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Capillary Assembly of Anisotropic Particles at Cylindrical Fluid–Fluid Interfaces

  • Jack L. Eatson
    Jack L. Eatson
    Department of Physics & Mathematics, University of Hull, Hull HU6 7RX, U.K.
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  • Jacob R. Gordon
    Jacob R. Gordon
    Department of Chemistry & Biochemistry, University of Hull, Hull HU6 7RX, U.K.
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  • , 
  • Piotr Cegielski
    Piotr Cegielski
    AMO GmbH, Otto-Blumenthal-Str. 25, Aachen 52074, Germany
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  • , 
  • Anna L. Giesecke
    Anna L. Giesecke
    AMO GmbH, Otto-Blumenthal-Str. 25, Aachen 52074, Germany
    University of Duisburg-Essen, Bismarckstr. 81, Duisburg 47057, Germany
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  • , 
  • Stephan Suckow
    Stephan Suckow
    AMO GmbH, Otto-Blumenthal-Str. 25, Aachen 52074, Germany
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    Orcidhttps://orcid.org/0000-0002-1116-169X
  • , 
  • Anish Rao
    Anish Rao
    Center for Cooperative Research in Biomaterials (CIC BiomaGUNE), Basque Research and Technology Alliance (BRTA), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain
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  • Oscar F. Silvestre
    Oscar F. Silvestre
    Center for Cooperative Research in Biomaterials (CIC BiomaGUNE), Basque Research and Technology Alliance (BRTA), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain
    Centro de Investigación Biomédica en Red, Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain
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  • Luis M. Liz-Marzán
    Luis M. Liz-Marzán
    Center for Cooperative Research in Biomaterials (CIC BiomaGUNE), Basque Research and Technology Alliance (BRTA), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain
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    Orcidhttps://orcid.org/0000-0002-6647-1353
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  • Tommy S. Horozov
    Tommy S. Horozov
    Department of Chemistry & Biochemistry, University of Hull, Hull HU6 7RX, U.K.
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  • , and 
  • D. Martin A. Buzza*
    D. Martin A. Buzza
    Department of Physics & Mathematics, University of Hull, Hull HU6 7RX, U.K.
    *Email:[email protected]
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    Orcidhttps://orcid.org/0000-0002-9728-7858
Cite this: Langmuir 2023, 39, 17, 6006–6017
Publication Date (Web):April 18, 2023
https://doi.org/10.1021/acs.langmuir.3c00016

Copyright © 2022 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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The unique behavior of colloids at liquid interfaces provides exciting opportunities for engineering the assembly of colloidal particles into functional materials. The deformable nature of fluid–fluid interfaces means that we can use the interfacial curvature, in addition to particle properties, to direct self-assembly. To this end, we use a finite element method (Surface Evolver) to study the self-assembly of rod-shaped particles adsorbed at a simple curved fluid–fluid interface formed by a sessile liquid drop with cylindrical geometry. Specifically, we study the self-assembly of single and multiple rods as a function of drop curvature and particle properties such as shape (ellipsoid, cylinder, and spherocylinder), contact angle, aspect ratio, and chemical heterogeneity (homogeneous and triblock patchy). We find that the curved interface allows us to effectively control the orientation of the rods, allowing us to achieve parallel, perpendicular, or novel obliquely orientations with respect to the cylindrical drop. In addition, by tuning particle properties to achieve parallel alignment of the rods, we show that the cylindrical drop geometry favors tip-to-tip assembly of the rods, not just for cylinders, but also for ellipsoids and triblock patchy rods. Finally, for triblock patchy rods with larger contact line undulations, we can achieve strong spatial confinement of the rods transverse to the cylindrical drop due to the capillary repulsion between the contact line undulations of the particle and the pinned contact lines of the sessile drop. Our capillary assembly method allows us to manipulate the configuration of single and multiple rod-like particles and therefore offers a facile strategy for organizing such particles into useful functional materials.

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1. Introduction

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Colloidal particles adsorbed at fluid–fluid interfaces are of great importance for a wide range of applications, including emulsification, (1,2) encapsulation, (3) food and pharmaceuticals, (4) reconfigurable biomimetic systems, (5) and surface patterning. (6,7) They are also an ideal system for studying self-assembly due to a number of attractive features in this system. For example, because of the very high adsorption energies for particles with sizes greater than 10 nm, (8) colloids at liquid interfaces are highly confined, allowing us to study self-assembly in two-dimensions. (9,10) In addition, the fact that fluid–fluid interfaces are soft means that they can be easily deformed to generate strong and long-ranged capillary interactions between the particles, providing a powerful handle with which to control and harness self-assembly. (11−15) A well-known example of capillary assembly is the so-called “Cheerios effect” which occurs when cereals in a bowl of milk spontaneously aggregate together. (16) This effect is due to the deformation of the flat fluid–fluid interface by the gravitational force acting on the adsorbed particles so that, when the deformations from neighboring particles overlap, the particles experience a strong capillary attraction to reduce interfacial area and therefore aggregate together.
In the simple case where the adsorbed particles are spherical, the meniscus deformations due to gravity are circularly isotropic and the resultant capillary interactions are monopolar in nature. (17) Such monopolar capillary interactions are generally negligible for spherical particles with sizes less than 10 μm. (2) However, particles in this size range can still undergo significant capillary interactions if they are anisotropic, either in terms of particle shape or chemical heterogeneity. For example, when a chemically homogeneous rod-shaped particle such as an ellipsoid or a cylinder is adsorbed at a fluid–fluid interface, the constant contact angle requirement at the three-phase contact line around the particle cannot be satisfied by a flat interface for non-neutrally wetting rods and the meniscus spontaneously deforms to create quadrupolar deformations consisting of two rises and two falls in the contact line. (18−20) These deformations lead to orientationally dependent quadrupolar capillary interactions, (19−21) driving ellipsoidal particles to assemble side-to-side (21−23) and cylindrical particles to assemble tip-to-tip. (20,23) In the case of chemically heterogeneous particles such as Janus ellipsoids and cylinders (where the two “hemispheres” across the long axis of the particle have different surface chemistries), both shape and chemical anisotropy leads to more complex hexapolar contact line undulations and capillary interactions between the particles. (24−26)
The deformation of the fluid–fluid interface not only generates capillary interactions between adsorbed colloidal particles, but can also be used as an external field to direct self-assembly. For example, when rod-shaped particles are adsorbed at a curved interface, the rods will rotate until their quadrupolar rise axis is aligned with the principal axis of curvature where the interface is concave up. (27,28) Indeed, Lewandowski et al. have shown that, when assembling two cylindrical particles at the curved interface, it was possible to suppress tip-to-tip assembly in favor of side-to-side assembly, by increasing the curvature of the interface so that the energy penalty for the cylinders to deviate from the orientation imposed by the host interface becomes too prohibitive. (28) In addition, when rod-shaped particles are adsorbed at a fluid–fluid interface with a non-uniform curvature, the particles migrate toward regions of high curvature and simultaneously align themselves along the principal axis of curvature of the host interface. (29)
In this paper, we study the self-assembly of anisotropic rod-shaped particles at a cylindrical interface formed by a sessile liquid drop. Specifically, using the finite element method Surface Evolver, (30) we study the assembly of single and multiple rods as a function of drop curvature and particle properties such as shape (ellipsoid, cylinder, and spherocylinder), contact angle, aspect ratio, and chemical heterogeneity (homogeneous and triblock patchy). The simple curved geometry of cylinders (constant finite curvature transverse to the cylinder and zero curvature along the cylinder) allows us to elucidate the interplay between interfacial curvature and particle properties in determining the configuration of single and multiple rods. Surprisingly, we find that, although the lateral dimension of the cylindrical drop is larger than the length of the rods in all cases studied, the curved interface allows us to effectively control the orientation of the rods so that they lie parallel, perpendicular, or oblique, with respect to the cylindrical drop. In addition, by tuning particle properties to achieve parallel alignment of the rods, we show that the cylindrical drop geometry favors tip-to-tip assembly for two rods, not just for cylinders, but also for ellipsoids and triblock patchy rods. Finally, although there are no curvature gradients in the host interface that can be used to control particle position, we can still achieve spatial confinement of the rods transverse to the cylindrical drop by using the capillary repulsion from the pinned contact lines of the sessile drop. (31,32)

2. Theoretical Model and Methods

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In this section, we describe the geometry and thermodynamics of the composite system consisting of rod-like particles adsorbed at a sessile cylindrical liquid drop and the Surface Evolver method used to study this system theoretically.
For rod-like particles, we consider three different particle shapes, namely, ellipsoids, cylinders, and spherocylinders (Figure 1). For ellipsoids and cylinders, we use the super-ellipsoid equation (23,33)
(x′2a2)+(y′2+z′2b2)η=1
(1)
to define the particle shape, where x′, y’, z’ are the Cartesian coordinates in the particle reference frame (see below in this section), a, b are the semi-major and semi-minor lengths of the rod, respectively, and η is a sharpness parameter that defines the sharpness of the super-ellipsoid edge. We use η = 1 for ellipsoids and η = 4 for cylinders (i.e., we consider cylinders with rounded edges; see Figure 1a). For spherocylinders, we use
f(x′,y′,z′)={y′2+z′2=b2,|x′|<a−b(x′−x′|x′|(a−b))2+y′2+z′2=b2,|x′|≥a−b
(2)
Note that in all cases, the particle aspect ratio is a/b.

Figure 1

Figure 1. (a) Geometry of the simulated rod-like particles; (b) geometry of a rod-like particle adsorbed at a cylindrical sessile drop.

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In terms of surface chemistry, we consider ellipsoids, cylinders, and spherocylinders with homogeneous surface chemistry, as well as spherocylinders with triblock patchy surface chemistry (Figure 1a).
For the cylindrical sessile drop, we consider a drop with a rectangular base of width W = 20b and length L = 2a + 10b. For convenience, we refer to the top and bottom fluid phases as oil and water, respectively (i.e., the fluid making up the drop is water), though our model is in fact general and applies to any fluid–fluid interface. Assuming the origin of the lab frame in Cartesian coordinates to be at the center of the base with z perpendicular to the base and x, y parallel and perpendicular to the long axis of the cylinder, respectively, we assume that the contact lines of the cylindrical drop at y = ±10b are pinned and apply reflecting boundary conditions for the interface at x = ±(a + 5b) (see Figure 1b). Note that the width of the drop is greater than the particle length for all cases studied in this work (W > 2a), and we have chosen L to be large enough so that for a particle positioned at the center of the drop (x, y = 0), the effect of the reflecting boundary conditions is small and the particle is effectively isolated. We fix the curvature of the cylindrical drop by applying a Laplace pressure of γow/R across the interface, where γow is the oil–water interfacial tension and R is the radius of the cylinder in the absence of any adsorbed particles. Although the behavior of adsorbed rods is controlled by the curvature of the cylindrical drop, (27−29) it is easier to control and measure the height of the drop experimentally. For convenience, we therefore parameterize the curvature of the drop using the drop height in the absence of adsorbed particles, h, which is related to R and W according to R=h/2+W2/8h. Note that we consider drop heights h > b, so that the substrate does not play a critical role in determining particle behavior, that is, the adsorbed particles are in the flotation rather than the immersion regime. (11,12)
In order to specify the center-of-mass position of the adsorbed particle, we use cylindrical polar coordinates, where the position transverse to the cylindrical drop is given by the polar angle θp, the radial distance from the long axis of the cylindrical drop is given by rp = R + Δr, and the position along the long axis is given by xp (see Figure 2a).

Figure 2

Figure 2. Degrees of freedom of a rod adsorbed at the cylindrical interface; the red dot represents the center of mass of the rod. (a) Cylindrical polar coordinates used to specify the position of the rod; (b) bond angle θb and tilt angle θt (defined with respect to the interfacial frame u, v, w) used to specify the orientation of the rod.

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In order to specify the orientation of the adsorbed particle, we first define a Cartesian coordinate system in the interfacial frame (u, v, w), with the origin coinciding with the particle center and u, v, w pointing in the directions of change for the cylindrical coordinates x, θ, r, respectively (see Figure 2b); the interfacial frame coordinates are related to the lab frame coordinates (x, y, z) by
(uvw)=(1000cos⁡θp−sin⁡θp0sin⁡θpcos⁡θp)(x−xpy−ypz−zp)
(3)
where yp = rpsinθp, zp = rpcosθp are the center-of-mass coordinates of the particle in the lab frame. Since we are considering axisymmetric rod-like particles, the orientation of the particle can be specified relative to (u, v, w) using two angles that we call the bond angle θb and the tilt angle θt, as defined in Figure 2b.
Finally, we can define a Cartesian coordinate system in the particle frame (x′,y′,z′), with x′ aligned along the semi-major axis of the particle and y′, z′ aligned along the semi-minor axes of the particle. These coordinates are related to the interfacial frame (u, v, w) coordinates by the following rotational transformations (34)
(x′y′z′)=(cos⁡θt0−sin⁡θt010sin⁡θt0cos⁡θt)(cos⁡θbsin⁡θb0−sin⁡θbcos⁡θb0001)(uvw)
(4)
For micron or sub-micron particles, which is the focus of this work, gravity is negligible and the energy of the composite system is primarily due to the interfacial energy, which is given by (33,35)
Eint=γowAow+γpoApo+γpwApw
(5)
where γow, γpo, and γpw are the interfacial tensions and Aow, Apo, and Apw are the areas of the oil–water, particle–oil, and particle–water interfaces, respectively. We have neglected line tension contributions in eq 5 because these are sub-dominant compared to interfacial tensions for the particles that we are considering, where a, b > 10 nm. (36) We can simplify eq 5 by eliminating either the particle–water or the particle–oil interface from the problem. For example, using Apw = A – Apo (where A is the total area of the particle) and Young’s equation γowcosθw = γpo – γpw (where θw is the contact angle) and dropping irrelevant constant terms, we can eliminate the particle–water interface from eq 5 to obtain
Eint=γow(Aow+cos⁡θwApo)
(6)
For a given particle configuration, the energy given by eq 6 (or its equivalent obtained by eliminating the particle-oil interface) is calculated using Surface Evolver. (30) This is a finite element software that represents each interface as a mesh of triangles. The resultant vertices are then displaced to minimize the total interfacial energy, subject to the constraints of the boundary conditions of the simulation box, particle geometry, and the Laplace pressure. (30,33−35) Note that the constant contact angle requirement around the particle is automatically satisfied within this scheme since Young’s equation arises from minimization of the total interfacial energy. We work with length and energy units such that b = 1 and γow = 1, and we use a variable triangular mesh edge length between 0.02b and 0.1b and quadratic edges to capture the shape of the fluid–fluid interface and three-phase contact line more accurately. The Surface Evolver script that we used to generate the results in this paper is available as part of Supporting Information.
In this paper, we only consider particles with aspect ratio ≥2.5, where the equilibrium tilt angle at a flat interface is θt = 0°, that is, particles are in the “side-on” state. (33) For the curved cylindrical interfaces and representative rod-shaped particles we are considering in this paper, we have checked that this result remains true, independent of the bond angle θb; we have therefore set θt = 0° in all simulations. In a typical simulation, we specify the position variables θp, xp, and the bond angle θb of the rod-like particles, but allow the radial coordinate Δr to equilibrate for a given particle configuration.

3. Results and Discussion

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3.1. Single Rods at a Flat Interface

In order to establish a baseline for our simulations of particles at cylindrical interfaces, we first analyze the behavior of single rods with homogenous surface chemistry at a flat interface (we consider the behavior of patchy rods in Section 3.5 below). In Figure 3 we show contour plots of the meniscus deformation for particles with aspect ratio a/b = 2.5, for different particle shapes (ellipsoids, cylinders and spherocylinders) and contact angles (θw = 70, 90, 110°). As expected, no meniscus deformations are observed for spherocylinders or neutrally wetting ellipsoids or cylinders (θw = 90°). On the other hand, for non-neutrally wetting ellipsoids and cylinders, we see quadrupolar meniscus deformations, in agreement with the literature. (18−20) Good agreement with the literature is also obtained for the direction of the contact line curvature, relative to the long axis of the particle. For ellipsoids, the quadrupolar rise axis lies parallel and perpendicular to the long axis, for hydrophobic (θw = 110°) and hydrophilic (θw = 70°) particles, respectively. In contrast, for cylinders, the quadrupolar rise axis lies parallel and perpendicular to the long axis, for hydrophilic and hydrophobic particles, respectively. This means that the contact line curvature can be controlled by tuning particle shape and wettability, providing an effective way to control the orientation of rod-like particles at a cylindrical drop.

Figure 3

Figure 3. Contour plot of meniscus deformation around rod-like particles with aspect ratio 2.5 and homogeneous surface chemistry, adsorbed at a flat fluid–fluid interface for different particle shapes and contact angles.

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3.2. Single Rods at a Cylindrical Interface–Particle Orientation

Having studied the behavior of single rods with homogenous surface chemistry at a flat interface, we next analyze their behavior at a cylindrical interface. Specifically, in this section, we study the impact of particle shape, contact angle, aspect ratio, and droplet curvature on particle orientation, as specified by the bond angle θb. As discussed in Section 2, since we are considering particles with aspect ratio ≥2.5, we set the tilt angle to θt = 0°. (33) In addition, since we are interested in the effect of interfacial curvature on the orientation of isolated rods, we set θp = 0°, xp = 0 (i.e., particle at apex of cylindrical drop, in the center of simulation box) to minimize the impact of the pinned contact line and reflecting boundary conditions of the cylindrical drop. We then calculate the energy of the system as a function of bond angle from θb = 0 to 90° in increments of 1°, noting that the energy only needs to be calculated within this range due to the symmetry of the energy with respect to θb.
We first discuss the effect of contact angle on the orientation of rods with different shapes for relatively short rods (aspect ratio = 2.5) and a cylindrical drop height h = 5b. In Figure 4a, we plot interfacial energy (relative to the minimum energy state) as a function of bond angle θb for ellipsoids with contact angles θw = 70, 90, 110°. The equilibrium orientation of the ellipsoids (i.e., θb corresponding to the energy minimum) clearly depends on the contact angle: for hydrophilic ellipsoids (θw = 70°), the particles are aligned perpendicular to the cylindrical drop, that is, θb = 90°, whereas for hydrophobic ellipsoids (θw = 110°), the particles are aligned parallel to the cylindrical drop, that is, θb = 0°. This result can be readily understood from the fact that when particles with a capillary quadrupole are adsorbed at a curved interface, the particle will rotate to try to align its quadrupolar rise axis to the principal axis of curvature of the host interface (where the interface is concave up) to minimize the distortion to the host interface. (27−29) Since hydrophilic and hydrophobic ellipsoids have their rise axis perpendicular and parallel to the long axis, respectively (see Figure 3), but the principal axis of curvature for a cylindrical drop is parallel to its long axis, it is not surprising that hydrophilic ellipsoids align perpendicular to the cylindrical drop whereas hydrophobic ellipsoids align parallel to it.

Figure 4

Figure 4. Interfacial energy as a function of bond angle for relatively short rods with different contact angles, adsorbed at a cylindrical interface, for (a) ellipsoids; (b) cylinders; and (c) spherocylinders. All rods have θp = 0°, xp = 0, θt = 0°. Note that in (c), the curve for θw = 70° is not visible as it lies underneath the curve for θw = 90°.

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We additionally found (Figure 4a) that neutrally wetting ellipsoids (θw = 90°) preferentially align parallel to the cylindrical drop. Since these ellipsoids do not possess an intrinsic capillary quadrupole (see Figure 3), the primary driving force for such an alignment is not contact line curvature but shape anisotropy. Specifically, because of the curvature of the cylindrical interface, an anisotropic particle such as an ellipsoid removes a larger area of the energetically unfavorable oil–water interface when it is parallel, rather than perpendicular, to the cylindrical drop and therefore the parallel orientation has a lower energy. Shape anisotropy also has an impact on the orientational energy of the non-neutrally wetting ellipsoids (Figure 4a). For hydrophobic ellipsoids, where both contact line curvature and particle anisotropy favor parallel alignment, that is, where the two effects are synergetic, both the depth and curvature of the energy minima (the latter being proportional to the “spring constant” of the potential confining the rod to its equilibrium orientation) are greater compared to the case of hydrophilic ellipsoids, where contact line curvature favors perpendicular alignment but particle anisotropy favors parallel alignment, that is, where the two effects are antagonistic.
In Figure 4b, we plotted the interfacial energy as a function of bond angle for cylinders with contact angles θw = 70, 90, 110°. In this case, the dependence of particle orientation on the contact angle is opposite to that for ellipsoids, with hydrophilic and hydrophobic cylinders aligning parallel and perpendicular to the cylindrical drop, respectively. This difference is not surprising because the orientation of the quadrupolar rise axis relative to the long axis of the particle is opposite for cylinders compared to ellipsoids for a given contact angle (see Figure 3). However, similar to ellipsoids, the depth and curvature of the energy minima are greater for cylinders in the parallel orientation, compared to those in the perpendicular orientation, due to the synergistic effect of contact line curvature and shape anisotropy. Unexpectedly, we see that neutrally wetting cylinders preferentially align perpendicular to the cylindrical drop. We believe that this counterintuitive result is due to the short cylinders considered here (a/b = 2.5) removing a larger area from the oil–water interface in the perpendicular orientation than in the parallel orientation; as discussed in this section below, for longer cylinders, particle anisotropy always favors the parallel orientation. Note that in Figure 4, the energy scale for the orientational energy is slightly larger for cylinders compared to ellipsoids and we attribute this difference to the larger amplitude for contact line undulations in cylinders compared to ellipsoids. For example, for the particles shown in Figure 3, the difference between maximum and minimum height of the contact line is approximately 0.15b for cylinders and 0.11b for ellipsoids.
In Figure 4c, we plot interfacial energy as a function of bond angle for spherocylinders with contact angles θw = 70, 90, 110°. For all contact angles, spherocylinders are found to lie parallel to cylindrical drop. This is not surprising since spherocylinders do not possess an intrinsic capillary quadrupole (Figure 3) and particle orientation is therefore primarily determined by particle anisotropy, which favors the parallel orientation. We also note that the energy scale of the orientational energy for spherocylinders is almost one order of magnitude smaller than that for ellipsoids and cylinders. This effect is again due to spherocylinders not possessing an intrinsic capillary quadrupole, so that the only interfacial deformations are those induced by the curvature of the host interface, which are much weaker.
Next, we study the effect of aspect ratio on the orientation for particles of different shapes and contact angles. In Figure 5 we plot the orientational energies for ellipsoids for a cylindrical drop of height h = 5b. When the contact line curvature favors parallel alignment (θw = 110°), increasing the aspect ratio of the ellipsoids only leads to an increase in the energy scale of the potential well, but does not change the equilibrium orientation. This is as we would expect because the effect of particle anisotropy on particle orientation is synergistic to the effect of contact line curvature in this case.

Figure 5

Figure 5. Interfacial energy as a function of bond angle for ellipsoids with different aspect ratios and contact angles, adsorbed at a cylindrical interface. All rods have θp = 0°, xp = 0, and θt = 0°.

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However, when the contact line curvature favors perpendicular alignment (θw = 70°), particle anisotropy is antagonistic to contact line curvature. As the aspect ratio of the ellipsoid is increased from a/b = 2.5 to 5 and 7.5, particle anisotropy simultaneously leads to the formation of high energy barriers at θb = 90, 270° and suppresses the energy barrier due to contact line curvature at θb = 0, 180, 360°. Intriguingly, the competition between particle anisotropy and contact line curvature means that the ellipsoid does not align either parallel or perpendicular in this case, but instead aligns obliquely to the long axis of the cylindrical drop, where the bond angle of the oblique orientation is determined by the length of the particle relative to the radius of curvature of the drop.
For cylindrical particles (Figure 6), we see the same trends as those for ellipsoids, except that the parallel orientation is now observed for θw = 70° (where particle anisotropy and contact line curvature are synergistic), whereas the novel oblique orientation is observed for θw = 110° (where particle anisotropy and contact line curvature are antagonistic). Note that the effect of particle anisotropy is stronger for cylinders compared to ellipsoids, so that the energy barrier at θb = 0, 180, 360° is strongly suppressed for a/b = 5 and essentially disappears for a/b = 7.5.

Figure 6

Figure 6. Interfacial energy as a function of bond angle for cylinders with different aspect ratios and contact angles, adsorbed at a cylindrical interface. All rods have θp = 0°, xp = 0, and θt = 0°.

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For spherocylinders, which do not have an intrinsic capillary quadrupole so that contact line curvature essentially plays no role in determining particle orientation, we find that the particles are in the parallel orientation for both θw = 70 and 110° but the depth of the confining potential well increases with increasing particle aspect ratio (Figure S1).
It should be noted that the oblique orientations seen for ellipsoids and cylinders in Figures 5 and 6 only arise when the particle is long enough relative to the radius of curvature of the host interface. For example, in Figure 5, the oblique orientation is observed for rods with aspect ratios a/b = 5 and 7.5, where 2a/R = 0.8 and 1.2, respectively, but not for particles with aspect ratio 2.5, where 2a/R = 0.4. This explains why the oblique orientation was not observed in previous studies, for example, in Lewandowski et al. (28) for cylindrical particles adsorbed at a plane–parabolic interface, where 2a/R = 6 × 10–3.
In Supporting Information, we also studied the effect of changing droplet curvature on the orientation of rod-shaped particles and find that increasing drop height (and hence interfacial curvature) at a fixed rod length (Figure S2) produces the same trend for the orientation of the rods as increasing rod length at a fixed interfacial curvature (Figures 5 and 6). This result can be readily understood by recognizing that the competition between particle anisotropy and contact line curvature depends primarily on the ratio between particle length and interfacial radius of curvature so that increasing particle length is essentially equivalent to decreasing the radius of curvature (i.e., increasing curvature). However, for the parameter ranges explored in this paper, we find that changing particle aspect ratio has a much bigger impact on orientational energy compared to changing droplet height (compare Figure S2 to Figure 5).
Note that even though the lateral dimension of the cylindrical drop is larger than the length of the rods in all cases studied above (W > 2a), the potential energy well confining the orientation of rods is very large. For example, for a hydrophobic ellipsoid with aspect ratio a/b = 5 and cylindrical height h = 5b, the potential energy well depth is ΔEint/γowb2≈0.8 (Figure 5). For an oil-water interface with γow = 30 mN/m, this translates into ΔEint ≈ 6 × 106 kT for a micron-sized rod with b = 1 μm and ΔEint ≈ 600 kT for a nanorod with b = 10 nm. This result is consistent with what has been found by Lewandowski et al. who were able to control the orientation of a cylindrical microparticle using a curved interface, which has a radius of curvature much greater than the particle length. (28)

3.3. Single Rods at a Cylindrical Interface–Spatial Confinement

Having studied particle orientation in the previous section, in this section, we study the impact of the cylindrical interface on the spatial confinement of rod-like particles. Although no curvature gradients are present in a cylindrical sessile drop, which could provide control over particle position, (29) spatial confinement of the rods transverse to the cylindrical drop can still be achieved due to steric repulsion from the substrate and capillary repulsion from the pinned contact lines. (31,32) Both effects will be considered in this section. We restrict our analysis to rods in the parallel alignment as this is the most favorable alignment toward achieving the tip-to-tip assembly considered in the next section.
To estimate the spatial confinement due to steric repulsion from the substrate, we make the simplifying assumption that the fluid–fluid interface is the unperturbed cylindrical interface and that this interface goes through the center of the rod. In this case, from simple geometry, the maximum displacement of the rod from the origin in the y direction, ys, is given by (see Figure 7a)
ys=R2−(R−h+b)2
(7)

Figure 7

Figure 7. (a) Simplified model for calculating the maximum displacement of adsorbed rods lateral to a cylindrical drop, due to steric repulsion from the substrate. (b) Interfacial energy as a function of lateral displacement for adsorbed rods with different shapes and surface chemistry, which are aligned parallel to the cylindrical drop. The dashed black line indicates maximum lateral displacement allowed by steric repulsion with the substrate.

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Substituting R=h/2+W2/8h into eq 7, we can rewrite eq 7 as
ys=12(1−b/h)(W2+4bh)
(8)
Note that in the limit where the short axis length of the rod b → 0, we recover ys = W/2 as expected. From eq 8, we find that ys decreases as we reduce the drop height h, that is, the rods become increasingly confined laterally, and indeed ys = 0 for h = b. However, for the relatively large drop heights considered here, ys is close to W/2, that is, the degree of lateral confinement due to steric repulsion from the substrate is insignificant. This point is illustrated in Figure 7b for the case h = 5b, where ±ys is represented by the vertical dashed lines.
Next, we consider the lateral confinement due to capillary repulsion from the pinned contact line of the sessile drop. This repulsion arises because the meniscus around the particle would like to deform because of the particle’s contact line undulations, but this deformation is suppressed when the particle approaches the pinned contact line, causing the energy of the system to increase. (31,32) To calculate this effect, we set θt = 0°, xp = 0, and θb = 0° and calculate the energy of the system as a function of the polar angle, from θp = 0° to θmax, in increments of 1°, where θmax is approximately the polar angle corresponding to y = W/2. Note that we only need to calculate the energy for positive θp because the energy is symmetric about θp = 0°.
In Figure 7b, we plot the energy of the system (relative to the energy at θp = 0°) as a function of yp = Rsinθp, for ellipsoids with θw = 110°, cylinders with θw = 70°, and spherocylinders with θw = 90°, where the contact angles have been chosen to ensure that the rods are in the parallel orientation (triblock patchy rods will be discussed in Section 3.5 below). All particles have an aspect ratio of a/b = 5 and the drop height is h = 5b. We see that all particles are repelled by the pinned contact line at y = W/2. The repulsion is strongest for the ellipsoidal particle because this shape has the largest contact line undulation at the sides (see Figure 3). The repulsion is smaller for the cylindrical particle because the contact line undulation at the side is smaller, but the repulsion is weakest for spherocylinders, which do not have an intrinsic capillary quadrupole (see Figure 3).
However, our main conclusion from Figure 7b is that, for rod-shaped particles with homogeneous surface chemistry, the potential well around yp = 0 is very flat, so that the lateral spatial confinement due to capillary repulsion from the pinned contact lines is weak. For example, for ellipsoids with b = 10 nm, the range of yp values for which ΔEint < 3 kT is −8.4b ≤ yp ≤ 8.4b (3 kT is a reasonable estimate for the point where the confining potential starts to become significant compared to thermal energy), which corresponds to 91% of 2ys, the maximum lateral spatial range available to the ellipsoid due to steric repulsion by the substrate (i.e., the range between the dashed vertical lines in Figure 7). The spatial confinement for cylinders and spherocylinders with b = 10 nm is even weaker, with ΔEint < 3 kT at yp = ±ys. We therefore conclude that the lateral spatial confinement due to capillary repulsion from the pinned contact lines of the sessile drop is weak for rods with homogeneous surface chemistry.

3.4. Capillary Interaction and Self-Assembly of Rods at a Cylindrical Interface

Having considered the orientation and spatial confinement of single rods at a cylindrical interface in previous sections, we now consider the interaction and self-assembly of two rods at a cylindrical interface. We restrict our analysis to rods with parallel alignment and primarily consider the tip-to-tip assembly of such rods. However, we point out that it is also possible to use our method to orient the rods perpendicular to the cylindrical drop, which would favor side-to-side assembly.
For the sake of simplicity, we first consider the case where the long axes of the two parallel rods are aligned to each other, and they are both at the apex of the cylindrical drop, that is, θb, θp = 0°; later in this section, we will consider the case where the rods are not aligned to each other. We also exploit the fact that the energy of one rod approaching the reflecting boundary of the simulation cell is equal to half the energy of a two-particle system where both rods are approaching each other. When calculating the tip-to-tip interaction between two rods, we therefore just need to consider a one-rod simulation where we vary the distance of the rod to the reflecting boundary. Note that when two rods in the tip-to-tip orientation are close to each other, they may induce a capillary dipole in each other, so that θt is no longer zero. To check the size of this effect, we performed Surface Evolver simulations of two ellipsoids in a mirror symmetric configuration at the smallest surface-to-surface separation we studied and calculated the interfacial energy of the system as a function of the tilt angle of the rods θt. We found that the equilibrium tilt angle θt < 1°, suggesting that this capillary polarization effect is very small. We therefore set θt = 0° for both interacting rods in our calculations.
In Figure 8, we plot the tip-to-tip interaction energy between two rods (i.e., energy relative to the energy of the two rods at maximum separation) as a function of the surface-to-surface separation between the two rods, for ellipsoids with θw = 110°, cylinders with θw = 70°, and spherocylinders with θw = 90° (triblock patchy rods will be discussed in Section 3.5 below). The contact angles have been chosen, so that all rod shapes are in the parallel alignment since, as explained at the start of this section, we are primarily interested in the tip-to-tip assembly of rods. All particles have an aspect ratio of a/b = 5 and the drop height is h = 5b. We observe that tip-to-tip attraction is strongest for cylindrical particles and weakest for spherocylinders. These results are similar to what has been observed at flat interfaces (23) and is due to cylinders having the largest contact line undulation around their tips, compared to ellipsoids and spherocylinders (see Figure 3). Specifically, for nanoscale rods with b = 10 nm, at an oil-water interface with γow = 30 × 10–3 N/m, the interaction energy for the tip-to-tip contact is 25 kT for cylinders, 8 kT for ellipsoids, and 0.05 kT for spherocylinders. The capillary interaction for nanoscale cylinders and ellipsoids is therefore significant, whereas that for spherocylinders is negligible compared to thermal energy.
Note that, at a flat interface, ellipsoids tend to approach each other tip-to-tip initially and then “roll-over” into the side-to-side configuration because of its lower energy. (19,23,37) However, for ellipsoids at a cylindrical interface, the reduction in capillary interaction energy when going from the tip-to-tip to the side-to-side configuration is generally much smaller than the increase in orientational energy incurred in making this transition. For example, for ellipsoids with an aspect ratio a/b = 3 and contact angle θw = 80 or 100°, studied in Botto et al., (23) the reduction in capillary interaction energy is ≈ 0.001γowb2, whereas the increase in the orientational energy for the similar ellipsoid at a cylindrical interface in Figure 5 (a/b = 2.5, θw = 110°) is ≈ 0.1γowb2. Therefore, for two ellipsoids aligned tip-to-tip at a cylindrical interface of high enough curvature, the roll-over into the side-to-side configuration is suppressed and the ellipsoids remain assembled tip-to-tip. A similar phenomenon has also been observed in Lewandowski et al. (28) for cylinders where for high enough curvatures of the host interface, the cylinders assemble side-to-side because the reorientation of the cylinders that would allow them to assemble tip-to-tip is suppressed. In the Supporting Information, we estimate the minimum cylindrical drop height (and hence minimum curvature of the host interface) required to suppress the roll-over transition and find that the minimum height is much smaller than b for typical rods (Figure S3). Since we are interested in the flotation regime (i.e., h > b), we can safely neglect the roll-over transition when studying the self-assembly of rod-like particles at a cylindrical interface.

Figure 8

Figure 8. Tip-to-tip capillary interaction for adsorbed rods with different shapes and surface chemistry, which are aligned parallel to the cylindrical drop.

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Up to now, we have considered the capillary interaction between two parallel rods, which are already aligned tip-to-tip. However, from our previous discussion, we know that rods with homogeneous surface chemistry are only weakly confined transverse to the cylindrical drop. Therefore, the most usual case is in fact where the two interacting rods are not aligned initially. We now consider this more general case. Since the energy scale for rotating the rods away from their preferred parallel alignment is so high, we set θt, θb = 0° for both particles. However, even with this restriction on particle configuration, a full analysis of the two-particle problem is very expensive in Surface Evolver as it would require us to calculate the capillary interaction for different lateral positions of one of the rods and different positions of the second rod relative to the first. In order to make the problem numerically tractable, we therefore fix the position of the first rod and study the trajectory of the second rod at different starting positions relative to the first one. Specifically, to obtain an upper bound estimate for the effect of non-alignment on self-assembly, we fix the position of the first rod to be close to the edge of the cylindrical drop (roughly one rod diameter away, along the interface, from the pinned contact line of the cylindrical drop).
In Figure 9a,b, we plot the interaction energy of the two-particle system (i.e., energy relative to the energy of the two rods at maximum separation) as a function of the position of the second particle, expressed in terms of the coordinates X = xp,Y = Rθp, for ellipsoids with a/b = 2.5, θw = 110° (Figure 9a) and cylinders with a/b = 2.5, θw = 70° (Figure 9b) on a cylindrical drop with height h = 5b. The shaded-out region on the bottom left of each plot represents the region excluded to the second particle due to steric repulsion with the first one. Note that the energy landscape in Figure 9 was obtained by first calculating the energy for X, Y values on an approximately 30 × 30 grid. Since the capillary interaction between the rods is small, the resultant discrete energy landscape was quite noisy. Therefore, in order to obtain the full energy landscape for any X, Y, we use a radial basis function to interpolate the data with the Python package SciPy. Using the radial basis function allows us to use a smoothing parameter that makes the interpolation smoother by not insisting that it fits each discrete data point exactly. This procedure resulted in a smoother energy landscape, which we could use to calculate the trajectory of the second particle.

Figure 9

Figure 9. Capillary interaction energy of a two-particle system as a function of the position of the second particle relative to the first one, which is positioned at the edge of the cylindrical drop. The black lines are the trajectories of the second particle for different starting positions, the red curve is the separatrix that separates trajectories ending up in the tip-to-tip or side-to-side configurations, and the yellow lines are the dynamical attractors to which the trajectories converge at the later stages of their evolution. The energy landscape and trajectories are shown for (a) ellipsoids with a/b = 2.5, θw = 110°; (b) cylinders with a/b = 2.5, θw = 70°. The shaded-out white region on the bottom left of each plot represents the region excluded to the second particle due to steric repulsion with the first particle (colored yellow).

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Assuming that the trajectory of the second particle follows paths of steepest descent in the interaction energy landscape, (38,39) we plot the trajectories of the second particle for different initial positions on the edge of a rectangular region around the first one (black lines in Figure 9a,b). We see that, for initial positions closer to the tip of the first particle, the final state of the system is the tip-to-tip configuration, whereas for initial positions closer to the side of the first particle, the final state of the system is the side-to-side configuration. The red line shown in Figure 9a,b is the “separatrix” which demarcates the boundary between two different types of trajectories; (39,40) all trajectories originating from points to the left of the separatrix will flow toward the side-to-side configuration but all trajectories originating from points to the right of the separatrix will flow toward the tip-to-tip configuration. Interestingly, trajectories on either side of the separatrix each converge to their own “dynamical attractor” in the later stages of their evolution (yellow lines), a feature that is commonly seen in many dynamical systems. (38−40)
Since the adsorption of rods onto the cylindrical drop is essentially a random process, the probability that an adsorbed rod will end up in the side-to-side or tip-to-tip configuration relative to a fixed rod at the edge of the cylindrical drop is proportional to the areas of the “domain of influence” of the fixed rod that are on either side of the separatrix (we define the domain of influence as the region where the capillary interaction energy ≳kT). In the far-field, we can approximate the capillary interaction energy between two rods with center-to-center separation r as Vcap(r)≈−γowH2(2rc/r)4, (19−21) where H is the amplitude of the contact line undulation and rc ≈ (a + b)/2 is the average radius of the contact line. It is useful to estimate the radius of the rod b where the domain of influence of the fixed rod extends across the entire width of the cylindrical drop, that is, where |Vcap(W=20b)|≈kT. For a typical rod-like particle, for example, an ellipsoid with aspect ratio a/b = 2.5 and contact angle θw = 110° where H ≈ 0.05b (see Figure 3), we find this radius to be b ≈ 50 nm. This means that for rods with b ≳ 50 nm, the domain of influence of the fixed rod extends beyond the width of the cylindrical drop, and the drop width acts as a cut-off length limiting the area to the left of the separatrix. In contrast, there is essentially no such cut-off along the length of the cylindrical drop and the area to the right of the separatrix therefore increases indefinitely as we increase b. These results show that for rods with b ≳ 50 nm, the cylindrical drop geometry favors tip-to-tip assembly for both ellipsoids and cylinders, even when the long axes of the two interacting rods are not lined up initially. Finally, if more than two rods are present at the cylindrical interface, our analysis suggests that capillary interaction will lead to the formation of a long chains of rods that are connected to each other tip-to-tip.

3.5. Triblock Patchy Rods

Up to now, we have considered rod-like particles with homogeneous surface chemistry and controlled contact line undulations through particle shape. In this section, we consider the assembly of patchy rods, which have heterogeneous surface chemistry. In order to minimize the effect of particle shape on self-assembly, we consider patchy spherocylinders. As discussed in the introduction, diblock patchy rods (i.e., Janus rods) have hexapolar contact line undulations. (24−26) In order to compare the results in this section with those in the previous sections on particles with capillary quadrupoles, we therefore consider spherocylinders with triblock patchy geometry as this is the simplest patchy-particle geometry that possesses a capillary quadrupole; with advances in synthetic chemistry, such patchy particles can now be readily synthesized. (41) In order to achieve parallel alignment of the rods, we consider spherocylinders with hydrophilic hemispherical caps (θw = 70°) and hydrophobic cylindrical sides (θw = 110°), so that the contact line is concave upward along the long axis of the particle (Figure 1a).
In Figure 10, we show a contour plot of the meniscus deformation around the patchy particles at a flat plane. We can see that despite the spherocylinder shape, the triblock patchy particle has much larger contact line undulations both at the sides and the tips compared to rods with homogeneous surface chemistry (compare the range of interfacial heights in Figures 3 and 10). We anticipate that these large undulations will lead to much stronger spatial confinement and the tip-to-tip capillary interactions for the patchy rod compared to the non-patchy rods.

Figure 10

Figure 10. Contour plot of meniscus deformation around a triblock patchy particle with aspect ratio 5 adsorbed at a flat fluid–fluid interface.

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In Figure 7b, we plot the energy as a function of yp = Rsinθp for the triblock patchy rods with aspect ratio a/b = 5 at a cylindrical drop with height h = 5b compared to the corresponding non-patchy rods. As anticipated, the patchy rod experiences a much stronger spatial confinement lateral to the cylindrical drop. For example, the potential well depth is ΔEint ≈ 0.14γowb2 for the patchy rod but only ΔEint ≈ 0.02γowb2 for non-patchy ellipsoids; for γow = 30 × 10–3 N/m, these well depths correspond to 100 and 15 kT, respectively, for nanoscale particles with b = 10 nm.
In Figure 8, we plot the tip-to-tip interaction energy as a function of separation between two patchy rods with aspect ratio a/b = 5 at a cylindrical drop with height h = 5b compared to the corresponding non-patchy rods. Once again, as anticipated, the capillary interaction is much greater for patchy rods compared to the non-patchy rods. For example, the interaction energy at contact is −0.08γowb2 for the patchy spherocylinder but only −0.04γowb2 for non-patchy cylinders; for γow = 30 × 10–3 N/m, these contact energies correspond to −50 kT and −25 kT, respectively, for nanoscale particles with b = 10 nm. In addition, the range of the interactions is much greater for the patchy rod (≈6b) compared to the non-patchy rods (≈2b for the non-patchy cylinder). Since both lateral spatial confinement and tip-to-tip attraction are much greater for patchy rods, we conclude that tip-to-tip assembly at a cylindrical interface is much more likely for triblock patchy rods compared to non-patchy rods.

4. Conclusions

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We have used the finite element method Surface Evolver to study the capillary assembly of rod-shaped particles adsorbed at a sessile liquid drop with cylindrical geometry. We considered the flotation regime where the drop height is greater than the diameter of the rods and studied the assembly of rods as a function of interfacial curvature, particle shape (ellipsoid, cylinder, and spherocylinder), contact angle, aspect ratio, and chemical heterogeneity (homogeneous and triblock patchy).
For rods with homogeneous surface chemistry, we can achieve very strong localization of particle orientation using cylindrical drops with a lateral width much greater than the length of the rods. By changing particle shape, contact angle, and aspect ratio, we can tune the interplay between interfacial curvature, particle contact line curvature, and particle anisotropy and control the rod not only to align parallel or perpendicular to the long axis of the cylindrical drop but also to align in novel oblique orientations. In contrast, we can only achieve weak spatial confinement of the rods transverse to the cylindrical drop because of the weak repulsion between the capillary quadrupole of the particle and the pinned contact lines of the sessile drop.
For ellipsoids and cylinders oriented parallel to the cylindrical drop, the capillary interaction is strong when the rods are oriented tip-to-tip, even at the nanoscale. In contrast, the capillary interaction between spherocylinders in the parallel orientation is extremely weak because these particles do not possess an intrinsic capillary quadrupole. Since in the confined geometry of the cylindrical drop, rods in the parallel orientation are more likely to approach each other in a tip-to-tip orientation, whereas interfacial curvature suppresses the transition from the tip-to-tip to the side-to-side configuration, the cylindrical drop favors the tip-to-tip assembly of rod in the parallel orientation, not only for cylinders but also for ellipsoids.
Finally, for triblock patchy rods, which possess much larger contact line undulations compared to non-patchy rods, the stronger capillary quadrupole leads to stronger lateral spatial confinement and tip-to-tip capillary attraction, resulting in an even stronger tendency for patchy rods in the parallel orientation at a cylindrical interface to assemble tip-to-tip compared to non-patchy rods.
The proposed capillary assembly mechanism allows us to manipulate the configuration of single and multiple rod-like particles and therefore offers a facile strategy for organizing such particles into useful functional materials.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.langmuir.3c00016.

  • Orientational energy landscape for spherocylinders; orientational energy landscape for ellipsoids for different droplet heights; and energy barrier for the roll-over transition for ellipsoids (PDF)

  • Surface Evolver script used for this paper (TXT)

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Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Author
  • Authors
    • Jack L. Eatson - Department of Physics & Mathematics, University of Hull, Hull HU6 7RX, U.K.
    • Jacob R. Gordon - Department of Chemistry & Biochemistry, University of Hull, Hull HU6 7RX, U.K.
    • Piotr Cegielski - AMO GmbH, Otto-Blumenthal-Str. 25, Aachen 52074, Germany
    • Anna L. Giesecke - AMO GmbH, Otto-Blumenthal-Str. 25, Aachen 52074, Germany;  University of Duisburg-Essen, Bismarckstr. 81, Duisburg 47057, Germany
    • Stephan Suckow - AMO GmbH, Otto-Blumenthal-Str. 25, Aachen 52074, Germany;  Orcidhttps://orcid.org/0000-0002-1116-169X
    • Anish Rao - Center for Cooperative Research in Biomaterials (CIC BiomaGUNE), Basque Research and Technology Alliance (BRTA), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain
    • Oscar F. Silvestre - Center for Cooperative Research in Biomaterials (CIC BiomaGUNE), Basque Research and Technology Alliance (BRTA), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain;  Centro de Investigación Biomédica en Red, Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain
    • Luis M. Liz-Marzán - Center for Cooperative Research in Biomaterials (CIC BiomaGUNE), Basque Research and Technology Alliance (BRTA), Paseo de Miramón 182, Donostia-San Sebastián 20014, Spain;  Orcidhttps://orcid.org/0000-0002-6647-1353
    • Tommy S. Horozov - Department of Chemistry & Biochemistry, University of Hull, Hull HU6 7RX, U.K.
  • Author Contributions

    J.E. performed all the theoretical modeling; J.E. and D.M.A.B. wrote the manuscript; all the other authors commented on the manuscript; D.M.A.B. supervised the study; and D.M.A.B. and T.S.H. conceptualized the study.

  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 861950, project POSEIDON. J.E. and D.M.A.B. acknowledge the Viper High Performance Computing facility of the University of Hull and its support team. J.G. acknowledges funding from the University of Hull PhD Scholarship Scheme. O.F.S. acknowledges the support from the Provincial Council of Gipuzkoa under the program Fellows Gipuzkoa.

References

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    Defined core-shell particles as the key to complex interfacial self-assembly
    Menath, Johannes; Eatson, Jack; Brilmayer, Robert; Andrieu-Brunsen, Annette; Buzza, D. Martin A.; Vogel, Nicolas
    Proceedings of the National Academy of Sciences of the United States of America (2021), 118 (52), e2113394118CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    The two-dimensional self-assembly of colloidal particles serves as a model system for fundamental studies of structure formation and as a powerful tool to fabricate functional materials and surfaces. However, the prevalence of hexagonal symmetries in such self-assembling systems limits its structural versatility. More than two decades ago, Jagla demonstrated that core-shell particles with two interaction length scales can form complex, nonhexagonal min. energy configurations. Based on such Jagla potentials, a wide variety of phases including cluster lattices, chains, and quasicrystals have been theor. discovered. Despite the elegance of this approach, its exptl. realization has remained largely elusive. Here, we capitalize on the distinct interfacial morphol. of soft particles to design two-dimensional assemblies with structural complexity. We find that core-shell particles consisting of a silica core surface functionalized with a noncrosslinked polymer shell efficiently spread at a liq. interface to form a two-dimensional polymer corona surrounding the core. We controllably grow such shells by iniferter-type controlled radical polymn. Upon interfacial compression, the resulting core-shell particles arrange in well-defined dimer, trimer, and tetramer lattices before transitioning into complex chain and cluster phases. The exptl. phase behavior is accurately reproduced by Monte Carlo simulations and min. energy calcns., suggesting that the interfacial assembly interacts via a pairwise-additive Jagla-type potential. By comparing theory, simulation, and expt., we narrow the Jagla g-parameter of the system to between 0.9 and 2. The possibility to control the interaction potential via the interfacial morphol. provides a framework to realize structural features with unprecedented complexity from a simple, one-component system.
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  8. 8
    Pieranski, P. Two-Dimensional Interfacial Colloidal Crystals. Phys. Rev. Lett. 1980, 45, 569– 572,  DOI: 10.1103/physrevlett.45.569
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    8
    Two-dimensional interfacial colloidal crystals
    Pieranski, Pawel
    Physical Review Letters (1980), 45 (7), 569-72CODEN: PRLTAO; ISSN:0031-9007.
    Polystyrene spheres (2450 Å in diam.) are trapped in a surface energy well at the water/air interface. Because of asymmetry of charge distribution, elec. dipoles are assocd. with each interfacial particle. The dipole-dipole repulsive interactions organize the polystyrene spheres into a 2-dimensional triangular lattice. The direct microscopic observations of such an interfacial colloidal crystal are reported.
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  9. 9
    Law, A. D.; Buzza, D. M. A.; Horozov, T. S. Two-Dimensional Colloidal Alloys. Phys. Rev. Lett. 2011, 106, 128302,  DOI: 10.1103/physrevlett.106.128302
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    9
    Two-Dimensional Colloidal Alloys
    Law, Adam D.; Buzza, D. Martin A.; Horozov, Tommy S.
    Physical Review Letters (2011), 106 (12), 128302/1-128302/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We study the structure of mixed monolayers of large (3 μm diam.) and small (1 μm diam.) very hydrophobic silica particles at an octane-water interface as a function of the no. fraction of small particles ξ. We find that a rich variety of two-dimensional hexagonal super-lattices of large (A) and small (B) particles can be obtained in this system due to strong and long-range electrostatic repulsions through the nonpolar octane phase. The structures obtained for the different compns. are in good agreement with zero temp. calcns. and finite temp. computer simulations.
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  10. 10
    Law, A. D.; Auriol, M.; Smith, D.; Horozov, T. S.; Buzza, D. M. A. Self-Assembly of Two-Dimensional Colloidal Clusters by Tuning the Hydrophobicity, Composition, and Packing Geometry. Phys. Rev. Lett. 2013, 110, 138301,  DOI: 10.1103/physrevlett.110.138301
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    10
    Self-assembly of two-dimensional colloidal clusters by tuning the hydrophobicity, composition, and packing geometry
    Law, Adam D.; Auriol, Melodie; Smith, Dean; Horozov, Tommy S.; Buzza, D. Martin A.
    Physical Review Letters (2013), 110 (13), 138301/1-138301/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We study the structure of binary monolayers of large (3 μm diam.) very hydrophobic (A) and large (3 μm diam.) hydrophilic (B) or small (1 μm diam.) hydrophilic (C) silica particles at an octane-water interface. By tuning the compn. and packing geometry of the mixed monolayer, we find that a rich variety of two-dimensional hexagonal superlattices of mixed A/B or A/C clusters are formed, stabilized by short-ranged electrostatic induced dipole interactions. The cluster structures obtained are in excellent agreement with zero temp. calcns., indicating that the self-assembly process can be effectively controlled.
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  11. 11
    Kralchevsky, P. A.; Nagayama, K. Capillary forces between colloidal particles. Langmuir 1994, 10, 23– 36,  DOI: 10.1021/la00013a004
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    11
    Capillary forces between colloidal particles
    Kralchevsky, Peter A.; Nagayama, Kuniaki
    Langmuir (1994), 10 (1), 23-36CODEN: LANGD5; ISSN:0743-7463.
    A special kind of capillary interaction is considered, which differs from the common lateral capillary forces between floating particles. It appears between particles protruding from a liq. film and its phys. origin is the capillary rise of the liq. along the surface of each particle. Special attention is paid to the case when the position of the contact line is fixed. The resulting capillary force is compared with that at fixed contact angle. The 2 alternative approaches to the calcn. of capillary interactions (the force and the energetic one) are equiv. When the liq. films is thin, the disjoining pressure affects the capillary interactions between particles attached to the film surfaces. The appearance of this effect is studied quant. for 2 specific systems modeling globular proteins in an aq. film on a Hg substrate and membrane proteins incorporated in a lipid bilayer. For both systems the capillary forces appear to be strong enough to engender 2-dimensional particle aggregation and ordering. This is a possible explanation of a no. of exptl. observations of such effects.
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  12. 12
    Kralchevsky, P. A.; Nagayama, K. Capillary interactions between particles bound to interfaces, liquid films and biomembranes. Adv. Colloid Interface Sci. 2000, 85, 145– 192,  DOI: 10.1016/s0001-8686(99)00016-0
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    12
    Capillary interactions between particles bound to interfaces, liquid films and biomembranes
    Kralchevsky, P. A.; Nagayama, K.
    Advances in Colloid and Interface Science (2000), 85 (2-3), 145-192CODEN: ACISB9; ISSN:0001-8686. (Elsevier Science B.V.)
    A review, with 116 refs., is given. This article is devoted to an overview, comparison and discussion of recent results (both theor. and exptl.) about lateral capillary forces. They appear when the contact of particles or other bodies with a fluid phase boundary causes perturbations in the interfacial shape. The capillary interaction is due to the overlap of such perturbations which can appear around floating particles, vertical cylinders, particles confined in a liq. film, inclusions in the membranes of lipid vesicles or living cells, etc. In the case of floating particles the perturbations are due to the particle wt.; in this case the force decreases with the 6th power of the particle size and becomes immaterial for particles smaller than ∼10 μm. In all other cases the interfacial deformations are due to the particle wetting properties; the resulting immersion capillary forces can be operative even between very small particles, like protein globules. In many cases such forces can be responsible for the exptl. obsd. two-dimensional particle aggregation and ordering. An analogy between capillary and electrostatic forces enables one to introduce capillary charges of the attached particles, which characterize the magnitude of the interfacial deformation and could be both pos. and neg. Also, the capillary interaction between particle and wall resembles the image force in electrostatics. When a particle is moving bound to an interface under the action of a capillary force, one can det. the surface drag coeff. and the surface viscosity supposedly the magnitude of the capillary force is known. Alternative (but equiv.) energy and force approaches can be used for the theor. description of the lateral capillary interactions. Both approaches require the Laplace equation of capillarity to be solved and the meniscus profile around the particles to be detd. The energy approach accounts for contributions due to the increase of the meniscus area, gravitational energy and/or energy of wetting. The 2nd approach is based on calcg. the net force exerted on the particle, which can originate from the hydrostatic pressure, interfacial tension and bending moment. In the case of small perturbations, the superposition approxn. can be used to derive an asymptotic formula for the capillary forces, which was found to agree well with the expt. Capillary interactions between particles bound to spherical interfaces are also considered taking into account the special geometry and restricted area of such phase boundaries. A similar approach can be applied to quantify the forces between inclusions (transmembrane proteins) in lipid membranes. The deformations in a lipid membrane, due to the inclusions, can be described theor. in the framework of a mech. model of the lipid bilayer, which accounts for its hybrid rheol. (neither elastic body nor fluid). In all considered cases the lateral capillary interaction originates from the overlap of interfacial deformations and is subject to a unified theor. treatment, despite the fact that the characteristic particle size can vary from 1 cm down to 1 nm.
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  13. 13
    Botto, L.; Lewandowski, E. P.; Cavallaro, M.; Stebe, K. J. Capillary interactions between anisotropic particles. Soft Matter 2012, 8, 9957,  DOI: 10.1039/c2sm25929j
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    13
    Capillary interactions between anisotropic particles
    Botto, Lorenzo; Lewandowski, Eric P.; Cavallaro, Marcello; Stebe, Kathleen J.
    Soft Matter (2012), 8 (39), 9957-9971CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    Micro and nanoparticle adsorption to and assembly by capillarity at fluid-fluid interfaces are intriguing aspects of soft matter science with broad potential in the directed assembly of anisotropic media. The importance of the field stems from the ubiquitous presence of multiphase systems, the malleability of fluid interfaces, and the ability to tune the interactions of the particles adsorbed on them. While homogeneous spherical particles at interfaces have been well studied, the behavior of anisotropic particles - whether the anisotropy originates from shape or chem. heterogeneity - has been considered only very recently. We review recent advances in the field of anisotropic particles at fluid interfaces, by focusing on particles in the micron and submicron range. We discuss capillary adsorption, orientation, migration, and self-assembly, on planar and curved interfaces, and the rheol. of particle-laden interfaces. Prospects for future work and outstanding challenges are also discussed.
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  14. 14
    Davies, G. B.; Krüger, T.; Coveney, P. V.; Harting, J.; Bresme, F. Assembling Ellipsoidal Particles at Fluid Interfaces Using Switchable Dipolar Capillary Interactions. Adv. Mater. 2014, 26, 6715– 6719,  DOI: 10.1002/adma.201402419
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    14
    Assembling Ellipsoidal Particles at Fluid Interfaces using Switchable Dipolar Capillary Interactions
    Davies, Gary B.; Krueger, Timm; Coveney, Peter V.; Harting, Jens; Bresme, Fernando
    Advanced Materials (Weinheim, Germany) (2014), 26 (39), 6715-6719CODEN: ADVMEW; ISSN:0935-9648. (Wiley-VCH Verlag GmbH & Co. KGaA)
    Self-assembly of ellipsoidal particles at fluid interfaces using switchable dipolar capillary interactions is considered. By using magnetic anisotropic particles interacting with external magnetic fields, the authors show how to dynamically tune dipolar capillary interactions between particles by varying the dipole-field strength, and how to switch these dipolar capillary interactions on and off by making the particles undergo a first-order orientation transition. Simulations reveal self-assembled structures that depend on the surface coverage of particles and the dipole-field strength. The formation of "capillary caterpillars" is obsd. in which particles align in side-side configurations, and "capillary couples" where particles in individual caterpillars align in tip-tip chains with particles in other caterpillars, due to the anti-sym. menisci formation.
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    Dasgupta, S.; Auth, T.; Gompper, G. Nano- and microparticles at fluid and biological interfaces. J. Phys.: Condens. Matter 2017, 29, 373003,  DOI: 10.1088/1361-648x/aa7933
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    15
    Nano- and microparticles at fluid and biological interfaces
    Dasgupta, S.; Auth, T.; Gompper, G.
    Journal of Physics: Condensed Matter (2017), 29 (37), 373003/1-373003/41CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
    A review. Systems with interfaces are abundant in both technol. applications and biol. While a fluid interface separates 2 fluids, membranes sep. the inside of vesicles from the outside, the interior of biol. cells from the environment, and compartmentalize cells into organelles. The phys. properties of interfaces are characterized by interface tension, those of membranes are characterized by bending and stretching elasticity. Amphiphilic mols. like surfactants that are added to a system with 2 immiscible fluids decrease the interface tension and induce a bending rigidity. Lipid bilayer membranes of vesicles can be stretched or compressed by osmotic pressure; in biol. cells, also the presence of a cytoskeleton can induce membrane tension. If the thickness of the interface or the membrane is small compared with its lateral extension, both can be described using 2-dimensional math. surfaces embedded in 3-dimensional space. Here, we review recent work on the interaction of particles with interfaces and membranes. This can be micrometer-sized particles at interfaces that stabilize emulsions or form colloidosomes, as well as typically nanometer-sized particles at membranes, such as viruses, parasites, and engineered drug delivery systems. In both cases, we 1st discuss the interaction of single particles with interfaces and membranes, e.g., particles in external fields, non-spherical particles, and particles at curved interfaces, followed by interface-mediated interaction between 2 particles, many-particle interactions, interface and membrane curvature-induced phenomena, and applications.
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    Nicolson, M. M. The interaction between floating particles. Proc. Cambridge Philos. Soc. 1949, 45, 288– 295,  DOI: 10.1017/s0305004100024841
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    Loudet, J. C.; Yodh, A. G.; Pouligny, B. Wetting and Contact Lines of Micrometer-Sized Ellipsoids. Phys. Rev. Lett. 2006, 97, 018304,  DOI: 10.1103/physrevlett.97.018304
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    18
    Wetting and Contact Lines of Micrometer-Sized Ellipsoids
    Loudet, J. C.; Yodh, A. G.; Pouligny, B.
    Physical Review Letters (2006), 97 (1), 018304/1-018304/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We exptl. and theor. investigate the shapes of contact lines on the surfaces of micrometer-sized polystyrene ellipsoids at the water-air interface. By combining interferometry and optical trapping, we directly observe quadrupolar symmetry of the interface deformations around such particles. We then develop numerical solns. of the partial wetting problem for ellipsoids, and use these solns. to deduce the shapes of the corresponding contact lines and the values of the contact angles, θc(k), as a function of the ellipsoid aspect ratio k. Surprisingly, θc is found to decrease for increasing k suggesting that ellipsoid microscopic surface properties depend on ellipsoid aspect ratio.
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    Lehle, H.; Noruzifar, E.; Oettel, M. Ellipsoidal particles at fluid interfaces. Eur. Phys. J. E 2008, 26, 151– 160,  DOI: 10.1140/epje/i2007-10314-1
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    Ellipsoidal particles at fluid interfaces
    Lehle, H.; Noruzifar, E.; Oettel, M.
    European Physical Journal E: Soft Matter (2008), 26 (1-2), 151-160CODEN: EPJSFH; ISSN:1292-8941. (EDP Sciences)
    For partially wetting, ellipsoidal colloids trapped at a fluid interface, their effective, interface-mediated interactions of capillary and fluctuation-induced type are analyzed. For contact angles different from 90°, static interface deformations arise which lead to anisotropic capillary forces that are substantial already for micrometer-sized particles. The capillary problem is solved using an efficient perturbative treatment which allows a fast detn. of the capillary interaction for all distances between and orientations of two particles. Besides static capillary forces, fluctuation-induced forces caused by thermally excited capillary waves arise at fluid interfaces. For the specific choice of a spatially fixed three-phase contact line, the asymptotic behavior of the fluctuation-induced force is detd. anal. for both the close-distance and the long-distance regime and compared to numerical solns.
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    Lewandowski, E. P.; Cavallaro, M., Jr.; Botto, L.; Bernate, J. C.; Garbin, V.; Stebe, K. J. Orientation and self-assembly of cylindrical particles by anisotropic capillary interactions. Langmuir 2010, 26, 15142– 15154,  DOI: 10.1021/la1012632
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    20
    Orientation and Self-Assembly of Cylindrical Particles by Anisotropic Capillary Interactions
    Lewandowski, Eric P.; Cavallaro, Marcello, Jr.; Botto, Lorenzo; Bernate, Jorge C.; Garbin, Valeria; Stebe, Kathleen J.
    Langmuir (2010), 26 (19), 15142-15154CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
    In this research, we study cylindrical microparticles at fluid interfaces. Cylinders orient and assemble with high reliability to form end-to-end chains in dil. surfaces or dense rectangular lattices in crowded surfaces owing to capillary interactions. In isolation, a cylinder assumes one of two possible equil. states, the end-on state, in which the cylinder axis is perpendicular to the interface, or the side-on state, in which the cylinder axis is parallel to the interface. A phase diagram relating aspect ratio and contact angle is constructed to predict the preferred state and verified in expt. Cylinders in the side-on state create distortions that result in capillary interactions. Overlapping deformations by neighboring particles drive oriented capillary assembly. Interferometry, electron microscopy, and numerical simulations are used to characterize the interface shape around isolated particles. Expts. and numerics show that "side-on" cylinders have concd. excess area near the end faces, and that the interface distortion resembles an elliptical quadrupole a few radii away from the particle surface. To model the cylinder interactions for sepns. greater than a few radii, an anisotropic potential is derived based on elliptical quadrupoles. This potential predicts an attractive force and a torque, both of which depend strongly on aspect ratio, in keeping with expt. Particle trajectories and angular orientations recorded by video microscopy agree with the predicted potential. In particular, the anal. predicts the rate of rotation, a feature lacking in prior analyses. To understand interactions near contact, the concd. excess area near the cylinder ends is quantified and its role in creating stable end-to-end assemblies is discussed. When a pair of cylinders is near contact, these high excess area regions overlap to form a capillary bridge between the particles. This capillary bridge may stabilize the end-to-end chains. Finally, on densely packed surfaces, cylinder-covered colloidosomes form with particles arranged in regular, rectangular lattices in the interface; this densely packed structure differs significantly from assemblies reported for colloidosomes or particle-stabilized droplets in the literature.
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    Loudet, J. C.; Alsayed, A. M.; Zhang, J.; Yodh, A. G. Capillary Interactions Between Anisotropic Colloidal Particles. Phys. Rev. Lett. 2005, 94, 018301,  DOI: 10.1103/physrevlett.94.018301
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    21
    Capillary Interactions Between Anisotropic Colloidal Particles
    Loudet, J. C.; Alsayed, A. M.; Zhang, J.; Yodh, A. G.
    Physical Review Letters (2005), 94 (1), 018301/1-018301/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    The authors report on the behavior of micron-sized prolate ellipsoids trapped at an oil-H2O interface. The particles experience strong, anisotropic, and long-ranged attractive capillary interactions which greatly exceed the thermal energy kBT. Depending on surface chem., the particles aggregate into open structures or chains. Using video microscopy, the authors ext. the pair interaction potential between ellipsoids and show it exhibits a power law behavior over the length scales probed. Observations can be explained using recent calcns., if the authors describe the interfacial ellipsoids as capillary quadrupoles.
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    Zhang, Z. K.; Pfleiderer, P.; Schofield, A. B.; Clasen, C.; Vermant, J. Directed Self-Assembly of Patterned Anisometric Polymeric Particles. J. Am. Chem. Soc. 2011, 133, 392– 395,  DOI: 10.1021/ja108099r
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    Synthesis and Directed Self-Assembly of Patterned Anisometric Polymeric Particles
    Zhang, Zhen-Kun; Pfleiderer, Patrick; Schofield, Andrew B.; Clasen, Christian; Vermant, Jan
    Journal of the American Chemical Society (2011), 133 (3), 392-395CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    A simple and versatile method for making chem. patterned anisotropic colloidal particles is proposed and demonstrated for two different types of patterning. Using a combination of thermo/mech. stretching followed by a wet chem. treatment of a sterically stabilized latex, both patchy ellipsoidal particles with sticky interactions near the tips as well as particles with tunable fluorescent patterns could be easily produced. The potential of such model colloidal particles is demonstrated, specifically for the case of directed self-assembly.
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    Botto, L.; Yao, L.; Leheny, R. L.; Stebe, K. J. Capillary bond between rod-like particles and the micromechanics of particle-laden interfaces. Soft Matter 2012, 8, 4971,  DOI: 10.1039/c2sm25211b
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    23
    Capillary bond between rod-like particles and the micromechanics of particle-laden interfaces
    Botto, L.; Yao, L.; Leheny, R. L.; Stebe, K. J.
    Soft Matter (2012), 8 (18), 4971-4979CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    Rod-like microparticles assemble by capillarity at fluid interfaces to make distinctively different microstructures depending on the details of the particle shape. Ellipsoidal particles assemble in side-to-side orientations to form flexible chains, whereas cylinders assemble end-to-end to form rigid chains. To understand these differences, we simulate the near-field capillary interactions between pairs of rod-like particles subject to bond-stretching and bond-bending deformations. By comparing ellipsoids, cylinders, and cylinders with smooth edges, we show that geometric details dramatically affect the magnitude and shape of the capillary energy landscape. We relate this energy landscape to the mechanics of the chains, predicting the flexural rigidity for chains of ellipsoids, and a complex, non-elastic response for chains of cylinders. These results have implications in the design of particle laden interfaces for emulsion stabilization and encapsulation, and for oriented assembly of anisotropic materials.
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    Rezvantalab, H.; Shojaei-Zadeh, S. Role of Geometry and Amphiphilicity on Capillary-Induced Interactions between Anisotropic Janus Particles. Langmuir 2013, 29, 14962– 14970,  DOI: 10.1021/la4039446
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    24
    Role of Geometry and Amphiphilicity on Capillary-Induced Interactions between Anisotropic Janus Particles
    Rezvantalab, Hossein; Shojaei-Zadeh, Shahab
    Langmuir (2013), 29 (48), 14962-14970CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
    Capillary interactions between ellipsoidal Janus particles adsorbed at flat liq.-fluid interfaces have been studied. 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 equil. The interface deforms around ellipsoids with tilted orientations and thus results in energetic interactions between neighboring particles. The authors 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. 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. The effect of the particles aspect ratio and the degree of amphiphilicity on the interparticle force and the capillary torque was evaluated. The energy landscapes enable prediction of micromechanics of particle chains, which has implications in predicting the interfacial rheol. of such particles at fluid interfaces.
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    Park, B. J.; Choi, C.-H.; Kang, S.-M.; Tettey, K. E.; Lee, C.-S.; Lee, D. Double Hydrophilic Janus Cylinders at an Air–Water Interface. Langmuir 2013, 29, 1841– 1849,  DOI: 10.1021/la304829s
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    25
    Double Hydrophilic Janus Cylinders at an Air-Water Interface
    Park, Bum Jun; Choi, Chang-Hyung; Kang, Sung-Min; Tettey, Kwadwo E.; Lee, Chang-Soo; Lee, Daeyeon
    Langmuir (2013), 29 (6), 1841-1849CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
    The interfacial behavior of asym. hydrophilic Janus cylinders (or double hydrophilic Janus cylinders with two different hydrophilic regions) trapped at an air-water interface has been investigated. These double hydrophilic Janus cylinders with aspect ratios of 0.9, 1.2, and 2.4 adopted both end-on and tilted configurations with respect to the interface. Numerical simulations showed that the coexistence of these configurations was a result of multiple energy min. present in the attachment energy profile that can be represented as a complex energy landscape. Double hydrophilic Janus cylinders with tilted orientations induced hexapolar interface deformation, which accounted for the pair interactions between the particles as well as the nondeterministic assembly behaviors of these particles at the interface.
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    Park, B. J.; Choi, C.-H.; Kang, S.-M.; Tettey, K. E.; Lee, C.-S.; Lee, D. Geometrically and chemically anisotropic particles at an oil–water interface. Soft Matter 2013, 9, 3383,  DOI: 10.1039/c3sm27635j
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    Geometrically and chemically anisotropic particles at an oil-water interface
    Park, Bum Jun; Choi, Chang-Hyung; Kang, Sung-Min; Tettey, Kwadwo E.; Lee, Chang-Soo; Lee, Daeyeon
    Soft Matter (2013), 9 (12), 3383-3388CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    We present the behavior of particles with chem. and geometrical anisotropy at a planar oil-water interface. We find that Janus cylinders with a small aspect ratio adopt an upright configuration, whereas the particles with a large aspect ratio exhibit both the upright and tilted configurations, which can be explained by the presence of two min. in the attachment energy profile. Such unique configurations significantly affect their assembly structure and lateral interactions. In particular, we observe strong capillary attractions between two tilted Janus cylinders and show that the scaling behavior of the interaction depends on the lateral alignments of two cylinders. Consequently, this capillarity leads to a variety of assembled structures, which we attribute to the quasi-quadrupolar interface deformation obsd. around each particle.
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    Dominguez, A.; Oettel, M.; Dietrich, S. Capillary Attraction of Colloidal Particles at an Aqueous Interface. J. Chem. Phys. 2008, 128, 12
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    Lewandowski, E. P.; Bernate, J. A.; Searson, P. C.; Stebe, K. J. Rotation and alignment of anisotropic particles on nonplanar interfaces. Langmuir 2008, 24, 9302– 9307,  DOI: 10.1021/la801167h
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    Rotation and Alignment of Anisotropic Particles on Nonplanar Interfaces
    Lewandowski, E. P.; Bernate, J. A.; Searson, P. C.; Stebe, K. J.
    Langmuir (2008), 24 (17), 9302-9307CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
    We study the alignment of micron-scale particles at air-water interfaces with unequal principle radii of curvature by optical microscopy. The fluid interface bends to satisfy the wetting conditions at the three phase contact line where the interface intersects the particle, creating deflections that increase the area of the interface. These deflections decay far from the particle. The far field interface shape has differing principle radii of curvature over length scales large compared to the particle. The deflections create excess area which depends on the angle of the particle with respect to the principle axes of the interface. We show that when particles create surface deflections with quadrupolar modes, the particles rotate to preferred orientations to minimize the free energy. In expt., we focus on uniform surface energy particles, for which quadrupolar modes are forced by the particle shape. Anal. expressions for the torque and stable states are derived in agreement with expt. and confirmed computationally.
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    Cavallaro, M., Jr.; Botto, L.; Lewandowski, E. P.; Wang, M.; Stebe, K. J. Curvature-driven capillary migration and assembly of rod-like particles. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 20923– 20928,  DOI: 10.1073/pnas.1116344108
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    Curvature-driven capillary migration and assembly of rod-like particles
    Cavallaro, Marcello, Jr.; Botto, Lorenzo; Lewandowski, Eric P.; Wang, Marisa; Stebe, Kathleen J.
    Proceedings of the National Academy of Sciences of the United States of America (2011), 108 (52), 20923-20928, S20923/1-S20923/10CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    Capillarity can be used to direct anisotropic colloidal particles to precise locations and to orient them by using interface curvature as an applied field. We show this in expts. in which the shape of the interface is molded by pinning to vertical pillars of different cross-sections. These interfaces present well-defined curvature fields that orient and steer particles along complex trajectories. Trajectories and orientations are predicted by a theor. model in which capillary forces and torques are related to Gaussian curvature gradients and angular deviations from principal directions of curvature. Interface curvature diverges near sharp boundaries, similar to an elec. field near a pointed conductor. We exploit this feature to induce migration and assembly at preferred locations, and to create complex structures. We also report a repulsive interaction, in which microparticles move away from planar bounding walls along curvature gradient contours. These phenomena should be widely useful in the directed assembly of micro- and nanoparticles with potential application in the fabrication of materials with tunable mech. or electronic properties, in emulsion prodn., and in encapsulation.
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    Guzowski, J.; Tasinkevych, M.; Dietrich, S. Free energy of colloidal particles at the surface of sessile drops. Eur. Phys. J. E 2010, 33, 219– 242,  DOI: 10.1140/epje/i2010-10667-2
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    Free energy of colloidal particles at the surface of sessile drops
    Guzowski, J.; Tasinkevych, M.; Dietrich, S.
    European Physical Journal E: Soft Matter and Biological Physics (2010), 33 (3), 219-242CODEN: EPJSFH; ISSN:1292-8941. (Springer)
    The influence of finite system size on the free energy of a spherical particle floating at the surface of a sessile droplet is studied both anal. and numerically. In the special case that the contact angle at the substrate equals π/2, a capillary analog of the method of images is applied in order to calc. small deformations of the droplet shape if an external force is applied to the particle. The type of boundary conditions for the droplet shape at the substrate dets. the sign of the capillary monopole assocd. with the image particle. Therefore, the free energy of the particle, which is proportional to the interaction energy of the original particle with its image, can be of either sign, too. The analytic solns., given by the Green's function of the capillary equation, are constructed such that the condition of the forces acting on the droplet being balanced and of the vol. constraint are fulfilled. Besides the known phenomena of attraction of a particle to a free contact line and repulsion from a pinned one, we observe a local free-energy min. for the particle being located at the drop apex or at an intermediate angle, resp. This peculiarity can be traced back to a non-monotonic behavior of the Green's function, which reflects the interplay between the deformations of the droplet shape and the vol. constraint.
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    Guzowski, J.; Tasinkevych, M.; Dietrich, S. Effective interactions and equilibrium configurations of colloidal particles on a sessile droplet. Soft Matter 2011, 7, 4189,  DOI: 10.1039/c0sm01440k
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    Effective interactions and equilibrium configurations of colloidal particles on a sessile droplet
    Guzowski, J.; Tasinkevych, M.; Dietrich, S.
    Soft Matter (2011), 7 (9), 4189-4197CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    We study the free energy landscapes of a pair of micron-sized spherical particles floating at the surface of a sessile droplet. The particles are subjected to radial external forces resulting in a deformation of the droplet shape relative to the ref. shape of a spherical cap. This deformation leads to tangential forces on the particles. For small deformations and for the contact angle θ0 at the substrate being equal to π/2, the corresponding linearized Young-Laplace equation is solved anal. The soln. is constructed by employing the method of images from electrostatics, where each of the particles plays the role of a capillary monopole and the substrate is replaced by a virtual drop with image charges and by imposing the conditions of fixed droplet vol. and vanishing total force on the droplet. The substrate boundary conditions det. the signs of the image capillary charges and therefore also the strength of the tangential forces on the particles. In the cases of an arbitrary contact angle θ0 these forces are calcd. numerically by employing a finite element method to find the equil. shape of the droplet for those configurations in which the particles are close to the local free energy min.
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    Newton, B. J.; Buzza, D. M. A. Magnetic cylindrical colloids at liquid interfaces exhibit non-volatile switching of their orientation in an external field. Soft Matter 2016, 12, 5285– 5296,  DOI: 10.1039/c6sm00136j
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    Magnetic cylindrical colloids at liquid interfaces exhibit non-volatile switching of their orientation in an external field
    Newton, Bethany J.; Buzza, D. Martin A.
    Soft Matter (2016), 12 (24), 5285-5296CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    We study the orientation of magnetic cylindrical particles adsorbed at a liq. interface in an external field using anal. theory and high resoln. finite element simulations. Cylindrical particles are interesting since they possess multiple locally stable orientations at the liq. interface so that the orientational transitions induced by an external field will not disappear when the external field is removed, i.e., the switching effect is non-volatile. We show that, in the absence of an external field, as we reduce the aspect ratio α of the cylinders below a crit. value (αc ≈ 2) the particles undergo spontaneous symmetry breaking from a stable side-on state to one of two equiv. stable tilted states, similar to the spontaneous magnetization of a ferromagnet going through the Curie point. By tuning both the aspect ratio and contact angle of the cylinders, we show that it is possible to engineer particles that have one, two, three or four locally stable orientations. We also find that the magnetic responses of cylinders with one or two stable states are similar to that of paramagnets and ferromagnets resp., while the magnetic response of systems with three or four stable states are even more complex and have no analogs in simple magnetic systems. Magnetic cylinders at liq. interfaces therefore provide a facile method for creating switchable functional monolayers where we can use an external field to induce multiple non-volatile changes in particle orientation and self-assembled structure.
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    Newton, B. J.; Mohammed, R.; Davies, G. B.; Botto, L.; Buzza, D. M. A. Capillary Interaction and Self-Assembly of Tilted Magnetic Ellipsoidal Particles at Liquid Interfaces. ACS Omega 2018, 3, 14962– 14972,  DOI: 10.1021/acsomega.8b01818
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    Capillary Interaction and Self-Assembly of Tilted Magnetic Ellipsoidal Particles at Liquid Interfaces
    Newton, Bethany J.; Mohammed, Rizwaan; Davies, Gary B.; Botto, Lorenzo; Buzza, D. Martin A.
    ACS Omega (2018), 3 (11), 14962-14972CODEN: ACSODF; ISSN:2470-1343. (American Chemical Society)
    Magnetic ellipsoidal particles adsorbed at a liq. interface provide exciting opportunities to create switchable functional materials, where self-assembly can be switched on/off using an external field (G.B. Davies, et al., 2014). To gain a deeper understanding of this novel system in the presence of an external field, capillary interaction and self-assembly of tilted ellipsoids using anal. theory and finite element simulations were assessed. We derive an anal. expression for the dipolar capillary interaction between tilted ellipsoids in elliptical polar coordinates, which exhibited a 1/r2 power law dependence in the far field (i.e., large particle sepns., r) was derived; it correctly captured orientational dependence of capillary interactions in the near-field. Using this dipole potential and finite element simulations, the particle cluster energy landscape, consisting of up to eight tilted, in-contact ellipsoids, were analyzed. For two particle clusters, the side-to-side configuration was stable; the tip-to-tip configuration was unstable. For more than three particle clusters, the circular loops of side-to-side particles became globally stable; linear chains of side-to-side particles became meta-stable. The energy barrier for the linear to loop transition decreased with increasing particle no. Results explained thermodynamically and kinetically why tilted ellipsoids assemble side-to-side locally, but have a strong tendency to form loops on larger length scales.
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    Newton, B. J.; Brakke, K. A.; Buzza, D. M. A. Influence of magnetic field on the orientation of anisotropic magnetic particles at liquid interfaces. Phys. Chem. Chem. Phys. 2014, 16, 26051– 26058,  DOI: 10.1039/c4cp04270k
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    Influence of magnetic field on the orientation of anisotropic magnetic particles at liquid interfaces
    Newton, Bethany J.; Brakke, Kenneth A.; Buzza, D. Martin A.
    Physical Chemistry Chemical Physics (2014), 16 (47), 26051-26058CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    The authors study theor. the influence of an external magnetic field on the orientation of an ellipsoidal magnetic particle adsorbed at a liq. interface. Using the finite element program Surface Evolver, the authors calc. the equil. meniscus shape around the ellipsoidal particle and its equil. tilt angle with respect to the undeformed interface θt when a magnetic field B is applied perpendicular to the interface. As the authors increase field strength, θt increases and at a crit. magnetic field Bc1 and tilt angle θc1, the particle undergoes a discontinuous transition to the perpendicular orientation (θt = 90°). The results agree qual. with the simplified theory of Bresme and Faraudo [F. Bresme and J. Faraudo, J. Phys.: Condens. Matter, 2007, 19, 375110] which assumes that the liq. interface is flat, while they agree quant. with recent lattice-Boltzmann simulations of Davies et al. [G. Davies et al., Soft Matter, 2014, 10, 6742] which account for the deformation of the liq. meniscus. Also upon reducing the external magnetic field, at a crit. magnetic field Bc2 < Bc1, the particle undergoes a 2nd discontinuous transition from the perpendicular orientation to a finite tilt angle θc2 < θc1. For micron-sized particles where the thermal energy kBT is negligible compared to the interfacial energy, the tilt angle vs. magnetic field curve exhibits hysteresis behavior. Due to the higher degree of accuracy of the Surface Evolver method, the authors are able to analyze the behavior of the particles near these orientational transitions accurately and study how the crit. quantities Bc1, Bc2, θc1 and θc2 vary with particle aspect ratio and contact angle.
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    Coertjens, S.; Moldenaers, P.; Vermant, J.; Isa, L. Contact Angles of Microellipsoids at Fluid Interfaces. Langmuir 2014, 30, 4289– 4300,  DOI: 10.1021/la500888u
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    Contact Angles of Microellipsoids at Fluid Interfaces
    Coertjens, Stijn; Moldenaers, Paula; Vermant, Jan; Isa, Lucio
    Langmuir (2014), 30 (15), 4289-4300CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
    The wetting of anisotropic colloidal particles is of great importance in several applications, including Pickering emulsions, filled foams, and membrane transduction by particles. However, the combined effect of shape and surface chem. on the three-phase contact angle of anisotropic micrometer and submicrometer colloids has been poorly investigated to date, due to the lack of a suitable exptl. technique to resolve individual particles. In the present work, we investigate the variation of the contact angle of prolate ellipsoidal colloids at a liq.-liq. interface as a function of surface chem. and aspect ratio using freeze-fracture shadow-casting cryo-SEM. The method, initially demonstrated for spherical colloids, is extended here to the more general case of ellipsoids. The prolate ellipsoidal particles are prepd. from polystyrene and poly(Me methacrylate) spheres using a film stretching technique, in which cleaning steps are needed to remove all film material from the particle surface. The effects of the prepn. protocol are reported, and wrinkling of the three-phase contact line is obsd. when the particle surface is insufficiently cleaned. For identically prepd. ellipsoids, the cosine of the measured contact angle is, in a first approxn., a linearly decreasing function of the contact line length and thus a decreasing function of the aspect ratio. Such a trend violates Young-Laplace's equation and can be rationalized by adding a correction term to the ideal Young-Laplace contact angle that expresses the relative importance of line effects relative to surface effects. From this term the contribution of an effective line tension can be extd. This contribution includes the effects that both surface chem. and topog. heterogeneities have on the contact line and which become increasingly more important for ellipsoids with higher aspect ratios, where the contact line length to contact area ratio increases.
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    Morgan, S. O.; Fox, J.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Stasiuk, G. J.; Horozov, T. S.; Buzza, D. M. A. Adsorption trajectories of nonspherical particles at liquid interfaces. Phys. Rev. E 2021, 103, 042604,  DOI: 10.1103/physreve.103.042604
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    Adsorption trajectories of nonspherical particles at liquid interfaces
    Morgan, S. O.; Fox, J.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Stasiuk, G. J.; Horozov, T. S.; Buzza, D. M. A.
    Physical Review E (2021), 103 (4), 042604CODEN: PREHBM; ISSN:2470-0053. (American Physical Society)
    The adsorption of colloidal particles at liq. interfaces is of great importance scientifically and industrially, but the dynamics of the adsorption process is still poorly understood. In this paper we use a Langevin model to study the adsorption dynamics of ellipsoidal colloids at a liq. interface. Interfacial deformations are included by coupling our Langevin dynamics to a finite element model while transient contact line pinning due to nanoscale defects on the particle surface is encoded into our model by renormalizing particle friction coeffs. and using dynamic contact angles relevant to the adsorption timescale. Our simple model reproduces the monotonic variation of particle orientation with time that is obsd. exptl. and is also able to quant. model the adsorption dynamics for some exptl. ellipsoidal systems but not others. However, even for the latter case, our model accurately captures the adsorption trajectory (i.e., particle orientation vs. height) of the particles. Our study clarifies the subtle interplay between capillary, viscous, and contact line forces in detg. the wetting dynamics of micron-scale objects, allowing us to design more efficient assembly processes for complex particles at liq. interfaces.
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    Morgan, S. O.; Muravitskaya, A.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Horozov, T. S.; Stasiuk, G. J.; Buzza, D. M. A. Using adsorption kinetics to assemble vertically aligned nanorods at liquid interfaces for metamaterial applications. Phys. Chem. Chem. Phys. 2022, 24, 11000– 11013,  DOI: 10.1039/d1cp05484h
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    Using adsorption kinetics to assemble vertically aligned nanorods at liquid interfaces for metamaterial applications
    Morgan, S. O.; Muravitskaya, A.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Horozov, T. S.; Stasiuk, G. J.; Buzza, D. M. A.
    Physical Chemistry Chemical Physics (2022), 24 (18), 11000-11013CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    Vertically aligned monolayers of metallic nanorods have a wide range of applications as metamaterials or in surface enhanced Raman spectroscopy. However the fabrication of such structures using current top-down methods or through assembly on solid substrates is either difficult to scale up or have limited possibilities for further modification after assembly. The aim of this paper is to use the adsorption kinetics of cylindrical nanorods at a liq. interface as a novel route for assembling vertically aligned nanorod arrays that overcomes these problems. Specifically, we model the adsorption kinetics of the particle using Langevin dynamics coupled to a finite element model, accurately capturing the deformation of the liq. meniscus and particle friction coeffs. during adsorption. We find that the final orientation of the cylindrical nanorod is detd. by their initial attack angle when they contact the liq. interface, and that the range of attack angles leading to the end-on state is maximised when nanorods approach the liq. interface from the bulk phase that is more energetically favorable. In the absence of an external field, only a fraction of adsorbing nanorods end up in the end-on state (.ltorsim.40% even for nanorods approaching from the energetically favorable phase). However, by pre-aligning the metallic nanorods with exptl. achievable elec. fields, this fraction can be effectively increased to 100%. Using nanophotonic calcns., we also demonstrate that the resultant vertically aligned structures can be used as epsilon-near-zero and hyperbolic metamaterials. Our kinetic assembly method is applicable to nanorods with a range of diams., aspect ratios and materials and therefore represents a versatile, low-cost and powerful platform for fabricating vertically aligned nanorods for metamaterial applications.
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    Chen, Q.; Yan, J.; Zhang, J.; Bae, S. C.; Granick, S. Janus and Multiblock Colloidal Particles. Langmuir 2012, 28, 13555– 13561,  DOI: 10.1021/la302226w
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    Janus and multiblock colloidal particles
    Chen, Qian; Yan, Jing; Zhang, Jie; Bae, Sung Chul; Granick, Steve
    Langmuir (2012), 28 (38), 13555-13561CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
    A review. Recent developments in the synthesis and self-assembly of Janus and multiblock colloidal particles, highlighting new opportunities for colloid science and technol. that are enabled by encoding orientational order between particles as they self-assemble, are considered. Emphasizing the concepts of mol. colloids and colloid valence unique to such colloids, the authors describe their rational self-assembly into colloidal clusters, taking monodisperse tetrahedra as an example. The authors also introduce a simple method to lock clusters into permanent shapes. Extending this to 2-dimensional lattices, the authors also review recent progress in assembling new open colloidal networks including the Kagome lattice. In each application, areas of opportunity are emphasized.
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  • Abstract

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    Figure 1

    Figure 1. (a) Geometry of the simulated rod-like particles; (b) geometry of a rod-like particle adsorbed at a cylindrical sessile drop.

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    Figure 2

    Figure 2. Degrees of freedom of a rod adsorbed at the cylindrical interface; the red dot represents the center of mass of the rod. (a) Cylindrical polar coordinates used to specify the position of the rod; (b) bond angle θb and tilt angle θt (defined with respect to the interfacial frame u, v, w) used to specify the orientation of the rod.

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    Figure 3

    Figure 3. Contour plot of meniscus deformation around rod-like particles with aspect ratio 2.5 and homogeneous surface chemistry, adsorbed at a flat fluid–fluid interface for different particle shapes and contact angles.

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    Figure 4

    Figure 4. Interfacial energy as a function of bond angle for relatively short rods with different contact angles, adsorbed at a cylindrical interface, for (a) ellipsoids; (b) cylinders; and (c) spherocylinders. All rods have θp = 0°, xp = 0, θt = 0°. Note that in (c), the curve for θw = 70° is not visible as it lies underneath the curve for θw = 90°.

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    Figure 5

    Figure 5. Interfacial energy as a function of bond angle for ellipsoids with different aspect ratios and contact angles, adsorbed at a cylindrical interface. All rods have θp = 0°, xp = 0, and θt = 0°.

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    Figure 6

    Figure 6. Interfacial energy as a function of bond angle for cylinders with different aspect ratios and contact angles, adsorbed at a cylindrical interface. All rods have θp = 0°, xp = 0, and θt = 0°.

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    Figure 7

    Figure 7. (a) Simplified model for calculating the maximum displacement of adsorbed rods lateral to a cylindrical drop, due to steric repulsion from the substrate. (b) Interfacial energy as a function of lateral displacement for adsorbed rods with different shapes and surface chemistry, which are aligned parallel to the cylindrical drop. The dashed black line indicates maximum lateral displacement allowed by steric repulsion with the substrate.

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    Figure 8

    Figure 8. Tip-to-tip capillary interaction for adsorbed rods with different shapes and surface chemistry, which are aligned parallel to the cylindrical drop.

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    Figure 9

    Figure 9. Capillary interaction energy of a two-particle system as a function of the position of the second particle relative to the first one, which is positioned at the edge of the cylindrical drop. The black lines are the trajectories of the second particle for different starting positions, the red curve is the separatrix that separates trajectories ending up in the tip-to-tip or side-to-side configurations, and the yellow lines are the dynamical attractors to which the trajectories converge at the later stages of their evolution. The energy landscape and trajectories are shown for (a) ellipsoids with a/b = 2.5, θw = 110°; (b) cylinders with a/b = 2.5, θw = 70°. The shaded-out white region on the bottom left of each plot represents the region excluded to the second particle due to steric repulsion with the first particle (colored yellow).

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    Figure 10

    Figure 10. Contour plot of meniscus deformation around a triblock patchy particle with aspect ratio 5 adsorbed at a flat fluid–fluid interface.

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    This article references 41 other publications.

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      Emulsions stabilised solely by colloidal particles
      Aveyard, Robert; Binks, Bernard P.; Clint, John H.
      Advances in Colloid and Interface Science (2003), 100-102 (), 503-546CODEN: ACISB9; ISSN:0001-8686. (Elsevier Science B.V.)
      A review. The prepn. and properties of emulsions, stabilized solely by the adsorption of solid particles at the oil-H2O interface, are reviewed esp. in the light of our own work with particles of well-controlled surface properties. Where appropriate, comparison is made with the behavior of surfactant-stabilized emulsions. Hydrophilic particles tend to form oil-in-H2O (o/w) emulsions whereas hydrophobic particles form H2O-in-oil (w/o) emulsions. Many of the properties can be attributed to the very large free energy of adsorption for particles of intermediate wettability (contact angle at the oil-H2O interface, for example, between 50 and 130°). This effectively irreversible adsorption leads to extreme stability for certain emulsions and is in contrast to the behavior of surfactant mols. which are usually in rapid dynamic equil. between the oil-H2O interface and the bulk phases. There is evidence that, in some systems, weak flocculation of the particles improves the emulsion stability. Phase inversion from w/o to o/w can be brought about by increasing the vol. fraction of H2O. Emulsions close to this inversion point tend to be the most stable, again in contrast to surfactant systems. The vol. fraction needed for inversion depends on the particle wettability and the nature of the oil and these effects were rationalized in terms of surface energy components. Stable multiple emulsions (w/o/w and o/w/o) can be made using 2 types of particles with slightly different wettability. Similar multiple emulsions prepd. with 2 types of surfactant tend to be much less stable. The possibility of prepg. novel solid materials by evapg. solid-stabilized emulsions is also proposed. Finally the authors report on some extensions to the work of Levine et al. who obtained expressions for the free energy of formation of emulsion drops covered with close-packed monolayers of monodisperse spherical particles. In particular in the light of the observations that nanoparticles can act as excellent emulsion stabilizers, the authors have considered potential effects on the free energy of emulsion formation of the action of small (phys. realistic) pos. and neg. line tensions in the 3-phase contact lines skirting particles adsorbed at the droplet interfaces. The authors also explore the possibility that curvature properties of close-packed particle monolayers can affect emulsion properties in much the same way that surfactant monolayer properties influence emulsion type and stability.
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      Dinsmore, A. D.; Hsu, Ming F.; Nikolaides, M. G.; Marquez, Manuel; Bausch, A. R.; Weitz, D. A.
      Science (Washington, DC, United States) (2002), 298 (5595), 1006-1009CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
      We present an approach to fabricate solid capsules with precise control of size, permeability, mech. strength, and compatibility. The capsules are fabricated by the self-assembly of colloidal particles onto the interface of emulsion droplets. After the particles are locked together to form elastic shells, the emulsion droplets are transferred to a fresh continuous-phase fluid that is the same as that inside the droplets. The resultant structures, which we call "colloidosomes," are hollow, elastic shells whose permeability and elasticity can be precisely controlled. The generality and robustness of these structures and their potential for cellular immunoisolation are demonstrated by the use of a variety of solvents, particles, and contents.
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      Dickinson, E. Food emulsions and foams: Stabilization by particles. Curr. Opin. Colloid Interf. Sci. 2010, 15, 40– 49,  DOI: 10.1016/j.cocis.2009.11.001
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      Food emulsions and foams: Stabilization by particles
      Dickinson, Eric
      Current Opinion in Colloid & Interface Science (2010), 15 (1-2), 40-49CODEN: COCSFL; ISSN:1359-0294. (Elsevier B.V.)
      A review. Recent advances in the stabilization of emulsions and foams by particles of nanoscale and microscopic dimensions are described. Ongoing research in this highly active field is providing insight into (i) the mol. factors controlling particle wettability and adsorption, (ii) the structural and mech. properties of particle-laden liq. interfaces, and (ii) the stabilization mechanisms of particle-coated droplets and bubbles. There is much potential for exploiting the emerging knowledge in new food product applications. The prepn. of cheap and effective colloidal particles based on food-grade ingredients, esp. proteins, is the key technol. challenge.
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      Forth, J.; Kim, P. Y.; Xie, G.; Liu, X.; Helms, B. A.; Russell, T. P. Building Reconfigurable Devices Using Complex Liquid–Fluid Interfaces. Adv. Mater. 2019, 31, 1806370,  DOI: 10.1002/adma.201806370
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      Rey, M.; Law, A. D.; Buzza, D. M. A.; Vogel, N. Anisotropic Self-Assembly from Isotropic Colloidal Building Blocks. J. Am. Chem. Soc. 2017, 139, 17464– 17473,  DOI: 10.1021/jacs.7b08503
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      Anisotropic Self-Assembly from Isotropic Colloidal Building Blocks
      Rey, Marcel; Law, Adam D.; Buzza, D. Martin A.; Vogel, Nicolas
      Journal of the American Chemical Society (2017), 139 (48), 17464-17473CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      Spherical colloidal particles generally self-assemble into hexagonal lattices in two dimensions. However, more complex, non-hexagonal phases have been predicted theor. for isotropic particles with a soft repulsive shoulder but have not been exptl. realized. We study the phase behavior of microspheres in the presence of poly(N-isopropylacrylamide) (PNiPAm) microgels at the air/water interface. We observe a complex phase diagram, including phases with chain and square arrangements, which exclusively form in the presence of the microgels. Our exptl. data suggests that the microgels form a corona around the microspheres and induce a soft repulsive shoulder that governs the self-assembly in this system. The obsd. structures are fully reproduced by both min. energy calcns. and finite temp. Monte Carlo simulations of hard core-soft shoulder particles with exptl. realistic interaction parameters. Our results demonstrate how complex, anisotropic assembly patterns can be realized from entirely isotropic building blocks by control of the interaction potential.
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      Menath, J.; Eatson, J. L.; Brilmayer, R.; Andrieu-Brunsen, A.; Buzza, D. M. A.; Vogel, N. Defined core–shell particles as the key to complex interfacial self-assembly. Proc. Natl. Acad. Sci. U.S.A. 2021, 118, e2113394118  DOI: 10.1073/pnas.2113394118
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      Defined core-shell particles as the key to complex interfacial self-assembly
      Menath, Johannes; Eatson, Jack; Brilmayer, Robert; Andrieu-Brunsen, Annette; Buzza, D. Martin A.; Vogel, Nicolas
      Proceedings of the National Academy of Sciences of the United States of America (2021), 118 (52), e2113394118CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      The two-dimensional self-assembly of colloidal particles serves as a model system for fundamental studies of structure formation and as a powerful tool to fabricate functional materials and surfaces. However, the prevalence of hexagonal symmetries in such self-assembling systems limits its structural versatility. More than two decades ago, Jagla demonstrated that core-shell particles with two interaction length scales can form complex, nonhexagonal min. energy configurations. Based on such Jagla potentials, a wide variety of phases including cluster lattices, chains, and quasicrystals have been theor. discovered. Despite the elegance of this approach, its exptl. realization has remained largely elusive. Here, we capitalize on the distinct interfacial morphol. of soft particles to design two-dimensional assemblies with structural complexity. We find that core-shell particles consisting of a silica core surface functionalized with a noncrosslinked polymer shell efficiently spread at a liq. interface to form a two-dimensional polymer corona surrounding the core. We controllably grow such shells by iniferter-type controlled radical polymn. Upon interfacial compression, the resulting core-shell particles arrange in well-defined dimer, trimer, and tetramer lattices before transitioning into complex chain and cluster phases. The exptl. phase behavior is accurately reproduced by Monte Carlo simulations and min. energy calcns., suggesting that the interfacial assembly interacts via a pairwise-additive Jagla-type potential. By comparing theory, simulation, and expt., we narrow the Jagla g-parameter of the system to between 0.9 and 2. The possibility to control the interaction potential via the interfacial morphol. provides a framework to realize structural features with unprecedented complexity from a simple, one-component system.
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      Pieranski, P. Two-Dimensional Interfacial Colloidal Crystals. Phys. Rev. Lett. 1980, 45, 569– 572,  DOI: 10.1103/physrevlett.45.569
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      Two-dimensional interfacial colloidal crystals
      Pieranski, Pawel
      Physical Review Letters (1980), 45 (7), 569-72CODEN: PRLTAO; ISSN:0031-9007.
      Polystyrene spheres (2450 Å in diam.) are trapped in a surface energy well at the water/air interface. Because of asymmetry of charge distribution, elec. dipoles are assocd. with each interfacial particle. The dipole-dipole repulsive interactions organize the polystyrene spheres into a 2-dimensional triangular lattice. The direct microscopic observations of such an interfacial colloidal crystal are reported.
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      Law, A. D.; Buzza, D. M. A.; Horozov, T. S. Two-Dimensional Colloidal Alloys. Phys. Rev. Lett. 2011, 106, 128302,  DOI: 10.1103/physrevlett.106.128302
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      Two-Dimensional Colloidal Alloys
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      Physical Review Letters (2011), 106 (12), 128302/1-128302/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We study the structure of mixed monolayers of large (3 μm diam.) and small (1 μm diam.) very hydrophobic silica particles at an octane-water interface as a function of the no. fraction of small particles ξ. We find that a rich variety of two-dimensional hexagonal super-lattices of large (A) and small (B) particles can be obtained in this system due to strong and long-range electrostatic repulsions through the nonpolar octane phase. The structures obtained for the different compns. are in good agreement with zero temp. calcns. and finite temp. computer simulations.
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      Law, A. D.; Auriol, M.; Smith, D.; Horozov, T. S.; Buzza, D. M. A. Self-Assembly of Two-Dimensional Colloidal Clusters by Tuning the Hydrophobicity, Composition, and Packing Geometry. Phys. Rev. Lett. 2013, 110, 138301,  DOI: 10.1103/physrevlett.110.138301
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      Self-assembly of two-dimensional colloidal clusters by tuning the hydrophobicity, composition, and packing geometry
      Law, Adam D.; Auriol, Melodie; Smith, Dean; Horozov, Tommy S.; Buzza, D. Martin A.
      Physical Review Letters (2013), 110 (13), 138301/1-138301/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We study the structure of binary monolayers of large (3 μm diam.) very hydrophobic (A) and large (3 μm diam.) hydrophilic (B) or small (1 μm diam.) hydrophilic (C) silica particles at an octane-water interface. By tuning the compn. and packing geometry of the mixed monolayer, we find that a rich variety of two-dimensional hexagonal superlattices of mixed A/B or A/C clusters are formed, stabilized by short-ranged electrostatic induced dipole interactions. The cluster structures obtained are in excellent agreement with zero temp. calcns., indicating that the self-assembly process can be effectively controlled.
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    11. 11
      Kralchevsky, P. A.; Nagayama, K. Capillary forces between colloidal particles. Langmuir 1994, 10, 23– 36,  DOI: 10.1021/la00013a004
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      11
      Capillary forces between colloidal particles
      Kralchevsky, Peter A.; Nagayama, Kuniaki
      Langmuir (1994), 10 (1), 23-36CODEN: LANGD5; ISSN:0743-7463.
      A special kind of capillary interaction is considered, which differs from the common lateral capillary forces between floating particles. It appears between particles protruding from a liq. film and its phys. origin is the capillary rise of the liq. along the surface of each particle. Special attention is paid to the case when the position of the contact line is fixed. The resulting capillary force is compared with that at fixed contact angle. The 2 alternative approaches to the calcn. of capillary interactions (the force and the energetic one) are equiv. When the liq. films is thin, the disjoining pressure affects the capillary interactions between particles attached to the film surfaces. The appearance of this effect is studied quant. for 2 specific systems modeling globular proteins in an aq. film on a Hg substrate and membrane proteins incorporated in a lipid bilayer. For both systems the capillary forces appear to be strong enough to engender 2-dimensional particle aggregation and ordering. This is a possible explanation of a no. of exptl. observations of such effects.
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    12. 12
      Kralchevsky, P. A.; Nagayama, K. Capillary interactions between particles bound to interfaces, liquid films and biomembranes. Adv. Colloid Interface Sci. 2000, 85, 145– 192,  DOI: 10.1016/s0001-8686(99)00016-0
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      12
      Capillary interactions between particles bound to interfaces, liquid films and biomembranes
      Kralchevsky, P. A.; Nagayama, K.
      Advances in Colloid and Interface Science (2000), 85 (2-3), 145-192CODEN: ACISB9; ISSN:0001-8686. (Elsevier Science B.V.)
      A review, with 116 refs., is given. This article is devoted to an overview, comparison and discussion of recent results (both theor. and exptl.) about lateral capillary forces. They appear when the contact of particles or other bodies with a fluid phase boundary causes perturbations in the interfacial shape. The capillary interaction is due to the overlap of such perturbations which can appear around floating particles, vertical cylinders, particles confined in a liq. film, inclusions in the membranes of lipid vesicles or living cells, etc. In the case of floating particles the perturbations are due to the particle wt.; in this case the force decreases with the 6th power of the particle size and becomes immaterial for particles smaller than ∼10 μm. In all other cases the interfacial deformations are due to the particle wetting properties; the resulting immersion capillary forces can be operative even between very small particles, like protein globules. In many cases such forces can be responsible for the exptl. obsd. two-dimensional particle aggregation and ordering. An analogy between capillary and electrostatic forces enables one to introduce capillary charges of the attached particles, which characterize the magnitude of the interfacial deformation and could be both pos. and neg. Also, the capillary interaction between particle and wall resembles the image force in electrostatics. When a particle is moving bound to an interface under the action of a capillary force, one can det. the surface drag coeff. and the surface viscosity supposedly the magnitude of the capillary force is known. Alternative (but equiv.) energy and force approaches can be used for the theor. description of the lateral capillary interactions. Both approaches require the Laplace equation of capillarity to be solved and the meniscus profile around the particles to be detd. The energy approach accounts for contributions due to the increase of the meniscus area, gravitational energy and/or energy of wetting. The 2nd approach is based on calcg. the net force exerted on the particle, which can originate from the hydrostatic pressure, interfacial tension and bending moment. In the case of small perturbations, the superposition approxn. can be used to derive an asymptotic formula for the capillary forces, which was found to agree well with the expt. Capillary interactions between particles bound to spherical interfaces are also considered taking into account the special geometry and restricted area of such phase boundaries. A similar approach can be applied to quantify the forces between inclusions (transmembrane proteins) in lipid membranes. The deformations in a lipid membrane, due to the inclusions, can be described theor. in the framework of a mech. model of the lipid bilayer, which accounts for its hybrid rheol. (neither elastic body nor fluid). In all considered cases the lateral capillary interaction originates from the overlap of interfacial deformations and is subject to a unified theor. treatment, despite the fact that the characteristic particle size can vary from 1 cm down to 1 nm.
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    13. 13
      Botto, L.; Lewandowski, E. P.; Cavallaro, M.; Stebe, K. J. Capillary interactions between anisotropic particles. Soft Matter 2012, 8, 9957,  DOI: 10.1039/c2sm25929j
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      Capillary interactions between anisotropic particles
      Botto, Lorenzo; Lewandowski, Eric P.; Cavallaro, Marcello; Stebe, Kathleen J.
      Soft Matter (2012), 8 (39), 9957-9971CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      Micro and nanoparticle adsorption to and assembly by capillarity at fluid-fluid interfaces are intriguing aspects of soft matter science with broad potential in the directed assembly of anisotropic media. The importance of the field stems from the ubiquitous presence of multiphase systems, the malleability of fluid interfaces, and the ability to tune the interactions of the particles adsorbed on them. While homogeneous spherical particles at interfaces have been well studied, the behavior of anisotropic particles - whether the anisotropy originates from shape or chem. heterogeneity - has been considered only very recently. We review recent advances in the field of anisotropic particles at fluid interfaces, by focusing on particles in the micron and submicron range. We discuss capillary adsorption, orientation, migration, and self-assembly, on planar and curved interfaces, and the rheol. of particle-laden interfaces. Prospects for future work and outstanding challenges are also discussed.
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      Davies, G. B.; Krüger, T.; Coveney, P. V.; Harting, J.; Bresme, F. Assembling Ellipsoidal Particles at Fluid Interfaces Using Switchable Dipolar Capillary Interactions. Adv. Mater. 2014, 26, 6715– 6719,  DOI: 10.1002/adma.201402419
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      Assembling Ellipsoidal Particles at Fluid Interfaces using Switchable Dipolar Capillary Interactions
      Davies, Gary B.; Krueger, Timm; Coveney, Peter V.; Harting, Jens; Bresme, Fernando
      Advanced Materials (Weinheim, Germany) (2014), 26 (39), 6715-6719CODEN: ADVMEW; ISSN:0935-9648. (Wiley-VCH Verlag GmbH & Co. KGaA)
      Self-assembly of ellipsoidal particles at fluid interfaces using switchable dipolar capillary interactions is considered. By using magnetic anisotropic particles interacting with external magnetic fields, the authors show how to dynamically tune dipolar capillary interactions between particles by varying the dipole-field strength, and how to switch these dipolar capillary interactions on and off by making the particles undergo a first-order orientation transition. Simulations reveal self-assembled structures that depend on the surface coverage of particles and the dipole-field strength. The formation of "capillary caterpillars" is obsd. in which particles align in side-side configurations, and "capillary couples" where particles in individual caterpillars align in tip-tip chains with particles in other caterpillars, due to the anti-sym. menisci formation.
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      Dasgupta, S.; Auth, T.; Gompper, G. Nano- and microparticles at fluid and biological interfaces. J. Phys.: Condens. Matter 2017, 29, 373003,  DOI: 10.1088/1361-648x/aa7933
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      Nano- and microparticles at fluid and biological interfaces
      Dasgupta, S.; Auth, T.; Gompper, G.
      Journal of Physics: Condensed Matter (2017), 29 (37), 373003/1-373003/41CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
      A review. Systems with interfaces are abundant in both technol. applications and biol. While a fluid interface separates 2 fluids, membranes sep. the inside of vesicles from the outside, the interior of biol. cells from the environment, and compartmentalize cells into organelles. The phys. properties of interfaces are characterized by interface tension, those of membranes are characterized by bending and stretching elasticity. Amphiphilic mols. like surfactants that are added to a system with 2 immiscible fluids decrease the interface tension and induce a bending rigidity. Lipid bilayer membranes of vesicles can be stretched or compressed by osmotic pressure; in biol. cells, also the presence of a cytoskeleton can induce membrane tension. If the thickness of the interface or the membrane is small compared with its lateral extension, both can be described using 2-dimensional math. surfaces embedded in 3-dimensional space. Here, we review recent work on the interaction of particles with interfaces and membranes. This can be micrometer-sized particles at interfaces that stabilize emulsions or form colloidosomes, as well as typically nanometer-sized particles at membranes, such as viruses, parasites, and engineered drug delivery systems. In both cases, we 1st discuss the interaction of single particles with interfaces and membranes, e.g., particles in external fields, non-spherical particles, and particles at curved interfaces, followed by interface-mediated interaction between 2 particles, many-particle interactions, interface and membrane curvature-induced phenomena, and applications.
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      Vella, D.; Mahadevan, L. The “Cheerios effect”. Am. J. Phys. 2005, 73, 817– 825,  DOI: 10.1119/1.1898523
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      Nicolson, M. M. The interaction between floating particles. Proc. Cambridge Philos. Soc. 1949, 45, 288– 295,  DOI: 10.1017/s0305004100024841
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      Loudet, J. C.; Yodh, A. G.; Pouligny, B. Wetting and Contact Lines of Micrometer-Sized Ellipsoids. Phys. Rev. Lett. 2006, 97, 018304,  DOI: 10.1103/physrevlett.97.018304
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      Wetting and Contact Lines of Micrometer-Sized Ellipsoids
      Loudet, J. C.; Yodh, A. G.; Pouligny, B.
      Physical Review Letters (2006), 97 (1), 018304/1-018304/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We exptl. and theor. investigate the shapes of contact lines on the surfaces of micrometer-sized polystyrene ellipsoids at the water-air interface. By combining interferometry and optical trapping, we directly observe quadrupolar symmetry of the interface deformations around such particles. We then develop numerical solns. of the partial wetting problem for ellipsoids, and use these solns. to deduce the shapes of the corresponding contact lines and the values of the contact angles, θc(k), as a function of the ellipsoid aspect ratio k. Surprisingly, θc is found to decrease for increasing k suggesting that ellipsoid microscopic surface properties depend on ellipsoid aspect ratio.
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      Lehle, H.; Noruzifar, E.; Oettel, M. Ellipsoidal particles at fluid interfaces. Eur. Phys. J. E 2008, 26, 151– 160,  DOI: 10.1140/epje/i2007-10314-1
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      Ellipsoidal particles at fluid interfaces
      Lehle, H.; Noruzifar, E.; Oettel, M.
      European Physical Journal E: Soft Matter (2008), 26 (1-2), 151-160CODEN: EPJSFH; ISSN:1292-8941. (EDP Sciences)
      For partially wetting, ellipsoidal colloids trapped at a fluid interface, their effective, interface-mediated interactions of capillary and fluctuation-induced type are analyzed. For contact angles different from 90°, static interface deformations arise which lead to anisotropic capillary forces that are substantial already for micrometer-sized particles. The capillary problem is solved using an efficient perturbative treatment which allows a fast detn. of the capillary interaction for all distances between and orientations of two particles. Besides static capillary forces, fluctuation-induced forces caused by thermally excited capillary waves arise at fluid interfaces. For the specific choice of a spatially fixed three-phase contact line, the asymptotic behavior of the fluctuation-induced force is detd. anal. for both the close-distance and the long-distance regime and compared to numerical solns.
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      Lewandowski, E. P.; Cavallaro, M., Jr.; Botto, L.; Bernate, J. C.; Garbin, V.; Stebe, K. J. Orientation and self-assembly of cylindrical particles by anisotropic capillary interactions. Langmuir 2010, 26, 15142– 15154,  DOI: 10.1021/la1012632
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      Orientation and Self-Assembly of Cylindrical Particles by Anisotropic Capillary Interactions
      Lewandowski, Eric P.; Cavallaro, Marcello, Jr.; Botto, Lorenzo; Bernate, Jorge C.; Garbin, Valeria; Stebe, Kathleen J.
      Langmuir (2010), 26 (19), 15142-15154CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
      In this research, we study cylindrical microparticles at fluid interfaces. Cylinders orient and assemble with high reliability to form end-to-end chains in dil. surfaces or dense rectangular lattices in crowded surfaces owing to capillary interactions. In isolation, a cylinder assumes one of two possible equil. states, the end-on state, in which the cylinder axis is perpendicular to the interface, or the side-on state, in which the cylinder axis is parallel to the interface. A phase diagram relating aspect ratio and contact angle is constructed to predict the preferred state and verified in expt. Cylinders in the side-on state create distortions that result in capillary interactions. Overlapping deformations by neighboring particles drive oriented capillary assembly. Interferometry, electron microscopy, and numerical simulations are used to characterize the interface shape around isolated particles. Expts. and numerics show that "side-on" cylinders have concd. excess area near the end faces, and that the interface distortion resembles an elliptical quadrupole a few radii away from the particle surface. To model the cylinder interactions for sepns. greater than a few radii, an anisotropic potential is derived based on elliptical quadrupoles. This potential predicts an attractive force and a torque, both of which depend strongly on aspect ratio, in keeping with expt. Particle trajectories and angular orientations recorded by video microscopy agree with the predicted potential. In particular, the anal. predicts the rate of rotation, a feature lacking in prior analyses. To understand interactions near contact, the concd. excess area near the cylinder ends is quantified and its role in creating stable end-to-end assemblies is discussed. When a pair of cylinders is near contact, these high excess area regions overlap to form a capillary bridge between the particles. This capillary bridge may stabilize the end-to-end chains. Finally, on densely packed surfaces, cylinder-covered colloidosomes form with particles arranged in regular, rectangular lattices in the interface; this densely packed structure differs significantly from assemblies reported for colloidosomes or particle-stabilized droplets in the literature.
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      Loudet, J. C.; Alsayed, A. M.; Zhang, J.; Yodh, A. G. Capillary Interactions Between Anisotropic Colloidal Particles. Phys. Rev. Lett. 2005, 94, 018301,  DOI: 10.1103/physrevlett.94.018301
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      Capillary Interactions Between Anisotropic Colloidal Particles
      Loudet, J. C.; Alsayed, A. M.; Zhang, J.; Yodh, A. G.
      Physical Review Letters (2005), 94 (1), 018301/1-018301/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The authors report on the behavior of micron-sized prolate ellipsoids trapped at an oil-H2O interface. The particles experience strong, anisotropic, and long-ranged attractive capillary interactions which greatly exceed the thermal energy kBT. Depending on surface chem., the particles aggregate into open structures or chains. Using video microscopy, the authors ext. the pair interaction potential between ellipsoids and show it exhibits a power law behavior over the length scales probed. Observations can be explained using recent calcns., if the authors describe the interfacial ellipsoids as capillary quadrupoles.
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      Zhang, Z. K.; Pfleiderer, P.; Schofield, A. B.; Clasen, C.; Vermant, J. Directed Self-Assembly of Patterned Anisometric Polymeric Particles. J. Am. Chem. Soc. 2011, 133, 392– 395,  DOI: 10.1021/ja108099r
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      Synthesis and Directed Self-Assembly of Patterned Anisometric Polymeric Particles
      Zhang, Zhen-Kun; Pfleiderer, Patrick; Schofield, Andrew B.; Clasen, Christian; Vermant, Jan
      Journal of the American Chemical Society (2011), 133 (3), 392-395CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      A simple and versatile method for making chem. patterned anisotropic colloidal particles is proposed and demonstrated for two different types of patterning. Using a combination of thermo/mech. stretching followed by a wet chem. treatment of a sterically stabilized latex, both patchy ellipsoidal particles with sticky interactions near the tips as well as particles with tunable fluorescent patterns could be easily produced. The potential of such model colloidal particles is demonstrated, specifically for the case of directed self-assembly.
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      Botto, L.; Yao, L.; Leheny, R. L.; Stebe, K. J. Capillary bond between rod-like particles and the micromechanics of particle-laden interfaces. Soft Matter 2012, 8, 4971,  DOI: 10.1039/c2sm25211b
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      Capillary bond between rod-like particles and the micromechanics of particle-laden interfaces
      Botto, L.; Yao, L.; Leheny, R. L.; Stebe, K. J.
      Soft Matter (2012), 8 (18), 4971-4979CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      Rod-like microparticles assemble by capillarity at fluid interfaces to make distinctively different microstructures depending on the details of the particle shape. Ellipsoidal particles assemble in side-to-side orientations to form flexible chains, whereas cylinders assemble end-to-end to form rigid chains. To understand these differences, we simulate the near-field capillary interactions between pairs of rod-like particles subject to bond-stretching and bond-bending deformations. By comparing ellipsoids, cylinders, and cylinders with smooth edges, we show that geometric details dramatically affect the magnitude and shape of the capillary energy landscape. We relate this energy landscape to the mechanics of the chains, predicting the flexural rigidity for chains of ellipsoids, and a complex, non-elastic response for chains of cylinders. These results have implications in the design of particle laden interfaces for emulsion stabilization and encapsulation, and for oriented assembly of anisotropic materials.
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      Rezvantalab, H.; Shojaei-Zadeh, S. Role of Geometry and Amphiphilicity on Capillary-Induced Interactions between Anisotropic Janus Particles. Langmuir 2013, 29, 14962– 14970,  DOI: 10.1021/la4039446
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      Role of Geometry and Amphiphilicity on Capillary-Induced Interactions between Anisotropic Janus Particles
      Rezvantalab, Hossein; Shojaei-Zadeh, Shahab
      Langmuir (2013), 29 (48), 14962-14970CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
      Capillary interactions between ellipsoidal Janus particles adsorbed at flat liq.-fluid interfaces have been studied. 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 equil. The interface deforms around ellipsoids with tilted orientations and thus results in energetic interactions between neighboring particles. The authors 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. 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. The effect of the particles aspect ratio and the degree of amphiphilicity on the interparticle force and the capillary torque was evaluated. The energy landscapes enable prediction of micromechanics of particle chains, which has implications in predicting the interfacial rheol. of such particles at fluid interfaces.
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      Park, B. J.; Choi, C.-H.; Kang, S.-M.; Tettey, K. E.; Lee, C.-S.; Lee, D. Double Hydrophilic Janus Cylinders at an Air–Water Interface. Langmuir 2013, 29, 1841– 1849,  DOI: 10.1021/la304829s
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      Double Hydrophilic Janus Cylinders at an Air-Water Interface
      Park, Bum Jun; Choi, Chang-Hyung; Kang, Sung-Min; Tettey, Kwadwo E.; Lee, Chang-Soo; Lee, Daeyeon
      Langmuir (2013), 29 (6), 1841-1849CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
      The interfacial behavior of asym. hydrophilic Janus cylinders (or double hydrophilic Janus cylinders with two different hydrophilic regions) trapped at an air-water interface has been investigated. These double hydrophilic Janus cylinders with aspect ratios of 0.9, 1.2, and 2.4 adopted both end-on and tilted configurations with respect to the interface. Numerical simulations showed that the coexistence of these configurations was a result of multiple energy min. present in the attachment energy profile that can be represented as a complex energy landscape. Double hydrophilic Janus cylinders with tilted orientations induced hexapolar interface deformation, which accounted for the pair interactions between the particles as well as the nondeterministic assembly behaviors of these particles at the interface.
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      Park, B. J.; Choi, C.-H.; Kang, S.-M.; Tettey, K. E.; Lee, C.-S.; Lee, D. Geometrically and chemically anisotropic particles at an oil–water interface. Soft Matter 2013, 9, 3383,  DOI: 10.1039/c3sm27635j
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      Geometrically and chemically anisotropic particles at an oil-water interface
      Park, Bum Jun; Choi, Chang-Hyung; Kang, Sung-Min; Tettey, Kwadwo E.; Lee, Chang-Soo; Lee, Daeyeon
      Soft Matter (2013), 9 (12), 3383-3388CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      We present the behavior of particles with chem. and geometrical anisotropy at a planar oil-water interface. We find that Janus cylinders with a small aspect ratio adopt an upright configuration, whereas the particles with a large aspect ratio exhibit both the upright and tilted configurations, which can be explained by the presence of two min. in the attachment energy profile. Such unique configurations significantly affect their assembly structure and lateral interactions. In particular, we observe strong capillary attractions between two tilted Janus cylinders and show that the scaling behavior of the interaction depends on the lateral alignments of two cylinders. Consequently, this capillarity leads to a variety of assembled structures, which we attribute to the quasi-quadrupolar interface deformation obsd. around each particle.
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      Dominguez, A.; Oettel, M.; Dietrich, S. Capillary Attraction of Colloidal Particles at an Aqueous Interface. J. Chem. Phys. 2008, 128, 12
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      Lewandowski, E. P.; Bernate, J. A.; Searson, P. C.; Stebe, K. J. Rotation and alignment of anisotropic particles on nonplanar interfaces. Langmuir 2008, 24, 9302– 9307,  DOI: 10.1021/la801167h
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      Rotation and Alignment of Anisotropic Particles on Nonplanar Interfaces
      Lewandowski, E. P.; Bernate, J. A.; Searson, P. C.; Stebe, K. J.
      Langmuir (2008), 24 (17), 9302-9307CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
      We study the alignment of micron-scale particles at air-water interfaces with unequal principle radii of curvature by optical microscopy. The fluid interface bends to satisfy the wetting conditions at the three phase contact line where the interface intersects the particle, creating deflections that increase the area of the interface. These deflections decay far from the particle. The far field interface shape has differing principle radii of curvature over length scales large compared to the particle. The deflections create excess area which depends on the angle of the particle with respect to the principle axes of the interface. We show that when particles create surface deflections with quadrupolar modes, the particles rotate to preferred orientations to minimize the free energy. In expt., we focus on uniform surface energy particles, for which quadrupolar modes are forced by the particle shape. Anal. expressions for the torque and stable states are derived in agreement with expt. and confirmed computationally.
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      Cavallaro, M., Jr.; Botto, L.; Lewandowski, E. P.; Wang, M.; Stebe, K. J. Curvature-driven capillary migration and assembly of rod-like particles. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 20923– 20928,  DOI: 10.1073/pnas.1116344108
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      Curvature-driven capillary migration and assembly of rod-like particles
      Cavallaro, Marcello, Jr.; Botto, Lorenzo; Lewandowski, Eric P.; Wang, Marisa; Stebe, Kathleen J.
      Proceedings of the National Academy of Sciences of the United States of America (2011), 108 (52), 20923-20928, S20923/1-S20923/10CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      Capillarity can be used to direct anisotropic colloidal particles to precise locations and to orient them by using interface curvature as an applied field. We show this in expts. in which the shape of the interface is molded by pinning to vertical pillars of different cross-sections. These interfaces present well-defined curvature fields that orient and steer particles along complex trajectories. Trajectories and orientations are predicted by a theor. model in which capillary forces and torques are related to Gaussian curvature gradients and angular deviations from principal directions of curvature. Interface curvature diverges near sharp boundaries, similar to an elec. field near a pointed conductor. We exploit this feature to induce migration and assembly at preferred locations, and to create complex structures. We also report a repulsive interaction, in which microparticles move away from planar bounding walls along curvature gradient contours. These phenomena should be widely useful in the directed assembly of micro- and nanoparticles with potential application in the fabrication of materials with tunable mech. or electronic properties, in emulsion prodn., and in encapsulation.
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      Brakke, K. A. The Surface Evolver. Exp. Math 1992, 1, 141– 165,  DOI: 10.1080/10586458.1992.10504253
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      Guzowski, J.; Tasinkevych, M.; Dietrich, S. Free energy of colloidal particles at the surface of sessile drops. Eur. Phys. J. E 2010, 33, 219– 242,  DOI: 10.1140/epje/i2010-10667-2
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      Free energy of colloidal particles at the surface of sessile drops
      Guzowski, J.; Tasinkevych, M.; Dietrich, S.
      European Physical Journal E: Soft Matter and Biological Physics (2010), 33 (3), 219-242CODEN: EPJSFH; ISSN:1292-8941. (Springer)
      The influence of finite system size on the free energy of a spherical particle floating at the surface of a sessile droplet is studied both anal. and numerically. In the special case that the contact angle at the substrate equals π/2, a capillary analog of the method of images is applied in order to calc. small deformations of the droplet shape if an external force is applied to the particle. The type of boundary conditions for the droplet shape at the substrate dets. the sign of the capillary monopole assocd. with the image particle. Therefore, the free energy of the particle, which is proportional to the interaction energy of the original particle with its image, can be of either sign, too. The analytic solns., given by the Green's function of the capillary equation, are constructed such that the condition of the forces acting on the droplet being balanced and of the vol. constraint are fulfilled. Besides the known phenomena of attraction of a particle to a free contact line and repulsion from a pinned one, we observe a local free-energy min. for the particle being located at the drop apex or at an intermediate angle, resp. This peculiarity can be traced back to a non-monotonic behavior of the Green's function, which reflects the interplay between the deformations of the droplet shape and the vol. constraint.
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      Guzowski, J.; Tasinkevych, M.; Dietrich, S. Effective interactions and equilibrium configurations of colloidal particles on a sessile droplet. Soft Matter 2011, 7, 4189,  DOI: 10.1039/c0sm01440k
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      Effective interactions and equilibrium configurations of colloidal particles on a sessile droplet
      Guzowski, J.; Tasinkevych, M.; Dietrich, S.
      Soft Matter (2011), 7 (9), 4189-4197CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      We study the free energy landscapes of a pair of micron-sized spherical particles floating at the surface of a sessile droplet. The particles are subjected to radial external forces resulting in a deformation of the droplet shape relative to the ref. shape of a spherical cap. This deformation leads to tangential forces on the particles. For small deformations and for the contact angle θ0 at the substrate being equal to π/2, the corresponding linearized Young-Laplace equation is solved anal. The soln. is constructed by employing the method of images from electrostatics, where each of the particles plays the role of a capillary monopole and the substrate is replaced by a virtual drop with image charges and by imposing the conditions of fixed droplet vol. and vanishing total force on the droplet. The substrate boundary conditions det. the signs of the image capillary charges and therefore also the strength of the tangential forces on the particles. In the cases of an arbitrary contact angle θ0 these forces are calcd. numerically by employing a finite element method to find the equil. shape of the droplet for those configurations in which the particles are close to the local free energy min.
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      Newton, B. J.; Buzza, D. M. A. Magnetic cylindrical colloids at liquid interfaces exhibit non-volatile switching of their orientation in an external field. Soft Matter 2016, 12, 5285– 5296,  DOI: 10.1039/c6sm00136j
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      Magnetic cylindrical colloids at liquid interfaces exhibit non-volatile switching of their orientation in an external field
      Newton, Bethany J.; Buzza, D. Martin A.
      Soft Matter (2016), 12 (24), 5285-5296CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      We study the orientation of magnetic cylindrical particles adsorbed at a liq. interface in an external field using anal. theory and high resoln. finite element simulations. Cylindrical particles are interesting since they possess multiple locally stable orientations at the liq. interface so that the orientational transitions induced by an external field will not disappear when the external field is removed, i.e., the switching effect is non-volatile. We show that, in the absence of an external field, as we reduce the aspect ratio α of the cylinders below a crit. value (αc ≈ 2) the particles undergo spontaneous symmetry breaking from a stable side-on state to one of two equiv. stable tilted states, similar to the spontaneous magnetization of a ferromagnet going through the Curie point. By tuning both the aspect ratio and contact angle of the cylinders, we show that it is possible to engineer particles that have one, two, three or four locally stable orientations. We also find that the magnetic responses of cylinders with one or two stable states are similar to that of paramagnets and ferromagnets resp., while the magnetic response of systems with three or four stable states are even more complex and have no analogs in simple magnetic systems. Magnetic cylinders at liq. interfaces therefore provide a facile method for creating switchable functional monolayers where we can use an external field to induce multiple non-volatile changes in particle orientation and self-assembled structure.
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      Newton, B. J.; Mohammed, R.; Davies, G. B.; Botto, L.; Buzza, D. M. A. Capillary Interaction and Self-Assembly of Tilted Magnetic Ellipsoidal Particles at Liquid Interfaces. ACS Omega 2018, 3, 14962– 14972,  DOI: 10.1021/acsomega.8b01818
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      Capillary Interaction and Self-Assembly of Tilted Magnetic Ellipsoidal Particles at Liquid Interfaces
      Newton, Bethany J.; Mohammed, Rizwaan; Davies, Gary B.; Botto, Lorenzo; Buzza, D. Martin A.
      ACS Omega (2018), 3 (11), 14962-14972CODEN: ACSODF; ISSN:2470-1343. (American Chemical Society)
      Magnetic ellipsoidal particles adsorbed at a liq. interface provide exciting opportunities to create switchable functional materials, where self-assembly can be switched on/off using an external field (G.B. Davies, et al., 2014). To gain a deeper understanding of this novel system in the presence of an external field, capillary interaction and self-assembly of tilted ellipsoids using anal. theory and finite element simulations were assessed. We derive an anal. expression for the dipolar capillary interaction between tilted ellipsoids in elliptical polar coordinates, which exhibited a 1/r2 power law dependence in the far field (i.e., large particle sepns., r) was derived; it correctly captured orientational dependence of capillary interactions in the near-field. Using this dipole potential and finite element simulations, the particle cluster energy landscape, consisting of up to eight tilted, in-contact ellipsoids, were analyzed. For two particle clusters, the side-to-side configuration was stable; the tip-to-tip configuration was unstable. For more than three particle clusters, the circular loops of side-to-side particles became globally stable; linear chains of side-to-side particles became meta-stable. The energy barrier for the linear to loop transition decreased with increasing particle no. Results explained thermodynamically and kinetically why tilted ellipsoids assemble side-to-side locally, but have a strong tendency to form loops on larger length scales.
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      Newton, B. J.; Brakke, K. A.; Buzza, D. M. A. Influence of magnetic field on the orientation of anisotropic magnetic particles at liquid interfaces. Phys. Chem. Chem. Phys. 2014, 16, 26051– 26058,  DOI: 10.1039/c4cp04270k
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      Influence of magnetic field on the orientation of anisotropic magnetic particles at liquid interfaces
      Newton, Bethany J.; Brakke, Kenneth A.; Buzza, D. Martin A.
      Physical Chemistry Chemical Physics (2014), 16 (47), 26051-26058CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      The authors study theor. the influence of an external magnetic field on the orientation of an ellipsoidal magnetic particle adsorbed at a liq. interface. Using the finite element program Surface Evolver, the authors calc. the equil. meniscus shape around the ellipsoidal particle and its equil. tilt angle with respect to the undeformed interface θt when a magnetic field B is applied perpendicular to the interface. As the authors increase field strength, θt increases and at a crit. magnetic field Bc1 and tilt angle θc1, the particle undergoes a discontinuous transition to the perpendicular orientation (θt = 90°). The results agree qual. with the simplified theory of Bresme and Faraudo [F. Bresme and J. Faraudo, J. Phys.: Condens. Matter, 2007, 19, 375110] which assumes that the liq. interface is flat, while they agree quant. with recent lattice-Boltzmann simulations of Davies et al. [G. Davies et al., Soft Matter, 2014, 10, 6742] which account for the deformation of the liq. meniscus. Also upon reducing the external magnetic field, at a crit. magnetic field Bc2 < Bc1, the particle undergoes a 2nd discontinuous transition from the perpendicular orientation to a finite tilt angle θc2 < θc1. For micron-sized particles where the thermal energy kBT is negligible compared to the interfacial energy, the tilt angle vs. magnetic field curve exhibits hysteresis behavior. Due to the higher degree of accuracy of the Surface Evolver method, the authors are able to analyze the behavior of the particles near these orientational transitions accurately and study how the crit. quantities Bc1, Bc2, θc1 and θc2 vary with particle aspect ratio and contact angle.
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      Coertjens, S.; Moldenaers, P.; Vermant, J.; Isa, L. Contact Angles of Microellipsoids at Fluid Interfaces. Langmuir 2014, 30, 4289– 4300,  DOI: 10.1021/la500888u
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      Contact Angles of Microellipsoids at Fluid Interfaces
      Coertjens, Stijn; Moldenaers, Paula; Vermant, Jan; Isa, Lucio
      Langmuir (2014), 30 (15), 4289-4300CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
      The wetting of anisotropic colloidal particles is of great importance in several applications, including Pickering emulsions, filled foams, and membrane transduction by particles. However, the combined effect of shape and surface chem. on the three-phase contact angle of anisotropic micrometer and submicrometer colloids has been poorly investigated to date, due to the lack of a suitable exptl. technique to resolve individual particles. In the present work, we investigate the variation of the contact angle of prolate ellipsoidal colloids at a liq.-liq. interface as a function of surface chem. and aspect ratio using freeze-fracture shadow-casting cryo-SEM. The method, initially demonstrated for spherical colloids, is extended here to the more general case of ellipsoids. The prolate ellipsoidal particles are prepd. from polystyrene and poly(Me methacrylate) spheres using a film stretching technique, in which cleaning steps are needed to remove all film material from the particle surface. The effects of the prepn. protocol are reported, and wrinkling of the three-phase contact line is obsd. when the particle surface is insufficiently cleaned. For identically prepd. ellipsoids, the cosine of the measured contact angle is, in a first approxn., a linearly decreasing function of the contact line length and thus a decreasing function of the aspect ratio. Such a trend violates Young-Laplace's equation and can be rationalized by adding a correction term to the ideal Young-Laplace contact angle that expresses the relative importance of line effects relative to surface effects. From this term the contribution of an effective line tension can be extd. This contribution includes the effects that both surface chem. and topog. heterogeneities have on the contact line and which become increasingly more important for ellipsoids with higher aspect ratios, where the contact line length to contact area ratio increases.
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      Loudet, J. C.; Pouligny, B. How do mosquito eggs self-assemble on the water surface?. Europhys. Lett. 2009, 25, 2718
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      Morgan, S. O.; Fox, J.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Stasiuk, G. J.; Horozov, T. S.; Buzza, D. M. A. Adsorption trajectories of nonspherical particles at liquid interfaces. Phys. Rev. E 2021, 103, 042604,  DOI: 10.1103/physreve.103.042604
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      Adsorption trajectories of nonspherical particles at liquid interfaces
      Morgan, S. O.; Fox, J.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Stasiuk, G. J.; Horozov, T. S.; Buzza, D. M. A.
      Physical Review E (2021), 103 (4), 042604CODEN: PREHBM; ISSN:2470-0053. (American Physical Society)
      The adsorption of colloidal particles at liq. interfaces is of great importance scientifically and industrially, but the dynamics of the adsorption process is still poorly understood. In this paper we use a Langevin model to study the adsorption dynamics of ellipsoidal colloids at a liq. interface. Interfacial deformations are included by coupling our Langevin dynamics to a finite element model while transient contact line pinning due to nanoscale defects on the particle surface is encoded into our model by renormalizing particle friction coeffs. and using dynamic contact angles relevant to the adsorption timescale. Our simple model reproduces the monotonic variation of particle orientation with time that is obsd. exptl. and is also able to quant. model the adsorption dynamics for some exptl. ellipsoidal systems but not others. However, even for the latter case, our model accurately captures the adsorption trajectory (i.e., particle orientation vs. height) of the particles. Our study clarifies the subtle interplay between capillary, viscous, and contact line forces in detg. the wetting dynamics of micron-scale objects, allowing us to design more efficient assembly processes for complex particles at liq. interfaces.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhtVyisLzJ&md5=57c0312d2605d2c36e2c9dee6dfe2914
    39. 39
      Morgan, S. O.; Muravitskaya, A.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Horozov, T. S.; Stasiuk, G. J.; Buzza, D. M. A. Using adsorption kinetics to assemble vertically aligned nanorods at liquid interfaces for metamaterial applications. Phys. Chem. Chem. Phys. 2022, 24, 11000– 11013,  DOI: 10.1039/d1cp05484h
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      Using adsorption kinetics to assemble vertically aligned nanorods at liquid interfaces for metamaterial applications
      Morgan, S. O.; Muravitskaya, A.; Lowe, C.; Adawi, A. M.; Bouillard, J.-S. G.; Horozov, T. S.; Stasiuk, G. J.; Buzza, D. M. A.
      Physical Chemistry Chemical Physics (2022), 24 (18), 11000-11013CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      Vertically aligned monolayers of metallic nanorods have a wide range of applications as metamaterials or in surface enhanced Raman spectroscopy. However the fabrication of such structures using current top-down methods or through assembly on solid substrates is either difficult to scale up or have limited possibilities for further modification after assembly. The aim of this paper is to use the adsorption kinetics of cylindrical nanorods at a liq. interface as a novel route for assembling vertically aligned nanorod arrays that overcomes these problems. Specifically, we model the adsorption kinetics of the particle using Langevin dynamics coupled to a finite element model, accurately capturing the deformation of the liq. meniscus and particle friction coeffs. during adsorption. We find that the final orientation of the cylindrical nanorod is detd. by their initial attack angle when they contact the liq. interface, and that the range of attack angles leading to the end-on state is maximised when nanorods approach the liq. interface from the bulk phase that is more energetically favorable. In the absence of an external field, only a fraction of adsorbing nanorods end up in the end-on state (.ltorsim.40% even for nanorods approaching from the energetically favorable phase). However, by pre-aligning the metallic nanorods with exptl. achievable elec. fields, this fraction can be effectively increased to 100%. Using nanophotonic calcns., we also demonstrate that the resultant vertically aligned structures can be used as epsilon-near-zero and hyperbolic metamaterials. Our kinetic assembly method is applicable to nanorods with a range of diams., aspect ratios and materials and therefore represents a versatile, low-cost and powerful platform for fabricating vertically aligned nanorods for metamaterial applications.
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      Chen, Q.; Yan, J.; Zhang, J.; Bae, S. C.; Granick, S. Janus and Multiblock Colloidal Particles. Langmuir 2012, 28, 13555– 13561,  DOI: 10.1021/la302226w
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      Janus and multiblock colloidal particles
      Chen, Qian; Yan, Jing; Zhang, Jie; Bae, Sung Chul; Granick, Steve
      Langmuir (2012), 28 (38), 13555-13561CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
      A review. Recent developments in the synthesis and self-assembly of Janus and multiblock colloidal particles, highlighting new opportunities for colloid science and technol. that are enabled by encoding orientational order between particles as they self-assemble, are considered. Emphasizing the concepts of mol. colloids and colloid valence unique to such colloids, the authors describe their rational self-assembly into colloidal clusters, taking monodisperse tetrahedra as an example. The authors also introduce a simple method to lock clusters into permanent shapes. Extending this to 2-dimensional lattices, the authors also review recent progress in assembling new open colloidal networks including the Kagome lattice. In each application, areas of opportunity are emphasized.
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