NATURE | VOL 403 | 10 FEBRUARY 2000 | www.nature.com 641

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.................................................................Air entrapment in coatingsby way of a tip-streaming meniscusP. G. Simpkins & V. J. Kuck

Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974, USA

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Entrapment of small air bubbles is a problem for continuousliquid-®lm coatings processes. The coating of any surface requiresthat the surrounding air in contact with it be displaced by anadvancing liquid interface. Studies of dynamic wetting suggestthat if the interface motion is too rapid, the air is not completelyremoved and it becomes entrained in the coating material1. Thisprocess, which can lead to undesirable ¯aws in the form ofbubbles, blemishes or voids, limits the speed at which the sub-strate can be moved in the production of uniform precisioncoatings. However, the entrapment process is not understood indetail. Here we report an experimental investigation of airentrapment in high-speed coating operations. Tip streamingÐaphenomenon well known in emulsi®cation technology2, involvingthe ejection of a ®ne ®lament from the cusped interface betweentwo immiscible ¯uidsÐis shown to be the precursor of airentrainment. We demonstrate that tip-streaming air ®lamentsemanating from the contact zone of a dynamic liquid interfacegive rise to minute (,10 mm) bubbles.

Considerable research has been carried out on air entrainment(see ref. 3 for a review). When the coating speed is less than a criticalvalue for a particular material, the contact line is generally normal tothe motion of the substrate. As the speed exceeds the critical valuethe contact line starts to become serrated. A qualitative explanationfor this behaviour in the application of coatings for photographic®lm has been reported4. Other works5,6 have subsequently reportedsimilar observations at supercritical coating speeds and all havenoted that bubble generation originates at the leading-edge verticesof the serrations, that is, those projecting into the liquid.

A recent study of free-surface cusps7 noted that powder sprinkledon the liquid interface was swept into the interior of the ¯uid whenthe motion speed was suf®ciently large. However, there was noevidence of air entrainment, so that the phenomenon was left

unexplained. That work was suggested by one8 that reported studiesof two-dimensional cusps in a roller apparatus reminiscent of thatoriginally used by Taylor9. An analogy between this type of ¯ow andthe motion of a dynamic contact line, however, is merely qualitative,because a cusp represents a singularity in the free-surface curvatureand only occurs in the absence of surface tension8. In contrast,imposition of the no-slip boundary condition in the immediatevicinity of a dynamic three-phase contact line leads to acontradiction10 and it appears that a cusp cannot be formed withoutintroducing slip at the boundary. Thus, although the contact-lineinterface appears to be cusp-like (see Fig. 1), analysis shows this notto be the case. Further analytical work on interfacial cusps has beenreported recently11,12.

Several works on nonlinear extensional ¯ows, beginning with ref.9, have reported that when a sheared drop becomes pointed, smallbubbles can be ejected from the cusp. Tip streaming, however, onlyoccurs when the viscosity ratio M of the two ¯uids involved is lessthan O (0.1), see ref. 2. Here, M � m2=m1 and the subscripts 1 and 2refer to the continuous phase and the dispersed phase, respectively.We note that this criterion is satis®ed by air in combination withalmost all liquids. Tip streaming is also discussed elsewhere13±16.Other observations17 suggest that surfactants play some role in tip-streaming events, but it is unclear whether they are essential for thephenomenon to occur. Recently, analytical studies (Siegal, M,personal communication, and ref. 18) have examined the role ofvariable surface tension on unsteady cusp formation.

The basic experimental apparatus consists of a length of ®bre thatis pulled from a supply reel onto a take-up reel, passing through atwo-dimensional, liquid-®lled test cell in the process. Appropriatelyarranged pulleys position the ®bre so that it is moving verticallydownward as it enters the cell. Alignment of the ®bre through thecell is achieved by micrometer adjustment on the pulleys above andbelow it. A stepper motor, controlled by a micro-step indexer,provides the drive to the take-up reel which moves the ®bre atspeeds of 0.01±3.5 m s-1.

Two-dimensional test cells are constructed from 10 ´ 10-cm2

Figure 1 The air±liquid interface near the contact line on a 240-mm ®bre. The ®bre is

moving vertically downward at 0.48 m s-1; Ca � 8:4, Red � 0:06. The arrows indicate

the edges of the ®bre.

A

B

Figure 2 Close-up of the contact region. An air ®lament (A) extending from a plunging

meniscus, and a separated bubble (B) are shown. The draw speed is 0.14 m s-1;

Ca � 2:4, Red � 0:02. The bubble is approximately 140 3 40 mm2. This high-speed

image was recorded at framing rate of 500 s-1 and the vertical ®eld of view is about

1.3 mm. Arrows at the bottom of the image indicate the edges of the ®bre.

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panels of display glass separated by 2- or 4-mm thick Perspexspacers. With these con®gurations the sidewall in¯uence on theobserved motion was found to be negligible. The top of the cell,where the ®bre enters the liquid, is open to the atmosphere and atthe bottom the ®bre leaves through a perforation in a silicone dam,which prevents excessive leakage. Observations of the air±liquidmeniscus shape at the contact region with the ®bre are made with aNikon stereo microscope mounted horizontally on an adjustabletraversing system. Typically, the ®eld of view with this arrangementis 2 ´ 2 mm2 at a magni®cation of ´50. A digital CCD (charge-coupled device) camera is used to obtain video recordings of themeniscus dynamics as a function of draw speed.

In all of the experiments we used ®bre that is nominally 240 mmin diameter and has a urethane-acrylate exterior. The surfacestructure, viewed at ´1,000 with a scanning electron microscope,appears to be smooth; any roughness is on a sub-micrometre scale.Before each test, we measured the ®bre diameter with an opticalcaliper (Imagen HR 1024) coupled to a Nikon microscope. Freshglycerine was used in all of the tests and the effect of waterabsorption has been neglected. The viscosity of the glycerine wasrecorded as 1,100 cP at ambient temperature with a Brook®eld DV-11viscometer.

Bubble generation was ®rst detected in the test cells at a drawspeed of approximately 0.12 m s-1. This speed corresponds to acritical capillary number Cac�� mU=g� > 2:1. Here, m is the viscos-ity, U is the draw speed and g is the surface tension. The equivalentReynolds number Re (� rUd=m, where r is the density), based onthe ®bre diameter d, is Red = 0.033. This result corresponds to oneparticular density ratio, and variations of the liquid-phase proper-ties are expected to modify the value of Cac. It has been suggested fora roll-coating system5 that an inertial group given by g�r=m4g�1=3,where g is the gravitational constant, is the other controllingparameter. Our result is in reasonably good agreement with thatcorrelation.

Air bubbles as small as 6 mm have been detected in videorecordings after the draw speed exceeds the critical value. At thatpoint the meniscus is dragged progressively further into the liquidand ®ne ®laments begin to appear at the unsteady contact line.Figure 1 shows the meniscus in close proximity to the ®bre at asupercritical speed. The cusp-like distortion of the free surfaceunder the shearing action of the ®bre can be clearly detected. Atip-streaming ®lament that exists along the left side of the ®bre nearthe apex of the meniscus is harder to discern. Streaks in thebackground of the picture indicate the trajectories of minutebubbles.

An enlargement of the dynamic contact region showing theinterface under conditions close to critical is displayed in Fig. 2.In the image a distorted bubble which has just separated from thecusp is readily apparent. From our observations the bubbles do not

seem to adhere to the ®bre surface for very long; they migrate intothe region about one ®bre diameter from it under the action ofvorticity. There is also some evidence to suggest that the extendedmeniscus itself is not always in contact with the ®bre.

A line pro®le intensity trace taken through a series of bubblesissuing from a tip-streaming meniscus at Ca = 4.2 is shown in Fig. 3.The presence of six bubbles is easy to identify on this trace.Measurements of the bubble diameter on the photograph fromwhich the trace was taken suggest a mean value of about 53 mm, witha separation wavelength of approximately 150 mm. The range ofbubble diameters created at a particular value of Ca is dif®cult toassess because of the agglomeration that occurs during recirculationin the test cell. However, it does appear that close to Cac the bubblediameters are greater than those observed at larger capillarynumbers.

Figure 4 displays the termination of a tip-streaming ®lament,again with a separated elongated bubble. This image, taken at a drawspeed of 0.29 m s-1 (Ca = 5.0), illustrates how much the air ®lamentcan be extended by the shear ¯ow. Here the ®lament length is at leasteighteen times its width; an edge-enhancing software measurementsuggests that the width is about 33 mm.

These experiments demonstrate for the ®rst time, to our knowl-edge, that coating ¯ows at supercritical speeds give rise to tip-streaming air ®laments. At the onset of air entrainment the criticalcapillary number is found to be approximately 2.1 for the glycerine±air case reported here. Tip streaming originates at the apex of thecontact region interface and air bubbles are dispersed into the bulkof the liquid phase. Aspect ratios greater than twenty have beenrecorded for the air ®laments observed in these experiments andtheir magnitude, although unsteady, appears to be to be speeddependent. A connection between air entrainment in coatingoperations and tip-streaming events has not, to our knowledge,been made previously. M

Received 14 June; accepted 21 December 1999.

1. Benjamin, D. F. et al. Coating ¯ows: form and function. Indust. Coating Res. 1, 1±37 (1991).

2. Grace, H. P. Dispersion phenomena in high viscosity immiscible ¯uid systems and the application of

static mixers as dispersion devices in such systems. Chem. Eng. Commun. 14, 225±277 (1982).

3. Kistler, S. F. in Wettability (ed. Berg, J. C.) 311±429 (Marcel Dekker, New York, 1993).

letters to nature

642 NATURE | VOL 403 | 10 FEBRUARY 2000 | www.nature.com

1 210

50

100

150

200

250

300

41 61 81

Axial distance (10–1 µm)

Arb

itrar

y un

its

101 121 141 161 181

Figure 3 Representative bubble distribution from a tip-streaming meniscus. The mean

diameter is ,53 mm and the draw speed = 0.24 m s-1; Ca � 4:2, Red � 0:03.

A

B

Figure 4 A tip-streaming ®lament (A) with an elongated bubble (B) on a ®bre moving at

0.3 m s-1; Ca � 5:0, Red � 0:04. The framing rate is 500 s-1. The dark spots are

bubbles that are not in the focal plane. Arrows at the bottom of the image indicate the

edges of the ®bre.

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4. Blake, T. D. & Ruschak, K. J. A maximum speed of wetting. Nature 282, 489±491 (1979).

5. Bolton, B. & Middleman, S. Air entrainment in a roll coating system. Chem. Eng. Sci. 35, 597±601

(1980).

6. Ghannam, M. T. & Esmail, M. N. Effect of substrate entry angle on air entrainment in liquid coating.

Am. Inst. Chem. Engrs J. 36, 1283±86 (1990).

7. Jeong, J. -T. & Moffatt, H. K. Free-surface cusps associated with ¯ow at low Reynolds numbers. J. Fluid

Mech. 241, 1±22 (1992).

8. Joseph, D. D., Nelson, J., Renardy, M. & Renardy, Y. Two-dimensional cusped interfaces. J. Fluid Mech.

223, 383±409 (1991).

9. Taylor, G. I. The formation of emulsions in de®nable ®elds of ¯ow. Proc. R. Soc. Lond. A 146, 501±523

(1934).

10. Dussan, V. E. B. & Davis, S. H. On the motion of a ¯uid±¯uid interface along a solid surface. J. Fluid

Mech. 65, 71±95 (1974).

11. Pozrikidis, C. Numerical studies of cusp formation at ¯uid interfaces in Stokes ¯ow. J. Fluid Mech.

357, 29±57 (1998).

12. Shiikhmurzaev, Y. D. On cusped interfaces. J. Fluid Mech 359, 313±328 (1998).

13. Sherwood, J. D. Tip streaming from slender drops in a nonlinear extensional ¯ow. J. Fluid Mech. 144,

281±295 (1984).

14. Smith, P. G. & Van de Ven, T. G. M. Shear induced deformation and rupture of suspended solid/liquid

clusters. Colloids Surf. 15, 191±210 (1985).

15. Stone, H. A. Dynamics of drop deformation and breakup in viscous ¯uids. Annu. Rev. Fluid Mech. 26,

65±102 (1994).

16. Eggers, J. Nonlinear dynamics and breakup of free-surface ¯ows. Rev. Mod. Phys. 69, 865±929 (1997).

17. de Bruijn, R. A. Tip streaming of drops in simple shear ¯ows. Chem. Eng. Sci. 48, 277±284 (1993).

18. Siegal, M. In¯uence of surfactant on rounded and pointed bubbles in 2-D Stokes ¯ow. SIAM J. Appl.

Math. 59, 1998±2007 (1999).

Acknowledgements

We thank R. Slagle (Oxford Lasers, Inc.) for help in obtaining the high-speed imagesdisplayed in Figs 2 and 4, and C. Pozrikidis for helpful comments on an earlier version ofthe manuscript.

Correspondence and requests for materials should be addressed toP. G. S. (e-mail: pgs@bell-labs.com).

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NATURE | VOL 403 | 10 FEBRUARY 2000 | www.nature.com 643

.................................................................Water exchange betweenthe subglacial Lake Vostokand the overlying ice sheetMartin J. Siegert*, Ron Kwok², Christoph Mayer³ & Bryn Hubbard§

* Bristol Glaciology Centre, School of Geographical Sciences, University of Bristol,

Bristol BS8 1SS, UK² California Institute of Technology, Jet Propulsion Laboratory, Pasadena,

California 91109, USA³ Alfred-Wegener-Institute for Polar and Marine Research, Department of

Geophysics, Bremerhaven, Germany§ Centre for Glaciology, Institute of Geography and Earth Sciences,

University of Wales, Aberystwyth SY23 3DB, UK

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It has now been known for several years that a 200-km-long lake,called Lake Vostok, lies beneath the ice sheet on which sits VostokStation in Antarctica1±5. The conditions at the base of the ice sheetabove this subglacial lake can provide information about theenvironment within the lake, including the likelihood that itsupports life2. Here we present an analysis of the ice-sheetstructure from airborne 60-MHz radar studies, which indicatesthat distinct zones of basal ice loss and accretion occur at the ice±water interface. Subglacial melting and net ice loss occur in thenorth of the lake and across its 200-km-long western margin,whereas about 150 m of ice is gained by subglacial freezing in thesouth. This indicates that signi®cant quantities of water areexchanged between the base of the ice sheet and the lake waters,which will enrich the lake with gas hydrates, cause sedimentdeposition and encourage circulation of the lake water.

Three 60-MHz radar lines are aligned parallel to the generaldirection of ice ¯ow, as indicated by interferometric-SAR (InSAR)

data (Fig. 1). These transects run across the northwestern margin ofthe lake (Fig. 2a), the southern central region of the lake (Fig. 2b),and from the southwestern margin of the lake to Vostok Station(Fig. 2c). InSAR data show the ice-velocity ®eld above the lake, andalong each of the transects, in metres per year (Fig. 1b). Severalisochronous internal radar layers, extracted from the raw 60-MHzradar data, were traced across each transect. The vertical distancebetween the highest and lowest traceable internal layers, and thatbetween the lowest layer and the ice-sheet base, were both thenmeasured along these transects. The resulting data (Fig. 3) clearlyindicate marked, spatially-coherent deviations in the dip of thebasal radar layers away from parallelism with the ice-sheet base.Such deviations could be caused by a number of processes. Forexample, basal topography is commonly translated into the over-lying ice column. However, such topography is negligible over LakeVostok. The ice base in contact with the lake surface is characterizedby a small (0.002) but steady gradient from the north of the lakewhere the ice is ,4,200 m in thickness, to the south where the ice is,3,800 m thick. There is very little variation in ice thickness acrossthe lake from west to east. This situation is therefore analogous toan ice shelf, where the absence of basal shear stress preventsdeformation of internal layers. For the same reason, the in¯uenceof compressive ¯ow, which would force internal layers apart, andextending ¯ow, which would bring them closer together, is con-sidered unlikely except near the lake margins, where an abrupttransition in basal drag is expected. Independent evidence for arelatively uniform stress ®eld across the lake is provided by theInSAR data (Fig. 1b), which indicate a smooth divergence of ice,similar to that in ¯oating ice shelves. Dipping internal layers could,however, still re¯ect oblique topographic in¯uences inherited fromgrounded sections of the ice sheet located up-¯ow of the transectunder consideration. In this case, the observed dip will vary with thesine of the angle between the transect and the ice ¯ow direction.Inherited dip will therefore be zero where transects are alignedparallel to ice ¯ow. Since no transect in the present study is alignedexactly parallel to the ice ¯ow (Fig. 1), this argument cannot be usedto constrain inherited layering precisely. However, the in¯uencemay be evaluated where two transects cross the same ice-¯ow path atdifferent angles. Here, a similar pattern of inherited internal layeringwill be present in both transects, and the amplitude of the patternwill vary with the sine of the angle that each transect makes with theice ¯ow direction. Finally, basal ice melting and accretion as the icesheet passes over the lake surface will cause ¯ow-parallel internallayers to dip downwards and upwards respectively. We believe thatat least three approximately ¯ow-parallel transects indicate thatsuch processes operate on a signi®cant scale above Lake Vostok.

In transect AB when ice is grounded (Fig. 2a, Fig. 3a), fourtraceable internal layers are observed to run parallel to the ice sheetbase, and are separated from each other by a relatively constantthickness along the transect. However, as ice ¯ows over the westernmargin of the lake, the distance between the ®rst and fourth internallayers increases by 200 m over a horizontal distance of 8 km, whilethat between the layer and the ice base decreases by 300 m (Fig. 3b).The rate of change of basal ice thickness with distance along thetransect is greatest (-140 m km-1) at 1 km downstream of the icegrounding line, decreasing to only a few metres per kilometre at8 km from the grounding line (Fig. 3c). Since high basal relief isabsent over the lake area, we interpret this thinning in terms of iceloss by basal melting. Here, the melt rate may be calculated fromthe InSAR-derived ice-surface velocities, since two-dimensionalnumerical modelling of ice ¯ow over the lake indicates negligiblechange in horizontal ice velocity down the vertical ice column6. Thismodel also calculates a vertical ice-velocity pro®le consistent withthe dipping of our internal layers (Fig. 2a), where the thicknessbetween layers increases down the ¯owline. Concurrent with thisenglacial thickening, the model predicts ice loss between the lowestlayer and the ice base only if sub-ice melting is accounted for. The

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