Geometric Effects of Colloidal Particles on Stochastic Interface

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Geometric Effects of Colloidal Particles on Stochastic Interface Adsorption Dong Woo Kang, Byung Gyu Park, Kyu Hwan Choi, Jin Hyun Lim, Seong Jae Lee, and Bum Jun Park Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01124 • Publication Date (Web): 09 Jul 2018 Downloaded from http://pubs.acs.org on July 11, 2018

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Geometric Effects of Colloidal Particles on Stochastic Interface Adsorption Dong Woo Kang1, Byung Gyu Park2, Kyu Hwan Choi1, Jin Hyun Lim1, Seong Jae Lee2*, 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

KEYWORDS: ellipsoid particle, heterogeneity, interface adsorption, optical laser tweezers, fluid-fluid interface

ACS Paragon Plus Environment

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ABSTRACT: The stochastic interface adsorption behaviors of ellipsoid particles were investigated using optical laser tweezers. The particles were brought close to the oil-water interface, attempting to attach forcefully to the interface. Multiple attempts of the particle attachments statistically quantified the dependence of the adsorption probability on the particle aspect ratio. It was found that the adsorption probability proportionally increased with the aspect ratio due to the decrease in electrostatic interactions between the charged particles and the charged interface for higher aspect ratio particles. In addition, the adsorption holding time required for the interface attachments was found to increase as the aspect ratio decreased. Notably, the probabilistic adsorption behaviors of the ellipsoid particles and the holding time dependence revealed that the particle adsorption to the interface occurred stochastically, not deterministically. We also demonstrated that the adsorption behaviors measured on a single particle scale were consistent with the gravity-induced spontaneous adsorption properties performed on a large scale with regard to the non-deterministic adsorption behaviors and the aspect ratio dependence on the adsorption probability.


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INTRODUCTION The surface energy of emulsions composed of two immiscible fluids can reduce via adsorption of molecular surfactants to the emulsion surface.1-2 Similarly, adsorption of typical colloidal particles to fluid-fluid interfaces or emulsion surfaces decreases the contacting interfacial area between the two phases, leading to a reduction in surface energy and thus preventing coalescence of the particle-stabilized emulsions.3-9 Such Pickering emulsion systems have been of great interest in many research and industrial fields (e.g., foods,10-11 pharmaceuticals,12-14 cosmetics,15-16 catalysis,17-18 and separation processes19-20) due to the relative ease of manufacture of colloidal particles with desired morphologies and surface functionalities as well as their low production cost. There are two main focuses on the study of interfacial phenomena of colloidal particles. One area of research is related to the configurations,21-23 interactions,24-30 assemblies,31-38 rheology,39-41 and stabilization of particles that are already trapped at fluid-fluid interfaces.6, 8, 42-45 The other focuses on the interactions and dynamics between particles and fluid interfaces prior to or upon interface adsorption of the particles.46-52 Particularly, in the latter case, a fundamental understanding of the interactions between charged colloidal particles and charged fluid interfaces is important to clarify the formation efficiency of Pickering emulsions, the rheological properties of emulsion solutions, and the stability of emulsions against coalescence processes. For example, Hartley et al. directly measured the surface forces between a silica particle and a hydrocarbon oil using an atomic force microscope.46-47 Wang et al. investigated the effects of charge-image charge interactions on the particle adsorption behavior at a fluid-fluid interface and the efficiency of Pickering emulsion formation.50 Kelleher et al. demonstrated that negatively charged particles dispersed in a nonpolar solution with a moderate dielectric constant created their image charges in the aqueous phase, and the charge-image charge interaction attracted the particles toward the oil-


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water interface.48-49 Park et al. proposed that the electrostatic forces between charged particles and a charged oil-water interface had an effect on the lateral interactions between the particles.52 The interface above the particles could be deformed due to the presence of electrostatic disjoining pressure and the lateral capillary attraction decreased the surface free energy leading to the formation of hexagonally packed colloidal crystals. Recently, we reported that the adsorption of polystyrene microspheres onto an oil-water interface occurred non-deterministically and that the adsorption probability decreased logarithmically as the ionic strength decreased.51 The heterogeneous interface adsorption could be attributed to surface charge heterogeneities.53-54 In the context of further clarifying the nondeterministic interface adsorption behaviors of colloidal particles to fluid-fluid interfaces, herein the effects of the geometric anisotropy of the colloidal particles were investigated quantitatively. The ellipsoid particles were fabricated with different values of aspect ratio (E, the ratio of the major axis to the minor axis) by mechanically stretching a polymer (polyvinyl alcohol) film containing embedded spherical polystyrene particles at an elevated temperature. The value of E could be tuned by how much the film was stretched. A planar oil-water interface was formed in a specially designed flow cell, in which the ellipsoid particles were dispersed in the aqueous subphase. Optical laser tweezers were employed to trap individual particles and then translate them upward to the oil-water interface. The probability of their interface adsorption was then determined by multiple attempts to attach the optically trapped particles with different E to the interface. It was further confirmed that the dependence of the measured adsorption behavior on E at a single particle scale showed a good agreement with the large-scale properties on the basis of gravityinduced spontaneous adsorption.


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EXPERIMENTAL SECTION Synthesis of polystyrene microspheres. Spherical polystyrene (PS) particles with a diameter of ~3 μm were prepared by the dispersion polymerization method.55 100 g of isopropanol (Sigma-Aldrich), 0.2 g of azobisisobutyronitrile (AIBN, Daejung Chemicals, Korea), and 5 g of ultrapure water (resistivity > 18.2 MΩ cm) were added to a reactor equipped with a reflux condenser. Upon magnetically stirring the solution, 20 mL of styrene (Samchun Chemicals, Korea) and 0.8 g of polyvinylpyrrolidone (PVP, Sigma-Aldrich) were added to the reactor, and the solution was then held overnight at a temperature of 70 °C under nitrogen. The resulting particle dispersion was purified by washing with water and isopropanol followed by centrifugation several cycles. The ζ-potential of the PS particles was approximately ‒47 mV, as measured by a Zetasizer (ZEN3600, Malvern Instruments). The obtained PS microspheres were used as seed particles for fabrication of the ellipsoid particles. Fabrication of ellipsoid particles. PS ellipsoids were prepared by the previously reported film stretching method.56-57 To prepare the PVA films with embedded PS microspheres, 15 g of PVA was dissolved in 150 mL of water at 80 °C by magnetically stirring for 5 h. After cooling at ambient conditions, 0.1 wt% PS spheres dispersed in water were added to the PVA aqueous solution. The mixture was poured on a horizontal tray and dried to obtain a PVA film with embedded PS particles. After cutting the film into a 2 cm × 8 cm rectangle, it was firmly fixed to the tensile grips of a universal testing machine (UTM). The gripped film was immersed in a silicone oil bath and the bath temperature increased to 135 °C, which was above the glass transition temperature of both the PVA film and PS particles. Then, the film and the embedded particles were stretched simultaneously at a rate of 0.5 mm/s, leading to the formation of ellipsoid particles. The E value of the ellipsoid particles could be readily tuned according to how much the film was


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elongated. After dissolving the PVA matrix in water at 80 °C, the released ellipsoid particles were collected and purified by repeated cycles of centrifugation and redispersion. The E values were determined by analyzing scanning electron microscopy images (SEM, AIS2000C, Seron Technologies Inc.), as shown in Fig. S1. Preparation of flow cell. The flow cell was composed of inner and outer cylinders (Fig. 1).27 The outer cylinder was constructed with a cylindrical glass ring with 25 mm diameter and 11 mm height. A circular coverslip (Marienfeld, no. 1.5H) was attached to one side of the glass ring using a UV adhesive (Norland Optical Adhesive 81, Norland) cured by UV exposure (365 nm) for >20 min. The inner cylinder was made of two cylindrical rings. A Teflon ring (12 mm diameter and 9 mm height) was inserted tightly into an aluminum ring (14 mm diameter and 8 mm height). The inner cylinder was placed on three small glass spacers attached to the coverslip surface of the outer cylinder. The glass pieces were cut from a circular coverslip and were glued with UV adhesive onto the coverslip surface of the outer cylinder. The flow cell was mounted on a motorized stage (Marzhauser Wetzlar) of an inverted microscope (Nikon, Ti-U) equipped with an optical laser tweezer apparatus.51, 58 To form an oil-water interface, 2 mL of ultrapure water (resistivity > 18.2 MΩ cm) and 0.2 mL of n-decane (Acros Organics) were consecutively added to the flow cell. The n-decane was filtered through aluminum oxide particles (Acros Organics, acidic activated, particle size 100−500 μm) to eliminate any polar impurities prior to use. Approximately 1.4 ‒ 1.5 mL of water was removed from the outside of the inner cylinder to pin the interface to the junction of the Teflon and aluminum rings. Then, 50 μL of a diluted particle solution was dropped onto the surface of the n-decane super-phase in the inner cylinder, leading to submergence of the particles in the aqueous phase.


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Optical laser tweezer apparatus. Optical laser tweezers were built with an inverted microscope (Nikon, Ti-U).51, 58 A 10W CW Nd:YAG laser was used to generate a laser beam with a wavelength of 1064 nm, which was passed through an acousto-optic deflector (AOD, Optoelectric DTSXY-400-1064 2D). The laser beam entered and filled the back aperture of a water immersion objective (Nikon, CFI Plan Apochromat VC 60× with a numerical aperture (NA) of 1.2 and a working distance of 300 μm). The beam leaving the objective was highly focused on a focal plane generating an optical trap. The x and y positions of the optical trap were manipulated by diffracting the laser beam entering the AOD, which was operated by the LabVIEW software. The laser power (P20× ) used in the work was measured by an optical power meter (Thorlabs, PM100D) above a lens with a low NA (0.45) objective (CFI Super Plan Fluor 20×). Details of the optical laser tweezers setup were provided in previous publications.58-59 Forced interface adsorption. For determining the adsorption probability of the colloidal particles to the oil-water interface, an ellipsoidal particle dispersed in the water phase was optically trapped and translated upward to the oil-water interface by manually adjusting a knob of the inverted microscope (Fig. 1). After it was observed with a CCD camera (KP-M1AN, Hitachi) installed on the microscope that the trapped ellipsoid particle reached the interface, the particle was held for a holding time (th) of ~5 s without further upward translation, and the optical trap was translated downward to determine if the particle was attached to the interface. Notably, when the ellipsoid particle dispersed in water was held by optical laser tweezers, the major axis of the particle was always aligned along the direction of the laser beam, and thus, the particle orientation was perpendicular to the oil-water interface before the interface adsorption occurred.60 More than 𝑁𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑒𝑑 ≥ 25 particles were attempted to attach forcefully to the interface at each condition depending on the aspect ratio (E), the laser power (P20× ), the NaCl concentration (I), and the


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holding time. The adsorption probability was then obtained by 𝑁𝑎𝑡𝑡𝑎𝑐ℎ𝑒𝑑 /𝑁𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑒𝑑 , in which 𝑁𝑎𝑡𝑡𝑎𝑐ℎ𝑒𝑑 was the number of particles attached to the interface. Flow cell for gravity-induced interface adsorption. Approximately 5 μL of particle solution was introduced into a glass capillary tube (WPI, I.D. = 0.58 mm).51 One side of the capillary tube was covered with a coverslip (Marienfeld, no. 1.5H) and sealed by vacuum grease to prevent water evaporation. The other side of the tube was immersed in n-decane, leading to the formation of an oil-water interface at the end of tube. The particles in the tube were allowed to migrate downward due to gravity and eventually reached the interface.

RESULTS AND DISCUSSION To statistically determine the interface adsorption probability of the ellipsoid particles as a function of E, it was required to establish a simple criterion to determine if the optically trapped particles were attached to the oil-water interface. As shown in Figs. 1 and 2, upon trapping an ellipsoid particle in the aqueous sub-phase, the particle was oriented vertically along the direction of the laser beam (Figs. 2a,d).60 The trapped particle was translated upwards, eventually reaching the oilwater interface. In this configuration, the tip region of the particle came closest to the interface. When the optical trapping force was weaker than that of the electrostatic repulsive force between the particle surface and the interface, the electrostatic repulsion hindered the particle attachment to the interface, and thus, the particle stayed beneath the interface. When the particle was pushed against the interface further, the particle was slightly tilted relative to the interface (Fig. 2b). To confirm that the particle was not attached to the interface, the optical trap was consecutively translated downward. In this case, the particle was re-orientated vertically when the separation


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between the particle and the interface was larger than the radius of the particle major axis (Fig. 2c). In the opposite case, when the optical trapping force was stronger than that of the electrostatic force, the particle was transferred and attached to the interface. The particle adsorbed at the interface was laid horizontally to increase the cross-section area displaced by the interface (Fig. 2f).21, 23 The adsorption moment was noticeable when the brightness of the particle suddenly changed (Figs. 2e,f). The gray value obtained from the image analysis using the ImageJ software increased sharply after adsorption occurred, as shown in Fig. 2h. The gray value could be calculated by RGB values along the dotted line in Figs. 2e,f using the formula of gray = 0.299R + 0.587G + 0.114B.61 The optical force in the experiment was not sufficiently strong to detach the particle from the interface.51,

60

Therefore, upon translating the optical trap downward, the

horizontal configuration of the particle at the interface remained unchanged and the shape of the particle blurred because the focal plane moved away from the particle (Fig. 2g). It was previously reported that the detachment energy of an ellipsoid particle with E = 4.28 was extremely strong (~108 kBT) and the corresponding force was approximately 100 times greater than the optical trapping force in typical trapping conditions, consequently demonstrating that the interface adsorption process was irreversible.60 Note that the attachment or detachment energy was an important factor to determine the equilibrium configurations of the particles that were already located at the interface,21 while the interface adsorption probability was independent of the strength of the attachment energy. In addition, when the particle approached the oil-water interface, the optical trapping force in the range of the laser powers used in the current work was not sufficiently strong to deform the oil-water interface, and therefore, any effects of the local interface deformation on the adsorption behaviors could be negligible.59 On the basis of the interface adsorption criteria, attachment to the oil-water interface of


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more than 25 ellipsoid particles at each condition depending on E, P20×, and I was attempted. As shown in Fig. 3a, the adsorption probability proportionally increased as E increased and I decreased. We postulated that the adsorption probability depending on E and I could be attributed to the geometric effects on the electrostatic interactions between the particles and the oil-water interface that were both charged negatively.51-52 It was previously reported that the oil-water interface could be charged due to the formation of molecular dipoles of hydroxyl ions near the interface.62 Qualitatively, the electrostatic repulsion between a curved surface (e.g., ellipsoid) and a plane (e.g., interface) likely possessed a proportional relationship with the average value (〈𝐻𝑖 〉) of the distance between any points in the upper half of the ellipsoid surface and the corresponding points projected to the interface, as shown in Fig. S2. As E increased, 〈𝐻𝑖 〉 decreased, and therefore, the magnitude of the electrostatic interactions decreased. Consequently, the particles with larger E could more closely approach the oil-water interface, increasing the adsorption probability. More quantitatively, the electrostatic repulsion between the ellipsoid particle and the interface could be numerically calculated by the Derjaguin approximation.52, 63 The electrostatic double layer interactions between two charged planar surfaces were determined by the nonlinear Poisson-Boltzmann (PB) equation,63

̂ 𝑑2 𝜓 ̂ 𝑑ℎ 2

= sinh 𝜓̂,

(1)

where 𝜓̂ = 𝑒𝜓/𝑘𝐵 𝑇 is the nondimensionalized potential of the charged planar surface, ℎ̂ = 𝜅ℎ is the nondimensionalized surface-to-surface distance, kB is the Boltzmann’s constant, and T is the 1000𝑒 2 𝑁𝐴2𝐼

temperature. 𝜅 = √

𝜀 𝑊 𝜀 0 𝑘𝐵 𝑇

is the inverse Debye length, where e is the elementary charge, 𝜀0


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and 𝜀𝑊 are the dielectric constants of vacuum and water, respectively, NA is Avogadro’s number, and I is the ionic strength. To solve the non-linear PB equation, Eq (1) was separated into the two ̂

𝑑𝜓 first order ordinary differential equations (ODEs) by substituting 𝜓1 = 𝜓̂ and 𝜓2 = 𝑑ℎ̂ = 𝜓̂′,

𝜓1′ = 𝜓2 and 𝜓2′ = sinh 𝜓1

(2)

The numerical method to solve the PB equation between two planes (a and b) with different surface potentials 𝜓̂𝑎 and 𝜓̂𝑏 was proposed by Chan et al.64-65 The potential 𝜓1 = 𝜓̂ was integrated numerically from ℎ̂ = 0, at which the corresponding potential and the potential gradient were 𝜓1 = 𝜓̂𝑚 and 𝜓2 = 0 due to symmetry (initial value problem, IVP). When the integrated potential reached the value of 𝜓̂𝑎 , the distance |ℎ̂𝑎 | between the plane-a and ℎ̂ = 0 was determined. Similarly, |ℎ̂𝑏 | was obtained for 𝜓̂𝑏 at the plane-b, and the separation distance between the two planes was given by ℎ̂ = |ℎ̂𝑎 | + |ℎ̂𝑏 |, which was related to 𝜓̂𝑚 . The same procedure was repeated to further calculate ℎ̂ as a function of 𝜓̂𝑚 , and the electrostatic disjoining pressure between the two planes was obtained by substituting 𝜓̂𝑚 into the equation,

̂ 𝐼𝑉𝑃 = Π𝐼𝑉𝑃 = 2000𝑁𝐴 𝐼𝑘𝐵𝑇 (cosh 𝜓̂𝑚 − 1). Π 𝛾𝜅 𝛾𝜅

(3)

For large separations (i.e., ℎ̂ ≫ 1) between two parallel surfaces with identical surface potentials, the superposition approximation yielded the analytical expression of the electrostatic disjoining pressure,63, 66


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̂

𝛱𝑒𝑙 = 64 × 103 𝑁𝐴 𝐼𝑘𝐵 𝑇Υ02 𝑒 −ℎ ,

(4)

̂ 𝜓

where Υ0 = tanh 4 is the Gouy-Chapman parameter. In addition, the negative charge present in the water phase could generate its image charge in the oil phase that was negative.50 The magnitudes of the two negative charges in the two phases were similar to each other. The formation of the image charge also induced the electrostatic repulsion between the two surfaces with a separation of 2h,

̂

𝛱𝑖𝑚𝑔 = 64 × 103 𝑁𝐴 𝐼𝑘𝐵 𝑇Υ02 𝑒 −2ℎ .

(5)

The disjoining pressure due to the van der Waals interaction was expressed as,66

𝐴𝑒𝑓𝑓

𝛱𝑣𝑑𝑤 = − 6𝜋ℎ3.

(6)

𝐴𝑒𝑓𝑓 is the non-retarded Hamaker constant for two macroscopic phases (PS(1) and n-decane(2)) interacting across a medium (water(3)) given by,

3

𝜀 −𝜀

𝜀 −𝜀

𝐴𝑒𝑓𝑓 = 4 𝑘𝐵 𝑇 (𝜀1 +𝜀3) (𝜀2 +𝜀3) + 1

3

2

3

3ℎ𝑃 𝜈𝑒

(𝑛12 −𝑛32 )(𝑛22−𝑛32)

8√2

√𝑛12 +𝑛23 √𝑛22 +𝑛32 {√𝑛12 +𝑛32+√𝑛22+𝑛23 }

,

(7)

where ℎ𝑃 = 6.626 × 10−34 Js is the Plank’s constant, 𝜈𝑒 ≈ 3 × 1015 s −1 is the plasma frequency of the free electron gas, and 𝑛j (j = 1,2,3) is the refractive indices.67 The electrostatic force between a charged ellipsoid and a charged, non-deformable plane


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could be estimated by the Derjaguin approximation.63, 67-68 The ellipsoid surface was divided into a stack of infinitesimal annuli with surface area dAi. The total electrostatic repulsion between the ellipsoid surface and the planar surface could be obtained from the summation of electrostatic repulsions 𝛱𝑖 between each annulus on the ellipsoid surface and the corresponding annulus projected to the planar plane, expressed as 𝐹 = ∑𝑛𝑖=1 𝛱𝑖 × 𝑑𝐴𝑖 , where n is a sufficiently large ̂ 𝐼𝑉𝑃 in Eq. number (e.g., n = 5×105). We used the numerical solution of non-linear PB equation Π (1-3) to improve the calculation accuracy,

𝐹𝐼𝑉𝑃,𝐷𝑒𝑟𝑗 = ∑𝑛𝑖=1 𝛱𝐼𝑉𝑃,𝑖 × 𝑑𝐴𝑖 .

(8)

For comparison, Eq. (4) was also used to numerically calculate the corresponding force between the ellipsoid surface and the planar interface based on the Derjaguin approximation,

𝐹𝑒𝑙,𝐷𝑒𝑟𝑗 = ∑𝑛𝑖=1 𝛱𝑒𝑙,𝑖 × 𝑑𝐴𝑖 .

(9)

The analytical expression for Eq. (9) could be obtained by replacing 𝑑𝐴𝑖 = 2𝜋𝑥𝑑𝑥, where 𝑑𝑥 is the thickness of each annulus and x is the distance from the major axis to the annulus (Fig. 4e). Differentiation of the geometric relationship of ℎ − 𝑠 = 𝑐 − 𝑧 = 𝑐(1 − √1 − 𝑐−ℎ+𝑠 𝐸2

𝑑ℎ, and thus, 𝑑𝐴𝑖 = 2𝜋 𝑥2

𝑐−ℎ+𝑠 𝐸2

𝑥2 𝑎2

) yielded 𝑥𝑑𝑥 =

𝑐

𝑑ℎ; the equation of an ellipse with 𝐸 = 𝑎 on the x-z Cartesian

𝑧2

plane is 𝑎2 + 𝑐 2 = 1, where an arbitrary point is given by (x,z). As shown in Fig. 4e, s is the closest distance between the ellipsoid and the interface, and h is the distance between each annulus and the interface. After substituting Eq. (4) into Eq. (9) and integrating from ℎ = 𝑠 to ℎ = 𝑖𝑛𝑓., the


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resulting interaction force was

𝜅𝑐−1

𝐹𝑒𝑙,𝑎𝑛𝑎𝑙 = 128 × 103 𝜋𝑁𝐴 𝐼𝑘𝐵 𝑇Υ02 𝐸2 𝜅2 𝑒 −𝜅𝑠 .

(10)

The upper limit ℎ = 𝑖𝑛𝑓. for the integration was likely rational because the disjoining pressure decayed exponentially as the distance increased. For a spherical particle with E = 1 and a radius c = Rs, Eq. (10) became,

𝐹𝑒𝑙,𝑎𝑛𝑎𝑙 = 128 × 103 𝜋𝑁𝐴 𝐼𝑘𝐵 𝑇Υ02

𝜅𝑅𝑠 −1 −𝜅𝑠 𝑒 . 𝜅2

(11)

Similarly, the force resulting from the charge-image charge interaction in Eq. (5) was calculated numerically using the Derjaguin approximation,

𝐹𝑖𝑚𝑔,𝐷𝑒𝑟𝑗 = ∑𝑛𝑖=1 𝛱𝑖𝑚𝑔,𝑖 × 𝑑𝐴𝑖 .

(12)

After substituting of Eq. (5) into Eq. (12) and integrating from ℎ = 𝑠 to ℎ = 𝑖𝑛𝑓., the analytical expression of the corresponding force was,

𝜅𝑐−1

𝐹𝑖𝑚𝑔,𝑎𝑛𝑎𝑙 = 64 × 103 𝜋𝑁𝐴 𝐼𝑘𝐵 𝑇Υ02 𝐸2 𝜅2 𝑒 −2𝜅𝑠 .

(13)

To compare the relative strengths of the van der Waals interactions (Eq. (6)) and the electrostatic repulsions (Eq. (3) and (4)) between two planar planes, it was assumed that the two planes with an equal surface potential (𝜓𝑎 ≈ 𝜓𝑏 ≈ −47 mV) were made of PS and n-decane


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Langmuir

interacting with each other across water (I = 2.2 × 10-6 M). The non-linear PB equation (Eq. (1) and (2)) was solved numerically and the resulting potential profiles as a function of κh at given values of 𝜓𝑚 were shown in Fig. 4a. The distance between the two planes ℎ̂ = |ℎ̂𝑎 | + |ℎ̂𝑏 | was related to each 𝜓𝑚 (red squares in Fig. 4b) and, consequently, the electrostatic disjoining pressure Π𝐼𝑉𝑃 (black circles in Fig. 4b) was obtained using Eq. (3). The numerical solution Π𝐼𝑉𝑃 was consistent with the result from Eq. (4) over the range of large separations (ℎ̂ ≫ 1), whereas a small deviation was found at ℎ̂ ≲ 1 (Fig. 4c). The magnitude of the van der Waals interactions (Eq. (6)) with the effective Hamaker constant Aeff = 7.35 × 10-21 J (Eq. (7)) was negligibly small compared to that of the electrostatics, and thus, the van der Waals interaction was not considered (Fig. 4c). For the spherical PS particle (E = 1 and Rs = 1 µm) and the oil-water interface (assuming 𝜓𝑃𝑆 ≈ 𝜓𝑖𝑛𝑡𝑒𝑟𝑓 ≈ −47 mV), the repulsive force between the two surfaces was calculated using the analytical expressions (Eq. (11) and (13)) and the numerical calculations based on the Dejaguin approximation (Eq. (8), (9), and (12)). As shown in Fig. 4d, the force curves of 𝐹𝑒𝑙,𝑎𝑛𝑎𝑙 (Eq. (11)) and 𝐹𝑒𝑙,𝐷𝑒𝑟𝑗 (Eq. (9)) showed an excellent agreement with each other, demonstrating the validity of the numerical calculation via the Dejaguin approximation. 𝐹𝐼𝑉𝑃,𝐷𝑒𝑟𝑗 was found to be slightly less compared to 𝐹𝑒𝑙,𝐷𝑒𝑟𝑗 due to the less magnitude of Π𝐼𝑉𝑃 than Π𝑒𝑙 (Fig. 4c). The charge-image charge forces obtained from Eq. (13) (𝐹𝑖𝑚𝑔,𝑎𝑛𝑎𝑙 ) and Eq. (12) (𝐹𝑖𝑚𝑔,𝐷𝑒𝑟𝑗 ) were also consistent with each other, and their magnitude was found to be ~O(1) weaker than that of the 𝐹𝑒𝑙,𝐷𝑒𝑟𝑗 or 𝐹𝐼𝑉𝑃,𝐷𝑒𝑟𝑗 . Therefore, to understand the effect of electrostatics on the particle adsorption behaviors to the oilwater interface, we only considered 𝐹𝐼𝑉𝑃,𝐷𝑒𝑟𝑗 hereafter. Note that Eq. (1) - (3) was useful for calculating the electrostatic disjoining pressure and the corresponding force when two planes carried different surface potential values. For the experimental condition, the surface potential of PS particles was measured to be 𝜓𝑎 = 𝜓𝑃𝑆 ≈ −47


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mV and the surface potential of the oil-water interface was reported as 𝜓𝑏 = 𝜓𝑖𝑛𝑡𝑒𝑟𝑓 ≈ −80 mV due to the spontaneous adsorption of hydroxyl ions.62 After calculating 𝜓 as a function of |ℎ̂| at different values of 𝜓𝑚 (Figs. 5a,b), Π𝐼𝑉𝑃 was obtained as a function of ℎ̂ by substituting 𝜓𝑚 into Eq. (3) (Fig. 5b). The corresponding force 𝐹𝐼𝑉𝑃,𝐷𝑒𝑟𝑗 using the Derjaguin approximation (Eq. (8)) was found to increase as E decreased, as shown in Fig. 5c. For instance, the magnitude of 𝐹𝐼𝑉𝑃,𝐷𝑒𝑟𝑗 for the particle with E = 8.13 was approximately ten times lower than that with E = 1. The calculated results sufficiently confirmed the postulation that as E increased, 〈𝐻𝑖 〉 decreased, thereby decreasing the magnitude of the electrostatic interactions. The decreased electrostatic repulsion for particles with larger E consequently promoted their interface adsorption probability, consistent with the experimental observations in Fig. 3a. To further demonstrate the adsorption phenomena that depended on the relative strength between the optical trapping force and the electrostatic repulsion, similar experiments were performed with different laser powers. As shown in Fig. 3a, the adsorption probabilities for all values of E were found to increase consistently as the laser power increased from 𝑃20× = 12.9 to 60.1 mW; the stronger laser power could deliver the particles closer to the interface. Importantly, the non-unity of the adsorption probability justified that the adsorption behavior of a particle to the interface was stochastic, not deterministic. Note that the orientation angle change of the particle when it approached from water to the interface prior to adsorption was approximately

Geometric Effects of Colloidal Particles on Stochastic Interface

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