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Particuology 9 (2011) 1–13

Contents lists available at ScienceDirect

Particuology

journa l homepage: www.e lsev ier .com/ locate /par t ic

nvited paper

articles dispersion on fluid–liquid interfaces

athish Gurupathama, Bhavin Dalala, Md. Shahadat Hossaina, Ian S. Fischera,ushpendra Singha,∗, Daniel D. Josephb,c,∗

Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USADepartment of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USADepartment of Mechanical and Aerospace Engineering, University of California Irvine, Irvine, CA 92697, USA

r t i c l e i n f o

rticle history:eceived 9 July 2010eceived in revised form 9 October 2010ccepted 18 October 2010

eywords:dsorption

nterfacial tensionarticle dispersionluid–liquid interfaceapillary forceiscous drag

a b s t r a c t

This paper is concerned with the dispersion of particles on the fluid–liquid interface. In a previous study wehave shown that when small particles, e.g., flour, pollen, glass beads, etc., contact an air–liquid interface,they disperse rapidly as if they were in an explosion. The rapid dispersion is due to the fact that the capillaryforce pulls particles into the interface causing them to accelerate to a large velocity. In this paper we showthat motion of particles normal to the interface is inertia dominated; they oscillate vertically about theirequilibrium position before coming to rest under viscous drag. This vertical motion of a particle causes aradially-outward lateral (secondary) flow on the interface that causes nearby particles to move away. Thedispersion on a liquid–liquid interface, which is the primary focus of this study, was relatively weakerthan on an air–liquid interface, and occurred over a longer period of time. When falling through an upperliquid the particles have a slower velocity than when falling through air because the liquid has a greaterviscosity. Another difference for the liquid–liquid interface is that the separation of particles begins in theupper liquid before the particles reach the interface. The rate of dispersion depended on the size of the

particles, the densities of the particle and liquids, the viscosities of the liquids involved, and the contactangle. For small particles, partial pinning and hysteresis of the three-phase contact line on the surface ofthe particle during adsorption on liquid–liquid interfaces was also important. The frequency of oscillationof particles about their floating equilibrium increased with decreasing particle size on both air–water andliquid–liquid interfaces, and the time to reach equilibrium decreased with decreasing particle size. These

ith ociety

21

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results are in agreement w© 2010 Chinese So

. Introduction

In recent years, there has been much interest in the absorbedtate of particles at fluid-liquid interfaces because of their impor-ance in a range of applications. The behavior of particles trapped atuid–liquid interfaces is important in many applications and chal-

enges our understanding of the fundamental flow physics at play.ome topics which fall into this frame are the self-assembly of par-icles at fluid–fluid interfaces, the stabilization of emulsions, theollination in hydrophilous plants, the flotation of insect eggs, theispersion of viruses and protein macromolecules, etc. (see Aubry,

ingh, Janjua, & Nudurupati, 2008; Balzani, Venturi, & Credi, 2003;owden, Terfort, Carbeck, & Whitesides, 1997; Cox & Knox, 1989;rzybowski, Bowden, Arias, Yang, & Whitesides, 2001; Murray,agan, & Bawendi, 2000; Nudurupati, Janjua, Singh, & Aubry,

∗ Corresponding authors.E-mail addresses: singhp@njit.edu (P. Singh), joseph@aem.umn.edu

D.D. Joseph).

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674-2001/$ – see front matter © 2010 Chinese Society of Particuology and Institute of Process Eoi:10.1016/j.partic.2010.10.002

ur analysis.of Particuology and Institute of Process Engineering, Chinese Academy of

Sciences. Published by Elsevier B.V. All rights reserved.

010; Tang, Zhang, Wang, Glotzer, & Kotov, 2006; Wasielewski,992).

It was recently shown by Singh, Joseph, Gurupatham, Dalal, andudurupati (2009) that when small particles like flour, pollen, etc.,ome in contact with a liquid surface they immediately dispersealso see Singh, Joseph, and Aubry (2010) which contains a com-rehensive review of this work). When particles contact a liquidurface they are pulled strongly towards their equilibrium posi-ion and are accelerated to a relatively-large velocity normal to thenterface. This creates a lateral flow on the interface which induceshe observed dispersion. Sprinkled particles, in fact, disperse so vio-ently that the images of dispersion appear strikingly similar to thatf an explosion, except that the dispersion of particles occurs on aurface as the particles remain trapped at the interface (see Fig. 1nd the movies showing dispersion which are available in Singh

t al. (2009) as the supporting information for this paper in theournal’s website).

This relatively-violent phase, which lasts for about one secondr less on mobile liquids like water, is usually followed by a phaseominated by attractive lateral capillary forces during which parti-

ngineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

2 S. Gurupatham et al. / Partic

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ttapdtwcetswtby contamination, the experiment was repeated many times using

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ig. 1. Sudden dispersion of glass particles sprinkled onto water in a dish. Streak-ines formed due to the radially-outward motion of the particles emanating fromhe location where they were sprinkled. The size of glass particles was ∼12–50 �m.

les slowly come back to cluster (see Aveyard & Clint, 1996; Binks,002; Chan, Henry, & White, 1981; Fortes, 1982; Gifford & Scriven,971; Katoh, Fujita, & Imazu, 1992; Kralchevsky, Paunov, Ivanov, &agayama, 1992; Nicolson, 1949; Singh & Joseph, 2005). However,

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ig. 2. Dispersion and clustering of two plastic beads on the air–water interface. The diamhe experiment was repeated more than 30 times with fresh Millipore water, and the Pnsure that contamination was not a factor and that this behavior of the beads did not chhe gap (D) between the beads and the velocity (v) with which they are moving apart aftenitially increased as the beads moved apart and then decreased as they clustered under

oved apart was about six times larger than the maximum velocity with which they cam

uology 9 (2011) 1–13

nce micron- and nano-sized particles are dispersed, they mayemain dispersed since attractive capillary forces for particles thatmall are insignificant. Small particles may experience other lateralorces, e.g., electrostatic, Brownian, etc., which may cause them toluster or form patterns (see Bresme & Oettel, 2007; Kralchevsky

Denkov, 2001; Kralchevsky et al., 1992; Lehle & Oettel, 2007;aunov, Kralchevsky, Denkov, & Nagayama, 1993; Stamou, Duschl,Johannsmann, 2000, and the references therein).To illustrate the phenomena described above, let us consider

he case of two plastic beads as shown in Fig. 2, which were simul-aneously dropped onto the water surface. The beads first movedpart and then came back together. The former phase, which is therimary focus of this paper, is discussed below. The latter phase isue to attractive capillary forces that arise because of the deforma-ion of the interface by the trapped beads, as they are heavier thanater. More specifically, the floating beads experience attractive

apillary forces because the interface height between them is low-red due to the interfacial tension. Notice that the speed with whichhe beads dispersed was about six times larger than the maximumpeed attained during the clustering phase. The time duration forhich the beads moved apart was about one third of the time they

ook to cluster. To ensure that these results were not influenced

he same beads while the water used in the test was changed.The modeling of interactions among floating particles is a

ormidable challenge because of the complexity of the interac-ions and forces involved, i.e., the fluid dynamics of the interface

eter of beads was 4.46 mm. The beads were carefully washed in water many times.etri dish used in the experiment was rinsed with Millipore water every time, to

ange with time. (a) A photograph sequence showing dispersion and clustering. (b)r they came in contact with the interface are shown as a function of time. The gapthe action of lateral capillary forces. The maximum velocity with which the beadse together.

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S. Gurupatham et al. /

otion, the contact-angle condition on the surface of the parti-les, the contact-line motion, etc. Recently, a DNS approach waseveloped for particles trapped at fluid–fluid interfaces (see SinghJoseph, 2005; Singh et al., 2009). This approach provides not onlycapability for resolving the motion of a particle as well as clus-

ers of particles, but also the ability to address the rapidly-changingynamics of the particles. It is necessary to resolve the particle-leveletails for particles trapped at fluid–fluid interfaces because theeformation of the interface in between the particles determineshe strength of the lateral capillary forces between them, which isne of the main driving forces for their motion. The DNS schemeas used to study two different cases of constrained motions ofoating spherical and cylindrical particles. In the first case, the con-act angle of floating spheres was fixed by the Young-Dupré lawhile the contact line was allowed to move to meet the contact-

ngle requirement. In the second case, the contact line was pinnedt the sharp edge of disks (short cylinders) with flat ends, while theontact angle at the sharp edge was allowed to change within theimits specified by the Gibbs extension to the Young-Dupré law.

In the remainder of this paper, we will discuss the forces thatct on a particle when it comes in contact with a liquid surface, andhe equation governing its motion, which will be followed by a dis-ussion of the results for the dispersion of particles on fluid–liquidnterfaces.

. Forces balance and particles motion normal to auid–liquid interface

In this section we summarize some of the results of the analy-is presented in Singh et al. (2009) for the adsorption of particlest fluid–liquid interfaces that are compared with the experimentalesults obtained in this study. When a particle comes in contactith a fluid–liquid interface it is pulled inwards to its equilib-

ium position within the interface by the normal component of theapillary force. The equilibrium position of the particle after it isdsorbed in the interface is determined by the balance of the par-icle’s buoyant weight, the vertical capillary force, and any otherorce (with a vertical component) that acts on the particle. Thearticle is in stable equilibrium in the sense that if it is movedway from its equilibrium position, a restoring capillary force actso bring it back. However, if the capillary force is not sufficiently-arge to overcome the buoyant weight, a balance of the forces in theirection normal to the interface is not possible and the particle isot trapped in the interface. This is normally the case for millimeternd larger particles that are heavier than the liquid below. Micron-nd nanoparticles, on the other hand, for which the buoyant weights negligible compared to the capillary force, are readily trapped athe interface.

Furthermore, the vertical capillary and pressure forces must alsovercome the momentum of the particle, which it possesses beforeoming in contact with the liquid surface. Owing to the fact that theapillary force acting on a particle varies linearly with the particleize, and the buoyant weight and the momentum vary as the thirdower of the particle radius, smaller particles are more readily cap-ured at the interface. In experiments, particles were released onlyfew millimeters from the interface to reduce the momentum at

ontact.The motion of a particle during adsorption is dominated by iner-

ia, and so it overshoots the equilibrium height (see Fig. 3). Although

he viscous drag causes the particle to slow down, its magnitude isot large enough to stop the particle completely and consequentlyhe momentum of the particle carries it below the equilibriumeight. When the particle moves below its equilibrium height, theapillary force reverses its direction and acts in the same direction

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uology 9 (2011) 1–13 3

s the drag. Hence, after moving down some additional distance,he particle reverses its direction, leading to several oscillationsnd the generation of interfacial waves before the particle comes toest. This behavior of the particle is similar to that of under-dampedass–spring–dashpot systems.The strength of the force holding a particle at the interface can

e quantified in terms of the interfacial energy released when thearticle is trapped. It can be shown that when the deformationf the interface due to the trapped particle is negligible, which ishe case for micron and nanoparticles, the adsorption energy, i.e.,he decrease in the interfacial energy due to the adsorption of apherical particle of radius R, is given by (see Aveyard, Binks, andlint (2003)):

a = �R2�(1 + cos ˛)2, (1)

here ˛ is the contact angle, as defined by Young-Dupré law:os ˛ = (�p2 − �p1)/� , and �pi is the interfacial tension between theth fluid and the particle, and � is the interfacial tension betweenhe upper and lower fluids. For larger, heavier particles, the inter-ace between the two fluids deforms to balance the buoyant weightf the particles which increases the area between the two fluids,nd thus the adsorption energy holding the particles at the interfaces smaller than the above value.

The main driving forces for the motion of a particle normal to thenterface (after it comes in contact with the interface) are the ver-ical capillary force and the particle’s buoyant weight. The viscousrag resists the particle’s motion. The acceleration of the particlender the action of these forces can be written as:

dV

dt= Fst + FD + Fg, (2)

here m is the effective mass of the particle which includes thedded mass contribution, V is the particle velocity, Fst is the verticalapillary force, FD is the drag, and Fg is the gravity force.

It is noteworthy that since for R > 10 nm the adsorption energys several orders of magnitude larger than kT, where k is the Boltz-

an constant and T the temperature, the capillary force acting onuch particles is much stronger than the Brownian force, and thushe latter need not be included in Eq. (2). The Brownian force on arapped particle, which arises because of the random thermal fluc-uations in the fluid, can influence the adsorption process of a smallarticle when Wa is comparable to kT. For the air–water interface,

f the contact angle is 90◦, Wa = 109kT for R = 10 �m, Wa = 105kTor R = 100 nm, and Wa = 10kT for R = 1 nm. Therefore, since Wa

ecreases as the square of the particle radius, the Brownian forcehould be considered only for nanometer sized particles.

From Fig. 4, it is apparent that the vertical component of theapillary force (FC) depends on the particle radius R, the interfacialension � , the filling angle �c and the contact angle ˛, and is giveny (see Singh and Joseph (2005) for details)

C = −2�R� sin �c sin(�c + ˛) (3)

he above expression holds for all values of the contact angle, i.e.,or both the hydrophobic and hydrophilic cases.

The right-hand side of Eq. (2) was approximated in Singh et al.2009) by assuming that the particle is spherical, the contact angles equal to its equilibrium value, the drag force is given by the Stokesormula, and the buoyancy force depends on the particle’s verticalosition, to obtain

dV

dt= −2�(R sin �c)� sin(�c+˛)−6�R�V�(s) + Q (�p − �c)g. (4)

ere �(s) is a parameter that accounts for the dependence of therag on the fraction s of the particle that is immersed in the lowernd upper fluids and on the viscosities of the fluids, Q is the particle

4 S. Gurupatham et al. / Particuology 9 (2011) 1–13

Fig. 3. Trapping (or adsorption) of a particle at an interface (left and middle). The test partforce (right). The particle oscillates about the equilibrium height. These oscillations of thmove away. Our experiments show that tracer particles remain trapped at the interface ansmaller than the test particle’s maximum velocity normal to the interface.

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ig. 4. Schematic of a heavier-than-liquid hydrophilic (wetting) sphere of radius Ranging on the contact line at �c. The point of extension of the flat meniscus on thephere determines the angle �1 and h2 is defined as h2 = R (cos�c − cos�1). The angleis fixed by the Young-Dupré law and �c by the force balance.

olume, �c is the effective fluid density which changes with s whilehe particle moves in the direction normal to the interface, and �p

s the particle density.Eq. (4) was solved numerically in Singh et al. (2009) to obtain the

article’s position z as a function of time (see Fig. 5). The obtainedolution was qualitatively similar to that obtained using the DNSpproach and in experiments discussed in this paper, i.e., the par-icle oscillates about its equilibrium height several times while themplitude of the oscillations decays with time because of viscousrag (see Fig. 6 and Singh et al. (2009) for details).

Eq. (4) can be linearized about the particle’s equilibrium positiono show that it is equivalent to a mass–spring–dashpot system (seeingh et al., 2009). Assuming that ˛ = �/2, �c = �/2 and �(s) = 1/2,

fter linearization, we obtained

43

R3�pd2Z

dt2+ 3R�

dZ

dt+ 2�Z + R2(�p − �c) g Z = 0, (5)

ig. 5. The z-coordinate of the particle center obtained numerically by solving Eq.4) is shown as a function of time. The particle oscillates about the equilibriumosition (z = 0) before coming to rest. The amplitude of oscillations decreases with

ncreasing time. The parameters are the same as in Fig. 6. Taken from Singh et al.2009).

V

Iaampi∼i∼

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Tca

icle comes in contact with the interface and is pulled downwards by the interfaciale particle give rise to a flow on the interface that causes small tracer particles tod move away from the test particle with a velocity which is an order of magnitude

here Z is the particle’s position. The solution of this ordinary dif-erential equation can be written as:

= Z0ekt, (6)

here k = (−3R� ± √D)/((8/3)R3�p) and

= 9R2�2 − (16/3)R3�p(2� + R2(�p − �c)g). The nature of theolution depends on the sign of the discriminant D. If the sign isositive, then k is real and negative for both of the roots. In thisase, the solution decays exponentially with time to zero. Thiss expected to be the case when the fluid viscosity is sufficientlyarge. If the sign of D is negative, then k is complex and the solutions oscillatory. In this case, the frequency of the oscillation is giveny

=3√

−9R2�2 + (16/3)R3�p

(2� + R2

(�p − �c

)g)

16�R3�p. (7)

he real parts of both roots are negative and so both of the solutionsecay exponentially to zero. The time constant � of the solution, i.e.,he time taken by the solution to decay by a factor of e−1, is giveny

= 8R2�p

9�. (8)

t is clear that time constant decreases with decreasing particleize and with increasing viscosity. Therefore, the vertical oscilla-ions of a trapped particle decay faster when the radius is smallernd the viscosity is larger. Notice that as R becomes small, there iscritical value of R for which D becomes positive.

Eq. (4) can be integrated to obtain the particle velocity when itasses through its equilibrium position for the first time (see Singht al. (2009) for details).

=−(9/4)� +

√(81/16)�2 + 4R�p

((3/2)� + 2R2

(�p − �c

)g)

2R�p.

(9)

n obtaining above equation we have assumed that the contactngle is 90◦ and that in equilibrium the center of the particle ist the undeformed interface. The above equation implies that theaximum velocity attained by a particle increases with decreasing

article radius. For example, a particle of diameter 200 �m (whichs roughly the size of a sand particle) can accelerate to a velocity of1 m/s at the water surface, and a particle of diameter 10 nm, which

s roughly the size of a virus or a protein molecule, to a velocity of

40 m/s. In the limit of R approaching zero, the velocity is given by

= 2�

3�. (10)

his is the maximum velocity that can be attained by a parti-le under the action of the vertical capillary force which for their–water interface is 46.7 m/s.

S. Gurupatham et al. / Particuology 9 (2011) 1–13 5

Fig. 6. The velocity distribution on the midplane of the computational domain; the deformed interface around the particles is also shown. The particle radius was 0.1 cmand the contact angle was 85◦ . The initial height of the two particles was 0.95R above the undeformed fluid interface. The initial lateral distance between the particles was3.2R. The densities of the particle, the upper fluid and the lower fluid were 0.5, 0.1 and 1.0 g/cm3, respectively, and the interfacial tension was 10.0 dyn/cm. The viscositiesof the upper and lower fluids were 0.1 and 1.0 cP. The interface near the particles deformed to meet the contact angle condition. The resulting vertical capillary force pulledthe particles downwards. The particles motion also caused the interface to deform and waves to develop. The resulting flow caused the two particles to move apart. For thefi etersn e fluida ve api

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nal figure, the distance between the particles was 6.19R. The dimensionless paramumber Re = 14.6, G = 368.2, Capillary number Ca = 7.3 × 10−4. (i) The particles, and thre moving upwards and away from each other. They continued to oscillate and monformation for the paper at the journal’s website. Taken from Singh et al. (2009).

. Experimental setup

The liquids used in this study were Millipore water, corn oilnd decane. Millipore water was used to ensure that contami-ants were not present as even when their concentration on their–water interface is very small they might change the interfa-ial tension and the contact angle of the liquid. Furthermore, glassarticles were thoroughly rinsed in water and then dried for sev-ral hours at the temperature of 70 ◦C in an oven to overcome thenfluence of any residual moisture which could influence the con-act angle and hence the position of the three-phase contact linen the particle’s surface. It may be noted that water and decane,nd water and corn oil, are immiscible, and that the three liq-ids have different densities which resulted in the formation oforizontal liquid layers. For example, since decane is lighter thanater, the decane was in the upper layer and the water in the

ower layer. The densities of water, corn oil and decane are 1000,22 and 726 kg/m3, respectively. The viscosities of water, corn oilnd decane are 1.0, 65.0 and 0.92 cP, respectively. The interfacialension was 72.4 mN/m between air–water, 51.2 mN/m betweenecane–water, and 33.2 mN/m between corn oil–water.

The horizontal positions of particles were recorded using aigital video camera connected to a Nikon Metallurgical MEC600icroscope and the vertical positions of particles were recorded

sing a high-speed camera (Casio Exilim F1) mounted on the side, ashown in Fig. 7. The latter positions evolved much more rapidly and

herefore a high speed camera was needed to resolve the motion.or example, the frequency of oscillation during adsorption for thearticles investigated in this study was approximately between 20nd 120 Hz.

ig. 7. Schematic of the experimental setup used to study the dispersion of particlesn a fluid–liquid interface.

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based on the numerically computed lateral velocity of the particles are: Reynoldsaround them, are moving downwards and away from each other. (ii) The particles

art. An animation showing this is available in Singh et al. (2009) as the supporting

The distance between the particles was measured by analyz-ng the movies frame-by-frame with a calibrated digital ruler. Thearticles were released very close to the surface of the upper liq-id (about 1 mm above the surface). This was done to ensure thatheir speed before touching the interface was small. The verticalnd horizontal positions of particles were measured as a functionf time by analyzing the video recordings. The fluid velocity at thenterface was measured by tracking small tracer particles trappedn the interface.

. Transient motion of particles during their adsorption

Our analysis and direct numerical simulations show that whilearticles are being trapped on the surface of a mobile liquid theyscillate about their equilibrium positions before reaching a statef rest, and that this results in a radially-outward flow on the inter-ace away from the particle which causes tracer and other particlesn the interface to move away. To investigate these oscillations andhe resulting interfacial flow, we analyzed the video recordings ofhe motion of particles after they came in contact with the inter-ace. The particle size was varied between approximately 5 �m andmm. The behavior was investigated for the air–water, oil–waternd decane–water interfaces. We first describe the case when aingle particle comes in contact with a fluid–liquid interface whichs followed by a discussion of the cases when two particles and aluster of particles come in contact with the interface.

.1. Adsorption of a single particle

The motion of a 2 mm spherical plastic bead from the time itame in contact with the decane–water interface is shown in Fig. 8.he bead released in the upper liquid slowly sedimented to theecane–water interface, and once it came in contact with the inter-ace it was pulled downwards by the vertical capillary force. Theead continued to move downward even after reaching the equi-

ibrium height. However, when this happened the vertical capillaryorce reversed its direction, and thus after travelling some addi-ional distance the direction of bead’s motion also reversed. Theead oscillated three times about its equilibrium position before

ts motion became indiscernible. Since the bead overshoots andscillates about the equilibrium position before stopping, we mayonclude that its motion is inertia-dominated and similar to that ofn underdamped mass–spring–dashpot system. The motion of the

ead also caused ring-shaped interfacial waves that moved awayrom the bead and slowly dissipated.

A similar behavior was observed for a plastic bead releasedbove the air–water interface (see Fig. 9) and a mustard seedeleased above the decane–water interface (see Fig. 10). Notice that

6 S. Gurupatham et al. / Particuology 9 (2011) 1–13

Fig. 8. Trapping of a spherical plastic bead of 2 mm diameter on the decane–water interface. The bead oscillated about its equilibrium position before its motion stopped.The sequence shows the phenomenon from the time the bead touched the interface to the time it reached the equilibrium position.

Fig. 9. Trapping of a plastic bead of 2 mm diameter on the air–water interface. The bead oscillated about its equilibrium position before its motion stopped. Notice that theequilibrium floating height of the bead is lower than in Fig. 8 for the decane–water interface.

S. Gurupatham et al. / Particuology 9 (2011) 1–13 7

Fig. 10. Trapping of a mustard seed of 1.36 mm diameter on the decane–water interface. The mustard seed oscillated about its equilibrium position before its motion ceased.

Fig. 11. Trapping of a 650 �m glass particle on the air–water and decane–water interfaces. (a) The dimensionless vertical positions (Z/R) as a function of time. Here Z ismeasured from the undeformed interface. Notice that the oscillatory behavior is similar to that in Fig. 5. (b) The contact lines on the air–water and decane–water interfacesfor the particle.

8 S. Gurupatham et al. / Particuology 9 (2011) 1–13

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Fig. 14. The velocity of tracer particles on the air–water interface plotted as afunction of the distance (d) from the center of a test glass particle. The velocitydistribution plotted here is at a time 0.033 s after the particle was trapped at theid

coiaTwtTft

tiicTtd

ig. 12. The dimensionless vertical positions (Z/R) as a function of time after the par-icles come in contact with the interface. (a) Air–water interface, (b) decane–waternterface.

ince the mustard seed was hydrophobic its floating height waselatively greater than that of the plastic bead. The behavior of aighter plastic bead (lighter than the lower liquid) which rose tohe air–water interface, was also similar. Notice that the frequencyf oscillation for these cases was 20 Hz or larger, and therefore aigh speed camera was needed to see and analyze the motion.

The photographs shown in Figs. 8–10 were taken from high-peed movies of particles undergoing adsorption at the fluid–liquidnterfaces. These movies were also analyzed frame-by-frame tobtain the dimensionless distance of the center of particles (Z/R)rom the undeformed interface as a function of time. The lat-

er results for a 650 �m glass bead are shown in Fig. 11 for their–water and decane–water interfaces. Fig. 11(a) shows that thequilibrium height of the center of the particle relative to thendeformed interface is lower on the air–water interface. This

ig. 13. The frequency of oscillation of spherical glass particles on the decane–waternd air–water interfaces versus the particle diameter. Both experimentally mea-ured frequencies and that given by Eq. (7) (indicated by “th”) are shown.he parameter values in Eq. (7) are assumed to be: �p = 2600.0 kg/m3 andP − �c = 16000 kg/m3; for the air–water interface � = 0.001 Pa s, �12 = 72.4 mN/m;nd for the decane–water interface � = 0.001 Pa s, �12 = 51.2 mN/m.

wfac

aka

Fto

nterface. The data were taken for 3 different test particles of the same approximateiameter of 550 �m.

an be also seen in Fig. 11(b) which shows that the particle floatsn the decane–water interface such that a smaller fraction of its immersed in the water, whereas on the air–water interface

larger fraction of its lower surface is immersed in the water.his is expected since for the same floating height the buoyanteight of the particle on the air–water interface is larger, and

hus to balance its weight it is more immersed in the lower liquid.he angle of the three-phase contact line on the particle’s sur-ace is another important parameter, but its value is not knowno us.

It is noteworthy that even after the vertical oscillations ofhe particle subsided, its floating height on the decane–waternterface slowly decreased before reaching a constant value. Thiss due to the partial pinning of the contact line on the parti-le’s surface and the contact angle hysteresis (see Fig. 11(a)).his issue is discussed below in more detail. Also notice thathe amplitude of oscillation of the particle was larger on theecane–water interface. This is because the densities of decane andater are closer than the densities of air and water, and there-

ore the restoring buoyant force resulting from a displacementway from the equilibrium position is smaller for the decane–waterase.

When the diameter of glass particles in our experiments waspproximately 650 �m or larger a significant fraction of the sprin-led particles were not captured on the corn oil–water interfacend those that were captured did not disperse. This was also the

ig. 15. The velocity of a tracer particle on the air–water interface initially at a dis-ance of 2.31 mm from a glass test particle of diameter 550 �m shown as a functionf time. The velocity became negligibly small at t = ∼0.24 s.

S. Gurupatham et al. / Particuology 9 (2011) 1–13 9

F of twoi

cgwoo

ig. 16. A sequence of photographs showing the trapping, dispersion and clusteringnterface.

ase for millimeter-sized mustard seeds and plastic beads. Smallerlass particles were captured, and as discussed below, after theyere captured they dispersed on the interface. The vertical motion

f these particles during their trapping is not discussed becauseur present experimental setup did not allow us to monitor their

ms

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mustard seeds on fluid–liquid interfaces. (a) Decane–water interface, (b) air–water

otion in the direction normal to the interface on account of theirmall size.

Fig. 12 shows the vertical motion of glass particles with diame-ers between 580 �m and 2 mm. The floating height depends on theensities of the particles and the liquids, and on the contact angle.

10 S. Gurupatham et al. / Partic

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ig. 17. The gap (D) between two mustard seeds, shown as a function time after theyame in contact with the air–water and decane–water interfaces. The diameter ofustard seeds was ∼1.3 mm. The maximum gap and the maximum velocity were

arger on the air–water interface.

he figure shows that the floating height increased with decreasingiameter, and the time taken to reach equilibrium decreased withecreasing diameter.

The floating height on the decane–water interface of particlesmaller than approximately 1 mm decreased slowly before reach-ng a constant value. This was not the case for the same particlesn the air–water interface. We believe that this is due to the facthat when a particle smaller than 1 mm moved downward in theecane–water interface, the three-phase contact line became par-ially pinned on the particle’s surface increasing the contact anglebove the equilibrium value. This in turn increased the vertical cap-llary force making the net vertical force on the particle zero evenhough the particle was above its equilibrium height. As the con-act line slowly moved down on the particle’s surface, the contactngle was reduced and the particle moved downward.

A comparison of the two cases described in Fig. 11(a) also showshat the time taken by the particle to oscillate once about thequilibrium height, i.e., the inverse of which is the frequency ofscillation, is larger on the decane–water interface. Since the effec-ive interfacial viscosity is larger when the upper fluid is decane andhe interfacial tension and the buoyant weight of the particle forhe decane–water interface are both smaller than the correspond-ng values for the air–water interface, the experimental result forhe frequency of oscillation is consistent with Eq. (7) according tohich the frequency decreases with decreasing interfacial tension,ecreasing buoyant weight, and increasing viscosity.

We also investigated the dependence of the frequency of oscilla-ion of glass particles on the air–water and decane–water interfacesn their diameter. These results are shown in Fig. 13. For both inter-aces the frequency increased with decreasing particle size. Therequency of oscillation was larger on the air–water interface thanf the same particle on the decane–water interface. These resultsre in agreement with our analysis presented in Section 2 (see Eq.7)). It is noteworthy that Eq. (7) contains only the fluid and particleroperties, and that there are no adjustable parameters.

.1.1. Flow induced on the interfaceTo investigate the fluid motion induced at the interface due to

he adsorption of a test particle, the interface was seeded with00 �m sized glass tracer particles. The tracer particles were small

ompared to the test particle so that they did not significantly influ-nce the fluid motion caused by the test particle. The velocity ofracer particles decreased with increasing distance from the testarticle (see Fig. 14) and also decreased with time (see Fig. 15). Theotal distance traveled away from the test particle depended on its

iwtpi

uology 9 (2011) 1–13

nitial distance from the test particle. From Figs. 12 and 15 we notehat even after the vertical oscillations of an ∼580 �m particle sub-ided at t = ∼0.03 s, tracer particles on the interface continued toove apart for t < ∼0.22 s. This shows that the flow induced at the

nterface by a test particle persists even after the particle attainsertical equilibrium.

The same trend for interfacial fluid velocity was observed whenwo identical test particles were dropped simultaneously onto their–water interface, but the velocity was almost double that ofhen a single glass particle was dropped (see Singh et al. (2009)

or details).

.2. Simultaneous adsorption of two particles

Two particles released simultaneously and near each otherbove a fluid–liquid interface were trapped at the interface by theame mechanism by which a single particle was trapped. Specifi-ally, they were pulled into the interface and oscillated verticallyeveral times before the amplitude of oscillation became negligiblymall (see Fig. 16). In addition, the particles moved apart from eachther along the line joining their centers.

Fig. 16(a) shows the adsorption of two mustard seeds of theame approximate size on the decane–water interface. After theyouched the interface, they were pulled inwards by the vertical cap-llary force which was followed by vertical oscillations about theirquilibrium positions. During this time they also started to movepart because of the hydrodynamic force and the interfacial flowesulting from the particles motion normal to the interface. Thescillations decayed after some time, but the particles continuedo move apart because of the induced interfacial flow.

After the particles stopped moving apart, they clustered backogether under the action of lateral capillarity forces which ariseecause of the deformation of interface caused by the particles.he behavior of two mustard seeds released onto the air–waternterface, as shown in Fig. 16(b), was qualitatively similar, excepthat the velocity with which they moved apart and their maxi-

um separation were larger than on the decane–water interface.lso notice that in both cases the particles started to move apart

mmediately after they came in contact with the interface, but thisecame apparent only after approximately one vertical oscillation.

The gap between the particles for the above two cases is showns a function of time in Fig. 17. Notice that the particles reachedheir maximum lateral velocity shortly after they came in con-act with the interface. The figure also shows that the velocityith which they initially moved apart was larger than the velocityith which they later approached each other. This implies that the

orces that cause the initial dispersion are stronger than the lat-ral attractive capillary forces that arise because of their buoyanteight.

From Figs. 16 and 17, we also note that the time interval forhich the particles oscillated vertically after coming in contactith a fluid–liquid interface was much smaller than the time inter-

al for which they moved apart. Thus, the interfacial flow causedy the particles persisted, and continued to move nearby particlespart, even after their vertical oscillations become indiscernible.

.3. Adsorption of particle clusters

We next describe dispersion of small clusters of particles whenhey come in contact with the corn oil–water and decane–water

nterfaces. Glass particles of diameter ranging from 5 to 120 �m

ere used in this study. Particles were sprinkled onto the surface ofhe upper liquid where they were allowed to cluster, and then wereushed downward making them sediment to the liquid–liquid

nterface.

S. Gurupatham et al. / Particuology 9 (2011) 1–13 11

Fig. 18. Dispersion of 45 �m glass spheres on the corn oil–water interface. The figure shows that particles reach the interface at different times and that they disperseviolently as they come in contact with the interface. The particles trapped on the interface are in focus and those above the interface are out of focus. Initially, particles areabove the interface, at t = 10 s some of the particles have reached the interface, and at t = 23 s most of the particles are trapped at the interface.

Fig. 19. Dispersion of 20 �m glass spheres on the corn oil–water interface. The figure shows that particles disperse as they come in contact with the interface. After all ofta

he particles were trapped on the interface and the dispersive forces subsided, the particls capillary forces are relatively weaker.

Fig. 20. Dispersion of 5–8 �m glass spheres on the corn oil–wat

es clustered under the action of lateral capillary forces. The cluster is rather porous

er interface. Particles disperse as they reach the interface.

12 S. Gurupatham et al. / Particuology 9 (2011) 1–13

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ig. 21. Dispersion of 45 �m glass spheres on the decane–water interface. In the firhe decane–water interface which makes them out of focus. Subsequently, they bec

Fig. 18 shows dispersion of 45 �m glass spheres on the cornil–water interface. This situation is different from when sprinkledarticles fall rapidly through the air onto a liquid surface. Sincehe particles of the cluster sediment at different velocities, theyo not reach the corn oil–water interface at the same time, but

nstead over a few seconds (also see Figs. 19–21). The moment par-icles came in contact with the interface of corn oil and water, theyispersed radially into an approximately circular region. Since par-icles falling through the corn oil took several seconds to reach thenterface, the dispersion process on the interface continued for aonger time than on the air–water interface. Also, the speed with

hich particles dispersed was smaller than on the air–water inter-ace because of the higher viscosity and because the particles didot reach the interface at the same time. Particles already trappedn the interface remained dispersed until additional particles fellnto the interface.

When all of the particles were captured at the interface, theylustered under the action of lateral capillary forces. The speed withhich particles clustered was smaller than the speed with which

hey dispersed (see Fig. 19).Furthermore, the packing density of particles within a cluster

ecreased with decreasing particle size (see Figs. 18–20). This isecause the strength of capillary forces decreases with decreasingarticle size and so particles do not pack tightly leaving many voidpaces within the cluster. The smaller-sized particles disperse moreeadily, but since they are smaller it is difficult to observe themndividually. Also, the smaller-sized particles took longer to reachhe interface and thus the fall time onto the interface was longer.

e also observed that when more particles reached the interfaceogether, the dispersion speed and the radius of the area into whichhey spread were larger because each particle contributed to theutward dispersion of the cluster and hence the resulting flow onhe interface was stronger.

Fig. 21 shows dispersion of glass particles on the decane–waternterface. As the viscosity of decane is an order of magnitudemaller than that of corn oil, the particles sedimented relativelyuickly and all of the particles reached the interface in a shorterime. As a result, the velocity with which they dispersed after com-ng in contact with the interface was larger. After the dispersiveorces subsided, the particles clustered under the action of lateralapillary forces.

. Conclusion and discussion

Experiments reported in this paper show that when particlesome in contact with a liquid–liquid interface they spontaneouslyisperse as they do on an air–liquid interface. Specifically, weonducted experiments in which glass and other particles with

iameters ranging from 5 �m to 4 mm were sprinkled onto

iquid–liquid interfaces. The upper liquid in these studies wasecane or corn oil and the lower liquid was water. Particles fellhrough the upper liquid onto the interface where they dispersedhile remaining trapped at the interface. All of the particles men-

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tograph most of the particles are on the decane surface, but the camera focus is onisible as they reached the decane–water interface where they dispersed.

ioned above were captured and dispersed on the decane–waternterface. However, on the corn oil–water interface, only glass par-icles smaller than ∼650 �m dispersed; larger glass and plasticarticles and mustard seeds did not disperse. In fact, a significantraction of the larger particles were not even captured at the inter-ace. This perhaps is due to a smaller interfacial tension of the cornil–water interface.

Analyses of high-speed video recordings show that the time forhich a test particle oscillated vertically after coming in contactith the interface was much shorter than the time interval forhich tracer particles placed on the interface for flow visualizationoved away from the test particle. Furthermore, the lateral veloc-

ty of tracer particles was an order of magnitude smaller than theest particle’s velocity in the normal direction to the interface. Thishows that the test particle causes a flow on the interface away fromhe test particle and that this flow persists even after the particleeaches vertical equilibrium.

When the upper fluid was a liquid, and not a gas, particles fell tohe interface slowly due to the higher viscosity of the upper liquid,nd did not all reach the interface around the same approximateime. Since particles reached the interface over an interval of time,he dispersion occurred over a longer time interval and was rela-ively weaker than for the case when the upper fluid was a gas. Theate of dispersion on the corn oil–water interface was weaker thann the decane–water interface as the decane viscosity is smallerhan the corn oil viscosity.

The frequency of vertical oscillation of a particle increased withecreasing particle size on both air–liquid and liquid–liquid inter-aces. The frequency on the decane–water interface was slightlymaller than on the air–water interface. For a ∼500 �m particle therequency on the decane–water interface was around 100 Hz. Ourxperimental technique did not allow us to measure the frequencyf particles that were smaller than ∼500 �m. The results for therequency of particles between 500 �m and 3.0 mm diameter weren agreement with our analytical result for the frequency given byq. (7). This agreement is noteworthy since the only parametersontained in Eq. (7) are the properties of the fluids and the particle,.e., it contains no adjustable parameters.

Our experiments have also shown that for small particles theartial pinning of the contact line on the particle’s surface is impor-ant. When this happened the particle did not oscillate verticallybout its equilibrium floating height, but instead about a heighthat was higher while continuing to slowly move downward in thenterface as the contact line receded downward on its surface. Theinning of the contact line occurred only when the particle size wasmall, and since our present experimental setup did not allow us totudy the motion of particles smaller than ∼500 �m in the normalirection to the interface, this issue will be investigated in a future

tudy.

After the forces causing dispersion subsided, particles clusterednder the action of lateral capillary forces. For the two-particle case,he time taken to cluster was about three times larger than the timeaken by them to move apart and the maximum velocity for the

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atter was about six times larger, indicating that the forces causingispersion are stronger than those causing clustering. Similarly, thepeed of dispersion is much larger than the speed of clustering.

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