Wicking through a confined micropillar array

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by  Baptiste Darbois Texier

Noname manuscript No.
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Wicking through a confined micropillar array
Baptiste Darbois Texier · Philippe Laurent · Serguei Stoukatch · St´ephane Dorbolo
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Received:date/Accepted:date
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Abstract ThisstudyconsidersthespreadingofaNew- etal(2007)studiedexperimentallyhowawettingliquid

5 tonian and perfectly wetting liquid in a square array of propagates along solid surfaces decorated with a forest
2
cylindric micropillars confined between two plates. We of micropillars. This problem leads to numerous exper-

] show experimentally that the dynamics of the contact imentalderivations(Kimetal(2011);Maietal(2012);
n
line follows a Washburn-like law which depends on the Spruijtetal(2015);XiaoandWang(2011)),numerical
y
characteristics of the micropillar array (height, diam- contributions (Semprebon et al (2014)) and theoretical
d
- eter and pitch). The presence of pillars can either en- modelizations (Hale et al (2014a,b)). In the case of mi-
u
hanced or slow down the motion of the contact line. A crochannels,studieshaveinspectedtheeffectofsurface
l
f theoretical model based on capillary and viscous forces patterned by posts or pillars on the dynamics of liquid
.
s has been developed in order to rationalize our observa- insidethemicrochannel(Gamratetal(2008);Mognetti
c
i tions. Finally, the impact of pillars on the volumic flow andYeomans(2009);MohammadiandFloryan(2013)).
s
rate of liquid which is pumped in the microchannel is Besides, the liquid spreading into a confined micropil-
y
h inspected. lararrayisinvolvedinvariousapplicationssuchaslab-
p on-a-chip chromatography (Op de Beeck et al (2012)),
[ Keywords Microchannel wicking · micropillar array ·
isolation of tumor cells (Nagrath et al (2007)) or flow
liquid impregnation
2 regulation in microfluidic (Vangelooven et al (2010)).
v If the problem of liquid impregnation in a micropillar
6
array with a free surface has been extensively studied
1
Introduction
9 Bocquet and Barrat (2007), the effect of confinement
7 on this phenomenon is still an open question that we
0 Wicking can be defined as the spreading of a liquid address in this article.
. in the tiny structures of a porous media due to capil-
1
0 lary forces. The pioneer work concerning the spreading
6 of a viscous liquid in a capillary tube is attributed to
1 Washburn(1921).Thewickingphenomenonhasalarge
:
v spectra of applications such as textile science (Kissa
i (1996))orheatpipedesign(TienandSun(1971)).The
X Inthisworkweconsiderthecaseofasquarearrayof
problemwasrevisitedbecauseofmicroelectronicscom-
r cylindric micropillars confined between two plates. We
a ponents bonding (Schwiebert and Leong (1996)). One
study experimentally the spreading of a perfectly wet-
way to model the wicking process is to consider the
ting liquid as a function of the pillars characteristics
spreading of a liquid into a micropillar array. Ishino
(height, pitch, diameter). The measurements are com-
paredwithatheoreticalmodelbasedoncapillaryforces
B.DarboisTexier ·S.Dorbolo
Grasp,PhysicsDept.,ULg,Li`ege,Belgium andviscousresistance.Finally,theconsequencesofthis
E-mail:[email protected] work for capillary pumping is discussed. In addition,
P. Laurent·S.Stoukatch more complex micropillars arrangements are examined
Microsys,MontefioreInstitute,ULg,Li`ege,Belgium such as non-square and non-uniform lattices.
2 BaptisteDarboisTexier, PhilippeLaurent,SergueiStoukatch,St´ephane Dorbolo
(a)
Fig.1 Sketchoftheexperimentalsetup.Thechannelheight
is h, its width is (cid:96). The channel contains an array of pillars
ofdiameterdspacedbyapitchp.Theheightofthepillarsis
thesamethattheoneofthechannel.Thedynamicviscosity
of the fluid which spreads in the channel is denoted η. The
meanpositionofthecontactlineis x.
(b)
1 Experiments Fig.2 Superimpositionoftop-viewpicturesofliquidimpreg-
nationofmicrochannelsovertime.Thetimestepbetweentwo
frames is 1 s. The fluid is Newtonian silicone oil V100. (a)
Microchannels were obtained by negative SU-8 pho-
Empty microchannel of height h=80µm. (b) Microchannel
tolithography made on a silicon wafer (Del Campo and ofheighth=80µmfilledwithpillarsofdiameterd=400µm
Greiner (2007)) prior to a PDMS molding (Folch et al separatedbyapitch p=800µm.
(1999)).ThePDMScavitywasbondedonasubstrateof
thesamematerialbyoxygenplasmaexposure.Byvary-
16
ing the design of a the mask used in the photolithogra-
phy,wechangedthegeometryofthemicropillarsarray 14
inside the channel [Fig. 1]. The channel height h could 12
be changed between 50µm and 100µm, the pillar di-
)10
ameter d from 80µm to 1000µm and the lattice pitch m
m 8
pbetween150µmand1000µm.Thus,thepillaraspect ( 250
ratio h/d may vary between 0.05 and 1.2 and the ra- x 6 200
2m)150
tio p/d between 1.5 and 10. The transverse dimension 4 x2(m100
of the channel (cid:96) was chosen in order to keep the ratio 50
2
(cid:96)/h larger than 100. These PDMS microchannels were 00 10 20 30 40 50
0 t(s)
put into contact with a liquid reservoir containing a 0 10 20 30 40 50
Newtonian silicone oil V100 of density ρ = 980kg/m3, t(s)
dynamic viscosity η = 100mPa·s and surface tension Fig. 3 Mean position of the contact line x as a function of
γ (cid:39) 23±0.3mN/m at a temperature T = 20◦C. The time t for a silicone oil V100 (ρ=980kg/m3, γ =23mN/m
and η = 100mPa·s) which spreads in two different mi-
mean position of the contact line was recorded from
crochannels.Eachchannelhasthesameheighth=80µmbut
above and reported in Fig. 2. The analysis of these ex-
a different density of pillars. The blue dashed line shows the
periments provides the position of the contact line x case of an empty microchannel and the red solid line stands
over time as shown in Fig. 3. foramicrochannelwithpillarsofdiameterd=400µmsepa-
rated by a pitch p = 800µm. The inset shows the square of
Oneobservesthat,whatevertheconsideredmicrochan-
the mean contact line position x2 as a function of time for
nel,themeanpositionofthecontactlinepropagatesac-
thetwopreviousexperiments.
cording to the square root of time. This dependency is
underlined by the linearity of x2 with time as shown in
the inset of Fig. 3 and corresponds to a Washburn-like Thereafter,theeffectofthemicropillararrayonthe
law expected for the spreading of a viscous fluid in- spreading dynamics is quantified. We measure the pre-
√
duced by capillary forces (Washburn (1921)). Besides, factorDofthesquarerootlawx= Dtformicrochan-
one notices the influence of the pillars array on the dy- nels as a function of the lattice properties (pillar diam-
namicsofthecontactlinebyachangeofthepre-factor eter d, height h and pitch p). This pre-factor defines
ofthesquarerootlaw.Intheconsideredcase,thepres- the diffusivity of the liquid in the micro-channel and is
ence of pillars in the channel slows down the liquid dy- measured by fitting the slope of x2(t) by the way of
namics. Additionally, the introduction of pillars in the a least mean squares method. The analogous diffusiv-
microchannel induces a stick-slip behavior in the dy- ity of the liquid in the empty microchannel is denotes
namics of the contact line which appears by the way of D . Symbols in Fig. 4a and 4b show the evolution of
0
steps in the red curves in Fig. 3. the experimental ratio D/D as a function of the pillar
0
Wickingthroughaconfinedmicropillararray 3
density φ=πd2/4p2 where d and h are kept constants ence of pillars enhances the spreading of the liquid in
andonlypisvaried.Twosetsofexperimentshavebeen the channel.
done for a different pillar aspect ratio of h/d=0.2 and
h/d=1 and are plotted with dots.
2 Model
(a)
2.1 Empty cavity
1 Thegoalofthissectionistodevelopatheoreticalmodel
in order to rationalize the experimental observations.
0.8
First, we evaluate the dimensionless numbers associ-
ated to previous experiments in order to determine the
0.6
DD0 mainforcesinpresence.Accordingtoexperimentaldata
0.4 presented in Section 1, the spreading velocity reaches
x˙ ∼1mm/s in less than 20 ms. This allows to estimate
0.2 theReynoldsnumberassociatedtothefluidflowatthis
particulartime:Re=ρhx˙/η ∼103×10−4×10−3/0.1∼
00 0.2 0.4 0.6 10−3.Therefore,theliquidflowisdominatedbyviscous
φ
frictionandnoinertialtermswillbetakenintoaccount
(b)
in this study. The experimental Bond number, which
compares the relative effect of gravity and surface ten-
1 sion, is Bo = ρgh2/γ ∼ 103×10×10−8/10−2 ∼ 10−2.
Also, we estimate the capillary number Ca = ηU/γ (cid:39)
0.8
0.1×10−3/2×10−2 (cid:39) 5×10−3. The small value of
the capillary number compared to one signifies that
0.6
DD0 thecontactangleremainsclosetoitsequilibriumvalue
0.4 (Thompson and Robbins (1989)). Finally, the experi-
mental data presented in Section 1 will be approached
0.2 byamodelwhichbalancescapillaryandviscousforces.
In the case where the microchannel is empty, the
0
0 0.2 0.4 0.6 surface wetted by the fluid at a positon x is equal to
φ
S =2xl (neglectingthechannelheighthwhichismore
Fig. 4 Diffusivity ratio D/D0 as a function of the pillars than 100 times smaller than its width (cid:96)). The resulting
density φ = πd2/4p2 for two different pillar aspect ratios
capillaryforceinthexdirectionisf =γ∂S/∂x=2γl.
h/d. Dots correspond to the flow of silicone oil V100 (ρ = γ
980kg/m3, γ = 23mN/m and η = 100mPa·s) into micro- The non-slipping boundary conditions of the fluid on
channels of height h = 80µm with pillars of diameter d = the two microchannel plates impose a Poiseuille flow
400µm (h/d=0.2) in figure (a) and pillars of diameter d= which can be expressed as: v = v (cid:0)1−4z2/h2(cid:1) [Fig.
x 0
80µm (h/d = 1) in figure (b). Solid lines correspond to the
5a]. The flow conservation imposes v = 3x˙/2. The
theoretical model developed further in Section 2 and show 0
viscous force along the x-direction is equal to f =
thesolutionofequation(12)fordifferentpillarsaspectratio x
(h/d=0.2fortheredsolidlineinfigure(a)andh/d=1for η(cid:82)x(cid:48)=x(cid:82)y(cid:48)=l/2 (cid:82)z(cid:48)=h/2 ∆v dx(cid:48)dy(cid:48)dz(cid:48) which provides
x(cid:48)=0 y(cid:48)=−l/2 z(cid:48)=−h/2 x
thebluesolid lineinfigure(b)).
f = −12ηxx˙l/h. Balancing the capillary force with
x
the viscous contribution provides
As expected, when the pillar density tends to zero, γh
x2(t)= t (1)
thefluiddiffusivityD recoverstheoneofanemptymi-
3η
crochannel D . In the case for which the pillars aspect
0
ratio h/d = 0.2, the diffusivities ratio D/D decreases This approach predicts the square root evolution of
0
monotonically with the pillar density φ. The situation thecontactlinepositionwithtimeobservedexperimen-
ismorecomplicatedforthecaseofh/d=1becausethe tally in Section 1 and provides a theoretical value of
ratio D/D first increases up to pillar density of about the diffusivity D = γh/3η in the case of an empty
0 0
0.06andthendecreasestoreachvalueslowerthanunity microchannel. Note that, in practice this law is used
for large pillar densities (φ > 0.15). One remarks that to estimate the flowing time of the underfill process in
in the situation where h/d=1 and φ<0.15, the pres- microelectronics (Wan et al (2008)).
4 BaptisteDarboisTexier, PhilippeLaurent,SergueiStoukatch,St´ephane Dorbolo
(a) betweenmicropillars.Wesupposethatthefluidvelocity
profilein-betweenpillarsisv =v (cid:0)1−4y2/(p−d)2(cid:1)
xp 0p
(cid:0)1−4z2/h2(cid:1)whereasthefluidvelocityprofileelsewhere
remains v = v (cid:0)1−4z2/h2(cid:1) [Fig. 5b]. The flow mass
x 0
conservation yields v =9px˙/4(p−d) and v =3x˙/2.
0p 0
Finally, the viscous force resulting from the fluid flow
between two pillars reads
(b)
(cid:18) (cid:19)
p h p−d
f =−12ηx˙d + (5)
xp p−d p−d h
The total viscous force in-between pillars when the
contact line is located in x can be approached by
(cid:18) (cid:19)
d p h p−d
f =−12ηx˙xl + (6)
xptot p2 p−d p−d h
Fig.5 (a)Sketchoftheliquidflowinanemptymicrochannel The previous equation yields
viewed from the side. (b) Sketch of the assumed liquid flow
in-betweenfourmicropillarsviewedfromabove.  (cid:115) 
2
12ηx˙xl h φ
f =−  + (7)
xptot h (cid:113) (cid:16)(cid:113) (cid:17)2 φ 
2.2 Effect of the pillar array φm φm −1 m
φ φ
When the pillars are present in the microchannel, the withφm =π/4(cid:39)0.78themaximalpillardensitywhen
surface of the cavity wetted by the fluid reads p=d.Theviscouscontributionresultingfromthefluid
flow out of inter-pillar areas is
(cid:18)d(cid:19)2 x l x l
S =2xl−2π +πdh (2) (cid:18) (cid:19)
12ηx˙xl d
2 pp pp
f =− 1− (8)
xtot h p
where x/p is the mean number of pillar rows wetted by
the fluid when its interface is located in x and l/p the This equation reduced to
meannumberofwettedpillarrowsalongthetransverse (cid:32) (cid:115) (cid:33)
12ηx˙xl φ
directionofthemicrochannel.Theglobalapproachused
f =− 1− (9)
hereisjustifiedbecauseweconsiderthemeandynamics xtot h φm
of the contact line over a distance x much larger than
Balancing the capillary and viscous forces, f +
the characteristics of the pillar array (p and d). xptot
f +f =0, gives
The resulting capillary force is xtot γ
(cid:34) (cid:35)
f =2γl(cid:20)1− πd2 + hπd2(cid:21) (3) 12ηxx˙l 1+ h2 =
γ 4p2 d 2p2 h (cid:113)φm (cid:16)(cid:113)φm−1(cid:17)2 (10)
φ φ
(cid:2) (cid:0) (cid:1) (cid:3)
2γl 1+ 2h−1 φ
which reduces to
(cid:2) (cid:0) (cid:1) (cid:3) This relation simplifies as
f =2γl 1+ 2h−1 φ (4)
γ
(cid:0) (cid:1)
with φ = πd2/4p2 the pillar density and h = h/d the x˙2 = γh 1+ 2h−1 φ (11)
pillar aspect ratio. 3η 1+h2/(cid:113)φm (cid:16)(cid:113)φm −1(cid:17)2
Thepresenceofpillarsinsidethemicrochannelmod- φ φ
ifies the fluid velocity profile inside the cavity and thus Inpresenceofpillars,theliquidstillfollowsaWashburn-
√
theviscousforceexperiencedbythefluid.Variousstud- like law (x= Dt) but the diffusivity now depends on
iesinspectedexperimentallyandtheoreticallythepres- thepillararraycharacteristics(φandh).Wedefinethe
suredropresultingofaliquidflowthroughasquarear- ratio of the diffusivity D of the channel in presence of
raysofcylindersembeddedinsideamicrochannel(Gunda pillars and the diffusivity D = γh/3η for an empty
0
et al (2013); Sadiq et al (1995); Tamayol and Bahrami channel. This ratio is expressed as
(2009); Tamayol et al (2013)). However, these studies
(cid:0) (cid:1)
only consider the situation where the vertical confine- D 1+ 2h−1 φ
= (12)
ment is negligible. In this paper, we estimate the vis- D0 1+h2/(cid:113)φm (cid:16)(cid:113)φm −1(cid:17)2
cous force by assuming a Poisseuille flow in the space φ φ
Wickingthroughaconfinedmicropillararray 5
2.3 Comparison with experiments 3 Discussion
PredictionsofEq.(12)arecomparedwithexperimental 3.1 Fluid pumping
data in Fig. 4 for different pillars density φ and vari-
In our experiments, the capillary forces pump the fluid
ousnormalizedpillarheighthbythewayofsolidlines.
inside the microchannel. The liquid flow rate is defined
Thegoodagreementbetweenexperimentsandtheoret-
as Q = dV/dt with V the volume of fluid inside the
ical predictions validates the assumptions made for the
microchannel. When the contact line has the position
profileoftheflowinthemodeldevelopedinsection2.2.
x, the volume of fluid inside the cavity is V = lhx−
Thereafter,weconsiderthepredictionsofthemodel √
π(cid:0)d(cid:1)2hl x. Substituting the relation x = Dt in the
in a broader range of parameters compared to experi- 2 pp
previous definition leads to
ments. Figure 6 shows D/D as a function the pillar
0
density φ and the normalized height h as predicted by hl(1−φ)(cid:114)D
Eq. (12). One notices that whatever the pillar aspect Q= (13)
2 t
ratio, the diffusivity ratio D/D tends to one for small
0
The ratio of the flow rate Q in a cavity with pillars
pillar densities (φ → 0) and falls to zero for large pil-
and the flow rate Q in an empty cavity is
lar densities (φ → φ ). If h > 0.5, the diffusivity ra- 0
m
tio is slightly larger than one for small density values Q (cid:114) D
(φ (cid:46) 0.15) as delimited by the white solid line in Fig. Q =(1−φ) D (14)
0 0
6. The analytical determination of the domain where
Inserting equation (12) in the previous ratio yields
D/D > 1 is presented in the Appendix A. The maxi-
0
mal value of the diffusivity ratio D/D0 increases with Q (1−φ)(cid:0)1+(cid:0)2h−1(cid:1)φ(cid:1)1/2
thenormalizedaspectratioh.Forh=3,D/D reaches = (15)
a maximal value of about 1.1 for φ=0.06. Thi0s behav- Q0 (cid:18)1+h2/(cid:113)φm (cid:16)(cid:113)φm −1(cid:17)2(cid:19)1/2
ior can be understood by the fact that when the liquid φ φ
wetsapillar,thegainofsurfaceenergyisγπdhwhereas
iftherewasnopillarthegainofenergywouldhavebeen
γπd2/2.Ifh>d/2,thepresenceofpillarsincreasesthe
3 1
energetic benefit of the liquid impregnation and fasten
its dynamics.
2.5
0.8
2
0.6
3 1 /d1.5
h
2.5 0.4
0.8 1
2
0.2
0.5
0.6
d
/1.5
h

0.4 0 0.2 0.4 0.6
1 φ
Fig. 7 RatioQ/Q asafunctionofthepillardensityφand
0.2 0
0.5 thenormalizedheightofthechannelh=h/d.

0 0.2 0.4 0.6 Figure7presentstheflowrateratioQ/Q asafunc-
φ 0
tion of the pillar density φ and the pillar aspect ratio
Fig. 6 RatioD/D0 asafunctionofthepillardensityφand h. Whatever the ratio h/d, if the pillar density is small
thenormalizedheightofpillarsh=h/daspredictedbyequa-
(φ → 0), the flow rate ratio Q/Q tends to one. In
tion(12).Thewhitesolidlineseparatestheregionwherethe 0
diffusivity ratio D/D is larger than one. The white dashed the opposite situation where the pillar density is large
0
line shows the pillar density φmax which maximizes the ra- (φ→φm),theratioQ/Q0fallstozero.Onenoticesthat
tio D/D0 for a given aspect ratio h larger than 0.5. The the presence of pillars only slows down the flow rate Q
whitedottedlineindicatesthepillaraspectratioh which
max inacavitywithpillarscomparedtothecaseofanempty
maximes the ratio D/D for a given pillar density. The the-
0
oretical predictions of these three lines are detailed in the cavity. Thus, the addition of micro-structures in a mi-
AppendixA. crochannel does not enhance its pumping properties.
6 BaptisteDarboisTexier, PhilippeLaurent,SergueiStoukatch,St´ephane Dorbolo
This conclusion is of first importance for the develop- tion pillars, presence of defect or random distribution
ment of capillary pumps in the context of autonomous of pillars.
microfluidic.Insuchasituation,thepresenceofmicro-
structures in the capillary pump increases its efficiency
by inducing large interface curvatures which produce 3.3 Model limitations
a large overpressure and drive the fluid (Zimmermann
We propose in this section to discuss the limitation of
et al (2007)). According to our conclusions, this effect
the model developed in Section 2. This latter is based
works only if the hydraulic resistance of the system is
on a basic assumption of the fluid velocity profile in-
imposed by an element placed before the pump. In the
betweenpillars.Thisvelocityprofileisclearlynon-physical
context of autonomous microfluidic, the determination
because it does not respect the non-slipping conditions
of the hydraulic resistance of the capillary pump com-
of the fluid velocity along walls. In order to solve ex-
pared to the one of upstream elements is essential to
actly the problem of the fluid flow inside a confined
maximize the fluid flow rate.
micropillarsarray,theStokesequationhastobesolved
with the corresponding boundary conditions and thus
the resulting viscous force can be deduced. We expect
3.2 Other lattices
ourmodeltobeclosefromtheexactsolutioninthecase
of low pillar densities but to be less accurate for large
Thissectionaimstodiscusshowtheconclusionsdrawn
pillardensitieswheretheconfinementmodifiesthefluid
previously for square lattices of cylindrical pillars de-
flow far from the assumed profile. Such an effect may
pend on the pillars geometry and arrangement. With
explain the discrepancy between experiments and the-
the procedure described in Section 1 for microchannels
oreticalpredictionsobservedforlargepillardensitiesin
fabrication,wecreateoriginalpillarsarrangementssuch
Fig. 4b.
as square lattices of square pillars [Fig. 8a], square lat-
Also, we assumed in Section 2.2 that the fluid flow
tices of cylindric pillars with defects [Fig. 8b] and ran-
adopts everywhere a fully developed velocity profile.
domlydistributedlatticesofcylindricmicropillars[Fig.
The time needed for the fluid to reach such a state can
8c].
beestimated.Whenanon-slippingboundarycondition
For all these cases, we recorded the wicking of a
is imposed to a fluid flow, it spreads in a lateral direc-
√
Newtonian silicone oil V100 inside the microchannels.
tion z according the relation: z = νt where ν = η/ρ
The diffusivity coefficient D was measured and com-
is the kinematic viscosity of the liquid. In the case of
pared with the one of a reference case that we choose
the microchannels studied previously, the non-slipping
to be a square lattice of cylindric micropillars with the
conditions on the top and bottom walls spread in the z
same pillar density. For lattices that were non-square (cid:112)
direction according z = ν/D x. The fully developed
0
with cylindrical pillars, the density of pillars is defined
velocity profile is reached when z = h/2 which occurs
by the ratio between the cross section of all the pillars (cid:112)
for a critical traveled distance x /h = D /ν/2.
min 0
and the total section of the microchannel. The results For η =100mPa·s, ρ=980kg/m3, γ =23mN/m and
of these experiments are listed in Table 1. h = 80µm we estimate that x /h ∼ 10−3. As the
min
First, one notices that in the three situations con-
microchannel is much longer than its height in our ex-
sidered previously, the diffusivity modification induced
periments,theassumptionofafullydevelopedvelocity
by a change of pillar geometry or lattice arrangement
profile is justified.
are moderate. Indeed, the relative variation of the dif-
fusivity stays below 3 % in the three situation consid-
ered above. Thus, the dynamics of the contact line in 3.4 Comparison between open and confined
a microchannel is mainly ruled by the micro-structure micropillar arrays
density φ and their aspect ratio h/d. In details, we ob-
serve that the change in the geometry of the pillars Thepresenceofatopwallmodifiestheliquidimpregna-
from cylindrical to square section reduces the dynam- tionbehaviorcomparedtothecaseofanopenmicropil-
ics of the liquid impregnation. The presence of defects lar array. Such a wall introduces two counteracting in-
in a square lattice of cylindric micropillars enhances gredients on the capillary flow: an additional surface
slightlythefluidwicking.Finally,arandomdistribution towetandanotherno-slipboundarycondition.Follow-
ofcylindricmicropillarsisabitmoreefficientrelatively ing the same procedure as in Section 2.2 with a free
toasquarearrangement.Obviously,asystematicstudy boundary condition at the top, we deduce a theoreti-
asafunctionofthelatticeparametershastobeleadin calpredictionforthediffusivityD inthecaseofan
open
order to conclude on the precise impact of square sec- openmicrochannel.Theratiobetweenthediffusivityin
Wickingthroughaconfinedmicropillararray 7
(a) (b) (c)
Fig. 8 Examples of the progression of a Newtonian silicone oil of dynamic viscosity η = 100mPa·s in original lattices of
micropillars: (a) Square lattice of square pillars of width d = 250µm. The pitch between pillars is p = 750µm. (b) Square
lattice of cylindric micropillars of diameter d = 250µm with defects. The pitch of the lattice is p = 600µm. (b) Lattice of
randomly distributed cylindric micropillars of diameter d = 250µm. In these three experiments, the time step between two
positionsofthecontact lineis1s.
Lattice d(µm) p(µm) φ h(µm) D(m2/s) D (m2/s) D/D
ref ref
square section pil- 250 750 0.11 80 5.82×10−6 5.94×10−6 0.98
lars
square lattice of 250 600 0.12 80 6.06×10−6 5.88×10−6 1.03
cylindric pillars
with defects
random lattice of 250 - 0.05 80 6.2×10−6 6.07×10−6 1.02
cylindric pillars
Table 1 Measurements of diffusivity coefficients D for a Newtonian silicone oil V100 (ρ = 980kg/m3, γ = 23mN/m and
η = 100mPa·s) in original lattices (with square section pillars, square lattice with defects and random lattice of cylindrical
pillars). The diffusivity coefficient is compared with a reference case defined as a square lattice of cylindrical pillars with the
samedensityofpillars.
an open micropillar array and the diffusivity in a close
3
one reads
Dopen =(cid:32)1+(cid:0)4h−1(cid:1)φ(cid:33) 1+h2/(cid:113)φφm (cid:16)(cid:113)φφm −1(cid:17)2 (16)2.5 1
D 1+(cid:0)2h−1(cid:1)φ 1+2h2/(cid:113)φm (cid:16)(cid:113)φm −1(cid:17)2 2 0.9
φ φ
d
Figure 9 shows D /D as a function the pillar /1.5 0.8
open h
density φ and the normalized height h as predicted by
1
Eq. (16). One notices that depending on the pillar as- 0.7
pect ratio and the pillar density, the diffusivity ratio
0.5
D /D is lower or larger than one. Thus, the two 0.6
open
antagonistic effects, i.e. capillary pumping and viscous

0 0.2 0.4 0.6
resistance,causedbytheintroductionofatopwallcan
φ
overcomeeachotherregardingthepropertiesofthemi-
Fig. 9 Ratio D /D as a function of the pillar density φ
cropillar array. For dense arrays of slender pillars, the open
andthe normalizedheight ofpillarsh=h/daspredicted by
liquid impregnation in a confined micropillar array is
equation(16).Thewhitesolidlineseparatestheregionwhere
faster than in a similar open array. In such situations, thediffusivityratioD /D islargerthanone.
open
the confinement enhances the liquid spreading.
the presence of pillars can either enhance or slow down
Conclusions the dynamics of the contact line. However, the pillars
only slow down the flow rate of the liquid which pen-
The wicking of a micropillar array confined between etrates into the microchannel. A model based on the
two plates by a perfectly wetting and viscous liquid estimationofcapillaryforcesandviscousfrictioninside
has been investigated experimentally. We showed that the microchannel was developed. This latter provides
8 BaptisteDarboisTexier, PhilippeLaurent,SergueiStoukatch,St´ephane Dorbolo
a fair agreement with experimental data. In the end, maximizes the diffusivity ratio D/D for a given pillar
0
this study predicts the fall of the efficiency of the liq- density verifies
uid impregnation with increasing pillar density. This
result has implications for the underfill process in mi- φ(cid:18)(cid:113)φm (cid:16)(cid:113)φm −1(cid:17)2+h2 (cid:19)=
φ φ max (18)
croelectronics packaging where the liquid filling time is
(cid:0) (cid:1)
h 1+(2h −1)φ
a limiting factor of the production (Wan et al (2008)). max max
A perspective of this work is to refine the basic
The solution of Eq. (18) is presented by a white
model proposed to describe the liquid dynamics. The
dotted line in Fig. 6.
exactdeterminationoftheviscousforceinthemicrochan-
nel can be addressed by the way of numerical simu-
lations. Besides, this study only considers pillars with
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