Transient Flow Induced by the Adsorption of Particles

Abstract

The paper describes the physics of particle adsorption and the spontaneous dispersion of powders that occurs when they come in contact with a fluid-liquid interface. The dispersion can occur so quickly that it appears explosive, especially for small particles on the surface of mobile liquids like water. Our PIV measurements show that the adsorption of a spherical particle at the interface causes an axisymmetric streaming flow about the vertical line passing through the particle center. The fluid directly below the particle rises upward, and near the surface, it moves away from the particle. The flow, which develops within a fraction of second after the adsorption of the particle, persists for several seconds. The flow strength, and the volume over which it extends, decrease with decreasing particle size. The streaming flow induced by the adsorption of two or more particles is a combination of the flows which they induce individually. The flow not only causes particles sprinkled together onto a liquid surface to disperse, but also causes a hydrodynamic stress which is extensional in the direction tangential to the interface and compressive in the normal direction. These stresses can cause the breakup of particle agglomerates when they are adsorbed on a liquid surface.

1. Introduction

The focus of the paper is to describe the physics of particle adsorption and the spontaneous dispersion of powders that occurs when they come in contact with a fluid-liquid interface. While past studies have been concerned with understanding the mechanisms by which particles already trapped on fluid-liquid interfaces interact leading to their self-assembly into monolayered patterns (Kralchevsky and Nagayama, 2000; Kralchevsky et al., 1992; Nicolson, 1949), (Singh et al., 2010, and the references therein), the sudden dispersion of particles coming into contact with a fluid-liquid interface described in this paper has not been considered prior to our recent study (Singh et al., 2009) and (Gurupatham et al., 2011, 2012).

It was shown in (Singh et al., 2009) that (i) particles sprinkled over a small area almost instantaneously spread over an area that can be several orders of magnitudes larger (see Fig. 1); (ii) a newly-adsorbed particle causes particles already trapped on the interface to move away creating a particle-free region around itself (see Fig. 2); and (iii) dispersion influences the nature, e.g., structure and porosity, of the monolayer clusters that are formed. These phenomena have importance in a wide range of applications, such as pollination in hydrophilous plants, transportation and spreading of microbes and viruses, and the self-assembly of particles leading to the formation of novel nano-structured materials, stabilization of emulsions, etc. (Aveyard et al., 2003; Binks and Horozov, 2006; Cox, 1988; Cox and Knox, 1989; Dryfe, 2006; Gust et al., 2001; Pickering, 1907; Tang et al., 2006; Wasielewski, 1992).

Fig. 1

Sudden dispersion of flour sprinkled onto water in a dish. Streaklines were formed due to the radially-outward motion of the particles emanating from the location where they were sprinkled. The size of flour particles was ∼2–100 μm. Taken from (Gurupatham et al., 2011)

Fig. 2

Trapping (or adsorption) of particles at an interface. (a) The particle comes in contact with the interface. (b) The particle is pulled downwards by the interfacial force (γ₁₂). (c) The particle oscillates about the equilibrium height within the interface causing a radially outward flow on the interface. Both the side and top views are shown. The lengths of the arrows are not proportional to the flow strength. (d) (Left) A glass sphere of diameter 1.1 mm being dropped onto a monolayer of 18 μm tracer glass particles on the surface of a 60% glycerin in water. (Right) The flow on the surface causes all of the nearby tracer particles to move away so that a roughly circular, particle-free region is created.

The dispersion can occur so quickly that it appears explosive, especially on the surface of mobile liquids like water. An experiment showing this can be performed easily in a household kitchen by filling a dish partially with water and then sprinkling a small amount of a finely-ground powder such as wheat or corn flour onto the water surface. The moment the flour comes in contact with the surface it quickly disperses into an approximately-circular shaped region, forming a monolayer of dispersed flour particles on the surface (see Fig. 1). The interfacial forces that cause this sudden dispersion of flour particles are, in fact, so strong that a few milligrams of flour sprinkled onto the surface almost instantaneously covers the entire water surface in the dish.

It was shown by (Singh et al., 2009) and (Gurupatham et al., 2011) that the initial dispersion of particles is due to the fact that when a particle comes in contact with the interface, the vertical capillary force pulls it into the interface, thereby causing it to accelerate in the direction normal to the interface (see Fig. 2). The maximum velocity increases with decreasing particle size; for nanometer-sized particles, e.g., viruses and proteins, the velocity on an air-water interface can be as large as ∼47 m/s. Also, since the motion of a particle on the surface of mobile liquids like water is inertia dominated, it oscillates vertically about its equilibrium height before the viscous drag causes it to stop. This gives rise to a streaming flow on the interface away from the particle (see Fig. 2c).

The energy needed to pull a particle into the interface and to induce the streaming flow comes from the net decrease in the interfacial energy (W_a) due to the adsorption of the particle (Singh et al., 2009). By assuming that the particle floats without significantly deforming the interface, it can be shown that W_a for a spherical particle of radius R is   

$$W_{\text{a}}=\pi R^2{\gamma }_{12}{\left(1+\text{cos}\theta \right)}^2$$(1)

where θ is the contact angle and γ₁₂ is the interfacial tension between the upper and lower fluids (see Binks and Horozov, 2006 and Tsujii, 1998).

Therefore, when two or more particles are simultaneously adsorbed at the interface, each of the particles causes a streaming flow on the interface away from itself thereby causing the other particles to move away. When two particles are adsorbed the maximum distance by which they move apart is about a few diameters. However, as the number of particles being adsorbed increases, the distance travelled by a particle, especially if it is near the outer periphery of the cluster, can be very large—several orders of magnitude larger than any dimension of the area over which particles are sprinkled. For example, as noted above, a few milligrams of flour sprinkled over a very small area on the water surface almost spontaneously disperses to cover the entire water surface in the dish.

The dispersion phase, which lasts for a short period of time (about one second for the case described in Fig. 1), is followed by a phase that is dominated by attractive lateral-capillary forces during which particles slowly come back to cluster. The latter phase has been a focus of many past studies (Fortes, 1982; Kralchevsky et al., 1992; Lucassen, 1992; Nicolson, 1949). Particles trapped at a fluid-fluid interface generally interact with each other via attractive capillary forces that arise because of their weight. A common example of this capillarity-driven self-assembly is the clustering of breakfast-cereal flakes floating on the surface of milk. This mechanism is widely used for two-dimensional assembly of particles at liquid surfaces. However, if the buoyant weight of particles is negligible, as is the case for colloidal particles, then the particles will only disperse since the attractive capillary forces between them are negligible (Aubry et al., 2008; Bresme and Oettel, 2007; Singh and Joseph, 2005; Stamou and Duschl, 2000).

One may postulate that the gradient of particle concentration gives rise to a (Maringoni) force that causes particles to disperse because the concentration of particles in the region in which they are sprinkled is larger than in the surrounding region. However, this is clearly not the case, as the dispersed particles cluster again under the action of lateral capillary forces. This shows that the gradient of particle concentration does not give rise to a dispersion force and that particles disperse only when they come in contact with the interface for the first time (during which each of the particles gives rise to a flow away from itself as they are pulled into the interface).

Also, one may postulate that the dispersion is because of the presence of contaminants on the surface of particles which are released into the liquid when they come in contact with the interface and their presence in fact causes particles to disperse. If this is the case, then the intensity with which the particles disperse should diminish when they are washed. We ruled out this possibility by repeatedly washing the particles and showing that their dispersive behavior did not change when the experiments were repeated (Gurupatham et al., 2011).

Particles also disperse on liquid-liquid interfaces. In fact, as Fig. 3 shows, the dispersion forces can break apart agglomerates of micron-sized particle that remain intact in the upper liquid (Gurupatham et al., 2012). This breakup and spreading of particle clumps (agglomerates) on liquid surfaces is important in various processes in the pharmaceutical and food industries such as wet granulation and food processing (Nguyen et al., 2010; Tüskea et al., 2005; Zajic and Buckton, 1990).

Fig. 3

Breakup and dispersion of agglomerate on the interface of corn oil and water, looking down from above (500× mag.). The size of glass particles of the agglomerate was ∼4 μm. (left) An agglomerate sedimented through corn oil and was captured at the interface. (right) After coming in contact with the interface it breaks apart explosively dispersing radially-outward into an approximately-circular region. Notice that some of the particles remained agglomerated. Taken from (Gurupatham et al., 2012).

The sudden dispersion of particles plays an important role also in some physical processes occurring on fluid-fluid interfaces, including the rate at which germs/microbes disperse on a water surface. An example in botany is the formation by hydrophilous (water-pollinated) plants of floating porous pollen structures called “pollen rafts” (Cox, 1988; Cox and Knox, 1989). A crucial first step in their formation is the initial dispersion of pollen that occurs after it comes in contact with the water surface (if it did not disperse, it would remain clumped/agglomerated).

In the next section, a description of the forces that act on a particle during adsorption and of the pertinent dimensionless parameters is given. The experimental set-up and results of our PIV measurements are presented in Sections 3 and 4. Finally, we summarize and conclude in Section 5.

2. Governing equations and dimensionless parameters

As discussed above, when a particle comes in contact with a fluid-liquid interface it is pulled inwards from the upper fluid into the interface with the lower fluid by the capillary force to its equilibrium position in the interface. It is crucial to understand this motion of the particle in the direction normal to the interface, as it gives rise to the streaming flow on the interface away from the particle.

The motion of the particle can be obtained by solving the governing equations for the two fluids and the momentum equation for the particle, which are coupled, along with the interface stress condition and a condition for the contact line motion on the particle surface. This is a formidable problem because the capillary force at the line of contact of the three phases on the particle surface depends on the slope of the interface which in general requires the solution of the aforementioned equations (Pillapakkam and Singh, 2001; Singh et al., 2003; Singh and Joseph, 2005). However, a decoupled momentum equation for a particle can be derived by modeling the forces that act on the particle (Singh et al., 2011; Singh et al., 2009). These forces are: the vertical capillary force (F_st), the buoyant weight (F_g), the Brownian force (F_B), and the viscous drag (F_D). A decoupled equation for the motion of the particle under the action of these forces can be written as:   

$$m\frac{\text{d}V}{\text{d}t}={\mathbf{F}}_{\text{st}}+{\mathbf{F}}_{\text{D}}+{\mathbf{F}}_{\text{g}}+{\mathbf{F}}_{\text{B}}$$(2)

where m is the effective mass of the particle which includes the added mass contribution (Currie, 1974), V is the velocity, F_st = 2πRγ₁₂sin(θ_c)sin(θ_c+α) is the capillary force, α is the contact angle, and θ_c is the particle position in the interface. The Brownian force in Eq. (2) is negligible compared to the capillary force (Singh et al., 2011; Singh et al., 2009).

The added mass and the drag calculations are complicated by the fact that the fraction of the particle that is immersed in the upper and lower fluids changes as the particle moves in the direction normal to the interface. We will assume that the added mass is one half of the mass of the fluid displaced, but this result is for a particle fully immersed in a fluid (Currie, 1974). The drag force will be assumed to be given by F_D = 6πμRV f_D, where μ is the viscosity of the lower liquid, and f_D is a parameter that accounts for the particle being immersed in both upper and lower fluids.

2.1 Governing dimensionless parameters

Let the characteristic velocity, length and time be given by U = γ₁₂/μ, R, and R/U, respectively. Then, Eq. (2) can be nondimensionalized to give (Singh et al., 2011):   

$$\begin{array}{l}\mathit{We}•m^{\prime }\frac{{\rho }_{\text{p}}}{\rho }\frac{\text{d}{V}^{\prime }}{\text{d}{t}^{\prime }}=\text{sin}\left({\theta }_{\text{c}}\right)\text{sin}\left({\theta }_{\text{c}}+\alpha \right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+3V^{\prime }{\text{f}}_{\text{D}}+\frac{2}{3}B\frac{{\rho }_{\text{p}}-{\rho }_{\text{c}}}{\rho }{f}_{\text{b}}.\end{array}$$(3)

Here the primed variables are dimensionless. f_b is the dimensionless buoyancy which is O(1) but depends on the profile of the deformed interface. ρ and ρ_a are the densities of the lower and upper fluids, ρ_c is the effective density of the volume displaced by the particle, and ρ_p is the particle density. The dimensionless parameters in the above equation are: the Weber number $\mathit{We}=\frac{2}{3}\frac{\rho R{\gamma }_{12}}{{\mu }^2}$, the Bond number B = ρR²g/γ₁₂, $\frac{{\rho }_{\text{p}}}{\rho }$, and the contact angle α. The Weber number is the ratio of inertia forces to capillary forces, and the Bond number is the ratio of gravitational forces to surface tension forces.

For an air-water interface, the parameters have the values: μ = 0.001 Pa.s, ρ = ρ_p = 1000 kg/m³, $\frac{{\rho }_{\text{p}}-{\rho }_{\text{c}}}{\rho }=0.1$ and γ₁₂ = 0.07 N/m. Let us assume that μ_a = ρ_a = 0, m′ = 1.5, f_D = 0.5 and f_b = 1. Then, We = ∼10⁸R and B = ∼10⁵R², where R is in meters. Therefore, the role of particle inertia becomes negligible only when R is much smaller than 10 nm because only then We is much smaller than one. The influence of gravity becomes negligible when R < ∼1 mm in the sense that such small particles float so that the interfacial deformation is negligible. However, even a negligibly-small deformation of the interface gives rise to attractive lateral capillary forces which, even though small, cause floating particles to cluster. This happens because a particle floating on a liquid surface is free to move laterally. The only resistance to its lateral motion is the hydrodynamic drag which can slow the motion but cannot stop it. Consequently, only very-small particles, for which lateral capillary forces are smaller than Brownian forces, do not cluster.

3. Experimental Setup

The setup consisted of a square Petri dish which was partially filled with Millipore water (see Fig. 4). The cross-section of Petri dish was 10 × 10 cm, and the depth was 1.5 cm. PIV measurements were performed in a vertical plane (normal to the camera axis) illuminated by a laser sheet. The vertical position of the camera was in line with the water surface, providing an undistorted view of the volume directly below the water surface. The test particles were released very close to the liquid surface, about 1 mm from the surface, in an area near the intersection of the laser sheet and the camera axis.

Fig. 4

Schematic diagram of the experimental setup.

A high-speed camera was used to record the motion of seeding particles visible in the laser sheet. A Nikon 1 series V1 camera equipped with a 30 mm Kenko automatic extension tube and a Tamron SP AF 60 mm 1:1 macro lens was used to provide the required magnification. The laser sheet was generated using a ZM18 series 40 mw solid state diode laser of wavelength 532 nm (green color). Movies were recorded at a resolution of 1280 × 720 pixels. For the particle size range considered (∼500 μm to 2 mm), the optimal recording speed for performing the PIV analyses was found to be 60 frames per second. This was determined by a trial and error procedure.

The water was seeded with silver-coated hollow glass spheres of density around 1 gm/cc and average size of around 8–12 μm. The density of seeding particles closely matched the water density, but there was a small particle-to-particle variation. Consequently, some particles sedimented and some rose slowly giving us ample time to record their motion when a flow was induced due to the adsorption of one or more test particles. The seeding particles were silver coated which ensured that the intensity of the scattered light was sufficient to track their motion.

An open-source code, PIVlab, was used for performing the time-resolved PIV analysis. PIVlab is a MatLab-based software which analyzes a time sequence of frames to give the velocity distribution for each of the frames. A MatLab code for post-processing and plotting results was written.

4. Results

We first discuss our PIV measurements of the transient flow on a water surface that was induced due to the adsorption of a single test particle. Glass particles of three different diameters, 2.0, 1.1, and 0.65 mm were used to obtain the qualitative nature of the flow, and determine how the strength and time duration of the induced flow vary with the particle size.

In agreement with the analytic results obtained in (Gurupatham et al., 2011), test particles in all cases oscillated vertically before reaching their equilibrium positions in the interface. The frequency of oscillation increased, and the adsorption time decreased, with decreasing particle size in agreement with the analytic results. For example, the frequencies for the diameters 2.0, 1.1 and 0.65 mm were 25 Hz, 50 Hz and 85.71 Hz respectively.

The adsorption of a test particle caused a flow on the air-water interface, which caused tracer particles trapped on the surface to move away from the adsorbed test particle. Consequently, the water surface near the test particle had few tracer particles which made fluid velocity measurement at and near the water surface difficult. Also, the air-water interface near the test particles was deformed since their density was larger than the water density. In fact, the center of particles was a fraction of radius below the position of the undeformed interface. The deformation of the interface made viewing of the interface by a camera mounted on a side difficult (see Fig. 4). Therefore, in our PIV measurements, the velocity was measured only in the region below a horizontal line passing through the point of contact of the interface with the particle which was a fraction of the particle radius below the undeformed interface (see Fig. 5).

Fig. 5

(Left) The origin of the coordinate system was at the intersection of the vertical lines passing through the center of the test particle and the horizontal line passing through the point of contact of the interface with the particle. The center of the test particle was a fraction of particle radius below the position of the undeformed interface. (Right) Velocity vectors for the streaming flow induced by a 2 mm test particle 0.67s after it came in contact with an air-water interface. The particle was dropped in a vertical plane illuminated by a thin sheet of laser light. The velocity vectors of tracer particles have been superimposed on the PIV image, and the interface is marked by a horizontal white colored line. A purple-colored mask was used in the PIV analysis to define the region occupied by the test particle. The velocity distribution was approximately axisymmetric about the vertical passing through the center of the particle.

Although the water near the test particle started to move as soon as the particle came in contact with the surface, the adsorption-induced streaming flow intensity developed over a period of time. The intensity reached a maximal strength after a fraction of a second and then it decreased. In the time interval after which the streaming flow reached its maximal strength, the vertical oscillations of the test particle were already negligible. The PIV measurements show that the streaming flow was approximately axisymmetric about the vertical line passing through the center of the test particle (see Fig. 5). Tracer-particles in the region below the test particle moved upwards, and those near and in the water surface moved away from the test particle. The trajectories of fluid particles were qualitatively similar to that for a stagnation point flow, with the center of the test particle being the stagnation point. This implies that the stress on the test particle due to the induced flow was extensional in the horizontal plane near the water surface, and compressive in the direction normal to the surface.

Since the induced velocity field was approximately axisymmetric, it can be conveniently quantified in terms of its y-component along the vertical line passing through the center of the test particle and the x-component along the x-axis as defined in Fig. 5. Time was measured from the instant at which the test particle came in contact with the water surface and the distance was measured from the origin of the coordinate system. The former was identified by a frame-by-frame analysis the movie images. Fig. 6a shows the y-component of velocity for a 650 μm test particle along the vertical line passing through its center at five different times; the x-component of velocity along this line was relatively small. The y-component of velocity was positive, indicating that the flow was in the upward direction towards the particle. The fluid velocity near the surface of the test particle was small, as the test particle was not moving, and increased with increasing distance from the particle reaching a maximal strength at a distance of about one particle radius from the surface. The velocity then decreased with increasing distance from the particle, but remained significant for a distance of several diameters.

Fig. 6

Temporal evolution of the streaming flow induced by the adsorption of a 650 μm particle. The velocity is shown at five different time intervals after the test particle came in contact with the water surface. (a) The vertical component of velocity (V) is shown as a function of −y/R. (b) The horizontal component of velocity (U) is shown as a function of x/R.

The figure also shows that the fluid velocity did not develop instantaneously after the particle was adsorbed. For example, at t = 0.167 s, the maximum velocity of 3.10 mm/s was at a distance of 0.78 mm from the particle, and the velocity at a distance of 5.5 mm was only 0.09 mm/s. The velocity increased with time to reach the maximum value of 8.9 mm/s at t = 0.37 s and y = −1.04 mm. At y = −5.5 mm, the velocity at this time was 1.07 mm/s. After reaching the maximum strength, the fluid velocity started to decrease. The decrease first occurred closer to the test particle, while it was still increasing farther away from the particle. For example, the maximum velocity at t = 0.82 s was 7.12 mm/s at y = −1.3 mm, and at t = 1.25 s and y = −1.82 it was 5.47 mm/s. The velocity at larger distances from the particle continued to increase for a longer time interval before starting to decrease. At a distance of y = −5.5 mm, the velocity at t = 1.58 s was 1.60 mm/s, which was larger than the velocity at this location at t = 0.37 s. The streaming flow slowly reduced in strength but continued for several seconds.

Fig. 6b shows the x-component of velocity for a 650 μm test particle along a horizontal line at five different times; the y-component of velocity along this line was negligible. The velocity was positive which means that the water near the surface was moving away from the particle. As in Fig. 6a, the fluid velocity near the test particle was small because it was not moving, and increased with increasing distance from the particle and then after reaching a maximal value it decreased with increasing distance. However, the maximal velocity was at a distance of about two particle diameters from the particle surface, whereas below the particle the maximum was reached at a distance of one particle radius. The velocity remained significant for a distance of several diameters. The maximum water velocity near the surface was comparable to the maximum velocity below the test particle.

The temporal evolution of streaming flow near the water surface was qualitatively similar to that in the water below the test particle. The flow started when the particle touched the water surface, and reached a maximal value after a time interval which was comparable to that below the particle. At t = 0.167 s, the maximum velocity of 1.91 mm/s was at a distance of 2.1 mm from the particle, and the velocity at a distance of 5.5 mm was 1.1 mm/s. The velocity increased with time to reach the maximum value of 8.2 mm/s at t = 0.37 s and x = 1.82 mm. At this time, the velocity at x = 5.5 mm was 4.4 mm/s. This shows that the velocity near the water surface was more intense over a larger area than below the particle. The strength of the streaming flow then decreased with time with the decrease first occurring closer to the test particle.

The time interval after which the streaming flow attained the maximal intensity and the volume over which the flow extended varied with the particle size. The velocity distribution for a 2 mm particle is shown Fig. 7. The figure shows that the streaming flow evolution was qualitatively similar to that for a 650 μm particle described above. However, it developed relatively more slowly. The maximum velocity of 13.5 mm/s was reached 1.58 s after the particle came in contact with the water surface. For a 650 μm particle, on the other hand, the maximal velocity was reached at t = 0.38 s. For a 2 mm particle, not only the maximum velocity was larger, it occurred at a larger distance of 1.38 mm from the test particle, and so the volume over which the flow was intense was larger than for a 650 μm particle. Our PIV measurements of the three particle sizes considered show that the time interval after which the maximal flow strength was attained, the volume over which the flow intensity extends, as well as the time interval for which the flow persists, increase with increasing particle size.

Fig. 7

Temporal evolution of the streaming flow induced by the adsorption of a 2 mm particle. The velocity is shown at five different time intervals after the test particle came in contact with the water surface. (a) The vertical component of velocity (V) is shown as a function of −y/R. (b) The horizontal component of fluid velocity (U) is shown as a function of x/R.

4.1 Adsorption of two or more particles

We next consider the case in which two 650 μm particles were simultaneously adsorbed at an air-water interface. The particles were dropped onto the interface such that the line joining their centers was in the plane of the laser sheet. This ensured that the induced flow was approximately symmetric about the vertical plane passing though the centers of the two particles, and also about the vertical plane bisecting the line joining their centers. Fig. 8 shows that each of the particles induced a streaming flow which was similar to that which was induced by a single particle. The combined flow below the particles was in the upward direction and near the water surface the flow was away from the particles. In the region between the particles, the horizontal flow contributions approximately cancelled and the vertical contribution added. Thus, the combined streaming flow was approximately the sum of the flows induced by the two particles individually, and thus stronger than the flow induced by one particle. The combined flow developed in about 0.4 s which was comparable to the time in which the streaming flow developed for a single 650 μm particle.

Fig. 8

The figure shows the streaming flow induced by two 650 μm test particles after they came in contact with an air-water interface. The particles were dropped in a vertical plane illuminated by a thin sheet of laser light. The velocity vectors have been superimposed on the PIV images. The flows induced by the particles caused them to move apart. The size of velocity vectors is arbitrary, and for clarity a larger magnification is used in the plot at t = 0.033 s than in the later plots.

For the case shown in Fig. 8, the two particles were initially close to each other and so the streaming flow induced by the first particle caused the second one to move away, and vice versa. The symmetry of the streaming flow with respect to the laser sheet ensured that the two particles remained in the plane of the laser sheet while they moved apart. However, when the particles were dropped so that the angle between the line joining their centers and the laser sheet was not small, the streaming flow carried them away out of the plane of the laser sheet, and so they were visible only for the time duration for which they were illuminated. For the case shown in Fig. 8, the speed at t = 0.167 s was approximately 12.3 mm/s. The speed decreased as they moved farther apart, and also with time as the streaming flow intensity diminished with time. In fact, after the distance between them was about 10R their speeds became negligible.

We also considered cases where about 10–30 particles were dropped onto an air-water interface in and near the laser sheet (see Fig. 9). Again, each of the particles induced a streaming flow in the water that was similar to the flow induced by a single particle. These flows induced by the particles caused neighboring particles to move away, and so the net result was that particles moved radially outward from the location where they were dropped. Experiments show that clusters of particles disperse radially outward from the center (see Fig. 1 which shows streak lines), and that when the cluster size was larger the radius of the approximately-circular area over which its particles dispersed and the dispersion velocity were larger. This increase in the dispersion velocity with increasing number of particles was also seen in our direct numerical simulations (Singh et al., 2009). The PIV measurements show that the water rises in the region below the particles and on the surface it moves away from the particle.

Fig. 9

The figure shows the streaming flow induced by about 20, 650 μm particles after they came in contact with an air-water interface. The particles were dropped in and near a vertical plane illuminated by a thin sheet of laser light, and towards the left side of the photographs. The figure shows the motion of the particles that moved to the right side (some of the particles moved in other directions and so are invisible in the photographs). The velocity vectors have been superimposed on the PIV images. The size of velocity vectors is arbitrary.

5. Conclusion and discussion

When a particle comes in contact with a fluid-liquid interface the vertical capillary force pulls it into the interface which gives rise to a transient streaming flow. The PIV measurements show that the liquid below a newly-adsorbed particle rises upwards and the liquid near the surface moves away from the particle. The induced flow for a spherical particle was axisymmetric about the vertical line passing through the particle center. Also, the flow strength is not established immediately after the particle comes in contact with the interface, but builds up over a short time interval. For a 650 μm glass sphere the maximum flow strength occurred about 0.4 s after the particle come in contact, and for a 2 mm sphere after about 1.5 s. We also considered 1.1 mm, 850 mm and 550 mm glass spheres for which the maximal flow strength occurred after 0.75 s, 0.47 s and 0.18 s, respectively. These results show that the time interval after which the maximal flow strength occurred decreased with decreasing particle size.

When two or more particles were simultaneously adsorbed, the streaming flow was a combination of the flows induced by the particles individually and so the flow strength increased with increasing number of particles. Consequently, the distance travelled by the particles near the outer periphery of a cluster sprinkled on a liquid surface can be several orders of magnitude larger than any dimension of the area over which the particles were sprinkled. This can be important for some physical processes occurring on a water surface, such as the pollination of hydrophilous plants, and the transportation and rate of spread of microbes and viruses on a water surface. Furthermore, the streaming flow can break apart agglomerates of particles when they are adsorbed at a fluid-liquid interface which is important in various processes in the pharmaceutical and food industries such as wet granulation and food processing.

Acknowledgements

We gratefully acknowledge the support of the National Science Foundation (CBET-1236035).

Author’s short biography

Naga Musunuri

Naga A Musunuri is a graduate student in the department of Mechanical and Industrial Engineering at New Jersey Institute of Technology (NJIT), New Jersey, USA. He received his B.S. from Jawaharlal Nehru Technological University (JNTU), India in 2009 and M.S. from New Jersey Institute of Technology (NJIT) in 2011. His current research interests include PIV measurements, adsorption, fluid mechanics, biological fluid mechanics, microfluidics, and Micro Electro Mechanical Systems (MEMS).

Bhavin Dalal

Bhavin Dalal completed his Ph.D. in Mechanical Engineering from the New Jersey Institute of Technology in 2012. He received his M.S. from Bradley University, Illinois, USA and his B.S. from Gujarat, India.

Daniel Codjoe

Daniel Codjoe graduated for St. Augustine’s College at Cape Coast in Ghana. He received his B.S. in Mechanical Engineering from New Jersey Institute of Technology in 2013. He is interested in working on the water and electricity issues in his native country Ghana, and other related mechanical engineering related problems.

Ian S. Fischer

Ian S. Fischer is Professor of Mechanical Engineering at New Jersey Institute of Technology. He received his B.S. from Columbia University in 1970, his M.S.E. from Princeton University in 1973, and his D. Eng. Sci. from Columbia University in 1985. His research interests include the flow of solid particles on fluid interfaces, and the dual-number modeling of mechanical systems. He is a member of the American Society of Mechanical Engineers.

Pushpendra Singh

Pushpendra Singh is Professor of Mechanical Engineering at New Jersey Institute of Technology. He received his B.S. form IIT Kanpur in 1985, and M.S. and Ph.D. from the University of Minnesota in 1989 and 1991, respectively. His research interests include Newtonian and non-Newtonian fluid mechanics, biological fluid mechanics, microfluidics, direct numerical simulations and modeling of multiphase flows, including those on fluid-liquid interfaces, and electrohydrodynamics. He is a fellow of the American Physical Society and the American Society of Mechanical Engineers.

References

•  Aubry  N.,  Singh  P.,  Janjua  M.,  Nudurupati  S., Assembly of defect-free particle monolayers with dynamically adjustable lattice spacing. In Proceedings of the National Academy of Sciences (Vol. 105, pp. 3711–3714), 2008.
•  Aveyard  R.,  Clint  J.H.,  Horozov  T.S., Aspects of the stabilization of emulsions by solid particles: Effects of line tension and monolayer curvature energy. Phys. Chem. Chem. Phys., 5 (2003) 2398–2409.
•  Binks  B.P.,  Horozov  T.S., Colloidal particles at liquid interfaces: An introduction:, Cambridge University Press, 2006.
•  Bresme  F.,  Oettel  M., Nanoparticle at fluid interfaces, J. Phys.: Condens. Matter, 19 (2007).
•  Cox  P.A., Hydrophilous pollination, Annu. Rev. Ecol, Syst, 19 (1988) 261–280.
•  Cox  P.A.,  Knox  R.B., Two-dimensional pollination in hydrophilous plants: Convergent evolution in the genera Halodule (Cymodoceaceae), Halophila (Hydrocharitaceae), Ruppia (Ruppiaceae), and Lepilaena (Zannichelliaceae), Amer. J. Bot, 176 (1989) 164–175.
•  Currie  I.G., Fundamentals mechanics of fluids, New York, McGraw Hill, 1974.
•  Dryfe  R.A.W., Modifying the liquid/liquid interface: pores, particles and deposition, Phys. Chem. Chem. Phys., 8 (2006) 1869–1883.
•  Fortes  M.A., Attraction and repulsion of floating particles, Can. J. Chem, 60 (1982) 2889.
•  Gurupatham  S.K.,  Dalal  B.,  Hossain  M.,  Fischer  I.,  Singh  P.,  Joseph  D.D., Particles Dispersion on Fluid-Liquid Interfaces, Particuology, 9 (2011) 1–13.
•  Gurupatham  S.K.,  Dalal  B.,  Hossain  M.,  Fischer  I.,  Singh  P.,  Joseph  D.D., Breakup of particle clumps on liquid surfaces, Powder Technology, 217 (2012) 288–297.
•  Gust  D.,  Moore  T.A.,  Moore  A.L., Mimicking Photosynthetic Solar Energy Transduction, Acc. Chem. Res., 34 (2001) 40.
•  Kralchevsky  P.A.,  Nagayama  K., Capillary interactions between particles bound to interfaces, liquid films and biomembranes, Advances in Colloid and Interface Science, 85 (2000) 145–192.
•  Kralchevsky  P.A.,  Paunov  V.N.,  Ivanov  I.B.,  Nagayama  K., Capillary Meniscus Interactions between Colloidal Particles Attached to a Liquid—Fluid Interface, J. Colloid Interface Sci, 151 (1992) 79–94.
•  Lucassen  J., Capillary forces between solid particles in fluid interfaces, Coll. Surf, 65 (1992) 131.
•  Nguyen  T.H.,  Eshtiaghi  N.,  Hapgood  K.P.,  Shen  W., An analysis of the thermodynamic conditions for solid powder particles spreading over liquid surface, Powder Technology, 201 (2010) 306–310.
•  Nicolson  M.M., The interaction between floating particles, Proc. Cambridge Philosophical Soc., 45 (1949) 288.
•  Pickering  S.U., Emulsions, Journal of the Chemical Society, London, 91 (1907) 2001.
•  Pillapakkam  S.B.,  Singh  P., A Level Set Method for computing solutions to viscoelastic two-phase flow, J. Computational Physics, 174 (2001) 552–578.
•  Singh  P.,  Hesla  T.I.,  Joseph  D.D., A Modified Distributed Lagrange Multiplier/Fictitious Domain Method for Particulate Flows with Collisions, Int. J. of Multiphase Flows, 29 (2003) 495–509.
•  Singh  P.,  Joseph  D.D., Fluid dynamics of Floating particles, J. of Fluid Mech., 530 (2005) 31.
•  Singh  P.,  Joseph  D.D.,  Aubry  N., Dispersion and Attraction of Particles Floating on fluid-Liquid Surfaces, Soft matter, 6 (2010) 4310–4325.
•  Singh  P.,  Joseph  D.D.,  Fischer  I.,  Dalal  B., The role of particle inertia in adsorption at fluid-liquid interfaces, Physical Review E83 041606 (2011).
•  Singh  P.,  Joseph  D.D.,  Gurupatham  S.K.,  Dalal  B.,  Nudurupati  S., Spontaneous dispersion of particles on liquid surfaces, Proceedings of the National Academy of Sciences, 106 (2009) 19761–19764.
•  Stamou  D.,  Duschl  C., Long-range attraction between colloidal spheres at the air-water interface: The consequence of an irregular meniscus, Physical Review (2000) 5263–5272.
•  Tang  Z.,  Zhang  Z.,  Wang  Y.,  Glotzer  S.C.,  Kotov  N.A., Self-Assembly of CdTe Nanocrystals into Free-Floating Sheets, Science, 314 (2006) 274–278.
•  Tsujii  K., Surface Activity: Principles, Phenomena, and Applications, San Diego, Academic Press, 1998.
•  Tüskea  Z.,  G  R.J.,  Erős  I.,  Srčičb  S.,  Pintye-Hódi  K., The role of the surface free energy in the selection of a suitable excipient in the course of a wet-granulation method, Powder Technology, 155 (2005) 139–144.
•  Wasielewski  M.R., Photoinduced electron transfer in supramolecular systems for artificial photosynthesis, Chem. Rev, 92 (1992) 435.
•  Zajic  L.,  Buckton  G., The use of surface energy values to predict optimum binder selection for granulations, International Journal of Pharmaceutics, 55 (1990) 155–164.