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Journal of Colloid and Interface Science219,81–89 (1999)Article ID jcis.1999.6422, available online at http://www.idealibrary.com on

ontrolled Formation of Low-Volume Liquid Pillars between Plates witha Lattice of Wetting Patches by Use of a Second Immiscible Fluid1

Jonathan Silver,*,2 Zihou Mi,* Keiji Takamoto,* Peter Bungay,† James Brown,†† and Adam Powell‡

*Laboratory of Molecular Microbiology, Building 4/338, NIAID, National Institutes of Health, Bethesda, Maryland 20892;†Bioengineering and PhysicalScience Program, Building 13/3N17, MSC 5766, National Institutes of Health, Bethesda, Maryland 20892;††Cytonix Corporation, Beltsville, Maryland

20705; and‡Metallurgy Division (855) MATLS B164, National Institute of Standards and Technology, Gaithersburg, Maryland 20899

E-mail: jsilver@nih.gov, bungay@box-b.nih.gov, jfbrown@cytonix.com, adam.powell@nist.gov.

Received February 19, 1999; accepted July 12, 1999

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We describe a method for forming an array of microdropletsetween two plates, at least one of which is patterned with a latticef wetting patches, using a second immiscible fluid to controlroplet formation. The method may be useful for performingultiple, small-volume biochemical reactions in parallel. We an-

lyze the forces responsible for droplet formation, describe resultsf a computer simulation using Surface Evolver, and derive annalytic criterion for droplet formation in terms of the contactngles of the droplet:second fluid interface on the wetting patchesnd surrounding surface, the diameter of the wetting patches, theistance between wetting patches, and the distance between thelates. © 1999 Academic Press

Key Words: microdroplets; arrays; wetting patches; Surfacevolver.

INTRODUCTION

There is growing interest in fluid handling methodserforming large numbers of chemical reactions in parallelds such as combinatorial chemistry, pharmaceuticalcreening, and DNA diagnostics. When the number of sams measured in the thousands, methods for dispensing comeagents in parallel have obvious advantages. Miniaturizs often desirable to conserve reagents, but miniaturizoses new challenges with respect to evaporation and diation of submicroliter volumes. We faced these problemeeking a way to perform multiple submicroliter polymerhain reactions (pcrs) in parallel for DNA diagnostic applions (1). In the case of pcr, the problem of evaporatioarticularly acute because the aqueous samples haveepeatedly cycled between;50 and 90°C. Here we describeethod for forming an array of submicroliter droplets betwicroscope slides that largely avoids evaporation and e

1 The use of commercial product names does not imply a recommenr endorsement by the National Institute of Standards and Technologoes it imply that the products are necessarily the best available fourpose.

2 To whom correspondence should be addressed.

81

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ates the need for individual pipetting steps. The methodifferential wettability of one or more surfaces to hold dropf one liquid while a second, immiscible, fluid is usedisplace excess portions of the first liquid from regions

ween wetting patches. For our application the second fluidUV-curable prepolymer so that aqueous droplets,

ormed, could be encased in a solid matrix to prevent evation. In applications where evaporation is not critical, aative second fluids, including gases, can be used.To understand the determinants of droplet formation in

ystem we used publicly available computer software forace energy minimization, Surface Evolver (2). The equium shapes of liquids in contact with materials with differurface energies is a classical problem described by soluo Laplace’s equation (3). However, numerical solutionaplace’s equation may, by themselves, provide little ins

nto the major determinants of a system’s behavior. Theace Evolver analysis suggested that, at a critical point dulling of our devices, the fluid interfaces were very closeurfaces of revolution. Making use of this approximation,erived a relation between material and geometric paramthe contact angles of the liquid interface on the hydrophnd hydrophilic surfaces, the distance between plates, anadius and distance between wetting patches). This relatiorovides insight into the determinants of droplet formationhould be helpful in designing new devices.

MATERIALS AND METHODS

To create a patterned wetting surface, we printed a hyhobic coating on glass using screen printing technologyscreen that left an array of 1-mm bare spots devoid of co

Fig. 1A). A typical coating consisted of 100 parts by weirethane acrylate (CN964, Sartomer Company, Exton, PAarts ethoxylated trimethylolpropane triacrylate (SR454,

omer), 2 parts silane methacrylate (Aldrich Cemical Cany, Milwaukee, WI), 1 part cobalt blue, and 1 part beimethyl ketal (BDK) UV catalyst (First Chemical Corpo

onor

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0021-9797/99 $30.00Copyright © 1999 by Academic Press

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82 SILVER ET AL.

ion, Pascagoula, MS). The rationale for these ingredientss follows. A wide variety of acrylates with different surfanergies are commercially available, so coating propeould be easily varied. The urethane acrylate made theolymer viscous, which was important to prevent printedolymer from “running” onto the bare glass areas before

ng. The triacrylate improved structural rigidity through croinks. Silane methacrylate was included to promote bondin

FIG. 1. (A) Schematic diagram of droplet forming device. (B) Deviceollowed by 100 ml of displacing fluid. Shadow at upper left is fromqueous-displacing fluid interface before and just after isolation of drop

as

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he glass surface. Cobalt blue was added to make the coisible to facilitate aligning of wetting patches. BDK is a Unitiator for acrylate polymerization.

The printed coating was cured by exposure to a meramp for 2 min under nitrogen. An adhesive strip consistin00 parts urethane acrylate (Sartomer SR966 J75), 25

sooctyl acrylate (Sartomer SR440), and 1 part BDKrinted around a block of wetting patches to form a

ng filled with 40ml of 1% food coloring in water (to visualize the aqueous phding pipette, and ruler units are cm. (C) and (D), 53 photomicrographs o

beiloalet.

83MICRODROPLET ARRAYS ON WETTING PATCHES

FIG. 1—Continued

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84 SILVER ET AL.

etween plates and cured as above. The plates were bakh at 100°C, immersed in dilute hydrofluoric acid to cleanare glass areas, diced into 13 3 cm units, and assembled insandwiches” with coated sides facing inward and bare getting patches aligned. The spacing between glass plate0–200mm, depending on the thickness of the glue trackAqueous solution pipetted onto the loading port entered

pace between the cover slips by capillarity. An immiscydrophobic “displacing fluid” was then pipetted onto

oading port from which it also entered the device by capity. The displacing fluid pushed the aqueous solution forwasily displacing it from the hydrophobic surface, but wh

he aqueous phase overlay bare glass it tended to adhere.he interface between the two liquids encountered a weatch, it wrapped around the edges of the wetting p

orming a small column of aqueous solution that eventuas completely surrounded by displacing fluid. This proas easily visualized by adding food coloring to the aquehase (Fig. 1B). Low magnification images show the conf the interface when a nascent column is about three-quurrounded (Fig. 1C) and just after it is completely isolaFig. 1D). The small scale undulation of the interface surrong the wetting patch shown in Fig. 1C is probably duerregularity in the edge of the surface coating. The lackircularity of the contour just after capture (Fig. 1D) is mikely a kinetic effect as video images showed that the droontour oscillated briefly after capture before relaxing tircle.For the pcr experiments we used acrylate solutions as

lacing fluids and polymerized them with UV light. A displang fluid that worked reasonably well consisted of 5 partseight) propoxylated neopentyl glycol diacrylate (SartoR9003), 3 parts ethoxylated trimethylolpropane triacry

Sartomer SR499), and 1% BDK catalyst. For applicathere evaporation is not a serious problem, many otherlacing fluids could be used, including mineral oil, silicoils, and nonpolar solvents.For computer modeling of droplet formation we used S

ace Evolver version 2.01 (The Geometry Center, 1300 Second Street, Minneapolis, MN 55454), available at htww.geom.umn.edu. Numerical calculations were perforsing Mathematica (Wolfram Industries, 100 Trade Cerive, Champaign, IL 61820-7237).

RESULTS

Several parameters were noted to affect whether the disng fluid formed and captured aqueous droplets on the weatches or just pushed the aqueous phase off both wettinonwetting surfaces. Droplets failed to form if the plates w

oo far apart, if the wetting patches were not very wettablhe coating was not very hydrophobic, or if the displacing flas viscous or flowed too rapidly.To understand quantitatively how these parameters influ

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roplet formation one must see how they affect the shapiquid interfaces. The calculation of minimal energy surfaor liquids in contact with nonhomogeneous surfaces is clicated by the fact that contact angles at boundaries betaterials with different surface energies are not uniquely

ned. Thus, the contact angle of a liquid interface at the rimwetting patch can be anywhere between the contact an

he interface on the coating and the contact angle ofnterface on the wetting patch (4). This freedom to assifferent contact angles is related to the ability of the inter

o “bend” around the edges of the surface discontinuity.To better understand the determinants of droplet formae began with a computer program, Surface Evolver,pproximates liquid interfaces as a set of triangular facetsrogram uses the steepest descent method to iteratively

he vertices toward the minimal energy configuration foiven triangulation topology. Evolver’s native scripting lauage was used to program an algorithm that cycles bet

his energy minimization and dynamic remeshing with graesh refinement, until a close approximation of the minnergy surface shape is reached. Figure 2A shows a scheiagram of a partially filled device with relevant geomearameters labeled. Inputs for the Evolver analysis wereontact angles of the aqueous–organic interface on the hhobic coating (wc) and on the glass wetting patch (wg), theadius of the wetting patch (r ), the vertical distance betwehe glass plates (2h), the center-to-center distance betwetting patches (2d), and the volume of displacing fluid in thevice.We assumed that gravity could be neglected in calculainimal energy surfaces since when devices were on a

urface, the maximal vertical displacement was less tham. We also assumed that at the low displacing velocity,

nertia, and contact line drag could be neglected, so at anstant the interface assumes the equilibrium shape fourrent volume of displacing fluid. Since we could fill tevices with incremental amounts of displacing fluid and a

he interface to come to equilibrium, the equilibrium assuion is reasonable. For a given volume of displacing fluid inevice, Evolver calculated the minimum energy interfhich was then displayed with an accompanying progeomView. By incrementing the volume of displacing flue could step through the filling process and follow chang

he predicted minimal energy surface during filling.As an example, Fig. 2B displays the Evolver-predicted a

us–organic interface when enough displacing fluid hasdded so that droplets are approximately half-formed.implicity, the figure shows only the portion of the interfaying between the lower glass plate and the midplane bethe glass plates. By symmetry, the upper and lower halfaces, and the left and right half surfaces, must be mmages. The interface curvatures are qualitatively consiith those shown in Fig. 2A.For givenr , h, d, andw , Evolver predicted that for value

c

iple radio withr

FIG. 2. (A) Schematic diagram of pcr-displacing fluid interface in partially filled device. Insets show vertical planes in regions 1 and 2, with princif curvatureR1 andR2, coordinatesR andw, and device parametersr , d, h, wr, andwc. (B) Interface calculated by Surface Evolver for half-formed droplet5 0.05 cm,d 5 0.1 cm,h 5 0.005 cm,w 5 81°, andw 5 20°.

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86 SILVER ET AL.

f wg below a critical value,wcrit, the displacing fluid woultay off the wetting patch, while for values ofwg greater thahe critical value the interface would march over the wetatch as displacing fluid was added, implying failure of dro

ormation. Lower values ofwg correspond to a more hydrhilic wetting patch. Interestingly, for values ofwg slightly less

han the critical value, Evolver predicted that the interould begin to encroach on the wetting patch as dropproached the half-formed state, but then retreat back oetting patch as more displacing fluid entered. Increaseswc,orresponding to increased hydrophobicity of the coating,avored droplet formation. Increases inh, the gap between thlates, led to failure of droplet formation, as we observeeal devices.

Evolver made several additional predictions for whichad no empirical data. It indicated that increases ind or r ,eeping other parameters constant, would favor droplet foion, i.e., allow it to occur at larger values ofwcrit. It alsondicated that the pressure drop across the interface wasst when the droplets were approximately half-surroundeisplacing fluid, and that at this time the contour of the in

ace in region 1 (see Fig. 2A) was very close to circular inorizontal plane. Near circularity of the interface at this p

s important as it allows mathematical simplification (seeow). As a measure of the circularity of the Evolver-predicnterface in region 1, we calculated the root-mean-squareerence between the radial coordinate of each Evolver verthe plane of the glass plate, and the radius of a circle of ra-r , using a coordinate system centered at the midpoinween wetting patches. This root-mean-square difference.0005 for d-r 5 0.05, so theEvolver surface was withibout 1% of the idealized circular interface in region 1.ontour of the interface in region 2 (Fig. 2A) also was circs it followed the contour of the circular wetting patch.If one assumes that the interfaces in regions 1 and 2 are

o circular in horizontal planes when the pressure drop ahe interface is greatest, i.e., that they form surfaces of rution about thez1 andz2 axes, respectively (Fig. 2A), then oan derive a relationship for the range of values ofr , d, h, wc,ndwg compatible with droplet formation. By Laplace’s eq

ion, the pressure drop across an interface is given by

p 5 g~1/R1 1 1/R2!, [1]

hereR1 andR2 are principal radii of curvature andg is thenterfacial surface tension. For surfaces of revolution, theiple radii of curvature can be written in terms of the raoordinateR and heightz of the interface (see Fig. 2A)ollows (5):

1/R1 5 2@d2R~ z!/dz2#$1 1 @dR~ z!/dz# 2% 23/ 2 [2]

1/R 5 @1/R~ z!#$1 1 @dR~ z!/dz# 2% 21/ 2. [3]

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When the distance between plates is small compared timensions of the wetting patches, as is the case in our dene can obtain a perturbation solution forp in Eq. [1] in termsf the parametersec 5 h/(d-r ) ander 5 h/r . Substitutingp 5

0 1 e p1 1 « 2 p2 1 . . . andR( z) 5 5 0( z) 1 e 5 1( z) 12 5 2( z) 1 . . . into [1], [2], and [3], and equating terms in liowers ofe, gives differential equations for5 0( z) and5 1( z)

hat can be solved subject to appropriate boundary condin region 1,dR/dz 5 0 at z 5 0 (the midplane between tlass plates), whereas atz 5 h (the top plate),R 5 d-r andR/dz 5 2cot wc. In region 2,dR/dz 5 0 atz 5 0, whereatz 5 h, R 5 r anddR/dz5 cot wr, wherewr is the angle tha

he interface makes with the surface at the rim of the weatch. The equations for5 0(h) and 5 1(h) in region 1 andegion 2 can then be solved forp0 and p1 leading to theollowing formulas, correct to first order inec ander:

pc 5 ~g/h!$coswc 1 ~ec/ 2!

3 @~p/ 2 2 wc!/coswc 1 sin wc# 1 . . .% [4]

pr 5 ~g/h!$cosw r 2 ~e r / 2!

3 @~p/ 2 2 w r!/cosw r 1 sin w r# 1 . . .%. [5]

ere,pc is the pressure drop in region 1 andpr is the pressurrop in region 2.The pressure drop across the interface in region 1 mu

he same as in region 2 at equilibrium. The anglewr for whichr 5 pc is the angle the interface must make with the surt the rim of a wetting patch in order for the pressure in reto equal that in region 2. Equations [4] and [5] implic

efinewr in terms ofg, h, wc, r , andd. The anglewr starts ouqual to wc when the interface first makes contact withetting patch and gradually decreases as the interfaceround the wetting patch. It reaches a minimum, correspon

o maximum pressure drop across the interface when theets are half-formed. Butwr cannot be less thanwg, or else thenterface would wash over the wetting patch. Thus, requhat wg # wr provides a criterion for droplet formation.

This result can be understood in the following manneegion 1, both principal curvatures 1/R1,region1 and 1/R2,region1

ncrease pressure in the displacing fluid for the contact ahown in Fig. 2A, whereas in region 2 the curvature 1/R1,region2

ncreases the pressure while 1/R2,region2 decreases it. Thus, fhe pressures to be equal, 1/R1,region2 must be numericallreater than 1/R2,region2. The greater the curvature 1/R1,region2, themaller the contact anglewr. The pressure is greatest whroplets are approximately half-formed because at this/R1,region2 is largest. Hence, at this pointwr will be at a mini-um. Butwr cannot be less thanwg. Thus, a limiting criterion

s that wg must be less than the smallestwr necessary tqualize the pressure when the droplets are half-formed.To evaluate the accuracy of the perturbation solution in

4] and [5], we used the fact that for surfaces of revolution,

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87MICRODROPLET ARRAYS ON WETTING PATCHES

1] can be solved exactly in terms of elliptical integrals. Tteps are as follows: the principal radii of curvature canewritten in terms of the radial coordinate,R, and angle olevation,w, as 1/R1 5 d(sinw)/dR and 1/R2 5 sin w/R (6)iving p 5 (g/R) zd(Rsin w)/dR. Integrating once givesRin w 5 ( p/ 2g)[R2 2 (d-r ) 2] 1 (d-r ) sin wc, where theonstant of integration is chosen so thatw 5 wc whenR 5 d-ron the coated surface). Sincedz/dR 5 2tan w, h is thentegral of 2tan w(dR/dw)dw from p/2 to wc. This can beewritten in terms of elliptic integrals (7) as

5 ~g/p!@coswc 1 ~kc 2 1/kc! F~kc, p/ 2 2 wc!

1 ~1/kc! E~kc,p/ 2 2 wc!#, [6]

here F(kc, p/ 2 2 w c) and E(kc, p/ 22w c) are ellipticntegrals of the first and second kind, and

kc 5 @1 1 ~ p/g! 2~d-r ! 2 2 2~ p/g!~d-r ! sin wc#21/ 2. @7#

quation [6] was solved forp and the resulting expressiubstituted into [7]. Equation [7] was then solved numericor kc for given values ofg, d, r , h, andwc, using Mathematicaquation [6] was then used to plotp as a function ofwc, shown

n the solid curves in Fig. 3.The same approach applied to the interface at the rim oetting patch (region 2) gives similar formulas, withd-r

eplaced byr , wc replaced bywr, kc replaced bykr, and signhanges resulting from the opposite direction of curvabout thez2 axis,

h 5 ~g/p!@cosw r 2 ~kr 2 1/kr! F~kr, p/ 2 2 w r!

2~1/kr! E~kr, p/ 2 2 w r!# [8]

kr 5 @1 1 ~ pr/g! 2 1 2~ p/g!r sin w r#21/ 2. [9]

quations [8] and [9] were solved numerically in the saanner as [6] and [7] and results plotted as the dashed c

n Fig. 3. A related approach to this problem has beenished (8).

One can use Fig. 3 graphically to find the maximum conngle on the wetting patch compatible with droplet formator a given contact angle on the coating,wc, a vertical line

ntersects the solid curve at the maximum pressure drop ahe interface for region 1. Tracing the horizontal line acroshe dashed curve and then down gives the contact angle oim required to produce this pressure in region 2; this isaximum allowable contact angle on the glass,wcrit. From thisrocedure it is clear that smallerh favors droplet formation b

ncreasing the slopes (Fig. 3A), leading to largerwcrit for aiven wc. Figure 3B shows that increasingd-r lowers theressure drop in region 1 while increasingr raises the pressurop in region 2. Thus, increases inr and d-r favor drople

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ormation by bringing the pressure drop curves closer togeaisingwcrit for any wc.

The pressure drop curves in Figs. 3A and B resemble cunctions, with amplitudes inversely proportional toh, dis-laced up or down by amounts proportional to the curvatu

he horizontal plane, as predicted by the approximate form4] and [5].

The error introduced by neglecting higher order terms inerturbation solution was estimated by comparing the pures calculated numerically (as in Fig. 3) to those calcurom formulas [4] and [5] for a series of contact angles.c 5 er 5 0.1 the perturbation result was within 1% ofumerical result, while forec 5 er 5 0.2 the difference wa2–3%. We conclude that Eqs. [4] and [5] give a satisfacpproximation that should be useful for designing drop

orming devices.Requiring that the interface be radially symmetric in regand 2 neglects the problem of how the solutions in these

egions are joined. To estimate the effect of neglectingoining region, we calculated the critical contact angle on gor typical values of our parameters,3 r 5 0.05 cm,d 5 0.1m,h 5 0.005 cm, andwc 5 81°. The circular approximatioredicted a critical angle of 56° while Evolver predicteritical angle of 55°, indicating that for estimation of tritical angle, this approximation is reasonable.

DISCUSSION

The method described here for forming droplets ofiquid on a surface of varying wettability is novel in that it u

second fluid to gently displace excess quantities of theiquid. It is important to note that the second fluid can also

gas. When we made devices with extremely hydrophoatings so that the aqueous phase had to be forced betwelates under pressure, releasing the pressure resulted in

3 We estimated the parameters in our system as follows. The capillarf pcr solution, displacing fluid, and distilled water in NaOH-rinsed gapillaries with inner diameter of 0.056 cm was 3.7, 2.0, and 5.2 cm, reively. The density of each fluid is approximately 1.0 g/ml. The contact af each liquid on glass was estimated using a video camera to record

mages. These angles werewpcr,g 5 24°, wdf,g 5 18°, andww,g 5 20°, with anstimated error of 20%. Using the formulag 5 r g hgt rad/ (2 coswg) whereis the gravitational constant, hgt is the capillary rise, rad the capillary randwg the contact angle on glass, givesgpcr 5 50 dyn/cm,gdf 5 27 dyn/cmnd gw 5 70 dyn/cm. The contact angle of the pcr solution-displacing

nterface in a glass capillary,wg, was estimated as 30° (measured from gdge through the pcr solution to the liquid–liquid interface). Lettingg 5 the

nterfacial energy between pcr solution and displacing fluid, from Youaw, g coswg 5 gpcr coswpcr/g 2 gdf coswdf/g f g 5 23 dyn/cm. The contangles of pcr solution and displacing fluid on the hydrophobic coatingstimated from video pictures of droplets on open devices aswpcr,c 5 55° and

df,c 5 22°, respectively. Substituting these values into Young’s law,g cos

c 5 gpcr cos wpcr/c 2 gdf cos wdf/c, gives wc 5 81°. Given the large errossociated with the estimates of contact angles, these values must bered fairly uncertain.

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88 SILVER ET AL.

uid being spontaneously expelled, leaving an array of drourrounded by air.The second fluid is important in that it provides a w

radually to form and isolate droplets, minimizing surfnergy at each step along the way. Calculating the ener

ntermediate configurations allowed us to derive a neceondition for droplet formation. The simplest criterionroplet formation is that the free energy of a droplet oetting patch surrounded by displacing fluid must be less

he free energy of the alternative configuration with displa

FIG. 3. (A) Numerically calculated interfacial pressure drop going from5 0.05 cm,d 5 0.1 cm, and indicated values ofh (in cm); (solid line) pr

or h 5 0.0025 cm andindicated values ofr andd-r .

ts

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uid on the wetting patch. An interesting example of this tf analysis for droplets on an open surface was recentlyented (9). However, this approach does not take into acow the system arrives at the state of isolated droplets. If

s no path to the final state along which the free energontinuously decreased, the system may be trapped at aather than a global minimum. In our system, a criterionroplet formation based solely on comparing the initialnal free energies misses the fact that droplet formatioritically dependent on the distance between wetting pat

splacing fluid to pcr solution for radially symmetric interfaces withg 5 23 dyn/cmure drop in region 1; (dotted line) pressure drop in region 2. (B) Pressu

diess

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89MICRODROPLET ARRAYS ON WETTING PATCHES

his distance affects the energy barrier of the intermediaten which the displacing fluid “squeezes” between adjaetting patches.As a first step toward understanding the parameters

ffect droplet formation in our devices we used the compodel Surface Evolver. This model captured the major feae observed including the need for a large difference in

ability and a small gap between plates. It also indicatedhe maximum pressure drop across the interface occursroplets are half-formed. The calculated pressure usuallyroad maximum as a function of the amount of displacing

n the device, with small fluctuations probably due to detaihe numerical calculation. The contour of the interfaceicted by Evolver was very close to circular at this poinlling, which justified a simplified model in which the interfaas approximated as a surface of revolution in each regi

his point in filling. This led to an implicit formula for the ranf values of r , d, h, wc, and wg compatible with drople

ormation.Droplet formation in our devices appeared to be someore restrictive than predicted by this formula, although

nability to measure the contact angleswc and wg accuratelyrecluded a precise test. We estimatedwc from Young’s for-ula, the contact angles of droplets on coated surfaces

ounded by air, and results of capillary rise experimentsootnote 3). However, in our application there was an aional complication. We needed to include bovine serum ain (BSA) in our pcr mixtures in order to prevent adsorpf enzyme or DNA to the glass surface. We observed thatxposure of hydrophobic and hydrophilic surfaces to theolution altered their wettability, reducing the contrast betwetting patches and surrounding hydrophobic coating. T

he contact angles we measured with single droplets onurfaces may not accurately reflect the contact angles in aevices.Our simplified model also does not take into account

amic aspects of the filling process. We observed thapeed with which the displacing fluid moved had a signififfect on droplet capture. The filling speed could be contro

o some extent by tilting a device so that gravity eitheranced or retarded the flow. For many devices, droplets wot form at faster filling speeds, indicating that we were oting in a regime where viscous drag was of the same ordagnitude as the surface forces tending to form dropiscous forces should be more important than inertial forceur devices since the Reynolds number,NRe, a measure of thatio of these forces, is about 0.06 (NRe 5 rvh/m, r ; 1 gm23, v ; 0.2 cm sec21, h ; 0.01 cm,m ; 0.03 g cm21

ec21). The problem of viscous forces might be mitigated

tet

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sing displacing fluids of lower viscosity, an extreme ceing the use of a gas. The viscosity of air is approxima.02 centipoise, compared to 3 centipoise for the acrolutions we used. If the filling rate can be controlled,xample with a syringe, gaseous displacing fluids might th

ore be advantageous in certain applications.Despite its limitations, the model described here prov

nsights that may guide further experiments. For exampleodel predicts that devices with a pattern of wetting spotnly one side should act like two-sided devices with a laap. In the simplest case where the interface makes a 90°ith the unpatterned surface, a one-sided device should actwo-sided device with twice the gap, since by symmetry

ngle made by the interface with the midplane between ps 90°. Preliminary experiments with single-sided devicesn agreement with this prediction. It is much easier to mevices with only one patterned surface since this elimin

he need for wetting spot alignment. For many purposould be desirable to further miniaturize droplets. Sevethods for patterning surfaces on a scale of 1 to 100mm haveeen reported (10–12). Our model predicts that droplet fo

ion should be unchanged so long as the ratios ofh to r andho d-r remain unchanged. As diagnostic devices becmaller and the need for increasingly parallel analyses grays of trapping and controlling liquids on a microscaleecome increasingly important.

ACKNOWLEDGMENTS

We thank Drs. Robert Miura and Jim Warren for insightful discusshilip McQueen for assistance with Mathematica, Seth Goldstein foreasuring contact angles, and Rick Coker for printing hydrophobic coa

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