Assembling Ellipsoidal Particles at Fluid Interfaces using Switchable Dipolar Capillary Interactions Gary B. Davies,1, a) Timm Kr¨uger,2, 1, b) Peter Coveney,1, c) Jens Harting,3, 4, d) and Fernando Bresme5, 6, e) 1)Centre for Computational Science, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom. 2)Institute for Materials and Processes, School of Engineering, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JL, Scotland, United Kingdom. 3)Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. 4)Faculty of Science and Technology, Mesa+ Institute, University of Twente, 7500 AE Enschede, The Netherlands. 5)Department of Chemistry, Imperial College London, London, SW7 2AZ, United Kingdom. 6)Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway.

The fabrication of novel soft materials is an important cles adsorbed at fluid-fluid interfaces contort the interface scientific and technological challenge. We investigate the around them in a quadrupolar fashion purely because of response of ellipsoidal particles adsorbed at fluid-fluid in- their shape:8,17 the interface is depressed more at the terfaces to external magnetic fields. By exploiting pre- tips than it is elevated at the sides, leading to orientation- viously discovered first-order orientation phase transi- dependent interaction potentials between particles.8,15–17 tions,1–3 we show how to switch on and off dipolar capil- The resulting capillary interaction energies can be sev- lary interactions between particles, leading to the forma- eral orders of magnitude larger than the thermal energy, 5 tion of distinctive self-assembled structures and allowing E ∼ 10 kBT , providing a strong driving force for self- dynamical control of the bottom-up fabrication of novel- assembly.8,17 However, neither the monopolar nor the structured materials. quadrupolar interactions are dynamically tunable. Particles adsorb strongly at fluid-fluid interfaces: de- Dipolar capillary charges are created by anisotropic tachment energies of spherical particles can be orders of particles influenced by torques. External torques can be 4,5 magnitude greater than the thermal energy, kBT . Once caused by particles with uneven mass distributions in- particles are adsorbed at an interface, competing hydro- teracting with gravity, complex surface chemistries (e.g. dynamic, electromagnetic and capillary interactions can Janus particles) interacting with fluids,18–20 or — the fo- lead to particles self-assembling into materials with spe- cus of this Communication — embedded dipoles interact- cific mechanical, optical, or magnetic properties.6,7 ing with external magnetic fields, opening up new routes Capillary interactions8 have attracted interest for their for the manipulation of particle monolayers. role in assembling mosquito eggs adsorbed at air-water Materials science advances6,21 have enabled the pro- interfaces,9 suppressing the coffee ring effect,10 and ag- duction of anisotropic particles with embedded ferromag- gregating Cheerios.11,12 Capillary interactions occur be- netic dipoles22 or (super)-paramagnetic dipoles23,24 so cause of the creation of menisci around particles, and that particles can interact with external magnetic fields. these particle-induced interface deformations, called cap- Bresme et al.2 investigated the behaviour of magnetic illary charges, interact with one another analogously to ellipsoidal particles adsorbed at fluid-fluid interfaces un- 13,14 electric charges. der the action of a magnetic field, predicting that parti- The height of the interface deformation, h, due to the cles undergo a discontinuous, first-order phase transition 2 presence of a particle obeys Young’s equation, ∇ h = 0, from a tilted state to a vertical state if a critical dipole- 8,15,16 which can be solved using a multipole expansion. field strength is reached.3 Davies et al.1 provided further Heavy particles deform the interface symmetrically in a monopolar fashion leading to log r interaction poten- tials between particles, where r is the inter-particle dis- µ arXiv:1408.3140v1 [cond-mat.soft] 13 Aug 2014 tance.8,11,12 However, for micron-sized particles, gravi- tational forces are usually small compared to surface- H tension forces.8 Non neutrally-wetting micron-sized ellipsoidal parti-

FIG. 1: Anti-symmetric meniscus formation (solid red a)Electronic mail: [email protected] b) lines) for a single magnetic ellipsoidal particle with Electronic mail: [email protected] dipole moment, µ, under the influence of an applied c)Electronic mail: [email protected] d)Electronic mail: [email protected] magnetic field, H, directed normal to the undeformed e)Electronic mail: [email protected] interface (blue dashed). 2

H Before First-Order Transition After First-Order Transition

1 (a) B = 0, t = 0 (b) B = 2 Bc, t → ∞ (c) B > Bc, t = 0 (d) B > Bc, t → ∞ FIG. 2: (a) The particles are distributed randomly in their equilibrium orientations with surface fraction φ = 0.38. (b) Applying a magnetic field parallel to the interface normal, H, causes them to self-assemble due to dipolar capillary interactions. (c) Once the critical field strength is reached, particles transition to the vertical state, halting dipolar capillary interactions. (d) Illustration: once capillary interactions have been turned off, the particles may order randomly if magnetic dipole-dipole and van der Waals interactions are weak compared to thermal fluctuations. evidence that the transition exists and also showed that dependent and will give rise to torques orthogonal to the particle-induced interface deformations significantly af- interface causing in-plane rotation and ordering. Further, fect the transition. since the magnitude of the interface deformations depend When a magnetic prolate spheroidal particle with on the dipole-field strength, it is possible to tune the dipole moment, µ, is adsorbed at a fluid-fluid inter- strength of these dipolar capillary interactions by chang- face and subjected to an external magnetic field, H, di- ing the strength of the external field, particle dipole mo- rected normal to the interface, it experiences a torque, ment, or both. T = µ × H, which attempts to align the particle with We employed lattice-Boltzmann simulations1,25–36 to the external field. Surface-tension forces oppose the mag- investigate magnetic prolate spheroidal particles with netic torque, and so for a given dipole-field strength, aspect-ratio α = 2 adsorbed at a liquid-liquid interface B = |µ||H|, the particle is tilted with respect to the under the influence of a magnetic field applied paral- external field, rather than aligned with it. When the par- lel to the interface normal, implemented in our LB3D ticle is tilted with respect to the interface, the constant code37 with the same model and parameters as described contact-angle condition stipulated by Young’s equation in Davies et al.1 In our simulations, the dipole-field means that the particle deforms the interface:1 the inter- strength, B, is dominated by a strong external magnetic face is depressed on one side and elevated on the other, field and weak dipole moment so that magnetic dipole- as shown in Figure 1. dipole interactions are neglected. This means that the When a critical dipole-field strength, Bc, is reached, structures we observe are purely a result of dipolar cap- however, the magnetic torque overcomes surface-tension illary interactions between particles. forces and the particle undergoes a first-order phase tran- In Figure 2 we show the particles undergoing a first- sition and flips from a tilted orientation to a vertical ori- order transition into the vertical orientation by exceeding entation, with respect to the interface. In the vertical the critical dipole-field strength for a single particle, Bc, orientation, interface deformations are absent because of demonstrating the unique phenomenology of the dipolar the rotational symmetry of the three-phase contact-line capillary interaction mode. The particles begin randomly in this configuration. oriented on the interface (Figure 2a). An external field is The interface deformations that occur before the par- applied such that the dipole-field strength is B = 0.5Bc, ticle transitions to the vertical state are important be- creating capillary charges and causing the particles to in- cause they are analogous to electrostatic charges, with a teract with each other (Figure 2b). The particles assem- twist; depressions attract depressions, elevations attract ble into “capillary caterpillars”, which we analyse further elevations, and depressions repel elevations: opposites re- in Figure 3. Finally, the dipole-field strength is increased pel, rather than attract. Therefore, if more than one beyond the single particle critical dipole field, B > Bc, particle is adsorbed at a fluid-fluid interface and under where the particles transition into the vertical configu- the influence of an external magnetic field such that the ration (Figure 2c and 2d) and capillary interactions are dipole-field strength is less than the critical dipole-field spontaneously switched off. strength, B < Bc, we expect these capillary charges to After the particles have transitioned to the vertical ori- interact with each other. entation, they order according to a complex balance of Due to the anti-symmetric, dipolar nature of the in- other forces, such as magnetic dipole-dipole interactions, terface deformations, these interactions are orientation thermal fluctuations, and van der Waals forces, which de- 3

0.0Bc 0.2Bc 0.5Bc 0.8Bc 1.2Bc 38 . = 0 φ

(a) (b) (c) (d) (e) 53 . = 0 φ

(f) (g) (h) (i) (j) 60 . = 0 φ

(k) (l) (m) (n) (o) FIG. 3: Snapshots of self-assembled structures as the external field, B, and surface fraction of particles, φ, is varied, showing a strong dependence on both parameters. At small applied fields, B = 0.2Bc, we see some particle ordering where particles prefer to be in a side-by-side or tip-to-tip state (b), (g), (l). For B = 0.5Bc we observed the formation of curved “capillary caterpillars”, in which the particles are oriented side-by-side and separate caterpillars prefer to face each other tip-to-tip (c), (h), (m). As the field strength is increased to 0.8Bc, the caterpillars begin to prefer sharper, 90◦ corners instead of curved chains (d), (i), (n). We see a further increase in corner sharpness for field strengths of B = 1.2Bc, and we observed small numbers of flipped particles in the vertical state for φ = 0.38 (e). As the surface fraction is increased to φ = 0.53 (j) and φ = 0.60 (o), fewer flipped particles are observed. pend strongly on different combinations of external field i.e. the particles are tilted with respect to the interface, strength, dipole moment, particle size and shape, surface for several surface-fractions, φ = NAp/A, where N is the packing fraction and fluid properties, making final struc- number of particles adsorbed at the interface, Ap is the tures in the absence of capillary interactions difficult to interface cross sectional area of a single particle, and A predict. Figure 2d illustrates a situation in which ther- is the area of the interface. mal fluctuations are greater than magnetic dipole-dipole We find that the self-assembled structures depend interactions, and the particles order randomly. strongly on the dipole-field strength. For weak fields, The above mentioned parameters can be varied to ob- B = 0.2Bc, particles show some ordering, orienting in the serve the transition experimentally. For a typical system side-by-side and tip-to-tip configuration (Figure 3b, 3g, with a super-paramagnetic particle of long-axis radius and 3l). As the dipole-field strength is increased to 2 −1 Rk ≈ 1µm, saturated magnetic moment µ ≈ 80 Am kg B = 0.5Bc the particles begin to form long, curved or- and surface-tension γ ≈ 0.05 Nm−1, a magnetic field dered chains, or “capillary caterpillars”, in which parti- strength H < 1T should suffice to observe the first-order cles strongly prefer to orient side-by-side (Figure 3c, 3h, phase transition.2 and 3m). Different capillary caterpillars face each other In Figure 3 we present a systematic investigation of in a tip-tip configuration, as can be seen in the upper- the intermediate dipole-field strength regime before the right of Figure 3m. The particles in this region align in particles have transitioned to the vertical configuration “capillary couples” (Figure 4). 4

B > Bc side-side orientations. Only once individual capillary caterpillars have formed do the particles in each cater- pillar arrange tip-tip with particles in other caterpillars. Fully understanding dipolar capillary-induced chain for- mation is a priority for future study. For quadrupolar capillary interactions, tip-tip configurations occur when

B electrostatic repulsion between particles exists, and cap- 0 < B < Bc illary arrows can form when the particles are of un- equal size.9,17,38 The effect of particle size, aspect-ratio, contact-angle, charge and magnetic moment remain un-

Increasing explored for the dipolar capillary interactions we report here. By using magnetic anisotropic particles interacting B = 0 with external magnetic fields, we showed how to dynami- cally tune dipolar capillary interactions between particles by varying the dipole-field strength, and how to switch FIG. 4: Sideview illustrations. (Lower) The particles these dipolar capillary interactions on and off by mak- are randomly distributed on the interface, with B = 0 ing the particles undergo a first-order orientation transi- (Figure 2a). (Middle) Due to capillary charges — like tion. Our simulations reveal novel self-assembled struc- menisci attract, unlike menisci repel — the particles tures that depend on the surface coverage of particles and arrange into pairs called “capillary couples” because of the dipole-field strength. We observed the formation of the anti-symmetric nature of the meniscus formation “capillary caterpillars”, in which particles align in side- (Figure 2b). (Upper) Particles have transitioned to the side configurations, and “capillary couples” where parti- vertical configuration, aligning with the external field. cles in individual caterpillars align in tip-tip chains with Interface deformations are absent due to the rotational particles in other caterpillars, due to the anti-symmetric symmetry of the three-phase contact-line and capillary menisci formation. In addition to providing motivation interactions are turned off (Figure 2c and 2d). for new experiments on the bottom-up fabrication of new materials, the novel assembly behaviour reported here should also be observable in colloid-liquid crystal mix- tures,40,41 and has implications for liquid crystals in gen- For dipole-field strengths of B = 1.2B , the parti- c eral. Further, the discontinuous transition of the prolate cles should be flipped into the vertical state having ex- spheroidal particles and the on-off tunability of capillary ceeded the critical dipole-field strength required to tran- interactions could find applications in photonic systems sition a single particle into the vertical state (Figure 4 that require dynamic control of optical properties, such and 2c). However, for low surface fractions φ = 0.38 as e-readers.42 Finally, the sensitivity of the first order only a small number of particles are flipped (Figure 3e), orientation transition, and hence the assembly process, and the number of particles in the flipped state decreases to the particle size and shape, external field strength as the surface-fraction increases to φ = 0.53 (Figure 3j) and interface surface tensions has potential applications and φ = 0.60 (Figure 3o). This suggests that many-body in colloidal metrology for sensors that respond to small effects inhibit the first-order phase transition and shift changes in interface properties. the critical dipole-field required to flip the particles into the vertical state to larger values. Additionally, for dipole-field strengths of B = 0.8Bc ACKNOWLEDGMENTS (Figure 3d, 3i, and 3n) and B = 1.2Bc (Figure 3e, 3j, and 3o) the capillary caterpillars favour straight chains with sharp corners rather than curved chains as seen for GBD & PVC are grateful to EPRSC and Fujitsu Lab- B = 0.5Bc (Figure 3b, 3g, and 3l). A possible expla- oratories Europe for funding GBD’s PhD studentship. nation is as follows. For a larger dipole-field strength, JH acknowledges financial support from NWO/STW the particles’ tilt-angle with respect to the interface in- (VIDI grant 10787 of J. Harting). FB thanks EPSRC creases.1 Particles prefer to align with their dipole axes for awarding a Leadership Fellowship and TK acknowl- parallel with one another. The sharper corners are simply edges the University of Edinburgh for awarding a Chan- changes to the in-plane components of the dipole-dipole cellor’s Fellowship. We ran our simulations on HEC- angle due to the larger tilt-angles, however, a more thor- ToR and ARCHER, the UK’s HPC service, via CPU ough investigation of the formation and properties of in- time awarded by grants EPSRC “Large Scale Lattice dividual capillary caterpillars is needed to confirm this Boltzmann for Biocolloidal Systems” (EP/I034602/1), hypothesis. “2020 Science” (EP/I017909/1), “UK Consortium on Compared with the self-assembled structures observed Mesoscale Engineering Sciences” (EP/L00030X/1) and due to quadrupolar capillary interactions,8,9,17,38,39 we EU-FP7 CRESTA grant (Grant No. 287703). find that the dipolar capillary interactions also favour 5

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