Dimensional crossover in the coalescence dynamicsof viscous drops confined in between two platesMaria Yokota and Ko Okumura1

Department of Physics, Graduate School of Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Edited by Harry L. Swinney, University of Texas at Austin, Austin, TX, and approved February 23, 2011 (received for review November 16, 2010)

Coalescence of liquid drops is a daily phenomenon familiar toeverybody and is related to many fields from biology to astronomyand also related to a variety of practical problems in industry.However, the detailed physical understanding of the dynamicshas been revealed only recently with the aid of high-speed camera,high-performance computer, and scaling analysis. In this study,coalescence of a viscous drop to a bath of the same liquid is studiedin a confined space. This is because dealing with a small amount ofliquid drops becomes increasingly important (e.g., in industrial andbiological applications). Here, the aqueous drop and bath are sur-rounded by low-viscosity oil and sandwiched by two parallel platesof the cell. We quantify experimentally the width of a neck thatbridges the drop and the bath during coalescence. As a result, wefind that the neck width increases linearly with time at short times,but the dynamics slows down significantly at longer times. Thanksto simple and original scaling arguments, we clearly show that thistransition of the dynamicswith time in a single coalescence event isbrought about by a crossover from a three-dimensional viscousdynamics for a spherical drop to a quasi two-dimensional one fora disk drop. In addition, we report an unusual type of coalescencethat is possibly caused by naturally accumulated electric charge inthe confined geometry and whose dynamics seems self-similar.

From daily experiences, everybody knows a liquid drop fallingonto a bath of the same liquid merges to the bath. A specta-

cular aspect of this mundane phenomenon of coalescence ofdroplets was already studied as early as 1885 by Thomson andNewall (1). It is important in various problems, such as fusionof cells in biology and of galaxies in the universe, and in a largenumber of industrial applications, such as emulsion stability, ink-jet printing, and others (2). Accordingly, it is still an active area ofresearch (3–5).

In particular, the recent advances in technology, associatedwith high-speed camera, image analysis, and simulation, togetherwith theoretical development, are leading us to a new phase ofunderstanding. Recently, it has been established that the coales-cence dynamics driven by capillary force is balanced by viscousforce at shorter times (or in viscous drops) and by inertial forceat longer times (or in less-viscous drops) (6, 7). More recently,this has been confirmed also in two-dimensional coalescence(8) and the possibility of a new inertial regime is reported (9).Furthermore, still another new viscous regime, which can berather regarded as a film bursting, is reported in ref. 10.

Here, we study coalescence of a droplet of radius R in a con-fined geometry of a Hele–Shaw cell (in between two plates whosedistance D is smaller than the droplet size R), which will berelevant in many practical situations where a small amount ofliquid has to be manipulated (e.g., microfluidics and biologicalapplications). We followed the dynamics of the half of the neckwidth r (see Fig. 1). As a result, we find in a single coalescenceevent a crossover from the three-dimensional viscous dynamicsfor a spherical drop to a unique type of quasi two-dimensionalviscous one for a disk drop, which is potentially important invarious practical contexts. Original theoretical arguments, whichagree well with our observation, are developed in this paper. Thisis because, although pinch-off dynamics in the Hele–Shaw cellhas been studied frequently (11, 12) the “reverse” coalescence

dynamics of drops in the Hele–Shaw cell has not been exploredthoroughly enough.

Results and DiscussionWe observed a quasi two-dimensional aqueous drop, surroundedby low-viscosity oil in a Hele–Shaw cell, merging into a liquid bathof the same liquid, as illustrated in Fig. 1. We measured the halfof the neck width r as a function of time t, as in Fig. 2. The plot inlog scales demonstrates that r is linearly dependent on t at shorttimes, and the dynamics slows down at long times where r scalesas t1∕4.

We consider first the linear dynamics at short times wherer ≲D∕2. In this region, the neck is three-dimensional; it is likea cylinder of radius r (see Fig. 1). Thus, we can expect the recentlyestablished three-dimensional viscous dynamics (7) in this regimewhere the velocity V ≃ dr∕dt scales as γ∕η because the drops arefairly viscous (we did not observe, as in ref. 7, the logarithmiccorrection theoretically predicted in ref. 6). Here, γ and η repre-sent the surface tension and viscosity of the drop (and the bath).Indeed, the Reynolds number, ρVr∕η (≃ργr∕η2) in this case, isless than unity when 2r ≲D, where ρ is the density of the drop.This linear dynamics, consistent with the log–log plot in Fig. 2,can be expressed in a dimensionless form,

r∕D≃ t∕τi with τi ¼ Dη∕γ: [1]

This is well confirmed in Fig. 3: The data collapse well on astraight line with a slope of the order of unity at short times (r ≲Dand t≲ τi) in the Upper plot. Furthermore, in the Lower plot theconstant velocity V obtained from the data in the short-timeregime is shown to scale as γ∕η, as predicted by Eq. 1.

At this initial stage of coalescence, we did not observe appear-ance of a dimple surrounded by a rim at the drop bottom of thedroplet, although such possibility is implied in a Hele–Shaw cell(11, 12) and is explicitly demonstrated in droplet coalescence(13); in the Top images in Fig. 1, the dark parts at the contact arenot such rims but show the surfaces of a glycerol cylinder asshown in Movie S1.

We next consider the dynamics at long times where d≳D.Here, the neck length d (see Fig. 1) satisfies a geometricalrelation,

R2 ¼ ðR − dÞ2 þ r2: [2]

For this purpose, we note that the short-time viscous dynamics inEq. 1 can be obtained dimensionally by balancing a gain in surfaceenergy per unit time, dðγr2Þ∕dt, with a viscous dissipation ηðV∕rÞ2(per time and per volume) localized in a volume r3. Contrary tothis, at long times the gain in surface energy per unit time is

Author contributions: M.Y. and K.O. designed research; M.Y. and K.O. performedresearch; M.Y. and K.O. analyzed data; and K.O. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: okumura@phys.ocha.ac.jp.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1017112108/-/DCSupplemental.

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replaced by dðγrDÞ∕dt while the viscous dissipation ηðV∕DÞ2 islocalized in a volume rDd, to minimize the total dissipation (14).This form of dissipation associated with the gradient V∕D origi-nating from the Poiseuille flow between the cell plates separatedby a distance D is expected to be dominant when d≳D becausethen this dissipation becomes larger than that associated with thegradient V∕d. By using an approximation d≃ r2∕R to the geome-trical relation in Eq. 2 (6, 14), we obtain r ≃ ðγRD2t∕ηÞ1∕4. Thist1∕4 dynamics is consistent with the log–log plot in Fig. 2 and canbe expressed in the following dimensionless form:

r∕ffiffiffiffiffiffiffi

RDp

¼ ðt∕τf Þ1∕4 with τf ¼ Rη∕γ: [3]

This dynamics is limited by two conditions: (i) d≳D (as alreadymentioned) and (ii) d≲ R (for the approximated geometricalrelation). These two conditions can be expressed as r ≳

ffiffiffiffiffiffiffi

RDp

and r ≲ R, respectively, so that the conditionffiffiffiffiffiffiffi

RDp

≲ R is re-quired to observe this regime; the drop radius must be larger

than the cell thickness. This regime is well confirmed again inthe log–log plot in Fig. 4: The data collapse well on a straightline with a slope 1∕4 at long times (r ≳

ffiffiffiffiffiffiffi

RDp

and t≳ τf ) in theUpper plot (see also Fig. S1 for a detailed examination of thedata). Furthermore, in the Lower plot the constant r∕t1∕4 obtainedfrom the data in the long-time regime is shown to scale asðγRD2∕ηÞ1∕4, as predicted by Eq. 3.

The agreement of our theory with our experiment impliesthat the viscous dissipation in the thin film of oil existing betweenthe neck and the cell plates (see Methods) is virtually inhibited.Although the viscosity of the thin film is much smaller than theviscosity of the liquid forming the neck, the thickness is expectedto be much thinner than the cell thickness. Thus, it is natural thatthe development of the velocity gradient inside the thin filmis inhibited to avoid a large viscous dissipation inside the film;D

PDMS (oil)

glycerol

glycerol drop

g

2r

R

O

2r

R

d

R-d

O

Fig. 1. (Top) A quasi two-dimensional glycerol drop in a Hele–Shaw cell,surrounded by low-viscosity oil, merging into a bath of glycerol (seeMovie S1). The short-time regime (2r ≲ D) is shown with 4-ms frame separa-tion on the left, and the long-time regime (r ≫ D) is shown with 40-ms frameseparation on the right. Triangles in Fig. 2 below are generated from thisevent. (Middle) Illustrative geometry of the short-time (left) and long-time(right) regimes, in which the neck width 2r, the neck length d, and theradius of the droplet R are defined. (Bottom) Illustration of the experimentalsetup. The cell is placed vertically (g is the gravitational constant), and theglycerol drop goes down because of gravity before coalescence.

0 1 2 3 4 5

0 0.5 1 1.5 2 2.5

r [m

m]

t [s]

0.1

1

10

0.001 0.01 0.1 1 10

r [m

m]

t [s]

Fig. 2. The half of the neck width r is plotted at various time t (Upper)and the same dynamics in the log scales (Lower). The solid and dashed lines(Lower) indicate slopes 1 and 1∕4, respectively. Viscosity η, cell thickness D,and drop radius R for plot symbols are as follows: square, 62.9 mPa · s,0.7 mm, 5.62 mm; circle, 289 mPa · s, 0.7 mm, 5.56 mm; triangle,888 mPa · s, 1.0 mm, 4.13 mm; cross, 964 mPa · s, 1.0 mm, 4.32 mm.

0

1

2

3

4

5

6

7

8

0 50 100 150 200 250 300

r/D

t/τi

0

0.1

0.2

0 0.1 0.2 0.3 0.4

V [m

/s]

γ/η [m/s]

Fig. 3. (Upper) A plot between the half of the neck width r and time t,normalized by Eq. 1 for short-time dynamics. (Lower) Comparison of the con-stant velocity V obtained from the data in the short-time regime with theexperimental parameter γ∕η. The Upper plot demonstrates a global pictureor scaling view with clarification of the characteristic scales, whereas theLower provides the qualitative agreement.

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virtually no flow is generated inside the thin film. As a result, thespeed of liquid forming the neck becomes almost zero at thefilm–neck boundary while the velocity gradient V∕D is developedinside the neck. Similar inhibition of viscous dissipation in thinfilm was recently confirmed in ref. 15.

The pure two-dimensional coalescence dynamics for cylindri-cal drops in which velocity is constant along the cell-thicknessdirection is predicted to have the same scaling as the three-dimensional analogue (6). Compared with this, the present quasitwo-dimensional dynamics described by Eq. 3 results from thedisk (i.e., quasi two-dimensional) shape of the drop, and thedynamics is governed by the Poiseuille flow whose velocitychanges along the direction of the cell thickness.

We sometimes observed an unusual type of coalescence as inFig. 5. Although in Eq. 2 we assumed that the coalescence startsonly when a circular drop directly touches the horizontal surfaceof the bath, in general coalescence could start with a slight butfinite distance d0 between the drop bottom and the bath surface.Indeed, we occasionally observed that this distance is very large.In such a case, as seen in the three video frames of Fig. 5, singularhumps are formed both in the drop and bath before the initialcontact; the top hump (from the drop) and the bottom hump(from the bath) seem to be attracted to each other. The coales-cence events examined in the above analysis are limited to thecase where this initial distance d0 is much smaller than D toexclude this remarkable but more complex cases from the quan-titative analysis.

The initial distance d0 seems to be controlled by at least twofactors: (i) the descending velocity of the drop before coalescence(seeMethods) and (ii) naturally accumulated electric charge. Thesecond point is confirmed by examining the effect of dischargingthe setup. This is practically accomplished, although a completecontrol of natural charges is difficult in the present environment,by performing the same experiment in different seasons. In thedistrict where these experiments are performed, the humiditychanges significantly depending on seasons; electric dischargein daily life is very frequent in winter, whereas it is never observedin early summer. Accordingly, we observed a large d0 frequentlyin winter (a typical humidity is 30%), whereas we always observed

nearly zero d0 in early summer (a typical humidity is 70%). Thissuggests that the humps might be related to the Taylor cone dueto electric charge (16, 17). This kind of charge effect in all pre-vious studies has been always induced artificially by a high voltagesupply, and recently noncoalescence is induced by this artificialeffect (3, 4). In this present case, however, it is possible that

4

1

0.4

0.1

0.04 40 10 4 1 0.4 0.1 0.04

r/(R

D)1/

2

t /τf

10-3

10-2

10-10 10-9

r/t1/

4 [m/s

1/4]

γRD2/η[m4/s]

Fig. 4. (Upper) A plot in log scales between the half of the neck width r andtime t, normalized by Eq. 3 for long-time dynamics. (Lower) Comparison inlog scales of the constant r∕t1∕4 obtained from the data in the long-time re-gime with the experimental parameter γRD2∕η. The solid lines in the Upperand Lower plots both indicate the slope 1∕4. A scaling view and a quantita-tive comparison are demonstrated in theUpper and Lower plots, respectively.

Fig. 5. Video frames at the initial contact between the drop and bath with alarge initial neck length d0 (see Movie S2). Singular humps appear both fromthe drop and bath to attain the initial contact. The frames are separated by20∕3;000 s. η ¼ 991 mPa · s, D ¼ 1.5 mm, R ¼ 2.95 mm.

-0.2

-0.1

0

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5

y[m

m]

x[mm]

-2

-1

0

1

2

0 1 2 3 4 5

y/r

x/r

Fig. 6. Self-similar shape of the bridge between the drop bottom and thebath surfaces, obtained from the coalescence event shown in Fig. 5. Theorigin of the plots corresponds to the center of the bridge, or the startingpoint of coalescence; only the right half of the bridge is shown. The half-bridge shapes represented by circles, triangles, and squares are respectivelyseparated by 3∕3;000 and 7∕3;000 s (this initial dynamics is nonlinear). Thethree shapes in the original Upper plot in the units of millimeter collapse wellon to a single master shape in the Lower plot (especially near the tip of theshape) where both axes are rescaled by the neck radius r at each time.

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naturally accumulated electric charge causes this effect. Thishump dynamics should be investigated further from this view-point of charge effect by quantitatively controlling the amountof electric charge, and also from the viewpoint of the dependenceof the dynamics on the descending velocity of the drop beforecoalescence.

In addition, this dynamics should be examined further as aself-similar critical dynamics. This is because the self-similarityhas been studied well in the case of drop pinch-off (18–21) butrarely studied in the opposite process of coalescence (22). As amatter of fact, as demonstrated in Fig. 6, the drop shape nearsingularity is confirmed to be self-similar: The shape looks thesame when both the horizontal (x) and vertical (y) axes are re-scaled by the neck size r (although the geometry of the self-similarshape has yet to be explored). This situation is in contrast withthe case observed in ref. 19, where rescaling factors for a goodcollapse are different in the corresponding two axes.

ConclusionIn conclusion, we observed coalescence of a viscous drop to abath of the same liquid confined in a Hele–Shaw cell. The dropand the bath are surrounded by a less-viscous oil; direct contactof the drop and bath liquid with the cell plates is avoided becauseof thin oil film on the plates. We find that the initial dynamicsis described by the well-established three-dimensional viscousdynamics for a spherical drop and the final dynamics by a differ-ent quasi two-dimensional viscous dynamics for a disk drop. Ourtheoretical arguments can predict the ranges of these regimestogether with scaling laws of the dynamics, which agree well withour data. In other words, a dimensional crossover of the dynamicsis observed in a single event in the present study. This crossoverdynamics established here can be potentially important in variousfields, such as in microfluidics and biological applications, wheresmall amount of liquids should be manipulated. We also observedcoalescence starting with a finite distance between the dropbottom and the bath surfaces, which seem to be attracted to eachother to initiate coalescence. This remarkable effect revealed inthis study potentially opens up completely new ways to studycharge effect and self-similarity in liquid drop dynamics.

MethodsWe fabricated a Hele–Shaw cell from transparent acrylic plates of a few milli-meters thickness with the plates separated by a distance D (of the order of

millimeter) with spacers of homogeneous thickness as in our previous studies(23, 24). The width and depth of the cell are 10 and 15 cm, respectively. We fillthe vertically positioned cell with a low-viscosity silicone oil [polydimethylsi-loxane (PDMS)] of kinematic viscosity 1 cS (about 1 mPa · s) and then heavierglycerol; the two liquids are immiscible and hence cause a phaseseparation where the lighter oil is on top of the glycerol (see Fig. 1). Afterwaiting (at least 30 min) for the bath interfaces to reach equilibrium, weinsert with a syringe from the above oil phase a glycerol drop at the centerof the width of the cell (i.e., away from the edges). The drop, whose radius isabout several millimeters (much smaller than the cell width), gradually goesdown through the oil phase (of depth around several centimeters) due togravity before it starts coalescing to the lower phase of the same glycerol.The high-speed video images of coalescence are obtained by a camera, MEM-RECAM fx 6000 (Nac), in some cases with a microscope lens D-6MP (Degimo).

To change viscosity η of glycerol (of the drop and the bath) we addedwater, and each time we directly measured the viscosity of the aqueoussolution avoiding a large change in room temperature because the viscosityis sensitive to temperature (25). The viscosity of surrounding bulk oil (about1 mPa · s) can be always neglected compared with those of glycerol solutions(about 50–1000 mPa · s). The interfacial energy αγ of glycerol solutions con-tacting the oil can be assumed to be a constant in practice (10, 21). Here,γ ¼ 20 mN∕m for convenience with α a numerical coefficient of the orderof unity.

The density of glycerol solution ρ and that of oil are 1.21–1.26 (dependingon the viscosity) and 0.818 g∕cm3, respectively, and the difference results indescending motion of the glycerol drop (before coalescence) at velocity ofthe order of 1 mm∕s (15). This speed is dependent on the rate of flow forcreation of the drop from a syringe (inner radius 5 mm) with a needle (innerdiameter 0.55 mm). The observations demonstrated above are obtainedwhen the piston is pushed manually around at 5 mm∕s (i.e., at flow rateof the order of 0.1 cm3∕s). When the descending speed is too fast or slowcompared to this speed, slightly different dynamics is observed. When toofast, the flow might tend to be turbulent, and clear scaling laws reportedabove are contaminated; when too slow, the drop bottom tends to becomeflat before coalescence with a thin oil film formed between the bottom andthe bath surface (10).

Because the oil, which likes the acrylic plates, is poured first, the glycerolphase and the falling drop is always covered with a thin oil film; there is nodirect contact of glycerol with the plates, hence, no contact line for thefalling drop. This point is quite important to remove experimental contam-ination due to the intricate effect of the contact line.

ACKNOWLEDGMENTS. M.Y. is supported by the Japan Society for thePromotion of Science Research Fellowships for Young Scientists. K.O. thanksthe Japanese Ministry of Education, Culture, Sports, Science and Technologyfor a Grant-in-Aid for Scientific Research.

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