Surfactant effects on buoyancy-driven viscous interactions of deformable drops

Colloids and Surfaces A: Physicochem. Eng. Aspects 282–283 (2006) 50–60

Surfactant effects on buoyancy-driven viscous interactionsof deformable drops

Michael A. Rother a,∗, Alexander Z. Zinchenko b, Robert H. Davis b

a Department of Chemical Engineering, University of Minnesota, Duluth, MN 55812-3025, USAb Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309-0424, USA

Received 20 October 2005; received in revised form 29 November 2005; accepted 30 November 2005Available online 18 January 2006

Dedicated to Professor Ivan B. Ivanov (LCPE, University of Sofia) on the occasion of his 70th birthday.

Abstract

Buoyancy-driven interactions of two deformable drops with a bulk-insoluble, nonionic surfactant are studied with a curvatureless boundary-integral algorithm. At moderate drop-to-medium viscosity ratios with a linear equation of state, the surfactant causes breakup of the larger drop tobecome dominant over breakup of the smaller drop at O(0.1) values of the dimensionless gas constant β for a surface Peclet number Pes of 100. ForP

twta©

K

1

ilcapsuiriorTii

0d

es ≤ 10, the surfactant has little effect on breakup for β ≤ 0.2. For Pes = 1000, critical horizontal offsets for breakup of the larger drop exceedhose for breakup of the smaller drop at β = O(0.01). The influence of surfactant on the deformation and breakup of the smaller drop increaseshen it becomes closer in size to that of the larger drop. Surfactant effects are less significant when the drop-to-medium viscosity ratio is greater

han unity. For the case of bubbles, at Pes = 100, the surfactant is swept to the trailing edges early in the trajectories and leads to cusp formationt β = O(0.01). 2005 Elsevier B.V. All rights reserved.

eywords: Surfactants; Drops; Bubbles

. Introduction

We appreciate this opportunity to write an article for a specialssue in honor of Professor I.B. Ivanov, who has been a worldeader in the development of theory for surfactants and interfa-ial phenomena. Surfactants are important in a wide variety ofpplications, in both traditional areas such as detergent and therocessing of many materials [1] and in growing research areasuch as microfluidics [2,3], ‘smart’ drug delivery [4,5], and liq-id crystals [6]. Because surfactants are active at liquid–liquidnterfaces, their action can be decisive in determining the endesult of droplet interactions in an immiscible mixture, includ-ng coalescence, stability and breakup. Most theoretical studiesf surfactants to date have dealt with a single drop, and thus theesults from multidrop experiments have been mostly empirical.he focus of this paper is to study how a bulk-insoluble, non-

onic surfactant affects the interactions of two deformable dropsn buoyancy-driven Stokes flow.

∗ Corresponding author. Tel.: +1 218 726 6154; fax: +1 218 726 6907.E-mail address: [email protected] (M.A. Rothe).

Although non-linear equations of state, such as the Langmuirand Frumkin models, are important at high surfactant concen-trations and have been used to investigate the breakup of a singledrop in axisymmetric extensional flow [7–9], we employ the lin-ear model, which is valid at lower surfactant concentrations andexpressed in dimensionless form [10] as

σ = 1 − βΓ. (1)

The dimensionless interfacial tension σ, gas constant β, andsurfactant concentration Γ are found from

σ = σ∗

σ0, β = Γ0RT

σ0, Γ = Γ ∗

Γ0, (2)

where dimensional variables are marked with an asterisk, σ0 theinterfacial tension in the absence of surfactant, Γ0 the initiallyuniform surfactant concentration, R the gas constant, and T isthe absolute temperature, which is assumed constant. Non-linearmodels to replace (1) for high surface concentrations can beincorporated with modifications to our code.

Results from previous work on one deformable drop in exten-sional flow indicate the probable outcome if nonlinearities areaccounted for in the surface equation of state, or bulk solubility

927-7757/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.colsurfa.2005.11.086

M.A. Rother et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 282–283 (2006) 50–60 51

is included in the model. In general, saturation of surfactant onthe interface leads to strong Marangoni stresses which reducethe deviation in surface coverage from linear-model (1) results[7]. Stretching of the drop is inhibited, and smaller deformationis observed than in the linear case. When transport of the surfac-tant is possible between the bulk and interface, less deviation inthe surface coverage is also observed in comparison to the insol-uble limit [7,8], except for the cases of dilute bulk concentrationor slow adsorption–desorption exchange.

While the ideal gas equation of state is valid only at low sur-factant surface concentrations, this regime is important becauseeven small amounts of surfactant are sufficient to immobilizethe drop interfaces [11]. Moreover, the linear model avoids ad-ditional parameters and the complications of surface viscosity[12–14] and also holds for nondilute systems with small de-viations from the mean surface concentration [15]. As will beseen in the results section, interesting behavior is observed formoderate surface Peclet numbers at values of β less than 0.1where (1) is valid. We note that, with these restrictions, the re-sults presented here have practical relevance for modeling thebehavior of compatibilizers and of soluble surfactants, when thetime scale for diffusive exchange between bulk and interface isslow [11].

Following early pioneering work [16,17], many authors haveconsidered the gravitational motion of a single drop or bubblein the presence of surfactants [18–24]. The assumption in thesesibf

σ

rostRioiskwt

sohsihtsbbt[

and others on axisymmetric film drainage between two slightlydeformable drops with surfactants [11,38,39].

A single surfactant-covered drop that is allowed to deformglobally has been the subject of much investigation by boundary-integral methods, particularly for the axisymmetrical case ofbreakup in extensional flow [7,9,10,40–42]. Relevant results fortipstreaming from a single contaminated drop have been ob-tained by theory [8] for linear extensional flow and experimentfor non-linear extensional [43] and simple shear [44] flows, aswell.

Buoyancy interactions of two deformable drops or bubbleswith clean interfaces have been studied by experiment [45,46]and both axisymmetric [47] and three-dimensional [48] bound-ary integral calculations. Among the possible outcomes are cap-ture of the smaller drop by the larger one at low drop-to-mediumviscosity ratios and breakup of one drop when the viscosities ofthe drop and surrounding liquid are nearly equal. In addition,more complicated behavior, such as combined capture-breakupphenomena, may occur. In this work, we extend previous re-sults by considering the influence of nonionic, bulk-insolublesurfactant on binary droplet interactions in buoyancy. Becausedeformation is important and inertia is negligible, the work isrestricted to relatively large drops in a viscous fluid, and weemploy the boundary-integral method. In Section 2, the prob-lem and solution techniques are described. Results are given inSection 3, and concluding remarks in Section 4.

2

btiwwtl

idc

wmflidtP

bPttv

ingle-drop studies is that negligible or no deformation of thenterface occurs. It has been shown that an isolated drop or bub-le remains spherical only if the interfacial tension is of theorm

= σ0 + a1 cos(θ), (3)

egardless of whether interfacial viscosity is neglected [25,26]r included [27] in the analysis. However, when the drop ismall, so that the capillary number or Bond number is small,he assumption of sphericity is reasonable, independent of theeynolds number, as confirmed by experiments [28–30]. For

nstance, based on a perturbation analysis for the deformationf a single drop at low Reynolds and capillary numbers [31],t has been demonstrated that a droplet subjected to a moderateurface tension gradient remains nearly spherical [32]. To ournowledge, it remains an open question whether a single dropith a higher capillary or Bond number at higher interfacial

ension gradients will deform significantly.Binary interactions of spheres in buoyancy in the presence of

urfactants have received limited attention. The case of flotationf a small spherical particle by a contaminated spherical bubbleas been examined in the limit of small deviation in surfactanturface coverage [33]. Subsequently, the results were general-zed to two drops or bubbles [34]. Arbitrary surfactant coverageas not been considered for two spherical drops in gravity, al-hough the rheology of a dilute emulsion of surfactant-coveredpherical drops in linear [35] and time-dependent [36] flows haseen investigated when redistribution of surfactant was possibleut surface diffusion was neglected. Moreover, near-contact mo-ion of two spherical drops with surfactants has been examined37]. We also note that important work has been done by Ivanov

. Formulation

In Fig. 1, two deformable drops are shown interacting inuoyancy motion. Consisting of the same fluid, the drops movehrough a second immiscible fluid with negligible inertia undersothermal conditions in a gravitational field g that acts down-ard. The drop density and viscosity are ρ′ and µ′, respectively,hile those of the surrounding medium are ρe and µe, where

he undeformed radius of the smaller drop is a1, and that of thearger one is a2.

A bulk-insoluble, nonionic surfactant is present on the dropnterfaces whose transport obeys the time-dependent convective-iffusion equation. In dimensionless form, the surfactant con-entration Γ is governed by

dΓ

dt= w · ∇SΓ − Γ∇s · us − 2kmΓ u · n + 1

Pes∇2

s Γ, (4)

here d/dt is the time rate of change in the reference frameoving with the mesh-node velocity u + w, u the interfacialuid velocity with tangential component us = (I − nn) · u, w

s an additional tangential velocity used to prevent mesh degra-ation, ∇S = (I − nn) · ∇ the surface gradient operator, kmhe mean curvature, n the outward unit normal vector, andes = (�ρga3

2)/(µeDs) is the surface Peclet number with Dseing the surface diffusivity and �ρ = |ρ′ − ρe|. The surfaceeclet number describes the importance of convection relative

o diffusion of the surfactant along the drop interfaces. In (4),ime t∗ has been dimensionless as t = t∗/(µe/�ρga2), and theelocity is scaled with Vs = (�ρga2

2/µe).

52 M.A. Rother et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 282–283 (2006) 50–60

Fig. 1. Definition sketch for two deformable drops interacting in the presenceof a bulk-insoluble surfactant. Surfactant molecules are indicated on the dropinterfaces, and external flow relative to the sedimenting drops is also sketched.

In addition to the initial orientation of the two drops, the di-mensionless gas constant β and the surface Peclet number Pes,three dimensionless parameters govern the interactions: the dropsize ratio k = a1/a2, the drop-to-medium viscosity ratio µ =µ′/µe, and the gravitational Bond number Bo = �ρga2

2/σ0.An additional parameter, the Marangoni number Ma = β/Bo =(Γ0RT )/(µeVs), represents the ratio of Marangoni stresses toviscous stresses. Because β is assumed small and moderatelydeformable drops are being considered, where Bo = O(10), theMarangoni number is generally much less than unity for theconditions studied here. Analogous to previous work [10] on asingle deformable drop in linear extensional flow, using (1) asthe surface equation of state, we define a modified Bond numberBo∗:

Bo∗ = �ρga22

σ0(1 − β)= Bo

1 − β. (5)

The modified Bond number is calculated at the initial interfacialtension when the surfactant is uniformly distributed.

With the exception of the interfacial stress jump, the dimen-sionless form of the boundary-integral equations for the velocityfield remains unchanged from the case of drops with clean in-terfaces:

u�(y) =2∑

2µ − 1

µ + 1

∫u(x) · τ

(�)n(x)(x; y) dSx

was

interface becomes

f(x) = 2

[(1 − βΓ (x))

Bo+ z

]km(x)n(x) + β

Bo∇SΓ (x), (7)

where 1 − βΓ (x) is the now variable dimensionless interfacialtension (1).

Trajectory calculations require simultaneous solution of theconvective-diffusion Eq. (4) together with the boundary-integralformulation for the velocity field (6) and (7). Treating the veloc-ity field first, we note that the governing equations for buoyancy-driven motion in the presence of surfactant and combinedthermocapillary-gravitational motion with clean interfaces areidentical when a linear equation is employed for the dependenceof interfacial tension on surfactant concentration and tempera-ture, respectively. That is, with the substitution of the dimension-less gradient of the interfacial tension with respect to tempera-ture for β, (6) and (7) are the same as the corresponding equa-tions from our previous work on two sedimenting deformabledrops in a vertical temperature gradient [49] (see (2–3) ofref. [49]).

In order to proceed to long times and neck formation duringbreakup at moderate drop-to-medium viscosity ratios with fixedtopology, we have developed a curvatureless boundary-integralformulation coupled with passive mesh stabilization [48]. Forthe problem of variable interfacial tension, either due to sur-factant or changing temperature, it then becomes necessary tot∫

Abdircf

fifrninca

itiiatd

α=1 Sα

+2∑

α=1

2

µ + 1

∫Sα

f(x) · G(�)(x − y) dSx, (6)

here u� is the �th component of the velocity, and G(�)(x − y)nd τ

(�)n(x)(x; y) are the fundamental solution and corresponding

tress vector on the interface. The normal stress jump across the

ransform the inhomogeneous part of (6) and (7):

Sα

(2σk(x)n − ∇sσ) · G(x − y) dSx

= 1

8π

∫Sα

σrr3

[1 − (r · n)2

r2

]dSx. (8)

n efficient algorithm completely applicable to the problem ofuoyancy-driven motion with bulk-insoluble surfactant has beeneveloped to handle singular and near-singular behavior of thentegrands on the right side of (8), and the interested reader iseferred to our earlier work [49] on droplet interactions under theombined forces of gravity and a vertical temperature gradientor details.

In solving the convective-diffusion Eq. (4), local paraboloidtting was used to determine the surface gradients of the sur-actant concentration and tangential velocity. Derivation of theelevant equations may be found in Appendix A; however, weote here that the method is truly curvatureless. After rotationnto a coordinate system where the local z-axis is parallel to theormal vector at each node, the z-component of Γ∇s · us can-els identically with 2kmΓ u · n from (4) to eliminate explicitppearance of km.

As a check on the code, comparison was made with ax-symmetric calculations for a single deformable drop in ex-ensional flow [10]. In Fig. 2a, the deformation parameter D

s graphed versus the modified capillary number Ca∗ for var-ous values of β and γ , where D = (L − B)/(L + B) with Lnd B being the the drop half-length and half-breadth, respec-ively, Ca∗ = Ca/(1 − β) = µeγa/σ0(1 − β) at shear rate γ androp radius a, and γ = σ0(1 − β)a/(µeDs). The solid lines mark


Fig. 2. Comparison between the current 3D code and the axisymmetric workof Stone and Leal [10] on a single drop in extensional flow for (a) deformationparameter D vs. modified capillary number Ca∗ at various values of β and γ

and (b) the dimensionless surfactant concentration profile as a function of axialposition. The results of [10] are indicated as solid lines. In (a) the current resultsare marked by filled circles and in (b) by dashed lines.

previous results, while solid circles indicate points determinedby our 3D algorithm. For a wide range of parameters, excellentagreement is observed. In Fig. 2b, the dimensionless surfactantconcentration profile vs. the axial position is shown at severalcombinations of β, γ and Ca∗. Again, the new results (dashedlines) agree well with the earlier calculations (solid lines). Asa second check, we compared new calculations [50] for three-dimensional interactions of two spherical droplets in buoyancywith arbitrary surfactant surface coverage to our deformabledrop results and also obtained very similar surfactant concen-tration profiles and trajectories.

3. Results

Images from a typical trajectory are shown in Fig. 3 for thecase of small deformation (Bo∗ = 0.95) at a large surface Pecletnumber (Pes = 500). In the upper left, as pictured in Fig. 3a, ata dimensionless time of 55 with both drops falling, the largerdrop has begun to catch up to the smaller one having startedat a vertical separation of ten larger drop radii. Moving coun-terclockwise around the figure, by t = 90 in Fig. 3b, the dropsare at almost the same vertical position and reach a minimumdimensionless gap of 0.018. The larger drop continues to passthe smaller one, and, in Fig. 3d in the upper right, the distancebetween the drops is increasing. As described in our earlier work[51], the drop surfaces are discretized into triangles by inscribingeither an icosahedron or dodecahedron into an initially sphericalshape with refinement to the desired tessellation by repeated di-vision of the triangles into four at the midpoints and projectionof the new vertices onto the sphere. Throughout the trajectory inFig. 3 with 15,360 triangles on the larger drop and 5120 triangleson the smaller one, surfactant is conserved to within 0.14% forthe larger drop and 0.16% for the smaller one without enforcingconstant total concentration. In Fig. 3 and subsequent figures,the contours are projections of the level lines for the surfactantconcentration onto the plane of the drawing. For Pes = O(1000),

Fig. 3. Images from the interaction of two drops with k = 0.5, µ = 1, β =0.05, Bo∗ = 0.95, Pes = 500 and initial horizontal and vertical separations,x0/a2 = 0.5 and z0/a2 = 10, respectively, moving counterclockwise aroundthe image at t = (a) 55, (b) 90 and (d) 125. Isoconcentration lines are shown foreach drop interface. In (c), a close-up of the gap at t = 90 is shown, indicatingsurfactant redistribution.


high triangulation is needed on the larger drop to keep the sur-factant concentration from becoming negative on the leadingedge. This requirement makes systematic calculations at largersurface Peclet numbers difficult.

The combined effects of external movement of the continuousphase and the lubrication flow between the drops on the surfac-tant distribution are illustrated in Fig. 3. Because the drops arefalling, the external liquid is displaced upward, while the liquidbetween the drops is squeezed out as they come close together.Thus, as shown in the close-up in Fig. 3c, there is a small regionbelow the drops’ line of centers, where the surfactant concentra-tion on the smaller drop builds up due to the opposing effects ofexternal and lubrication flows. However, above the line of cen-ters, the surfactant concentration on the smaller drop decreasesmore than would be expected for an isolated drop, because both

the external and lubrication flows sweep surfactant out of thegap. The effect is more pronounced for the smaller drop becauseit moves more slowly than the larger one, and so the convectiveterm from (4) Γ∇s · us is weaker.

In Figs. 4 and 5, the effects of β and smaller Pes, respectively,on breakup of the smaller drop are illustrated. Images from tra-jectories are shown in Fig. 4a and b for β = 0.001 and Fig. 4cand d for β = 0.025. Because β is so small in Fig. 4a and b, sur-factant has little effect on the interfacial tension and the resultsare nearly identical to those for clean drops (not shown). In Fig.4b and particularly d, the increase in β has caused a measurabledecrease in the interfacial tension, leading to larger deformationin both drops. In Fig. 4d, the smaller drop is nearly 25% longerthan the smaller drop in Fig. 4b at the identical time. We notethat, in Fig. 4, for Pes = 100, 3840 triangles were used on each

Fz

d

ig. 4. Effect of β on breakup of the smaller drop with k = 0.7, µ = 1, Bo∗ = 9.2

0/a2 = 10, respectively, at t = 100 and 200 for β = 0.001 (a and b) and β = 0.025), only the smaller drop is shown.

6, Pes = 100 and initial horizontal and vertical separations, x0/a2 = 1.9 and(c and d). Isoconcentration lines are shown for each drop interface. In (b and


Fig. 5. Effect of small Pes on breakup of the smaller drop with k = 0.7, µ = 1, Bo∗ = 11.6, β = 0.2 and initial horizontal and vertical separations, x0/a2 = 1.9 andz0/a2 = 10, respectively, at t = 120 and 175 for Pes = 1 (a and b) and Pes = 10 (c and d). Isoconcentration lines are shown for each drop interface. In (b and d),only the smaller drop is shown.

drop, a number sufficient to maintain good surfactant conserva-tion and resolve most details in the surfactant contours up to apoint where breakup is evident.

In Fig. 5, images from trajectories are shown for Pes = 1 ina and b on the left, and Pes = 10 in c and d on the right. A largevalue of β = 0.2 was chosen to illustrate the negligible influenceof smaller Pes. In Fig. 5a, there has been little redistribution ofsurfactant due to the weakness of convection, and Γ is close tounity on both drops. In Fig. 5b, when breakup of the smaller drophas begun, the surfactant concentration is still nearly uniform,but Γ is approximately 0.8 over the surface of the smaller drop.The decrease in surfactant concentration is due to dilution, asthe interfacial area increases because of stretching. In Fig. 5c,for Pes = 10, there has been more redistribution of surfactant

and stretching of the larger drop at the trailing edge, where Γ

is higher and σ is lower. Similarly, convection is more evidentin Fig. 5d, but the dilution effect is also visible, and the lengthof the smaller drop at the larger Pes is less than 5% greaterthan for Pes = 1. Thus, even for large β, where small changescan significantly impact interfacial tension and deformation, thevalue of Pes has little influence on smaller drop breakup.

In Fig. 5, 1280 triangles were used on each drop. Becauseconvection is still not very strong, this discretization is sufficientto achieve good surfactant conservation. Some details in thedrop deformation are not well resolved; however, in order to dosystematic calculations, similar discretizations were sometimesused at Pes = 10 because the time-step required for stability inour explicit scheme was less then 0.01 of that required for clean


Fig. 6. Images from the interaction of two drops leading to breakup of the larger drop with k = 0.7, µ = 1, β = 0.075, Bo∗ = 9.26, Pes = 100 and initial horizontaland vertical separations, x0/a2 = 1.9 and z0/a2 = 10, respectively, at t = (a) 70 and (b) 87.5. Isoconcentration lines are shown for each drop interface. In (c), acloseup of the mesh on the larger drop is shown at t = 87.5.

drops [48]. No calculations were attempted at Pes < 1. For thelinear equation of state and the study of breakup considered here,smaller surface Peclet numbers have negligible effect.

In Fig. 6, breakup of the larger drop is illustrated at β =0.075 and Pes = 100. In calculations, it was found that, as β in-creased, the tip of the trailing edge of the larger drop became finer(Fig. 6a). As a result, high curvatures occurred at the point wherethe drop began to stretch, and very fine triangulation was re-quired for calculations to proceed. In Fig. 6, 20,480 triangleswere used on the larger drop and 3840 on the smaller one. InFig. 6b, a neck has started to form toward the end of the thin fil-ament, which is approximately 2.65 times longer than the orig-inal undeformed radius, and breakup is evident. It is not clear,however, how much more the drop will stretch before it breaks.In Fig. 6c, the mesh on the larger drop is pictured at t = 87.5,indicating triangle stretching in the region of the thin filament.

The effect of β on critical horizontal offsets for drop breakupis shown in Figs. 7 and 8 at various Pes. Breakup of both thelarger and smaller drop is considered. In Fig. 7, the size ratio is0.7, and the modified Bond number is 9.26. The initial vertical

separation is 10 larger drop radii to be consistent with our pre-vious studies of clean drops [48,49]. For k = 0.7, the influenceof surfactant on breakup of the smaller drop is visible but weak.On the other hand, breakup of the larger drop is very sensitive tothe presence of surfactant at Pes = 100 and 1000. Because theisolated drop velocity in buoyancy is proportional to the drop ra-dius squared, convection is much more important for the largerdrop. Surfactant is swept to the tail of the drop more quicklythan for the smaller drop, so that interfacial tension is lower anddeformation more important. In Figs. 7 and 8, error bars indi-cate the upper and lower bounds for the critical offset, at whichseparation and breakup occur, respectively.

The more significant influence of surfactant on the larger dropresults in a crossover between the critical offsets for breakupof the larger and smaller drops. For clean drops (β = 0), thecritical offset for smaller-drop breakup is greater; however, atβ ≈ 0.01 for Pes = 1000 and β ≈ 0.04 for Pes = 100, the off-sets cross, indicating that larger-drop breakup becomes dom-inant. For Pes = 10, only a marginal increase in both offsetsis visible for the range of β considered in Fig. 7. At β = 0.5


Fig. 7. Effect of β on the critical offset for breakup �x0,cr/a2 determined atan initial dimensionless vertical separation of 10 at Bo∗ = 9.26, k = 0.7, λ = 1and Pes = 10 (dotted lines), 100 (solid lines) and 1000 (dashed lines). Offsetsare shown for breakup of both the larger drop and the smaller drop.

(not shown), the critical offset for breakup of the larger drop is3.438 ± 0.063 at Pes = 10, much larger than the value of 1.17for clean drops. Although interesting mathematically, calcula-tions for such large β are beyond the validity of the model. Therapid increase in the critical offset for breakup of the larger dropat certain values of β in Figs. 7 and 8 leads to the open ques-tion, as discussed in the introduction, as to whether an isolated,surfactant-covered drop can break in buoyancy.

Faaa

Fig. 9. The dimensionless critical horizontal offset for breakup as a function ofthe modified Bond number Bo∗ for k = 0.7, µ = 1, and z0/a2 = 10 for cleandrops (solid lines) and in the presence of surfactant with Pes = 100 and β =0.025 (dashed lines).

In Fig. 8, a larger size ratio (k = 0.9) and smaller modifiedBond number (Bo∗ = 4.25) are considered. Similar trends tothose in Fig. 7 are observed; however, the effect of surfactanton the smaller drop is more pronounced. Because the size ra-tio is larger, the smaller drop moves faster than compared withFig. 7, so that convection is more important, resulting in lower

Fod0

ig. 8. Effect of β on the critical offset for breakup �x0,cr/a2 determined atn initial dimensionless vertical separation of 10 at Bo∗ = 4.25, k = 0.9, λ = 1nd Pes = 10 (dotted lines), 100 (solid lines) and 1000 (dashed lines). Offsetsre shown for breakup of both the larger drop and the smaller drop.

ig. 10. The dimensionless critical horizontal offset for breakup as a functionf the modified Bond number Bo∗ for k = 0.7, µ = 2, and z0/a2 = 10 for cleanrops (solid lines) and in the presence of surfactant with Pes = 100 and β =.025 (dashed lines).


Fig. 11. Images from the interactions of two bubbles leading to capture at k = 0.7, µ = 0.001, Bo∗ = 3, Pes = 100, β = 0.01, and initial horizontal and verticalseparations, x0/a2 = 1.5 and z0/a2 = 10, respectively, at t = (a) 65, (b) 75, and (c) 90. Isoconcentration lines are shown for each drop interface. In (d), at Bo∗ = 3.16and β = 0.05, a cusp forms on the larger bubble.

interfacial tension at the drop’s trailing edge and more defor-mation. Crossover again occurs between breakup of the largerand smaller drops, at β ≈ 0.04 for Pes = 1000 and β ≈ 0.125for Pes = 100, with a much weaker surfactant effect on breakupat Pes = 10. For the results in Figs. 7 and 8, 15,360 triangleswere used on the larger drop and 3840 on the smaller one atPes = 1000. At Pes = 100, 5120 triangles were used on thelarger drop and 3840 on the smaller one for most calculations.For Pes = 10, discretizations of 5120 (larger) and 1280 (smaller)triangles were often employed for larger-drop breakup and 1280(larger) and 3840 (smaller) triangles for smaller-drop breakup.

The effect of surfactant on critical offsets over a range ofmodified Bond numbers at fixed Pes and β is shown in Figs. 9and 10 for µ = 1 and 2, respectively. In Fig. 9, the stronger

surfactant influence on larger-drop breakup is again illustrated,with crossover between breakup of the larger and smaller dropsoccurring at Bo∗ ≈ 12 for Pes = 100 and β = 0.025. As theviscosity ratio is increased to 2 in Fig. 10, resistance to surfactantredistribution increases, and crossover takes place at a largerBo∗ (≈ 18). However, at µ = 2, the more significant effect ofsurfactant on the larger drop is clear.

Some preliminary results for bubble interactions are given inFig. 11. At a small modified Bond number and dimensionlessgas constant (Bo∗ = 3, β = 0.01), a result leading to captureof the smaller bubble by the larger one is shown at Pes = 100(Fig. 11a–c). When β is increased to 0.05 at a slightly largermodified Bond number (Bo∗ = 3.16), however, a cusp formson the larger bubble (Fig. 11d) with 3840 triangles, and it is


not possible to continue calculations. Simple attempts at cuspsmoothing [48] proved unsuccessful due to difficulties with thesurfactant calculation. When the node positions at high curvaturewere modified, it was not clear how to change the surfactantconcentration correspondingly. As a result, instability occurredwhen solving the convective-diffusion Eq. (4).

4. Concluding remarks

Interactions of two deformable drops in the presence of abulk-insoluble, nonionic surfactant have been studied by a cur-vatureless boundary-integral algorithm. Since the larger dropmoves more quickly than the smaller one under the influenceof gravity, convection is more important on the larger drop. Asa result, surfactant concentration increases more on the trail-ing end of the larger drop, causing lower interfacial tensionand increased deformation. Thus, breakup of the larger dropeventually becomes dominant over that of the smaller drop, asmeasured by critical horizontal offsets. The presence of surfac-tant also tends to produce a thinner filament of liquid as thedrop stretches and breaks, making calculations more difficult.Drops, both the larger and smaller, appear to stretch more be-fore breakup in the presence of surfactant than in its absence.These results concerning breakup of the larger drop suggestthat further study of the possibility of gravity-driven breakupof an isolated drop in the presence of surfactant should bem

A

MTNs

Ac

nFt

x

war(wc

e

e

Fig. A.1. Schematic for the calculation of the normal vector and surfactantconcentration at a node with the typical coordination number of six.

The components of the metric tensor, gij = ei · ej, are

g11 = 1 + (2Cξ1 + dξ2)2,

g12 = g21 = (2Cξ1 + Dξ2)(Dξ1 + 2Eξ2),

g22 = 1 + (Dξ1 + 2Eξ2)2, (A.3)

where gij = δij and ∂gij/∂ξk = 0 at ξ1 = ξ2 = 0. Similarly, forthe contravariant metric tensor, ‖gij‖ = ‖gij‖−1, gij = δij and∂gij/∂ξk = 0 at ξ1 = ξ2 = 0.

By general rule, the term ∇s · us from (4) is

∇s · us = 1√g

∂(uα√g)

∂ξα

, (A.4)

where uα = gαβuβ = gαβ(u · eβ) are contravariant componentsof the fluid velocity u (1 ≤ α, β ≤ 3). Since the surface gradientof the interfacial velocity is evaluated at the origin of the rotatedcoordinate system, where ξ1 = ξ2 = 0, calculation gives

∇s · us|O =(

∂u∂ξα

· eα + u · ∂e∂ξα

)ξ1=ξ2=0

. (A.5)

If u(ξ1, ξ2) = (ux, uy, uz) in Cartesian coordinates(x1, x2, x3), then

∇s · us|O =[∂ux

∂ξ1+ ∂uy

∂ξ2+ 2(C + E)uz

]ξ1=ξ2=0

. (A.6)

Acta

u

u

(

ade.

cknowledgements

MAR is grateful to the VDIL at UMD and the University ofinnesota Supercomputing Institute for computing resources.

his work was supported by NASA grants NAG3-2116 andNCO5GA55G, and grant 40430-AC from the Petroleum Re-

earch Fund of the American Chemical Society.

ppendix A. Details of the solution method for theonvective-diffusion Eq. (4)

As in our previous algorithm [48], at each node the surfaceormal n is determined and the x3-axis is directed along n (seeig. A.1). For discretization of the surfactant-distribution equa-

ion (4), the surface is fitted locally as

3 = Cx21 + Dx1x2 + Ex2

2, (A.1)

here the coefficients C, D, and E are found by least-squaresnd are taken as outputs from our previous curvature-normaloutine upon convergence of iterations. With x2

1 + x22 small, Eq.

A.1) is the local parametrization of the surface r = r(ξ1, ξ2)ith ξ1 = x1 and ξ2 = x2. The covariant basis vectors of the

urvilinear coordinate system are

1 = ∂r∂ξ1

= (1, 0, 2Cξ1 + Dξ2),

2 = ∂r∂ξ2

= (0, 1, Dξ1 + 2Eξ2). (A.2)

n interesting consequence of (A.6) is that the last term will beancelled by the curvature term in (4), so that the algorithm isruly curvatureless. A least-square quadratic fit is found for ux

nd uy:

x(ξ1, ξ2) = ux(0, 0) + axξ1 + bxξ2 + cxξ21

+ dxξ1ξ2 + exξ22, (A.7a)

y(ξ1, ξ2) = uy(0, 0) + ayξ1 + byξ2 + cyξ21

+ dyξ1ξ2 + eyξ22 . (A.7b)

Evaluating at ξ1 = ξ2 = 0 gives

∂ux

∂ξ1+ ∂uy

∂ξ2

)O

= ax + by. (A.8)


The terms ∇sΓ and ∇2s Γ from (4) are calculated in a similar

way by a quadratic fit of Γ :

Γ (ξ1, ξ2) = ΓO + aξ1 + bξ2 + cξ21 + dξ1ξ2 + eξ2

2 . (A.9)

As in our previous work on thermocapillary motion of de-formable drops [12], where the surface gradient of temperatureis required,

∇sΓ |ξ1=ξ2=0 = (a, b). (A.10)

For the surface Laplacian of Γ ,

(∇2s Γ )O = 1√

g

∂(∇αΓ√

g)

∂ξα

, (A.11)

where ∇αΓ = gαβ∇βΓ = gαβ(∂Γ/∂ξβ) are contravariant com-ponents. Simplification yields

∇2s Γ |ξ1=ξ2=0 =

(∂2Γ

∂ξ21

+ ∂2Γ

∂ξ22

)O

= 2c + 2e. (A.12)

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Surfactant effects on buoyancy-driven viscous interactions of deformable drops