Heterogeneous Capillary Interactions of Interface-Trapped Ellipsoid

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Heterogeneous Capillary Interactions of InterfaceTrapped Ellipsoid Particles using the Trap-Release Method Jin Hyun Lim, Jun Young Kim, Dong Woo Kang, Kyu Hwan Choi, Seong Jae Lee, Sang Hyuk Im, and Bum Jun Park Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03882 • Publication Date (Web): 12 Dec 2017 Downloaded from http://pubs.acs.org on December 19, 2017

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Heterogeneous Capillary Interactions of InterfaceTrapped Ellipsoid Particles using the Trap-Release Method Jin Hyun Lim1†, Jun Young Kim2†, Dong Woo Kang1†, Kyu Hwan Choi1, Seong Jae Lee2*, Sang Hyuk Im3*, and Bum Jun Park1*

1

Department of Chemical Engineering, Kyung Hee University, Yongin 17104, South Korea

2

Department of Polymer Engineering, The University of Suwon, Hwaseong, Gyeonggi 18323,

South Korea 3

Department of Chemical and Biological Engineering, Korea University, Seoul 02841, South

Korea

KEYWORDS: ellipsoid, heterogeneity, capillary force, pair interaction, optical laser tweezers, fluid-fluid interface

ACS Paragon Plus Environment

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ABSTRACT: Heterogeneous capillary interactions between ellipsoid particles at the oil-water interface were measured via optical laser tweezers. Two trapped particles were aligned in either tip-to-tip (tt) or side-to-side (ss) configurations via the double-trap method and were released from the optical traps, leading to particle-particle attractions due to the capillary forces caused by quadrupolar interface deformation. Based on image analysis and calculations of the Stokes drag force, the capillary interactions between two ellipsoid particles with the same aspect ratio (E) were found to vary with the particle pairs that were measured, indicating that the interactions were nondeterministic or heterogeneous. Heterogeneous capillary interactions could be attributed to undulation of the interface meniscus due to chemical and/or geometric particle heterogeneity. The power law exponent for the capillary interaction 𝑈𝑐𝑎𝑝 ~𝑟 −𝛽 was found to be β ≈ 4 and was independent of the aspect ratio and particle configuration in long-range separations. Additionally, with regard to the tt configuration, the magnitude of the capillary force proportionally increased with the E value (E >1) when two ellipsoid particles approached each other in the tt configuration.


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INTRODUCTION Typical colloidal particles attach to fluid-fluid interfaces and the adsorption leads to a reduction in surface free energy, consequently decreasing the surface tension and stabilizing the interface.1-2 The interface-trapped particles experience two competitive interactions: one being an electrostatic interaction due to the presence of surface charges and the other being a capillary interaction caused by interface deformation around the particles.3-12 Capillary interactions in particular are dominant when the interface-trapped particles are geometrically and/or chemically anisotropic.13-16 Intensive studies have been reported with regard to the effect of the anisotropic characteristics of colloidal particles on the capillary interactions that hierarchically affect assembly patterns, microstructural features, and consequently the rheological properties of particle-laden interfaces.17-24 There is little doubt that the chemical and geometric properties of individual colloidal particles are not perfectly identical, even if they are synthesized from the same batch or protocol. This fact likely leads to a heterogeneity in the small scale behavior of the particles (e.g., configurations of individual particles and interparticle interactions) that can lead to deviations between the experimental observations of large scale properties (e.g., assemblies and rheology) and the corresponding theoretical predictions based on an assumption of homogeneous small scale properties. For instance, Park et al. used optical laser tweezers to statistically measure the electrostatic repulsive forces for many particle pairs at an oil-water interface.25 The magnitude of the interaction forces varied depending on the particle pairs, and the heterogeneous repulsive interactions directly affected the assembled microstructures of two-dimensional suspensions. In context of the interaction heterogeneity of particles trapped at the fluid-fluid interface, the nonidentical geometric and chemical characteristics of individual anisotropic particles can change the magnitude or morphology of surrounding interfacial deformation, consequently leading to


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heterogeneous capillary interactions that are likely responsible for forming diverse assembly structures. From this point of view, quantitative measurements of attractive forces over many pairs of anisotropic particles are important because the results can be directly related to large scale behaviors composed of many particles. However, the heterogeneity of capillary interactions between anisotropic particles (i.e., ellipsoids) has not yet been quantitatively investigated. Loudet et al. reported a pair of interaction potentials by tracking the motion of two attractive ellipsoid particles at an oil-water interface; the approaching velocity could be converted to a capillary force using the Stokes drag force.14 Note that this indirect or passive method is extremely time-consuming because one must wait for the approach of two sufficiently close arbitrary particles that are able to spontaneously commence. The shape interface deformation around an ellipsoid particle was a quadrupole and the quadrupolar capillary interaction between two particles scaled as 𝑈𝑐𝑎𝑝 ~𝑟 −𝛽 with 𝛽 = 4 when the particles were found to approach in a tipto-tip (tt) manner. With regard to the side-to-side approach (ss), the obtained capillary force decayed slowly with 𝛽 = 3 as a function of the interparticle separation (r). However, the power law exponents for each approach configuration were obtained from experimental results that were performed only once for each configuration. Lehle et al. theoretically derived an asymptotic expression of the capillary interaction between two ellipsoids for large separations; the power law exponent of the capillary interaction was 𝛽 = 4, regardless of the manner of approach (tt and ss).26 They insisted that inconsistent results between the experiment and theoretical calculations could be attributed to separation dependence with regard to the scaling behavior of the capillary interactions; that is, the tt and ss approach in small separations led to a further increase and decrease in the 𝛽 value, respectively. In the context of clarifying the scaling behavior of capillary interactions between interface-trapped ellipsoid particles, further experimental investigations are


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required with regard to the effects of particle configurations and aspect ratios on their lateral capillary interactions. Compared to the passive method for measuring colloidal forces that are both inefficient and time-consuming, optical laser tweezers are a powerful tool that can directly measure interaction forces between colloidal particles with pico-newton resolution (i.e., active method).2729

For example, when a laser beam hits an object dispersed in a fluid medium, the reflection and

refraction of the beam at the particle-medium interface can generate changes in momentum that cause the gradient force to move the object toward the laser focus.28, 30 To optically trap the object, the refractive index of the object needs to be slightly higher than that of the medium; additionally, the laser beam should be highly focused on a focal plane. These two conditions can result in a gradient force that is larger than the radiation force pushing the object in the direction of the beam propagation. To measure the interaction force between two particles, for example, one trapped particle is translated toward the other particle that is held by a stationary trap. The stationary particle is displaced from an equilibrium position depending on attraction or repulsion between the particles. When the particles are spherical, the drag calibration is typically used to determine the trap stiffness (𝜅𝑡 ) that can convert the displacement (Δx) of the stationary particles to an interaction force using the following equation: 𝐹𝑝𝑎𝑖𝑟 = 𝜅𝑡 𝛥𝑥.27 In this case, an optically trapped particle is subjected to a surrounding fluid with constant velocities under creeping flow regimes, leading to displacement (Δx) of the particle. At mechanical equilibrium, the Stokes drag force (Fd) is balanced with the optical trapping force, 𝐹𝑡 = 𝜅𝑡 𝛥𝑥 ; thus, the trap stiffness can be obtained from the relationship 𝜅𝑡 = 𝐹𝑑 /Δ𝑥. For anisotropic particles, the trap stiffness depends on the polar orientation angle between the axis of symmetry and the fluid velocity; thus, it is challenging to experimentally attain the trap


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stiffness values over all orientation angles using the drag calibration method. This technical difficulty hampers the direct measurement of interaction forces between anisotropic particles. In this work, we combined the active and passive methods (i.e., trap-release method) to measure the capillary interaction forces between ellipsoid particles confined at an oil-water interface.31 The use of time-sharing optical traps generated multiple traps that could modulate the initial alignment (side-to-side (ss) or tip-to-tip (tt)) of two particles within an active regime. Then, the two particles were released from the optical traps in a passive region. The movements of the particles were recorded using a high-speed camera to analyze their trajectories and determine the interaction forces between them. Note that the use of multiple traps within the active regime enabled quantification of the initial configuration effects on the interaction behaviors. Additionally, the trap-release method allowed us to statistically measure the interaction forces over different particle pairs with varied geometric factors (i.e., aspect ratio values (E)). In doing so, we could quantitatively investigate heterogeneity in the capillary interactions and clarify the deviation with regard to interaction scaling behaviors between prior experimental measurements and theoretical predictions.

EXPERIMENTAL SECTION Materials. Styrene (Samchun Chemical, Korea) was purified by distillation under reduced pressure prior to use. The initiator, 2,2’-azobisisobytyronitrile (AIBN) (Daejung Chemicals, Korea), was purified via recrystallization from methanol (Daejung Chemicals, Korea). Poly(vinyl pyrrolidone) (PVP) (MW = 4.0 × 104 g/mol), used as a steric stabilizer, and poly(vinyl alcohol) (PVA) (MW = 1.46 × 105 − 1.86 × 105 g/mol, hydrolysis: 87 - 89%), used as a matrix polymer, were purchased from Sigma-Aldrich. Ultrapure water (resistivity = 18.2 MΩ·cm) was used in all


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experiments. Synthesis of spherical polystyrene particles. Monodisperse polystyrene (PS) microspheres with a diameter of ~3 μm were synthesized via dispersion polymerization, following a modified procedure presented in a prior study.32 Briefly, 20 mL of styrene and 0.8 g of PVP were placed inside a reactor containing 100 g of isopropanol, 0.2 g of AIBN, and 5 g of water; the reactor was equipped with a stirrer, reflux condenser, and a nitrogen inlet. The polymerization was performed at 70 °C for 24 h with an agitation speed of 120 rpm. The product was washed with water and isopropanol followed by centrifugation. This procedure was repeated more than 3 times to obtain purified PS microspheres. Synthesis of ellipsoid particles. Ellipsoidal PS particles were prepared by a film stretching method following a procedure reported in a prior study.33-34 First, PVA films with embedded PS microspheres were fabricated as follows. 15 g of PVA was dissolved in 150 mL of water at 80 °C and magnetically stirred for 5 h. After cooling under ambient conditions, 0.1 wt. % PS seed spheres were added to the PVA solution. The mixture was poured onto a horizontal tray and dried to obtain a flexible PVA film embedded with PS microspheres. Next, the rectangular cut film was firmly fixed to the tensile grips of a universal testing machine (UTM). Prior to stretching, the chamber temperature was set to 135 °C, which was above the glass transition temperature of both the PVA film and PS particles. Film stretching was performed at a rate of 0.5 mm/s with a suitable range of elongation depending on the desired aspect ratio (E) of the ellipsoid particles. The stretched film was dissolved in water at 80 °C to remove the PVA matrix. The freed ellipsoidal particles were collected via repeatedly performed centrifugation and washing steps. Aspect ratio values were determined by analyzing scanning electron microscopy images (SEM, AIS2000C, Seron Technologies Inc.) (Fig. 1). A schematic illustration for stretching of the PVA film embedded with


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spherical PS can be seen in Fig. S1. Preparation of a flow cell. To achieve a planar oil-water interface, a specially designed flow cell was constructed in which a Teflon ring with a 12 mm diameter and 9 mm height was inserted into an aluminum ring with a 14 mm diameter and 8 mm height (Fig. 2).35 The Teflonaluminum inner cylinder was placed onto a small piece of cover slip attached to a circular cover slip (Marienfeld, no. 1.5H) that provided a space for the passage of water. The circular cover slip was attached to an outer glass ring with a 25 mm diameter and 11 mm height using epoxy glue (Norland) by irradiating UV light (λ = 365 nm) for 20 min. The flow cell was mounted to the mechanical stage of an inverted microscope (Nikon, Ti-U) equipped with an optical laser apparatus.36 2 mL of ultrapure water and 0.2 mL of n-decane (Acros Organics) were consecutively added to form an oil-water interface that was pinned at the junction between the Teflon and aluminum in the inner cylinder of the flow cell. Note that prior to use, any polar impurities containing n-decane were removed by filtering through aluminum oxide (Acros Organics, acidic activated, particle size 100 – 500 μm).37 The ellipsoid particles were then inserted into the oilwater interface by dropping a dilute aqueous particle solution containing 30 vol. % isopropanol (Sigma-Aldrich) that was used as a spreading solvent. After waiting for several minutes, a small amount of water was carefully and repeatedly removed from the outer cylinder until the particles at the oil-water interface were visible on a computer screen connected to a charge-coupled device (CCD) camera (Hitachi, KP-M1AN). Another circular cover slip covered the top of the flow cell and was sealed with vacuum grease (Dow corning) to prevent the evaporation of fluids and convective flows. Optical laser tweezer apparatus. Optical laser tweezers were constructed with an inverted microscope (Nikon, Ti-U).36, 38 A laser beam generated from a 10 W CW Nd:YAG laser


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(Coherent Matrix, λ = 1064 nm) passed through an acousto-optics deflector (AOD, Opto-electronic DTSXY-400-1064 2D) that controlled the diffraction of the laser beam. The beam was subsequently incident to the back aperture of a water immersion objective (Nikon, CFI Plan Apochromat VC 60× with a numerical aperture of NA = 1.2 and a working distance of ~300 μm) and was highly focused on a focal plane to generate optical traps. The trap positions were controlled using a LabVIEW program that modulated the operation of the AOD. The laser power used in this study was measured from the back aperture (P = 31 mW) using an optical power meter (Thorlabs, PM100D). Drag calibration for a spherical particle. The drag calibration method was used to determine the trap stiffness (κt).27, 30 A spherical particle with a diameter of d ≈ 3 μm at the oilwater interface within the flow cell was optically trapped. Unidirectional surrounding flows were applied to the trapped particle with a diameter d by moving the mechanical microscope stage with a constant velocity onto which the flow cell was mounted. A sequence of microscope particle images was monitored and recorded using the CCD camera at a rate of 30 frames per minute (fps) while the fluid velocity (u) was varied in creeping flow regimes. Particle displacement values (Δx) from the equilibrium position when u = 0 were calculated as a function of fluid velocity using the ImageJ software.39 Under mechanical equilibrium conditions, the trapping force 𝐹𝑡 = 𝜅𝑡 Δ𝑥 was balanced with the Stokes drag force 𝐹𝑑 = 3𝜋𝑑𝜂𝑒𝑓𝑓 𝑢 , where 𝜂𝑒𝑓𝑓 is the effective viscosity depending on the surface area exposed to each fluid phase, given by 𝜂𝑒𝑓𝑓 = [𝜂𝑜 (1 − cos 𝜃) + 𝜂𝑤 (1 + cos 𝜃)]/2.35 The three-phase contact angle (θ = 130°) of a spherical particle at the oilwater interface was measured using the gel trapping method;40 the viscosities of oil and water were 𝐹

used as ηo = 0.92 and ηw = 1.0 mPa·s, respectively. The trap stiffness 𝜅𝑡 = 𝛥𝑥𝑑 was then obtained from a linear regression of a plot of Fd versus Δx.


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Active measurements of pair interactions between spherical particles. Two spherical particles at the oil-water interface were optically trapped with a sufficiently large separation in which interparticle interactions were negligible (i.e., r > 30 μm); thus, the center of the trapped particle was consistent with the trap position.35 One particle was fixed and the other was translated stepwise toward the stationary particle until one of the particles was released from the trap or the two particles came into contact. The displacements Δx of the stationary particle from the equilibrium position were calculated using the ImageJ software and converted to an interaction force using the previously obtained value of the trap stiffness, 𝐹𝑝𝑎𝑖𝑟 = 𝜅𝑡 Δ𝑥 . The force was subsequently related by the particle separation r. Trap-release method. Two optical traps were used to hold one ellipsoid particle within the active regime; the relative configurations (i.e., tt or ss alignment) of the two trapped ellipsoid particles were manipulated by controlling the positions of the two optical traps on each particle. At particular separation lengths whereupon the capillary attractions between particles were sufficiently strong, the traps were removed and the released particles spontaneously approached one another within the passive regime. A high-speed camera (AMETEK) was used to monitor and record the movements of the particles at 200 fps. The recorded image sequence was analyzed to obtain the trajectories of the particles using ImageJ.

RESULTS AND DISCUSSION During the first phase of the investigation, we justified use of the trap-release method. Provided the trap stiffness (𝜅𝑡 ) was identified, the direct or active measurement method was extremely convenient and powerful to obtain interaction force profiles as a function of particle separation. In


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this case, one particle optically held by a single translational trap was translated stepwise toward another particle held by a single stationary trap (i.e., single-trap direct measurement method). The displacement ( 𝛥𝑥 ) of the stationary particle was subsequently analyzed to convert the pair interaction force that was related to the separation, 𝐹𝑝𝑎𝑖𝑟 = 𝜅𝑡 𝛥𝑥. In general, the trap stiffness was determined by the drag calibration with the following relationship: 𝜅𝑡 = 𝐹𝛥𝑥𝑑, where 𝐹𝑑 is the Stokes drag force. To measure the interaction force between ellipsoid particles that depended on the relative configurations (e.g., tt, ss, or random configurations) of each particle, the drag calibration should be performed in a similar manner by trapping one ellipsoid particle with one optical trap. However, because the trapped ellipsoid particle tended to rotate along the flow stream parallel to the particle’s axis of symmetry, only the 𝜅𝑡 value for the major axis could be obtained. Therefore, when the interactions between two ellipsoid particles were measured using the single-trap method, the obtained 𝜅𝑡 value was only applicable when they were aligned in the tt configuration. Alternatively, two optical traps could be used to hold one ellipsoid particle (i.e., doubletrap direct measurement method), particularly when particles resided at the oil-water interface. In this case, the separation between two traps was shorter than the major axis length of the particle. Note that the optical trapping force led to particle rotation along the direction of the beam propagation when dispersed within a single fluid medium, as can be seen in Figs. 3a and 3b (see Movie S1). Thus, it was challenging to simultaneously hold and manipulate the polar orientations of the particle on a two-dimensional plane perpendicular to the beam propagation using two optical traps. When the ellipsoid particle was located at the oil-water interface, the particle adopted a horizontal orientation to decrease the attachment energy of the particle by increasing the displaced area (SI) intercepted by the interface (see the detailed calculation method for the attachment energy in the SI).41-42 Strong attachment of the particle at the oil-water interface hindered vertical rotation


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across the interface (Fig. 3c), thereby enabling the two traps to steer the angle (θd) between the particle’s axis of symmetry and the fluid velocity for the drag calibration. Ideally, once the trap stiffness values are defined over the range of 0° ≤ 𝜃𝑑 ≤ 180°, an identical process with the pair interaction measurements between spherical particles can be employed to directly measure the pair interactions between the ellipsoid particles using the double-trap method when approaching with ss or tt alignment or with any random angle. Notably, based on numerical calculations of surface areas using the Hit-and-Miss Monte Carlo method,41 the decrease in attachment energy from the upright (𝜃𝑟 = 0°) to the horizontal (𝜃𝑟 = 90°) configuration was approximately −1.0 × 108 kBT for an ellipsoid particle with an aspect ratio (E = 4.28) and minor axis diameter (~1.82 μm) perpendicular to the axis of symmetry (see the full attachment energy profile in Fig. S2). For example, the torque (τ) required to rotate the horizontally oriented ellipsoid particle with Δ𝜃𝑟 = 10° could be approximated by 𝛥𝐸𝑎𝑡𝑡 (𝜃𝑟 = 90°

90°) − 𝛥𝐸𝑎𝑡𝑡 (80°) = ∫80° 𝜏 𝑑𝜃𝑟 ≈ 𝜏𝛥𝜃𝑟 = 𝐹𝑟𝑜𝑡 𝐿𝛥𝜃𝑟 , assuming that the torque and angular displacement were in the same direction. Substituting the length of the particle’s major axis with the moment arm (L), the corresponding force required to rotate the particle Δ𝜃𝑟 = 10° was approximately Frot ≈ 2 × 103 pN, which was significantly greater than the magnitude of the typical optical trapping force in the current experiment (e.g., Ft = O(1) pN). Additionally, it was previously demonstrated that the radiation force that pushed the interface-trapped particle in the direction of the beam propagation was negligibly small compared to the surface tension force of the interface under typical optical trapping conditions; therefore, interface deformation was unlikely to occur. In short, the use of optical tweezers barely led to any undesired interparticle interactions (e.g., capillary interactions caused by interface deformation).30 Prior to demonstrating the infeasibility of the double-trap direct measurement method to


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measure the interaction force between ellipsoid particles at the interface, it was checked that particles preferred to approach exclusively or precisely in the manner of either the tt or ss configurations using the single-trap method, otherwise they aligned in non-deterministic manner. In this case, particles trapped by a single trap could freely rotate to adopt configurations corresponding to low energy minimum states during their approach. Hereafter, the absolute value of the difference in the polar orientation angles (ω1 and ω2) between two ellipsoid particles, ω12 = |𝜔1 − 𝜔2 |

(Fig. 4a), was used to define the interparticle configurations; the ss and tt

configurations corresponded to the ranges of 0 ≤ ω12 (𝑠𝑠) ≤ 45° and 135° ≤ ω12 (𝑡𝑡) ≤ 180°, respectively, as well as the intermediate configuration between them, 45 < ω12 (𝑖𝑛𝑡) < 135°. As shown in Fig. 4, multiple runs of the experiments using the single-trap method showed that the configurations between two ellipsoid particles with E = 4.28 changed depending on the pairs and separation between them. For instance, a translational particle (the right one in Fig. 4a) approached a stationary particle (the left one) until they became attached. The initial tt configuration over large separations was maintained after making contact. When the right particle was reversely translated, the left particle was released from the trap and the tt configuration was still preserved. The pair in Fig. 4b underwent a transition from the tt to int configuration upon releasing the left particle from the trap. The slightly angled tt configuration in Fig. 4c was aligned along the two particle’s axis of symmetry prior to being attached and a strong attraction led to ss attachment between the sides indicated by the red marks, in which one particle jumped to the other. As the pair in Fig. 4d with an initial int configuration came closer to one another, they consecutively transformed from the ss to tt configurations. After making contact with tt, they adopted ss upon the release of one particle from the trap. Some pairs initially exhibited the ss configuration at large separations, as can be seen in Figs. 4e-4g. Interestingly, the pair in Fig. 4e demonstrated reversible transitions between


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ss and int while being attached and detached repeatedly by moving the right particle forward and backward, respectively. A similar transition also occurred from ss to either int or tt for the pairs in Figs. 4f and 4g; the later pair was found to be flipped when they were attached. Overall, the diverse and non-deterministic configuration behaviors depending on particle pairs and separations demonstrated that the capillary attractions were heterogeneous. Additionally, it was important to note that when two ellipsoid particles made contact with the tt configuration, the contact point between the two tips of each particle slightly deviated from the geometry involving symmetric attachment, as shown in the magnified images in Figs. 4a, 4b, and 4d. We believe that heterogeneity in the geometric and chemical properties of the particle could lead to undulation of the interface meniscus and an uncertainty of orientation dependence with regard to capillary interactions, consequently resulting in heterogeneous interaction behaviors. Notably, it has been well known that the ellipsoid particle formed quadrupolar interface deformation and the resulting capillary interactions scaled with 𝑈𝑐𝑎𝑝 ~𝑟 −4.14, 26 With regard to the undulated interface deformation around spherical particles due to non-homogeneous surface properties, the meniscus shape could be fitted via multipole expansion.8, 43 The lowest stable pole was a quadrupole that dominated capillary interactions and decayed with 𝑈𝑐𝑎𝑝 ~𝑟 −4 . We believe that capillary interactions caused by undulated interface deformation were likely responsible for the heterogeneous capillary interactions between ellipsoid particles at the oil-water interface. The heterogeneous capillary interactions revealed by the single-trap method demonstrated the infeasibility of the double-trap direct measurement method that fixed the configurations of two ellipsoid particles while they approached each other. Due to the configuration dependence of the capillary interactions (Fig. 4), the measured interactions with the fixed configuration were not likely to be accurate. To further demonstrate the infeasibility of the double-trap measurement, two


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ellipsoid particles at the oil-water interface were held by the double-trap method and were aligned with the tt configuration. The left particle was stationary and the right particle approached by translating the double-traps toward the stationary particle. As can be seen in Fig. 5a, the tt configuration of the pair was maintained until making contact and the contact point was approximately symmetric. By contrast, other pairs in Figs. 5b-5d showed that the trapped particles tended to rotate to overcome the optical trapping force while the optical traps were translated laterally. For instance, the right particle in the pair in Fig. 5b rotated clockwise slightly at moderate separations (yellow dashed box) and the pair was attached in an asymmetric manner between two tips (magnified image). The pair in Fig. 5c was found to be misaligned from the initial state (yellow dashed box) and became aligned as the pair approached in a line. Then, the particles were rotated (yellow arrows) and attached to form the tt configuration, followed by switching to the ss configuration after the right particle was released from the trap. Interestingly, the right particle in Fig. 5d did not move forward (red dashed box); however, it rotated clockwise slightly (yellow dashed box). Upon laterally translating the traps on the right particle, the particle rotated vertically and finally the tip-to-side (ts) configuration formed. This unusual behavior was likely because the factor of the undulated meniscus significantly affected capillary interactions, resulting in heterogeneous properties. Overall, the active measurement method was inappropriate to accurately measure the capillary forces between ellipsoid particles at the oil-water interface due to the challenge of obtaining the trap stiffness, which depended on the polar orientation angle of the single-trap method and heterogeneous capillary interactions, which varied due to particle separation and polar orientation angles for the double-trap method. These results justified the use of a combined approach of active and passive methods or the trap-release method. During the next phase of the investigation, quantitative measurements of heterogeneous


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capillary interactions with various aspect ratio values (E) using the trap-release method were focused on, where the initial configurations (tt and ss) were modulated via the double-trap method. Prior to describing the results, interactions between the PS seed microspheres used to prepare the ellipsoid particles were directly measured using the active method. The trap stiffness value was obtained by the conventional drag calibration method (Fig. 5a). When the Stokes drag force was balanced with the optical trapping force, the trap stiffness could be obtained using the following relationship: 𝜅𝑡 =

6𝜋𝑅𝜂𝑒𝑓𝑓 𝑢 Δ𝑥

(see detailed methods in the experimental section). As a translational

particle approached a stationary particle, the interaction force was obtained as 𝐹 = 𝜅𝑡 𝛥𝑥, in which Δx was the displacement of the stationary particle from the equilibrium position (Fig. 6b). As can be seen in Fig. 6c, some particle pairs were found to experience energy barriers prior to forming dimers (black curves in the left inset figure). The presence of such energy barriers demonstrated that non-negligible electrostatic repulsive interactions between the spherical particles likely existed due to the polar surface groups (i.e., PVP) that were used as surfactants during fabrication of the PS seed particles. The power law exponents in the force profiles averaged over the two attractive pairs were found to be 〈𝛽〉 = 3.2 within the linear region on a log-log scale (the right inset figure in Fig. 6c). According to the scaling exponents of 𝛽 = 3 for electrostatic repulsion and 𝛽 = 4 for quadrupolar capillary attractions, deviations from capillary attractions in the experimental exponent likely suggested that electrostatic repulsive interactions were still effective in the spherical pairs. Notably, heterogeneous pair interactions between the spherical particles were consistent with the microstructure. As can be seen in Fig. 6d, some particles formed aggregates while other particles remained as stable monomers due to the presence of energy barriers between them. Note that the interaction heterogeneity for the spherical particles at the oil-water interface could be attributed to the nonhomogeneous surface charge distribution and/or triboelectrification


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caused by rubbing between the particles and oil when the spreading solvent was used.12, 44 To investigate the capillary interactions between ellipsoid particles at the oil-water interface, spherical seed particles were mechanically stretched above the glass transition temperature of PS. In marked contrast to the spherical particles, the majority of the ellipsoid particles with E = 1.29, 1.56, 1.85, and 4.28 formed aggregates (Fig. S3) when spread over the oil-water interface. The aggregated microstructures revealed that capillary interactions were dominant compared to electrostatic repulsive interactions. These ellipsoid particles were used to evaluate the capillary interactions via the trap-release method. As can be seen in Fig. 7, two particles were optically trapped using single or double traps to manipulate the tt or ss configurations in the active regime. As similarly observed in Fig. 5, despite using double traps that were aligned as either tt or ss, the initial configurations were not found to be completely tt or ss due to long-range capillary interactions that likely overcame the optical trapping force. The two particles that were optically trapped within the active zone in Fig. 7 were released from the traps at moderate separations, upon which the capillary interactions were sufficiently strong to draw one another. Diverse orientation behaviors were found in the passive zone where particles could freely rotate and follow a route that lowered the total interaction energy as they approached each other. For instance, the pair with the tt configuration in Fig. 7a gradually rotated, where the left particle moved clockwise and the right particle moved counter-clockwise while maintaining tt; the two particles subsequently rotated in the reverse direction to form the int configuration upon attachment. The ss configuration of the pair in Fig. 7b was preserved until the particles made contact. To quantitatively investigate the interaction behaviors, data was collected, which satisfied the conditions such that the particle configurations did not significantly change prior to attachment as the particles approached each other. For instance, the case shown in Fig. 7c was excluded, whereupon the configuration transition


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of the pair occurred from ss-int-tt-int with regard to determining the power law exponents and the magnitude of the capillary interactions. To statistically measure the capillary interactions and scaling behaviors of ellipsoid particles with different E values, the movement of more than ten particle pairs for each E value were monitored by controlling the initial pair configurations within the active zone (i.e., tt and ss). Among the pairs, cases were selected and analyzed in which either the tt or ss configurations within the passive zone were maintained without significant configurational changes. As can be seen in Fig. 8, the separation (r) curves for the tt configuration were plotted as a function of time (tmax – t), where tmax indicates the time at which the two particles made contact. As the value of tmax – t decreased, the separation decreased due to capillary attraction within the passive zone. The different colored curves indicated pairs with different E values; different curves with the same color demonstrated heterogeneity within the magnitude of capillary attraction for the tt configuration. To examine the scaling behavior of 𝑈𝑐𝑎𝑝 ~𝑟 −𝛽 , linear regression using the equation log 𝑟 = 𝑎 + 𝑏 log(𝑡𝑚𝑎𝑥 − 𝑡) was used to fit the separation curves within the linear regime on a log-log scale, as indicated by the light yellow area in Fig. 8b. The mean values 〈𝑏〉 of the slopes for the linear fit averaged over different particle pairs for each E value were converted to the corresponding mean value of the power law exponent via the following relationship: −〈𝛽〉 = 2 − 〈𝑏〉−1 (see the detailed derivation in SI).14, 16 As shown in the plot of 〈𝛽〉 versus E in Fig. 8c, the value of 〈𝛽〉 was not found to significantly depend on the E value; however, the error bars for each data point demonstrated the interaction heterogeneity. The b and β values averaged over the entire pairs regardless of E values were 〈𝑏〉𝑎𝑙𝑙 = 0.169 ± 0.013 and 〈𝛽〉𝑎𝑙𝑙 = 3.96 ± 0.48, respectively, and the latter was indicated with a dashed line in Fig. 8c. Note that the obtained power law exponent for the tt configuration was consistent with the theoretical prediction for quadrupolar


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capillary interactions (β = 4) over long-range separations. Based on the parameter values obtained from the linear regression, the capillary force 𝑑𝑒𝑓𝑓 1+𝛽

profiles could be calculated using the equation 𝐹𝑐𝑎𝑝 = −𝐹𝑚𝑎𝑔 (

𝑟

)

, where Fmag is the 1

magnitude of the capillary force. The effective diameter was given by 𝑑𝑒𝑓𝑓 = 𝑑𝑎 𝐸 3 for an ellipsoid, where da indicates the diameter of an axis of symmetry.45 Based on the fit parameters (〈𝑎〉 and 〈𝑏〉) that were averaged over the various pairs in Fig. 8b, the separation curves were regenerated via the following relationship: 〈𝑟〉𝑓𝑖𝑡 = 10〈𝑎〉 (𝑡𝑚𝑎𝑥 − 𝑡)〈𝑏〉 for various E values, as can be seen in Fig. 8d. The Stokes drag force was used to obtain the capillary force, 𝐹𝑐𝑎𝑝 = −𝐹𝑆 = 𝑑(〈𝑟〉𝑓𝑖𝑡 )

−3𝜋𝑑𝑒𝑓𝑓 𝜂𝑒𝑓𝑓 𝑢(𝑟)𝑓𝐸 , where the velocity could be calculated by 𝑢(𝑟) = 𝑑(𝑡

𝑚𝑎𝑥 −𝑡)

in the plot in

Fig. 8d. The fE term is the Stokes correction for ellipsoid particles within the creeping flow regime, where 𝑓𝐸 =

4 2 (𝐸 −1) 3

2𝐸2 −1 ln[𝐸+√𝐸 2 −1]−𝐸 √𝐸2 −1

and 𝑓𝐸 =

8 2 (𝐸 −1) 3

2𝐸2 −3 ln[𝐸+√𝐸 2 −1]+𝐸 √𝐸2 −1

when two ellipsoids approach each

other with tt and ss alignment, respectively.45 As shown in Fig. 8e, the calculated forces that were only effective at long-range separation (i.e., 𝑡𝑚𝑎𝑥 − 𝑡 ≥ 0.1 s) demonstrated that the magnitude of force (𝐹𝑚𝑎𝑔 ) increased as the E value increased. Each force curve was quantitatively fitted using 𝑑𝑒𝑓𝑓 5

𝐹𝑐𝑎𝑝 = −𝐹𝑚𝑎𝑔 (

𝑟

) and the fitting parameter Fmag was plotted as a function of the E value. It

was found that Fmag proportionally increased with E that could be qualitatively explained by the interface deformation magnitude. It was previously reported that as the aspect ratio increased, the interface deformation became more significant.46 In this case, as the aspect ratio increased, the reduction of surface free energy increased when one particle approached another along the tip-totip alignment, leading to the increase in the capillary interaction. To compare the interaction behavior between the tt and ss configurations, the separation


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versus 𝑡𝑚𝑎𝑥 − 𝑡 curves for six ss pairs and three tt pairs were plotted for the ellipsoid particles with E = 4.28. As shown in Fig. 9a, the separation curves for the tt configuration were an upper bound in the plot, qualitatively demonstrating that tt interactions were longer ranged compared to ss interactions. The mean slope of the separation curve for the ss configuration was 〈𝑏〉 = 0.171 ± 0.006 within the linear region on a log-log scale, indicated by a light yellow region in Fig. 9b, which was averaged over six different pairs. The corresponding power law exponent for the capillary interactions (𝑈𝑐𝑎𝑝 ~𝑟 −𝛽 ) was 〈𝛽〉 = −3.86 ± 0.20, demonstrating that ss interactions could be described by quadrupolar interactions over a long separation range (i.e., 𝑡𝑚𝑎𝑥 − 𝑡 ≥ 0.1 s), similar to that of tt interactions. To quantify the relative strength of the capillary force between the tt and ss configurations over long-range separations, the separation curves were regenerated (Fig. 9c) using fitting parameter values (〈𝑎〉 and 〈𝑏〉) averaged over multiple pairs; the approach velocity was obtained as a function of separation based on a similar method described in Fig. 8d. The capillary force was subsequently obtained from the Stokes drag force with the Stokes correction fE when particles approached with the ss configuration. As can be seen in Fig. 9d, the magnitude of the capillary force (Fmag) for the tt configuration was found to be slightly stronger (i.e., 16%) than that of the ss configuration.

CONCLUSIONS We investigated the heterogeneous capillary interactions between ellipsoid particles at the oilwater interface. To fabricate ellipsoid particles with different aspect ratios (E), a unidirectional tensile force was applied to a PVA film embedded with polystyrene microspheres at an elevated temperature. Optical laser tweezers were subsequently utilized to quantitatively measure the capillary attractions between ellipsoid particles at the oil-water interface. The key findings in this


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work can be described as follows. First, the active measurement method via optical laser tweezers (i.e., both the single- and double-trap methods) could not be employed due to capillary interactions that depended on the configuration and separation between particles. This fact demonstrated that the capillary interactions were heterogeneous. The heterogeneous interactions could be attributed to undulated interface deformation due to chemical and/or geometric particle heterogeneity. Second, using the trap-release method, the power law exponent for capillary interactions 𝑈𝑐𝑎𝑝 ~𝑟 −𝛽 was found to be β ≈ 4 in long-range separations, regardless of E values and the relative configurations between two particles (i.e., tip-to-tip (tt) or side-to-side (ss) configurations). The experimental results were consistent with the theoretical predictions for the asymptotic form of the capillary interactions.26 Third, the magnitude of the capillary force proportionally increased with the E value (E >1) when two ellipsoid particles approached each other with the tt configuration. Lastly, capillary interactions for the tt configuration were statistically stronger than those with the ss configuration. It was well-known that the presence of an ellipsoid particle at a fluid-fluid interface leads to the quadrupolar interface deformation.14,26 If the ellipsoid particles at the interface adopt accurately and smoothly quadrupolar interface deformation, they should be attractive in the manner of either tip-to-tip or side-to-side configurations deterministically. However, based on the experimental results obtained from the single- and double-trap methods, the relative configurations between two trapped ellipsoid particles were found to be non-deterministic, but varied depending on the particle pairs and their separations. Additionally, the use of the trap-release method resulted in the heterogeneity in the capillary interaction magnitude that also varied in particle pairs. These results consistently demonstrated that the interface deformation was not accurately quadrupolar and additional factors (i.e., undulated meniscus) should be involved.8, 43 The undulated interface


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meniscus around the ellipsoid particles might be attributed to chemically and/or geometrically heterogeneous particle properties, similar to the case of spherical particles.8, 43 To clarify the exact link between the interaction heterogeneity and the nonhomogeneous chemical/geometric particle characteristics, further investigation will be required; performing the multipole expansion8, 26 to mathematically fit randomly undulated meniscus around an interface-trapped ellipsoid particle, solving the Laplace equation using the meniscus boundary condition, and calculating the surface free energy and the corresponding capillary interaction between two ellipsoid particles. Furthermore, we also focus on quantifying the capillary interactions and the corresponding scaling behaviors over relatively shorter separations between two ellipsoid particles. In this case, the traprelease method is likely inappropriate because rotational motions of the two particles would become more significant. Alternatively, the double-trap direct measurement method can be employed; the position and orientation of one particle are fixed by two stationary traps, and the interaction force is measured by rotating the other particle over 0 to 360° and approaching stepwise.

ASSOCIATED CONTENT Supporting Information Supporting Information is available free of charge on the ACS Publications website at DOI: xxx. Schematic fabrication of ellipsoid particles, calculations of the attachment energy for ellipsoid particles at the oil-water interface, ellipsoid particle microstructures at the oil-water interface, and movie showing optical trapping of an ellipsoid particle in water.


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AUTHOR INFORMATION Corresponding Author *E-mail: [email protected], [email protected], [email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. †These authors contributed equally to this work.

Notes The authors declare no competing financial interests.

ACKNOWLEDGMENTS This work was supported by the National Research Foundation (NRF) of Korea, funded by the Korea government (MSIP) (NRF-2017R1A2B4003123), and the Engineering Research Center of Excellence Program, funded by the MSIP (NRF-2014R1A5A1009799).

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Table of Contents


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Figures

Figure 1. SEM images of (a) polystyrene (PS) microspheres and ellipsoid particles with the following aspect ratios: (b) E = 1.29, (c) 1.56, (d) 1.85, and (e) 4.28. All ellipsoid particles were prepared from the same batch of spherical PS particles.


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Figure 2. Schematic for a flow cell used to measure the pair interactions between particles at the oil-water interface. Particles are not drawn to scale.


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Figure 3. Optical microscope images and corresponding schematics for optically trapping an ellipsoid particle (E = 4.28) in (a,b) an aqueous phase and (c) at the oil-water interface.


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Figure 4. Demonstration of the heterogeneous interactions of different particle pairs with E = 4.28 as the right particle approached a stationary left particle using optical laser tweezers. Each particle was trapped by a single optical trap.


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Figure 5. Heterogeneous interactions over four different ellipsoid pairs with E = 4.28 at the oilwater interface. The tt configuration was maintained using the double trap method while the right particle approached the left particle by translating the optical traps.


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Figure 6. The direct measurement of PS seed microspheres using optical laser tweezers. (a) Schematic illustration of the drag calibration of a spherical particle. (b) Optical microscope images captured during pair interaction measurements between two PS spheres residing at the oil-water interface. (c) Heterogeneous pair interaction forces that depended on the particle pairs. Different symbols indicate different pairs. The inset on the left-hand side is a magnified plot showing repulsive energy barriers at moderate separations. The inset on the right-hand side is the force profile of two attractive pairs on a log-log scale. (d) Optical microscope image of a twodimensional suspension composed of PS particles dispersed at the oil-water interface.


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Langmuir

Figure 7. Representative optical microscope images showing the trap-release method. The initial configurations of two ellipsoid particles with E = 4.28 were modulated using the double-trap method within the active region. Particles were released from the optical traps and attractive motions subsequently occurred within the passive regions. Each panel indicates a different transition behavior for each pair configuration while transferring from the active to passive region: (a) tt (initial)-tt-int (final), (b) ss (initial)-ss-ss (final), and (c) ss (initial)-int-tt-int (final). Red dots within the active region indicate the position and number of optical traps.


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Figure 8. The capillary attraction of two ellipsoid particles as they approached with the tt configuration. (a) The measured interparticle separation (r) as a function of E and tmax – t. (b) The corresponding plot on a log-log scale. A linear regression was used to fit curves within the linear regime indicated by the yellow boxed area. (c) The average value of the power law exponent (β) as a function of E. (d) Regenerated plot of r versus tmax – t based on the average value of the fitting parameters obtained in panel b. (e) The capillary force converted by the Stokes drag force in which the velocity was obtained from the plot in panel d. The force was only effective for long-range separations that corresponded to a range over tmax – t ≥ 0.1. (f) The magnitude of the capillary force as a function of E.


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Figure 9. Capillary interactions of two ellipsoid particles with E = 4.28 as they approached with the ss configuration. (a) Interparticle separation (r) as a function of tmax – t. For comparison, the profile for the tt configuration with the same E value in Fig. 8a was also inserted. (b) The corresponding plot on a log-log scale. (c) Regenerated plot of r versus tmax – t based on the average value of the fitting parameters obtained in panel b. The curve for the tt configuration was also inserted. (d) The capillary force converted by the Stokes drag force, in which the velocity was obtained from the plot in panel c. The force was only effective for long-range separations that corresponded to a range over tmax – t ≥ 0.1.


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Heterogeneous Capillary Interactions of Interface-Trapped Ellipsoid

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