RESEARCH ARTICLE

Evelyn N. Wang Æ Shankar Devasenathipathy

Hao Lin Æ Carlos H. Hidrovo Æ Juan G. Santiago

Kenneth E. Goodson Æ Thomas W. Kenny

A hybrid method for bubble geometry reconstructionin two-phase microchannels

Received: 7 July 2005 / Revised: 11 January 2006 / Accepted: 15 January 2006� Springer-Verlag 2006

Abstract Understanding bubble dynamics is critical tothe design and optimization of two-phase microchannelheat sinks. This paper presents a hybrid experimentaland computational methodology that reconstructs thethree-dimensional bubble geometry, as well as providesother critical information associated with nucleatingbubbles in microchannels. Rectangular cross-sectionsilicon microchannels with hydraulic diameters less than200 lm were fabricated with integrated heaters for theflow experiments, and the working liquid used was wa-ter. Bubbles formed via heterogeneous nucleation andwere observed to grow from the silicon side walls of thechannels. Two-dimensional images and two-componentliquid velocity field measurements during bubble growthwere obtained using micron-resolution particle imagevelocimetry (lPIV). These measurements were combinedwith iterative three-dimensional numerical simulationsusing finite element software, FEMLAB. The three-dimensional shape and location of the bubble werequantified by identifying the geometry that provided thebest match between the computed flow field and thelPIV data. The reconstructed flow field through thisprocess reproduced the experimental data within an er-

ror of 10–20%. Other important information such ascontact angles and bubble growth rates can also beestimated from this methodology. This work is animportant step toward understanding the physicalmechanisms behind bubble growth and departure.

1 Introduction

Interest in cooling microprocessors using microchannelheat sinks has significantly increased in the past decade,due to advancements in micromachining technology andincreases in chip power density (International Technol-ogy Roadmap for Semiconductors, http://www.pub-lic.itrs.net). Two-phase microchannel heat sinks providea potential solution where large heat generation ispresent, because latent heat during the phase-changeprocess can be leveraged to capture and transfer highheat fluxes. Implementation of two-phase microchan-nels, however, is challenging due to the instability of thevapor bubbles and slugs, leading to local dry-out andsubsequently poor cooling in such regions (Kennedyet al. 2000; Hetsroni et al. 2001). Efforts have thereforefocused on the understanding of flow patterns duringmicrochannel boiling, which are different than theirmacroscale counterparts (Jiang et al. 2001; Zhang et al.2002, 2005; Peng and Wang 1993; Hetsroni et al. 2002;Kandlikar 2004; Mukherjee and Kandlikar 2005). In-sights provided by such studies are essential for thedevelopment and optimization of two-phase micro-channel cooling solutions.

As a step toward understanding microchannel boilingbehavior, incipient boiling has been examined in whichbubbles nucleate, grow, and depart from nucleation siteson channel walls. Many studies have used flow visuali-zation with high-speed cameras and white light micros-copy to aid in understanding mechanisms and theassociated forces exerted during bubble nucleation anddeparture (Kandlikar and Stumm 1995; Lee et al. 2004;Qu and Mudawar 2002). Lee et al. (2004) reported

E. N. Wang (&) Æ S. Devasenathipathy Æ H. LinC. H. Hidrovo Æ J. G. Santiago Æ K. E. Goodson Æ T. W. KennyDepartment of Mechanical Engineering, Stanford University,Stanford, CA 94305-3030, USAE-mail: enwang@stanford.eduTel.: +1-650-7360044Fax: +1-650-7233521

Present address: S. DevasenathipathyDepartment of Molecular Pharmacology, Stanford University,Stanford, CA 94305, USA

Present address: E. N. WangBell Laboratories, Lucent Technologies,Murray Hill, NJ 07974, USA

Present address: H. LinDepartment of Mechanical and Aerospace Engineering,Rutgers University, Piscataway, NJ 08854, USA

Experiments in Fluids (2006)DOI 10.1007/s00348-006-0123-z

bubble growth rates, departure sizes, and departurefrequencies for a single trapezoidal microchannelgeometry with Dh=41.3 lm (where Dh denoteshydraulic diameter) using a high-speed camera andcommercial edge detection software. They reported thatbubble diameter was often linear with time and bubbledeparture radii were in good agreement with a modifiedLevy model (Levy 1967). Their work provides importantdata regarding bubble dynamics during incipient boilingas well as comparisons with classical models.

Recent modeling developments have contributed tounderstanding growth of vapor bubbles in microchan-nels. Mukherjee and Kandlikar (2005) used the level settechnique to numerically model the growth dynamics. Inaddition, various modeling efforts have also been com-bined with white light imaging techniques to enablebetter predictions relevant to microscale incipient boil-ing processes. Qu and Mudawar (2002) developed amodel predicting incipient heat flux based on visualiza-tion experiments performed in microchannels ofDh=350 lm. Kandlikar and Stumm (1995) on the otherhand, measured bubble departure radii and contact an-gles in channels with Dh=5.7 mm to supplement acontrol volume based bubble departure model. In thelast two cases, model predictions relied on assumptionsof bubble shape, location, liquid velocity fields, andassociated surface forces since experimental data onthese quantities were limited.

Liquid velocity fields in general two-phase systemscan be obtained using particle image velocimetry (PIV)(Qiu and Dhir 2002; Lindken and Merzkirch 2002; Liuet al. 2004; Park et al. 2004). Qiu and Dhir (2002) usedPIV to examine the local flow fields surrounding slid-ing bubbles on heated inclined surfaces: vorticalstructures were observed in the rear region of thebubble which confirmed their temperature fringe pat-tern results. Lindken and Merzkirch (2000, 2002)developed a technique using PIV and shadowgraphy tomeasure velocity fields in bubbly two-phase flow. Theyused this measurement technique to investigate turbu-lent structures induced by gas bubbles. Park et al.(2002, 2004) suggested the possibility of using confocalmicroscopy with micron-resolution PIV (lPIV) tocapture velocity fields surrounding slowly moving airslugs at several depths to reconstruct the three-dimensional liquid film and velocity field. However, thelatter technique would be very limited by the lowtemporal resolution of confocal imaging for both im-age capture and variations of the location of the focalplane. Despite these limitations, these studies suggestthat velocity field measurements can potentially pro-vide valuable data and insight that can complementwhite light imaging and shed more insight into bubbledynamics and departure.

In this work, we present a hybrid experimental andcomputational methodology to reconstruct the three-dimensional bubble geometry in microchannels with theuse of lPIV and numerical simulations. This workcontributes toward the understanding of dynamics and

mechanisms for bubble growth and departure by pro-viding quantitative information on bubble diameter,location, contact angles, growth rate, and bubble mo-tion. This information can be integrated to provideestimates of the three-dimensional flow field which canbe used for force analyses.

We first used lPIV measurements in microchannelswith Dh=190 lm to obtain two-dimensional liquid-phase velocity fields surrounding nucleating bubbles.To avoid difficulties associated with real-time controlof the location of the measurement volume, we choseto obtain velocity fields at a single channel depthduring one nucleation-to-departure cycle (see theexemplary result in Fig. 2). These transient measure-ments provide two-component velocity fields in two-dimensional planes of the three-dimensional flow field.Nonetheless, by combining this information with iter-ative numerical simulations (FEMLAB, http://www.comsol.com), it is possible to approximatelyreconstruct the three-dimensional bubble geometry aswell as the flow field. This hybrid experimental andcomputational methodology is the primary focus ofthe current paper.

This paper is organized as follows: The experimentalsetup is summarized in Sect. 2, together with typicallPIV measurement results. In Sects. 3 and 4, we presentthe hybrid experimental and numerical methodologyand results, respectively. In Sect. 5, we draw conclusionsand make suggestions for future work. Estimates ofuncertainties associated with the measurement andmethodology are presented in Appendix.

2 Experimental setup

2.1 Device fabrication and flow delivery

A schematic of the experimental setup and microchanneltest device is shown in Fig. 1. The device was fabricatedusing standard micromachining techniques. The micro-channel was deep reactive ion etched (DRIE) on a sili-con substrate and was 2 cm long, 200 lm wide, and180 lm deep (Dh=190 lm). These dimensions corre-spond to the coordinate directions x, y, and z, respec-tively. The DRIE process created fine features on theside walls with approximate roughness ranging from 0.1to 0.4 lm and occasionally larger defects ranging from 2to 5 lm, which probably acted as nucleation sites forbubbles (Zhang et al. 2005). A localized titanium heaterof 1 mm2 area was patterned in the middle of thechannel back side. Titanium thin-film temperature sen-sors were fabricated 1 mm upstream and 1 mm down-stream of the heater to measure approximately 1 mm2

area-averaged wall temperatures. A glass slide (Pyrex)was anodically bonded to the silicon channel which al-lowed for flow visualization during the bubble growthprocess. Wire bonds electrically connected the heaterand temperature sensors to the printed circuit board,where an input heater voltage could be applied and

sensor voltages measured to determine wall temperature.Fluidic inlet and outlet ports (Upchurch Scientific) wereglued to the backside of the chip.

The flow was delivered by a constant flow rate syringepump (PhD 2000, Harvard Apparatus). The workingfluid (deionized water) was seeded with 1 lm fluorescentpolystyrene particles to a volume density of 0.05%. Theparticles (Molecular Probes) had a peak excitationwavelength of 540 nm and a peak emission wavelengthof 560 nm. Triton X-100 surfactant (Sigma Corpora-tion) was added at a concentration of 0.005% by volumeto the particle solution to decrease the effects of particleflocculation and adsorption to channel walls (the criticalmicelle concentration for this surfactant is 0.01% byvolume). We applied typical flow rates of 0.1 ml/min(ReD=10, where ReD is Reynolds number based onhydraulic diameter). These flow rates are within therange used in microchannel cooling research (Zhanget al. 2002).

2.2 Flow visualization and lPIV measurements

An epi-fluorescent microscope (Nikon TE300) wasintegrated with a dual-pulsed Nd:YAG laser (NewWave, Minilase System) as an illumination source(Devasenathipathy et al. 2003). The 532 nm frequencydoubled emission of the laser was routed through anoptical fiber (Oriel Instruments) to the sample for illu-mination of the entire device test section volume. A filtercube assembly containing a 532 nm exciter filter, a di-chroic beam splitter with a cut-on wavelength of545 nm, and a 555 nm long-pass barrier emission filterwas used to filter and separate the excitation light from

the emission signal. The images were captured with a20· objective (NA=0.45) and a 12-bit interline transfercooled CCD camera (Roper Scientific CoolSnap HQ).We used air immersion lenses in effort to be thermallydecoupled from the experiment. The camera had a1,300·1,040 pixel array with square pixels 6.45 lm on aside. We used 2·2 on-chip pixel binning during imageacquisition. The bubble diameters grew at a typical rateof 5 lm/s for an applied power of 0.58 W. The corre-sponding wall temperature for these nucleating bubbleswas measured to be approximately 60�C. (Local abso-lute pressure at the bubble site is estimated to be1.0 atm.) Previous studies have shown a decrease inincipient boiling temperature by increasing the air con-tent in water (Murphy and Bergles 1972; Steinke andKandlikar 2004). However, the low measured tempera-tures of our study strongly indicate that the bubblenucleation and growth we observed might be mostlyattributed to the outgassing of dissolved gases (e.g.,carbon dioxide, nitrogen, and oxygen, etc.), not boiling.We also hypothesize that residual gases trapped in cav-ities on the silicon channel walls provided nucleationsites for such outgassing. As discussed by Jones et al.(1999), such nucleation sites reduce the energy barrierfor nucleation to occur at relatively low temperatures.Similar behavior has been observed at comparabletemperatures and pressures (Jiang et al. 2001; Li andCheng 2004; Klein et al. 2005). The distinction andinteraction between outgassing and incipient boiling inmicrochannels is an important area that warrants fur-ther investigation, and may lead to different growthrates and departure characteristics. Local wall temper-ature and pressure measurements are critical to suchstudy.

Fig. 1 Schematic ofexperimental setup withfabricated microchannel devicewith integrated, thin-filmheaters and temperaturesensors: a schematic andb image of microfabricatedsilicon channel test deviceviewed from the backside.A white horizontal linesuperimposed on the imageindicates the location of thechannel on the front side

Liquid velocity fields around growing bubbles weremeasured using lPIV (Santiago et al. 1998). A customPIV algorithm was used to interrogate the images anddetermine time-averaged velocity fields using the averagecross-correlation method (Meinhart et al. 2005). Typicaltime between frames was 200 ls for 0.1 ml/min flowrates. Three pairs of images were used in the correlationalgorithm and each pair of images was separated by222 ms. We assumed that the bubbles were approxi-mately quasi-stationary during acquisition of these threeimage pairs. The measurement depth defined as twice thedistance from the focal plane where particle-imageintensity is sufficiently low such that it will not signifi-cantly influence the velocity measurement is 13.5 lmbased on diffraction limits, geometrical optics, and thefinite size of the particle (Meinhart et al. 2000).

2.3 Velocity field measurement

An example experimental image for the lPIV measure-ment is shown in Fig. 2a. The imaged channel sectionwas 450 lm long (in the x, streamwise direction) and200 lm wide (in the y, spanwise direction). This imagewas taken at a depth of d2=17.5 lm, where d2 denotesthe distance of the focal plane from the inner side of theglass channel wall (see Fig. 3b). The imaging depth intothe channel was determined by first focusing on particlesadsorbed to the inner glass wall of the channel and thenmoving the focal plane using the microscopes stage. The

total channel depth (into the page) was 180 lm. Thecorresponding measured velocity field is shown inFig. 2b; note that this velocity field is a two-componentvelocity measurement of the full three-dimensionalvelocity field at a particular focal plane. The measure-ments can be interpreted as projections of the three-dimensional velocity field onto the focal plane. Similarly,the cross-section of the three-dimensional bubble at thischannel depth was a truncated disk with a radius ofa0=48.5 lm. Uncertainties associated with the flow fieldmeasurements and the methodology were approximately10%. Details of these errors and associated uncertaintiesare given in Appendix.

Velocity datawere obtained at various times (includingvarious stages of bubble growth) for single bubblenucleation-to-departure cycles. We also varied the loca-tion of the focal plane to obtain velocity fields for bubblesoriginating from the same nucleation site at differentdepths of the channel, specifically, at d2 = 10, 17.5, and30 lm. In some cases, the bubble nucleation and depar-ture cycle was reproducible enough to warrant a com-parison across imaging plane locations (i.e., acrossrealizations of the bubble nucleation anddeparture cycle).

3 Bubble geometry and flow field reconstruction method

The bubble image and lPIV measurements discussedabove provide limited information. They yield a two-component velocity measurement in a two-dimensional

Fig. 2 Example image (a) andmeasured lPIV velocity fieldsurrounding the bubble (b). Thefocal plane was a distance ofd2=17.5 lm from the glasschannel wall (see Fig. 3). Theimage shows a 450 lm section(in the streamwise, x direction)of a 200 lm wide (in thespanwise, y direction) channel,and the channel depth (into thepage) is 180 lm. The bubbleimage is a truncated disk ofa0=48.5 lm which representsthe cross-section of its fullthree-dimensional geometry atthe focal plane. The coordinatesin graph (b) are scaled with thehalf-channel width ofL=100 lm. Note that themeasurement shows the two-component velocity field of thethree-dimensional flow field inthe microchannel

slice of the three-dimensional flow. In this section wepresent a hybrid experimental and numerical method toreconstruct the full three-dimensional flow field as wellas bubble geometry.

Our basic approach is as follows. Shown in Fig. 3aare two possibilities of the instantaneous three-dimen-sional bubble geometry, both with the same cross-sec-tional area matching the truncated disk with radius a0(as shown in Fig. 2a, b) at the focal plane P at a distanced2 from the inner glass channel wall. Although these twoexample cases have identical bubble cross-sections, thethree-dimensional flow fields surrounding the bubbleshave distinctly different two-component velocity fields atthe two-dimensional slice of interest. We can thereforepredict a variety of three-dimensional flow solutions forbubbles with varying diameters (and varying centerlocations) but identical bubble cross-sections. The in-plane projection of the predicted three-dimensionalvelocity vectors at the focal plane can be compared tothe respective lPIV measurements and the best matchbetween the predicted and measured velocity fields givesan estimate of the actual bubble geometry.

3.1 Assumptions

We shall employ three important assumptions for ourbubble geometry and flow field reconstruction method-ology:

1. The bubbles are assumed to be truncated spheres.Typical Reynolds numbers based on bubble diameter

are on the order of Re�10. The Weber number,which signifies the relative importance of inertialforces to surface tension, is typically on the order ofWe�0.01. This scaling suggests that the bubbleundergoes minimal shape deformation and remainsnearly spherical. However, at later stages of bubblegrowth, when bubbles often occupy a large portion ofthe channel, deformation may be significant due tosubstantial surface force variations, non-uniformpressure fields in the flow, and contact angle hyster-esis. In these situations, the sphericity assumptionmay not be valid and a more complex geometry maybe needed.

2. Bubble location and diameter are assumed to beapproximately fixed during the 444 ms duration ofour three image pair acquisition. For the currentexperiments, bubble diameters typically grow at arate of 5 lm/s; a change of approximately 2 lmduring image acquisition. The flow field is also as-sumed to be quasi-steady. The Strouhal number,based on the growth rate of the bubble (5 lm/s) andthe characteristic velocity (40 mm/s) is on the orderof St � 10�4, suggesting unsteady flow effects areinsignificant.

3. The liquid viscosity is assumed to be uniform andthat of water at 60�C (the measured wall tempera-ture). The residence time scale, computed as the li-quid residence length in the heated region before itreaches the bubble divided by the bulk flow velocity,is approximately 2.5 s. This time scale is much greaterthan the heat diffusion time scale (approximately

Fig. 3 Schematic for bubblegeometry reconstruction.Shown in a are two possiblebubble geometries in themicrochannel where they haveidentical cross-sectional areas atfocal plane P matching thetruncated disk shown in Fig. 2.These geometries producedistinctly different flow fieldswhich are compared with thelPIV data to determine the bestmatch. Part b shows aschematic of the trial bubblegeometry. The bubble maycontact both the glass (in thepositive z direction) and silicon(in the positive y direction)channel walls when R>d and/or R>dy, respectively. Thebubble radius R and center-to-image-plane distance d1 need tosatisfy Eq. 1, leaving R to be thesingle degree of freedom

0.067 s), the time required for liquid to reach uniformtemperature via heat diffusion from the channelwalls. The liquid temperature in the simulated regioncan thus be assumed to be that of the channel wall(60�C), resulting also in relatively uniform liquidproperties such as viscosity.

3.2 Trial geometry construction

Following our first assumption above, we present aschematic of the bubble geometry in Fig. 3b. This con-struct is used to determine the three-dimensional bubblegeometries for a given bubble cross-section at planeP. The bubble cross-sectional shape is first determinedby an image-processing algorithm where points out-lining the shape of the bubble are selected based on localpeak intensities at the interface. These points are thenbest fit to a circle to determine the center (x0, y0) andradius a0 of the circular segment that outlines the bubblecross-section (see also Fig. 2). The z coordinate of theimage center zp in Fig. 3b is known from the image-to-glass-slide distance d2. The location of the bubblecenter (x0, y0, z0) and the (three-dimensional) bubbleradius R then completely describe the bubble geometry.Note that the distance of the bubble center (x0, y0, z0) tothe image center (x0, y0, zp), d1, can be related to thebubble radius as

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d21 þ a20

q

; ð1Þ

where d1=zp�z0. This constraint implies that a singlefree parameter, d1 or R, can be chosen to fully describethe bubble geometry. Note also that the bubble maycontact two channel walls, when R>d and/or R>dy,where d and dy are the distances of the bubble centerfrom the glass and silicon channel walls, respectively, asillustrated in Fig. 3b.

3.3 Numerical iteration scheme with FEMLAB

We vary the single free parameter (R or d1) and computethe corresponding flow field for an initial trial geometry.The best approximating value for the actual bubble ra-dius R=R0 is chosen based on the closest match betweenthe computed and experimental velocity fields. Ournumerical procedure is summarized in Fig. 4. A trialbubble radius of R ‡ a0 is chosen and the geometry ofFig. 3b is used to determine d1 and d. A trial pressuredifference is then imposed on the flow.

We use FEMLAB to solve the incompressible three-dimensional Navier–Stokes equations using the finiteelement method for the liquid-phase only. The flow fieldis considered steady as per Assumption 2 above. Thegeometry is created and meshed with the boundaries ofthe computational domain located at least two diametersupstream and downstream of the nearest bubble surface.(A larger computational domain was used to verify that

the pressure gradient at the upstream and downstreamboundaries approximately relaxes to the constant valuefar from the bubble.) The prescribed boundary condi-tions are as follows: (1) Flow was introduced by anapplied pressure difference between the inlet and theoutlet; (2) No-slip boundary conditions were prescribedat the channel walls; (3) No-slip boundary conditionswere prescribed at the liquid-vapor interface. (Theseemingly unconventional choice of a no-slip boundarycondition for the bubble interface is discussed in Sect. 4.)

The correct pressure difference across the computa-tional domain was also determined using an interativeprocedure within each iteration of trial bubble geometry.Upon solving the flow field using a trial pressure dif-ference, the velocities at the inlet are integrated todetermine the flow rate. When the inlet flow rate con-

Fig. 4 Iteration scheme used for numerical simulation. Thecommercial finite element software FEMLAB was used to solvethe three-dimensional, steady-state Navier–Stokes equations in themicrochannel with trial values of R for each a0. The appliedpressure difference was varied until flow rate matched that of theexperiments. For each R, the computed flow field was comparedwith lPIV data and an RMS error was obtained using Eq. 2. Thebest approximating radius R=R0 was identified from the minimaof the eRMS(R) curves (Fig. 6)

verges to within ±2% of the experimentally measuredflow rate, the pressure iteration is completed. The pro-jected two-dimensional flow field at plane P is thencompared with the experimental velocity data using aroot mean square (RMS) error analysis, defined as

eRMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P

n

iðuexp;i � unum;iÞ2 þ ðvexp;i � vnum;iÞ2

P

n

iðu2

exp;i þ v2exp;iÞ

v

u

u

u

u

u

t

; ð2Þ

where u and v denote the velocities in the x and ydirections, respectively. The subscript exp denotes theexperimental lPIV results, num the numerical results,and the subscript i is the index of points where PIVvelocity data is obtained. The sum is over all of the nsampled points.

For each image depth d2 and instantaneous imageradius a0, we solve for the three-dimensional velocityfield, and the function eRMS(R) is obtained for a range oftrial bubble radii R. The best approximating geometrywhere R=R0 is identified by locating the minimum ofthe eRMS(R) curve (Fig. 6), i.e., R0 satisfies

eRMSðR0Þ ¼ minR½eRMSðRÞ�: ð3Þ

In Sect. 4, we present and analyze our results based onvarious available data sets.

4 Results

Figure 5 shows an overlay of best-matched experimentaland simulated velocity fields for the case shown in

Fig. 5 Exemplary best matched experimental and simulated veloc-ity fields at the focal plane. For this case, the bubble image has aradius of a0=48.5 lm at d2=17.5 lm from the glass channel wall(Fig. 2). The best approximating bubble radius R0=53 lm iscaptured from the minima of the a0=48.5 lm curve in Fig. 6b.

Shown in the figure are a overlay of experimental and simulatedvelocity field, and b u-velocity slices as a function of y at varying xlocations. The length and velocity scales are L=100 lm andV=40 mm/s, respectively. For this case, the calculated RMS erroris 12%

Fig. 2, where a0=48.5 lm at a focal plane location ofd2=17.5 lm. R0=53 lm is chosen from the minimumof the a0=48.5 lm curve in Fig. 6b. The coordinates arenondimensionalized by the half-channel widthL=100 lm, and the velocity is nondimensionalized by acharacteristic velocity scale V=40 mm/s. The u-velocityprofiles at various nondimensionalized x locations in thechannel are shown in Fig. 5b. Fairly good agreement isattained between the computed and experimentalvelocity data, and the RMS error (i.e., discrepancy be-tween measurement and computation) for this particularcase is approximately 12%.

Figure 6 shows the RMS error as a function of thetrial radius R. Plots 6a, b correspond to various mea-surement depths (and therefore individual bubblenucleation-to-departure cycles or realizations) ofd2=10 lm (left) and d2=17.5 lm (right), respectively.Each plot contains four curves corresponding to variousa0 values (i.e., different instances during the growth ofthe bubble). The minimum RMS error of each curve ingeneral ranges from 10 to 20%, and deteriorates as a0increases. However, the clear presence of a minimum foreach RMS error curve evidently suggests that there is abest approximating bubble geometry and three-dimen-sional velocity field for each growing bubble at eachinstant in time. Note the RMS curves cannot be

extended further toward the left as the bubble radius Rmust be equal to or larger than the image radius a0.

We have also evaluated the efficacy of differentboundary conditions in the numerical simulations at theliquid–vapor interface, and the results are shown inFig. 7. Both slip and no-slip boundary conditions wereprescribed for the bubble interface for depths ofd2=10 lm (left) and d2=17.5 lm (right) and for variousvalues of a0. The RMS errors presented are the mini-mized error min

R½eRMSðRÞ� (or eRMS(R=R0)) as deter-

mined from the curves in Fig. 6. The results suggest thatthe no-slip boundary condition consistently results insmaller values of eRMS. This result at first seems to beinconsistent with bubble/liquid interaction, as liquid–gasinterfaces are typically modeled using slip boundaryconditions (Batchelor 1967). We attribute this result tothe presence of surfactants in our experiments (see Sect.2). While the surfactants decrease the surface tension ofthe bulk liquid, the important parameter is the hydro-dynamic conditions of the interface. Surfactants areknown to effectively immobilize bubble surfaces andresult in a liquid–vapor interface that acts more like asolid surface (Clift et al. 1978; Levich 1962). Otherpossibilities such as surface tension spatial variations or

Fig. 6 RMS error (Eq. 2) as a function of the trial bubble radius Rfor image depths of a d2=10 lm and b d2=17.5 lm. Each value ofa0 corresponds to a unique bubble nucleation-to-departure cycle.The best approximating radius R=R0 was extracted from theminima of the curves for each value of a0

Fig. 7 Minimal RMS errors computed with slip and no-slipboundary conditions prescribed at the liquid–vapor interface, forthe cases of a d2=10 lm and b d2=17.5 lm. Here, RMS errors arethe minimal errors eRMSðR0Þ ¼ min

R½eRMSðRÞ� for each value of a0

(see Fig. 6). The results suggest the no-slip boundary conditionprovides a better agreement between the experimental data andnumerical simulations

surfactant accumulation at the rear region of the bubble(Levich 1962) may also complicate the boundary con-ditions. The issue of liquid–vapor hydrodynamicboundary conditions in boiling (and outgassing bubble)flows is a critical area where more work is needed.

Figure 8 examines the movement of the bubble in thedepth direction, z, from the glass channel wall as afunction of the best approximated bubble radius R0, forthree nucleation realizations measured at d2=10, 17.5,and 30 lm. The total distance of the bubble center fromthe glass channel wall d is obtained where d=d1+d2(Fig. 3b). The results indicate that for R0 £ 55 lm, thereis an approximately linear relation between the distanceof the bubble center from the glass channel wall and thebubble radius; whereas for R0 ‡ 55 lm no obviousrepeatable trend is observed. Most importantly, theagreement of trajectories in the range of R0 £ 55 lmacross values of d2 (focal depth) suggest a trend in thevalue of contact angles as discussed below.

The linear dependence of R0 versus d for R0 £ 55 lmsuggests consistent values of contact angles. As shown inFig. 3b, the bubble may contact both the channel walls,when R0>d and/or R0>dy, respectively. The contactangles are given by

hg ¼ cos�1dR0

� �

; hs ¼ cos�1dy

R0

� �

; ð4Þ

where hg and hs denote the contact angles on the glassand silicon channel walls, respectively. (Note that acontact angle computed according to hs=cos�1(dy/a0),

which may be obtained from Fig. 2 alone without thefull geometry reconstruction, does not correctly repre-sent the contact angle of the windward and leeward sidesof the three-dimensional bubble.) A linear fit (intersect-ing d=R0=0) of the initial d–R0 curve for R0 £ 55 lm ispresented in Fig. 9a; a fitted value of d/R0=0.90 gives avalue of hg=26�, which is in close agreement with re-ported contact angles values (Cubaud and Ho 2004).Similarly, we also plot and fit the dy–R0 curves in Fig. 9band obtain dy/R0=0.73 and hs=43�. This value is withinthe range of contact angle values for water on silicon.More precise comparisons with past work are difficult asthe channel walls used here were etched using the DRIEprocess which may significantly alter surface roughness.

The agreement of the d–R0 and dy–R0 relations forR0 £ 55 lm (see Figs. 8, 9) provides evidence that themethodology is consistent. For three measurement planes(d2=10, 17.5, and 30 lm), and therefore three indepen-dent realizations of bubble nucleation-to-departure cy-cles, we observed consistent d–R0 and dy–R0 relations aswell as similar contact angle values on both glass andsilicon channel walls. Because (R0, d, dy) completelyspecify the three-dimensional bubble geometry as well as

Fig. 8 Bubble center distance in the depth (z) direction, d, as afunction of the best approximated radius R0. The distance d is thedistance of the bubble center from the glass channel wall (seeFig. 3). In general, bubble centers move away from the glasschannel wall as the bubbles expand. The agreement among data ofdifferent depths (various and independent nucleation-to-departurecycles) provides evidence that bubbles repeatedly nucleate at thesame site and grow in a repeatable fashion. The initial linear slopessuggest constant contact angle values as shown in Fig. 9

Fig. 9 Parameters determining contact angle for a water–glasscontact and b water–silicon contact. Data for R0 £ 55 lm fromFig. 8 are extracted and fitted with a straight line (a). Thecomputed contact angle (Eq. 4) gives a value of 26o which is inclose agreement with reported values in the literature (Cubaud andHo 2004). Plot b shows contact angle extraction for the water–silicon contact in the y direction (see Fig. 3b). Here, the computedvalue is 43� which is within the range of reported values

location, this agreement among data from differentnucleation cycles suggests that the bubbles consistentlynucleate at and grow anchored to the same site.

The departure from linearity of the d–R0 curves forR0 ‡ 55 lm in Fig. 8 may be attributed to various cau-ses such as contact angle hysteresis, and/or breakdownof the spherical and steadiness assumptions on which thecurrent methodology is based. Identifying andaccounting for the effects of flow-induced deformationof bubble shape and unsteady bubble growth and fluidflow may be needed for the development of more com-prehensive bubble departure criteria.

Finally, in Fig. 10 we present the best approximatedbubble radius R0 as a function of time. The growth ratesare also approximately linear, and this agrees with theobservations reported by Lee et al. (2004). For our flow,this linear growth rate may be particular to the dynamicsgoverning the transport between dissolved gases and theliquid. Further quantitative analysis of growth dynamicsneeds to be performed to account for effects of outgas-sing, boiling, and confined geometry flow dynamics, thelatter of which is unique to bubble growth patterns inmicrochannels.

5 Conclusions and future work

We have presented a hybrid methodology that can beused to approximately reconstruct three-dimensionalbubble geometry, location, and the associated three-dimensional velocity field during bubble growth andnucleation in a microchannel. We use two-dimensionalimages, velocity field measurements from lPIV, andnumerical simulations from FEMLAB. A measured lo-cal bubble nucleation temperature of 60�C suggests most

of the nucleation and growth of bubbles we observedwas due to outgassing of the liquid. Liquid velocity fieldmeasurements were performed at various depths of thechannel during bubble growth using lPIV. These imagedata and two-component velocity data in two-dimen-sional planes provide only limited information. Weperformed three-dimensional numerical flow simulationsusing a range of trial bubble geometries. The bestapproximating bubble geometry was identified bydetermining the closest match between computed andexperimental velocity fields at the focal plane. The in-plane projection of the predicted three-dimensionalvelocity field and the lPIV data showed agreement towithin 10–20% of our global RMS error scale. Contactangles and bubble growth rates were also estimated fromthese data. We found that a no-slip boundary conditionat the liquid–vapor interface provides a better matchbetween flow predictions and measurements than a slipcondition.

The results obtained from the current work suggestinteresting behavior of nucleating vapor bubbles in mi-crochannels. The distance of the bubble center from theglass channel wall increased (roughly linearly) withincreasing bubble radii up to approximately R0=55 lm,such that constant contact angles were maintained. Thecontact angles for the glass and silicon channel wallswere estimated to be hg=26� and hs=43�, respectively.The trajectory of the bubble center became less pre-dictable when the bubble grew beyond R0=55 lm; thismay be attributed to contact angle hysteresis or break-down of the assumptions used in this analysis. Bubbleradius also varied linearly in time. The agreement of ourbubble center trajectories and bubble growth rate acrossdifferent measurement depths also suggests our meth-odology is consistent.

The current work provides useful insight to modelingbubble dynamics in two-phase microchannel convection.The information obtained contributes toward theunderstanding of dynamics and mechanisms for bubblegrowth and departure. Our future work will aim tofurther obtain quantities such as viscous, pressure, aswell as surface forces, with an ultimate goal of estab-lishing a quantitative departure criterion for nucleatingbubbles in microchannels.

Acknowledgements This work is supported by the National DefenseScience and Engineering Graduate Fellowship (Evelyn N. Wang)and the Microelectronics Advanced Research Corporation(MARCO). The project made use of the National NanofabricationUsers Network facilities funded by the National Science Founda-tion under award number ECS-9731294.

6 Appendix: uncertainties of the method

In this section, we discuss the estimated uncertaintiesassociated with our hybrid method. In particular, wediscuss measurement uncertainty due to perspectiveprojection, Brownian motion, and correlation peak

Fig. 10 Best approximated bubble radius R0 as a function of timeand for three positions of the measurement volume. Each data set isshown with a linear regression fit that is provided in the legend

center determination; all of which influence velocitymeasurements. We also briefly discuss uncertainties inthe methodology due to image depths (d2) as well asbubble image radius (a0) determination.

Perspective projection error may contribute to theuncertainty in measurements due to our use of a lownumerical aperture objective (NA=0.45). This objectivecorresponds to a relatively large measurement depth of13.5 lm. Perspective error results in a z-velocity com-ponent that cannot be decoupled from the in-plane (x, y)velocity measurement in the experimental results. Usingthe analysis described by Raffel et al. (1998), a largethird-dimensional velocity can lead to an upper boundvelocity bias of 6 mm/s which is equivalent to an error ofapproximately 13% for a particle furthest from thecenter of the objective. This error, however, was mini-mized by positioning the channel such that the largestout-of-plane velocities are closest to the center of theCCD array. Typical velocity uncertainties due to thiseffect are approximately 6% in the region near thebubble.

Two additional quantifiable sources of uncertaintyin determining the velocity field are Brownian motionand identification of the center of the cross-correlationpeak. The uncertainty associated with Brownian mo-tion was ±1% (Devasenathipathy et al. 2003).Uncertainty associated with the identification of thelocation of the cross-correlation peak was mitigated byensuring that the peak was distributed over 3–4 pixels,so the center of the cross-correlation peak could bedetermined to within approximately one-tenth of theparticle image diameter (Prasad et al. 1992). The par-ticle image diameter in the focal plane was 1.8 lm,which corresponded to a normalized measurementuncertainty of 180 nm or ±2.2%.

Determination of the focal depth and bubble radiusgives rise to uncertainties in bubble location and shape,and contributes to the RMS error difference defined inSect. 3. Uncertainties associated with determining thelocation of the focal plane are minimized by examiningresults from iterations performed at neighboring d2values from the measured focal plane. Selection of afocal plane 2.5 lm from d2 can lead to an RMS errordifference of 5–10%. RMS error is also affected byuncertainties in the image radius a0. The location of theliquid–vapor interface was not always clearly defined asparticles that accumulate at the interface are typicallyout of focus due to light scatter effects such as refraction(see Fig. 2a). The image processing algorithm selectedpoints at the interface based on the peak intensity andlocated the boundary within variations of 5 lm in ra-dius. We estimate that this yields velocity RMS errordifferences of 5%.

We can estimate the total contribution to the globalRMS error scale by assuming each of these sources ofuncertainty are uncorrelated (and taking the square rootof the sum of the mean square value of uncertainties).This analysis results in a typical value of about 10% forthe value of eRMS as defined in Sect. 3.

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