1 Nanoliter Droplet Behaviour inMicr omachinedWells

K.T. HJELT�, G.W. LUBKING

�, M.J. VELLEKOOP

�,

L.J.VAN VLIET�, L.R. VAN DEN DOEL

�,

A. GREINER�, andJ.G.KORVINK

��ElectronicInstrumentationLaboratory, DIMES, Delft Universityof

Technology, Meckelweg 4, NL-2628CD Delft, TheNetherlands�PatternRecognitionGroup,Delft Universityof Technology,

Lorentzweg 1, NL-2628CD Delft, TheNetherlands�Institutefor MicrosystemTechnology, AlbertLudwigUniversityFreiburg,Georges-

Koehler-Allee 103,D-79085Freiburg, Germany

Chemicalandbiochemicalanalysisstandtogaintremendouslyin costanefficiencythroughminiaturisation. In fact, this trendis alreadysteadilyunderway. open � -reactorsor � -vial aredesirablebecauseof their relative easeof manufacture,andthe easeof accessingthe vials for the purposeof filling. They also do not needliquid handlingpumps,etc. A disadvantage,with respectto closedreactors,is theevaporationof liquid from theopensurface.Sincethis will bea recomingthemeinsimilar equipment,it is necessaryto understandtheevaporationprocesssufficientlywell, to characteriseit experimentally, andto developtechniquesto enablereal-timemonitoringof thevolumeof liquid still present.

By reducingthe dimensionsof liquid dropletsdown to nanolitervolumesweencounterphenomenathatarestronglyrelatedto theincreasedratiobetweensurfaceandvolume. Surfacetensionandshapeplay a dominantrole, while volumeeffectslike gravity are of minor importance. Appropriateexperimentaltechniquesandsimulationmethodsarerequiredfor thedescriptionof nanoliterdropletbehaviour.

In thischapterwediscussthesethreeaspects.Section1.1describesthesubnano-liter volumedetectionprinciples1.1.1,thedevice fabrication1.1.2andthemeasure-mentsetup1.1.3aswell astheperformedmeasurements1.1.4.Section1.2.1of thischapterexaminesthe temporalandspatialresolutionof threedifferentmicroscopictechniquesto studythedynamicbehaviour of liquid samplesin wells. Section1.2.2briefly introducestheopticalmodelbehindinterference-contrastmicroscopy. Withthis type of microscopy, imagesof interferencepatternsshowing fringesof equalheight in the liquid arerecorded.Section1.2.3presentsa signal-processingalgo-

i

ii NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

Fig. 1.1 A siliconsubnanoliterarray(chipsize1x2cm) leaningagainstaclassicalplate.

rithm to analysethesefringe patternsin time. This algorithmresultsin the heightprofilesof themeniscusduringtheevaporationprocess.Fromtheseheightprofileswe computethe importantfeaturesof the meniscus,suchas the evaporationrateandvariouscurvatures.Resultsof thesecomputationsareshown in Section1.2.4.Section1.3 describesthecomputationof thesurfaceshape,andits evolution underevaporation. In section1.3.1thephysicalmodelsareexplainsandin section1.3.2thecomputationof theflow field insidethedroplet,andtheimpedanceof thewell asafunctionof theliquid dropletshapewith respectof two electrodesembeddedin thewell baseis presented.A conclusionwill bedrawn in section1.3.3

1.1 SUBNANOLITER VOLUME DETECTION

Miniaturisationof (bio)chemicaltestequipmentyields considerablesavings in thecost of reagentsand compounds. It also allows more teststo be carriedout perunit time. Re-engineeringof existing measurementinstrumentationis requiredtoenable(sub)nanoliteranalysisof arrays.In ordertoobtaintheintendedmeasurementspeed,miniaturisationof analysissystemsbysiliconorglassintegrationisanobviouschoice.Additionally,byaddingintegratedelectronics,thefunctionalityof thesystemcanbeenhanced(e.g. by incorporatingauto-calibrationor compensationcircuitry).A largeamountof datageneratedby thesensorshasto beprocessedandshouldleadto thedesiredanalysisspecification.In Fig. 1.1,a first-generationsiliconarraychipis shown.

Dispensingandmaintainingsmallsample-volumesin (sub)nanoliterwells intro-ducesnew aspectswhichshouldbeconsideredin thedevelopmentof new instruments.Unwantedevaporationof the samplefrom the wells is suchan aspect.In ordertomonitortheevaporationbehaviourfromthewells,animpedancebasedliquid-volumesensorhasbeendevelopedaspartof theanalysissystem.Themeasurementsof thissensorarecomparedwith analternativeopticalvolumedetectionsystem.

SUBNANOLITER VOLUME DETECTION iii

Fig. 1.2 Electricalequivalentcircuitry of thereactor, insulationlayers( ��� ��� and ���� � ),andtheSi substrate.

1.1.1 The measurementprinciple

Silicon integratedarraysmake it possibleto useanelectricalmethodto measuretheamountof liquid in awell or reactor. Thepresenceof theliquid alterstheimpedancebetweentheelectrodes,which canbecorrelatedwith thevolumeof the liquid. Theuseof asiliconsubstrateopensupthepossibilityto integratethevolumemeasurementsensortogetherwith theelectronicsononechip.

In anidealmeasurementsystemthecurrentthroughtheliquid wouldbedeterminedsolelyby theelectricalpropertiesof theliquid. In practice,theimpedance�� of thesystemis notonly formedby thecapacitance��� andresistance��� of theliquid. Theelectricalequivalentcircuit of thereactorsystemincludingthecapacitancesformedby theinsulationlayers( �� � � and ���� � ) is depictedin Fig. 1.2Thecurrentthroughthe Si substrateis negligible becauseof the small valueof the capacitance������� � ,and the groundingof the substrate. By inspectingFig. 1.2, it can be seenthatthe impedance! is formedby thecapacitances�"�#�%$'&)( , andtheresistance* � andcapacitance+ � throughtheliquid:

! -, . * � �/"021 � + � � * � � 043 51 + ��� $ & (76 � (1.1)

where1 , 598;: , and: isthefrequency of theACcurrent. Therangeandsensitivity

of theelectricalliquid volumedetectionisgreatlyaffectedby thecapacitance+��#�%$'&)( .By using(1.1),thecasesfor non-conductiveandconductiveliquidscanbeinspected.It is straightforward to seethat when the resistanceR� decreases(i.e. the caseof a highly conductive liquid), the impedanceasymptoticallyapproachesthe valueof 5=<?> 1 + ��� $ & ( @ , which determinesthe maximumvaluefor the current. It is thusimportant to maximise + �#� $ & ( , for example,by minimising the thicknessof theinsulating �?�� �A� layer. The sensitivity of the measurementalso decreaseswithdecreasingliquid resistanceasthecurrentsaturates.

To illustrate the considerationspresentedabove, a simplified calculationin thecaseof a moderatelyconductive anda highly conductive liquid canbe made. Theresistanceof aliquid layerin asquarereactorcanbeapproximatedto dependonly ontheresistivity B � andthedepthC of theliquid layer * �-D B � <EC . Thisapproximatein-

iv NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

verseproportionalitywasvalidatedwith finite elementsimulations.Thecapacitance+���� $ &)( isof theorderof 0.2FHG in the60 FHI reactor. By insertingtheabovevaluesinto(1.1)we cancalculatethecurrentchangeJLK andthecorrespondingvoltagechangeJNMPO for ourset-upwhentheheightof theliquid layerincreases,say, from 3 �)Q to 6�)Q . We performedtestswith waterandwater/glycolmixtures.Thevoltagechangefor a typicalglycol/watermixture( B?� ,SR=T U U=V Q ) is about JNM O ,WUYX /[Z M , whereasit reducesto about J\M O ,SUPX U / M whencalculatedfor liquid with B?� ,SR=T V Q . Theliquidsusedin ourexperimentshaveresistivitiesrangingfrom

/ V Q toZ U U=U=V Q . For

example,a solutionof 0.1M Glucose-6-phosphatein water, and0.2M ammoniumchloride in 60% /40% (v/v) glycol/watermixture hasa resistivity of about

/ V Q ,whereas90%/10%(v/v) glycol/watermixturehasaresistivity of about

Z U U=U=V Q . Forthecaseof anon-conductiveliquid, theimpedance�] isdeterminedalmostpurelybytheliquid capacitance�^� , ascanbeseenfrom(1.1). Themeasurementis thenmainlylimited by theresolutionof themeasurementelectronics.finite elementsimulationswere conductedin order to estimatethe penetrationdepthof the electric field intheliquid whenmeasuringnon-conductiveliquids, for exampleglycol. Simulationsweremadefor differentreactorgeometries.Thecalculatedcapacitancefor our6 �)Qdeep60 F_I reactorvariesbetween2 fF (empty)and10 fF (filled). Thechangeof thecapacitancecanbeseento saturatewhentheliquid layerthicknessincreases.Fromour simulationsit was estimatedthat the penetrationdepthand thus the sensitivemeasuringrangeis approximatelyhalf of theelectrodedistance.Therequirementsfor theelectronicsweresetto enableus to measurebothnon-conductive liquids aswell asconductive liquids. This requiresa resolutionof about10 pA for thecurrentmeasurementin orderto achieve a 1 FHI resolutionfor thevolumedeterminationfornon-conductiveliquids.

1.1.2 Device fabrication

A subnanoliterarray chip with integratedelectrodesis shown in Fig. 1.3. Thesizeof thechip is 5x5 Q`Q � andit has

/ 5 differentreactorswith differentelectrodegeometries.Thedepthof thereactorsis 6 �)Q andthesizevariesfrom 100x100�)Q �to 500x500�)Q � correspondingto volumesfrom 60 FHI to 1.5 a I , respectively. Thelight areaswithin thedarkrectanglesarethealuminiumelectrodes.The light area,which separatesthedark rectanglesis theelectricalguard. This guardis groundedby themiddlewire of eachreactorset-up.A schematiccrosssectionof a reactorisshown in Fig. 1.4. Thereactorsarefabricatedon a silicon wafer. First thewafer iscoveredby a2 �)Q thick layerof SiO� . ThisSiO� layeractsasanelectricalinsulatorandreducesthecapacitancebetweenthesilicon andtheelectrodes,thusdecreasingtheparasiticcurrent. The300 a)Q thick aluminiumelectrodesarepatternedon theSiO� layer and coveredby a 500 a)Q thick Si N � layer for electrical insulation.Thereactorsareetchedin a 6 �)Q thick PECVD-depositedSiO� layer. TheSi N �insulationlayerenablesusto conductimpedancemeasurementswithout interferingwith thechemistryin thereactors(thereis no directcontactbetweentheliquid andthemetalelectrodes).

SUBNANOLITER VOLUME DETECTION v

Fig. 1.3 Picoliterarraywith integratedelectrodes.Thearrayis madeup of avarietyof wellandelectrodegeometries.Thechipmeasures5x5 QbQ � .

Fig. 1.4 Schematiccrosssectionof thereactorwith integratedelectrodes.

vi NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

Fig. 1.5 Impedancemeasurementset-up.

An aluminiumlayer was sputteredon the backsideof the Si wafer to form anohmic contact. The Si substratewasconnectedto groundusingthis contact. Thegroundingof the Si wafer is beneficialfor several reasons:it reducesthe effect ofexternalelectricalinterferences,andit minimisesthe parasiticcurrentbetweentheelectrodesflowing throughtheSi substrate.

1.1.3 Measurementelectronicsand setup

Well readout. The impedanceof the systemwas determinedby usinga floatingimpedancemeasurementmethodthatminimisesthe effect of parasiticcomponentsmainly causedby theparasiticcapacitanceCp of thecoaxialcablesusedto connectthe chip with the equipment. The schematicset-upof the electricalmeasurementis shown in Fig. 1.5. An AC voltagesource(20 kHz) is usedfor steeringand acurrent-to-voltageconverterto determinethecurrent. By applyingvoltagesteeringandkeepingtheoutputvoltagelow (suitablecurrent-to-voltageconverter),theeffectof theparasiticcomponentsis negligible. TheoutputvoltageM_c is loggedby usingadesk-topcomputer. In ourcasetheimpedanceundermeasurement,! , is of theorderof 100 d V , whichin ourset-upresultsin acurrentof about100 a e . Theimpedanceis determinedby the capacitive andresistive componentsand is dependenton thereactorgeometry, andtheconductivity andthedielectricconstantof theliquid. Theglycol/water(90%/10%,v/v) mixtureusedhasaresistivity of about3500 V Q , whichresultsin an impedanceof the orderof 1f V . In our set-upthis correspondsto acurrentof about10 a e . Themeasurementelectronicswasdesignedto giveanoutputvoltageof 20 M"<gQhe with a resolutionof about0.1 QiM anda maximumvoltageofabout10 M .

Integrated readoutelectronics. Thesimultaneousreadoutof a largenumberofwells, up to thousands,demandsa dedicatedreadoutdesign. If we assumethat thewells will bepositionedin a matrix of nxn we canusea column/row approach.Inthis setupwe excite all a wells in onecolumnby usinga demultiplexer. Theoutputcurrentof eachwell is measuredsimultaneouslyby a currentto voltageconverters;

SUBNANOLITER VOLUME DETECTION vii

Fig. 1.6 Architecturefor a3x3array.

Fig. 1.7 Photographof apartof asiliconchipcontainingCMOSliquid volumesensors.

by usinga multiplexer theseoutputscanbecombinedsothatfor furtherprocessing(suchasrectificationandfiltering) only onecircuit is required.

Thearchitecturefor a 3x3 arrayis givenin figure1.6. Besidestheninewells wehavethreeswitchesfor theselectionof thecolumnandthreeamplifier/buffer/switchesfor thereadoutof thethreerows; thisunderthecontrolof two 2-bit selectsignals.Aphotoof apartof a siliconchipshowing 5 channelsis givenin figure1.7.

1.1.4 Measurements

Well dispensing. The liquids weredispensedusinganEppendorftransjector5246(Eppendorf-Netheler- Hinz GmbH,22331Hamburg, Germany) with an EppendorfFemtotip. To achieve reproducibleandreasonablyaccuratedispensing,liquid mix-tureswith relativelyhighviscosityarepreferred.In themeasurementspresentedhere,pureglycol wasused.Thedispensingwasdoneby touchingthebottomof thereactor

viii NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

Fig. 1.8 Measuredvoltageduringtheliquid evaporationfrom a550 FHI reactor. Time j ,SUcoinsideswith theruptureof theliquid film.

with the tip, which causesthe liquid to be drawn out from the capillarydueto thesurfaceforces.Thespeedof thefilling processis determinedby theliquid viscosityandthesurfacetension.Thereactoris filled completelyandthe tip is removed. Amoredetaileddescriptionof thefilling procedurecanbefoundin [6].

Electrical evaporation measurements. Optical measurementswereconductedsimultaneouslywith electricalmeasurements.This enabledusto link themeasuredvoltageto the real volumeof the liquid in the well. Fig. 3.8 shows the measuredvoltagein a550 FHI reactorrecordedduringtheevaporationof theliquid. Time j ,SUcoincideswith themomentof theliquid film break-up(seefig. 1.26). Thebreakageof theliquid film caneasilybeidentifiedfrom themeasurements:thevoltagedropssuddenlyat the time of breakage.Thevolumeof the liquid beforethebreakageoftheliquid film wasdeterminedfrom thefluorescencemeasurementsto beabout220FHI .

Fig. ??showsthemeasuredvoltageasafunctionof liquid volume.Theresolutionin a measurementis as high as a few pl. In the 60 FHI reactorthe resolutionisbetter than 1 F_I [7]. The suddendecreaseof currentthroughthe well is causedby the increaseof impedanceasthe liquid is drawn to the walls of the well. Thisphenomenoncanbeusedin automatedsystemsasameasurefor excessevaporation.The optical concentrationmeasurementbecomesunreliableafter film rupture[8].Simultaneouslywith the electricalmeasurementpresentedin Fig. ??, the volume

is determinedby optical interferencemeasurements.Four CCD cameraimagesrecordedduring the evaporationare shown in Fig. 1.10A-D. The correspondingvolumesaremarkedwith lettersA-C: A (full well, 550 FHI ); B (440 FHI ); C (330 F_I ); D(afterliquid film rupture,220 FHI ). Thecurrentflowing throughtheliquid introducesheat,which may influencethe evaporationprocess. To estimatethe influenceof

SUBNANOLITER VOLUME DETECTION ix

Fig. 1.9 Measuredvoltageasa functionof liquid volumein a 550 F_I reactor. Arrows withlettersfrom A-C mark the volumescorrespondingto the CCD-cameraimagespresentedinFig. 1.10A-D.

Fig. 1.10 FourCCD-cameraimagesrecordedduringtheevaporationof glycol form a550F_Iwell. A (full well, 550 F_I ); B (440 FHI ); C (330 FHI ); D (afterliquid film rupture,220 F_I ).

x NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

Fig. 1.11 Glycol amountasa functionof time in a 550 F_I well. The two thick linesshowtheelectricalmeasurementresults;thedashedline is obtainedasa linearfit from theopticalmeasurements(i.e. without simultaneouselectricalmeasurements).Theslopesof 1.76 F_Ik<Eland1.55 F_Ik< l areobtainedfor the electronicmeasurementin the presenceof illumination(simultaneousopticalmeasurement)andwithout illumination,respectively.

this additionalheating,a simplified “worst case”calculationcanbe made. Let usconsiderthe heatingeffect in the 60 F_I well as an example. If we assumethatall the power is transformedinto heat,we canestimatethe dissipatepower to bemonqpsrtr ,vu K , wherethe averagevoltage u is about7 V. The averagecurrentI inthe caseof about1 f V impedanceis K , / U a e , which resultsin heatingpowerm nqpsrtr ,xw9y / UNz|{~} . If heattransferto the surroundingsis low, severeheatingcouldtakeplace(upto UYX Z�� ��< l ) duringimpedancemeasurement.Thiswouldresultin increasedevaporationrates. However, asmoreliquid is evaporated,the heatingeffect decreaseslinearly with thedecreasingcurrent. In our experimentswe foundthat the evaporationrate is acceleratedabout35 % if we measurethe impedancecontinuously. In thenext generationliquid volumesensorsthesteeringvoltagewillbeloweredby a factorof 10, resultingin anevaporationrateincreaseof only a fewpercent.Theopticalmeasurementmethodresultsin anincreasedevaporationrateofabout14%. Themeasuredevaporationratesunderdifferentmeasurementconditionsareshown in Fig. 1.11.

1.2 QUANTIT ATIVE MICR OSCOPIC IMA GING

An experimentalstudyof thedynamicalbehaviour of dropletsin sub-nanoliterwellsrequiresquantitative measurementsof physical quantitiesas a function of time.Someof thesephysicalquantitiescanbederivedfrom microscopicimagesequencesacquiredduringevaporationof theliquid in thewells. Fromanengineeringpointofview, animportantfeatureis theevaporationrateof theliquid. Thisallowscalibration

QUANTIT ATIVE MICR OSCOPIC IMA GING xi

of a dedicatedvolumesensorfabricatedat thebottomof eachwell. Thevolumecanbemeasuredvia the impedanceof the liquid. On theotherhand,from a theoreticalpoint of view dynamicsurfacefeaturesof the meniscusareof interest. They arerequiredfor modelling the evaporationprocessof small dropletsin sub-nanoliterwells. Thesefeaturesinclude isoheightcurvatureand principal curvaturesof themeniscus. Quantitative microscopicimagingand digital imageprocessingof theacquiredimagesequencesshouldprovidetherequiredsurfaceproperties.

1.2.1 Micr oscopicImaging of Evaporation in Sub-Nanoliter Wells

Opticalmonitoringof theevaporationprocessin sub-nanoliterwells requiresa typeof microscopy that allows reconstructionof the shapeof the liquid volume,espe-cially from theair-liquid interface,duringevaporation.This implies that the three-dimensionalvolumeneedsto beproperlysampledin space(both lateralandaxial)aswell asin time.

Videomicroscopy with a conventionalwide-fieldmicroscopecompliesonly withthe secondrequirement. In a wide-field microscopea three-dimensionalvolumeis imagedonto a two-dimensionalimagesensorafter which depthinformation islost. Considerthecasethata well is filled with a homogeneousfluorescentsolution.Variationsin thelocalfluorescentintensityindicateheightdifferencesin theliquid. Itis notstraightforward,however, to retrievethethree-dimensionalshapeof theliquidfrom theseintensitydifferences,asis shown in Figure1.12.

A secondwell-establishedmethodto analysethree-dimensionalmicrovolumesisconfocalmicroscopy. Thiskind of microscopy, however, compliesonly with thefirstrequirement.In aconfocalmicroscopesystemapinholein theexcitationpathcausespoint illumination,whereasasecondpinholein front of thedetectorpreventsout-of-focuslight from reachingthedetector. This way only light from a smallvolumeinthesampleis collected.As a resultanimageof a three-dimensionalvolumecanbeacquiredby scanningastackof slicesatdifferentdepths.Dueto thetime-consumingscanningthetemporalresolutionof confocalmicroscopy ispoor. Figure1.13showsacross-sectionof athreedimensionalconfocalimageof awell filled with afluorescentsolution.Thisfigureclearlyshowstheshapeof themeniscusaswell astheshapeofthewell.

A third type of microscopy that complieswith both requirementsis known asinterference-contrastmicroscopy. Unlike thefirst two typesof microscopy, in whichthe observed intensity is proportionalto the local concentrationof the fluorescentmoleculesin thesample,themeasuredcontrastin interference-contrastmicroscopyis anextrinsic propertyof thesampleunderobservation. In this typeof microscopytheevaporationprocessof theliquid is observedwith epi-illuminatedimaging.Partof the incidentlight is reflectedat thebottomof thewell. Thereflectedpartof theincident light interfereswith the direct part of the incident light. This resultsinconstructive anddestructive interferenceasa functionof thedistanceto thebottomof thewell. Theheightvaryingmeniscuscrossesthesemodulationsof theelectricfield of the incident light. It is this interactionthat resultsin the observed fringepattern.Evaporationof theliquid changestheprofileof themeniscus.As aresultof

xii NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

Fig. 1.12 This figure shows an image ofan evaporatingfluorescentliquid sampleinasquarewell.

54� o

Fig. 1.13 This figureshows a cross-sectionin a confocalimagethrougha liquid samplein a squarewell. The shapeof themeniscusis clearlyvisible.

this the interferencepatternsaredynamic. Imagingof themeniscusprofile yieldsafringepatternin whichtheisophotescorrespondto isoheightcurvesof themeniscus.Figure1.14 shows two imagesof an acquiredimagesequenceof the evaporationprocessin oneof thewells.

136 pixels

258

pixe

ls

Fig. 1.14 Theseimagesshow theinterferencepatternsasrecordedin a � X U �)Q deepvial ofamicroarray. Eachisophotecurvescorrespondsto aniso-heightcurveof themeniscus.In theright imagethe densityof the fringesis higherthanin the left image. This implies that themeniscusis steeperin theright image.

1.2.2 ImageFormation: Interfer ence-ContrastMicr oscopy

Interferencecontrastmicroscopy is basedon the optical model that describesthegenerationof fringepatternsin thin films [1]. After refractionat theair-liquid inter-face,a planewave from an incoherentlight source,propagatestowardsa reflecting

QUANTIT ATIVE MICR OSCOPIC IMA GING xiii

surface. After reflectionat thebottomof thewell, in this case,thereflectedpartofthe incidentplanewave interfereswith the direct part of the incidentplanewave.Whetheror not this interferenceis constructive or destructive dependson the theheight � with respectto thebottomof thewell. Thesumof all thesesingle-quantumdistance-dependentmodulationsgivesriseto theobservedinterferencepattern[2].

Theopticalpathdifference����� betweenthedirectpartandthereflectedpartoftheincidentplanewave at a height � abovethereflectingbottomof thevial is givenby ����� , 5�a�� ���)�����=�[>k��@�� (1.2)

wherea�� ��� is therefractiveindex of theliquid, and � is theangleof incidencein theliquid. Thephasedifference� correspondingto thisopticalpathdifferenceis

� , Z 8�a�� ���)�����El~>k��@� � (1.3)

where�

is thewavelengthof theincidentwave.The recordedinterferencepatternsshow the modulationsof the intensityof the

localelectricfield ���� >��@#� � of theincidentlight asafunctionof theheight � abovethereflectingbottomof the vial. The measuredintensityis averagedover all possibleanglesof polarisation� , wavelengths

�, andanglesof incidence� . With appropriate

weight functions for the wavelengthand the angleof incidenceof the light theintensityof thelocalelectricfield ata height � canbecomputedas:

� �� >��~@[� � ,��¡ ;¢=£¥¤ ;¦ O �¡§¨ ¦ O ��ª©« ¦ O;¬ � >k��@ ¬ � > � @[� �� >k���­�®�'��@#� � � �7�?��� � � (1.4)

with � asdefinedin Equation1.3, �� is the electricfield vectorin Cartesiancoor-dinates. The direct part of the incidentplanewave aswell asthe reflectedpart ofthe incidentplanewave contribute to the electricfield vector. The function ¬ � isthe weight factor for the angleof incidence,and ¬ � is the weight factor for thewavelengthof the incidentlight. This weight factoris definedby the transmissionspectrumof a narrow bandpassfilter which is placedin theillumination pathof themicroscope.The maximumangleof incidence�°¯^±t² is definedby the NumericalAperture³�´ of theobjective.

Equation1.4isevaluatednumerically asafunctionof theheight � abovethebottomof thevial. Theresultof thiscomputationis shown in Figure1.15.Thiscomputationiscomparedtomeasurementsin adigital recordingof anevaporatingliquid sampleina � X U �)Q deepwell etchedin silicondioxide. In eachframe,includingthetwo framesin Figure1.14,the intensityin thecentreof thewell is measured.Themodulatingintensity, measuredasafunctionof time,is analysedwith theunwrappingalgorithmto bepresentedin Section1.2.3. Theunwrappingalgorithmgivestheheightof theliquid ata certainpositionin thewell asa functionof time. Thecombinationof thecomputedheightandthemeasuredintensityis alsoshown in Figure1.15.As canbeseenin Figure1.15theperiodof themeasuredmodulationequalstheperiodof the

xiv NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

1 2 3 4 5 6 7 8

Frame1040

Height in µm

Simulated modulation

Measured modulation

Fig. 1.15 This graphshows the modulationof the strengthof the local electricfield asafunction of the height � above the reflectingbottom of the vial. One curve is basedonevaluationof Equation1.4,whereastheothercurve is experimentallyacquired.Frame1040correspondsto thetime in therecordingof theinterferencepatternwherethemeniscusis flat.

simulatedmodulation.Thelengthof themodulationsJ\� in Figure1.15equals

J\� , �Hµ5oa � ��� �#�=�#>¶�\·¥¸�@ � (1.5)

where�_µ

is the central wavelengthof the transmissionspectrumof the narrowbandpassfilter and � ·¥¸ is theeffectiveangleof incidence.

The optical modelof interference-contrastmicroscopy compliesvery well withthe experimentalresults. Section1.2.3will describethe algorithmto analysetherecordedinterferencepatternsasa functionof time. Thisalgorithmgivestheheightof the liquid at every position in the well in time. From theseheightprofilesthephysicalquantitiesof interestcanbecomputed.

1.2.3 ImageReconstruction: From Interfer ogramsto Surfaces

Theacquiredinterferogramsasshown in Figure1.14canbedescribedby:K_¹ Qº�­a�»�j¶¼ , e-¹ Q½��a�»­j¶¼¾�#�=�[>k��¹ Q½��a�»­j¶¼�@ 0À¿ ¹ Qº�­a�»�j¶¼ 0¡Á ¹ Qº�­a�»�j¶¼k� (1.6)

where eL¹Ã ¼ and¿ ¹¥Â ¼ arethe space/ time varying amplitudeandbackground.The

argument ��¹ Q½��a�»­j¶¼ lies in the range z 8ÅÄÆ��¹ Qº�­a�»�j¶¼LÇv8 and is the wrappedphasemap of the interferogram. From the absoluteor unwrappedphasefollowsthenumberof modulations.This numberof modulationsis relatedby Formula1.5to the meniscuslevel. The signal

Á ¹ Qº�­a�»­j¶¼ is an additive noisesignal. The goalof the algorithmhereis first to derive the phasemap ��¹ Q½��a�»­j¶¼ from the observedinterferogramby measuringtherelative phaseat every time. Thesecondstepis toretrievetheabsolutephaseateverypointby adding 598 to therelativephaseateverylocated598 discontinuity. The 598 discontinuitiesareknown asphasewraps. Thesealgorithmsarethereforeknown asunwrappingalgorithms.

Severalalgorithmsexiststhatunwrap2D phasemapsderivedfrom a staticinter-ferencepattern,suchastheonesshown in Figure1.14. Thesealgorithmsspatiallycomparephasevaluesat neigh-boringpixels. Spatialunwrappingalgorithmsmight

QUANTIT ATIVE MICR OSCOPIC IMA GING xv

suffer from thefollowing problems.First,phaseerrorsoriginatingin regionswith ahighnoiselevel propagateoutsidetheseregionsandcorrupttheremainingpartof theimage.Anotherproblemaretruediscontinuitiesin thephasemap,causedby heightdifferences.Here,heightdifferencesareintroducedin somewells by electrodesonthebottomof thewell. To avoid theseproblemstheacquiredinterferencepatternswill beunwrappedin timepointby point.

In theremainderof this sectionthespatialdependency of thephysicalquantitiesin Formula1.6will bedropped: K_¹ j¶¼ , K_¹ Qº�­a�»�j¶¼ andsoforth. Thesquarebracketsindicatethatthesignalsarediscretesignals.Figures1.16–1.19show thesuccessivestepsof the temporalphaseunwrappingalgorithm. The following recipegivesashortoverview of thedifferentsteps.

1. Subtractthe background¿ ¹ j¶¼ from the interferogramKH¹ j¶¼ . The resultof this

computationis denotedas K~ȶ¹ j¶¼ , which equals eL¹ j¶¼g���=�[>k��¹ j¶¼�@ 0ÉÁ ¹ j¶¼ . Seefigure1.16.

2. Computea subsamplingfactor : rËÊ9Ì in orderto subsampleK È ¹ j¶¼ in sucha waythatapproximatelytwo periodsof the subsampledinterferogram,denotedasK=ÈÍ�ÎEÏ ¹ j¶¼ fall into 128datapoints.

3. Multiply a128datapointwideregionof K ÈÍ�ÎEÏ ¹ j¶¼ with a128datapointswidewin-dow function(seeFigure1.17)andgetthewrappedphasefrom thefrequencyspectrum.Thewrappedphaseis denotedas � Í�ÎEÏ ¹ j¶¼ . SeeFigure1.18.

4. Retrieve theabsolutephase, Ð� Í�ÎEÏ ¹ j¶¼ , by unwrappingthewrappedphases.In-terpolatethe absolutephaseto its original length. This is denotedas Ð��¹ j¶¼ .Finally, convert theabsolutephaseto theheightof themeniscusasa functionof time. Theheightof themeniscusis denotedas C�¹ j¶¼ . Seefigure1.19.

The first stepto computethe phasemap ��¹ j¶¼ from the interferogramK_¹ j¶¼ is tosubtractthebackground

¿ ¹ j¶¼ fromtheinterferogram. Thetimedependentbackgroundisestimatedbylow-passfiltering K_¹ j¶¼ with aGaussiankernelwith averylargestandarddeviation. Theresultof thisoperationis denotedas K=Èk¹ j¶¼ andis shown in Figure1.16.Thesignal K~Èk¹ j¶¼ consistsof thecosinewith varyingamplitudeandcorruptedby noise.Thenext stepis to computethe1D phasemap ��¹ j¶¼ for all timepoints.Onewayto dothis is applyinganFFTalgorithmto aproperlywindowedpartof K~ȶ¹ j¶¼ . Thewindowfunctionavoidsdiscontinuitiesin theperiodicrepresentationof K=Èk¹ j¶¼ computedwiththe discreteFourier transform. The phaseto be estimatedequalsthe phaseof thebin with themaximumamplitudein thefrequency spectrum.Thewindow functionhasa fixed width of 128 points. The low frequentsignal K~È�¹ j¶¼ is subsampledwitha factor : ÍkÎ9Ï in sucha way that approximatelytwo periodsof K~Èk¹ j¶¼ fall within thewidth of the window } ¹ j¶¼ . The subsampledsignal is denotedas K=ÈÍ�ÎEÏ ¹ j¶¼ . Thisway of subsamplingis necessaryto avoid that Ñb¹ K~ÈÍkÎ9Ï ¹ j¶¼ } ¹ j¶¼s¼ is mixedwith theDCcomponentof the frequency spectrum.TheoperatorÑh¹Ã ¼ denotesthe FastFourierTransformand } ¹¥Â ¼ is the window function. The window function } ¹ j¶¼ and theproductK ÈÍkÎ9Ï ¹ j¶¼ } ¹ j¶¼ areshown in Figure1.17.Thephaseto beestimatedfrom K ÈÍ�ÎEÏ ¹ j¶¼is denotedas � ÍkÎ9Ï ¹ j¶¼ .

xvi NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

1000 1250 1500

-10

-5

5

10

Inte

nsity in

#A

DU

'sFrame number

Fig. 1.16 Thisgraphshows thebackgroundcorrecteddata Òtȶ¹ Ó­¼ in thecenterpoint of thevial in theframeseries750– 1500.

20 40 60Ô

80Õ

100 120

-10

-7.5

-5

-2.5

2.5

5Ö7.5× 10

Inte

nsity

in #

AD

U's Window

functionØ

Fig. 1.17 This graph shows the windowfunction } ¹ j¶¼ andtheproductK~ÈÍ�ÎEÏ ¹ j¶¼ } ¹ j¶¼ X

1000 1250 1500

−π

− πÙ2

πÙ2

π

Frame number

Ph

ase

Fig. 1.18 This graphshows the estimatedphasesÐ� ÍkÎ9Ï ¹ j¶¼ in theframeseries750–1500.

1000 1250 1500

2

4

6

8

Frame number

He

igh

t in

µm

Fig. 1.19 This graphshows theheightpro-file C�¹ j¶¼ , which followsby unwrappingof theestimatedwrappedphases.

Thesubsamplingfactor : ÍkÎ9Ï is definedastheratiobetweentheaveragelengthofthe modulationsin K~È�¹ j¶¼ andthe requiredaveragelengthof themodulationswithinthewidth Ú�Û of thewindow, i.e.

�­��Ü� , in formula:

: rËÊ9Ì , ¹ 5ÚÝÛ 5"Þ;ß#à~á â�ã;>�K~Èk¹ j¶¼�@äÝå�æ ¼¶� (1.7)

where ¹Ã ¼ is theroundoperator. Thesubsamplingfactor : Í�ÎEÏ is estimatedasfollows.Thenumberof zero-crossings

äçå"æis countedin K=ȶ¹ j¶¼ . Then K~È�¹ j¶¼ is filteredwith a

Gaussiankernel fç¹ j'»�è ,êé 5¾¼ andthe numberof zero-crossingsis countedagain.Thisfiltering is repeatedwith thesizeof fç¹ j'»ªè_¼ increasedby a factorof é 5 until thenumberof zero-crossingsdoesnot changein two successive filterings. Thenumberof zero-crossingsvariesfrom 1 at the sidewall of the vial to 84 in the centreofthevial. Underassumptionthat theevaporationrateremainsconstant,theaveragenumberof periodswithin Ú Û canbe computedwith Formula1.7. As a resultofthe resamplingthe averagenumberof modulationswithin Ú Û of the subsampledinterferogramK Í�ÎEÏ ¹ j¶¼ variesfrom 1.62to 2.39modulations.

After thesetwo stepsthewrappedphasemapis computed.Thewrappedphasemap,denotedas � Í�ÎEÏ ¹ j¶¼ is shown in Figure1.18. Thefinal stepof thealgorithmisto unwrapthe wrappedphasemap. The unwrappingis donewith an unwrappingoperatorfound in literature [3]. Finally, the subsampledunwrappedphasemap

QUANTIT ATIVE MICR OSCOPIC IMA GING xvii

0

2

4

6

0

2

4

6

sidewall of well

ësidewall of wellì

center of well

heig

ht o

f the

men

iscu

s in

µm

150 µm150 µm

sidewall of well

ësidewall of well

0

2

4

6

center of well

heig

ht o

f the

men

iscu

s in

µm

150 µm150 µm

Fig. 1.20 Thecomputedheightprofilescorrespondingto the interferogramsshown in Fig-ure1.14. Becauseof symmetryonly a quarterof thewell is shown. Comparethe isoheightcurvesin thesefigureswith the isophotecurvesin the interferograms.For clarity, only halfthenumberof isoheightcurvesis shown in theright figure.

Ð� ÍkÎ9Ï ¹ j¶¼ is interpolatedby a factor : ÍkÎ9Ï to theoriginal lengthof K_¹ j¶¼ . Theunwrappedphasemap Ð��¹ j¶¼ is scaledwith 598 , resultingin themeasurednumberof modulations.Multiplying with the length JL� per modulationfrom Equation1.5 converts thenumberof modulationstoarelativeheightin micrometers.Usingtwomeasurements,namelythatin frame1040themeniscusisflatandthatin frame4572theliquid reachesthebottomin thecentreof thewell, theabsolutedepthof thewell is calculated:thedepthof thewell is � X / R �)Q . Theheightof theperfectlyflat meniscuswith respectto thebottomof thewell in frame1040is indicatedin Figure1.15. Theheightasafunctionof time is shown in Figure1.19.

Thephaseunwrappingalgorithmin timein asingledatapointasdescribedaboveis appliedto all datapointsin a regionof interest.Figure1.20shows two computedheightprofilesof themeniscusduringevaporation.Notethatthecomputationof theheightprofilesarespatiallyuncorrelated,but correlatedin time! This correlationintimeis inverselyproportionalto thechangeof themeniscuslevel duringevaporation.

1.2.4 ImageAnalysis: Computing GeometricSurfaceFeatures

The result of the temporalphaseunwrappingalgorithmexplainedin the previoussectionis aseriesof menisciasafunctionof time. Theseheightprofilesallow quan-titativemeasurementsof anumberof physicalquantitiesasmentionedin Section1.2.

A straightforwardmeasurementis the remainingliquid volumein the well asafunctionof time. Of course,directlyrelatedto thatis theevaporationrateof theliquid.Theresultsof thesecomputationsareshown in Figure1.21. ThedifferentcurvesinFigure1.21show how theremainingliquid volumein differentsizedwells,all withadepthof � X / �)Q , changeasa functionof time. Theslopeof eachof thesecurvesistheevaporationrate.Thestraightcurvesin Figure1.21indicatethattheevaporationrateis constantduringevaporation.Theevaporationrateasa functionof thewidth

xviii NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

of the well is shown in Figure1.22. This graphshows that the evaporationrateisproportionalto the width of the well! This is quite a surprisingresult. Intuitivelyonewouldexpectthattheevaporationrateis proportionalto theareaof theair-liquidinterface,which is roughlyproportionalto thesquareof thewidth of thewell.

-50 50 100 150 200 250

100

200

300

400

500

600

700

552 pl

0

300x300µm2

time in s.

remaining liquid volume in pl.

size of wellvolume of well

246 pl

139 pl

61 pl35 pl

200x200µm2

150x150µm2

100x100µm2

75x75µm2

1.32 pl/s

0.83 pl/sí

0.62 pl/s0.41 pl/s

0.31 pl/s

Fig. 1.21 This graphshows the remainingliquid volumeasafunctionof timefor squarewells of varioussize. Thedepthof thewellsis � X / �)Q . The slope of the curves is theevaporationrate.

50 100 150 200 250 300

0.25

0.5

0.75

1

1.25

1.5

width of wells in î µm

evap

orat

ion

rate

in p

l/s

Fig. 1.22 Theevaporationrateasafunctionof thewidth of thewells.

Section1.3 will explain that the evaporationrateof the liquid dependsvia thevapourpressureon the curvatureof the air-liquid interface. Variousdefinitionsofcurvatureexists. Here,we areinterestedin the iso-heightcurvature ï_� Ítð#ñ)ò · ��ó ò#ô ofthe meniscus,andthe principal curvaturesï � and ï � . The iso-heightcurvatureisdefinedasfollows[4]: ï � Ítð#ñ)ò · ��ó ò#ô ,õzÝö ÷ùøúöû ö û � � (1.8)

where ö is the Hamiltonianor contourvector, and ø is the Hessianmatrix. Theprincipalcurvaturesarethe eigenvaluesof the following matrix product,which isknown astheWeingartenmappingmatrix [5]:3 /"0¡ü y � ü y üçýü y üçý /"0Àüçý � 6 ñ � øþ /"0Àü y � 0¡üçý � � (1.9)

whereü y and

üçýarethefirst derivativeoperators.All vectorandmatrix elements

of ö and ø , and the partial derivativesü y and

üçýarecomputedby convolution

of the height profileswith first and secondorder derivatives-of-Gaussiankernels.Figure1.23shows thereciprocalof theiso-heightcurvature,which equalsthelocalradius,of themeniscusatacertainmomentof theevaporationprocess.Darkregionsin this figure correspondto a small radius,which meansthat the meniscuslocallyhasa high curvature,whereasbright regionscorrespondto a largeradiusor a smallcurvature. The lower left cornerof this figure is thecentreof thewell. Theradiusalongthediagonalfrom thecentreof thewell to theupperright cornerin thefigureis muchsmallerthantheradiusalonga horizontalor verticalline from thecentreof

SIMULA TION xix

Fig.1.23 Thisfigureshowsthecontourlinesof the reciprocalof the iso-heightcurvature.Dark regions correspondto a small radius,whereasbright regionscorrespondto a largeradius. The lower left corneris thecentreofthewell.

-50 50 100 150 200

-0.02

0.02

time in s.

mean curvature

convex air-liquid interface

concave air-liquid interface

Fig. 1.24 This graphshows how the meancurvature of the meniscuschangesin timeduringtheevaporationprocess.Duringevap-orationthemeniscustransitesfrom a convexshapeto aconcave shape.

the well to the side-walls of the well. This implies that the iso-heightlines of themeniscuschangefrom a circular shapein the centreof the well to a squareshapetowardstheborderof thewell. Thiscanalsobeseenin Figure1.14.

Thesecondcurvaturethat is evaluatedis themeancurvature,which is definedasthesumof theprincipalcurvaturesï � and ï � . Themeancurvatureis positive for aconvex air-liquid interface(theshapeof themeniscusbefore it getsflat), whereasfora concave air-liquid interface(theshapeof themeniscusafter it wasflat) themeancurvatureis negative. A positivemeancurvaturewill acceleratetheevaporationrate,whereasa negative meancurvaturewill decreasethe evaporationrate. The meancurvatureshows very little variation over the areaof the meniscus. This can beexplainedby thefact that themeniscusis a very smoothsurfaceandthat theaspectratio of the heightandthe width is small: the width of the differentwells is 10 to50 larger thanthe depth. Figure1.24shows the meancurvatureaveragedover theair-liquid interfaceasa functionof time. This figureshows thepredictedbehaviourof themeancurvature.Fromthegraphsin Figure1.21weconcludethattheeffectofthemeancurvatureon theevaporationrateis negligible.

1.3 SIMULA TION

The physicaldescriptionandnumericalmodelhave to be chosenaccordingto therequirementsof thesmall surfaceto volumeratio. Themethodsfor computingthetime-evolution of a nanoliter-sizedliquid dropletcontainedin the micromachinedwell aredescribedin thefollowing.

1.3.1 Physicalmodelsfor small droplets

For our purposes,the relevant propertiesfor the simulationof a nanoliterdropletare the surfacetension,the evaporationrate, the inner fluid flow, and the dropletimpedance.

xx NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

Surfacetension. Increasingthesurfaceareaof a liquid requiresanamountofenergy ÿ } , � ÿ e-� (1.10)

whichis proportionalto thechangeÿ e in thesurface'sarea.Thevalueandstructure

of thesurfacetension� dependsonthematerialsoneithersideof theinterfaceformedbetweentwo phases.

In a stationarystate,wherea liquid dropletis sitting on a flat solid surfacein anotherwisegaseousenvironmenttherewill bea characteristiccontactangle

�at the

pointwhereall threephasestouch.Theangle�

isgivenasafunctionof therespectiveinterfacetensions � ,���� �[���~� 3o� r�� z � r�� �� 6 � (1.11)

wheretheindex " l¾I " denotesthesolid-liquid interface," l�� " thesolid-gasinterface,and" I � " theliquid-gasinterface,seefigure1.25. Thecontactanglemaybeusedasa constraintfor thecalculationof thedroplet'ssurfaceshape.

Sincethe well forms a sharpedgewith the chip surface, the droplet that weconsideris effectively pinnedat this edge. Pinning is energetically advantageousbecausea rangeof anglesareequallypermissible.Usually, a smoothsurfacewouldrequirethedroplet'sedgeto beconstrainedto stayonthesurface.In fact,it turnsoutthat this is often an idealisationfor sediment-containingdropletson a flat surface.As liquid evaporatesfrom thedroplet,andfasterat theedgeswherethecurvatureishigher, sedimentis transported(includingtheanalytemolecules)to theedgeandisdepositedthere.This roughensthesurfaceat theedge,creatingexcellentconditionsfor furtherpinning. As aresult,suchdropletstendto flattenasthey losevolume,andwouldcausearing-likestainif thesedimentwascoloured[13].

Evaporation process. Theevaporationprocessof smalldropletsis drivenby thenon-equilibriumthatarisesbetweenthe liquid phaseandthegaseousenvironment.Giventhetemperatureandthetotalpressureof thegas,theequilibriumpartialpressureof theliquid moleculesin thegasphaseis determined.If thepartialpressureof thegasmoleculesis not at its equilibrium value it will give rise to materialtransportacrosstheliquid-gassurfaceandthusleadto eithercondensationor evaporation.

Understationaryconditionsa flat boundary, separatingtwo infinite half-spacesof the liquid phaseand the gasatmosphere,will experiencea constantcurrentofmoleculesfrom onephaseto the other. In addition, the currentdensity j will beconstanteverywhereon the surface, which in the caseof infinite half-spacesisthereforetheonly meaningfulvariablefor thedescriptionof theevaporationprocess.Thecurrentdensityj is fully determinedby thedifferenceof theequilibriumpartialpressureF r of theliquid moleculesin thesaturatedgasphaseandtheactualvalueforthevapourpressureF n [14]

� , /é 5E8 � µ >%F r z F n @ � d���c �������� � � (1.12)

SIMULA TION xxi

wheren is thenormalvectorpointingfromtheliquid surfaceto thegasphase,d���c �����is themolarmassof themolecules,� is thegasconstant,� denotesthetemperature,and � µ is thecondensationcoefficient for theliquid.

Theabovelocaldescriptionholdsfor aflatsurfaceonly. Fordropletswith acurvedsurfacethe pressurecomponentdueto curvaturechangesthe situationthe smallertheradiusgets,which is of coursethecasefor nanoliterdroplets.For a convex or aconcave liquid surfacethevapourpressureincreasesor decreasesrespectively. Bothof thesecasesoccurfor liquid dropletscontainedin amicromachinedwell. Therefore(1.12)mustbe modifiedwith respectto the local saturationpressure.The vapourpressureF r of the saturatedgasphaseabove a curved surfaceis a function of thecurvatureï�> x @ at thespecificsurfacepointx [10]

F r >�ï_@ , F r > U @oÂ������ 3�5�QW� ���� �bB ï 6 � (1.13)

whereF r > U @ is thevapourpressurefor a flat surfacewith curvature ï , U , Q is themolecularmass,

�! is the Boltzmannconstant,� is the temperature,and B is the

massdensityof theliquid. Foranarbitrarysurfacewemaytake ï , > / <9� � 0i/ <9� � @ ,which is theaveragecurvaturewith theorthogonalradii � � and � � . Theradii aremeasuredfrom thecentreof circlesorthogonallytangentto thesurface. If a circlecentrelies in the liquid domainwe take the radiusas positive, else its negative.Inserting(1.13) in (1.12)we seethat the evaporationratefor a small dropletwithstronglycurvedsurfaceis enhanced. This meansthat theevaporationof a droplet,

Solid"

θ#

LiquidGas$

n

γ%sl

γ%lg

γ%sg

Solid"θ#

LiquidGas$

n&γ%sl

γ%lg

γ%sg

Fig. 1.25 Definitionof thedroplet's liquid-solid interfaceangle.Theangleis definedby theresultantforceat thepointwhereliquid, solid andgasintersect.If theangleis lessthan 8�<E5 ,theliquid is saidto wet thesolid.

in anenvironmentwhereatequilibriumno liquid phasecoexistswith thegasphase,is increasinglyacceleratedthesmallerthedropletgets.Ontheotherhand,aconcavesurfacehasnegative curvatureand its vapourpressureis reducedwith respecttheequilibriumvapourpressureabovea flat surface.

Theeffect of curvatureis of greatimportancein thedynamicalbehaviour of anevaporatingnanoliterdroplet,sinceit givesrise to differentevaporationrates,andthusdifferentmaterialfluxesacrosstheliquid surface,dependingwhetherthesurface

xxii NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

Timea' b

(c) d

*e+ f

Droplet breaks up(

Fig.1.26 Configurationstatesfor theevaporatingdroplet.Theassumptionsfor thesimulationholdfor theprocessesto theleft andright of theverticalgraybar. At thebar, thedropletformsinto a ring. Thisprocesshappensvery rapidly, asconfirmedby observations.

10−8

10−7

10−6

droplet radius [m],1.0

1.2

1.4

1.6

1.8

Enh

ance

men

t fac

tor

water-mercury.

Fig. 1.27 Enhancementfactors�����0/ � � «�1324�5 ÷76 ï98 for mercuryandwateratT = 300K.

curvatureis larger (convex) or lower (concave) thanzero. Thecurvatureitself is afunctionof geometryandsurfacetension.

Fluid transport. Theequationof motion for thevelocity field : of a liquid isgivenby [11] B �!:�=j , div ;�� (1.14)

SIMULA TION xxiii

where ; is thestresstensor. Note that theconvectiontermis still containedin thetotal derivateof : with respectto time on the l.h.s.of (1.14),dueto the definition�?<9�=j ,=< < < j 0 > :hÂ?>N@ .

For an inviscid fluid in motion,or any fluid at rest,the stresstensoris diagonalandfully-determinedby thethestaticfluid pressureF , whichis thetraceof thestresstensorF , z �� tr ; andfor this thestresstensorreads

; , z F I (1.15)

whereI is theidentitymatrix.For a viscid fluid in motion, we needto add a non-isotropicpart to ; which

contributesthe tangentialstresses;A@ . This is dueto thefact that friction insidethefluid existsonly for arelativemotionof infinitesimalfluid volumes.Thus ;A@ dependson the gradientof velocity. Termsindependentfrom the gradientgrad : mustbezero,so that ;A@ vanishesfor thefluid at rest. Furthermore,;A@ mustvanishfor thecaseof arigid rotationof thefluid. Themostgeneralform of stressthatfulfils all theaboverequirementsis givenby [11]

; , z F I0CB

grad : 0 > � 0 B R @ I > div :�@ (1.16)

whereB

and�

arescalarvelocity-independentmaterialcoefficients.They dependin generalonpressureandtemperature. In ourcasewetaketheasconstantparameters.Inserting(1.16)into (1.14)givestheequationof motion

B �!:�=j ,Éz gradF 0CBdivgrad : 0 > � 0 /R B @ graddiv : (1.17)

In an incompressiblefluid div : ,õU and thus the last term in (1.17) vanishes.Equation(1.17)canbe further simplified for the casewherethe viscousdiffusiontime D�E , B~I � < B is much lower than the characteristicresidencetime D � , Ik<GF(the time a fluid particleon averagewhiles in a region of length I ), where I is thecharacteristiclengthof the liquid domainand F a typical velocity of thestreamingmotion. Thissituationis characterisedby theReynoldsnumber

�IH , D ED � , BJF_IB (1.18)

It representsthe importanceof inertia relative to viscosity for processesthat aretypifiedby a velocity ratherthana time. A smallReynoldsnumber( �IHIK /

) andasteadyflow allowsustoneglectthetotalderivativeof thevelocityfieldonthel.h.s.ofequation(1.17)andthuswegetU-,úz gradF 0CB

divgrad : (1.19)

Thegradientof thepressureis negligible aswell, sothatweareleft with theLaplaceequations B

divgrad : ,WU (1.20)

xxiv NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

for thecomponentsof thevelocity field.B

in (1.20)mustbe taken into accountasa scalingfactor for the flux boundaryconditionsonly, otherwiseit doesnot affectthe calculationof : . For the situationconsideredhere(seeTable 1.1), we have�IHML T Â / U ñ � K /

. We also have that D�ENL / U ñ � lOK � U l , the typical valuemeasuredfor fastevaporation[12, 13]. Thismeansthatthesystemis in aquasi-staticdomainandoursimplifying assumptionsfor thefluid transportarereasonable.

Parameter PRQ=jSHUT <E� �UVXW?Y�� > /[ZP/ �]\?<G\@B RPX%w9Uw Z_^½/ U ñ � m Q�lB / U � Z � �Y<gQ ��I`E �]a c UPX { a Ik<9Q0b¶aF / U=U �)Q <ElD E UYX U / /[Z { lD � 5�l�IH TPX�w Z  / U ñ �Table1.1 Fluid parameters,characteristic times,Reynoldsnumber

Droplet impedance. The droplet impedanceis a volume-dependentvariablesuitableto monitor theevaporationprocess,seefig. 1.8. At the instantthedropletcentretouchesthebottomof thewell (seefigure1.26.d-e)it shows a rapidchange.This is dueto the fact that thedroplet's surfaceassumesits new equilibriumshapeon a short time salenot resolved by our model. Nevertheless,for a given surfaceconfigurationat a specifictime in our decoupledsimulationwe may calculatethecapacitanceand/orresistanceof theconfiguration.For thispurposeweagainconsidera rectangularreactorwith two electrodesat thebottomplaneandtheliquid surfaceat thecurrentconfigurationascomputedby thesurfaceevolutionalgorithm,togetherwith thedielectricconstantand/ortheresistivity of thefluid andsolidmaterials.

1.3.2 Computational method

Oursimulationgoalis to computea droplet'sphysicalbehaviour for a rangeof wellgeometries,andto extractusefuldesigninformationfrom thecomputationalresults.In a subsequentstep,we wish to usethemethodto obtainthebestpossibledesignwith respectto a rangeof externaloperatingconstraints.This includesconsideringmethodsfor keepingtheevaporationratelow, detectingthevolumeat every instant,andthusfinding the mostsuitablewell geometry. An additionalobjective wastoavoid developingnew computercode,but insteadto use,asfar aspossible,existingsimulators.

It is instructive to analysetheprocessestakingplace,soasto obtaina qualitativedescriptionof theresultingsimulationprocesswhichonewill require:

1. A liquid dropletis insertedinto thewell by meansof a transjector, seesection1.1.4technique,andassumesits equilibrium shapel~>�j­@ in a time negligiblew.r.t. otherprocessesof interestto ushere.

SIMULA TION xxv

2. Basedon its instantaneousshapel~>�j­@ , liquid escapesfrom thedropletacrossits boundaryto thegaseousenvironment.Themassflux j varieswith position.

3. The total massrate escapingfrom the surfaceat a particular instant is theintegralof themassflux densityover thedropletsurfacearea.

cd , ��dUegf �Jh , ��dUe BJ\�i��~eù� (1.21)

where\ i is thevelocityof themoleculesperpendicularto thesurfaceleavingthedroplet'svolume.

4. At a particularinstant,the droplet compensatesfor its contentloss over itssurfaceby adjustingits surfaceshapel~>�j­@ . At thesametime, theliquid flowsto replenishthenon-uniformlossof massover thedropletsurface.

Thereplenishingflow instantaneouslyfollows thereshapingprocess.Intuition sug-geststhatatmostflow takesplaceonly in theregion immediatelyunderthesurface.Observationsshow the opposite. Fluorescentlytaggedmoleculesare transportedtowardstheedgesof thedropletasit is drying. Thesurfaceshapel~>�j­@ is, accordingto the four statementsabove,givenasa functionof thedroplet's volume M andthevelocity \�i . A simplefirst orderTaylorexpansionin timegives

l~>�j 0 JLj­@ , l~>�M�>�j 0 J-j­@�@ , l~>kMÝ>�j­@�@ 0 < l< M cM-JLj 0 X[X#X (1.22)

cM , cdÉ<gB is givenby themassflux (1.21),which is anintegralquantityof \ i overthewholesurface. \ i itself dependsonthesurfacecurvatureandthuson l~>�j­@ , whichreadsfor termsupto first orderin time

\�i >�l~>�j 0 JLj­@­@ , \�i�>�j­@ 0 < \�i< l < l< M cMLJLj 0 X#X[X (1.23)

Thefirst ordertermin (1.23)is thesameasin (1.22)exceptfor thefactor < \ i < < l ,which is, roughly speaking,given by the changeof the enhancementfactor withsurfacecurvatureasshown in figure 1.27. Figure1.27 is a worst casestudyfor asphericaldroplet. On the lengthscalesof interestandconsideringthe fact that thesurfaceis not sphericalthefirst ordertermin (1.23)is evensmaller. This makesusassumethat \�i followsthechangein thesurfacecurvatureadiabaticallyandtheflowinsidethedropletinstantaneouslyadoptsthenew boundaryconditions.

To avoid solving for the fully-coupledphenomenagoverninga droplet's surfaceevolution, its evaporationand its flow, we will assumethat the surfaceevolutionprocessis sufficiently slow. Choosingthetime-stepJLjAK I¶<jF , wherel is a typicallength,effectivelydecouplessurfaceevolutionfrom thefluid flow problem.Thuswesuggestthefollowing algorithm:

1. Startwith massd > U @ andcomputethefluid dropletshapel~> U @ at time t=0.

2. For discretetimestepsJLjkK / l :

xxvi NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

al albm

albm

al bm

An

Bo

Cp

cq

Fig. 1.28 Computationalboundaryconditionsandconstraints.A) For thesurfaceevolutionprocess,thedropletis constrainedat thecornersmarkedby a. At thesametime, thevolumeis constrainedto a fixed value. B) For the Laplacesolver of the fluid flow, we usenonzeroNeumann(flux) boundaryconditionsat a. At b we usehomogeneousNeumann(zeroflux)boundaryconditions.Soasto fix thesimulation,weaddasingleDirichlet boundarycondition(zerovelocity) for the lower-left cornerc of the well, wherethis is also realistically to beexpectedto hold. C) For the impedanceextractionby the Laplacesolver, we useDirichletboundaryconditions(two specifiedpotentialswith 1 Volt difference)at the two electrodesmarkedby aandb.

(a) j , j 0 JLj .(b) Computetheaveragecurvatureof thesurfaceusing ï � E�` >�j­@ , > / <E� � 0/ <E� � @ .(c) Computemassflux densityf >�lg@ maponthedropletsurfaceusing(1.12).

(d) Computenew mass:d >�j 0 JLj­@ , d >�j­@ 0 J-jsr dUe f �Jh(e) Using f >klg@ computeinstantaneousflow field :]>�j­@ using(1.20)andstore.

(f) Using new mass d >�j­@ , computeinstantaneousdroplet shapel~>�j­@ andstore.

(g) If massis closeto zero, d >�j­@çÇ tolerance, or dropletshapebecomessingular, stop.

(h) Computetheliquid droplet's instantaneousimpedance�->�j­@ .Thealgorithmhasthreecomputationallyintensiveoperations,thecomputationof

adroplet'sinstantaneousshapel~>�j­@ givenitspropertiesandsuspensiongeometry, thecalculationof theflow field :]>�j­@ , andthecomputationof theensemble's impedance.Sincetheseoperationsareperformedbyseparatecomputerprograms,wealsorequirea methodto convertdatabetweentheprograms.

WehaveusedtheEvolver [16] to computethesurfaceshape.Thealgorithmusedforms an expressionfor the energy } of the dropletasa function of its geometryl~>�j­@ , thesurfacetension� , andthedensityB .

} , � dUe �7�~e 0 � e B���t~�~M (1.24)

Thealgorithmfor surfaceevolution is:

SIMULA TION xxvii

1. Minimise theenergy } subjectto:

(a) M , Mvu , thecurrentdropletvolume.

(b) Theboundaryconditionsandmovementconstraints.

The surfaceis discretisedinto flat, triangularpanelswhoseverticesandedgesareeitherfreeto move in space,or aresubjectto a varietyof user-definableconstraintssuchasmovementconstraintto theplanesformedby thewalls of thecontainer, asshown for atypicalgeometryin figure1.29.a.At convergencethedroplet'senergy is

a b

Fig. 1.29 Well and liquid surfacediscretization. a) initial meshfor a given volume of 1nanoliter, b) final surfaceconfiguration.

minimised.Thenext stepis to computethepositiondependentcurvaturemap ï;>kl¾@of thedropletsurface. This is donewithin theEvolver program. In fact,sincethesyntaxof theprogramallowstheadditionof user-definedexpressionsto beevaluatedacrossthe surface,we alsocomputethe evaporationflux f >�lg@ , its surfaceintegralr dUe :]>�j­@­�Jh , andstoreall computedvaluesin text files.

At this point the geometryis extracted[17] and storedin a form that can bereadby our in-houseLaplaceequationsolver BEMModule [15], which we useforthe computationof the flow field. This computationis basedon the multipole-acceleratedboundaryelementmethod,and is formally straightforward. However,since the evaporationflux f >kl¾@ at the last converged configuration l~>�j­@ forms aspatially-varyingboundaryconditionfor theflow field, awordon thecomputationalstrategy is in orderhere.

Thenumberof surfacepanelsthatcover thedroplet's surface,asrequiredby thesurfaceevolver, canbecomequite large (seefigure 1.29.b). Naively, eachsurfacepanelwouldbeassociatedwith aseparateflux boundarycondition,whichstressesourLaplacesolver in its capabilities.In addition,it is questionablewhethersucha fineresolutionof thisboundaryconditionqualitatively improvesthenumericalsolutionswe obtain. To alleviate this problem,we have computedthe discretecontoursofsurfaceflux density, between10 and20of them,andassociatedpanelsbelongingtoa contourgroupwith a singleboundarycondition. Varyingthenumberof contoursc (if c is equalto thenumberof surfacepanels,we have theoriginal situation),wewereable to monitor the effect andfound ten contoursto be morethansufficientfor accuratesimulations.For theimpedancecalculationsweagainuseBEMModule.Thegeometryis thesame,but now theboundaryconditionshavechanged.

Dependingon thegeometryof thewells,wemayexploit symmetryto reducethesizeof the dropletshapecomputation.Thusfor rectangularwells, only 1/4 of the

xxviii NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

geometryneedbecomputed,for squarewellsonly 1/8(seefigure1.30),andcircularwells reduceto a 2D computation(1D surface).For subsequentLaplacesolvestepsof theimpedance,however, thefull geometrymustberestoredbecausetheelectrodesdonotmatchthesymmetryof thedropletsurface.

For monitoringpurposesthesensorlosesits usefulnesswhenthe liquid volumehasdecreasedsufficiently for its surfaceto reachthebottomof thewell (seefigure1.26.d). At this point the surfacebreaksup to form a hole at its centre,and isobservedto rapidly retractto its equilibriumpositionat thebottomof thewell (seefigure1.26.e).

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c�Fig. 1.30 For mostnumericalstudiesof the surfaceevolution, the well a) waspartitionedalongits four symmetryplanes.The remaining

/ < { volumeb) wasmeshedandsimulated.Whenpoint 6 touchedpoint 5 of b), themeshwasmodifiedasin c) creatinga holeto enablethesimulationof thefinal dryingprocess.

Our modeldoesnot correctlydescribethedynamicsof breakingup. Clearly, webriefly entera regimewherethevelocity of particlesis very high, until thedropletagainassumesits equilibriumshape.Despitethedeficiency, wewishedto follow theevolutionprocessright up to drying,whichweachievedasfollows:

1. Monitoring the position of the centreof the droplet (seefigure 1.30.b),westoppedtheevolution theinstantthis point waswithin a smalldistanceof thelowerwell surface.Thedropletvolumewasrecorded.

2. All dropletpanelscoincidingwith themonitoredpositionweredeleted,leavinga hole in the droplet's surface. The hole createdwas small comparedtothe droplet's horizontalextent, madepossibleby the meshrefinement,andsubsequentunrefinementfacilitiesof theprogram(seefigure1.30.c).

3. All meshnodescoincidingwith theholewhereassignedacoordinateconstraintto tie themto thewell bottom.

4. Usingtherecordedvolume,thesurfaceevolution processwasnow continuedasbefore.

For thislatterpartweskippedthecomputationof thefluid flow field. Veryclosetothispointwherethevolumereducesto zero,thesimulationmustbestopped.At this

SIMULA TION xxix

point, thenumericalerrorsin positionbecomeof theorderof thecontainedvolumeconstraint,leadingto smallmovementsof thedropletnodesaboutthewell wallsandconvergencedifficultiesin thenumericalsolution.

1.3.3 Conclusionsand Discussion

A silicon-integratedsubnanoliterliquid volumedetectorwasrealized. The reactorvolumesrangefrom 60pl to 1.5nl. Theapplicablerangeandaccuracy of thevolumedetectiondependson the electricalparametersof the liquid. The resolutionin ameasurementcanbeashighas1pl. Theimpedancemeasurementsutilisingintegratedelectrodeson the bottomof the wells make it possibleto monitor the evaporationof liquids accurately. The electricalmeasurementsclearly show evaporationrateand film rupture. Dissipationdue to the impedancemeasurementresultedin anincreaseof theevaporationrateof about35%. By usinglower steeringpower, thiscan be reducedto only a few percent. The presentedtechniquecan be usedinautomatedhigh-speedscreeningsystemsto monitor unwantedevaporationand tocontroldispensedliquid amounts.Thefabricationof theelectricalvolumedetectoriscompatiblewith standardIC technology, whichenablesusto fabricatereactorarrayswith integratedvolumesensorsandmeasurementelectronics.

The evaporationprocessof a liquid samplein a sub-nanoliterwell needsto beproperlysampledin timeaswell asin spacein ordertodoquantitativemeasurements.Interference-contrastmicroscopy is a techniquethatsatisfiesbothsamplingrequire-ments. With this techniquean interferencepatternis generatedshowing fringesofequalheight.A temporalphaseunwrappingalgorithmis developedto analysethesemodulationsin timepointby point. Thestrengthof thisalgorithmlies in thefactthatthecomputedheightof themeniscusis spatiallyuncorrelated.Givena time seriesof retrievedheightprofiles,astraightforwardmeasurementis theevaporationrateoftheliquid. A strikingresultis thattheevaporationrateof theliquid is linearlydepen-denton thewidth of thewell. Thereciprocalof thecomputediso-heightcurvatureindicateshow the iso-heightlineschangefrom a circularshapein thecentreof thewell to asquareshapetowardstheborderof thewell. Thecomputedmeancurvatureshows little variationin spaceaswell asin time.

Thesimulationwasperformedby a strategy decouplingtheevaporationprocessfrom thesurfaceevolution. Thebreakingup of thesurface,whentheliquid touchesthe bottomof the containeris not describedexactly by our model. The time scaleon which thecreatedhole relaxesto its equilibriumshapeis smallcomparedto theon going evaporation,so that this part could be modelledas instantaneous.Fromthe experimentalresultswe concludethat the effect of the meancurvatureon theevaporationrateis negligible. This canbe confirmedby simulation,sincenotableeffectswill ariseonly on length-scalesmuchsmallerthanin this investigation.

xxx NANOLITER DROPLET BEHAVIOUR IN MICR OMACHINED WELLS

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15. Bächtold,M., Korvink, J. G. andBaltes,H., EnhancedMultipole AccelerationTechniquefor theSolutionof LargePoissonComputations,IEEE TransactionsonComputer-AidedDesignof IntegratedCircuitsandSystems,12, 1541(1996)

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