Transverse electrokinetic and microfluidic effects in micro-patterned channels: lubrication analysis for slab geometries

9 Pages·0.19 MB·English

by  Armand Ajdari

Transverse electrokinetic and microfluidic effects in micro-patterned channels:
lubrication analysis for slab geometries
Armand Ajdari
Laboratoire de Physico-Chimie Th´eorique, Esa CNRS 7083, ESPCI, 10 rue Vauquelin, F-75231 Paris Cedex 05, France
submitted to Phys. Rev. E. dec. 2000
Off-diagonal(transverse)effectsinmicro-patternedgeometriesarepredictedandanalyzedwithin
1
thegeneralframeoflinearresponsetheory,relatingappliedpresuregradientandelectricfieldtoflow
0
andelectriccurrent. Theseeffectscouldcontributetothedesignofpumps,mixersorflowdetectors.
0
2 Shapeandchargedensitymodulationsareproposedasameanstoobtainsizeabletransverseeffects,
as demonstrated by focusing on simple geometries and using thelubrication approximation.
n
a
PACS: 47.65.+a ; 83.60.Np, 85.90.+h
J

9
2 I. INTRODUCTION

]
t The development of microfluidic devices and studies
f
o has prompted the quest for various strategies to achieve y x z
θ
s pumpinginmicro-geometries[1–7]. Pressuredrivenflows e2
.
t is the first obvious possibility, with the inconvenience of
a e1
important Taylor dispersion due to the parabolic flow
m
profiles [8]. Electro-osmosis has been proposed and de-
- FIG.1. Slabgeometryconsideredinthispaper: Twopar-
veloped as a way to generate almost perfect plug-flows,
d
allel walls bear on their inner faces periodic micro-fabricated
n thereby reducing dispersion in various devices, which re-
patterns of principal axes x and y. The slab can be limited
o sultsinlimiteddilutionofsamples,andprocessabilityfor
c separation purposes [7]. This solution implies relatively sideways by walls, theaxis of theresulting capillary e2 at an
[ angle θ with the pattern. From section IV on, we focus on
high voltages applied between the ends of the channels.
patternsperiodic along x and invariant along y.
1 In this paper, micro-patterned channels are proposed
v as a way to generate a large class of effects using pres-
8 sure gradientsor electric fields. In particularvarious off- plicit realization is described and estimates for the ef-
3 diagonal effects can be obtained in which a cause along fects given. The calculations are performed using the
4
one direction leads to a measurable or useful effect in a lubrication approximation, and assuming weak surface
1
perpendiculardirection. Theseeffectscouldbe exploited potentials, which gives a convenient analytic handle on
0
1 for the realization of transverse pumps, mixers, flow de- these complex systems. Results for sinusoidal modula-
0 tectors,etc... Aproposedpatternistheperiodicmodula- tions of the shape and charge densities are reported in
/ tionoftheshapeofthechannels,whichcanbe improved Appendix 2. A brief discussion closes the paper (section
t
a by a combined modulation of the surface charge density. V), pointing out directions for future studies.
m (in line with an earlier study that dealt with transverse
- electro-osmosis on such surfaces [9]). The aim is here to
d
explore the ensemble of transverse effects achievable. II. LOCAL LINEAR RESPONSE
n
To reach this goal, the linear response regime is con-
o
c sidered,which permits the use of a verygeneralOnsager We consider the Hele-Shaw quasi-planar geometry of
: formalism, to relate fields (pressure and electric poten- Figure 1, where the two sides of the slab bear patterns
v
i tial gradients) to fluxes (hydrodynamic flow and electric of principal axes (x,y), and set the formalism relating
X current). For the sake of clarity, we further restrain our locally 2D fields to 2D fluxes (i.e. integrated over z).
r analysis to a simple class of geometries, with the fluid
a confinedbetweentwoparallelplates,homogeneouslyand
periodically patterned (Fig. 1). Parallel walls may be A. In the principal axes (x,y)
present so as to form a capillary.
Thegeometryandthematrixrepresentationofthelin-
In the linear response regime, the 2D currents (hydro-
earOnsagerformalismareproposedinSectionII.Insec- dynamic flow J and electric current Jel) are related to
tion III the off-diagonal effects are described at a phe-
the pressure gradient ∇p and to the electrostatic poten-
nomenological level, for the case of passive walls (a few tial gradient ∇φ=−E, by a matrix equation [11]:
situations where electrodes are embedded in the walls
are considered in Appendix 1, and the corresponding J=Kˆ.(−∇p)+Mˆ.(−∇φ) (1)
wall generated effects analyzed). In section IV an ex- Jel =Mˆ.(−∇p)+Sˆ.(−∇φ). (2)
1
where the matrices Fˆ with F equal to K, M or S are
diagonal in the (x,y) basis:
Fˆ = Fx 0 (3)
(cid:18)0 Fy (cid:19)
The permeation matrix Kˆ describes the flow induced by e2
e1
pressuredifferences(theDarcylawinaporousmedium).
The conduction matrix Sˆ relates the electric current to
the electric field (the medium’s Ohm’s law). The ma- y x
trix Mˆ describes the electro-hydrodynamiccoupling (so- θ
calledelectrokineticeffects). Inequation(1)itquantifies
electro-osmosis, i.e. the hydrodynamic flow induced by
the electric field. This effect stems from the presence of
thin diffuse layers in the vicinity of charged walls where
the fluid is non-neutral, and thus dragged by the local
electric field. In (2) Mˆ measures the electric current in- FIG.2. Top view of theslab geometry of Figure 1
duced by the presence of a net hydrodynamic flow. This
isduetotheconvectivetransportoftheabovementioned where the 2x2 matrices K, M, S, k, m and s are now
chargedlayerwhichleadsto anioniccurrent. The result non-diagonal and given by F = R .Fˆ.R−1, where R is
θ θ θ
ishydrodynamicallygeneratedelectrostaticpotentialdif- the rotation matrix between (x,y) and (e ,e ).
1 2
ferences (“streaming potentials”) and electric currents
(“streaming currents”). cosθ −sinθ
R = (10)
Equivalently, the inverse system (the resistance ma- θ (cid:18) sinθ cosθ (cid:19)
trix), gives the fields as functions of the currents:
so that the generic matrix
−∇p=kˆ.J+mˆ.Jel (4)
−∇φ=mˆ.J+ˆs.Jel. (5) F= F11 F12 (11)
(cid:18)F21 F22 (cid:19)
wherethematriceskˆ,mˆ andˆs,arediagonal2x2matrices
is symmetric and given by:
of the form (3) with, for i=x,y :
F cos2θ+F sin2θ (F −F )sinθcosθ
ksi ==KSi//∆∆i,,m∆i==−KMSi/−∆Mi 2 ((67)) F=(cid:18) (Fxx−Fy)sinyθcosθ Fyxcos2θy+Fxsin2θ (cid:19) (12)
i i i i i i i
III. NON-DIAGONAL EFFECTS
B. In an arbitrary (e1,e2) coordinate system
Non diagonal or transverse effects (a cause along e
1
All the above is standard in the 1D geometry of
inducesaneffectalonge orthe reverse)occurifsomeof
2
cylindrical capillaries, or for homogeneous and isotropic
thelocaltransversecoefficientsK ,M ,orS arenon-
12 12 12
porous media, where K, S and M are scalars. Here, we
zero. From the above geometric formula, this requires
explore the phenomena occurring in the present 2D ge-
K 6=K ,M 6=M ,orS 6=S ,i.e. thattheanisotropy
x y x y x y
ometrywith non-equivalentpropertiesalongthe xandy
in the pattern of the plates has translated into different
axes, allowing for the existence of off-diagonal effects. If
susceptibilities along the two principal axes x and y.
thepatternisdisposedatanangleθ tothemeasurement
We postpone to section IV a description of a way to
geometry,theabovelinearrelationsarebestexpressedin
achievethisasymmetry,andstartwithagenericdescrip-
thenewsystemofcoordinatesrelatedtothebasis(e ,e )
1 2 tion of the effects expected if local anisotropyis present.
(Figs. 1 and 2). This leads to 4x4 matrices describing
the generalized conductance of the system:
A. Open homogeneous geometries
J K M −∇p
= . (8)
(cid:20) Jel (cid:21) (cid:20)M S(cid:21) (cid:20) −∇φ(cid:21)
In anopen homogeneousgeometry(no walls in Figure
and its generalized resistance: 1), the two simplest situations are the following:
No pressure differences - Then an electric field E
−∇p k m J generates an electric current Jel = S.E and an electro-
= . (9)
(cid:20)−∇φ(cid:21) (cid:20) m s (cid:21) (cid:20) Jel (cid:21) osmoticflowJ=M.E,atfiniteangleswithrespecttothe
2
applied field E, if S and M are non-diagonal. Explicit
ways to obtain non-diagonal electro-osmotic flows were
proposed in [9], and are revisited in section IV within a
∆φ
simple lubrication picture.
R 1
No electrostatic potential differences - If all poten-
tial differences are short-circuited by an appropriate de-
sign of electrodes connected by low-resistance wiring,
then a pressure gradient creates not only a hydrody-
namic flow J = K.(−∇p), but also an electric current L
Jel =M.(−∇p) throughthe wiring circuit. This current
in a simple 1D geometry is called the streaming current.
Bothhydrodynamicandelectriccurrentsareagaingener- J J
el2 2
ically not aligned with the applied pressure gradient.
As this remainsformal,geometriesarenowconsidered
where the fluid is confined to a rectangular channel.
e2
e1
B. Channel with passive walls
d
TakeachanneloflengthLalonge2 andwidth dalong FIG. 3. Passive walls: Neither hydrodynamic flow nor
e1,boundedbyimpermeablewalls. Wesupposeherethat electric current is possible along the e1 axis. A potential dif-
no electric current can flow from one side to the other, ference ∆φ1 can be measured using a high input impedence
sobothfluxesindirectione1 areonaveragezero: J1 =0 voltmeter.
andJ =0. Potentialdifferencesbetweenthe twowalls
el1
can however be measured using a set of electrodes con-
(−∆p /d)=k J =(k /k )(−∆p /L) (15)
nected by a high resistance in series with a voltmeter 1 12 2 12 22 2
(Fig. 3). In this geometry it is straightforward to quan- (−∆φ1/d)=m12J2 =(m12/k22)(−∆p2/L) (16)
tify the effects generatedbya forcingalongthe lengthof
The pressure difference ∆p exists if k 6= 0, in which
the channel (direction e ). 1 12
2
case it indicates that recirculation is taking place in the
e direction: the pressure drop along e entrains fluid in
1 2
the e direction, which, due to the presence of the walls,
1. Pressure-driven effects 1
then has to recirculate. A certain amount of shearing
and mixing is thus induced.
Supposeapressuredrop∆p isappliedalongthechan-
2 The electric potential difference ∆φ is a “transverse
nel so that the average value of ∂ p is ∆p /L. We fur- 1
2 2 streaming potential “ proportional to the non-diagonal
thermoreconsiderthatthereisnoexternalelectricalpath
electrokinetic coefficient m . It is a transverse elec-
connectingthe twoendsofthechannel(inparticularthe 12
tric measure of the hydrodynamic flow along e . Note
fluid flow does not carry in or out a net current - no 2
that, even if the local off-diagonal coupling m /k is
streaming current carried by the convection of the neu- 12 22
large,this potentialdropisintrinsicallysmallerthanthe
tral fluid at the inlet and outlet of the channel-) so that
streaming potential along the channel by a factor d/L
J =0: anystreamingcurrent(alongthe walls)iscom-
el2 due to the geometry of the system.
pensated by a back-current in the bulk.
The application of the pressure drop ∆p then results
2
in longitudinal effects along e : a fluid flow J and a
2 2 2. Electrically-driven effects
potential drop ∆φ given by:
2
(−∆p2/L)=k22J2 (13) SupposenowthatanelectriccurrentJel2,oranaverage
electric field E , is applied along a channel connected to
(−∆φ /L)=m J (14) 2
2 22 2 large open reservoirsso that ∆p =0.
2
This yields classical effects along the channel, which
The potential drop ∆φ =(m /k )∆p is the classical
2 22 22 2
are simply Ohm’s law and the “electro-osmotic flow”: a
“streamingpotential”,proportionalinthesimplestcases
net flow J induced by the electric field:
to the average surface potential of the walls. The flow 2
J is given by a Darcy law modified by electric effects:
2 (−∆p /L)=k J +m J =0 (17)
J =k−1(−∆p /L). 2 22 2 22 el2
2 22 2 (−∆φ /L)=m J +s J =E (18)
More interestingly, transverse effects are also gener- 2 22 2 22 el2 2
ated in the e direction: a pressure difference ∆p and a
1 1 The electro-osmotic flow is thus J = −(m /k )J =
potential drop ∆φ : 2 22 22 el2
1 − m22 E while the resistivity of the channel is
k22s22−m222 2
3
given by E2 =(s22−m222/k22)Jel2. µ µ(x)
Inaddition,transverseeffectsaregeneratedintheper- 2 2
pendicular e1 direction: a pressure difference ∆p1 and a h /2 z h(x) z
potential drop ∆φ . 0 x x
1
m k −k m
12 22 12 22
(−∆p /d)=k J +m J = E (19)
1 12 2 12 el2 s22k22−m222 2 µ µ (x)
s k −m m 1 1
12 22 12 22
(−∆φ /d)=m J +s J = E (20)
1 12 2 12 el2 s k −m2 2
22 22 22
a b
Equation(19)indicatesthat the electric fieldE induces
2
an electro-osmotic flow along e (if m 6=0), which due
1 12
FIG. 4. Geometry considered in Section IV : a) case of
to the presence of the walls, leads to a pressure differ-
auniform flat system, b)theslip parameters describing elec-
ence ∆p thatinduces recirculationofthe fluid alonge .
1 1 trokinetic effects and thethickness vary along x only.
Again this induces shear and favors mixing. This effect
is due both to the non-diagonalpermeation (k ) and to
12
transverse electro-osmosis per se (m12). IV. SPECIFIC CALCULATIONS WITHIN
Equation(20)describesan“electrokineticHalleffect”: LUBRICATION APPROXIMATION
anelectricfieldappliedalonge resultsinanelectrostatic
2
potential difference along e : it consists of a term due
1 A. Geometry and approximations
to the anisotropic conductivity (s ) and to transverse
12
streaming current (m ).
12
In this section, modulating the shape and the charge
densityontheplatesisproposedasanefficientpatterning
to obtain transverse effects. Explicit calculations of the
C. Comments
matrixcoefficientsintroducedinsectionIIareperformed,
which provide an estimate of the off-diagonal effects as
Many other geometries can actually be envisaged. In
listed in section III. For the analysis to remain simple, I
Appendix 1 we consider situations where electrodes are
focusonthespecificcasewherethechargeand/orsurface
embedded in the walls, which permits, for example, the
pattern is periodic along the direction x and invariant
generationof flow along the channel from transverse po-
along the perpendicular direction y. To set notations,
tential differences between the walls. Let us stress here
the channelhas a local thickness h(x)=h +δh(x), and
a few general points: 0
modulated charge densities σ (x) on the bottom plate
-Evaluationoftheeffectiseasyonlyifthegeometry(im- 1
andσ (x)onthetopone(Figure4). Further,thefollow-
posed boundary conditions) permits homogeneous solu- 2
ing assumptions are made:
tions for the gradients and currents. Otherwise one has
(i) the modulation wavelength is larger than the gap
to solve the 2D conservation equations for the currents.
so that lubrication approximationholds [10].
- This obviously would also be the case with heteroge-
(ii) the ionic strength is high enough for the typical
neous patterns on the plates. We will return to this in
gapthicknessh tobemuchlargerthantheDebyelength
the discussion section. 0
λ =κ−1,
- A more serious (less obvious) problem has to do with D
(iii) the surface potentials (or charge densities) are
electrodes: the specificities of electrochemistry at their
weak enough for the double layers to be well described
surface may require a description in terms of the fluxes
by the Debye-Hu¨ckel theory. All effects are calculated
of the various ions rather than the simple description in
to first order in these surface charge densities, so that
termsofanelectriccurrentofAppendix1,inadditionto
terms proportional to products of σ are neglected alto-
themanyexperimentalproblemsinvolved(surfacecorro- i
gether. This restriction comes in addition to (and is dis-
sion, generation of gas bubbles, etc ...).
tinct from) the fact that we focus on the linear response
- In the channel geometries considered here, modifying
of fluxes to applied fields.
orcontrollingthe electricalconnectionbetweenthewalls
Althoughtheseassumptionslimitthequantitativepre-
affects the longitudinal coefficients K ,M ,S (if off-
22 22 22 cision of the results, they permit analytic calculations
diagonal coefficients are non-zero).
and thus a clear discussion of important qualitative fea-
Clearly several coefficients describe the various cou-
tures at a reasonably simple level.
plingsallowedbysymmetryinthislineartheory. Togive
quantitative estimates for these coefficients and thus for
the corresponding effects one has to deal with notori-
B. Flat and uniform surfaces
ously difficult (even at low Reynolds number) electro-
hydrodynamic calculations. However a useful guide can
To get a physical insight, it is useful to start with
be developed using the lubrication approximation, as
the simple geometry of uniform flat surfaces (Fig. 4a):
shown in the next section.
4
h(x)=h0,σ1 andσ2 constant. Thecomputationofelec- J = h3(x).(−∂ p(x))− h(x)µ1+µ2(x).E (x) (23)
x x x
trokinetic effects is then simple (given (ii) and (iii) [12]), 12η 2
and can be found in textbooks (e.g. [11]). µ +µ
1 2
J =−h(x) (x).(−∂ p(x))+σ h(x).E (x) (24)
elx x el x
2
Electro-osmosis - If an electric field E is present in
the channel, it will exert a drag on the thin charged Then,neglectingtermsproportionaltotheproductofσis
Debye layer in the vicinity of the surfaces. The actual (or µis), and performing integrals along x over a period
no-slip boundary condition for the solvent flow on the of the modulation, gives
real surface of the plates can then be replaced by an
eµliec=trσoi-λosDm/oηt,icansdlipηvieslothcietyvivsico=si−tyµoiEf woanteprl.atTehii,swlehaedres (cid:18)hh−−∂∂xxφpii(cid:19)= 1σ2eηl (cid:18)σeµl12(cid:10)+hh3µ132(cid:11) (cid:10)1µ2112η+h3µh12(cid:11)(cid:19)(cid:18)JJxelx(cid:19) (25)
to a simple flow profile v (z) = −µ E(1 −2z/h )− (cid:10) (cid:11) (cid:10) (cid:11)
EO 1 0
µ E(1+2z/h ) and a net electro-osmotic flow through where hfi is the average of the function f(x) over a pe-
2 0
the channel J =−(µ +µ )h E/2. riod. With the same approximations,inversion yields:
EO 1 2 0
Poiseuilleflow- Inadditionthereisnaturallythepres- Jx =∆ 121η h1 − µ12+h3µ2 h−∂xpi
sure driven flow vPoiseuille = (z2 −h20/4)∂xp/2η which (cid:18)Jelx(cid:19) x(cid:18)− µ(cid:10)12+h(cid:11)3µ2 (cid:10)σel h13(cid:11) (cid:19)(cid:18)h−∂xφi(cid:19)
gives a current J =−h3∂ p/12η. (cid:10) (cid:11) (cid:10) (cid:11)
Poiseuille 0 x (26)
Flow-induced current - This pressure-driven flow in- with ∆−1 = 1 1 .
duces transport of the charged fluid of the Debye lay- x h3 h
(cid:10) (cid:11)(cid:10) (cid:11)
ers. If the local shear rate in the vicinity of the i-
th plate is γ˙, the induced current is (per unit length D. Fields and currents along the perpendicular
in the y direction) Jeli = −σiλDγ˙ = −ηµiγ˙. Given direction y
that γ˙ = −h ∂ p/2η, this leads to a total current
0 x
J = (µ +µ )h ∂ p/2. Note that in prin-
elFlow−induced 1 2 0 x I now take fields and flows along the perpendicular
ciple, electro-osmotic flows also induce convective trans-
direction y. In an infinite geometry, the electric field
portofthe chargedregions,butthiseffectisthenclearly
E and the pressure gradient ∂ p are independent of x,
y y
proportional to σ σ and neglected here along (iii).
i j whereasthe currents are modulated alongthis direction.
Ohm’s law - Of course the flow-induced current is of- h3(x) µ +µ
1 2
ten a minor correction to the main conduction current Jy(x)= .(−∂yp)− h(x) (x).Ey (27)
12η 2
described by Ohm’s law J =σ h E.
elOhm el 0 µ +µ
1 2
J (x)=−h(x) (x).(−∂ p(x))+σ h(x).E (28)
ely y el y
2
Gatheringallcontributions,wearethusledtoascalar
(in this isotropic case) version of equations (1) and (2): Averaging over a period along x here gives:
J = h30 .(−∇p)− h0µ1+µ2.(−∇φ) (21) hJyi = 121η h3 − µ1+2µ2h −∂yp
12η 2 (cid:18)hJelyi(cid:19) (cid:18)− µ(cid:10)1+2µ(cid:11)2h (cid:10) σelhh(cid:11)i(cid:19)(cid:18)−∂yφ(cid:19)
µ +µ
1 2 (cid:10) (cid:11)
Jel =−h0 .(−∇p)+σelh0.(−∇φ). (22) (29)
2
The symmetry of the Onsager matrix is here clear. or upon inversion
−∂yp =∆−112η σelhhi µ1+2µ2h hJyi
C. Fields and currents along the modulation (cid:18)−∂yφ(cid:19) y σel (cid:18) µ1+2µ2h (cid:10)121η h3(cid:11)(cid:19)(cid:18)hJelyi(cid:19)
direction x (cid:10) (cid:11) (cid:10) (cid:11)
(30)
Returning to a channel periodically modulated along with ∆−1 = h3 −1hhi−1.
x,Iconsiderfirstthecasewherethefluxesandgradients y
(cid:10) (cid:11)
areappliedalongthisaxis. Inthisgeometry,thecurrents
J andJ areconstants,whereasthelocalvaluesofthe
x elx E. Summary for arbitrary modulations
pressuregradient∂ p(x)andelectricfieldE (x)vary(al-
x x
though slowly enough in the lubrication approximation
We can now write down the local coefficients of the
to consider them independent of z).
responsematrixtomaketheanisotropyin(x,y)explicit.
Inthelubricationpicturewecantranscribetheresults
The coefficient of the permeability matrices (section II)
of the previous subsection:
giving fluxes as functions of gradients are:
5
a
K = 1 1/h3 −1 ; K = 1 h3 (31)
x 12η y 12η
(cid:10) (cid:11) (cid:10) (cid:11) − − −
S =σ h1/hi−1 ; S =σ hhi (32) + +
x el y el
z
µ1+µ2 x
M =− 2h3 ; M =− µ1+µ2h (33)
x (cid:10)1 1(cid:11) y 2
h3 h1 (cid:10) (cid:11) + + h(x)
(cid:10) (cid:11)(cid:10) (cid:11) − − −
The anisotropy appears in a similar form in the “resis-
tance matrix” :
b c d
k =12η 1/h3 ; k =12η h3 −1 (34)
x y
+ + +
1(cid:10) (cid:11) (cid:10) (cid:11) − − −
s = h1/hi ; s = 1 hhi−1 (35)
x σ y σel + + +
el
− − −
6η µ +µ
mx = σ (cid:28) 1h3 2(cid:29) ; my = σ6eηlh(µhh1+ihµh23)ihi (36) + + +
el e
2 x
y
The coefficients describing off-diagonal effects (section E θ E E
III) are then obtained from these matrices expressed in x e y
1
the (e ,e ) frame (using equations (11) and (12)) .
1 2
FIG. 5. Generation of off-diagonal effects (here elec-
tro-osmosis)fromasinusoidalmodulationofshapeandcharge
F. Discussion
densities. The case depicted here (a) corresponds to α > 0,
δµ>0, Θ =π/2, and a zero average charge density µ0 =0.
To investigate semi-quantitativelythe consequencesof A schematic top view is then used in (b),(c), and (d), with
the formulas derived above, a simple but insightful ex- θ = π/4. (b): The field Ex generates a flow along x result-
ample is sinusoidal modulations of the shape and charge ingfromtheoppositeaction ofthenarrowpositivelycharged
density (Figure 5a) sections (tending to slip the fluid backwards along x) and of
thewider negatively charged sections (which inducea slip in
h(x)=h (1+αcos(qx)) (37)
0 the positive direction). The thin sections dominate because
µ +µ =2(µ +δµcos(qx+Θ)). (38) of their stronger hydrodynamic resistance so the net flow is
1 2 0
opposite to the applied field. (c): A field Ey applied along
Analyticalresultstakeasimpleform(seeAppendix2)in
y induces a stratified flow with thin and wide sections again
the limit of weak modulation amplitudes for the shape, pushing in opposite directions. As the field is uniform, the
whichallowstocalculatetheresultthroughanexpansion slip velocities in thetwo directions are of similar amplitudes.
inα. Fromthisandfrominspectionoftheformulaeofthe Thus the net flow is dominated by the wider sections, and
subsectionabove,afewimportantpointscanbederived: points in the direction of the applied field. (d): Thanks to
the linearity of the problem, the net flow created by a field
• a simple modulation of the charge pattern is not
E applied along e2 is obtained by superimposing those ob-
sufficient to induce off-diagonal effects. This is the
tainedin(b)and(c),resultinghereinadominanttransverse
consequenceofthelineardescriptionintermsofthe
component.
surface charge densities and of the +/− symmetry
(see e.g. [9] for a discussion of this)
the specific caseof electro-osmosis. Ashematic de-
• a modulation of the shape at homogeneous charge
scription is given in figure 5.
densityisenoughtoproduceoff-diagonaleffectsfor
all phenomenologies ( permeation k 6=0, electro- • extrapolation of these results to the case α → 1
12
osmosis and streaming potential/current m 6= 0, semi-qualitatively suggests that the off-diagonal
12
conductance s 6= 0). However for a slight modu- componentscanbeofthesameorderthanthediag-
12
lation δh(x) around a mean h , the coupling coef- onalone. Favorableingredientsareamarkedshape
0
ficients will be proportional to (δh/h )2. modulation (δh/h ∼ 1), and a rather contrasted
0
charge modulation (δµ/µ ∼1).
0
• a correlated coupling of the charge pattern and of
the shape leads to stronger O(δh/h ) amplitudes Indeed, the results of Appendix 2 show that the off-
0
fortheoff-diagonalelectro-hydrodynamiccouplings diagonalmatrix coefficients(k12,m12,s12,K12,M12,S12)
m and M . These off-diagonal couplings are are smaller by a factor ∼ α2 than their diagonal equiv-
12 12
proportional to δµ, and dominate the longitudinal alents if only the shape is modulated (δµ = 0). If in
ones if δµ/µ ≫ 1. This synergy between shape addition the charge density is commensurately modu-
0
and charge modulation stems from the shape- lated the off-diagonal electro-hydrodynamic coefficients
induced symmetry breaking between plus and mi- m12 and M12 are increased by a factor αδµµ0 cos(Θ) rela-
nus charges, and is described at length in [9] for tive to their diagonal equivalents.
6
microelectrodes [3,4,14,15]. This great diversity of ge-
ometries imposes a hand-in-hand development of theo-
retical proposals and experimental realizations that we
neutral
wish to pursue in the coming years.
− − + + Acknowledgments: This work owes much to a continu-
− − + + ous interaction with Abe Stroock, and I thank him for his
− − + + permanent support and for many useful suggestions. Re-
− − + + lated discussions with Y. Chen, D. Long, A. P´epin and
P. Tabeling are gratefully acknowledged.
neutral
E
∗
Electronic address: [email protected]
[1] Micro Total Analysis Systems 2000, Eds. A. van den
Berg, W. Olthuis, P. Bergveld, Kluwer Academic Pub-
FIG.6. Asimpleheterogeneousgeometrytocreateavor-
lishers 2000.
tex (top view). No shape modulation, and a simple charge
[2] A.D. Stroock, G. Whitesides, Physics Today in press.
pattern : a positively charged zone aside a a negatively
[3] G. Fuhr, T. Schnelle, B. Wagner J. Micromech. Micro-
charged one (ideally both plates are patterned in the same
eng., 4, 217-226 (1994).
way, i.e. floor and ceiling). The rest of the capillary is neu-
[4] T. Mu¨ller, et al., Electrophoresis, 14, 764-772 (1993).
tral. The electro-osmotic slip generated by an electric field
[5] M.Washizu,IEEEtransactionsonindustryapplications,
along thechannel leads to a recirculation vortex as shown.
34, 732-737 (1998).
[6] B.S. Gallardo et al., Science, 283, 57-60 (1999).
V. CONCLUSION [7] D.J. Harrison et al., Science, 261, 895 (1993).
[8] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal
dispersions, Cambridge University Press, Cambridge
Wehaveshownthatvariousoff-diagonaleffectscanbe
1989.
generated using micro-patterned geometries allowing for
[9] A. Ajdari, Phys. Rev. Lett., 75, 755-758 (1995); A. Aj-
variousfunctionalities: transversepumping,mixing,flow
dari, Phys. Rev. E, 53, 4496-5005 (1996).
detection, etc... The correspondingcouplings canlocally
[10] J. Happel, H. Brenner, Low Reynolds Number Hydrody-
be of same order as the usual (diagonal) electro-kinetic namics, Martinus Nijhoff Publishers, The Hague (1983).
coefficients(sometimeslarger). Theiractualglobalvalue [11] R.J.Hunter,FoundationsofColloidScience,OxfordUni-
depends on the geometry of the given device (through versity Press, New York 1991.
geometricalratios such as e.g. L/d). A detailed analysis [12] J.L.Anderson,Annu.Rev.FluidMech.,21,61-99(1989).
of the flow geometries (variations in the third (z) direc- [13] A.D. Stroock, et al., Phys. Rev. Lett., 10, 3314-3317
tion)andthusofthemixingcapabilitieswillbepublished (2000).
elsewhere. [14] A. Ajdari, Phys. Rev. E, 61, R45-R48 (2000).
Realization of charge-patterned surfaces at the 10- [15] A.B.D. Brown, C.G. Smith, A.R. Rennie, Phys. Rev. E,
micron scale has already been experimentally explored 63, 016305 (2001).
in simple geometries [13]. Its combination with shape
modulations is clearly within the range of current mi-
crofabrication technologies, and is the topic of ongoing
studies.
Naturally, more efficient devices for pumping, mixing
or flow detection should be realizable by playing with a
largerclassofpatterns,includinginparticularinhomoge-
neous ones. A simple example is schematized in Figure
6 where a vortex can be created by application of an
electric field. Additional features or control can result
from the increase of the third dimension : dealing with
thicker fluid layers allows, for example, to produce rolls
with axesparallelto the surface [9,13]. Eventuallyletus
stress that although the focus of the present paper is on
steady-state effects obtained with d.c. fields, interesting
effects are also expected using a.c. fields and arrays of
7
d
In addition, this pressure drop creates transverse ef-
fects: apressuredrop∆p alongthe e axisthatinduces
1 1
recirculation, as well as a transverse streaming current
I =LJ given by:
1 el1
d s k −m m
11 12 11 12
−∆p = ∆p (40)
1 L s k −m2 2
11 22 12
m
12
I =LJ = ∆p (41)
1 el1 s k −m2 2
11 22 12
L
Electrically driven flow - In the case where no pres-
sure drop exists between the two ends of the channel
(∆p =0), effects can be generated by applying an elec-
2
e2 triccurrentJ 6=0(oranelectricfieldE alonge (due
el2 2 2
e1 to the short-circuited walls we have J1 = 0, ∆φ1 = 0).
The applied field creates an electro-osmotic flow along
the channel:
x x
y θ y θ
M K −K M
22 11 21 12
J = E (42)
2 2
K
a b 11
that is different from the case where the walls were not
connected (equation below (18)), if off-diagonal coeffi-
FIG. 7. (a): connected electrodes; (b): the continuous
cients are non-zero. The electric conduction law is here
electrodesshort-circuitpotentialdifferencesalongthechannel J = S22K11−M122E . Transverse electro-osmosis results
el2 K11 2
in a pressure difference ∆p = d(M /K )E and a to-
1 12 11 2
APPENDIX 1: WALLS SHORTCUT BY tal current intensity I1 = LJel1 = LS12K11K−1M112M11E2
CONNECTED ELECTRODES through the connecting wires.
We now consider the geometry of sectionIII, but with
B. Continuous conducting electrodes
the walls perpendicular to e now covered by connected
1
electrodes, so as to shortcut any electrostatic potential
difference between the two walls. Duetothepresenceoftheelectrodes,anypotentialdif-
Ourpicturehereisverysimplisticaswedonotwantto ference between the entranceand the exit ofthe channel
enter the detail of electrochemicalreactions. Let us only isshort-circuited(i.e. abackwardcurrentrunsalongthe
make the following distinction (Figure 7). First, case electrodes in the e2 direction) so that ∆φ2 = 0 (Figure
(a), we will consider that these walls actually consist of 7b).
a series of electrode pairs not connected between them, Pressuredriven flow- Apressuredropalongthechan-
so that although the potential drop is on average zero nel (∆p2 6= 0) in this situation (∆φ2 = 0, J1 = 0,
in the e1 direction, potential differences can nevertheless ∆φ1 =0)againcreatesatransversepressuredropacross
existalongthee2 direction. Inthesecondsimplisticcase thechannel∆p1synonymousofrecirculationalongthee1
(b), the walls are coated with continuous electrodes in axis,aswellasatransverse streaming currentI1 =LJel1
which currents can circulate also along the e direction given by:
2
so that the electrostatic potential is essentially constant
d K
∆φ1 =∆φ2 =0. −∆p1 =− 12∆p2 (43)
LK
11
M K −M K
12 11 11 12
I =− ∆p (44)
A. Series of connected electrodes 1 K 2
11
Transverse electro-osmotic pumping - But we can
Pressure driven flow - In this geometry (Figure 7a)
also consider the situation where an electric current J
(J = 0, J = 0, ∆φ = 0), an applied pressure differ- el1
el2 1 1 is applied between the two walls. Formally, this case
ence along the channel ∆p 6= 0 results in a streaming
2 (∆φ =0,J =0)allowselectro-osmoticpumpinginthe
potential ∆φ that differs from the one obtained with 2 1
2 e direction, even in the absence of a driving pressure
passive walls: 2
gradient in that direction (∆p =0). The resulting flow
2
s m −s m is
11 22 21 12
∆φ = ∆p (39)
2 s k −m2 2
11 22 12 M21K11−K21M11
J =( )E (45)
2 1
K
11
8
or J = (M21K11−K21M11)J . This is simply a manifes- charge is almost zero so that δµ/µ is very large, then
tatio2noftheK1t1rSa1n1s−vMer12s1eelecelt1ro-osmosismentionedinthe in the limit αδµ ≫1, equations (56)0 and (59) become:
µ0
firstsubsectionofthissection. Itallowsthedesignoflat-
eral electro-osmotic pumps that do not require a global 4sinθcosθ
m /m ≃M /M ≃ (60)
potential difference between the entrance and the exit of 12 11 12 11 3cos2θ−sin2θ
the channel(i.e. ∆φ proportionalto L)butratherlocal
2
potential drops (∆φ proportional to d). so that longitudinal effects are totally dominated by
1
transverse ones for a “magic angle” θ =π/3.
APPENDIX 2: SINUSOIDAL MODULATIONS OF
SHAPE AND CHARGE DENSITIES
Weproposehereexplicitformulaeforthevariouscom-
ponents of the conductance and resistance matrices, ob-
tained from the lubrication approximation in section
(IV), and summarized in IV-E, for the particular case
of sinusoidal modulations of the height of the chan-
nel h = h (1 + αcos(qx)) and of the charge densities
0
(µ +µ )/2=µ +δµcos(qx+Θ).
1 2 0
The lubrication approximation requires that αh q ≪
0
1. We here present explicit formulae that correspond to
an expansion in α up to order α2, assuming explicitely
α≪1. This provides a useful guide. The results are for
the conductance matrix:
K = h30 (1−3α2); K = h30 (1+ 3α2) (46)
x 12η y 12η 2
1
S =σ h (1− α2); S =σ h (47)
x el 0 y el 0
2
M =−µ h (1− 3αδµcos(Θ)− 1α2) (48)
x 0 0 2 µ0 2
M =−µ h (1+ 1αδµcos(Θ)) (49)
y 0 0 2 µ0
and for the resistance matrix
12η
k = (1+3α2); k = 12η(1− 3α2) (50)
x h3 y h30 2
0
1 1
s = (1+ α2); s = 1 (51)
x σ h 2 y σelh0
el 0
12ηµ
m = 0(1− 3αδµcos(Θ)+3α2) (52)
x σ h3 2 µ0
el 0
12ηµ
m = 0(1+ 1αδµcos(Θ)− 3α2) (53)
y σ h3 2 µ0 2
el 0
Thus, the ratios of transverse (12) to longitudinal (11
or 22) coefficients are
k /k =k /k ≃ 9α2 sinθcosθ (54)
12 11 12 22 2
s /s =s /s ≃ 1α2 sinθcosθ (55)
12 11 12 22 2
m /m =m /m ≃(−2αδµcosΘ+ 9α2) sinθcosθ (56)
12 11 12 22 µ0 2
K /K =K /K ≃ − 9α2 sinθcosθ (57)
12 11 12 22 2
S /S =S /S ≃ − 1α2 sinθcosθ (58)
12 11 12 22 2
M /M =M /M ≃(−2αδµcosΘ− 1α2) sinθcosθ (59)
12 11 12 22 µ0 2
where terms of order α2(δµ)2 have been omitted in the
µ0
equations for m and M . Actually, if the average
12 12
9