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:
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:
Mohammadali Safavieh, Mohammad A. Qasaimeh, Ali Vakil, David
Juncker et Thomas Gervais
Date: 2015
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:
Safavieh, M., Qasaimeh, M. A., Vakil, A., Juncker, D. & Gervais, T. (2015). Two-Aperture Microfluidic Probes as Flow Dipoles: Theory and Applications. Scientific Reports, 5. doi:10.1038/srep11943
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Two-Aperture
M
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fic
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as
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D
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App
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Mohammadalfi Safavfieh1 ,2, Mohammad A. Qasafimeh3, Alfi Vakfil4, Davfid Juncker2 ,5 ,6& Thomas Gervafis7 ,8
A mficrolufidfic pro e M fis a mofile channel-less mficrolufidfic sstem under whfich a lufidfisfinected from an aperturefinto an open space, hdrodnamficall conned a surroundfin lufid, and entfirel re-aspfiratedfinto a second aperture. Varfious M s have een developed, and have een used for applficatfions ranfin from surface patternfin of photoresfists tolocal perfusfion of oranotpfic tfissue slfices. owever, the hdrodnamfic and mass transfer propertfies of the low under the M have not
een anal ed, and the low parameters are adusted empfirficall. ere, we present an analtfical model descrfi fin the ke transport propertfiesfin M operatfion,fincludfin the dfimensfions of the hdrodnamfic low connement area, dfiusfion roadenfin, and shear stress as a functfion of fi proe eometr fifi aspfiratfion-to-finectfion low rate ratfio fififi ap etween M and sustrate and fiv reaent dfi usfivfit. Analtfical results and scalfin laws were valfidated aafinst numerfical sfimulatfions and eperfimental results from pulfished data. These results wfill e useful to ufide future M desfin and operatfion, notal to control the M rush stroke whfile preservfin shear -sensfitfive cells and tfissues.
Mficrofufidfics referstothe manfipulatfion of mfinute amounts oflfiqufids, whfichtypficallytakes place wfithfin the confnement of closed mficrochannels1–3. However, one ofthe drawbacks of “closed” mficrofufidficsfis the challengeto process objectslfikelytofinduce cloggfing orthat are sfimplytoolargeto ftfin a channel network, such astfissue sectfions,large cells and partficles. Open mficrofufidfic systems overcome some of these problems by usfing hydrodynamfic fow confnement (HFC) to confne a fufid stream such as fin mficrofufidfic probes (MFP)4–6 or capfillary efectsto confne droplets between a probetfip and a surface, such as fin the “chemfistrode” confguratfion10,7,8. Usfing the HFC property of MFPs, reagents finjected by the probe can be confned wfithfin close proxfimfity ofthe probe, allowfingthefir delfiveryto a surface wfith hfigh spatfial resolutfion andlower shear stressthanfin channel-based mficrofufidfics systems.
In fits most general defnfitfion, a MFP fis an open mficrofufidfic system consfistfing of a fat, blunt tfip wfithtwo aperturesforfinjectfion and re-aspfiratfion of alfiqufid streamfin a gap betweenthe probe and a substrate whfile hydrodynamfically confnfingfit wfiththe surroundfing fufid. Ffig. 1 shows a schematfic of a two-aperture MFPfin operatfion. Reported MFP applficatfionsfinclude surface gradfients generatfion9, hfigh resolutfion addfitfive and subtractfive bfio-patternfing of bfiomolecules and sfingle cellsfin a hfighly-controlled
1Instfitut natfional delarecherchescfientfifique(INRS)- Énergfie Matérfiaux Télécommunficatfions(EMT), Unfiversfité
du Québec, 1650 Boulevard Lfionel-Boulet, Varennes,J3× 1S2, Québec, Canada.2Department of Bfiomedfical
Engfineerfing, McGfill Unfiversfity, Montréal, Quebec, H3A 1A4, Canada.3Dfivfisfion of Engfineerfing, New York Unfiversfity
Abu Dhabfi, Abu Dhabfi, Unfited Arab Emfirates.4Department of Mathematfics, Unfiversfity of Brfitfish Columbfia,
Vancouver, Brfitfish Columbfia, V6T1Z2, Canada.5Department of Neurology and Neurosurgery, McGfill Unfiversfity,
Montréal, Quebec, H3A 1A4, Canada.6McGfill Unfiversfity and Genome QuebecInnovatfion Centre, Montreal,
Quebec, H3A 1A4, Canada.7Department of Engfineerfing Physfics, Polytechnfique Montréal, Montreal, Quebec, H3C
3A7, Canada.8Centre derecherche du Centre hospfitalfier del’Unfiversfité de Montréal, andInstfitut ducancer de
Montréal, Montreal, Quebec, H2L 4M1, Canada. Correspondence and requests for materfials should be addressed to T.G. (emafil: Thomas.Gervafis@polymtl.ca)
Recefived: 18 December 2014 Accepted: 26 May 2015 Publfished: 14 July 2015
fashfion4, maskless lfithography by dfissolvfing photoresfist on a coated surface10, lysfis of human breast cancer cells and collectfing and analysfis of mRNA released from the lysate11, pharmacokfinetfic studfies on cells12, stafinfing of tfissue sectfions13,analysfis of neutrophfil dynamfics durfing chemotaxfis14,15, and per -fusfion of organotypfic brafin slfices16. In all these studfies17, the MFP desfign, fow rates and gap hefight
Ffigure 1. Schematfic representatfion of a two-aperture MFP. (A) Te MFPfis alfigned by usfingthe probe holder (not shown here) andfis set closetothetransparent substrate atthetop ofthefinverted mficroscope20. Te fufidfisfinjected and aspfiratedfromthe capfillarfies atthetop, whfich makesthefinjected
fow hydrodynamfically confned. (B) Pficture ofthe MFP21 composed of a sfilficon chfip bondedto a PDMS
block wfithfinserted capfillarfies. Te mesa andthe mficrofufidfic apertures are enclosedfinthe whfite dashed box. (C) A close up schematfics ofthe fow underthe MFP andthe resultfing HFC. (D) Mficroscopficfimage ofthe MFP showfingthe mesa andthe mficrofufidfic apertures. Te scale barfis 50 µ m. (E) Schematfic of a two-aperture MFPforthe analysfis of fow proflesfincludfing a referenceframe. Te apertures are separated by a dfistance d and are of square sfidea. Inthe case of round aperture probes,the dfimensfiona corresponds tothe aperture dfiameter. Te orfigfin was set atthe center ofthe connectfinglfine betweenthetwo apertures. (F) A fnfite element model of streamlfines under atwo-apperture MFP clearly showsthe HFC area as well asthe stagnatfion pofint (SP)located onthefar sfide ofthefinjectfion aperture. Te stagnatfion posfitfion (R)fis defned asthe dfistance betweenthe center ofthe MFP andthe SP.
were optfimfized empfirfically and calfibrated experfimentally based on a trfial and error method. Prevfious analyses fin other probe geometrfies (mficrofufidfic quadrupoles) have demonstrated that the steadfiness and hfigh reproducfibfilfity of MFP patterns are due to the presence of Hele-Shaw fow patterns at suf-cfiently small gap sfizes. Underthese condfitfions,the fow underneaththe probe can be consfidered quasfi two-dfimensfional and readfily analyzed usfing potentfial fow theory drawfing upon the strong analogy between 2D fow felds and electrostatfic felds9. Chrfist and Turner have studfied convectfion fin varfious two-aperture MFP confguratfions18. Tefir hydrodynamfic pressure measurements at several pofints under -neath the probe, supported by numerfical models and relevant pressure scales thoroughly confrm the valfidfity of potentfial fow assumptfions when modelfingthe fow behavfior underthe probe. Yet,thefir ana l-ysfis does not provfide an analytficalframeworkto quantfify nefither advectfion nor dfifusfionfintwo-aperture MFPs. Advectfion and dfifusfion are essentfial parametersto operatfing MFPs and control dfifusfion grad fi-ent generatfion, shear stress atthe surface, confnement area, andthe efect of port geometry.
Here, wefintroducethe relevant physficaltheory descrfibfingthe varfious desfign and operatfion param-eters wfithfin a MFP. Te two-aperture probe fis analyzed as a fow dfipole (or doublet)19 usfing the same formalfism asfor 2D electrostatfic dfipoles. Te model also goesfurtherthanthe conventfional Hele-Shaw formalfism by alsotakfingfinto account dfifusfive masstransfer. We extract several analytfical results as well as scalfinglawsto controlfimportant parameters such as HFC zone dfimensfions, shear stress and dfifusfion gradfients wfithfinthe gap belowthe mesa. Te resultfing model was usedto establfish unfiversal rulesfor systematfic probe desfign and operatfion. Te valfidfity ofthe analysfis wastested agafinst numerfical models and experfimental data. Ffinally, the shear stress on the substrate was calculated for varfious operatfing condfitfions and a workfing range based onthetolerance of cellsto shear stress was establfished.
esu
lts
Inthefir most generalform, dfipolar mficrofufidfic probes comprfisetwo apertures of dfimensfion a separated by a dfistance d from center–to-center. Te apertures are madefinto a fat mesa of sfide dfimensfionsL1× L2, whfich fis suspended above another fat surface, formfing a gap G (Ffig. 1). Flufid fis finjected through one ofthe apertures at a fow rate Qfinj and aspfirated back bythe second aperture at a fow rateQasp= αQfinj, where α fis a parameter expressfing the ratfio between aspfiratfion and finjectfion fow rates (>1 for con-fnementto exfist). Detafiled experfimental proceduresfor MFP operatfions are explafined elsewhere20,21.
HFC data descrfibed here was obtafined from prevfiously publfished experfimental results usfing a partficular MFP geometry (d = 50 µ m, square apertures wfith a = 25 µ m, and mesa dfimensfions of (550µ m× 447 µ m)4. Te gap sfize and ratfio of finjected to aspfirated fow rates were changed from one experfimentto another.
Te sfize of the HFC was determfined experfimentally by finjectfing a solute, namely fuorescently labelled IgG molecules underthe MFP, whfich was atop a substrate wherethe IgGs readfily adsorbed. Te sfize ofthe adsorptfion areafollowfing a 10 s exposure was determfined by fuorescentfimagfing asfillustrated fin Ffig. 2.
Leakage-free HFC was achfievedfor fow ratfios α ≥ 2.5forthe partficular MFP descrfibedfin Ffig. 2. For lower values ofα,thefinjected fufid fow patternfis sufcfientlylargeto extendfintothe area outsfidethe mesa andfis nolonger confned. Te bfindfing adsorptfion patterns of IgGs onthe substrate were controlled by the fow ratfio α and strongest for fow ratfios between α = 2.5 to α = 4. For fincreasfing fow ratfios, the surface patterns gradually vanfished,findficatfingthatthefinjected IgGs werefinstead recaptured bythe aspfiratfionfinlet wfithout reachfingthe substrate surface.
Te confnement as a functfion of the gap between the mesa of the MFP and the substrate was also assessed experfimentally (Ffig. 2B). For gaps varyfingfrom 4 µ mto 50 µ m andfor fow ratfios of 2.5to 4, the HFC fis relatfively finsensfitfive to the gap sfize. However, for smaller gap sfizes and large fow ratfios, a “wfing efect” can occur, by whfichthe dfifusfion gradfientfislocated underneaththefinjectfion aperture (See Ffigs. 2B(7) and 2B(10)). For a fow ratfio of 4,the sfize ofthe adsorbed pattern shrfinks wfithfincreasfing gap sfize. Atlarge enough α values,the adsorptfion pattern dfisappeared completely whenthe gap becametoo large. Tusfor a range of fow ratfios,the surface pattern shrfinksforfincreasfing gap sfizes, but atthe same tfimethefintensfity ofthe fuorescence weakens refectfingthatthelocal concentratfion ofthe moleculesfin solutfion nearthe substrate dfimfinfishes asthe MFPfis movedfurther away (Ffig. 2B). Quantfitatfive crfiterfia have been derfived, based on the theory presented below, to determfine the crfitfical values of α and G under whfichthe adsorptfion patternfis stable (see Supplementary Informatfion).
Theor ormulatfion of the eneral M prolem.Te general confguratfion descrfibfing fowfin a gap of hefight G rfight below a MFPfisfintroduced schematficallyfin Ffig. 1E. Whenthe gapG betweenthe platesfis sufcfiently small,the fow profle betweenthe plates can be descrfibed as a quasfitwo-dfimensfional fow, or Hele-Shaw fow22,23 (See supplementaryfinformatfion), mathematfically akfinto Darcy’slaw24.
υ η → (,, )≈− − ∇→ (, ) () − xyz G z2 G 1 Gz px y 1 H S 2
wfith z befing the vertfical dfistance above the bottom fat surface finsfide the probe where the no-slfip boundary condfitfions are satfisfed at z = 0 and z = G Telow Reynolds number assumptfion behfindthe Hele-Shaw fow approxfimatfion requfires smallfinertfialforces comparedtothe pressure and vfiscousforces
fin the fow. Te Reynolds number of the fow underneath the probe fis Re = ρUL/η, where the charac -terfistfic length L = G2/d under the lubrficatfion approxfimatfion and U fis the characterfistfic velocfity found underthe probe. Fromthe workfing condfitfions yfieldfingthe hfighest Re numberfinthfis paper, we obtafin Remax< 0.11 (a detafiled derfivatfionfis provfidedfinthe supplementaryfinformatfion).
Inthfis confguratfion,the fow exhfibfits a parabolfic fow proflefinthe dfirectfion oflocalfin plane fufid fow. A pseudo two-dfimensfional fow profle arfisfing from a scalar potentfial functfion p(x, y) can be obtafined by defnfingthe hefight-averaged velocfity underthe probe19.
Ffigure 2. HFC patterns for dfiferent fow ratfios and gap sfizes. (A) (1–4) Fluorescence mficrographs ofthe adsorbed protefin pattern are shownfor varfious fow ratfios. (5–8) Correspondfing HFC sfimulatfion resultsfor varfious fow ratfios. Gap between MFP and substratefis kept at G = 10 µ m. Te scale barfis 50 µ m. (B) (1–3, 7–9) Fluorescence mficrographs ofthe adsorbed protefin onthe surface are shownfor dfiferent gap sfizes. Images arefrom Juncker, D., Schmfid, H. & Delamarche, E. Multfipurpose mficrofufidfic probe. Nat. Mater. 4, 622–8 (2005). (4–6, 10–12) Numerfical sfimulatfions showthe concentratfion offinjected protefin atthe bottom substrate. Te fow ratfios are (1–6) 2.5 and (7–12) 4. Scale bar: 50 µ m. In all experfiments and sfimulatfions, thefinjectfion fow rate was set atQfinj= 0.44 nL/s, andthe IgG dfifusfion was setto D = 40 µ m2/s.
∫
υ υ η →(, )= → (,, )=− → (, )∇ () = = − xy G xyz G px y 1 12 2 z z G H S 0 2where η fisthe fufid dynamfic vfiscosfity andp(x, y)fisthe pressure dfistrfibutfion. In open mficrofufidfics,the Hele-Shaw condfitfionfis generally met asfit requfires both creepfing fow (Re ≪ 1) andthatthe fow profle varfies smoothly over a dfistance L muchlargerthanthe gapG (specfifcally (L2≫ G2)) where L fisthe char -acterfistficlengthfin (x, y) ofthe observed fow behavfior. When Re ≪ 1,the superposfitfion prfincfiple can be applfiedto add upthe fow potentfial ofthefinjectfion and aspfiratfion aperturesto compute a doublet19, a fow profle wfith streamlfines mathematficallyfidentficaltothe feldlfinesfin atwo-dfimensfional electrostatfic dfipole, or even more physfically accurate,tothe current densfitylfinesfin atwo-dfimensfional current dfipole.
Anal sfis of thefideal M .Unlfike electrfic charges, no fow apertures can be practfically reducedto a pofint source of streamlfines. Instead,they must be vfiewed as openfings (round, square, or others) of char-acterfistfic dfimensfion α. Yet, whenthe dfistance betweenthe aperturesfis muchlargerthanthe apertures, the pofint source approxfimatfionfis very efectfivefin modelfing multfipolar fow behavfior. We consfider here the cfircular aperture as a general case. Sfince we arefinthe presence of a potentfial fow, Gauss’stheorem9 (Eq. (3)) can be readfily usedto readfily calculatefin polar coordfinatesthe velocfity profle outsfide a round aperture ofthfickness G and radfiusa/2:
∇⋅ = π / < / > / () v Q aGr a r a 4 2 0 2 3 2
In polar coordfinates,the mass balance equatfion provfided above yfieldsthe sfimple result:
υ π π → = , < / , ≥ / () ˆ ˆ Qr aGr r a Q Grr r a 2 2 2 2 4 2
Te fow profle under the MFP can be calculated as a superposfitfion of two monopoles of dfiferent fow rateslocated at ± d/2fromthe orfigfin onthe x-axfis asfillustratedfin Ffig. 1E. When a ≪ d,the pofint source approxfimatfionfis accurate, andthe dfimensfion a can be neglected. Tefull velocfity profle under the probefisthen gfivenfin Cartesfian coordfinates by:
( )
( )
( )
( )
( )
( )
υ π α α →(, )= − − + − + + + + − + − + + . () ˆ ˆ xy Q G x x y x x y x x y x y yy 2 1 5 finj d d d d d d 2 2 2 2 2 2 2 2 2 2 2 2 2 2Stagnatfion pofint and hydrodynamfic confnementlfimfit
When α > 1, a pofint of zero velocfity (stagnatfion pofint) occurs outsfide the finjectfion aperture at a dfistance χSP ofthe center ofthe probe as shownfin Ffig. 1F. Telocatfion ofthfis stagnatfion pofint satfisfes the condfitfion:
(
) (
)
υ π α υ ( , )= − − + , ( , )=. () x Q G x x x 0 2 1 0 0 6 x SP finj SP d SP d y SP 2 2Telatter condfitfion on υyfis satfisfedfor ally duetothe symmetry ofthe problem alongthex-axfis. Te condfitfion on υx specfifesthe posfitfion ofthe stagnatfion pofint:
α α ≡ = + − . () x R d 2 1 1 7 sp
Tfis key result descrfibesthe maxfimum dfistance R fromthe probe center at whfichthefinjected analyte wfilltravel before befing recaptured bythe aspfiratfionfinlet. In other words,finjected analytes are confned wfithfin closed streamlfines as representedfin Ffig. 1F. Te defnfitfion of R fis hfighly analogoustothe con -fnementlength defned earlfierfor mficrofufidfic quadrupoles,fin whfichR d 2= / ( + )/( − )α 1 α 19. In both cases, when α = 1,finjectfion and aspfiratfion fow rates are equal andthe stagnatfion pofintfisfound
atfinfnfity,just asfin a regular dfipoles and quadrupoles. In Ffig. 3A,B,the numerfical values of R are plo t-ted agafinstthefirtheoretfical value obtafinedfin Eq. (7) asfuntfions ofα and G. Coefcfients of determfina -tfion R2 are lfisted fin Table 1. As expected, the best match fis obtafined for the probes where the ratfio of aperture sfizetofinteraperture dfistance (a/d)fis mfinfimum (R2= 0.99for a/d = 1/6) andthe worstforthe largest a/d ratfios (R2= 0.96 and 0.92for a/d = ½). Tusthe R2 value provfidedfin Table 1 can be vfiewed as a qualfitytest ofthe pofint source analytfical model when comparedtothefull 3D numerfical solutfions. An exact expressfion for the wfidth of the HFC underneath an fideal doublet can also be calculated usfing a sfimfilar methodfinvolvfing streamfunctfions (see supplementaryfinformatfion).
Ffigure 3. Scalfing of confnement area and efect of dfifusfion broadenfing on HFC. Te set of sfimulatfion data (shownfin black dots), and scalfinglaws has been shownfin each graphs wfith (a) d = 150 µ m (a = 25 µ m, round), (b) d = 100 µ m (a = 25 µ m, round), (c) d = 80 µ m (a = 25 µ m, round), (d) d = 50 µ m (a = 25 µ m, square). Mesa sfizes were sufcfientlylarge nottofinduce sfignfifcant efect onthe fow. Efect of (A) fow ratfio and (B) gap sfize onthe stagnatfion posfitfion R. Te dfifusfionlength varfies wfith (C)the fow ratfio (~Qfinj−1/2)
and (D)the gap sfize (~G1/2). (E) Te confnement area varfies as afunctfion of fow rate (E) and gap sfize (F)
accordfingtothe scalfinglawsfin A–D. Te gap sfizefin (A) (C) (E)fis 10 µ m. Te fow ratfiofin (B) (D) (F)fis setto α = 2.5. Te default valuesforfinjectfion fow rate was 0.44 nL/s,forthe fow ratfio was α = 2.5,the gap betweenthe MFP andthe substrate was G = 10 µ m, andforthe dfifusfion coefcfient ofthefinjected fow was D = 40 µ m2/s, and were varfied one at atfimefin each experfiment. Te ft coefcfientsfin (E) and (F) are shown
Scalfin laws of dfiusfion roadenfin alon the area.In most practfical sfituatfions, analytes wfithfin the probe wfill have a non-neglfigfible dfifusfivfity and wfill finduce a concentratfion gradfient on the outskfirts ofthe HFC area. Tfis broadenfing wfill yfield sfignfifcantlylarger surface adsorptfion patternsthan expected from the HFC alone. Furthermore, thfis dfifusfive broadenfing can be explofited, as descrfibed elsewhere,to generate rapfidlytunable, hfighly reproducfible sharp foatfing concentratfion gradfients9. Te general advectfion-dfifusfion equatfion of a dfifusfive specfies C(x,y) wfithfin underthe MFPfis descrfibedfin cartesfian coordfinates bythe equatfion:
∂ ∂ = ∇ − ⋅ ∇ = ∂ ∂ + ∂∂ − ∂∂ − ∂∂ () C t D C v C D C x C y v Cx x v Cy y 8 2 2 2 2 2
where v fisfoundfin Eq. (5). Whfile easfily solved usfing fnfite element methods,the unwfieldy Eq. (8) ofers lfittle physfical finsfight fin fits current form. However, a scalfing law descrfibfing the characterfistfic dfifusfion length ofthe HFC can be obtafined bylfinearfizfingthe velocfity profle nearthe stagnatfion pofint usfing a Taylor serfies expansfion of Eq. (5). Terefore:
π α α ≈− ( − ) ( − )+ (( − )) − + ( ) .ˆ ˆ () v Q dG x x O x x x y Oy y 2 1 {[ ] [ ]} 9 finj SP SP 2 3 2 2
To extractthe Péclet number nearthe stagnatfion pofint we make Eq. (8) and Eq. (9) dfimensfionless, by usfingthefollowfing scales:
π π α α = , = , = , = ( − ) = ( − ), = , = , = ( ) v Gd Q v x x d y y d Pe Q DG
x Pex x y Peyt DPe
d t C CC 2 2 8 1 4 10 x finj x finj SP 3 2 0 Tus, afer sfimplfifcatfions, Eq. (8) can be expressed underfits canonficform:
∂ ∂ = ∂∂ + ∂∂ + ∂∂ − ∂∂ ( ) C t C x C y x Cx y Cy 11 2 2 2 2
Although Eq. (11) has separable varfiables, weleavetofuture workthe determfinatfion offitsfull so lu-tfion, whfich would needtotakefinto accountthe curvature ofthe dfifusfionfinterface nearthe stagnatfion pofint, whfichfitself depends on several operatfion varfiables. In a frst approxfimatfion, analogoustothe one performed on quadrupoles elsewhere9, we make Eq. (11) one dfimensfional by settfingthe y-components ofthe concentratfion derfivatfivesto 0. In most cases,thfis approxfimatfionfis hfighlyjustfifed sfincethefin ter-face radfius of curvaturefis muchlargerthanthe dfifusfionlength scale. Furthermore,the dfifusfion broad -enfing regfion, fin almost all cases fin Ffig. 2, dfisplay a unfiform thfickness even far from the SP, only
Scalfinglaw/analytfical Solutfion a b c d A) R vs α R=d2
( )
αα+−11 R2= 0.96 R2= 0.97 R2= 0.98 R2= 0.99 B) R vs G fidem R = 58.33 R = 93.33 R = 116.66 R = 175 R2= 0.92 R2= 0.97 R2= 0.98 R2= 0.99 C) DL vs α =. π α α ( −) DL 128d DG Qfinj 8 13 R 2= 0.91 R2= 0.96 R2= 0.99 R2= 0.96 D) DL vs G fidem R2= 0.98 R2= 0.95 R2= 0.91 R2= 0.97 E) CA vs α = + α α αα + − ( −) CA cd cdDG Qfinj 1 11 2 13 2 c1= 0.44 c1 = 0.35 c1 = 0.29 c1 = 0.23 c2= 15.43 c2 = 22.77 c2 = 17.71 c2 = 25.46 R2= 0.99 R2= 0.99 R2= 0.98 R2= 0.99 F) CA vs G CA c=( +1 c G2 )2 c1= 121.21 c1 = 244.22 c1 = 284.24 c1 = 471.87 c2= 28130 c2 = 10617 c2 = 11900 c2 = 830.31 R2= 0.92 R2= 0.78 R2= 0.98 R2= 0.98Table 1. Analytfical results and scalfinglaws for posfitfion of stagnatfion pofint, confnement area and dfifusfionlength (Ffig. 3).
decreasfing fin thfickness when nearfing the aspfiratfion aperture. Te approxfimatfion fis further justfifed when usfing square apertures asthey fattenthe dfifusfion profle atthe SP even at small curvature radfifi (e.g. Ffigs 2A(7) and 2B(11)). Ffinally,the symmetry plane alongthe y-axfisfimplfiesthat∂ /∂ =C y 0 near y = 0. When consfiderfingthe steady state solutfion only and settfing ∂ /∂ =2C y 2 0,the sfimplfifed Eq. (11) wfithfits boundary condfitfions becomes:
∂ ∂ + ∂∂ =, ( >> )=, ( << )= ( ) C x x Cx 0Cx 1 0Cx 1 1 12 2 2
Te boundary condfitfions fin Eq. (12) are valfid only for hfigh Peclet number, fi.e. when the dfifusfion broadenfing areafisthfin an outsfide ofthefinjectfion aperture, a condfitfiontypfically metfin probe operatfion. Solvfing Eq. (12) yfields:
∫
π ξ ()= = − = ( ) ξ −∞ − Cx 2 e d 1 erf x erfc x 21 2 1 2 2 13 x 2 2where erfc(x) fisthe complementary errorfunctfion25.
ractfical consfideratfionsfin M operatfion.Tetfime scale t0= d2/(4DPe)fisthe characterfistfictfime for whfichthe steady state dfifusfion proflefis achfieved whenthe systemfis perturbed efither by a move -ment ofthe probe or by a changefinthe fow rates applfied. Itfis offinterestto notethatthe settlfingtfime decreases wfith an fincreasfing fow ratfio α . When α = 1, the settlfing tfime becomes theoretfically finfnfite asthe HFC zone extendstofinfnfity. Te characterfistficlength scale = / = /(x xx d0 2Pe)fisthe char -acterfistficlength ofthe dfifusfive broadenfing (concentratfion gradfient) atthe edge ofthe HFC. A dfifusfion length scale can be experfimentally defned asthe dfistance betweenthe pofints of concentratfion 0.1 and 0.9. Applyfingthfis crfiterfionto Eq. (13) yfields:
∆ π α α = = . / =. ( − ). ( ) x DL d Pe d DG Q 128 128 8 1 14 finj 3
Usfing typfical experfimental condfitfions for MFP operatfion9, Q
finj= 0.44 nL/s, G = 10 µ m, d = 50 µ m, α = 2.5, D = 40 µ m2/s (IgG), we fnd the specfifc values of Pe = 60, t
0= 260 ms, and x0= 8 µ m. Tus, the settlfing tfime fis short (<1 s ) fincreasfing lfinearly as t0 ~G and the dfifusfion length fis fafirly steady, fincreasfing as x0 ~G1/2. Ffinally, the area of the HFC scales as the square of the characterfistfic length Lc2 ~[(c1*R + c2*∆ x/2)]2, wherec1 andc2 are proportfionalfity constants. As both R and∆ xlfinearly depend on d,the area ofthe HFC scales as d2 (see supplementaryfinformatfion). Te valfidfity ofthe scalfinglaws derfived fin the prevfious sectfion fis verfifed by comparfing wfith full 3D numerfical sfimulatfions fin Ffig. 3. Strong agreement wasfound betweenthe sfimulatfion andtheoryforthe stagnatfion pofintlocatfion, con-fnement area as well as dfifusfionlength, as descrfibed bythe R2 values compfiledfin Table 1. Calculatfions of scalfing laws for the confnement area has been explafined fin further detafils fin the supplementary finformatfion.
Shearstress analsfis. Te MFP has been used to expose cells and tfissues wfith varfious chemficals, and anfimportant parameterthat canfinfuencethetreatment outcomefis shear stress. Shear stress plays a major rolefin many bfiologfical phenomena brfingfing about a number of cellular/fintracellular events26–28. For example, endothelfial cells and neutrophfilstransducethe applfied shear stress stfimulusfintofintrace l-lular bfiochemfical responses, by whfichthey regulatethe vessel structure29,30. Sfimfilarly,these stfimulfi also afectthe orfientatfion of osteoblasts and neurons resultfingfin cellular mfigratfion and matrfix outgrowth31,32. Anfimportant applficatfion ofthe MFPfis detachfing shear sensfitfive cells wfithout damagfingthe cell mem-brane, andthus calfibratfion of shear stressfis necessary.
In Hele-Shaw fows,the shear stressfis dfirectly proportfionaltothe normalfized velocfity profle:
τ ηυ τ υ τ η π (, )= →(, )= .→(, )xy , = . ( ) G xy xy Q Gd 6 6 15 finj 0 0 2
Te maxfimum value τ maxfisfound wherethe maxfimum value ofν lfies,fi.e. at (x = −(1 − a/d),y = 0) nearthefinner edge ofthe aspfiratfion aperture (Ffig. 4A). To refectthfis observatfion, we rescale τ max usfing the aspfiratfion fow rateα Qfinj asthe new characterfistfic fow rate and usfingthe aperture sfizea finstead of probelength d asthe new characterfistficlength. It ensuesthat when = /a ad 1 orα ≫ 1,the contr fi-butfion ofthefinjectfion aperturetothe maxfimum velocfity becomesfinsfignfifcant, gfivfingthe sfimple scales τmax≈ατ0/a andτmax≈.0866ατ0/a, respectfivelyforthe round and square aperture probesforthe maxfimum shear rate under the probe (see supplementary finformatfion). Tus maxfimum shear stress fis finversely proportfional to gap G (~1/G2, Ffig. 4B) and varfies lfinearly wfith the aspfiratfion fow rate Q
asp. Hence, we expect the maxfimum shear stress to vary lfinearly wfith the fow ratfio, as shown fin Ffig. 4C. Maxfimal shear stressfis anfimportant experfimental parameter sfincefin bfiologfical systems cells are afected
by shear, andfor examplethe growth rate of endothelfial cell was shownto be susceptfibleto shear aslow as 1 Pa33.
Hfigh shear stresses can cause damage to the cell membrane, and consequently, lead to cell death34. To account for thfis fissue fin MFP desfign and operatfion, the maxfimal shear stress at the bottom of the substratefis characterfized frst by varyfingthe fow ratfio andthen by varyfingthe gap sfize whfile keepfing the other parameter constant. Combfinfingthe knowledgefromformer studfies of cell detachment usfing low shear stress (around 0.5Pa)32to detach PC12 neural cells and maxfimum shear stress that can be tolerated by neurons (around 1.5Pa)32, we performed a set of sfimulatfions wfith dfiferent gaps and fow rate ratfios. By consfiderfing both lfimfits descrfibed above, we outlfined the safe operatfional parameters of the MFPto work wfith neurons (Ffig. 4D).
contour and wfidth.Anotherfimportant applficatfion ofthe current MFP modelfisto allow pre-cfise control overthe shape and dfimensfions ofthe probe wrfitfing head based on user-controlled parame-ters. When usedfin surface patternfing mode,the dfimensfions ofthe HFC area wfill dfictatethe shape ofthe “brush stroke”thatthe MFPleaves whenfit wrfites on a surface. One can proceedto vfisualfizethe shape of the hydrodynamfic fow confnement (HFC) area by determfinfing the assocfiated stream functfion22 of the velocfity profle. In generalterms,the streamfunctfionfis defned as:
ψ ψ (, )=∂ ∂ , (, )=−∂∂ ( ) v xy y v xy x 16 x y
Tus,fintegratfingthe y-velocfity componentfin normalfized coordfinates of Eq. (5) yfields:
Ffigure 4. Shear stress dfistrfibutfions under the two apertures MFP. (A) Dfimensfionless shear stress dfistrfibutfion profles alongthe mfiddlelfine atthe bottom substrate showfingthat max shear stress occurs at thefinner edge ofthe aspfiratfion aperture (α = 2.5, Qfinj= 0.44 nL/s). (B) Maxfimum shear stress atthe bottom
ofthe substrate as afunctfion of gap sfize G (α = 4, Qfinj= 0.44 nL/s). Te dashlfine representsthe shear stress
approxfimatfionfound by neglectfingthe contrfibutfion ofthefinjectfion aperture (monopole approxfimatfion), whfich slfightly underestfimates Eq. (15) by a maxfimum of 15% when d/a > 2 and α > 2. (C) Maxfimum shear stress atthe bottom surface as afunctfion ofthe fow rate ratfio α (G = 10 µ m,Qfinj= 0.44 nL/s). Dashlfine:
monopole approxfimatfion ofthe shear stress. (D) Calfibratfion of MFP’s operatfion condfitfions wfith respectto two parameters of fow ratfio and gap sfizeto apply shear stress on neurons. Te upper andlowerlfimfits are 1.5 Pa and 0.5 Pa, respectfively. Te hatched zone represents approprfiate condfitfion of fow ratfio and gap sfize forlocal cell detachment bytrypsfinfizatfion whfile avofidfing shear-finduced damagetothe surroundfing cells.
∫
π ψ(, )=− ( , ) =− α − + + , ( ) G Q xy v xydx x y x y2 arctan 1 arctan 1
17
finj y
wherethefintegratfion constant can be setto zero. Asthe streamlfines defnethetrajectory of anfinfnfi tes-fimal partficlefinthe fow feld,the streamlfine passfing bythe stagnatfion pofint (2xSP/d= (α + 1)/(α − 1), 0) wfill descrfibethe envelope ofthe HFC (neglectfing dfifusfivfity). In Eq. (11),the streamlfine correspondfing tothfis condfitfionfis
α α π − − + + =( − ) . ( ) x y x y
arctan 1 arctan 1 1
2 18
Hence, Eq. (18) fis the functfion descrfibfing the upper half (y > 0) of the HFC contour. Whfile the varfiables cannot be separated,the shape ofthefimplficfit streamfunctfionfis plottedfin Ffig. 5Aforvarfious values of α.
Te HFC area or sfimply confnement area (C. A.),fi.e.the area ofthe spotlef bythe probefin surface patternfing mode, can befound by plottfing Eq. (18)for a specfifc value of α and numerficallyfintegratfing under the curve. One exceptfion occurs when α = 2 where Eq. (18), once sfimplfifed, yfields (2x/d − 1)2+ (2y/d)2= 4, whfichfisthe equatfion of a cfircle of radfius x = d centered atthefinjectfion aper -ture (x = d/2) (Ffig. 5A). Te confnement area ofthe HFCfinthfis partficular casefis C.A. = πd2. Another exceptfion occurs when α = 3, whfich also has a sfimple solutfion: C.A. = 3 3 4d2/= 1,299 d2. Other cases can befound numerfically,for example C.A. = 8,712 d2 (α = 1.5), C.A. = 0,8264 d2 (α = 4), C.A. = 0,6130
Ffigure 5. HFC shape and dfimensfion. (A) Streamfunctfions representfingthe contour ofthe HFCfinthe symmetrfical upper half planeforthefideal pofint source probe. Plots are presentedfor varfious fow ratfios α. Te partficular case ofα = 2fis a half cfircle of radfiusfidentficaltothefinteraperture dfistance d. (B) Plot ofthe normalfized half wfidth W(α)/d andthe normalfizedlength2R/d ofthe HFC wfith respecttothe fow rate ratfio α = Qasp/Qfinj. Althoughthe wfidthfis systematfically shorterthanthelengthfin all cases,the
two dfimensfionsfollowthe same scale (lfinear wfithd, and varyfing as 1/(α − 1)). (C)Varfious brush strokes produced bythe movfing MFPfin surface patternfing mode. Movement (C) parallelto probe axfis, (D) perpendficularto probe axfis, (E) arbfitrary at an angle θ wfiththe probe’s axfis.
d2 (α = 5), C.A. = 0,4908 d2 (α = 6). In all cases, confnement area wfill be proportfionaltothe square of thefinter-aperture dfimensfiond and wfill decrease wfithfincreasfing fow ratfioα. Dfifusfionfis setto 0 every -wherefinthfis sectfionto hfighlfightthe contrfibutfion ofthe fow profle alonetothe C.A.
Theoretfical fidth.Perhapsthe sfimplest and most generally applficable result ofthfis paperfisthe posfitfions of the probe’s stagnatfion pofint R as descrfibed fin Eq. (7). Another useful measurement fis the HFC maxfimum wfidth W,the maxfimum extent ofthe HFCfinthe dfimensfion perpendficulartothe probe’s mafin axfis. Te HFC wfidth as a functfion of fow rate has been studfied numerfically and experfimentally elsewhere18, but wfithoutthefinsfight provfided by a complete mathematfical derfivatfion, whfichfis provfided hereforthefideal case of pofint sources underthe Hele-Shaw fow assumptfion.
Sfincethe contour ofthe HFC can be descrfibedtheoretfically bythe streamfunctfion descrfibedfin Eq. (18), we can compute the local maxfimum of the HFC wfidth fin the y dfimensfion by dfiferentfiatfing the streamfunctfion wfith respectto x suchthat
ψ ∂
∂x=− =.v 0y ( )19
Physfically,thfis representsthe pofint atthe edge ofthe HFC at whfichthe y component ofthe velocfity fisfidentfically zero. Usfingthe normalfized coordfinates, we get:
α α α α ( )= + − − ( − )− . ( ) xy 1 y 1 4 1 20 max 2 max2
Tfisfisthe condfitfionthat must be respected when vy= 0 andy = ymax= W/2, whereW fisthe HFC wfidth. Substfitutfing Eq. (20)fin Eq. (18) yfields:
( )
( )
α α π − − − − + − − +( − ) =. ( ) α α α α α α α α + − ( −) + − ( −) ∼ ∼ ∼ ∼arctan 1 arctan 1
1 2 0 21 W W W W 1 1 4 1 4 2 1 1 4 1 4 2 2 2 2 2
Tfis fis the transcendental equatfion that analytfically descrfibes the HFC’s normalfized full wfidth ( = /∼W 2W d). Te wfidthfunctfion α∼W()fis plottedfin Ffig. 5B. Notethat Eq. (21)fis a normalfizedfunctfion andthatfitfistherefore valfidfor any MFP geometry wherethe pofint source approxfimatfion can be made.
D
fiscuss
fion
Applficatfions to rush stroke analsfis and proe operatfion control. One ofthe mostfimportant applficatfions ofthe current modelfisto be ableto controlthe shape and dfimensfions ofthe probe wrfitfing head by specfifyfing user-defned parameters such asthefinjectfion and aspfiratfion fow rates orthe probe gap G. Te contourfunctfion descrfibedfin Eq. (18), combfined wfiththe HFC maxfimum radfiusR (Eq. (7)) and the HFC wfidth W (Eq. (21)) completely defnes the area and maxfimum dfimensfions of the MFPs wrfitfing area. To accountforthe dfifusfive broadenfing aroundthe contour, Eq. (14) can also be applfied. However, allthese parameters specfifythe shape ofthe probe’s wrfitfing under statfic condfitfions. Whenthe probefis movfing at constant velocfity U fin any gfiven dfirectfion wfith respectto a fxed substrate, vfiscous forces wfilltendto deformthe statfic fow profle. Tefully developed Hele-Shaw profle wfill be a super -posfitfion of a parabolfic fow profle stemmfing from the probe apertures and a Couette fow profle of mean velocfity U/2 finducedfinthe dfirectfion ofthe probe dfisplacement. Tfis fow behavfior can be reduced to a purely parabolfic fow profle as fin Eq. (1) by placfing oneself fin a referentfial movfing fin the dfirec -tfion ofthe probe atthe mean fufid velocfityU/2. In dofing so,the hefight-averaged fow proflefoundfin Eq. (2)fis kept unchanged andthe fowfinthfis referenceframefisfidentficaltothat of Eq. (5) (see proof fin supplementaryfinformatfion). Once solved,the velocfity proflefinthe probe’s movfingframe, usfingthe dfimensfionless scales descrfibedfin Eq. (10), nowtakestheform
α β θ α β θ →(, )= ( − ) ( − ) + − ( + ) ( + ) + − + ( − ) + −( + ) + − , ( ) ˆ ˆ vxy x x y x x y x x y x y yy 1 1 1 1 cos 1 1 1 sfin 22 2 2 2 2 2 2 2 2
where θ fisthe angle ofthe probe velocfity vector wfiththe probe axfis andβ = πGUd/2Qfinjfisthe dfimen -sfionless probe velocfity. When β > 1,the probe velocfity dfisturbsthe HFC contour sfignfifcantly. Yet when β ≪ 1, the contour can be assumed not to be perturbed by the probe movement. Usfing the typfical
operatfion parameters prevfiously descrfibed (Qfinj= 0.44 nL/s, G = 10 µ m, d = 50 µ m), we fnd that the probe perturbatfion duetofits velocfityfis neglfigfible when U ≪ 2Qfinj/πGd = 0.6 mm/s. Tfis crfitfical velocfity can be made arbfitrarfilylarge wfithout changfingthe HFC contour byfincreasfing Qfinj andQasp proportfion -ally whfile keepfingthe ratfio α constant.
Wfithfinthfis approxfimatfion of small β, we now defne a “brush stroke”forthe probe operatfion where the wfidth ofthe stroke wfill depend onthe user-controlled parameters,fincludfingthe probe’s velocfity and dfisplacement angle (Ffigs 5(C–E)). Te brush wfidth can be decomposed fin fits two prfincfipal axes: the wfidth ofthe probe when movfing parallel and perpendficulartothe probe’s axfis, respectfively W andW ,⊥ where W fis calculated usfing Eq. (21) and
α α = + = − . ( ) ⊥ W d R d 2 1 23
Te lfinear combfinatfion descrfibfing the brush stroke at low velocfitfies for an arbfitrary xy probe d fis-placementfis generally descrfibed as:
α α θ α θ
()= ( ) + ⊥() . ( )
Weff W cos W sfin 24
It fis fimportant to pofint out that the dfifusfion broadenfing has not been taken finto account when derfivfing Eq. (24) andto do so would requfire needlessly cumbersome mathematfical derfivatfions. However, the efect of dfifusfion can be sfimplytakenfinto account by makfingthe approxfimatfionthatthe dfifusfive broadenfingfis small comparedtothe HFC radfius andfincreasesfit unfiformly by a dfistance ∆ x, obtafined from Eq. (14), on each sfide except near the aspfiratfion aperture. Under thfis approxfimatfion,
α α ∆ α
(, )= ()+ ( , )
W D W 2x D andW⊥(, )=αD W⊥()+ ( , )α ∆ αx D and Eq. (24) can stfill be used.
onc
lus
fion
Inthfis paper, we have provfided atheoretfical analysfis offideal dfipolar probes (d ≫ a ≫ G) andtestedthe valfidfity of our transport model under probes wfith fnfite aperture sfizes and arbfitrary geometrfies. Te analysfis fis then valfidated wfith data obtafined from sfimulatfions and prevfiously publfished experfimental data. Te general set of analytfical results and scalfinglaws are summarfizedfin Table 2. Te sfimple results establfished wfill allow precfise control of HFC dfimensfions and confnement area, dfifusfion length and gradfient propertfies atthe HFC’s edges and shear stress control wfith respectto fow rate ratfio α and gap sfize G. Tey wfill provfide precfise desfign crfiterfia (d,a, and mesa dfimensfions, durfingfabrficatfion process) and control the probe’s brush stroke and velocfity durfing operatfion (α, G, Qfinj, D). For example, fif the user seeks a cfircular HFC of area A atthetfip,thefinter aperture dfistance d to obtafinthfis exact area can readfily be calculated (finthfis specfial case, when α = 2,the HFC areafis precfisely A = πd2, see supplemen -taryfinformatfion). Oncethe value of dfisfound,the dfimensfions ofthe mesa need onlyto be afewtfimes largerthanthe maxfimalR value ofthe HFC (wfithR = 3d/2 whenα = 2). Ten, usfingtheformulaforthe HFC outer edge (Table 2),the precfise shape ofthe HFC couldthen be calculatedfor any value of α for
Desfign Crfiterfia Formula Valfidfity Domafin HFC outer edge – Eq. (16) ψ xy(, )=( − ) α 1π
2 Ideal 2D MFP
Stagnatfion posfitfion (R) – Eq. (6) = +−
R 2d QaQasp Qfinjsp Qfinj R < < L1,LR ≫ 2 (mesaG, d d≫ fimensa fions), HFC Wfidth parallelto axfis – Eq. (20) W// Idem
HFC Wfidth perpendficularto axfis – Eq. (22) = + = α α ⊥ − W d R d 2 1 Idem Péclet Number – Eq. (9) = π α α ( −)
Pe 8QfinjDG 13 Vficfinfity of SP Dfifusfionlength scale (DL) – Eq. (12) DL d Pe∝ / R ≫ G,d ≫ a HFC Responsetfime – Eq. (9) t∝( −)ααG13 Under Hele-Shaw Condfitfion Shear stress (τ ) – Eq. (14) τ= →/6ηv G Idem
Max Shear stress
τmax≈ατ0da/ Round apertures τmax≈.086ατ0da/ Square apertures τ =0 6πηQfinjGd2 Characterfistfic scale
Table 2. Desfign crfiterfia, formula and valfidfity domafin of two apertures MFP. Tfis parameters and related formula provfide substantfial gufide to select best experfimental condfitfions for cell study applficatfions.
thfis probe. Ffinally,the outer edge dfifusfionlayer can be adjusted durfingthe experfiment by modulatfing the gap sfize G whfile mafintafinfing the HFC’s cfircular shape constant. In the prevfious example, we can also determfinefin advance whether we need square/round, orlarge/small apertures. Small apertures wfill be preferredfin sfituatfions where α fislargeto preventthe “wfing efect” (Ffig. 2B-10)that occurs whenthe stagnatfion pofint fis found finsfide the aperture. Large apertures are, on the other hand, preferable when the shear stress needsto be mfinfimal (Ffig. 4B,C).
We also provfide a complete characterfizatfion of the MFP’s “brush stroke” when operatfing fin surface patternfing mode. Te analysfis suggeststhatthe sfimplest wayto precfisely controlthe wfidth ofthe stroke fisto movethe MFP efither parallel or perpendficulartofits axfis sfincethe wfidth ofthe stroke ( ,W⊥ W )// fis then completely findependent of the fow velocfity and gfiven a sfimple formula (Table 2) that can be easfily modfifed to account for dfifusfion by addfing the dfifusfion broadenfing scale (DL, Table 2) to the wfidth.
Te results presented fin Table 2 can also be used to findfirectly measure and calfibrate probe hefight above a surface bylookfing at dfifusfionlength wfith respectto G (∝ √G) or shear stress as afunctfion of the applfied fow rates. Ffinally,the generaltheoretficalframework provfided here, andthe set of analytfical resultsthat stemfromfit, enables a more precfise controlfor MFPs of any shape orformfin a varfiety of applficatfions fincludfing neuron mfigratfion35, stem cell dfiferentfiatfion36,37 and shear-dependent cell pora-tfion38,39 wfithout usfing costly, complficatedtrfial and error experfimental method.
Methods
umerfical Modelfin. Te numerfical models were establfished usfing the fnfite element method wfith commercfial sofware (COMSOL Multfiphysfics 3.5a, USA) on a computer wfith 8 cores, 64-bfit CPU and 26 GB RAM. Te computatfional model was bufilt by combfinfing Stokes fow and steady state convectfion-dfifusfion mass transfer equatfions. Boundary condfitfions were assumed to be “no-slfip” con-dfitfions at solfid surfaces and “open boundarfies” at atmospherfic pressure atthe perfimeter ofthe MFP. To computethe masstransfer model,the MFP’s rfigfid walls and substrate surface are consfidered asfinsulators andthe fux was setto zero. Te concentratfion atthe perfimeter was setto zero refectfingthe assumptfion that under HFC no solute reachesthe edge, andthatthelarge volume constfitutes a sfink. Concentratfion atthefinjected aperture was normalfized and setto 1to provfide a relatfive scale. Atthe aspfiratfion aperture, dfifusfion was neglected owfingtothelocal domfinance of convectfive masstransport.
Numerfical calculatfions of the HFC were performed and the calculated concentratfion as a functfion of posfitfion atthe substrate surface was reported asfunctfion of fow ratfiofin Ffig. 2A, and asfunctfion of the gap sfize, Ffig. 2B and Ffig. 2B. Terefis a good vfisual agreement between experfimental and numerfical results, whfich was also quantfifed by measurfingthe surface areafor each case, and comparfingfitfor dfi f-ferent fow condfitfions (Ffig. S1). Te observed dfiscrepancy between numerfical and experfimental results can be mafinly attrfibuted to the findfirect measurement of protefin adsorptfion at the surface rather than the dfirect measurement ofthe fow profle. Uncertafinty onthe experfimental gap, exposuretfime, as well as vfibratfions of the MFP whfile runnfing the experfiment wfill cause the protefin footprfint on the surface to spread. As a consequence,the pattern onthe surfacefis not a record ofthe average pattern, but ofthe maxfimal extent ofthe fow proflethat may occur dueto varfious fuctuatfions durfingthe 10 s of streamfing acrossthe surface. Nevertheless,the numerfical models arefin good agreement wfiththe adsorbed protefin experfiments. To extend the valfidfity of our models to any probe geometry and to extract useful desfign crfiterfia based on scalfing laws (see sectfion “Teory”), numerfical sfimulatfions of the MFP wfith varfious geometrfies and gaps, and under dfiferent fow rates were performed usfingfull 3D fnfite-element models.
To test the lfimfits of our advectfion/dfifusfion models, we have proceeded to perform the followfing serfies of sfimulatfions emulatfing probe geometrfies already used fin the lfiterature, Table S1. Water was used as fufid (solvent) wfith a densfity of 998.2 kg/m3 and dynamfic vfiscosfity η = 0.001 N·m/s (at 20 °C). Immunoglobulfin G (IgG) was used as solute and a dfifusfion coefcfientfin water of D = 4 × 10−11 m2/s was used7. Te fow regfime was estfimated prfiortothe sfimulatfionsto help selectthe approprfiate models. Te Reynolds number fin all regfimes was on the order of Re ≈ O(10−2)≪ 1 findficatfing creepfing fow cond fi-tfionsto betreated usfing Stokesformalfism22. Sfimfilarly,the Peclet number representfingthe ratfio between convectfive and dfifusfive masstransport wasfoundto be hfigh (Pe ≫ 1)40for all MFP operatfion regfimes, findficatfingthat convectfive masstransport domfinates dfifusfion except very closetothe stagnatfion pofint.
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Acknow
led ments
Te authors wfishto acknowledge supportfrom FQRNT and NSERC. M.S.thanks supportfrom FRQNT scholarshfip and ISS NSERC-CREATE fellowshfip. D.J. acknowledges support from Canada Research Chafir. We thank Farhang Tarlan and Gufillaume Émond for carryfing out several sfimulatfions, and Roozbeh Safavfiehforfinsfightful dfiscussfions.
Author
ontr
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M.S. desfignedthe research, carrfied outthe FEM, and wrotethe manuscrfipt. M.A.Q and A.V. desfigned the research, wrotethe manuscrfipt. D.J. desfignedthe research, wrotethe manuscrfipt. T.G desfignedthe research, carrfied outthe mathematfical analysfis, and wrotethe manuscrfipt.
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Supplementaryfinformatfion accompanfiesthfis paper at http://www.nature.com/srep Competfing fnancfialfinterests: Te authors declare no competfing fnancfialfinterests.
How to cfite thfis artficle: Safavfieh, M.et al. Two-Aperture Mficrofufidfic Probes as Flow Dfipoles: Teory and Applficatfions. Scfi. Rep. 5, 11943; dofi: 10.1038/srep11943 (2015).
Tfis workfislficensed under a Creatfive Commons Attrfibutfion 4.0 Internatfional Lficense. Te fimages or otherthfird party materfialfinthfis artficle arefincludedfinthe artficle’s Creatfive Com -mons lficense, unless findficated otherwfise fin the credfit lfine; fif the materfial fis not fincluded under the Creatfive Commons lficense, users wfill need to obtafin permfissfion from the lficense holder to reproduce the materfial. To vfiew a copy ofthfislficense, vfisfit http://creatfivecommons.org/lficenses/by/4.0/
Erratum
: Two-Aperture
M
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Probes
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Mohammadalfi Safavfieh, Mohammad A. Qasafimeh, Alfi Vakfil, Davfid Juncker & Thomas Gervafis
Scfientfifc Reports 5:11943; dofi: 10.1038/srep11943; publfished onlfine 14 July 2015; updated on 08 October 2015
Te orfigfinal versfion ofthfis Artficle contafined an errorfinthetfitle ofthe paper, wherethe word “Dfipoles” wasfincorrectly gfiven as “Dfipole”.
In addfitfion,there weretypographfical errors.
Inthe Results sectfion under subheadfing‘Practfical consfideratfionsfin MFP operatfion’, “Usfingtypfical experfimental condfitfionsfor MFP operatfion7”
Now reads:
“Usfingtypfical experfimental condfitfionsfor MFP operatfion9” In reference 33: “Younga” now reads “Young”.
Tese errors have now been correctedfin boththe PDF and HTML versfions ofthe Artficle.
Tfis workfislficensed under a Creatfive Commons Attrfibutfion 4.0 Internatfional Lficense. Te fimages or otherthfird party materfialfinthfis artficle arefincludedfinthe artficle’s Creatfive Com -mons lficense, unless findficated otherwfise fin the credfit lfine; fif the materfial fis not fincluded under the Creatfive Commons lficense, users wfill need to obtafin permfissfion from the lficense holder to reproduce the materfial. To vfiew a copy ofthfislficense, vfisfit http://creatfivecommons.org/lficenses/by/4.0/
