Drop Traffic in Microfluidic Ladder Networks with Fore-Aft Structural Asymmetry

0.79 MB·English

by  Jeevan Maddala

Drop Traffic in Microfluidic Ladder Networks with Fore-Aft Structural
Asymmetry
Jeevan Maddala, William S. Wang, Siva A. Vanapalli and Raghunathan Rengaswamy1
Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79401-3121
(Dated: 30 January 2012)
We investigate the dynamics of pairs of drops in microfluidic ladder networks with slanted bypasses, which
break the fore-aft structural symmetry. Our analytical results indicate that unlike symmetric ladder net-
works, structuralasymmetryintroducedbyasingleslantedbypasscanbeusedtomodulatetherelativedrop
spacing, enabling them to contract, synchronize, expand, or even flip at the ladder exit. Our experiments
confirm all these behaviors predicted by theory. Numerical analysis further shows that while ladder networks
containingseveralidenticalbypassesarelimitedtonearlylineartransformationofinputdelaybetweendrops,
2 combinationofforwardandbackwardslantbypassescancausesignificantnon-lineartransformationenabling
1
coding and decoding of input delays.
0
2

n Understanding the spatiotemporal dynamics of con- ofslantedbypassesinladderscannon-linearlytransform
a fined immiscible plugs in interconnected fluidic paths the initial delay between drops. These advanced capa-
J
is essential for applications ranging from lab-on-chip bilities arise because slanted bypasses flexibly manipu-

7 technologies1,2 to physiological flows3 to porous media late (i) the locations in the channels where drop velocity
2
flows4. The traffic of drops or bubbles in even simple changes occur and (ii) the time drops spend with by-

networks such as bifurcating channels can be astonish- passes between them.
]
n ingly complex due to collective hydrodynamic resistive
y interactions in the branches5–7. Although such intricate
d
dynamics, in the case of lab-on-chip applications, make
-
u devicedesignchallenging,thecollectivebehaviorscanbe
l harnessedtoperformusefultaskssuchasdropletsorting8
f
. and storage9,10.
s
c Recently, the collective dynamics between pairs of
i drops have been harnessed in the so-called microfluidic
s
y ladder networks (MLNs) to control their relative drop
h spacing11. In MLNs, two droplet-carrying parallel chan-
p
nels are connected by narrow bypass channels through
[ FIG. 1. (a) Ladder with a vertical bypass (b) Three dis-
which the motion of drops is forbidden but the carrier
tinctconfigurationsarepossiblewhenapairofdropstraverse
3 fluid can leak. Current versions of ladders have fore-aft
throughasymmetricladdernetwork;Blackobjectsrepresent
v structural symmetry due to equally-spaced vertical by-
drops. Full and dashed lines denote transport and bypass
5 passes. Such symmetric ladders are limited in function- channels respectively. Arrows show flow direction.
4
ality because the distance between pairs of drops have
8
been shown to decrease at the exit only for constant in-
5 AsshowninFig. 1, thekeyframeworkforquantifying
let flow11–13. Sinceflexible manipulationof dropspacing
. drop spacing comes from understanding the variation in
1
innetworksiscrucialforpassivelyregulatingavarietyof
relativevelocity(u)betweendropsinthetopandbottom
1
tasks including drop coalescence14, detection and stor-
1 channelastheycrossnodesinthenetwork. Considerthe
age, there is a need to design microfluidic ladders with
1 simplecaseofasymmetricMLNwithoneverticalbypass.
: multiple functionalities. As shown in Fig. 1(b) left, when two drops driven by a
v
From a fundamental perspective, the dynamics of constantflowrateentertheladderwithaninitialsepara-
i
X dropsinMLNsisdistinctcomparedtothewidely-studied tion(∆x ),theymaintainthesameseparation,asu=0.
in
r microfluidic loops5,7,13,15. In loops, drops at junctions When the leading drop crosses the node, a new configu-
a choose a given branch. These discrete choices make such ration (see Fig. 1(b) center) is reached, and the relative
systems non-linear, enabling coding and decoding of in- velocitychangesasfluidleaksintothebypass,i.e. u<0,
put signals. Since drops do not typically make decisions causing a contraction in inter-drop distance. When both
at the bypass junctions in ladders, an open question is: drops cross the bypass, u=0 again as the pressure drop
isitpossibletodesignmicrolfuidicladdersthatyieldsig- acrossthebypassiszeroduetoequaldownstreambranch
nificant non-linear transformation of input signal? resistances (see Fig. 1(b) right). Thus, the outlet drop
In this Letter, we study the dynamics of spacing be- spacing,(∆x ),islessthantheinitialseparationandis
out
tween drop pairs in MLNs with slanted bypasses. We givenby∆x =∆x +u∆T,where∆T istheduration
out in
findthatbecausetheslantbreaksthefore-aftsymmetry, drops remain in the particular configuration of Fig. 1(b)
it provides significantly more control over drop spacing center.
than symmetric MLNs. We also discover that inclusion Innetworkswithmanybypasses,moredropconfigura-
2
tionsandrelativevelocitychangesarepossiblethanthat
ofFig. 1(b)asdropstraversethroughmultiplenodes. In
general, we find that
p
(cid:88)
∆x =∆x + u ∆T (1)
out in j j
j=1
where p is the number of distinct configurations of
droplets occurring in the network, u and ∆T are the
j j
associated relative velocities and time periods.
The degree of separation achieved at the ladder exit
depends on the contribution of the summation term in
Eqn (1) both with respect to magnitude and sign. The
strength of this contribution is modulated by the spe-
cificarchitectureoftheladdernetwork. Tocomputethis
contribution, weuseresistivenetworkmodeling13, where
FIG.2. Laddernetworkswith(a)backwardslant(b)forward
we assume each drop is a point object with the same
slant;(c)DynamicalregimesduetodroptrafficinMLNswith
hydrodynamic resistance (R ). Since the drop velocity
d asingle(i)verticalbypass(ii)backwardslantand(iii)forward
(V) is linearly dependent on liquid flow rate (Q)16, we
slant. M = 0.45.
have V =βQ/S , where S is the channel cross-sectional
area and 0 < β < 2. To preserve the relative separa-
tion when drops leave the ladder network, we choose the undergocontractionforthesamereasonasinthevertical
downstream channel resistances to be equal. bypass. Perfectsynchronizationcanalsoberealizedwith
We begin by discussing the effect of a single slanted just a single backward slant when ∆x = M∆L. More-
in M−1
bypass in regulating the dynamics of drop spacing. Fig. over, when ∆x < M∆L, a new regime emerges that we
2 (a, b) shows a representative ladder network with a in M−1
refer to as flipping. We observe that the leading droplet
slanted bypass. In contrast to the vertical bypass, a new
is initially ahead of the lagging droplet. However, when
controlparameter,∆Lisneededtodescribetwopossible
the leading drop crosses the bypass first, its velocity is
structural configurations—backward slant for ∆L < 0
reduced and the lagging drop has sufficient duration to
and forward slant for ∆L>0.
catch up and overtake it. Thus, the flipping behavior
To determine the drop spacing at the exit due to the
yields ∆x <0 as shown in Fig. 2(ii).
out
slantedbypass,weidentifytherelativevelocitiesthatare
Similartothebackwardslant,theforwardslant(where
non-zero and the corresponding durations as prescribed
∆L > 0) provides additional means of control. Interest-
byEqn(1). SimilartotheverticalbypassinFig. 1,non-
ingly, in contrast to the backward slant, the behavioral
zerouoccursonlyafteroneofthedropletscrossesanode.
transitions depend only on the value of input drop spac-
Solvingthehydrodynamiccircuitforthisdropconfigura-
ing relative to ∆L. When ∆x <∆L, the lagging drop
in
tion analytically, we obtain u=βQ/S·(R /(R +2R +
d b e crosses the bypass first, resulting in expansion as high-
R )) and ∆T =(|∆x −∆L|)S/(βQ), where R and R
d in b e lighted in Fig. 2(iii). Alternatively, if ∆x > ∆L, the
in
are the bypass and exit channel resistance respectively.
leading droplet crosses the bypass first, resulting in con-
Thus, Eqn (1) for the case of an MLN with a slanted
traction. If∆x =∆L,thenbothdropscrossthebypass
in
bypass transforms to
simultaneously,andtheinputspacingispreserved. Thus
forward slant allows drop pairs to expand, contract or
∆x =∆x −M(∆x −∆L) (2)
out in in
remain unchanged.
where M = R /(R +2R +R ) and 0 < M < 1. Note It would be misleading to visualize the slanted ladder
d b e d
inEqn(2),∆x >0correspondstothetopdropleading network as being equivalent to a vertical bypass network
in
over the bottom drop in the ladder. with the addition of ∆L to the inlet spacing as we have
Remarkably, Eqn (2) captures several dynamical alreadyshownthatverticalbypassescanonlyreducethe
regimes emerging from structural asymmetry due to a drop spacing for a fixed inlet flow. We assumed the exit
slanted bypass as illustrated inFig. 2(c). For thepartic- channels to have equal lengths to simplify the analysis;
ular case of a vertical bypass (∆L = 0), Eqn (2) reveals however,theanalysispresentedinthisletterextendstriv-
that drops can only undergo contraction (see Fig. 1(a)). ially to unequal exit lengths also.
Perfect synchronization of drop pairs, i.e. ∆x =0 is To confirm the different behaviors predicted by our
out
difficult to achieve with a single vertical bypass, as it theory, we sought to construct an MLN with a single
requires M to be unity. slant. However, the design space is large, requiring op-
In contrast to the vertical bypass, we find that the timization of upstream and downstream transport chan-
backward slant, where ∆L < 0, yields flexible control nelresistance, slantresistance, slantslope, andhydrody-
over drop spacing as illustrated in Fig. 2(ii). For large namicresistanceofdrops, whichitselfisacomplexfunc-
input drop spacing, ∆x > M∆L, the pairs at the exit tion of the flow conditions and fluid properties17,18. For-
in M−1
3
ladders with n bypasses, the maximum number of con-
figurations where u is non-zero is n(n+1)/2. However,
all these configurations need not be realized for a given
∆x , whichmakesthetheoreticalanalysiscomplex. We
in
therefore used resistive network based simulations13 to
fully quantify the exit drop spacing for arbitrary input
delay. AsshowninFig. 4(a),wefindthatadditionalver-
tical bypasses simply amplify the contraction effect due
to a single bypass, i.e., slope decreases with increasing
number of bypasses. A similar outcome also holds, as
shown in Fig. 4(b), for the particular case of an MLN
withforwardslants. Wealsofindthatthelargestchange
in drop spacing occurs in the first few bypasses. Ladders
withmulti-bypassescouldthereforebeusefultomaintain
the same behavior as their single bypass counterparts,
while dampening the effect of small fluctuations in input
drop spacing.
FIG.3. Experimentalconfirmationofthedynamicbehaviors
inMLNs: (a)Snapshotsshowingsynchronizationofadroplet
pair. Time interval between images is 0.08s; (b) Drop spac-
ing as a function of position in the ladder (|∆L| = 500µm).
Continuous phase was hexadecane, and dispersed phase was
aqueousdyesolution. Thetransportchannelsare100µmwide
and tall.
tunately, insights from Eqn (2) reduce the search space. FIG. 4. Ladder networks with multiple identical bypasses:
According to Fig. 3, the backward slant is the best can- (a) Vertical bypasses (b) Forward slants. R /R = 3
d b
didatetoachievemaximumcontraction,perfectsynchro- R /R =22,R /R =2,R =1.5kg/mm4s and β =1.4.
d ∆L d e d
nization and flipping. Moreover, Eqn (2) reveals that if
∆xin < 0 (.i.e., the top drop is lagging behind the bot- Our analysis of MLN designs with identical bypasses
tom drop in the ladder), then the forward slant becomes hasshownthatatsmallinputdelay,dropspacingmayei-
abackwardslant,allowingaccesstotheexpansionregime thercontractorexpand,whileitalwayscontractsatlarge
as well as the condition where the input separation does input delay (c.f. Fig. 4). The contraction at large input
not change. Thus, we chose the backward slant to test delay is expected because the leading drop is the first
our predictions. to cross the bypass and slow down. Initially, it appears
We incorporated two flow-focusing drop generators in that expansion is not possible at large input delays. To
polydimethyl(siloxane) devices to introduce drops at a further probe this notion, we developed an evolutionary
constantflowrateintotheladder. Toamplifytheeffects algorithm19 to search for ladder designs containing any
producedbythebackwardslant,wemaximizedthevalue combinationofslantand/orverticalbypassesthatmight
of M by minimizing Rb and Re. For example, as shown becapableofcontractionatlowinputspacingandexpan-
in Fig. 3(a), the bypass has an enlarged mid-section to sionatlargeinputspacing. Oursearchstrategyrevealed
minimizeRb. WealsoensuredRb andRe tobe∼O(Rd). that such networks do exist, and an example is shown in
InFig. 3(b),weshowexperimentalcurvescorresponding Fig. 5(a). The dynamics of drop spacing in this network
toeachofthedynamicalregimespredictedbyourtheory. cannot be rationalized from mere addition of functional-
Wefindthedropspacingtoberelativelyunchangedwhen ities of the single bypasses shown in Fig. 4. Instead, we
the drop pair is before or after the bypass. However, as findthatthefirstfivebypassescollectivelycausecontrac-
pairsofdropscrossthebypasssection,theirspacingmay tion of drop spacing, while the last two bypasses cause
expand, contract, flip, synchronize or remain unchanged drops to expand. However, the relative magnitudes of
depending on the initial separation. By comparing the contraction and expansion from these sets of bypasses
data of Fig. 3 with the expression for u, we estimate depends on the input drop spacing. We observe that at
Rd ≈ 1.2kg/mm4s; M ≈ 0.2. Taken together, our ana- small input delays, the first five bypasses dominate, re-
lytical results and experiments suggest that MLNs with sulting in contraction behavior (see Fig. 5(a)), whereas
slanted bypasses provide greater flexibility than ladders at large input delays, the last two bypasses dominate,
with vertical bypasses. yielding expansion.
To understand the flexibility due to additional by- AstrikingobservationfromFig. 5(a)isthatthecurve
passes, we investigated ladders containing several iden- is significantly nonlinear compared to the almost lin-
tical bypasses. In contrast to the single bypass case, in ear dependence observed in ladders with identical by-
4
mands that the relationship between input and output
delays remain bijective. In addition, this functional rela-
tionship cannot have maxima, minima, or saddle points
becauseithastobestrictlymonotonic. Thus,webelieve
reversibilityimposesboundsonthedegreeofnonlinearity
that can be achieved with microfluidic ladders.
In summary, we observe that MLNs with mixed by-
passes display rich dynamics in drop spacing as well
as nonlinear behavior. Such ladder networks in fact
also exist in natural systems including leaf venation20,
microvasculature3,21 andneuralsystems22. Interestingly,
the nonlinear shape of Fig. 5(a), resembles that of the
widely observed sigmoid function, which is essential for
coding/decoding neural signals23, as it produces an in-
vertible map. Finally, the framework described here can
be expanded to explore not only pairs of drops, but also
trainsofdropstofurtherprobecollectivehydrodynamics
in drop-based microfluidic networks.
We acknowledge National Science Foundation for par-
FIG. 5. (a) Nonlinear output delay in a ladder network
containing a mixed combination of slanted and vertical by- tial financial support (Grant No. CDI-1124814).
passes (structure of the network is shown in inset) (b) En-
1A. B. Theberge, F. Courtois, Y. Schaerli, M. Fischlechner,
coding and decoding signal using ladder network R /R =3
d b C. Abell, F. Hollfelder, and W. T. S. Huck, Angew. Chem.,
R /R =2.6, (R ) to (R ) is 1.01,R /R =20. R =
d ∆L e2 e1 e1 d d Int.Ed49,5846(2010).
1.5kg/mm4s and β =1.4. 2H.Song,D.L.Chen, andR.F.Ismagilov,Angew.Chem.,Int.
Ed45,7336(2006).
3R.S˙kalak,N.O¨zkaya, andT.C.Skalak,Ann.Rev.FluidMech.
21,167(1989).
passes. This result is significant because it implies non- 4W.L.Olbricht,Annu.Rev.Fluid.Mech.28,187(1996).
linear transformation of input delay without any droplet 5D. A. Sessoms, A. Amon, L. Courbin, and P. Panizza, Phys.
decision-making at bypass junctions. A unique conse- Rev.Lett105,154501(2010).
6W.Engl, M.Roche, A.Colin, andP.Panizza,Phys.Rev.Lett
quence of this nonlinear transformation is the capability
95,208304(2005).
to encode and decode input delays as shown in Fig. 5(b)
7M. J. Fuerstman, P. Garstecki, and G. M. Whitesides, Science
where the entrance delay between pairs of drops repre- 315,828(2007).
sents the input signal and the system of bypasses repre- 8G.Cristobal,J.-P.Benoit,M.jaonicot, andA.Ajdari,Applied
sents the encoder and decoder. First consider the curves PhysicsLetters.89,034104(2006).
9S.S.BithiandS.A.Vanapalli,Biomicrofluidics4,044110(2010).
in Fig. 3 where the input signal is ‘scrambled’ in the
10P. Abbyad, R. Dangla, A. Alexandrou, and C. N. Baroud, Lab
bypass section, but does not revert to its original value
Chip11,813(2011).
and therefore is not decoded. In striking contrast, we 11M. Prakash and N. Gershenfeld, Science 315, 832 (2007); Pro-
find in Fig. 5(a), that input signal gets encoded and de- ceedingsofmicroTAS (2007).
codedattwodifferentvaluesof∆x =4.5,7(∆L),where 12B.Ahn,K.Lee,H.Lee,R.Panchapakesan, andK.W.Oh,Lab
in
Chip.11,3956(2011).
∆x = ∆x . At these input delays, we find that the
in out 13M.SchindlerandA.Ajdari,Phys.Rev.Lett100,044501(2008).
above-discussed contraction and expansion effects intro- 14J.Hong,M.Choi,J.B.Edel, andA.J.deMello,LabChip10,
duced by sets of bypasses negate each other. Thus, our 2702(2010).
results highlight a new route to code and decode signals 15M.Belloul,W.Engl,A.Colin,P.Panizza, andA.Ajdari,Phys.
comparedtoearlierstudiesthatusedropdecisionmaking Rev.Lett102,194502(2009).
16S. Jakiela, S. Makulska, P. M. Korczyk, and P. Garstecki, Lab
eventsinnetworks7. Theencoder/decoderillustrationin
Chip11,3603(2011).
this letter is just an exemplar demonstration of the po- 17S.A.Vanapalli,A.G.Banpurkar,D.vandenEnde, andM.H.
tential of such networks in realizing complex behavior G.,Labchip9,982(2009).
without any active components. 18V.Labrot,M.Schindler,P.Guillot,A.Colin, andM.Joanicot,
Biomicrofluidics.3,012804(2009).
Given that ladders with mixed combinations of by- 19J. Maddala and R. Rengaswamy, in 22nd European Symposium
passes can display nonlinear behavior, we ask what de- on Computer Aided Process Engineering (Elsevier,2012).
greeofnonlinearitycanbeachievedinladders. Aqualita- 20A.Roth-Nebelsick,D.Uhl,V.Mosbrugger, andH.Kerp,Annals
tiveindicationcan be obtainedbyconsideringreversibil- ofBotany87,553(2001).
21L. T. Chau, B. E. Rolfe, and J. J. Cooper-White, Biomicroflu-
ityinladdernetworks. Reversibilityimpliesthattheorig-
idics5,034115(1)(2011).
inal input delay is recovered when the flow is reversed. 22Y. Hosoya, N. Okado, Y. Sugiura, and K. Kohno, Exp. Brain
Ladder networks are reversible because of the absence of Res.86,224(1991).
decision-making events13. This reversibility criterion de- 23C.KochandI.Segev,Natureneuroscience3,1171(2000).