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Flows in micro fluidic networks: from theory to simu lationDanny vam NoortEcology and Evolutionary Biology Dept., Guyot Hall,

Princeton University. Princeton. 08544 NJ ,US Adnnnvriiinrinceton.2dii

Abstract- When complex flow structures are de signed,such as in DN A computing [l], t is essential to be abl eto predict the flow pattern of the solutions in thefluidic network. A model based on the resistance ofthe channels and flow velocities of the inlets caneliminated re-iterative design steps. We haveconstructed a symbolic model (using Mathem aticam) odetermine the desired f low pattern based on theequations of Ohm and Kirchoff. The values from thissimulation were used in a flow simulation program .

1 IntroductionComplex flow structures. like the DNA-computer we aredeveloping [ l . Z ] . are difficult to construct without the useof calculation and simulation tools. Sim ple op ennetworks. i . e . . with one input and one output can beconstructed ad hoc. However. when there are multipleinputs and outputs. i t is a necessity to know the exact ratiobetween the flows and volume. The fluidic networkshould he balanced by determining the channel resistanceand the input velocities required to realize the directionand volume of f lows.

The resistance and flow properties are calculated in asimilar fashion to that used in electric circuit theory, i.e.the same laws apply. The resistance can he calculatedusing Ohm's law while the first law of Kirchhof statesthat the sum of currents is zero in nodes of the circuit.The analogy between fluidic and electric circuits is thatthe electric resistance (depending on the wire dim ension s.the dielectric properties of the material) corresponds tothe hydrodynamic flow resistance (depending on thechannel dimensions, the material properties of thechannel) while the current (the flow rate of electrons inthe wire) corresponds to the flow velocity (the flow rateof the liquid in the channels). The voltage drop o ve r awire corresponds to the pressure drop over a channel.The use of equivalent resistance circuits inmicrofluidic design was pioneered in the diploma thesisof Arne Bochman [3] (Univ. Jena. 1996). Other recentresearch use theoretical models to perform calculation,e.g., flow rates of a membrane pump or the imped ance formicrofluidic systems, were performed by Bardell andForster [4,5]. In our model the resistances of the microchannels were calculated algebraically using the computerpropram Mathem aticam (Wolfram Research In c, IL.US A). assuming an ideal situation. Visualisation of flow s

John S. McCaskillFraunhofer S ociety, BioMIP, lnstitutszentrumBirlinghoven, SchloP Birlinghoven, 537 54 SanktAugustin. Germanyi n c c n s k i l l [ ~ b i o m i n . ~ ~ . d e

were made with a microfluidic simulation softwarepackage from Coventor'" (Cov entor Inc.. NC. US A) .'First we present the necessary equations to calculate theresistance of a rectangular shaped tube after which thereactor modules and their working principle will heintroduced.

2 TheoryWhen flows are laminar. the non-linear term v.R in theNavier-Stokes equation becomes zero ( e . g . Happel andBrenner 161). Exact solutions of the remaining linearequations are then possible, especially if the flow issteady as is the case considered here. Indeed. themicrofluidic architecture was deliberately chosen to avoiddynam ic flow switching.For flow in the x-direction. the Navier-Stokes lawimplies:

(1 )Z U =--pP dxwith U is the f lowrate andp is the viscosity.The pressure grad ient 4 is constant for a steady statedxand is written as

*=-P ( 2 )d x Iwhere ~p > 0 is the pressure difference between theends of the channel of length 1. The end effects areneglected. So combining (1 ) an d (2 ) gives

( 3 )f u=--APP 'The volumetric flow rate Q is:i APQ = R I (4)where R is the specific resistance. Th e total resistance

over the channel isR , = RI ( 5 )

where I is the length of the channel. Substituting thisin eq . (4 ) gives - d p = R , Q (6 )

(which corresponds to Ohm's law Al'=RI ). In thecase of a rectangular channel with sides a an d h. thevolumetric flow rate is [SI:

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to determine the resistances and the desired inputs andoutput velocities.

To prevent the beads from flowing to other STMs. abead-barrier has been added to the design, which stops thebeads from disappearing down the channels but createsadditional hydrodynamic resistance.

o = * l u b Q 2 + b 2(7)

- pi 24 ( 1-__8 ;-

n 5 n=1 ( 2 n - l y

[ n " / u n ~ ( ~ ~ : + b ' / a n h ( ~ ; o : l A B CBy substituting eq 7 in eq 6, the resistance of the 1 I /j ;i j 1 :

channels can be calculated. ; I

3 Selection procedureIn DNA Computing [7]. DNA sequences are used asinformation carriers and as tools fo r the calculations. Inthe standard approach. each DNA sequence encodesbinary sequence ( S ) . Different DNA sub-sequences areselectors used to represent single bit values (0. 1) at acertain position in a sequence (S,, i=1.2,3 ... . After eachselection step. the sub-population is passed on to the nextselection step. Each selection module is coded with shortsingle stranded selection-DNA strand (ss DN A) from afinite set of predefined sequences. A selection is made bypicking ou t that word which has the desired bit necessaryfor further processing.

4 Microreactor StructureTh e selection procedure described ab ove can beimplemented in a microreactor. To this end we havedeveloped a selection transfer module ( STM) which isable to make positive selections from a population ofspecific DN A sequences. The principle of the selectionmodule is shown in Fig. 1. To transfer actively theselected DNA sequences to the appropriate output. theyare conveyed from one flow to another by movingparamagnetic beads, on which single stranded selector-DNAs (reverse complementary to a sub-sequence) areimmobilised [I] . The DNA strands in the templatesolution hybridise to the selector-strands and are thustransferred to another channel in the microflow reactorwhere they are denatured and passed on to the nextselection module (Fig. la). To optimise the transfer ofonly the appropriate sequences. a washing step has to beperformed so as to rinse off the non-specifically bou ndDNA sequen ces. Denaturation is performed by using analkali solution (NaOH), adjusted in concentration to thecommon melting temperature of the hybridised DNAstrands. Because of the change in pH at the denaturationstep, a subsequent neutralisation step is necessary aftereach selection stage before flowing the selected DNA intothe next module. Here the wash is performe d with theneutralisation solution. This is only possible if the flowshave the correct flow pattern to do an adequate mixing(Fig. lb ), which depends on the channel resistance in thesystem. Therefore a microreactor mode l has to be made

Wa outTi" N dH

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5 Microreactor ModelThe characteristics of the microfluidic system aredependent on the particular class of problems beingaddressed. In this case. u'e have designed a D N Acomputer to solve a particular NP-complete family ofproblems: the maxim um clique graph problem. Detailsfor this procedure can be found in McCaskill [ I ] and vanNoort er. a / [ 2 ] . The selection transfer module to bedesigned was shown in Fig. I .

3 2 1

I IF i yre 2

. " " ~ ,A sche,nnric rqwrsriirarion of rltc f i r i c r o f lm

~yncrorms i s r i q qf 7 r.esismrs. Ourpur 7 is con,irc~rerl u irzprir3 iri order- io ollou. colculnrions /o lw ,for.n seyrrence qf sac11ImClOlJ .

Figure 2 shows the module represented by resistorsused for the calculation of the channel resistances andflow velocities. There are 3 inputs (template.neutralisation. de-hybridisation) and 2 outlets (waste andtemplate). The horizontal part is the reaction chamber inwhich the selected DN A strands are transferred bymagnetic beads from left to right where they are releasedto the template output. Th e network has been madeiterative just for the calculations. which means thattemplate out is connected to template in (node 4 isconnected to node 3). In reality the problem is limited toa finite number of these reactors in sequence. To solvethe problem we are working on. we need at least 15 ofthese reactors in sequ ence.

Tablr I The cliannel resistances nnd flow i , ~ loc i r i~ i

Since we are working un der different buffer conditions(denaturation solution should be neutralised to allowhybridisation conditions again in the next microreactor),ratios between the different buffers should be fixed andhave been chosen l : l . This means that the velocity v ,and v3, should be equal. Oth er constrain ts are that thepressure p7 is zero (there is the assum ption that there is noresidual pressure on the output relative atmospheric). Thecalculation is being made with R34 as a variable and allthe other values fixed. If all the oth er resistors are set to 1(these values being arbitrary. since we are on ly interestedi n the ratio between the resistors and velocities) exceptRsh=3 and the velocity v2=4vI. then R,=4 and v,,=2vl.Using the values of these parameters gives us the correctfluidic network for the desired flow behaviou r. Table Ishows all the resistors and flo w v elocity valu es.

6 Flow SimulationTo visualise the calculated resistances an d velocities usedto design the microflow system, flow simulations weremade with CoventorTM. As mentioned before. the valuesof resistance were taken as ratios and not as absolutevalues.

Supply channels are needed to deliver the solutions tothe STMs. To ensure that the resistance o f the supply andwaste channels don't corrupt the overall resistance. theyshould be at least 3 imes wider with the same depth mainchannels.

A

B

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cliaf i i ie ls wirlr the ,sum widll i as the rnnirz rlmnnels: B s l i o r~ shecnsc w h o r r i r r supply c/iaiiriels ar e 3 rimes wider,This number is based on simulations concerning thedistribution over side channels with fixed width asfunction of the width of the supply channel (Fig. 3).Figure 4 shows the flow velocity for the given sidechannels as a function of the supply channel thickness.Where the points overlap, there is a uniform flowdistribution. The insert gives the dist anc e of the fir edside channels from the top of the network. as shown inFig. 3.

16 -14 --

3s2- 10--D 8 -2 -

a- : 1 2 :---E 6.

part of the STM . The cu rve C shows the ratios of thesevelocities. Because the flow in the reactor (v W) s 3 timesthat of the input I (vI). he ratio between these velocitiesin the ideal case should be 3 when the resistance value isone. When the resistan ce in the reactor changes. themixing ratio will alter. Howe ver. this ratio can becompensated with the inputflow velocities v, . v:. v3 (datanot shown). This gives a certain amount of flexibility inthe production stringency.

c c--- d', e

In general the design of the DNA-computer is determinedby the problem type and algorithm choice. which is alsoreflected in the numbe r of reactors. For short termconvenience, the number of STMs can be matched to theproblem studied, but this is not a principle limitation. Inorder to allow the magnetic beads to move uniformlymicroflow reactor, alterna te STM s are mirrored. Whenthe DNA-sequences denatures from the beads. the beadsin the next module will be in the correct place forhybridisation to tak e place.

To solve a 6-node clique instance as described in [2].

~ - . from left to right in all the STMs over the entire'

#? ' '/,//' *+E

+a 7 DN A computer+ b

1 , , , , , , , , , , , 15 modules are needed. A 16"' module has been added.200 400 600 ROO 1000 1200 as a dummy, to balan ce th e resistance of the 1 5 ' ~ odule..A l

6

I"

0.6 0.8 1.0 1.2 1.4 1.6Resistance [a.".]Figure 5 Tlrv ,olio fC i "f rile j7oo. velocities qr B i

rind r, ( A )nr,function qfrltr resisiancp change iit Rr .Simulations have shown that changes in the resistanceof the channels affect the distribution of the solutions.Figure 5 shows the flow velocities in the reactor (A ) andthe input I (B) s function of the resistance i n the reactor

which was found necessary when analysing thesimulation data. Figure 6sho ws the co mplete architectureof the DNA computer. Th e round pads are bead injectionpads, needed to insert into the reactor the paramagneticbeads functionalised with specific short selection DNA.The broad, vertical channels are supply channels whichare etched on the back of the wafer. while the squares onthese channels indicate the trough etc hing locations. Thetop layer of channels is a sequence of STM s as describedpreviously.

8 Structureand ProcessingThe array of cascading STMS were fabricated using 8multi-step wet etching procedure [SI [9] of 4"

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including the bead injection pads and the deliverychannels. are made in PDM S. The followingtechnological steps are perfo rme d. A multi-depth mas terwas fabricated by photo-lithographic patterning of anegative 'photoresist (SU 8-50, MicrolithographyChemical Corp. Newton. MA) using two different photo-emulsion masks. A specified amount of PDMSprepolymer (Sylgard 184, Dow-Co rning) is then castagainst the master. Cu ring the poly mer and releasing itfrom the master yields a replica containing the beaddelivery stmctures required. Th e structures are sealedirreversibly by oxidising the PD MS mould and the siliconwafer. after they are brough t into contact.

Figure 6 The romplerr design of r h u DNA-compurerf i x g,-aphs with up IO 6 irodm Tlic hluc cltmmds are rhr orqiurrirrd vqipl? ckarrnelr ,/OY rlw DNA cornpuler. erched 01 1 the hackqf rliu wfw. These ul y con,wcred on rhr bock OJ rlw waj@rOrile otirside world, Tlrr red circles UT? rlie bend i r r jecr iai I d s .while rlzc r l rhr Irorironralorange clmnnels con re^ rlzc bcods,f-omth e p d x O the mc r o r siruuted b ewee , ! rlre WO , ~ n ~ r ro l isar io r~clrunnels. The nDbreviatior~s m e as fo1loii.s: dH is dr-Ii?bridIsorior~ nler: N is rlir neutralisation inler: Ti,,s rhe inputre,nplure: Out is the o q z a rom /h e clique sorting ntodnles: Wais r1,r i , n S ! P .

The supply cliannels width on the front and thebackside, are wider t o red uce th e flow resistance a ndpressurr ;sops in the complete structure. The channelwidth in the STMs is 100 pm . To etch through the220 pm thick silicon wafer. etch ing pads on the front(400 x 400 pm j and on the back (1200 x 1200 pmj aremade as to obtain h oles of 4 00 x 40 0 pm, the same widthas the STM-supply channels on the front of the siliconsubstrate.

9 ConclusionsMathematical mode ls and flow simulation tools provide apowerful base to desig n microfluidic network s. Thisallows a prediction o f the flow patterns which isnecessary to predict controlled mixing of reactants whichis very important i n biochemical processes. A suitablecombination of analytical and simulation tools also savestime in the develop men t process for the network.AcknowledgmentsThe authors wish to acknowledge the support of the GMDand of the group's start-up grant (#01SF9952) from theGerman Ministry of Science (BMBF). We would like tothank Marlies Gohlke. and Patrick Wagler for theirdiscussions and th e m anufacturing of the microfluidicsnetwork and Harald Mathis for providing the Rhodamine6G and the laser set-up.Bibliography[I] McCaskill . J. S. (2001) Opr ica l l j progrmi t i r i r ig

D N A cor~rpritirrg rr rrricrqflou~ eucrors. Biosystems59 . 125-138.van Noort. D., Gast. F. -U . and McCaskill. J. S.(2002) DNA Coarputirrg in Microreactors. LNCS.in press.

[3 ] Bochman. A. 19 96 ). Diploma Thesis Univ. Jena.[4] Bardell. L. R.. Sharma. N. R.. Forster, F. K..Afromowitr. M. A. and Penney. R. J. (1997)Desigrlirig high-perjonrrarlce nricro-prinrps based 011r ~ o r i - i i r o ~ ~ i r ~ ~arrs. DSC-vol 621HTD-vol. 354.Microelectromechanical Systems (MEMS) ASME

[ Z ]

1997. pp. 47-53.Bardell. L.R. a nd Forster, F. K. (1998) I r i~pedur i re.~5]fo r desigrr of nricrofltridic syrenrs. Proc. of theMicro-TAS '98 Workshop. Banff. Canada. 13-16Oct 1998. (http://lettuce.me.washin_pton.edu /micropump/puplication/l998utas98wE.htm)Happel. J . and Brenner, H. (1983) in book:ha.Rewolds rruriiber h?.drodwarirics. Martinus NijhofPublishers, the Hag ue.171 Ptiun. P.. Rozenberg. G. and Salomaa, A. DN AConpi r i r rg : New Conipparirrg Paradignrs. SpringesVerlag. Heidelberg. Germany. October 1998. ISBN:3-540-64 196-3.

[SI Schmidt, K., Foerster. P.. Bochmann, A. andMcCaskill , J.S. (1997) A nr icrof lox reacfor for nrodinrerisiorial irrwstigariorts of in vit ro anrplif icarioris ~ s t e m s . In 1" international Conference onMicroreaction Technology. Book of abstracts.Dechema e.V., Frankfurt am Main.[9] Wagler, P., Gohlke, M., van Noort, D. andMcCaskill. J.S. (2001 Tlzree-diniertsiorialrrricrofliridir s p t e n i s fo r conrpnrufiorr ivith DN Aniolecules. 12"' Micromechanics Europe WorkshopM M E 2001, proceedings. Cork, Ireland.

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