Transport of a Sessile Aqueous Droplet over Spikes of Oil Based

May 29, 2019 - Here, we elucidate the transport of a sessile aqueous (diamagnetic) droplet placed over spikes of oil-based ferrofluid (FF) in the pres...
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Article Cite This: Langmuir 2019, 35, 8238−8245

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Transport of a Sessile Aqueous Droplet over Spikes of Oil Based Ferrofluid in the Presence of a Magnetic Field C. Mandal, U. Banerjee, and A. K. Sen* Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600036, India

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S Supporting Information *

ABSTRACT: Droplets can be used as carrier vehicles for the transportation of biological and chemical reagents. Manipulation of water- and oil-based ferromagnetic droplets in the presence of a magnetic field has been well-studied. Here, we elucidate the transport of a sessile aqueous (diamagnetic) droplet placed over spikes of oil-based ferrofluid (FF) in the presence of a nonuniform magnetic field. An oil-based FF droplet, dispensed over a rigid oleophilic surface, interacts with a magnetic field to get transformed into an array of spikes which then act as a carrier for the transportation of the aqueous droplet. Our study reveals that transportation phenomena is governed by the interplay of three different forces: magnetic force Fm, frictional force Ff, and interfacial tension force Fi, which is expressed in terms of the magnetic Laplace number (Lam) and magnetic Bond number (Bom) as Lam−1 = (Ff1/Fm,x) and BomLam−1 = (Ff2/Fi). Based on the values of the dimensionless numbers, three different regimes, steady droplet transport, spike extraction, and magnet disengagement, are identified. It is found that steady droplet transport is observed for Lam−1 ≤ 1 and BomLam−1 ≤ 1, whereas extraction of spikes is observed for Lam−1 ≤ 1 and BomLam−1 > 1 and magnet disengagement is observed for Lam−1 > 1. In the steady droplet transport regime, velocity of the aqueous droplet Uds was found to be dependent on the volumes of the aqueous droplet Vw and FF droplet VFF following Uds ∼ Vw−0.19VFF0.36. A simple model is presented that accurately predicts the aqueous droplet velocity Uds within 5% of the corresponding experimental data. In the spike extraction regime, the spike extraction distance Lse was found to vary with Vw, VFF, and the magnet velocity Ums following Lse ∼ Vw−1.75VFF0.75Ums−1.56.

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optical force,13 whereas passive methods employ the surface morphology to execute manipulations, e.g., Laplace pressure gradient,14 surface wettability,14,15 and chemically heterogeneous surfaces.16 Magnetic-field-based manipulation, being noncontact and biocompatible, exhibits prodigious advantages over other active manipulation techniques. Depending on the magnetic behavior of objects, magnetic manipulation can be classified as positive and negative magnetophoresis. Positive magnetophoresis17 embraces manipulation of paramagnetic or ferromagnetic objects inside a diamagnetic medium. On the contrary, negative magnetophoresis18 demonstrates manipulation of diamagnetic objects inside a paramagnetic or ferromagnetic

INTRODUCTION

Precise manipulation of aqueous droplets is receiving enormous attention from the microfluidic community due to its unique features such as low sample volume and cost effectiveness and for various applications such as cell sorting1 and biochemical analysis.2 Moreover, droplets can be used as the carrier vehicle for the transportation of biological3 and chemical reagents.4 The area of droplet microfluidics can be classified as open and closed channel microfluidics.5 Droplets in open geometries deal with the manipulation of individual droplets (of volume nL to μL) on engineered surfaces. On the other hand, in closed geometries, droplets (of volume pL to nL) are continuously generated and manipulated in a confined space such as inside a microchannel.6−9 Drop manipulation techniques can be classified into active and passive methods. Active methods involve external fields to facilitate droplet manipulation, e.g., electrowetting,10 magnetic field,11,12 and © 2019 American Chemical Society

Received: March 3, 2019 Revised: May 14, 2019 Published: May 29, 2019 8238

DOI: 10.1021/acs.langmuir.9b00631 Langmuir 2019, 35, 8238−8245

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Figure 1. (a) Schematic of an oil-based ferrofluid (FF) droplet dispensed on a rigid oleophilic substrate. (b) Formation of an array of FF spikes after incorporation of a normal magnetic field. (c) Aqueous droplet dispensed over the array of FF spikes, encapsulating the FF spikes, where ϕ is the angle between the axis of magnetization (magnet) and the axis of the droplet. (d) Force balance for the respective system in various regimes. (e) Various manipulation regimes based on the force ratios acting on the system.

magnetic force and surface tension force governs the shape change and splitting of FF droplets in the presence of a permanent magnet. Aqueous FF droplets dispensed on a superhydrophobic surface split28,29 when a perpendicular magnetic field (above a critical value) is applied. The droplets self-assemble (simple patterns) under a static magnetic field and can be switched to complicated patterns by applying a time varying magnetic field. A similar behavior is found where an oil based ferrofluid droplet splits30 into daughter droplets when a perpendicular magnetic field is applied at the bottom of the droplet. Since most of the techniques demonstrate manipulation of water- or oil-based ferrofluid droplets or droplets containing magnetic particles, the manipulation of aqueous droplets using magnetic fluids has not been addressed. Although several techniques32 have been reported for the manipulation of droplets using magnetic fields on various surfaces, from a practical application point of view, manipulation of aqueous droplets without the inclusion of magnetic particles in the aqueous medium is desired to prevent contamination.31 A recent study reported controlled motion of aqueous droplets33 immersed in oil based ferrofluid using two orthogonal rotating magnetic fields. The in-plane rotating magnetic field magnetizes the magnetic tracks which results in local magnetic field gradients, while the orthogonal field magnetizes the bulk ferrofluid. Kinematics of aqueous droplet motion and bulk fluid flow of ferrofluid was studied. However, the amount of bulk ferrofluid used in the above study is much higher thus making recovery of the aqueous droplets or contents present therein extremely challenging. Although, manipulation of aqueous droplets using aqueous based ferrofluid has been reported,34 in that case recovery of aqueous droplets is challenging. Here, we report the transport of aqueous droplets over spikes of oil-based FF in the presence of a normal magnetic field. We study the various forces that contribute toward the droplet motion and identify the various regimes based on the relative magnitudes of these forces. The effects of different parameters such as aqueous droplet volume, ferrofluid droplet volume and magnet velocity on the droplet transport behavior is studied.

medium. Magnetic medium can be synthesized by mixing magnetic particles or paramagnetic salts into a nonmagnetic medium (oil/water/organic solvents). Microdroplets containing magnetic particles19 or magnetic marbles20 can be manipulated on superhydrophobic (SH) surfaces using magnetic fields. It was found that, at concentrations ranging between 0.1 and 10% (by weight), these microparticles form chains, which is responsible for the droplet manipulation. Manipulation of droplets containing paramagnetic salts on SH surfaces using permanent magnets has been studied,21 where several kinds of paramagnetic salts of different molar magnetic susceptibilities have been used to show a parametric comparison. A detailed study22 reported magnetic manipulation of oil coated aqueous droplets in which magnetic beads are internalized; three important operating regimes were found which delineate droplet motion. These magnetic particles can also be used to fabricate magnetic platforms for droplet manipulation on open surfaces. A recent study12 reports the fabrication of a soot-wax coated SH surface in which iron particles are embedded in PDMS. A local deformation created due to the interaction between the permanent magnet and the iron particles drives the droplets on the surface. Due to the large size, magnetic particles agglomerate in the presence of magnetic field, and may get completely extracted from the carrier fluid, which is undesirable for practical applications. This issue can be conquered using ferrofluids in which magnetic nanoparticles are stabilized by surfactants to restrict the interparticle interactions. Ferrofluids (FF) are the colloidal suspensions of ferromagnetic particles (∼10 nm) suspended in oil or water exhibiting magnetization in the presence of a magnetic field. The high affinity of ferrofluids toward magnetic field opens up opportunity for several microfluidic applications.23−25 A detailed study26 reports the effect of magnetic field on shape, contact angle and motion of ferrofluid droplets on a planar surface. The apparent contact angle is found to reduce with increase in magnetic flux. Shape evolution and subsequent splitting of aqueous ferrofluid droplets on hydrophobic surface has been experimentally studied.27 Interplay between the 8239

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THEORETICAL ANALYSIS A schematic of the model system used for the present study is depicted in Figure 1(a-d). A droplet of an oil-based ferrofluid (FF) of volume VFF is dispensed on a rigid oleophilic substrate. Under the influence of an external magnetic field (due to the presence of a permanent magnet at a distance of g from the substrate), when the magnetic force overcomes the surface tension force, the FF droplet splits into an array of FF spikes.30 Since, the substrate is oleophilic, there is always a thin layer of oil (the base medium for the FF nanoparticles) between the FF spikes and the substrate. Next, an aqueous droplet of volume Vw is placed over the FF spikes that encapsulate the FF spikes. Further, when the magnet is moved horizontally at some velocity Um, the array of FF spikes tends to move along with it owing to the magnetic pull that carries the aqueous droplet placed over the spikes. We now analyze the various forces involved in the transport of the array of FF spikes and the aqueous droplet. The array of FF spikes present in a nonuniform magnetic field experiences magnetic force35 Fm, which is expressed as Fm =

VNPΔχ (B • ∇ B ) μ0

have estimated that Kf varies between 500 and 567 for aqueous droplet volumes in the range Vw = 2−8 μL. The movement of the FF spikes under the influence of a perpendicular magnetic field would also transport the encapsulating aqueous droplet along with it due to the interfacial tension force33 at the contact line between the FF spikes and the aqueous droplet, which can be expressed as Fi = 2Nlγi

where N is the total number of FF spikes present at the base of the aqueous droplet whose dependence on magnetic Bond number Bom and FF droplet diameter D is shown in Supporting Information section S.3 and Figure S.3a, l is the characteristic length of the FF spikes (taken as the slant height (l) of the FF spikes shown in Figure S.3a, which depends on the volume of the ferrofluid droplet VFF), and γi ≈ 48 mN/m is the interfacial tension between the aqueous phase and the ferrofluid. The interfacial tension force will tend to prevent any relative motion between the FF spikes and the aqueous droplet. When there is a relative motion between the FF spikes (due to the magnetic pull) and the aqueous droplet, there will be a friction force Ff2 at the contact line between the water droplet and the FF spikes, which will have a similar expression as that of Ff1 described above, i.e., Ff2 ≅ KfRbμFFUds, where Rb is the base radius of the aqueous droplet. It can be realized that when the horizontal component of the magnetic force Fm,x = Fm sin ϕ is greater than the friction force Ff1, the array of FF spikes will move to follow the magnet. With this condition satisfied, if the interfacial tension force Fi is greater than the frictional force Ff2, the aqueous droplet will get transported along with the FF spikes without any relative motion between them. Thus, the droplet transport regime can be characterized using the magnetic Laplace number Lam−1 = (Ff1/Fm,x) and magnetic Bond number BomLam−1 = (Ff2/Fi). The magnetic Laplace number (Lam) signifies the ratio of magnetic force to the friction force, while the magnetic Bond number (Bom) signifies the ratio of magnetic force to the interfacial tension force. Depending on the magnitude of the above two force ratios, different regimes will be observed, as shown in Figure 1e. The regime R1 indicates the “steady droplet transport regime” in which Lam−1 ≤ 1 and BomLam−1 ≤ 1. However, with Lam−1 ≤ 1 satisfied, if the interfacial tension force Fi is smaller than the frictional force Ff2, the FF spikes will get extracted out of the aqueous droplet and follow the magnet leaving the aqueous droplet behind. The regime R2 indicates the “spike extraction regime” in which Lam−1 ≤ 1 but BomLam−1 > 1. On the other hand, if the magnetic force Fm,x is smaller than the friction force Ff1, i.e., Lam−1 > 1, the FF spikes (and hence the aqueous droplets) will not follow the magnet, which is indicated by the regime R3, which is the “magnet disengagement regime”. The various regimes observed in our experiments and the corresponding force ratios are illustrated in detail in the Results and Discussion.

(1)

where VNP is the total volume of the nanoparticles present inside the array of FF spikes, which is calculated from the FF droplet volume and the nanoparticle concentration (11.8%), Δχ = χd − χf is the difference in magnetic susceptibility of aqueous droplet and ferrofluid ∼(− 10−5 − 6.79) ≈ − 6.79, B is the magnetic flux density (T), ∇B is the flux gradient (T m−1), and μ0 is the permeability of vacuum (4π × 10−7 H/m). To estimate the magnetic force Fm, the magnetic flux density B is obtained from the COMSOL multiphysics simulations, as presented in the Supporting Information (section S.1). The angle ϕ is defined as the angle between the axis of magnetization (magnet) and the axis of the droplet as shown in Figure 1c. Since the aqueous droplet encapsulates the FF spikes, the aqueous droplet can be carried with the FF spikes that are driven by the magnetic force. There will be a friction force, which is mainly attributed to the contact line pinning,22 acting at the contact line between the array of FF spikes (and the thin oil layer) with the substrate, which is expressed as Ff1 ≅ K f R dμFF Uds

(3)

(2)

where Kf is the friction constant, Rd is the base radius of the array of ferrofluid spikes present inside the aqueous droplet (shown in Figure 1c), μFF ≈ 8 mPa s is the viscosity of the ferrofluid, and Uds is the steady velocity of the aqueous droplet, which will be equal to the FF steady spike velocity in the stable droplet transport regime. The friction constant Kf is estimated 2πc(θ ) as36 K f = β ln(Λ /λ), where the dependence of c(θ) with

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contact angle θ is reported.36 For the surface considered in the present study, the value of c(θ) is 0.5 for θ = 70°. Here, Λ is a length scale and is a function of the FF base radius as Λ ∼ (Rd)n, where the value of n is obtained (n = 0.4) as a fitting parameter from our experimental data. We have taken λ ≈ 10, a cutoff molecular length scale which depends on the liquid molecules and the surface roughness and its variation does not considerably affect the experimental results as it appears in the logarithm, and β is a prefactor of ≈0.01. Using the above, we

EXPERIMENTAL SECTION

In our experiments, oil based ferrofluid (EMG901, Ferrotec) of particle (Fe3O4) concentration c = 11.8% (by volume), density ρ = 1430 kg/m3, and viscosity μ = 8 mPa s was used. The initial susceptibility of EMG 901 is χ = 6.79. A neodymium iron boron magnet (NdFeB, N52, K&J Magnetics Inc.) of dimension H = L = W = 0.47 cm and of residual flux Br = 1.48 T was used to provide the nonuniform magnetic field. To demonstrate our study, we have used an oleophilic and superhydrophobic substrate37 having an oil contact 8240

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Figure 2. (a) Time-lapse experimental images showing the different regimes for the magnetic manipulation of the FF spike−aqueous droplet system (also see Supporting Information Video S1). (b) Plot showing the different manipulation regimes based on magnetic Laplace number Lam and magnetic Bond number Bom, θ = 70°. angle θ = 70° for our experiments. In short, the PDMS monomer and curing agent (Sylgard 184, Silicone Elastomer Kit, Dow Corning) at a ratio of 10:1 (by weight) were mixed and degassed, and following that n-hexane (10% by weight) was stirred into the degassed mixture. Then, this mixture was spin coated at 4000 rpm for 10 s on a glass slide (sonicated in isopropyl alcohol followed by drying in N2) and was allowed to cure at 85 °C for 10 min. After curing, the glass slide was coated with soot particles using a candle flame for a duration of 5 min. The substrate was then cleaned by high speed water-jet to achieve a transparent superhydrophobic surface. The permanent magnet was mounted on a moving rack which was driven by a stepper motor through a pinion. The linear velocity of the rack was controlled by a linear speed regulator connected to the stepper motor. The permanent magnet was positioned by maintaining a gap g from the substrate using a nonmetallic z-stage (Thorlabs, Inc.). Micropipets (Eppendorf, Germany) of required capacity were used for dispensing various volumes of ferrofluid droplets (1−5 μL) and aqueous droplets (2−12 μL). The motion of ferrofluid, magnet, and aqueous droplets was captured by using a USB camera (Dinolite, Taiwan) and a highspeed camera (FASTCAM SA3 model, Photron, USA, Inc.) interfaced with a PC. To analyze the captured videos and images, ImageJ and PFV software were used. The surface tension of the oil-based FF and the aqueous phase and the interfacial tension between the two are measured as γFF = 23 mN m−1, γW = 72.8 mN m−1, γFF−W = 48 mN m−1, respectively. When an aqueous droplet is placed over the oil-based FF spikes, due to positive spreading parameter of the oil-based FF (SFF = (γW − γFF−W − γFF) > 0), the aqueous droplet is covered by a thin layer of oil-based FF. However, after the manipulation, the aqueous droplet can be easily recovered by using a polydimethylsiloxane (PDMS)-coated sponge, as illustrated from our experiments in the Supporting Information (Figure S.2) and described in detail elsewhere.38

droplet is characterized in terms of spike extraction distance Lse, and the magnet disengagement is briefly outlined. Magnetic Manipulation of the FF Spike−Aqueous Droplet System in Different Regimes. An aqueous droplet of volume Vw that needs to be transported is gently placed on top of the array of ferrofluid spikes. When the magnet is moved in the horizontal direction at some velocity Ums, depending on the competition between the various forces, namely, the magnetic force Fm,x, the interfacial tension force Fi, and the frictional force Ff, different regimes are observed. Contribution of various forces on the magnetic manipulation of the system is already introduced in the Theoretical Analysis section. As discussed, the various regimes are characterized based on the dimensionless numbers, Lam−1 = (Ff1/Fm,x) and BomLam−1 = (Ff2/Fi). Figure 2a shows time-lapse experimental images illustrating the various manipulation regimes. For a FF droplet volume VFF = 3 μL, aqueous droplet volume Vw = 4 μL, and magnet velocity Ums = 10.56 cm/s, steady transport of aqueous droplet is observed as shown in Figure 2a,i. In this case, the magnetic force Fm,x = 1.4 mN, interfacial force Fi = 0.69 mN, and frictional force Ff1 = Ff2 = 0.46 mN. The variation of angle ϕ with time t is shown in Figure S.4 in the Supporting Information. Here, we have used the smallest value of the angle, i.e., ϕ = 4° for the estimation of the minimum magnetic force required for the manipulation. We will discuss in the next section that the velocity of the FF spike Ud (which is equal to the aqueous droplet velocity) increases with time t (or distance x) until it becomes equal to the magnet velocity Um. Here, we have considered the maximum velocity of the FF spikes Uds (from experiments) for calculating the frictional force Ff. Our force calculations clearly show that the magnetic force Fm,x is adequate to overcome the frictional force Ff1 between the FF spikes and the surface. Also, the interfacial tension force Fi offered by the aqueous and FF interface is higher than the frictional force Ff2 at the contact line of the aqueous droplet and the FF spikes. For the present case, we calculate Lam−1 = 0.33 and BomLam−1 = 0.65, which satisfies the criteria Lam−1 ≤ 1 and BomLam−1 ≤ 1 (see Figure 1e) indicating steady droplet transport. Considering another case in which VFF = 3 μL, Vw = 8 μL, Ums = 10.03 cm/s, we obtain the magnetic force Fm,x = 2.31 mN, Fi = 0.8 mN, and Ff1 = 0.58 mN, which gives us Lam−1 = 0.25 and BomLam−1 = 0.73 and we obtain steady

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RESULTS AND DISCUSSION In this section, we first elucidate the various manipulation regimes exhibited by the aqueous droplet and FF system due to the motion of the magnet and illustrate the same based on the relevant forces acting on the FF spikes and aqueous droplet. Further, the effect of FF droplet volume VFF, aqueous droplet volume VW and the magnet velocity Um on the velocity of the aqueous droplet Ud is illustrated. Comparison between the predictions of a simple theoretical model for the aqueous droplet velocity and the corresponding experimental data is presented. Finally, the extraction of spikes from the aqueous 8241

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Figure 3. (a) Variation of the magnet velocity Um and aqueous droplet (Vw = 4 μL) velocity Ud (which is same as the FF spike velocity) for VFF = 3 μL with time t; variation of the location x of the magnet and the aqueous droplet (and the FF spikes) with time t is shown in the inset. (b) Variation of maximum velocity of aqueous droplets Uds with aqueous droplet volume Vw, for different ferrofluid volumes VFF, for VFF = 3 μL; a comparison between the model predictions and experimental data is also shown, θ = 70°.

1.13 mN, which gives us Lam−1 = 1.62 and BomLam−1 = 1.90, which again indicates the magnet disengagement regime. We can see here that a smaller ferrofluid volume and lower magnet velocity results in a lower magnetic force and friction force, respectively. By varying the volumes of FF and aqueous droplets, magnet velocity and contact angle, we observed that when Lam−1 > 1 (irrespective of the value of BomLam−1), magnet disengagement regime occurs, which is indicated by the regime R3 in Figure 3b. Dynamics of FF Spikes and Aqueous Droplets in Different Regimes. As discussed, the region Lam−1 ≤ 1 and BomLam−1 ≤ 1 in Figure 2 indicates the steady droplet transport regime (i.e., regime R1). Initially, at time t = 0, the magnet, FF spikes and the aqueous droplet are at rest. When the magnet starts moving at some velocity Um, if Lam−1 ≤ 1 and BomLam−1 ≤ 1, the FF spikes and the aqueous droplet will start moving and the velocity of the FF spikes and the aqueous droplet will gradually increase with time and attain a steady velocity Uds, which is equal to the magnet velocity Um. Figure 3a shows the variation of the magnet velocity Um and aqueous droplet velocity Ud (which is same as the FF spike velocity) with time t. The stepper motor used in our experiments for driving the magnet has rather a large time constant τ = 0.02 s due to which the magnet attains its set velocity of Ums = 22 cm/s only at t ≅ 0.1 s. The aqueous droplet (and the FF spikes) attains the steady velocity of Uds = 22 cm/s only at t ≅ 0.2 s. The variation of the location x of the magnet and the aqueous droplet (and the FF spikes) with time t are shown in in the inset of Figure 3a. The results show that, in the steady droplet transport regime, the aqueous droplet (and then FF spikes) safely follows the magnet. Next, we study the effect of aqueous droplet volume Vw and FF droplet volume VFF on the maximum droplet velocity Uds, which is same as the maximum allowable magnet velocity in the steady droplet transport regime. Obviously, this velocity would correspond to the points lying on the boundary of the regime R1. Figure 3b shows that for a fixed volume of the FF droplet VFF, the maximum aqueous droplet velocity Uds decreases with increase in the volume of the aqueous droplet Vw. In the steady droplet transport regime, considering that BomLam−1 ≤ 1 needs to be satisfied and the fact that Fi does not depend on the aqueous droplet volume Vw, the value of the frictional force Ff2 ≅ KfRbμffUd should not vary with Vw.

droplet transport. We varied the volume of the FF droplet in the range 1−5 μL, aqueous droplet in the range 2−12 μL, the magnet velocity in the range 10.03−24 cm/s and the contact angle θ = 70°. For all cases, we find that when Lam−1 ≤ 1 and BomLam−1 ≤ 1, steady droplet transport is achieved, which is indicated by the regime R1 in Figure 2b. Figure 2a,ii shows time-lapse experimental images for a system with a FF droplet volume VFF = 3 μL, aqueous droplet volume Vw = 4 μL, but a higher magnet velocity Ums = 25.9 cm/s. Here, we estimated the magnetic force Fm,x = 1.4 mN, interfacial force Fi = 0.69 mN, but a much higher frictional force Ff1 = Ff2 = 1.11 mN owing to the higher magnet velocity. We observe that the magnetic force Fm,x is still adequate to overcome the frictional force Ff1 between the FF spikes and the surface but the interfacial tension force Fi offered by the aqueous and FF interface is lower than the frictional force Ff2. We calculate Lam−1 = 0.80 and BomLam−1 = 1.6, that satisfies the criteria Lam−1 ≤ 1 and BomLam−1 > 1, indicating that the FF spikes would penetrate and move out of the aqueous droplet to follow the magnet, which we refer as the “spike extraction regime” (see Figure 1). By varying the volume of FF and aqueous droplets, magnet velocity, and contact angle, we observed that when Lam−1 ≤ 1 and BomLam−1 > 1, spike extraction takes place, which is indicated by the regime R2 in Figure 2b. Time-lapse experimental images for a system with a FF droplet volume VFF = 3 μL, aqueous droplet volume Vw = 4 μL, but a much higher magnet velocity Ums = 35.62 cm/s are presented in Figure 2a,iii. In this case, we estimated the magnetic force Fm,x = 1.4 mN, interfacial force Fi = 0.69 mN, but the frictional force Ff1 = Ff2 = 1.53 mN was found to be very high due to a much higher magnet velocity. We find that the frictional force Ff1 between the FF spikes and the surface is excessively high so the magnetic force is inadequate to overcome the frictional force. In spite of the fact that interfacial tension force is smaller than Ff2, we do not observe FF spike extraction since the ferrofluid spikes are unable to move due to the limitation that Fm,x < Ff1. We calculate Lam−1 = 1.10 and BomLam−1 = 2.20, which satisfies the criteria Lam−1 > 1, indicating that the FF spikes will not move, which we refer to as the “magnet disengagement regime” (see Figure 1e). Now, if we consider a case with VFF = 1 μL, Vw = 4 μL, Ums = 26.33 cm/s, we obtain Fm,x = 0.69 mN, Fi = 0.60 mN, and Ff1 = Ff2 = 8242

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Figure 4. (a) Variation in the velocity of the magnet (which is same as that of the FF spikes) and the aqueous droplet Ud with time t, VFF = 3 μL, Vw = 4 μL, and Ums = 24 cm/s; inset shows the variation of location x of the magnet (and FF spike) and the aqueous droplet with time t. (b) Variation of spike extraction distance Lse with velocity of the magnet Um for different volumes of ferrofluid VFF, Vw = 4 μL. (c) Variation of spike extraction distance Lse with velocity of the magnet Um for different volumes of aqueous droplets Vw, VFF = 1 μL. (d) Variation in the velocity of the magnet and the FF spikes Ud (which is same as that of the aqueous droplet) with time t, VFF = 3 μL, Vw = 4 μL, and steady magnet velocity Ums = 44 cm/s; inset shows the variation of location x of the magnet and the FF spike (aqueous droplet) with time t, θ = 70°.

intermediate velocity, the drag force Ff2 exceeds the interfacial tension force Fi, the FF spikes will get extracted out of the aqueous droplet and follow the magnet leaving the aqueous droplet behind, which is indicated by the regime R2. For a FF droplet volume VFF = 3 μL, aqueous droplet volume Vw = 4 μL and steady magnet velocity Um = 24 cm/s, the variation in the velocity of the magnet (which is same as that of the FF spikes) and the aqueous droplet Ud with time is shown in Figure 4a. The results show that at time t = 0.1 s, when the droplet velocity ∼21 cm/s, the frictional force Ff2 ∼ 0.91 mN, which exceeds the interfacial tension force Fi ∼ 0.7 mN. The FF spikes are thus extracted out of the aqueous droplet and the velocity of the aqueous droplet Ud sharply decreases thereafter. This behavior is also seen from the variation of location x of the magnet and the aqueous droplet with time t, which is shown as the inset of Figure 4a. The distance between the initial location of the aqueous droplet and the point at which the spike extraction takes place can be called as the ‘spike extraction distance Lse”. The dependence of the spike extraction distance Lse with the volume of the FF droplet, the volume of the aqueous droplet, and the magnet velocity is studied. For a fixed volume of aqueous droplet Vw = 4 μL, the variation of spike extraction distance Lse with velocity of the magnet Um for different volumes of ferrofluid VFF is depicted in Figure 4b. The results show that, for a fixed ferrofluid volume, the spike extraction

However, the base radius of the aqueous droplet Rb will increase when Vw increases (i.e., for a larger aqueous droplet); thus, theoretically, the droplet velocity Ud should decrease, which is observed in our experiments. It is also seen that, for a fixed volume of aqueous droplet Vw, the maximum droplet velocity Uds is higher for a larger volume of FF droplet VFF. This is because for a larger FF droplet, the magnetic force Fm becomes higher and thus the droplet velocity Uds will increase in order to ensure Lam−1 ≤ 1. From the experimental data, the maximum steady velocity of the aqueous droplet Uds is correlated with the volumes of the aqueous droplet Vw and the ferrofluid droplet VFF as Uds∼Vw−0.19VFF0.36. Using the simple model (i.e., eqs 1−3) presented in the Theoretical Analysis section, we predict the maximum droplet velocity Uds by equating the minimum of the magnetic and interfacial force i.e. min (Fm,x, Fi) with frictional force Ff. For a system with a FF droplet volume VFF = 1 μL, for different aqueous droplet volume in the range Vw = 2−8 μL, a comparison between the maximum droplet velocity Uds predicted using the simple model and measured in our experiments is also depicted in Figure 3b. A very good match, within 5%, is obtained indicating the validity of the simple model. The region Lam−1 ≤ 1 and BomLam−1 > 1 indicates the spike extraction regime (i.e., regime R2). As the velocity of the magnet is increased from zero to a steady value Ums, if at some 8243

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Langmuir

BomLam−1 > 1, and magnet disengagement is observed for Lam−1 > 1. Among the different regimes observed, the steady droplet transport regime would be of relevance for practical applications. We also demonstrated that the thin film of oilbased FF that covers the aqueous droplet surface can be easily cleaned using a PDMS-coated sponge to recover the aqueous droplet. In the steady droplet transport regime, aqueous droplet velocity Ud was found to be higher for a larger FF droplet but lower for a larger aqueous droplet, following the correlation Uds ∼ Vw−0.19VFF0.36. A simple model was able to accurately predict the aqueous droplet velocity Uds within 5% of the corresponding experimental data. In the spike extraction regime, the spike extraction distance Lse was found to be higher for a larger FF droplet but lower for a larger aqueous droplet and higher magnet velocity, following the correlation Lse ∼ Vw−1.75VFF0.75Ums−1.56.

distance Lse decreases with increase in the velocity of the magnet Um. This can be attributed to the fact that the velocity of the FF spikes will increase proportionally to the magnet velocity Um, and hence frictional force Ff2 will rapidly increase and exceed the interfacial tension force Fi over a shorter distance, thus giving rise to a smaller spike extraction distance Lse. With the aqueous droplet volume Vw and magnet velocity Um remaining fixed, it is observed that the spike extraction distance Lse increases with increase in the volume of the FF droplet VFF. This is because the height of the FF spikes hm increases with increase in the FF droplet volume VFF (see Figure S.3b in the Supporting Information), which gives rise to a higher interfacial tension force Fi (see eq 3). Since the interfacial tension force Fi increases with the volume of the FF droplet VFF, it would take a longer distance for the FF spikes to attain the critical velocity at which the frictional force Ff2 will exceed to surface tension force Fi thus offering a longer spike extraction distance Lse. For a fixed volume of FF droplet VFF = 1 μL, the variation of spike extraction distance Lse with velocity of the magnet Um for different volumes of aqueous droplets Vw, is depicted in Figure 4c. The results show that for a fixed magnet velocity Um and FF droplet volume VFF, the spike extraction distance Lse decreases with increase in the water droplet volume Vw. This may be attributed to the fact that a larger aqueous droplet gives rise to a larger contact radius Rb and hence a higher frictional force Ff2. Since the frictional force Ff2 increases with the increase in aqueous droplet volume Vw, the FF spikes would attain the critical velocity thus offering a smaller extraction length Lse. Further, for Lam−1 > 1 and BomLam−1 > 1, the frictional force Ff1 would exceed the magnetic force Fm,x and thus the magnet would get disengaged, leaving the FF spikes and the aqueous droplet behind (i.e., regime R3). For a FF droplet volume VFF = 3 μL, aqueous droplet volume Vw = 4 μL and steady magnet velocity Ums = 44 cm/s, the variation in the velocity of the magnet and the FF spikes Ud (which is same as that of the aqueous droplet) with time t is presented in Figure 4d. It is observed that since the frictional force Ff1 is higher than the magnetic force Fm, the FF spikes and the aqueous droplet do not move and the magnet gets disengaged. Similar behavior is also seen from the variation of location x of the magnet and the FF spikes (aqueous droplet) with time t, which is shown as the inset of Figure 4d.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.9b00631. Distribution of magnetic flux density of magnet; recovery of ferrofluid coated aqueous droplet using a PDMS coated sponge; variation of number of spikes N with magnetic Bond number Bom; variation of the maximum height of the FF spike hm and the critical gap g with FF droplet volume VFF and pictorial representation of ferrofluid spikes; variation of ϕ vs t (PDF) Different regimes for magnetic manipulation of the FF spike−aqueous droplet system (AVI)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

A. K. Sen: 0000-0001-6048-0091 Author Contributions

C.M. and U.B. have contributed equally. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank I.I.T. Madras (MEE/15-16/ 843/RFTP/ASHS) for providing the financial support which enabled this work.

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CONCLUSION We studied the transport of aqueous droplet placed on top of an array of oil-based ferrofluid (FF) spikes in the presence of a magnetic field. The spikes are formed when a FF droplet is placed on a rigid oleophilic substrate and exposed to a nonuniform normal magnetic field, due to the competition between surface tension and magnetic forces. Our experiments revealed that the behavior of the FF spikes−aqueous droplet system is governed by the competition between the magnetic force Fm,x, the interfacial tension force Fi, and the frictional force Ff. It was found that, depending on the values of the magnetic Laplace number and the magnetic Bond number, Lam−1 = (Ff1/Fm,x) and BomLam−1 = (Ff2/Fi), different regimes, i.e., steady droplet transport, spike extraction, and magnet disengagement, are observed. Our results show that steady droplet transport is observed for Lam−1 ≤ 1 and BomLam−1 ≤ 1, whereas extraction of spikes is observed for Lam−1 ≤ 1 and

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