Effective hydrodynamic boundary conditions for microtextured surfaces

1 MB·English

by  Anne Mongruel

Effective Hydrodynamic Boundary Conditions for Microtextured Surfaces
Anne Mongruel,1 Thibault Chastel,1 Evgeny S. Asmolov,2,3,4 and Olga I. Vinogradova2,5,6
1Physique et M´ecanique des Milieux H´et´erog`enes (PMMH),
UMR 7636 CNRS ; ESPCI ParisTech ; Univ. Pierre et Marie Curie (UPMC) ; Univ. Paris Diderot (Paris 7)
10 rue Vauquelin, 75231 Paris cedex 05, France
2A.N. Frumkin Institute of Physical Chemistry and Electrochemistry,
Russian Academy of Sciences, 31 Leninsky Prospect, 119991 Moscow, Russia
3Central Aero-Hydrodynamics Institute, 1 Zhukovsky str., Zhukovsky, Moscow region, 140180, Russia
4Institute of Mechanics, M. V. Lomonosov Moscow State University, 119992 Moscow, Russia
5Department of Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia
3
6DWI, RWTH Aachen, Forckenbeckstr. 50, 52056 Aachen, Germany
1
(Dated: January 11, 2013)
0
2
Understandingtheinfluenceoftopographicheterogeneities onliquid flowshasbecomeanimpor-

n tant issue with the development of microfluidic systems, and more generally for the manipulation
a of liquids at the small scale. Most studies of the boundary flow past such surfaces have concerned
J poorly wetting liquids for which the topography acts to generate superhydrophobic slip. Here we

0 focusontopographically patternedbutchemically homogeneoussurfaces,andmeasureadragforce
1 on a sphere approaching a plane decorated with lyophilic microscopic grooves. A significant de-
crease in the force compared with predicted even for a superhydrophobic surface is observed. To

] quantifytheforceweusetheeffectiveno-slipboundarycondition,whichisappliedattheimaginary
n
smoothhomogeneousisotropicsurfacelocated atanintermediateposition betweentopandbottom
y
ofgrooves. Werelateitslocationtoasurfacetopologybyasimple,butaccurateanalyticalformula.
d
Since groves represent the most anisotropic surface, our conclusions are valid for any texture, and
-
u suggest rulesfor therational design of topographically patternedsurfaces togenerate desired drag.
l
f
. PACSnumbers: 68.08.-p,68.35.Ct
s
c
i
s Introduction.–Theadventofmicrofluidicshasmoti-
y
vated the growing interest in understanding and model-
h
ing of flows at small scales or in tiny channels. In recent
p
[ yearsit has become clear that the no-slipboundary con-

dition at a solid-liquid interface is valid only for smooth
4
hydrophilic surfaces [1–4], and for many other systems
v
2 it does not apply when the size of a system is reduced.
3 Thusthehydrophobicityofsmoothsurfacescouldinduce
5 a partial slippage, v = b∂v/∂z, where v is the velocity
s s
6
at the wall, b the slip length, and the axis z is normal to
.
3 the surface [5]. This concept is now well supported by
0 nanorheology measurements [1, 2, 4].
2
1
However,only veryfew solids are molecularly smooth.
:
v Mostofthemarerough,oftenatamicrometerscale. This
i
X roughness may be induced by some processes of fabrica-
tion or coating, but microtextures are also found on the
r
a surfacesofmostplantsandanimals. Inparticular,many
solidsarenaturallystriatedbygrooves,whichcanalsobe FIG.1: Sketchofasphereapproachingamodelgrooved sur-
prepared for specific microfluidic purposes, such as pas- face (left) with the example of a typical experimental signal
sivechaoticmixing[6,7]. Moststudiesofflowpastrough and a scanning electron micrograph (taken under an angle)
surfaces have concernedpoorly wetting liquids for which of the surface obtained by a soft lithography (right)
thetopographyactstofavortheformationoftrappedgas
bubbles(Cassiestate),andtogeneratesuperhydrophobic
slippage [8, 9]. For rough wettable surfaces the situation rected, but rather for a separation, but not slip [13, 14].
isunclear,andoppositeconclusionshavebeenmade: one Another suggestion is to combine these two models [15].
is thatroughnessgeneratesextremelylargeslip[10], and In this Letter we describe how the boundary condi-
one is that it decreases the degree of slippage [11]. Re- tions can be modified by the surface texture. We focus
cent data (supported by simulations [12]) suggest that on the case of special interest where this model surface
the descriptionofflow nearroughsurfaceshas tobe cor- is decoratedby rectangularmicrogrooves,i.e. onthe sit-
2
uation of the largest possible anisotropy of the flow. We to 0.9), and e/L varies from 0.168 to 0.45, as displayed
analyzethe hydrodynamicinteractionbetweena smooth in Table 1. Contact angles against PDMS for all tex-
sphere and a grooved plane, as sketched in Fig. 1, and tureswerefoundtobebelow30◦,sothatsurfacescanbe
the texture parameters are systematically varied at the considered as lyophilic. Therefore, we expect PDMS to
micrometer level, in order to investigate their influence invade the surface texture (Wenzel state).
on a drag force. Our results do not support some previ- We measure the distance, h, which is defined from the
ous experimental conclusions on a large slip for similar top of the textures (contact) by using an interferometric
systems. Instead, we unambiguously prove the concept technique [16–18] with the accuracy 0.2 µm. The veloc-
of aneffective no-slipplane shifted downfromthe topof ity U(h) of the sphere is found by multiplying the ve-
roughness. To the best of our knowledge,this is the first locity of interference fringes displacement by a factor of
study, where experimentally found values of this shift λ/2n,whereλ=632.8nmisthewavelengthoftheHe-Ne
were quantified theoretically and related analytically to laser,andn=1.404therefractionindexofPDMS.After
controlled parameters of topographic patterns. opto-electronic conversion and amplification, the signal
Experimental.– We use a specially designed home- is recorded with a high frequency electronic oscilloscope
made setup [16–18] to measure on a microscale the dis- (DPO4032fromTektronics). Adecelerationofthesphere
placement of a sphere towards the corrugated wall at (Fig. 1) is reflected in the increase of the period of the
constant gravity force. The steel sphere of density ρ = signal, until contact occurs, and its position is defined
p
7.8×103kgm−3 and radii ranging from 3.5 mm to 6.35 from the recorded signal, at the time when the period
mm is embedded in a liquid contained in a cylindrical of the signal becomes very large indicating a vanishing
glass vessel with a 50mm diameter and a 40mm height. velocity. Note that the signal-to-noise ratio deteriorates
AsaliquidwehavechosenhighmolecularweightPDMS at vanishing frequency, because the low frequency limit
(silicone) oil (47V100000 Rhodorsyl oil, from Rhone- oftheoscilloscopeisreached. Themeasuredfrequencyis
Poulenc.),withdynamic viscosityµ=97.8Pasat25◦C, averagedover 7 to 8 periods, except just before the con-
which is Newtonian for shear rates up to 100s−1. Such tact, where no averaging is applied in order to capture
shear rates are never reached in our experiment. the rapid velocity variations occurring in that region.
Results and discussion.– Fig. 2 shows the drag F
Table 1 : Parameters of the textured samples and the shift (equal to the gravity force) scaled by the Stokes force
of effective hydrodynamic wall, s. FSt = 6πµaU(h), i.e. U(∞)/U(h). Solid line is a the-
N L δ φ e s, expe- s, theory oretical force (Taylor’s equation) predicted for a case of
riment, [Eqs.(4), (6), (7)], smooth wall and no slippage at the surface:
(µm) (µm) (µm) (µm) (µm)
F /F =a/h. (1)
1 100 50 1/2 45 5 ± 0.1 5.5 T St
2 150 50 1/3 45 4.2 ± 0.3 3.5 Alsoincludedaretheexperimentaldataforsampleswith
3 150 100 2/3 45 13 ± 2 11.8 similar e, but different φ and L, which show deviations
4 200 100 1/2 76 13 ± 3 10.4 from the behavior predicted by Eq. 1. Close to the wall,
5 200 100 1/2 45 9 ± 1.5 8.3 for a/h > 50, the drag is always significantly less than
6 250 25 1/10 42 0.5 ± 0.1 0.6 theforcenearasmoothwall,andthisreductionincreases
7 250 225 9/10 42 28.5 ± 0.5 23.5 with φ.
To examine these deviations we evaluate a correction
The microstructured surfaces were created by com- to the drag force,
mon soft lithography, in a three steps process, trans-
f∗(h)=F(h)/F (h) (2)
T
ferring geometric shapes from a mask: first to a sili-
con wafer coated with a (SU8) photoresist, second to a Note that in general case for a rough surface f∗ should
replica molding obtained by soft imprint of a thermo- also depend on the radius of the sphere [12]. However,
reticulable PDMS, and finally to a replica of the PDMS with our configuration geometry experimental data do
mold by soft imprint of a thiolene based resin (NOA 81, not vary with a. This is well illustrated in Fig. 3(a),
Norlandopticaladhesives)onglassmicroscopeslides(to where the experimental values of f∗ obtained with sam-
be fixed at the bottom of the vessel). This resin was ple #7 at different a and plotted as a function of h/L
chosen for its resistance to compression and to solvent- collapse into a single curve, which tends to unity at
swelling,andforitsgoodadhesiontoglass[19]. Thefinal large distances and decreases significantly when h be-
structures are checked by scanning electron microscopy comes of the order of L and smaller. Since at short
(see Fig. 1). The textures are characterized by spacing separations we observe f∗ → 0, one can conclude that
δ, height e and period L. The liquid fraction, φ = δ/L, slippage (which would lead to f∗ → 1/4 [20]) obvi-
can be precisely measured since it is the ratio of the up- ously does not mimic roughness when h is small, by
per surface of the crenellations over the total surface of overestimating the drag force. The same remark con-
the sample. It varies largely with the patterns (from 0.1 cerns effective superhydrophobic slippage where f∗ →
3
1000 1
U) (a)
F(St 500 f*0.5
F/
0 0
0 500 1000 1500 10−3 10−2 10−1 100 101
a/h
h/L
1
FIG. 2: Drag (squares) scaled by the Stokes force for walls
(b)
decorated with grooves of similar heights (e = 42−45 µm), *0.5
f
but different φ and L. From top to bottom the data sets for
samples #6, #2, #5, and #7 (see Table 1). Solid line shows
thetheoretical prediction for asmooth lyophilicwall, Eq.(1), 0
defined at the top of grooves. The dashed curves from top 10−3 10−2 10−1 100 101
h/L
tobottomarecalculations oftheforceexpectedforasmooth
lyophilic wall shifted from the textured wall to a distance
s=0.5,4.2,9 and 28.5 µm.
FIG.3: Measuredcorrectiontodrag(symbols)(a)forspheres
ofdifferentradii(a=3.5,5.75and6.35mm)interactingwith
2(4−3φ)/(8+9φ−9φ2) [21] and is equal to ≃ 0.3 for sample#7. Dashedlineshowsthecalculatedcorrection,using
s=28.5µminEq.(3),(b)forstripedwallshavingnearlysame
this particular sample.
heights e=42−45 µm , but different φ and period L; from
Therefore,our experimentalresults arenow compared
lefttoright: samples#6,2,5,7. Dashedlines: model,Eq.(3),
withtheoreticalcalculationsmadeforaneffectivesmooth
with (a) s = 28.5 µm, (b) from left to right: s = 0.5,4.2,9
plane shifted down from the top of the corrugations,i.e. and 28.5 µm.
by assuming F(h) = F (h+s) where s is the value of
T
constant, i.e. independent on h, shift. This implies that
1
F (h+s) h
f∗(h)= T = . (3)
(a)
F (h) h+s
T s/e0.5
Fig. 3(a) includes a calculation (dashed curve) in which
anadjustableparameter,ashiftofs=28.5µm, isincor-
0
porated into the Taylor equation. The fit is very good 0 0.2 0.4 0.6 0.8 1
φ
for all h, suggesting the validity of the model. Fig. 3(b)
shows another series of experiments made with the fixed 1
radius of the sphere, but different parameters ofthe tex-
(b)
ture. If similar fits are made to a variety of experiments
it is found that the shift of an equivalent plane required s/e0.5
to fit each run increases from 0.5 µm for sample #2 to
28.5 µm for sample #7. In Table 1 we present the ex- 0
0 0.2 0.4 0.6 0.8 1
perimentalvalues of s for different samples,andtheoret- φ
ical curves calculatedwith Eq.(3) are included in Figs. 2
and 3. The fit is excellent at all separations except as
very small, h/L ≤ 0.01. Thus our experiment shows FIG. 4: Experimental values of s (symbols) as a function of
thataneffective(scalar)shift, s,is auniquephysicalpa- φ for grooves with similar heights (e = 42−45 µm), but
rameter that fully quantifies drag reduction at a highly different φ and L. Lines show theoretical predictions, Eq.
anisotropic corrugated surface. This striking result indi- (4),wherebkeff andb⊥eff arecalculated byusingtheanalysisof
[22](solidlines),Eqs.(6)-(7)(dash-dottedlines),andEq.(5)
cates that in our experiment pressure remains isotropic
(dashed lines): (a) samples #6 and #7 (black), samples #2,
despite an anisotropy of the flow.
#3, (grey); (b) sample #5 (black) and sample #1 (grey).
Now we try to relate s to parameters of textured sur-
faces. As proven in [17, 21], for a large gap, h ≫ L, the
shift of the equivalent no-slipplane fromthe realsurface
Therefore, the problem of calculating s reduces to com-
is equal to the average of the eigenvalues of the effective
puting the two far-field eigenvalues, bk and b⊥ , which
slip-length tensor (at the slip plane defined at the top of eff eff
attainthe maximaland minimal directionalsliplengths,
asperities)
respectively.
bk +b⊥ In the limit e ≪ L ≪ h, the theory [23] predicts that
s≃ eff eff. (4) theeffectiveno-slipsurfaceforarbitrarysmoothperiodic
2
4
surfaces is at the average height: Eqs. (6) and (7) should be replaced by analytical or nu-
merical solutions for a corresponding texture, as will be
bk,⊥ ≃φe, (5) described in subsequent papers.
eff
We have also demonstrated that topographically pat-
sothats/eiscontrolledmainlybyφ. Toexaminethesig-
terned (Wenzel) surfaces could reduce a drag force more
nificanceofφmoreclosely,theexperimentalsnormalized
efficiently compared to expected even for slipping super-
by e are reproduced in Fig. 4. The measured data show
hydrophobic (Cassie) textures with trapped gas. There-
muchsmallers/ethanthetheoreticalpredictionofmodel
fore,ourresultssuggestedrulesandageneralstrategyfor
(5) shown by a dashed line. A possible explanation for
the rationaldesignoftopographicallypatternedsurfaces
thisdiscrepancyisthattheheightofasperitiesinourex-
to generate desired low drag.
perimentswasnotsmallenough,0.168≤e/L≤0.45.We
also compared our data with another calculation (solid
curves) for hydrophilic grooves with finite e/L based on
numerical results [22] for eigenvalues of the slip-length-
tensor. Evenatmoderate e/Ltheoreticalpredictions for [1] O.I.VinogradovaandG.E.Yakubov,Langmuir19,1227
s [22] are much smaller than measured values. (2003).
[2] C. Cottin-Bizonne, B. Cross, A. Steinberger, and
Analternativemodelcanbeobtainedifweusetheana-
E. Charlaix, Phys.Rev.Lett. 94, 056102 (2005).
lyticsolutionsforalternatingslipandno-slipstripes[24]:
[3] L. Joly, C. Ybert, and L. Bocquet, Phys. Rev. Lett. 96,
046101 (2006).
b|| ≃ L lnhsec(cid:16)π2φ(cid:17)i , (6) [4] Obo.isI,.PVhinyos.grRaedvo.vLa,etKt..1K0o2y,n1o1v8,3A02. B(2e0s0t,9)a.nd F. Feuille-
eff π 1+ πLelnhsec(cid:16)π2φ(cid:17)+tan(cid:16)π2φ(cid:17)i [5] O.I.Vinogradova,Int.J.MineralProces. 56,31(1999).
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H. A. Stone, and G. M. Whitesides, Science 295, 647
(2002).
ln sec πφ
L h (cid:16) 2 (cid:17)i [7] F. Feuillebois, M. Z. Bazant, and O. I. Vinogradova,
b⊥eff ≃ 2π1+ 2Lπelnhsec(cid:16)π2φ(cid:17)+tan(cid:16)π2φ(cid:17)i (7) [8] JP.hPy.s.RRoetvh.stEein8,2A, 0n5n5.3R01e(vR.)Fl(u2i0d10M).ech. 42, 89 (2010).
[9] O.I.VinogradovaandA.V.Belyaev,J.Phys.: Condens.
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is equal to the height of grooves. Fig. 4 shows that Eq. [10] E.Bonaccurso,H.-J.Butt,andV.S.J.Craig,Phys.Rev.
(4) together with Eqs. (6) and (7) (dashed curves) give Lett. 90, 144501 (2003).
[11] Y. Zhu and S. Granick, Phys. Rev. Lett. 88, 106102
almost quantitative agreement with experimental data.
(2002).
(WealsoincludetheoreticalvaluesofstoTable1toallow
[12] C.Kunert,J.Harting,andO.I.Vinogradova,Phys.Rev.
a direct comparison with experimental results.) There- Lett 105, 016001 (2010).
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thatourresultsdonotapplytoaverythingapsituation
and E. Bonaccurso, Microfluid Nanofluid 8, 653 (2010).
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[19] D.Bartolo,G.Degre,P.Nghe,andV.Struder,Lab.Chip
the imaginary smooth homogeneous isotropic surface lo-
8, 274 (2008).
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[20] O. I. Vinogradova, Langmuir 11, 2213 (1995).
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[21] E. S. Asmolov, A. V. Belyaev, and O. I. Vinogradova,
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ing simple formulae for effective slip lengths in the limit Mech. 658, 409 (2010).
[24] A. V. Belyaev and O. I. Vinogradova, J. Fluid Mech.
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mized,thesameconclusionwouldbevalidforothertypes
[25] See Supplemental Material at [URL will be inserted by
of anisotropic (e.g., sinusoidal, trapezoidal, and more)
publisher]fordetailsofthetheoreticalsolutionforathin
and/orisotropic(e.g. pillars,etc)textures,butofcourse, channel situation.