gas entrapment

8/7/2019 gas entrapment

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An experimental study of the gas entrapment processin closed-end microchannels

Ana V. Pesse, Gopinath R. Warrier *, Vijay K. Dhir

Mechanical and Aerospace Engineering Department, Henri Samueli School of Engineering and Applied Science,

University of California, Los Angeles, CA 90095-1597, United States

Received 24 May 2005; received in revised form 30 July 2005Available online 28 September 2005

Abstract

The physical mechanisms of the gas entrapment process in closed-end microchannels were investigated. Deionizedwater was the test fluid. The test pieces consisted of micromachined silicon squares with glass bonded on top. Themicrochannels had widths varying from 50 to 5 lm and had a mouth angle of 90. Experiments show two main fillingbehaviors: (1) A single meniscus at the entrance, (2) Two or more menisci: one at the entrance and the other near theclosed end. A single meniscus typically forms for higher contact angles (/ > 50), while two or more menisci form forlower contact angles (/ 6 30). For 30 6 / 6 50, one or two interfaces were observed. In all cases, after sufficient time(hours to days), the microchannel was completely flooded. In general, increasing the depth and/or width increases thetime taken to fill. On the other hand, decreasing the contact angle decreases the time taken to fill. Comparison of experi-

mental data with predictions based on a simple mass diffusion model shows reasonable agreement. 2005 Elsevier Ltd. All rights reserved.

Keywords: Microchannel; Gas entrapment; Flooding

1. Introduction

During boiling, vapor bubble formation starts frommicroscopic nucleation sites on the heated surface.These nucleating cavities have trapped gas or vapor in-side. The ability of a liquid to penetrate into these micro-

scopic cavities depends on such factors as the contactangle (/) of the liquid and the size and shape of the cav-ity. However, once the liquid penetrates into the cavity,it can either entrap the gas/vapor present in the cavity orfloods the cavity entirely.

Bankoff[1] was the first to propose a quantitative cri-terion for gas entrapment in a wedge by an advancingliquid front. The critical parameters affecting the gas/vapor entrapment in such a cavity were the contact angleand the wedge angle. Conical cavities were approxi-mated as wedge-shaped grooves. According to his crite-

rion, a wedge-shaped cavity on a surface will trap vapor/gas when the contact angle is greater than the wedgeangle.

Wang and Dhir [2] studied the effect of surface wetta-bility on the active nucleation site density during poolboiling of water. In their work, they related the cavitiesthat were present on a surface to those that actually be-came active. Based on the measured shape and size ofthe cavities present on the heater surface, they foundthat most of the deep cavities present on the surface were

0017-9310/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2005.07.020

* Corresponding author. Tel.: +1 310 825 9617; fax: +1 310206 4830.

E-mail address: [email protected] (G.R. Warrier).

International Journal of Heat and Mass Transfer 48 (2005) 51505165

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mailto:[email protected]:[email protected]


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spherical in nature. Although a large number of conicalcavities were present in the surface, these cavities werevery shallow, which according to Bankoffs criterionwere not expected to nucleate. It was found that as thewettability of the surface improves, the number densityof cavities that actually nucleate decreases. Wang andDhir [3] also evaluated the change in the Helmholtz freeenergy (DF) of a liquid droplet placed at the mouth of acavity, with the free surface of the droplet exposed to gasor vapor. The equilibrium position of the interface wascalculated by varying the location of the liquidgasinterface.

For a spherical cavity with mouth angle (wm) of 30they showed that for contact angles smaller than thecavity mouth angle (i.e., / < wm), the free energy de-creases as the interface moves to the bottom of the cav-ity. For these contact angles no gas will be trapped.However, for higher contact angles (/ > wm), the rela-

tive free energy reaches a minimum at as = 180 /and then increases to a maximum value aths = 90 /, where hs is the angular coordinate. Beyondhs = 90 /, the free energy decreases. This suggeststhat the interface will move from its position above thecavity to the location where DF is minimum and willhence entrap gas in the cavity. Based on their analysis,the condition for entrapment of gas could be stated as,

/ > wmin 1

where wmin is the minimum cavity side angle. The exper-imental results clearly showed that all the cavities that

entrapped gas and were hence active nucleation sites

satisfied the condition given by Eq. (1). It should benoted that the model of Wang and Dhir was a quasi-static model, which did not consider diffusion of gas intothe liquid. A similar analysis was done by Warrier [4] fora cylindrical cavity. For spherical, conical, and cylindri-cal cavities, wmin = wm.

Once a cavity entraps gas, increasing the temperatureof the cavity or the liquid may result in an increase in thevaporization at the liquidgas interface. According toCarey [5], the degree to which the entrapped gas helpsvaporization depends on the rate at which the entrappedgas diffuses into the liquid and is carried away from theinterface. Thus, once all the entrapped gas has diffusedinto the liquid, the cavity will be flooded.

Washburn [6] derived an equation to describe thespontaneous motion of a liquidgas interface in an openended capillary. His equation was based on the balancebetween the capillary force, the gravity force and the vis-

cous force given by Poiseuille [7]. Since the inertial forceswere neglected, the Washburn equation predicts an infi-nitely high initial velocity for the liquid during the earlystages of interface advancement. It was Szekely et al. [8]who removed this discontinuity by applying the correctmomentum balance for the entry flow. However sincethese studies were for open-end capillaries, they didnot consider the effect of the gas entrapped within thecapillary.

Yang et al. [9] measured the marching velocity ofcapillary menisci in microchannels (2 mm long). In theanalytical model developed, the velocity profile was

assumed to be parabolic and the pressure difference

Nomenclature

A interfacial areab microchannel breadthc molar concentration

CHe Henry constantDh hydraulic diameterD12 binary diffusion coefficient for chemical

species 1 and 2He Henry numberJ diffusive mass fluxL0 depth of microchannelM molecular weightn0 initial number of moles of airP pressureP0 pressure of airR radius of meniscus

R universal gas constantrc cavity mouth radiusT temperatureV volume

w cavity mouth widthxi mole fraction of species iX position of the liquidair interface inside the

cavity

Greek symbols

DF change in Helmoltz free energyq densityr surface tension/ contact anglewm cavity mouth angle

Subscripts

air airi species i

int interfaceliq liquidu liquid phases gas phase

A.V. Pesse et al. / International Journal of Heat and Mass Transfer 48 (2005) 51505165 5151


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across the liquidgas interface was calculated using theLaplace pressure drop. Neglecting the accelerationterm, they found that the position of the interfaceswas proportional to the square root of time. In general,their experiments showed that filling time increases withincreasing cross-sectional area of the microchannels.

However, there is no mention as to whether the micro-channels had an open or closed end. Also, no informa-tion is given regarding the filling mechanisms involvedor the role of mass diffusion.

Akselrud and Altshuler [10] proposed that the degreeto which the entrapped gas diffuses into the liquid sig-nificantly affects the nature of filling of a closed-endcapillary. Dovgyallo et al. [11] observed the phenome-non of bilateral filling of conical capillaries with aclosed end. They experimentally studied the filling ofclosed-end capillaries, with lengths varying from 30 to1000 lm and radii from 0.4 to 15 lm with various

liquids. They found that in a number of cases, the capi-llary is filled not only from the open end, but also fromthe closed end. However, for experiments with cylindri-cal capillaries, they found that the capillaries are filledonly from the open end. In the conical capillaries, theclosed end had a smaller cross-sectional area. This dif-ference between cylindrical and conical capillaries wasassociated with the presence of two menisci of differentcurvatures in the conical case. They theorized that thesecond meniscus forms at the closed end of the capillarydue to condensation from the liquid vaporized from thesurface of the meniscus with lower curvature. They alsofound that, in the case of cylindrical capillaries, the rateat which air dissolves into the liquid was considerablylower than that for the conical capillaries.

Wang [12] performed a few experiments to observethe filling of water and methanol in glass capillary tubesclosed at one end. The length of the capillary tubes wasabout 2 mm while the inner diameter varied from 10 to40 lm. His results showed that as the capillary diameterincreases the sorption rate increases. Additionally, it wasfound that the sorption rate for methanol (lower contactangle) was about three or four times larger than that forwater (higher contact angle).

Migun and Azuni [13] carried out experiments using

larger conical capillaries than those considered byDovgyallo et al. They observed the same double side fill-ing phenomena. In capillaries of radius, RP 50 lm theyalso observed individual liquid droplets on the channelwalls.

From the above discussion it is clear that for a givencavity size (and shape) liquids with certain contact an-gles will entrap gas in the cavity. Though the importantparameters that affect the dynamics of the interface andsubsequent entrapment of gas appear to be the liquidcontact angle, cavity size, cavity shape, and liquid prop-erties such as surface tension and viscosity, no quantita-

tive information is available regarding the effects of each

of these parameters. Additionally, no quantitative infor-mation is available regarding the effects of gas diffusionon the dynamics of the interface and the gas entrapmentprocess.

The objective of this study is to develop a basicunderstanding of the dynamics of liquidgas interfaces

in cavities and the process by which gas can be en-trapped in these cavities. This experimental work willfocus on the effects of the following parameters: cavitygeometry (depth, mouth width, and mouth angle), aswell as the static contact angle of the test liquid. Alsoof interest is the effect of gas diffusion on the interfacedynamics and gas entrapment process.

2. Experimental apparatus

The purpose of the experiments was to set up a rep-

resentation of a nucleation site or cavity to study thephysical mechanisms of the gas entrapment process. Inorder to allow observation of the moving liquid interfacein the cavity, closed-end microchannels with rectangularcross-sections were fabricated on a silicon substrate andcovered with glass. The experimental setup consisted ofa computer with image capturing software, a CCD cam-era, a microscope and the test pieces (Fig. 1).

Microchannels were fabricated with four entrancewidths (5, 15, 30 and 50 lm) and three depths (50,150 and 500 lm). The breadth (dimension going intothe plane of the pictures) was about the same asthe width and varied from 7 to 42 lm. Fig. 2(a)shows the schematic of a typical cavity and a simulatedcavity.

The test samples were fabricated on 100 mm siliconwafers, using standard micromachining techniques.The samples were etched to the required depth and thendiced into 2 2 cm pieces. Borofloat glass was thenanodically bonded to these samples. Fig. 2(b) shows

Fig. 1. Experimental setup.

5152 A.V. Pesse et al. / International Journal of Heat and Mass Transfer 48 (2005) 51505165


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the schematic of the test samples. Each test samplehad three to four microchannels on it with the openend (entrance of the microchannel) connected to areservoir of square cross-section with side 1 mm anddepth of around 300 lm. Further details can be found

in [14].

3. Experimental procedure

The first experimental run for each test piece wasconducted without any further surface cleaning, assum-ing that the microchannel remained clean during theanodic bonding process. The anodic bonding processsubjects the test pieces to temperatures of around300 C for 2 h.

Before starting each experimental run, the contactangle between the liquid droplet and the silicon surface

was measured, so as to quantify the condition of the sil-icon surface inside the cavity. Since it was not possible toplace a droplet in the microchannel, a small drop of thetest fluid was placed in the reservoir using a syringe anda photograph was taken. The contact angle was mea-

sured from this photograph. Fig. 3 shows a typical pho-tograph of a droplet in the reservoir and the contactangle measured.

After the first use of a test piece, it was thoroughlycleaned using a combination of the following processes:(1) Piranha bath (1:4, H2SO4:H2O2) at room tempera-ture for 50 min, followed rinsing with DI water rinse,blow drying with nitrogen and then to dehydrate, bakingat 110 C for 40 min. (2) Supercritical Dryer, with CO2and pure methanol for 1 h. (3) Bake at 160 C for90 min by placing the sample on a hot plate. The pur-pose of the surface cleaning was to modify the contactangle of the test fluid in contact with silicon. The contact

surface

interface

liquidliquid

airliquid

liquid

breadth (b)

depth (Lo)

cavity mouth radius

cavity depth

surface

width (w)

Sketch of a typical cavity Simulated cavity (closed-end microchannel)(a)

bCross Section B-B

Lo

microchannel

glass cover

BB

A

w

Top (experiment) viewLo

2 cm

reservoir

A

2 cm

microchannel

microchannel

glass cover

reservoir

Cross Section A-A

silicon

reservoir(b)

Fig. 2. Schematic of (a) an actual cavity and a simulated cavity and (b) test sample.



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angles corresponding to the cleaning procedures givenabove are listed as follows:

(a) After anodic bonding (before first use of newpiece): / $ 60.

(b) Used piece after Piranha bath (includes rinsingand dehydration bake): / 6 15.

(c) Used piece after Piranha bath followed by super-critical dryer: / $ 0.

(d) Same as (b) followed by hot bake at 160 C for 40minutes: / 6 25.

The contact angles corresponding to the cleaning proce-dures (b), (c), and (d), were obtained from the shape of

the meniscus inside the microchannel. This method wasadopted since it was very difficult to determine the con-tact angle by placing a droplet in the reservoir (due tothe low contact angle, the droplet wets the surface).Table 1 gives the range of contact angles measured onthe various test pieces.

Once the contact angle was recorded, the test piecewas placed under a microscope equipped with a CCDcamera. The test liquid was then placed in the reservoirand the entire filling process was recorded. Both theliquid and test sample were at room temperature (about23 C). At the beginning (first 20 images) of the experi-ments an image was captured every 10 s and afterwards

every 30, 60 and 120 s depending on the test liquid, con-tact angle and microchannel geometry.

4. Data reduction

Once a sequence of images was recorded, it was neces-sary to measure the distance from the entrance of the cav-ity to theliquidair interface. A Matlab program wasusedto read each image as a two-dimensional array of color

values and determine the location of the liquidair inter-face. The results were given in pixels (Fig. 4(a)), whichare then converted to microns. Once the liquid had pene-trated the microchannel, the contact angle inside themicrochannel was measured from the shape of the menis-cus. This was considered to be the actual static contactangle. Since two or more surfaces were in contact withthe liquid, a mean value was calculated for the staticcontact angle. This angle was measured by analyzingthe digital image with AutoCAD (Fig. 4(b)).

5. Measurement uncertainity

In these experiments only three quantities were mea-sured: position of the liquidair interface, time, and con-tact angle of the liquid in the microchannel. The positionof the interface was measured in pixels and which wasthen converted to microns using a conversion factor thathad its own uncertainty. The error in the pixel count was2 pixels which correspond to an error of 5 lm in themeasured microchannel depth. Thus the conversion fac-tor uncertainty was 2%, 6%, and 20% for the 500 lm,150 lm, and 50 lm long channels, respectively. The er-ror associated with the measurement of the position ofthe interface was 4 pixels. Hence the overall error asso-ciated with the position of the interface was 10 lm.The uncertainty of temporal measurements is 10 s,which is given by the frame rate of the camera. Theuncertainty for the contact angles measured in the reser-

Fig. 3. Photograph of a liquid droplet in the reservoir.

Table 1Dimensions and contact angles for various test pieces

Width [lm] Cavity depth, L0 [lm]

500 150 50

w = 9, b = 5 12 6 / 6 64 8 6 / 6 55 86 / 6 54w = 19, b = 15 86 / 6 62 15 6 /6 48 86 / 6 37w = 38, b = 24 86 / 6 66 8 6 / 6 90 236 / 6 90w = 56, b = 42 86 / 6 67 8 6 / 6 54 126 / 6 90

Fig. 4. Measurement of (a) interface position and (b) contact

angle inside the microchannel.



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voir of the test samples was estimated to be 5, whereasthe contact angles measured inside the microchannel hadan uncertainty of 8.

6. Results and discussion

In most of the experiments performed, only oneliquidair interface (moving from left to right, i.e., openend to closed end) was observed. Fig. 5(a) shows a se-quence of photographs of the single liquidair interface,while Fig. 5(b) shows the corresponding plot of theinterface location as a function of time. From Fig. 5(b)it can be seen that the motion of the liquidair interfaceis not smooth. The interface moves intermittently; theinterface moves, remains stationary and then movesagain. However, close to the closed end of the micro-channel, the interface moves quite rapidly. It must be

mentioned that in the ordinate in the plots does notbegin at zero because the first photograph was typicallytaken after a time interval of 10 s.

A variation of this filling behavior (i.e. only one inter-face) occurs when in addition of the single interface,liquid droplets are also present on the channel walls asshown in Fig. 6(a). It can be seen that the initial droplets

Fig. 5. Front advancing interface.

Fig. 6. Front advancing interface with droplets.



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merge to form larger droplets and sometimes even mergewith the advancing interface, which causes sudden jumpsin the interface location. These sudden jumps are clearlyseen in Fig. 6(b).

For experiments with low contact angles, a secondmechanism by which the microchannel can be filled

was observed. For these cases, two interfaces were ob-served; one at the entrance of the microchannel andthe other at the closed end. Referring to the sequenceof photographs shown in Fig. 7(a), initially only oneinterface is present but after 52,390 s a second interfaceappears near the closed end of the microchannel andstarts moving towards the entrance, finally meeting upwith the front interface and thus filling the microchan-nel. In most of the cases, when two interfaces were pres-ent, a thin film of liquid on the walls of the channelconnected the two menisci. Fig. 7(b) shows the corre-sponding interface locations as a function of time.

A variation of this second mechanism was also ob-served. In these cases, in addition to the two menisci, li-quid droplets were also present. This can be seen inFig. 8(a). The film connecting the two menisci can alsobe clearly seen. The droplets on the walls of the channelgrew and merged with each other and with the menisci.Fig. 8(b) shows a plot of the location of the interfaces asa function of time.

Additionally, in some instances, usually for longerchannels, more than two interfaces were seen. This gen-erally happens as a result of droplets growing and merg-ing to fill up the channel and forming an interface. Thisis the case shown in Fig. 9(a), which starts with only one

interface but after 23,385 s the liquid flowing along the

side walls feeds the droplets present in the middle ofFig. 7. Front and back advancing interfaces, with thin film.

Fig. 8. Front and back advancing interfaces, with thin film anddroplets.

Fig. 9. Front, middle, and back advancing interfaces.



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the channel with enough liquid to grow and merge form-ing two new interfaces. One of these interfaces moves to-wards the closed end of the microchannel while the otheradvances towards the first interface located closer to the

entrance of the channel. Later, at 28,425 s, the middleinterface that was moving towards the entrance mergeswith the original front interface. The described behaviorcan be seen clearly in Fig. 9(b), which shows the positionof the interfaces as a function of time. There are threedistinct stages: one corresponding to the original frontinterface (until 23,385 s), the second is the creation ofthe second and third interfaces, and finally the merging

of two interfaces (at 28,425 s) leaving only one interfacecloser to the end of the channel.

The experiments conducted, indicate that for /6 30, two interfaces connected by a thin liquid film

were always observed. On the contrary, for / > 50, onlyone interface was observed. In the range between 30and 50 both, one or two interfaces were observed. Thisis shown in Fig. 10 for different microchannels depths.Occasionally, for low contact angles (/ < 15), threeinterfaces were also observed.

All the microchannels with rectangular cross-sectionstested in this study were completely flooded given suffi-cient time. The time taken to completely flood the micro-channel varied from seconds, to minutes, to hours toeven days.

0

1

2

3

0 10 20 30 40 50 60 70 80 90

(deg.)

0 10 20 30 40 50 60 70 80 90

(deg.)

0 10 20 30 40 50 60 70 80 90

(deg.)

0

1

2

3

Numberofinterfaces

Numberofinterfaces

Numberofinterfaces

L0= 500 m L0= 150 m L0= 50 m

0

1

2

3

Fig. 10. Number of interfaces observed as a function of contact angle.

0 100 200 300 400 500 600L0 (m)

(a)

0 100 200 300 400 500 600L0 (m)

(b)

10

100

1000

10000

100000

1000000

totaltime(s)

10

100

1000

10000

100000

totaltime(s)

w = 57 m, b = 42 m

w = 35 m, b = 22 m

w = 19 m, b = 14 m

w = 9 m, b = 5 m

42 61

6 1923 3842 5866 90

w = 30 m, b = 22 m

Fig. 11. Flooding time as a function of microchannel depth.

10 20 30 40 50 60 70

width (m)

width (m)

(a)

(b)

1

10

100

1000

10000

100000

1000000

totaltime(s)

totaltime(s)

L0 = 500 m

L0 = 150 m

L0 = 50 m

10 20 30 40 50 600

10000

20000

30000

40000

50000L0 = 150 m

43 60

8 19

70 90

Fig. 12. Flooding time as a function of microchannel width.



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6.1. Effect of microchannel depth (L0)

As could be expected, for a given cross-section, deepermicrochannels take longer to flood. This can be seen inFig. 11(a) where the time, t, taken to flood a microchan-nel is plotted as a function of its depth. In general, it can

be seen that the time to flood increases nonlinearly withdepth, L0. For example, in Fig. 11(a), for w = 57 lm andb = 42 lm, the flooding time increases rapidly from1000 s to 30,000 s to 300,000 s as L0 increases from 50to 150 to 500 lm. Similar behavior is observed for micro-

channels of other sizes. Fig. 11(b) shows flooding time asa function of depth, for various contact angles and fixedw and b. Despite the scatter in the data, the nonlinear in-crease in time with L0 can be clearly seen.

6.2. Effect of microchannel width (w) and breadth (b)

In almost all the experiments performed, the timetaken to flood the microchannel increased slightly withincreasing cross-sectional (w and b) area as shown inFig. 12(a). For example, for L0 = 500 lm, the time taken

0

100

200

300

400

500

X(m)

0

100

200

300

400

500

X(m)

0

100

200

300

400

500

X(m)

0

100

200

300

400

500

X(m)

w = 57 m, b = 42 m

w = 37 m, b = 24 m

w = 19 m, b = 15 m

w = 9 m, b = 5 m

Lo= 500 m,17 20

0 20000 40000 60000 80000 100000 120000 140000 160000

t (s)

Fig. 13. Location of the waterair interface as a function of time for various microchannel widths and breadths (L0 = 500 lm and

176 / 6 20

).



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to fill the microchannel increases from 30,000 s to200,000 s as the width increases from 9 to 56 lm.Fig. 12(b) shows the variation of filling time as a functionof width, for L0 = 150 lm. From Fig. 12(b) it is clear thatfor a given contact angle, the filling time increases withincreasing width. The effect of the increase in the micro-

channel width and breadth can be further illustrated ifone plots the location of the waterair interface (relativeto the open end) as a function of time (Fig. 13 forL0 = 500 lm and / $ 20). The increase in the floodingtime with increase in microchannel cross-sectional areacan be clearly seen in Fig. 13. Similar data were obtainedfor the 150 and 50 lm deep microchannels.

6.3. Effect of contact angle (/)

The filling time is also affected by the contact angle.This is shown in Fig. 14(a). For example, for w =

38 lm and b = 24 lm, the time taken to fill the micro-channel increases from 38,000 s to 100,000 s as the con-tact angle increases from 8 to 58. Fig. 14(b) shows thevariation of the filling time with contact angle for a fixedw = 9 lm and L0 = 50, 150, and 500 lm. For L0 =500 lm, the flooding time increases from 15,000 s to30,000 s as the contact angle increases from 13 to 65.From Fig. 14(a) and (b) it can be seen that, in general,

as the contact angle increases, the time taken to floodthe microchannel also increases, for a given depth,mouth width and breadth.

Fig. 15 shows the location of the waterair interfacesas a function of time, for various contact angles(L0 = 500 lm, w = 9 lm and b = 5 lm). The increase

in the flooding time with increase in contact angle isclearly seen in Fig. 15. Similar results are obtained forthe shorter microchannels (L0 = 150 and 50 lm).

The increase in the contact angle also influences thetime taken for the back interface to appear (Fig. 16(a)).From Fig. 16(a), it is clear that as the contact angle in-creases, the time taken for the back interface to appearalso increases. For example, in Fig. 16(a), for L0 =500 lm, the time taken for the back interface to formincreases from 30 s to 300,000 s as the contact angleincreases from about 20 to about 68.

The formation of a back interface at the closed end

of the microchannels is due to a liquid film that flowsalong the corners of the channels. When this film hastraveled from the initial position of the front interfaceand reaches the closed end of the microchannel it mergesand forms the back interface. Quantitative measurementof this thin film was not possible in the setup used in theexperiments; however an attempt was made to calculatethe mean velocity of the thin film front. The mean veloc-ity of the leading edge of the film in the corners was cal-culated by dividing the distance traveled from the initialposition of the front interface (where the film starts) tothe position where the back interface was formed bythe time elapsed between these two events. The calcu-lated mean velocity of the film along the corners as func-tion of the contact angle is shown in Fig. 16(b). Fromthe figure it can be seen that the higher the contact angle,the lower is the mean velocity of the leading edge of thefilm. The results shown in Fig. 16(b) constitute anotherway to asses the effect of the contact angle on the flood-ing time. As the contact angle increases, the mean filmvelocity decreases, which means that it will take longerfor the back interface to appear and the microchannelswill take longer to fill. The mean velocity of the leadingedge of the thin film increases nonlinearly with contactangle. The observed behavior demonstrates the role

played by surface forces in sucking liquid along the cor-ners of the microchannels.

For a given width and contact angle, the large in-crease in flooding time with depth supports the fact thatthe time for which a boiling surface is exposed to the testliquid may be a parameter in determining the number ofcavities that become active.

7. Model development

A simple one-dimensional model was developed to

predict the movement of a single liquidgas interface

10 20 30 40 50 60 70 80 90 (deg.)

0

50000

100000

150000

200000

250000

totaltime(s)

totaltime(s)

L0 = 500 m

100

1000

10000

100000w = 9 m, b = 5 m

w = 56 m, b = 42 m

w = 38 m, b = 24 m

w = 19 m, b = 15 m

w = 9 m, b = 5 m

0

10 20 30 40 50 60 70 80 90 (deg.)

0

L0 = 500 mm

L0 = 150 mm

L0 = 50 mm

(a)

(b)

Fig. 14. Flooding time as a function of contact angle.



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as a result of gas diffusion. Fig. 17 shows details of thecontrol volume. In this model, the liquid moving from

the open end towards the closed end of the microchannelis assumed to represent a semi-infinite medium. Air is as-sumed to diffuse into the liquid, while the diffusion of theliquid into air is taken to be negligible. No evaporationis considered.

Assuming the air to be an ideal gas, the number ofmoles of air present in the microchannel at any giventime can be written as,

PairVairt

RT n0

Zt

0

J1;u dtAint 2

where Pair is the pressure of the air, Vair is the volume of

air at any time t, Aint is the surface area of the liquidair

interface, R is the universal gas constant, T is the tem-perature, n0 is the initial number of moles of air given

byP

0L

0wb

RT , P0 is the atmospheric pressure, L0 is the depthof the channel, w is the width and b is the breadth. In Eq.(2) the integral represents the total number of moles ofair diffused into water during the time t, through thefront interface.

The molar flux, Ji,u, of air diffusing into water (con-sidered to be a semi-infinite medium) through the frontinterface is given by,

J1;u cmix

ffiffiffiffiffiffiffiffiD12

p t

rx1;u x1;0 3

where cmix = q/M in kmol/m3 (q is density and M is the

molecular weight) is the molar concentration of the

0

100

200

300

400

500

X(m

)

X(m)

X(m)

= 45

= 30

L0 = 500 m, w = 9 m, b = 5 m

0

100

200

300

400

500

= 18

0 5000 10000 15000 20000 25000 30000 35000

t (s)

0

100

200

300

400

500

Fig. 15. Location of the waterair interface as a function of time for various contact angles (L0 = 500 lm, w = 9 lm and b = 5 lm).



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waterair mixture, D12 is the diffusion coefficient, x1,uand x1,0 are the mole fractions of air at the interfaceand the initial mole fraction of air in the liquid, respec-tively. The pressure in the air is taken as the liquid pres-sure plus the capillary pressure, i.e.,

Pair Pliq r

R;

1

R

1

R1

1

R2

2w b cos/

w b4

It is assumed that the initial concentration of air in water(x1,0) is zero, (i.e., x1,0 = 0) and that there is no liquidpresent in the air. Hence the concentration of air,x1,s = 1.

The mole fraction of air at the interface (x1,u) can becalculated as x1;u

x1;s

He 1

He, where He is the Henry num-

ber calculated as He CHeP0

, and CHe is the Henry con-stant in Pa. Using the numerical values for propertiesof water and air given by Mills [15], D12 was calculatedto be 2.77 109 m2/s.

Substituting Eqs. (3) and (4) into Eq. (2) and inte-grating from time 0 to t yields,

PairVairt

RT

P0L0wb

RT 2cmix

P0

CHeAint

ffiffiffiffiffiffiffiffiffiD12t

p

r5

The volume of air in the microchannel at time t is givenby,

Vairt wbL0 Xt Vcap 6

where the first term represents the volume of air in therectangular portion (X(t) is the position of the interface)and Vcap represents the volume of air in the curved capshaped portion. Since an exact calculation of Vcap andAint for the rectangular geometry of the microchannelis extremely difficult (the shape of the interface is suchthat the surface energy is minimum), we use an approxi-mation. In this approximation we assume the microchan-nel to be circular in cross-section, with the cross-sectionalarea of the circular microchannel equal to wb (cross-sec-

tional area of the rectangular microchannel). Now, for agiven contact angle, we can determine both Vcap and Aint.The resulting equations for Aint and Vcap are given as,

Aint pr2eq 1

1 sin/

cos/

2" #;

Vcap p

6r

3eq 3

1 sin/

cos/

1 sin/

cos/

3" #7

(deg.)

1

10

100

1000

10000

100000

Elapsedtimeuntilback

interfaceappears(s)

(a)

(b)

0.001

0.01

0.1

1

10

100

Meanvelocityoffilm(m/s)

1000

100 20 30 40 50 60 70 80 90

(deg.)

100 20 30 40 50 60 70 80 90

100

L0 = 500 m

L0 = 150 mL0 = 50 m

Fig. 16. (a) Time taken for the back interface to appear and (b)mean velocity of film in the corners as a function of the contact

angle.

Lo

w

Xf(t)

liqu2

id airu

Front interface

s Closed endOpen end1

Fig. 17. Control volume for the model.



13/16

where req is the equivalent radius of the circular micro-channel (req = (wb/p)

0.5). Substituting Eqs. (6) and (7)into Eq. (5) and solving for X(t) yields,

Xt L0 1 P0

Pair Vcap

wb 2cmix

RT

CHe

P0

Pair

Aint

wb ffiffiffiffiffiffiffiffiffiD12t

pr8

Eq. (8) is used to predict the position of the single inter-face as a function of time. The first experimental data

point was used as the initial position (at t = 0) for X(t)in the model.

Fig. 18(a) shows a comparison of the experimentaldata and the model predictions for varying cross-sec-tional area while Fig. 18(b) shows the comparison forvarious contact angles. From Fig. 18 it can be seen that

there is qualitative agreement between the experimentaldata and the model predictions. The step change in theinterface location observed in the experiments may bedue to the liquid flow in the corners of the microchannel.

0 2000 4000 6000 8000 10000

0

20

40

60

80

100

120

140

160

w = 30 m, b = 15 m, = 62o

X(m)

X(

m)

Time (s)

Time (s)

(a)

(b)

L0 = 151m

w = 9 m, b = 9 m, = 58o

0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

40

45

50

= 23o

L0 = 47 m, w = 24 m, b = 24 m

= 78o

Fig. 18. Comparison of experimental data and model predictions (a) effect of varying w and b and (b) effect of varying contact angle.



14/16

However, in general, the model predicts much longer fill-ing times than those observed in the experiments. We be-lieve that this is due to the fact that the liquid is suckedinto the microchannel along the corners as a thin film;this phenomenon is not accounted for in the model.The model appears to capture the effects of changing

w, b, and / on the interface location fairly well. Forgiven L0 and /, decreasing w and b results in a decrease

in the filling time. Similarly, for given L0, w and b,decreasing / results in a decrease the filling time.

The same analysis of the single interface was then ap-plied to predict the location of the front interface for lowcontact angles cases (when two interfaces are present). Acomparison of the experimental data with analytical

results, for the front interface, is shown in Fig. 19(a)and (b). It must be noted that, in this case, the capillary

0 1000 2000 3000 4000 5000 60000

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160L0 = 151 m, w = 19 m, b = 13 m, = 15

o

L0 = 500 m, w = 24 m, b = 11 m, = 17o

X(m)

X(

m)

Time (s)

Time (s)

(a)

(b)

Experiment - front interface

Experiment - back interface

Prediction

0 10000 20000 30000 40000 50000 60000 70000 80000 900000

50

100

150

200

250

300

350

400

450

500

Experiment - front interface

Experiment - back interface

Prediction

Fig. 19. Comparison of experimental data and model predictions for low contact angles.



15/16

pressure is multiplied by two because of the presence ofthe back interface. From Fig. 19 it can be seen thatthough the predicted position of the front interface asa function of time does not deviate much from that ob-served in the experiments.

The formation and growth of the liquid column at

the closed end of the microchannels may be explainedby the thin film liquid flow along the walls. Since thepresent experiments did not provide enough informa-tion, the details of the thin film cannot be discussed inthis work. Migun et al. [16,17] proposed a model for filmflow in a dead-end conic capillary. In their model, the ra-dius of curvature of the film naturally varied along thelength of the conic channel, which provides the drivingpotential for the film flow. They found that the growthof the back interface was proportional to the cubic rootof time. However, their model cannot be applied to thepresent geometry since there is no change in the cross-

sectional area of the channel.

8. Summary

Experiments were conducted to investigate the gas/vapor entrapment process in closed-end microchannelsof various sizes. These closed-end microchannels repre-sented cylindrical cavities (mouth angle of 90) on a hea-ter surface that may become active nucleation sitesduring boiling, provided the liquid entraps gas/vapor in-side. The contact angle between the test fluid (DI water)and the silicon microchannel was also varied usingdifferent surface cleaning procedures. The effect of thediffusion of the gas into the liquid has also beeninvestigated.

In all the experiments performed, the microchannelswere completely filled, given sufficient time (minutes tohours to days). The cavity mouth angles were about90, while the contact angles measured were less than90. Hence the condition for gas entrapment given byEq. (1) was not met. Two or more interfaces connectedby a thin liquid film were observed for low contactangles (/ 6 50). For / > 50, only one interface wasobserved. Occasionally, for / 6 15, three interfaces

were observed. In the range of parameters considered,the dimensions of the microchannels did not affect thenumber of interfaces observed. In general,

(i) For fixed width, breadth and contact angle, theflooding time increases nonlinearly (up to twoorders of magnitude) with increasing microchan-nel depth (L0).

(ii) For fixed depth and contact angle, the floodingtime increases with increasing microchannel width(w) and breadth (b).

(iii) For fixed depth, width and breadth, the flooding

time increases with increasing contact angle (/).

The most pronounced effect on the flooding time was thechange in the microchannel depth, followed by the effectof the microchannel width and breadth and last, thoughstill important, the effect of the variation of the contactangle.

A simple one-dimensional mass diffusion model for a

semi-infinite medium shows reasonable agreement forthe cases with only one interface. The model accountsfor the effects of varying w, b and / on the movementof the interface. However, this model cannot explainthe appearance and movement of the back interface.The movement of the back interface may be explainedby the liquid film flow along the corners of the rectangu-lar microchannels, but the investigation of the details ofthis flow was beyond the scope of this work.

Acknowledgement

This work received support from NASA under theFluid Physics Program.

References

[1] S.B. Bankoff, Entrapment of gas in the spreading of aliquid over a rough surface, AICHE J. 4 (1958) 2426.

[2] C.H. Wang, V.K. Dhir, Effect of surface wettability onactive nucleation site density during pool boiling of wateron a vertical surface, J. Heat Transfer 115 (1993) 659669.

[3] C.H. Wang, V.K. Dhir, On the gas entrapment andnucleation site density during pool boiling of saturated

water, J. Heat Transfer 115 (1993) 670679.[4] G.R. Warrier, unpublished notes, 2002.[5] V.P. Carey, LiquidVapor Phase-change Phenomena,

Taylor & Francis, Bristol, Pennsylvania, 1992.[6] E.W. Washburn, The dynamics of capillary flow, Phys.

Rev. 17 (1921) 273.[7] J.L. Poiseuille, Mem. de Savans etrangers 9 (1846) 433.[8] J. Szekely, A.W. Neuman, Y.K. Chuang, The rate of

capillary penetration and the applicability of the washburnequation, J. Coll. Interf. Sci. 35 (1971) 273278.

[9] L.J. Yang, T.J. Yao, Y.L. Huang, Y. Xu, Y.C. Tai,Marching velocity of capillary meniscuses in microchan-nels, in: 15th IEEE International Conference (MEMS 02),Las Vegas, USA, 2002.

[10] G.A. Akselrud, M.A. Altshuler, Introduction to Capillary-Chemical Technology, Khimia, Moscow, 1983.

[11] G.I. Dovgyallo, N.P. Migun, P.P. Prokhorenko, Thecomplete filling of dead-end conical capillaries with liquid,J. Eng. Phys. 56 (4) (1989) 395397.

[12] C.H. Wang, unpublished notes, 1993.[13] N.P. Migun, M.A. Azuni, Filling of one-side-closed

capillaries immersed in liquids, J. Coll. Interf. Sci. 181(1996) 337340.

[14] A.V. Pesse, Experimental study of the gas entrapmentprocess in closed-end microchannels, MS thesis, Universityof CaliforniaLos Angeles, Los Angeles, 2004.

[15] A.F. Mills, Basic Heat and Mass Transfer, Prentice Hall,

Upper Saddle River, New Jersey, 1999.



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[16] N.P. Migun, A.I. Shnip, L.E. Reut, Role of the diffusionmechanism of mass transfer processes in conic capillariessubmerged in a liquid, J. Eng. Phys. Thermophys. 75 (6)(2002) 14121421.

[17] N.P. Migun, A.I. Shnip, Model of film flow in a dead-endconic capillary, J. Eng. Phys. Thermophys. 75 (6) (2002)14221428.


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