Controllable Capillary Assembly of Magnetic Ellipsoidal Janus Particles into Tunable Rings, Chains and Hexagonal Lattices
CControllable Capillary Assembly of Magnetic Ellipsoidal JanusParticles into Tunable Rings, Chains and Hexagonal Lattices
Qingguang Xie and Jens Harting
2, 3, ∗ Department of Applied Physics, Eindhoven University of Technology,P.O. Box 513, 5600MB Eindhoven, The Netherlands Helmholtz Institute Erlangen-N¨urnberg for Renewable Energy (IEK-11),Forschungszentrum J¨ulich, F¨urther Str. 248, 90429 N¨urnberg, Germany Department of Chemical and Biological Engineering and Department of Physics,Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, F¨urther Str. 248, 90429 N¨urnberg, Germany
Colloidal assembly at fluid interfaces has a great potential for the bottom-up fabrication of novelstructured materials. However, challenges remain in realizing controllable and tunable assembly ofparticles into diverse structures. Herein, we report the capillary assembly of magnetic ellipsoidalJanus particles at a fluid-fluid interface. Depending on their tilt angle, i.e. the angle the particlemain axis forms with the fluid interface, these particles deform the interface and generate capillarydipoles or hexapoles. Driven by capillary interactions, multiple particles thus assemble into chain-,hexagonal lattice- and ring-like structures, which can be actively controlled by applying an externalmagnetic field. We predict a field-strength phase diagram in which various structures are presentas stable states. Owing to the diversity, controllability, and tunability of assembled structures,magnetic ellipsoidal Janus particles at fluid interfaces could therefore serve as versatile buildingblocks for novel materials.
Self-assembly of particles has received considerable at-tention with respect to their potential applications inadvanced technologies, such as displays, sensors andoptoelectronic devices. For certain industrial applica-tions including large-area coatings, the assembly of par-ticles can prove far more efficient than “top-down” ap-proaches such as lithography. However, controlling par-ticles into highly tunable and predictable structures re-mains a challenge. [1–4]
Herein, we report a highly control-lable and versatile strategy for the fabrication of orderedstructures via capillary assembly of magnetic ellipsoidalJanus particles at a fluid-fluid interface.Colloidal particles strongly attach at a fluid-fluidinterface [5] and deform the interface due to their weight,anisotropic shape and roughness. [6–9]
If neighbouringparticles generate deformations of the interface that over-lap, capillary interactions arise which drive the particlesto assemble into ordered structures. Such capillary in-teractions can be characterized by the modes of interfacedeformation. In the limit of small interface deformation,the interface height h around the particle can be de-scribed according to the Young-Laplace equation ∇ h = 0and it can be analysed by a multipole expansion in anal-ogy with electrostatics [8,10] . In the case of heavy particlesattached to the interface, the interface is pushed downby the particles resulting in a monopolar deformation [6] .For particularly small or light particles, the contact linewhere particle and fluid-fluid interface meet can undulatedue to anisotropic shapes or the roughness of the parti-cle surface and, thus, induce quadrupolar or hexapolarinterface deformations [7–9] . Driven by the capillary in-teractions, particles assemble into specific structures toreduce the total adsorption free energy. For example, ∗ [email protected] heavy particles tend to form a cluster [11] , while ellipsoidalparticles assemble into chains [12,13] . Furthermore, cubicparticles generate hexagonal or honeycomb lattices [14] .However, such structures are not dynamically tunablebecause the capillary interactions are dependent on theintrinsic properties of the particles, such as their weightand their precise shape.The synthesis of colloidal particles with specific phys-ical properties (e.g., electric or magnetic moments) andanisotropic chemical properties (e.g., amphiphilic Janusparticles) interacting with external fields allows for agreater control of the assembly process. For example,magnetic ellipsoids [15,16] and magnetic spherical Janusparticles [17] at fluid interfaces can generate dipolar cap-illary interactions. Those interactions can be preciselycontrolled by an external magnetic field. Yet, to the bestof our knowledge, so far the assembly of particles waslimited to create specific and regular structures, such aschains [16,18] or hexagonal lattices [19] .Here, we examine a combination of anisotropic physicalproperties and chemical properties of particles to achievethe formation of diverse structures with controllabilityand tunablity. We investigate the behavior of magneticellipsoidal Janus particles at a flat fluid-fluid interface,interacting with external magnetic fields. We find thatby varying its tilt angle due to the presence of an externalmagnetic field, a single particle generates a dipolar or ahexapolar interface deformation. Driven by the resultingdipolar or hexapolar capillary interactions, such particlesassemble into rings, chains, and hexagonal lattice struc-tures, where a dynamical transition between these assem-bly states can be obtained by dynamically manipulatingthe field.We consider an ellipsoidal Janus particle adsorbed at afluid-fluid interface, as illustrated in Figure 1a and Fig-ure 1b. The particle is composed of apolar and polar a r X i v : . [ c ond - m a t . s o f t ] S e p (a) Upright orientation (b) Tilted orientation -0.4 0 0.4 0.8 1.2 1.6 0 30 60 90 120 150 180 T o r que τ / ( A p γ ) Tilt-angle ϕ [ ° ] α = 3, β = 39 °α = 3, β = 30 °α = 3, β = 21 ° (c) Torque E ne r g y ∆ E / ( A p γ ) Tilt-angle ϕ [ ° ] β = 39 °β = 30 °β = 21 ° (d) Energy Figure 1: A single ellipsoidal Janus particle adsorbed at a fluid-fluid interface in an upright orientation a) and in atilted orientation b). c) Reduced torque τ /A p γ and d) free energy ∆ E/A p γ of a Janus ellipsoid with aspect ratio α = 3 as a function of tilt angle ϕ for β = 21 ◦ (black) , β = 30 ◦ (red) and β = 39 ◦ (green), where A p = πac and γ is the fluid-fluid interface tension.hemispheres of opposite wettability, represented by thethree-phase contact angles θ A = 90 ◦ + β and θ P = 90 ◦ − β ,respectively, where β indicates the amphiphilicity of theparticle. The boundary between these two hemispheresis called the Janus boundary. We denote the radii oflong- and short-axes of the Janus ellipsoid with c and a , respectively, and the aspect ratio α is defined as α = c/a . The magnetic moment m is perpendicularto the Janus boundary, and external magnetic fields H x and H z are applied in horizontal and vertical direction,respectively. We define magnetic dipole-field strengths B x = | H x || m | and B z = | H z || m | , which representthe magnitude of the interactions between the magneticdipole and the external fields. We apply lattice Boltz-mann simulations [17,20,21] to investigate the behaviour ofmagnetic ellipsoidal Janus particles adsorbed at a liquid-liquid interface. A detailed description of the method andsimulation parameters is provided in the SupplementaryMaterial. In our simulations, magnetic dipole-dipole in-teraction forces are six orders of magnitude smaller thanthe capillary interaction forces. Therefore, the dipole-dipole interaction is effectively negligible, and the assem-bling of structures is purely dominated by the capillaryinteractions between particles.In the absence of an external magnetic field, an iso-lated ellipsoidal Janus particle adsorbed at an interfacetakes its equilibrium orientation to minimize the totaladsorption free energy [22,23] . The total adsorption freeenergy is written as E = γ A + γ a A a + γ p A p + γ a A a + γ p A p , where γ ij are the interface tensions be-tween phases i and j and A ij are the contact surfaceareas between phases i and j , where i, j = {
1: fluid, 2:fluid, a : apolar, p : polar } . There is no exact analyticalexpression for the free energy of a tilted Janus ellipsoidat an interface, due to the difficulty in modelling theshape of the deformed interface and the segment area ofthe ellipsoid. Our lattice Boltzmann simulations are ca-pable of capturing interface deformations fully withoutimposing any assumptions about the magnitude of thedeformations or stipulating any particle-fluid boundary conditions [17] . We take the upright orientation θ = 0as a reference configuration, and numerically calculatethe free energy. To do this, we initialize the particle onthe interface with the desired tilt angle and then fix theposition and orientation of it. After equlibration of thefluids, we measure the torque subjected on the particlesfrom the fluid-fluid interface in the absence of magneticfields. We then obtain the free energy ∆ E = E ϕ − E ϕ =0 by integrating the torque on the particle as the particlerotates quasi-statically, ∆ E = (cid:82) ϕ tilt τ ϕ dϕ .Figure 1c shows the evolution of this torque τ ϕ versusthe tilt angle of a Janus ellipsoid (aspect ratio α = 3) fordifferent amphiphilicities β = 21 ◦ (black), β = 30 ◦ (red)and β = 39 ◦ (green). For all amphiphilicities, the torqueis zero at θ = 0 ◦ and increases linearly for small tilt angles( ϕ < ◦ ), in the direction resisting the rotation of theparticle. As the tilt angle increases further ϕ → ◦ , thetorque decreases followed by a sharp increase until thetilt angle ϕ → ◦ . Finally, the torque decreases to zerowhen the tilt angle approaches ϕ = 180 ◦ .Figure 1d shows the free energy ∆ E of the ellipsoidalJanus particle, as a function of the tilt angle for the sameamphiphilicities as in Figure 1c. For a large amphiphilic-ity β = 39 ◦ the free energy keeps increasing for the wholerange of tilt angles, indicating that the particle in the up-right orientation ϕ = 0 ◦ corresponds to the global energyminimum. The particle tends to reduce the free energyby increasing the interfacial area between the particleand its preferred fluid phase. For smaller amphiphilicities β = 21 ◦ and β = 30 ◦ , the free energy is not monotonic:for small tilt angles θ < ◦ , the free energy increases, fol-lowed by a decrease until the tilt angle increases furtherto θ ∼ ◦ , and afterwards, the free energy continuouslyincreases until the tilt angle reaches θ = 180 ◦ . For anamphiphility β = 21 ◦ , Figure 1d indicates the presenceof a local energy minimum for particles in the uprightorientation ( ϕ = 0 ◦ ) and a global energy minimum forparticles in the tilted state ( ϕ ∼ ◦ ). The Janus ellip-soid tends to occupy more of the fluid-fluid surface areaand in turn reduces the free energy. An energy barrier (a) ϕ = 80 ◦ (b) ϕ = 90 ◦ (c) ϕ = 120 ◦ Figure 2: Snapshots of an ellipsoidal Janus particle at a fluid-fluid interface at different tilt angles as obtained fromour simulations. The interface deformation appears to be hexapolar at ϕ = 80 ◦ (a), dipolar at ϕ = 90 ◦ (b), and alsodipolar shape at ϕ = 120 ◦ , but with a larger deformation (c). The insets depict the hexapolar/dipolar interfacedeformation.exists between the metastable upright orientated config-uration and the global energy minimum, which requiresa magnetic torque τ m stronger than the capillary torque τ /A p γ ∼ .
35 (as shown in Figure 1c), to rotate theparticle out of the tilted equilibrium orientation to theupright orientation.Next, we investigate the interface deformation inducedby the ellipsoidal Janus particle at different tilt angles.We find that the interface stays undeformed around theparticle with upright orientation, indicating the absenceof a torque at ϕ = 0 ◦ , which is consistent with our resultsin Figure 1c. Figure 2 shows how the three-phase contactline and interface deform around the ellipsoidal Janusparticle ( α = 3 , β = 21 ◦ ) for different tilt angles ϕ = 80 ◦ , ϕ = 90 ◦ and ϕ = 120 ◦ , respectively. At ϕ = 80 ◦ , the in-terface deforms around the particle in a hexapolar shape(Figure 2a), with three rises and three dips distributedaround the particle. The interface is raised up at the tipof the apolar hemisphere and depressed at the tip of thepolar hemisphere. When the particle aligns in the hori-zontal orientation ϕ = 90 ◦ , the interface shows a dipolardeformation (Figure 2b), with a dip around the apolarhemisphere and a rise around the polar hemisphere. Wenote that the magnitude of this dipolar deformation ismuch stronger than that of a hexapolar deformation. Thedeformation increases further with increasing tilt angleto ϕ = 120 ◦ and the deformed interface height is evenat the same order of particle short radius (Figure 2c).We also observe an unsymmetrical hexoplar, butterfly-like deformation at intermediate tilt angles (Supplmen-tary Material Figure S1). The rise and dip generatedaround the Janus ellipsoid result from the competitionbetween Janus and ellipsoidal properties of the particle:it is known that both, a Janus sphere and a homogeneousellipsoid, generate dipolar interface deformations. How-ever, the rise and dip areas are located oppositely [15,17] .The positioning and strength of the rises and dips can betuned by varying the aspect ratio and amphiphilicity ofthe particles, as observed in our simulations (Figure S2).If two or more particles are adsorbed at a fluid-fluid in-terface, the deformations induced by neighbouring par- ticles can overlap, leading to capillary interactions be-tween these particles. A pair of Janus ellipsoids inter-acting through hexapolar or dipolar capillary interac-tions prefers to align in a side-side configuration (Fig-ure S3), corresponding to a capillary energy minimumconfiguration [17,23] . However, many-body effects in thecapillary assembling of multiple Janus ellipsoids underexternal magnetic fields remains to be explored. In Fig-ure 3 we show the assembled structures for particle sur-face fractions Φ = 0 . , .
62 that form as we vary thedipole-field strengths B x and B z . We define the par-ticle surface fraction as Φ = N πac/A , where N is thetotal number of particles and A is the interface areabefore particles are placed at the interface. The parti-cles have an aspect ratio α = 3 and an amphiphilicity β = 21 ◦ . We define the normalized dipole-field strengthas ¯ B i = B i /A p γ , where i = x, z . Initially, the particlesare distributed randomly on the interface. In the absenceof external magnetic fields, they take their tilted equilib-rium orientation ϕ ∼ ◦ and introduce hexapolar inter-face deformations (as shown in Figure 2a). Then, the par-ticles assemble into locally-ordered structures (Figure 3aand Figure 3f). Small particle clusters with side-side, tip-tip, and side-tip alignments coexist, which indicates thatmultiple particles interacting through hexapolar capillaryinteractions have different (meta)stable configurations.When applying a downward magnetic field ¯ B z = − . ϕ ∼ ◦ and generate dipolar inter-face deformationis (as shown in Figure 2c). For a lowerparticle surface fraction Φ = 0 .
16, the particles form acircular ring (Figure 3b), instead of a chain structure pre-dicted by the pair-wise interactions, indicating the pres-ence of strong many-body interactions. Our results areconsistent with the theoretical prediction that a closedloop structure is the capillary energy minimum configu-ration for multiple tilted ellipsoidal particles interactingwith dipolar capillary interactions [24] . With increasingthe surface fraction Φ = 0 .
62, the particles form multiplerings and the rings are more curved due to geometricalrestriction. Along with ¯ B z = − .
3, additionally we applya horizontal magnetic field ¯ B x = 1 .
3. Then, the particles (a) ¯ B x = ¯ B z = 0 (b) ¯ B x = 0 , ¯ B z = − . B x = − ¯ B z = 1 . B x = 1 . , ¯ B z = 0 .
18 (e) ¯ B x = 0 , ¯ B z = 0 . B x = 0 , ¯ B z = 0 (g) ¯ B x = 0 , ¯ B z = − . B x = − ¯ B z = 1 . B x = 1 . , ¯ B z = 0 .
18 (j) ¯ B x = 0 , ¯ B z = 0 . Figure 3: Assembly of particles with different surface fractionr, a-e) Φ = 0 .
16 and f-j) Φ = 0 .
62 under differentmagnetic fields. The particle has an aspect ratio α = 3 and an amphiphilicity β = 21 ◦ .align into chain-like structures for both lower and highersurface fractions (Figure 3c and Figure 3h), consistentwith the prediction from pair-wise interactions, demon-strating that many-body effects are less relevant in thiscase. Here, the particles are forced to align in the di-rection parallel to the horizontal magnetic field and theyonly have 2 degrees of freedom (translation in x and y directions), which weakens the many-body effect. Onthe contrary, when only a vertical magnetic field ¯ B z isapplied, the particles have 3 degrees of freedom (transla-tion in x , y direction and rotation around the z axis) andcapillary torques can rotate the particles to form rings.If an upward magnetic field ¯ B z = 0 .
18 together with ahorizontal field ¯ B x = 1 . ϕ ∼ ◦ and generate hexapolar deformations.For lower particle surface fraction Φ = 0 .
16, the parti-cles align in a zigzag structure (Figure 3d). At highersurface fraction Φ = 0 .
64, the particles align in orderedhexagonal lattices (Figure 3i). With only the upwardmagnetic field ¯ B z = 0 . ϕ = 0 ◦ and align in disordered ar-rangements due to the absence of capillary interactions(Figure 3e and Figure 3j). The assembled structures canbe tuned by varying the directions of the magnetic fields(Supplmentary Material Movie S1), which has potentialapplications in sensor or display technology. To estimatethe time scale of the assembly and the structural transi-tion, we assume that colloidal particles of radius a = 4 µm are adsorbed at a decane-water interface, with a sur-face tension γ = 53 . mN/m , and the effective viscos-ity µ = 0 . mP a · s [25,26] . Based on our simulation re-sults, the estimated time scale of structural formationand transition is about t ∼ ms , which is sufficiently fast to satisfy the requirements of responsive materialsfor advanced sensor or display technologies. For a possi-ble experimental realization of our system, we note thatmagnetic spherical Janus particles have been experimen-tally fabricated [27–29] and investigated at a liquid-liquidinterface [25] . Such spherical particles may be stretchedmechanically to form ellipsoidal particles with various as-pect ratios [30,31] .In Figure 4 we construct the phase diagram for assem-bled structures of particles as a function of horizontal ¯ B x and vertical ¯ B z magnetic field-strengths, showing chains(diamonds), locally-ordered clusters (squares), rings (cir-cles), disordered alignments (triangles), and hexagonallattice structures (pentagons). Locally-ordered struc-tures are formed when external magnetic fields are turnedoff or only a very weak vertical magnetic field − . < ¯ B z < . B z > . B x . In this case, the par-ticles take small tilt angle ϕ < ◦ , where the deforma-tion of the interface is absent or negligible. The particlesform hexagonal lattices once a strong horizontal field isapplied along with an upward magnetic field satisfying¯ B x / ¯ B y > .
6. Chains are formed when a strong hori-zontal magnetic field and a downward magnetic field isapplied, in the range ¯ B x / | ¯ B z | > .
4. The particles as-semble into rings when the downward magnetic field ismuch stronger than the horizontal field | ¯ B z | / ¯ B x > . B x B z Figure 4: Field-strength phase diagram of assembledphases of Janus ellipsoids, showing chains (diamonds),locally-ordered clusters (squares), rings (circles),disordered alignments (triangles), and hexagonallattices (pentagons). The particles have an aspect ratio α = 3 and an amphiphilicity β = 21 ◦ . The interfacefrction covered by particle is Φ = 0 . Supporting Information
Supporting Information is available.
Acknowledgements
We thank Oscar Coppelmans for fruitful discussions. Fi-nancial support is acknowledged from the NetherlandsOrganization for Scientific Research (NWO) through aNWO Industrial Partnership Programme (IPP). This re-search programme is co-financed by Canon ProductionPrinting Netherlands B.V., University of Twente andEindhoven University of Technology. We are also gratefulfor financial support from the German Research Founda-tion (DFG) through priority program SPP2171 (grantHA 4382/11-1). We thank the J¨ulich SupercomputingCentre and the High Performance Computing CenterStuttgart for the technical support and allocated CPUtime.
Conflict of Interest
The authors declare no conflict of interest.
Keywords
Janus particles, ellipsoid, controllable assembly, particle-laden fluid interfaces [1] P. S. Weiss,
ACS Nano , ACS Nano , MRS Bulletin , Eur. Phys. J. E , Langmuir , J. Colloid. Interf. Sci. , Colloids Surf. , Phys. Rev. E , J. Colloid Interface Sci. , Eur. Phys. J. E , Science , Phys.Rev. Lett. , Phys. Rev. Lett. , Phys. Rev. Lett. , Soft Matter , Adv. Mater. , Soft Matter , Soft Matter , ACS Nano , Phys. Rev. E , Soft Matter , ACS Nano , Langmuir , ACS Omega , SoftMatter .[26] A. A. Adewunmi, M. S. Kamal,
Energy & Fuels , Advanced Materials , Lang-muir , Soft Matter , Proc. Natl.Acad. Sci. U.S.A. , Langmuir ,25