Role of Geometry and Amphiphilicity on Capillary-Induced Interactions

Nov 11, 2013 - Moreover, Janus ellipsoids in contact exhibit a larger capillary force at ... Even uncharged colloidal particles in the absence of grav...
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Role of Geometry and Amphiphilicity on Capillary-Induced Interactions between Anisotropic Janus Particles Hossein Rezvantalab† and Shahab Shojaei-Zadeh*,†,‡ †

Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, New Jersey 08854-8058, United States ‡ Institute for Advanced Materials, Devices and Nanotechnology, 607 Taylor Road, Piscataway, New Jersey 08854-8019, United States ABSTRACT: We study the capillary interactions between ellipsoidal Janus particles adsorbed at flat liquid−fluid interfaces. In contrast to spherical particles, Janus ellipsoids with a large aspect ratio or a small difference in the wettability of the two regions tend to tilt at equilibrium. The interface deforms around ellipsoids with tilted orientations and thus results in energetic interactions between neighboring particles. We quantify these interactions through evaluation of capillary energy variation as a function of the spacing and angle between the particles. The complex meniscus shape results in a pair interaction potential which cannot be expressed in terms of capillary quadrupoles as in homogeneous ellipsoids. Moreover, Janus ellipsoids in contact exhibit a larger capillary force at side-by-side alignment compared to the tip-to-tip configuration, while these two are of comparable magnitude for their homogeneous counterparts. We evaluate the role of particles aspect ratio and the degree of amphiphilicity on the interparticle force and the capillary torque. The energy landscapes enable prediction of micromechanics of particle chains, which has implications in predicting the interfacial rheology of such particles at fluid interfaces.

1. INTRODUCTION Colloidal particles adsorbed at interfaces between immiscible fluids play an important role in a variety of technological applications. They are used to stabilize droplets in Pickering emulsions in foods, cosmetics, and oil recovery1−3 and to form functional membranes, colloidosomes, or capsules in pharmaceutical industries.4−6 Colloidal particles trapped at an interface generally result in a deformed meniscus due to the action of gravitational force, surface roughness, or electrostatic charges.7,8 The overlap of deformed menisci leads to energetic interactions between neighboring particles. Even uncharged colloidal particles in the absence of gravitational effects or surface roughness will experience mutual interactions if they are slightly nonspherical.9 These so-called capillary interactions strongly depend on the shape of the particles and can be tuned by modification of surface chemistry.10 Therefore, they can be employed for designing self-assembling systems and fabrication of novel materials and structures.11−14 A variety of colloidal structures such as permeable hollow capsules made from partially fused polymer particles, colloid-based foams and emulsions, close-packed uniform monolayers, and cross-linked nanoparticle membranes have been reported.15,16 The anisotropic character of colloidal particles has been categorized by introducing patchiness, aspect ratio, and faceting.17 A particular example of anisotropy by patchiness is a Janus particle, i.e., a particle with two areas of different surface chemistry or polarity. Each area can have different hydrophobicity, giving rise to an amphiphilic character. When dispersed in a bulk liquid, Janus particles exhibit a variety of stable structures and phase behaviors.18,19 At the interface between two immiscible fluids, amphiphilic Janus particles can © 2013 American Chemical Society

function similar to surfactant molecules as stabilizing agent, giving rise to so-called Pickering systems.20 While homogeneous particles with proper size and contact angle can also act as interface stabilizers, introducing surface heterogeneity is found to improve the thermodynamic stability of such systems.21−23 Amphiphilic particles have been used in several studies as effective surface active agents.24−26 Because of their emerging applications, much attention has been recently paid on developing efficient approaches for production of large quantities of Janus particles.27−29 Recent progress has also enabled the synthesis of nonspherical Janus particles, providing an additional degree of freedom in tuning the particle characteristics.30−32 Janus particles with shape anisotropy have been produced using several established methods including mechanical stretching, droplet microfluidics, and metal evaporation on patterned surfaces.33−35 Moreover, surface modification of each side of Janus particles enables tailoring their orientation at the interface, which leads to a variety of self-assembled structures.36−38 Several theoretical works have been focused on understanding the behavior of spherical Janus particles at liquid−fluid interfaces, such as their equilibrium orientation and capillaryinduced interactions.22,39−43 However, little is known about the interactions of nonspherical Janus particles despite the significant role of particle geometry on their self-assembly at the interface. Recently, equilibrium orientation of isolated amphiphilic ellipsoids and dumbbells at a flat oil−water Received: October 10, 2013 Revised: November 6, 2013 Published: November 11, 2013 14962

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The mechanical response of particle chains to tensile forces and bending moments might be useful in determining the rheological properties of colloidal assembly at the interface. Finally, we summarize the results in section 4 and discuss the implications of this numerical analysis in designing complex interfaces containing Janus particles.

interface has been reported for a range of aspect ratio and amphiphilicity.44,45 It is demonstrated that the orientation and vertical displacement of nonspherical amphiphilic particles are significantly influenced by their shape, aspect ratio, and surface properties. Predicting the assembly behavior of such particles and, in turn, the stability of particle-covered fluid interfaces requires the analysis of interparticle forces. For uncharged particles of negligible weight and surface roughness, the dominant force is the interface-mediated capillarity which arises due to overlapping of deformed menisci. The interface distortion can be caused by surface/chemical heterogeneity, shape anisotropy, or a combination of the two.8,46−49 The interactions between anisotropic particles with homogeneous surface wettability have been studied for various geometries.50,51 For homogeneous ellipsoids, experimental observations indicate the existence of a long-ranged quadrupolar attraction bringing the particle tips in contact.52 However, the presence of chemical heterogeneity in Janus particles substantially changes the meniscus shape around the particles. This results in a deviation of the power law dependence of pair interaction potential from the quadrupolar form developed for homogeneous ellipsoids. The difference between the wettability of the two regions is an important parameter in defining the magnitude of capillary forces and the resulting assembly of these particles. In the present work, we investigate the capillary-induced interactions between ellipsoidal Janus particles adsorbed at a planar liquid−fluid interface. In this case, the interface deformation is induced by a combination of shape anisotropy and surface heterogeneity. We numerically calculate the surface area of an isolated particle in each fluid phase and the area of deformed oil−water interface for a given orientation to find the interfacial energy. We then address the energetic interactions between two neighboring particles and use the capillary energy to characterize the equilibrium alignment of particle pairs. Energy landscapes enable the calculation of the power law dependency of pair potential on the interparticle distance for pairs of similar ellipsoids. The influence of polydispersity and amphiphilicity on capillary force and torque acting between the particles is also evaluated. Our findings can provide a better understanding on the interactions and assembly of nonspherical Janus particles at liquid−fluid interfaces. This article is organized as follows: In section 2, we introduce the parameters governing the geometry and surface properties of the anisotropic particles. We quantify the surface free energy of a system comprised of such particles placed at an oil−water interface and demonstrate the calculation of the meniscus shape via minimization of the surface energy. In section 3.1, we briefly mention the effect of aspect ratio and amphiphilicity on the interface shape and equilibrium orientation for isolated particles. Having established the positioning of isolated particles at the interface, we then consider pairs of similar Janus ellipsoids in section 3.2. The nature of interparticle forces is evaluated based on the shape of neighboring menisci. We find the optimum alignment of the particles as they undergo a rotational relaxation within the interface plane in order to achieve the minimum energy state. In section 3.3, we extend our analysis to interactions between ellipsoids of dissimilar dimensions. Because of the strong dependence of the interactions on surface chemistry of Janus particles, we devote section 3.4 to analyze the influence of amphiphilicity on interparticle force and torque. In section 3.5, we use the results of pair interactions to predict the mechanics of particle chains.

2. SIMULATION PROCEDURE 2.1. Geometry and Surface Properties of Ellipsoidal Janus Particles. In this study, we investigate the interactions between prolate Janus ellipsoids adsorbed at a flat oil−water interface. The geometry of the considered Janus particles is shown in Figure 1a. The aspect ratio, AR, is characterized as the

Figure 1. (a) Geometry of an ellipsoidal Janus particle at a planar oil− water interface. (b) A Janus ellipsoid at a deformed oil−water interface resulting in four different particle−fluid regions.

ratio of axes lengths: AR = c/a. Unless otherwise stated, the surface area of all ellipsoidal particles is set to A = 4πrs2, where rs represents the radius of a corresponding spherical particle (AR = 1). Janus ellipsoids are comprised of apolar and polar regions of differing wettability. The line separating the two regions is called Janus boundary. For ellipsoidal particles, Janus boundary is the intersection of ellipsoid’s surface and the plane perpendicular to ellipsoid’s long axis and passing through its center of mass. The orientation angle θr is defined as the angle between the particle’s long-axis in the upright orientation (i.e., the long axis of ellipsoid perpendicular to the interface) and that in a rotated orientation. We characterize the surface wettability of apolar and polar regions of Janus ellipsoids by three phase contact angles θa and θp, respectively. We assume that the particles possess two regions of opposite wettabilities, represented by θa = 90° + β and θp = 90° − β, where β is a parameter defining the amphiphilicity of the particle. This ensures that the center of mass of these Janus ellipsoids will be located at the oil−water interface, regardless of the values of β and θr. 2.2. Surface Energy Calculations. Janus ellipsoids with a large aspect ratio or a small difference in the wettability of the two regions tend to have a tilted orientation at equilibrium.44 In 14963

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such case, the apolar and polar regions are partially wetted by both fluid phases. For a given orientation of the particle at the interface, four regions will thus appear as shown in Figure 1b. The corresponding surface areas depend on the shape of the interface, which is numerically calculated such that the total surface energy is minimized. As a result, the total surface energy of the system consists of four different particle surface−fluid regions in addition to the bare liquid−fluid interface and is expressed as Ei = γowAow + γaoA ao + γawA aw + γpoA po + γpwA pw

(1)

Figure 2. Interface shape around isolated ellipsoidal particles of AR = 3, adsorbed to the interface according to their equilibrium orientation for (a) a homogeneous particle with θ = 60° and (b) a Janus particle with β = 10°.

where γij is the interfacial tension between i and j phases, and the subscripts a, p, o, and w indicate apolar and polar surfaces of the Janus particle and oil and water phases, respectively. The surface free energy of the system when the particle is trapped at the interface is denoted as Ei . The interfacial tensions are correlated through Young’s law for apolar and polar surfaces as γaw = γao − γow cos θa ,

γpo = γpw + γow cos θp

homogeneous ellipsoids exhibit long-ranged interaction potentials that can be roughly expressed in terms of capillary quadrupoles.46,50 However, the interface shape can be significantly affected in presence of a surface heterogeneity. Figure 2b shows the interface shape for a Janus ellipsoid with AR = 3 and an amphiphilicity of β = 10°. We note that in this case the interface takes a hexapolar shape, with three rises and three dips distributed around the particle. The sign of meniscus height with respect to far-field elevation is changed as it passes through the Janus boundary, due to opposite affinity of the two regions to the fluid phases. This results in two additional poles around the particle compared to the homogeneous case. The extent and positioning of the rises and dips can be tuned through modification of particle amphiphilicity and aspect ratio, suggesting a complex variation of meniscus shape around Janus ellipsoids. In order to quantify the interactions between neighboring particles, we first need to evaluate how isolated Janus ellipsoids of various geometry and amphiphilicity orient at the interface. It has been shown that the equilibrium orientation of Janus ellipsoids strongly depends on their aspect ratio and amphiphilicity.44 The equilibrium orientation of such anisotropic particles has been quantified by assuming a flat interface shape around the particle. Here, we extend this analysis by considering the actual case of a deformed meniscus and including several amphiphilicities as required for investigating the capillary interactions between particle pairs. For each particle, we calculated the surface energy based on eq 1 as a function of the orientation angle θr, and the equilibrium orientation corresponds to minimized surface energy. Figure 3 shows the equilibrium orientation of isolated Janus ellipsoids as a function of their aspect ratio. The predicted equilibrium orientations for β = 30° closely match those reported for particles of similar geometry based on a flat interface assumption.44 As the aspect ratio is increased, the equilibrium orientation of Janus ellipsoids is dominated by the tendency to reduce the bare oil−water interfacial area and, in turn, the interfacial energy. Consequently, a tilted orientation is preferred at equilibrium. On the other hand, increasing the amphiphilicity enhances the affinity of apolar and polar regions to oil and water phases, respectively. As a result, the particle’s tendency to reduce the attachment energy by increasing the interfacial area between each region, and its preferred fluid phase has a dominant effect on inducing the upright orientation. 3.2. Capillary Interactions between Neighboring Janus Ellipsoids. We showed in previous section that single Janus ellipsoids with large aspect ratio or small amphiphilicity induce a hexapolar interface deformation. If the menisci formed

(2)

We numerically calculate the surface−fluid areas in eq 1 through an optimization algorithm, such that the total surface free energy Ei is minimized. The shape of the interface corresponding to minimum surface energy is obtained through gradient descent optimization method in Surface Evolver.53 In the case of capillary interactions between neighboring particles, we measure the surface energy with respect to that of the noninteracting far-located particles and then normalize it with γrs2 (where γ is the interfacial tension between the two fluid phases, e.g., γow in the case of an oil−water interface). This normalized surface energy is thus applicable to any liquid−fluid interface (such as air−water interface as in particle-stabilized foams) and is referred to as “capillary energy” in this article. The capillary force and torque are calculated based on variation of this energy via translation and rotation of the particles at the interface.

3. RESULTS AND DISCUSSION 3.1. Single Janus Ellipsoid at an Oil−Water Interface. For spherical Janus particles, the equilibrium orientation results in apolar and polar hemispheres fully exposed to oil and water, respectively. Thus, the sphere takes an upright orientation of θr = 0° irrespective of its degree of amphiphilicity. However, this is not the case for elongated Janus ellipsoids due to their geometric anisotropy. The contact angle requirement cannot be satisfied around the ellipsoid unless the interface is deformed. If the ellipsoid is homogeneous, the equilibrium orientation is where the particle makes the largest hole in the interface.51 Thus, the particle orients such that the larger axis is parallel to the liquid−fluid interface. For Janus ellipsoids, each region has a preferred affinity to one of the fluid phases. As a result, the interface shape depends on the difference between contact angles of the two regions. Figure 2a shows the interface shape around a homogeneous ellipsoid with a three-phase contact angle of θ = 60° and an aspect ratio of AR = 3, oriented with its equilibrium orientation of θr = 90° at the interface. We observe that the interface is quadrupolar, with rises along the sides and depressions on the tips. This is in agreement with the analytical solution to Young−Laplace equation based on multipole expansion of the interface shape, where it can be shown mathematically that a quadrupole is the lowest allowed term for such anisotropic particles.46,50 Experimental observations also suggest that 14964

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other in a tip-to-tip arrangement, they can initially have either of the configurations shown in Figure 4. The surface energy is minimized when particles possess opposite orientation angles, leading to overlapping of menisci with similar signs and thus energetic capillary attractions. Consequently, if two particles are oriented similarly as in Figure 4a, it is energetically favorable for one particle to flip toward the configuration shown in case b. Moreover, it is possible that overlapping of interface distortions forces the particles to slightly rotate out of the equilibrium orientation angle θr,eq that each one takes in an isolated configuration. We performed simulations to calculate the surface energy for various configurations by rotating either one particle at a time or both particles simultaneously around θr,eq. Our results indicate that for all aspect ratios and amphiphilicities considered the minimized surface energy state for particle pairs occurs once each particle is placed exactly according to its isolated equilibrium orientation angle. Any deviation from this arrangement results in unfavorable wetting of apolar and polar regions, thus leading to increased interfacial energy. As a result, a pair of Janus ellipsoids will be oriented at ±|θr,eq| in equilibrium. Under such configuration, attractive forces appear between the particles and they approach each other until they touch. 3.2.2. Rotational Relaxation of the Particles within the Interface Plane. In addition to the orientation normal to the interface plane, the particles can rotate about each other within the interface plane, giving rise to a preferred alignment. Figure 5a depicts the orientation of two neighboring Janus ellipsoids

Figure 3. Equilibrium orientation of an isolated ellipsoidal Janus particle at a liquid−fluid interface as a function of the aspect ratio (AR) and amphiphilicity (β).

around two such neighboring particles overlap, interfacemediated capillary interactions will arise between them. We quantify the interactions between two particles of similar shape and wettabilities by calculating the optimized interface shape and the corresponding surface energy as a function of the separation distance. It should be noted that in our analysis we assume that the particles are uncharged and also neglect the effect of gravity-induced deformation which is significant for particle sizes larger than ∼10 μm.7,54 Additionally, the effect of line tension is neglected since the magnitude of this force is negligible for smooth particles with a characteristic size larger than 1 nm.22,28 Capillary interactions between neighboring particles can be attractive or repulsive, depending on the sign of overlapping menisci.55 When the two particles possess similar orientation angles, the interacting menisci are of the opposite sign, as shown in Figure 4a. Consequently, the surface area is increased as the particles approach each other, and the free energy of the system increases correspondingly. This leads to a repulsive capillary force between the particles. On the other hand, when the particles possess orientation angles of reverse sign, the two menisci are in opposite directions as in Figure 4b, leading to capillary attraction. Generally, two Janus particles approaching one another adjust their orientations in order to minimize the total surface free energy of the system. 3.2.1. Adjustment of Orientations Normal to the Interface Plane. An isolated Janus ellipsoid takes the equilibrium orientation corresponding to minimum surface energy, as shown in Figure 3. Because of symmetry of rotation with respect to the normal to interface plane, the equilibrium orientation can be considered as ±|θr,eq|, meaning that the particle can be tilted in either directions at the interface. Therefore, considering two similar particles approaching each

Figure 5. (a) Parameters defining the interfacial configuration of two neighboring Janus ellipsoids within the interface plane (top view). (b, c) Schematic illustration of two Janus ellipsoids oriented in tip-to-tip and side-by-side alignments, respectively.

looking downward at the interface. The angle between long axes of the particles is called the bond angle ϕ, where ϕ = 0° and ϕ = 180° correspond to tip-to-tip and side-by-side particle alignments respectively, as shown in Figure 5b,c. The

Figure 4. Meniscus shape between two ellipsoidal Janus particles with AR = 5, β = 30° oriented at (a) θr1 = θr2 = 85° and (b) θr1 = 85°, θr2 = −85°. 14965

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interaction potentials is due to different shapes of overlapping regions near the Janus boundary (for side-by-side case) and far from the boundary (for tip-to-tip alignment). The results thus indicate a strong dependence of the interaction potential on the bond angle between the particles. In general, the pair potential between Janus ellipsoids can be expressed as

equilibrium configuration adopted by particle pairs corresponds to the bond angle ϕ, resulting in minimum surface energy. 3.2.3. Dynamics of Interactions toward the Equilibrium Configuration. Consider a pair of similar Janus ellipsoids with a = 2.5 μm, AR = 5, and β = 30° placed at the interface with equilibrium orientations of ±85°. We first evaluate the variation in capillary energy as the particles approach in a side-by-side alignment. A representative plot of the pair potential between such particles is shown in Figure 6a as a function of the center-

U /(γA) ∼ dcc m

(3)

where γ is the interfacial tension between the fluid phases, A is the ellipsoid surface area, and m = m(β,AR,ϕ) is a function of the amphiphilicity, aspect ratio, and bond angle between the particles. Therefore, the interaction potential between Janus ellipsoids has a complex form compared with that of their homogeneous counterparts. From energy landscapes, one can determine the interparticle capillary force as F = ∂U/∂dcc. Figure 6b shows the normalized capillary force as a function of minimum surface-to-surface separation d for both tip-to-tip and side-by-side configurations. The positive value of the force confirms the attractive nature of interactions in both cases. Additionally, the capillary force is finite in contact, decreases upon increasing the interparticle distance, and vanishes for separations larger than ∼4rs. The capillary force at contact for tip-to-tip ellipsoids is F ∼ 0.01γrs and for two ellipsoids aligned side-by-side is ∼0.15γrs, which is much stronger than that of the former configuration. Consequently, pairs of side-by-side ellipsoids are less susceptible to breakup under the action of external forces. This is in contrast to the case of homogeneous ellipsoids where the capillary forces for both tip-to-tip and side-by-side alignments are of comparable magnitude.52 In order to find the equilibrium alignment of the particles, the capillary energies for ellipsoids in contact are calculated by carrying out simulations for 0° ≤ ϕ ≤ 180° and small values of d and extrapolating the results to contact (d = 0). The dependence of capillary energy on bond angle for pairs of similar ellipsoids in contact is shown in Figure 7a. We demonstrate the variation for particles having aspect ratios of AR = 4, 5, and 6, assuming a zero value for capillary energy in side-by-side alignment (ϕ = 180°). The particle pairs are considered at their equilibrium orientation angles with respect to the interface (e.g., θr = ±85° for AR = 5). The energy landscape reveals that for all geometries capillary energy decreases monotonically with the bond angle ϕ, and the minimum occurs at side-by-side alignment (ϕ = 180°). Therefore, a side-by-side configuration is preferred for Janus ellipsoids in contact. A similar behavior has been found for homogeneous ellipsoids through experimental and numerical investigations.32,52 Further insights into transition of particle pairs toward the optimum alignment can be inferred from the capillary torque, calculated as T = −∂E/∂ϕ and evaluated as a function of the bond angle ϕ. The torque is defined so that T > 0 corresponds to a capillary torque resisting rotation from the equilibrium configuration. As shown in Figure 7b, the torque is positive for all bond angles, indicating that the particles cannot take an intermediate bond angle between 0° < ϕ < 180° and the capillary torque will force them to roll on each other within the interface plane toward tip-to-tip (ϕ = 0°) or side-by-side (ϕ = 180°) alignments. As a result, Janus ellipsoids do not show any metastable configuration at the interface. The capillary torque increases sharply in the neighborhood of the stable side-by-side configuration and then slowly drops by approaching the tip-totip alignment. The maximum capillary torque translates into ∼9

Figure 6. (a) Pair interaction potential between two Janus ellipsoids of AR = 5, β = 30° oriented side-by-side as a function of their center-ofmass separation (a logarithmic plot indicating the power law dependency is shown in the inset). (b) Capillary force between the particles in (a) as a function of the separation distance for tip-to-tip and side-by-side alignments.

to-center separation between the particles. The potential is expressed in units of thermal energy kBT in order to give an order of magnitude estimate of the interaction energy. Close to contact, the calculated capillary attraction is greater than 107kBT, which is much larger than the typical values due to electrocapillarity or gravity-induced attractions.7,56 The powerlaw dependency of the interactions between these two Janus ellipsoids is deduced from the logarithmic plot shown in the inset. We note that the long-ranged interactions are characterized by a potential varying as U ∼ dcc−2.5, which is different from the quadrupolar power law of dcc−4 for homogeneous ellipsoids. For a pair of side-by-side ellipsoids with aspect ratios varying in a range of AR = 3.5−7, the pair potential was consistent with a power law dependence of U ∼ dccm, with m = −2.5 ± 0.2. For the tip-to-tip configuration, the calculated potential is about 1 order of magnitude smaller than the side-by-side alignment, and the potential can be expressed as a power law with m = −7.5 ± 0.3. The large difference in the 14966

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Additionally, the magnitude of the capillary torque inducing this optimum configuration is very close for all cases. To compare the strength of interactions, we calculate the capillary force between particles in the preferred side-by-side alignment. Figure 8 depicts the variation of capillary force with

Figure 8. Capillary force between two dissimilar Janus ellipsoids as a function of the separation distance for β = 30°, AR1 = 5, and three different aspect ratios AR2.

surface-to-surface separation for pairs of dissimilar ellipsoids. All particles have equal surface area and amphiphilicity (β = 30°); the first particle in each pair is characterized by AR1 = 5, while the second particle adopts aspect ratios ranging from AR2 = 3.5 to 7. This range is selected since Janus ellipsoids with an amphiphilicity characterized by β = 30° tend to deform the interface and induce capillary interactions for AR > 3.3, as shown in Figure 3. We note that the capillary force at close spacings reduces as the aspect ratio is increased. For more elongated ellipsoids, the overlapping menisci are stretched over a longer axis length, such that the change in surface area of the interface is less sensitive to the variation of interparticle separation. Consequently, the slope of energy curve and thus the capillary force are reduced as AR2 increases. This indicates that for ellipsoidal particles with a constant surface area, one can tune the strength of capillary force in contact by modifying the particle’s aspect ratio using e.g. the mechanical stretching method.33,57 In addition to shape, the size of the neighboring particles also influences the pair potential as noted in eq 3. Basically, the magnitude and range of interparticle forces can be enhanced upon increasing the size of particles with similar aspect ratio, as shown previously for Janus spheres.43 The reason is that increased particle size distorts the interface to a larger extent which thus enhances the tendency of two neighboring menisci to overlap. 3.4. Effect of Particle Amphiphilicity on Interactions of Two Janus Ellipsoids. The interfacial assembly and effectiveness of colloidal particles as stabilizing agents can be tuned through modification of particle amphiphilicity. We consider ellipsoids of AR = 5 at equilibrium orientations and evaluate the interactions by enhancing the difference between the wettabilities of the two regions through increasing β. Figure 9a shows the capillary force as a function of normalized separation between side-by-side ellipsoids for three different values of β. We observe that increasing the amphiphilicity results in an enhancement in capillary forces between ellipsoidal particles. This is due to the increased meniscus depth as the wettability of each surface region deviates from 90°, leading to a

Figure 7. (a) Capillary energy in contact for ellipsoids of similar geometry and β = 30° as a function of the bond angle ϕ. (b) Capillary torque corresponding to the energy in (a).

× 105kBT for a particle with a = 2.5 μm and AR = 5 at an oil− water interface with γ = 50 mN/m. Moreover, a linear relationship between T and ϕ is a good approximation for angular deviations of up to 10°−15° from the equilibrium side-by-side configuration (ϕ = 180°), thus indicating a linearly elastic mechanical response to bond bending of side-by-side ellipsoids. We note that the capillary energy and torque are almost independent of the aspect ratio for Janus ellipsoids of similar surface area and amphiphilicity. It should be emphasized that for Janus spheres and those ellipsoids that take the upright orientation of θr = 0° at equilibrium according to Figure 3, the interface remains flat and the capillary energy and torque go to zero. 3.3. Influence of Particle Polydispersity on the Interactions of Two Janus Particles. As shown in previous section, the pair interaction potential between two Janus particles generally depends on their aspect ratio. We evaluated the capillary interactions between two Janus ellipsoids of identical geometry and surface properties. In practical cases such as in stabilization of Pickering systems, particles with a nonuniform size distribution may be present, and we would like to understand the nature of energetic interactions when particles of dissimilar geometry interact at the interface. Let us consider a typical Janus ellipsoid with AR1 = 5 approaching a particle with a different aspect ratio AR2. The particles will undergo a rotational relaxation within the interface plane in order to minimize the total surface free energy of the system. By evaluating the capillary energy as a function of the bond angle ϕ, we found a preferred side-by-side alignment for all pairs of ellipsoids irrespective of the difference in aspect ratios. 14967

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possible reduction in interfacial energy, thus leading to the formation of particle chains. We can deduce the mechanical response of particle chains formed at liquid interfaces to external forces based on calculated pair interactions. The interactions of colloidal particles at liquid interfaces can directly govern their interfacial rheology. The rigidity and rearrangement of colloidal monolayers under tensile forces and their mechanical response to bending moments are employed in calculating their rheological properties, such as the viscosity, yield stress, and elastic modulus.58,59 Consider an initially straight chain of Janus ellipsoids pulled along its axis by an external force Fext, as shown in Figure 10a.

Figure 9. (a) Capillary force between two ellipsoidal Janus particles of AR = 5.0 in side-by-side alignment as a function of the separation distance for several amphiphilicities. (b) Capillary torque as a function of the bond angle.

Figure 10. (a) A chain of side-by-side Janus ellipsoids under the application of a stretching force Fext and (b) bending of the particle chain out of the equilibrium alignment due to an external torque Text.

larger reduction in the interfacial area upon overlapping of neighboring menisci. For β = 0°, the particle becomes homogeneous with a contact angle of 90°, leading to a flat interface and thus no capillary-induced interactions. The amphiphilicity of the particle also influences the capillary torque that rotates the particles toward the equilibrium alignment. We found that the side-by-side configuration is preferable for Janus ellipsoids with a wide range of β. However, the magnitude of the capillary torque strongly depends on the amphiphilicity. Figure 9b shows the torque in contact as a function of the bond angle for pairs of ellipsoids with AR = 5 and equilibrium orientations of θr = 85°. As the difference between the wettability of the two regions is increased, the capillary torque on the particles increases, thus providing a stronger barrier against rotation out of the equilibrium alignment. The maximum capillary torque increases nearly by an order of magnitude as the amphiphilicity is increased from β = 10° to 30°. In addition, all the curves show a linear variation in T for small deviations δϕ from equilibrium, with an increased slope for larger β values. This indicates that the linearly elastic mechanical response to bond bending of ellipsoids from sideby-side alignment increases upon increasing the amphiphilicity. 3.5. Micromechanics of Janus Particle Chains. We showed in sections 3.1 and 3.2 that prolate Janus ellipsoids induce hexapolar interface deformations that lead to capillary attractions between neighboring particles. A pair of ellipsoids tends to align in a side-by-side configuration in order to minimize the total surface energy. If several particles are trapped at the interface, the overlapping of menisci of similar sign in the side-by-side configuration results in the maximum

The capillary force in contact can be calculated from the slope of energy curves as Fc = (∂E/∂d)|d=0. The chain will support the external force without stretching if Fext is less than the capillary force in contact. If Fext > Fc, the particle chain will break apart since the interparticle forces can not oppose the external field anymore. However, the position along the chain where breakage occurs cannot be predicted since the bonds between identical particles oppose the same capillary resistance and are subject to the same tensile force. Particle chains can also be subject to bond-bending deformations, as shown in Figure 10b. Chains of Janus ellipsoids are expected to behave as elastic elements in response to bending since the capillary interaction between pairs of ellipsoids is elastic for sufficiently small values of dϕ (see section 3.2). The curvature ρ of a particle chain is defined as the ratio of the infinitesimal change in angle dϕ to the corresponding length scale. The curvature in correspondence to each bond is therefore ρ = dϕ/(2a), where 2a is the distance between the bonds for side-by-side ellipsoids in contact. The flexural rigidity κ of the chain is simply the proportionality constant between T and ρ, i.e., T = κρ. Since the dimensionless capillary torque T/(γrs2) varies linearly with dϕ for small deviations, the flexural rigidity can be expressed as κ = 2aγrs2D, where D is the slope of the capillary torque curves near δϕ = 0 as in Figure 9b. For a micrometer-sized ellipsoidal particle with a = 2.5 μm, AR = 5, and β = 30° at an oil−water interface with γ = 50 mN/m, the flexural rigidity is calculated as κ ∼ 4.3 × 105kBT μm, which is converted to an elastic energy of 4.3 × 14968

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103kBT for chain of ellipsoids with a curvature of (100 μm)−1. Therefore, thermal fluctuations are unable to bend chains formed by Janus microellipsoids. However, for less amphiphilic and smaller particles, the elastic energy is comparable with the thermal energy and can bend the particle chains.

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4. CONCLUSIONS We have studied the interactions of ellipsoidal Janus particles at an oil−water interface. In contrast to homogeneous ellipsoids with quadrupolar deformations, Janus ellipsoids induce a hexapolar interface shape due to the change in meniscus sign by crossing the Janus boundary. The overlapping of such distortions for neighboring particles leads to long-ranged capillary interactions. For two neighboring Janus ellipsoids, the equilibrium orientation angle with respect to the interface plane is found to be a constant, irrespective of the interparticle spacing. However, the interacting particles undergo a rotational relaxation within the interface plane to induce overlapping of menisci with similar signs. Consequently, they approach due to capillary attractions until contact and then align side-by-side to minimize the overall surface energy. The capillary force in contact is reduced upon increasing the aspect ratio for particles with a polydisperse distribution. On the other hand, the capillary force and the torque inducing the side-by-side configuration is enhanced upon increasing the hydrophobicity of particles. Therefore, Janus ellipsoids exhibit stronger interactions compared to their homogeneous counterparts. Moreover, the pair potential shows considerable deviation from the quadrupolar interactions between homogeneous ellipsoids. We found that the power law form of the potential strongly depends on the amphiphilicity of the particles as well as their alignment in the interface plane. From the pair interaction, we also inferred the micromechanics of chains formed by several neighboring Janus ellipsoids. The results presented in this work can be used to predict the migration and oriented assembly of Janus ellipsoids of various geometrical and surface properties at liquid−fluid interfaces. They suggest the possibility of creating particle chains with upright and tilted orientations and thus tuning the properties of complex interfaces containing Janus particles.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (S.S-Z.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank German Drazer and Ken Brakke for helpful discussions and suggestions. We are grateful for support from Rutgers Institute for Advanced Materials, Devices, and Nanotechnology.

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REFERENCES

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