzschichten
,”
Ann. Phys. Chem.
,
243
(7), pp.
337
382
.
6.
Debye
,
P.
, and
Hückel
,
E.
,
1923
, “
Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen
,”
Phys. Z.
,
24
, pp.
185
206
.
7.
Burgreen
,
D.
, and
Nakache
,
F. R.
,
1963
, “
Electrokinetic Flow in Capillary Elements
,” Technical Report No. ASD-TDR-63-243.
8.
Burgreen
,
D.
, and
Nakache
,
F. R.
,
1964
, “
Electrokinetic Flow in Ultrafine Capillary Slits
,”
J. Phys. Chem.
,
68
(
5
), pp.
1084
1091
.
9.
Rice
,
C. L.
, and
Whitehead
,
R.
,
1965
, “
Electrokinetic Flow in a Narrow Cylindrical Capillary
,”
J. Phys. Chem.
,
69
(
11
), pp.
4017
4024
.
10.
Levine
,
S.
,
Marriott
,
J. R.
,
Neale
,
G.
, and
Epstein
,
N.
,
1975
, “
Theory of Electrokinetic Flow in Fine Cylindrical Capillaries at High Zeta-Potentials
,”
J. Colloid Interface Sci.
,
52
(
1
), pp.
136
149
.
11.
Tsao
,
H.-K.
,
2000
, “
Electroosmotic Flow Through an Annulus
,”
J. Colloid Interface Sci.
,
225
(
1
), pp.
247
250
.
12.
Kang
,
Y.
,
Yang
,
C.
, and
Huang
,
X.
,
2002
, “
Electroosmotic Flow in a Capillary Annulus With High Zeta Potentials
,”
J. Colloid Interface Sci.
,
253
(
2
), pp.
285
294
.
13.
Yang
,
D.
,
2011
, “
Analytical Solution of Mixed Electroosmotic and Pressure-Driven Flow in Rectangular Microchannels
,”
Key Eng. Mater.
,
483
, pp.
679
683
.
14.
Wang
,
C. Y.
,
Liu
,
Y. H.
, and
Chang
,
C. C.
,
2008
, “
Analytical Solution of Electro-Osmotic Flow in a Semicircular Microchannel
,”
Phys. Fluids
,
20
(
6
), p.
063105
.
15.
Wang
,
C. Y.
, and
Chang
,
C. C.
,
2007
, “
EOF Using the Ritz Method: Application to Superelliptic Microchannels
,”
Electrophoresis
,
28
(
18
), pp.
3296
3301
.
16.
Wang
,
C. Y.
, and
Chang
,
C. C.
,
2011
, “
Electro-Osmotic Flow in Polygonal Ducts
,”
Electrophoresis
,
32
(11), pp.
1268
1272
.
17.
Soong
,
C. Y.
,
Hwang
,
P. W.
, and
Wang
,
J. C.
,
2010
, “
Analysis of Pressure-Driven Electrokinetic Flows in Hydrophobic Microchannels With Slip-Dependent Zeta Potential
,”
Microfluid. Nanofluid.
,
9
(
2–3
), pp.
211
223
.
18.
Seshadri
,
G.
, and
Baier
,
T.
,
2013
, “
Effect of Electro-Osmotic Flow on Energy Conversion on Superhydrophobic Surfaces
,”
Phys. Fluids
,
25
(4), p.
042002
.
19.
Park
,
H. M.
, and
Choi
,
Y. J.
,
2008
, “
A Method for Simultaneous Estimation of Inhomogeneous Zeta Potential and Slip Coefficient in Microchannels
,”
Anal. Chim. Acta
,
616
(2), pp.
160
169
.
20.
Park
,
H. M.
,
2010
, “
A Method to Determine Zeta Potential and Navier Slip Coefficient of Microchannels
,”
J. Colloid Interface Sci.
,
347
(1), pp.
132
141
.
21.
Park
,
H. M.
,
2012
, “
Determination of the Navier Slip Coefficient of Microchannels Exploiting the Streaming Potential
,”
Electrophoresis
,
33
(6), pp.
906
915
.
22.
Chakraborty
,
S.
,
2008
, “
Generalization of Interfacial Electrohydrodynamics in the Presence of Hydrophobic Interactions in Narrow Fluidic Confinements
,”
Phys. Rev. Lett.
,
100
(
9
), p.
097801
.
23.
Chakraborty
,
S.
,
Chatterjee
,
D.
, and
Bakli
,
C.
,
2013
, “
Nonlinear Amplification in Electrokinetic Pumping in Nanochannels in the Presence of Hydrophobic Interactions
,”
Phys. Rev. Lett.
,
110
(
18
), p.
184503
.
24.
Yang
,
J.
, and
Kwok
,
D. Y.
,
2003
, “
Analytical Treatment of Flow in Infinitely Extended Circular Microchannels and the Effect of Slippage to Increase Flow Efficiency
,”
J. Micromech. Microeng.
,
13
(1), pp.
115
123
.
25.
Yang
,
J.
, and
Kwok
,
D. Y.
,
2003
, “
Microfluid Flow in Circular Microchannel With Electrokinetic Effect and Navier's Slip Condition
,”
Langmuir
,
19
(4), pp.
1047
1053
.
26.
Yang
,
J.
, and
Kwok
,
D. Y.
,
2003
, “
Effect of Liquid Slip in Electrokinetic Parallel-Plate Microchannel Flow
,”
J. Colloid Interface Sci.
,
260
(1), pp.
225
233
.
27.
Yang
,
J.
, and
Kwok
,
D. Y.
,
2004
, “
Analytical Treatment of Electrokinetic Microfluidics in Hydrophobic Microchannels
,”
Anal. Chim. Acta
,
507
(1), pp.
39
53
.
28.
Park
,
H. M.
, and
Kim
,
T. W.
,
2009
, “
Extension of the Helmholtz–Smoluchowski Velocity to the Hydrophobic Microchannels With Velocity Slip
,”
Lab Chip
,
9
(2), pp.
291
296
.
29.
Lim
,
J. M.
, and
Chun
,
M. S.
,
2011
, “
Curvature-Induced Secondary Microflow Motion in Steady Electro-Osmotic Transport With Hydrodynamic Slippage Effect
,”
Phys. Fluids
,
23
(10), p.
102004
.
30.
Squires
,
T. M.
,
2008
, “
Electrokinetic Flows Over Inhomogeneously Slipping Surfaces
,”
Phys. Fluids
,
20
(
9
), p.
092105
.
31.
Zhao
,
H.
,
2010
, “
Electro-Osmotic Flow Over a Charged Superhydrophobic Surface
,”
Phys. Rev. E
,
81
(
6
), p.
066314
.
32.
Zhao
,
C.
, and
Yang
,
C.
,
2012
, “
Electro-Osmotic Flows in a Microchannel With Patterned Hydrodynamic Slip Walls
,”
Electrophoresis
,
33
(6), pp.
899
905
.
33.
Belyaev
,
A. V.
, and
Vinogradova
,
O. I.
,
2011
, “
Electro-Osmosis on Anisotropic Superhydrophobic Surfaces
,”
Phys. Rev. Lett.
,
107
(
9
), p.
098301
.
34.
Ng
,
C.-O.
, and
Chu
,
H. C. W.
,
2011
, “
Electrokinetic Flows Through a Parallel Plate Channel With Slipping Stripes on Walls
,”
Phys. Fluids
,
23
(10), p.
102002
.
35.
Mas
,
N. d.
,
Gunther
,
A.
,
Schmidt
,
M. A.
, and
Jensen
,
K. F.
,
2003
, “
Microfabricated Multiphase Reactors for the Selective Direct Fluorination of Aromatics
,”
Ind. Eng. Chem. Res.
,
42
(4), pp.
698
710
.
36.
Fan
,
Z. H.
, and
Harrison
,
D. J.
,
1994
, “
Micromachining of Capillary Electrophoresis Injectors and Separators on Glass Chips and Evaluation of Flow at Capillary Intersections
,”
Anal. Chem.
,
66
(1), pp.
177
184
.
37.
Ziaie
,
B.
,
Baldi
,
A.
,
Lei
,
M.
,
Gu
,
Y.
, and
Siegel
,
R. A.
,
2004
, “
Hard and Soft Micromachining for BioMEMS: Review of Techniques and Examples of Applications in Microfluidics and Drug Delivery
,”
Adv. Drug Delivery Rev.
,
56
(2), pp.
145
172
.
38.
Shah
,
R. K.
,
1975
, “
Laminar Flow Friction and Forced Convection Heat Transfer in Ducts of Arbitrary Geometry
,”
Int. J. Heat Mass Transfer
,
18
(
7–8
), pp.
849
862
.
39.
Tamayol
,
A.
, and
Hooman
,
K.
,
2011
, “
Slip-Flow in Microchannels of Non-Circular Cross Sections
,”
ASME J. Fluids Eng.
,
133
(9), p.
091202
.
40.
Cervera
,
J.
,
García-Morales
,
V.
, and
Pellicer
,
J.
,
2003
, “
Ion Size Effects on the Electrokinetic Flow in Nanoporous Membranes Caused by Concentration Gradients
,”
J. Phys. Chem. B
,
107
(
33
), pp.
8300
8309
.
41.
Sadeghi
,
A.
,
Yavari
,
H.
,
Saidi
,
M. H.
, and
Chakraborty
,
S.
,
2011
, “
Mixed Electroosmotically and Pressure-Driven Flow With Temperature-dependent Properties
,”
J. Thermophys. Heat Transfer
,
25
(
3
), pp.
432
442
.
42.
Karniadakis
,
G.
,
Beskok
,
A.
, and
Aluru
,
N.
,
2005
,
Microflows and Nanoflows: Fundamentals and Simulation
,
Springer
,
New York
.
43.
Papadopoulos
,
P.
,
Deng
,
X.
,
Vollmer
,
D.
, and
Butt
,
H.-J.
,
2012
, “
Electrokinetics on Superhydrophobic Surfaces
,”
J. Phys. Condens. Matter
,
24
(46), p.
464110
.
44.
McLachlan
,
N. W.
,
1961
,
Bessel Functions for Engineers
,
Clarendon
,
London
.
45.
Yang
,
R. J.
,
Fu
,
L. M.
, and
Hwang
,
C. C.
,
2001
, “
Electroosmotic Entry Flow in a Microchannel
,”
J. Colloid Interface Sci.
,
244
(
1
), pp.
173
179
.
46.
Suli
,
E.
, and
Mayers
,
D. F.
,
2003
,
An Introduction to Numerical Analysis
,
Cambridge University
,
New York
.
47.
Golub
,
G. H.
,
1965
, “
Numerical Methods for Solving Linear Least-Squares Problem
,”
Numer. Math.
,
7
(3), pp.
206
216
.
48.
Tretheway
,
D. C.
, and
Meinhart
,
C. D.
,
2002
, “
Apparent Fluid Slip at Hydrophobic Microchannel Walls
,”
Phys. Fluids
,
14
(
3
), pp.
L9
L12
.
49.
Ren
,
Y.
, and
Stein
,
D.
,
2008
, “
Slip-Enhanced Electrokinetic Energy Conversion in Nanofluidic Channels
,”
Nanotechnology
,
19
(19), p.
195707
.
50.
Yan
,
Y.
,
Sheng
,
Q.
,
Wang
,
C.
,
Xue
,
J.
, and
Chang
,
H.-C.
,
2013
, “
Energy Conversion Efficiency of Nanofluidic Batteries: Hydrodynamic Slip and Access Resistance
,”
J. Phys. Chem. C
,
117
(
16
), pp.
8050
8061
.
You do not currently have access to this content.