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G.
Steinbach
*^{ab},
D.
Nissen
^{c},
M.
Albrecht
^{c},
E. V.
Novak
^{d},
P. A.
Sánchez
^{e},
S. S.
Kantorovich
^{de},
S.
Gemming
^{ab} and
A.
Erbe
*^{b}
^{a}Institute of Physics, Technische Universität Chemnitz, 09107 Chemnitz, Germany. E-mail: gabi.steinbach@physik.tu-chemnitz.de
^{b}Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany. E-mail: a.erbe@hzdr.de
^{c}Institute of Physics, University of Augsburg, 86159 Augsburg, Germany
^{d}Ural Federal University, Lenin av. 51, 620000, Ekaterinburg, Russia
^{e}University of Vienna, Sensengasse 8, 1090, Vienna, Austria

Received
28th November 2015
, Accepted 20th January 2016

First published on 21st January 2016

This paper presents a homogeneous system of magnetic colloidal particles that self-assembles via two structural patterns of different symmetry. Based on a qualitative comparison between a real magnetic particles system, analytical calculations and molecular dynamics simulations, it is shown that bistability can be achieved by a proper tailoring of an anisotropic magnetization distribution inside the particles. The presented bistability opens new possibilities to form two-dimensionally extended and flexible structures where the connectivity between the particles can be changed in vivo.

Following the idea of ‘colloidal Lego’,^{7,8} one simple approach for flexible modular design is to create colloids that can connect to each other in multiple ways, leading to various assembly patterns (minimum two) with different symmetry. To date, in most cases of directional self-assembly, colloidal particles can be bound together in only one specific position and orientation, which is set by the interactions at the contact points between the particles.^{9–12} For example, extensively studied ferromagnetic particles, which can be approximated by point dipoles,^{13–17} form exclusively linear structures due to the dominating head-to-tail minimum of the dipolar potential. Thus, the interparticle interaction defines only one specific type of connection based on which clusters can grow. In order to obtain more than one type of connection, non-uniform systems with at least two different types of particles have been used in previous reports.^{18–20} Here, we present a novel approach in which a single type of colloidal particle self-assembles through two symmetrically distinct types of connections, thus providing the system with a structural bistability. Flexibility/variability of this system stems from the fact that there coexist two almost equiprobable self-assembly scenarios. We show that these scenarios can be realized by magnetic particles with a permanent, anisotropic magnetization distribution, as the latter leads to essential deviations from the interaction landscape of particles that can be approximated by a single point dipole. In contrast to the approach of combining heterogeneous particles, the presence of a bistability in the system allows us to control the ratio between the available distinct connections by tuning the properties of a single type of colloid.

In particular, the system presented here exhibits a bistability already in clusters of three particles. We observed two stable states with distinct topologies: one cluster type is compact and has zero net magnetic moment, whereas the other one has the shape of a staggered chain, with a net magnetization perpendicular to the chain. The interconversion between those two stable states can be realized by applying external magnetic field pulses. Following the experimental observations, we develop a simplified model of spheres containing three point dipoles, which represent the magnetization distribution in the particles. We show, analytically and by molecular dynamics simulations, that the proposed model reproduces both stable configurations observed in the experimental system. Additionally, this model allows for the derivation of general design rules for the magnetization distribution inside the particles, which enables controlled tuning of the connection patterns. The detailed investigation of the stable connectivity between three particles provides a basis for the description of two-dimensionally extended self-assembled structures based on those connection patterns.

Fig. 1 Sketches and microscopy images of self-assembled structures from magnetic Janus particles. (a) Cross section sketch of a silica sphere with hemispherical metal coverage (yellow arc). The arrows indicate the magnetization distribution. (b) Sketch of two particles in contact, displaying the cap (dark blue) and the net magnetic moment (black arrow), which has an in-plane orientation measured by the angles φ and ψ. (c) The two-particle cluster exhibits a staggered antiparallel magnetic alignment. Three particles can assemble in two different structures, (d) a compact cluster, where the magnetic moments (red arrows) form a flux-closure ring or (e) a staggered cluster with antiparallel magnetic moments. (f) Image sequence showing the interconversion between both three-particle configurations by applying short pulses of external magnetic fields (at 2nd and 7th image) perpendicular to the image plane (see Movie, ESI†) (scale bar: 5 μm). |

In our system, the self-assembly of two particles always results in a dumbbell structure (Fig. 1b and c), characterized by the antiparallelly aligned caps φ − ψ = 180°, with angles φ and ψ defining the net magnetization direction of the first and the second particle, respectively. This remarkable magnetic feature of staggered antiparallel orientation has been presented before^{23} and is a consequence of the hemispherical metal coverage, by which the center of mass of the magnetic cap deviates from the geometric center of the particles.^{24} In order to explain this configuration, a model of so-called shifted-dipole particles (1sd-particles) has been introduced.^{23,25} In this model, every particle contains a dipole that is shifted radially away from its center. It has been shown that two interacting 1sd-particles reproduce the dumbbell shown in Fig. 1b and c. A similar model was used to describe the self-assembly of spherical colloids with embedded off-centred magnetic cubes.^{26}

Unexpectedly, during the self-assembly of three capped particles, two different stable cluster structures are observed. First, the particles can form a compact, equilateral triangle (Fig. 1d) with a three-fold rotational symmetry. Here, each pair of caps encloses an angle of φ − ψ = 120°. The caps form a flux-closure ring, which results in a vanishing net magnetic moment of the cluster. The transparent hemisphere of each particle is maximally visible, indicating that the particle-particle interaction favors an in-plane orientation. Second, three particles can form a staggered chain with mirror symmetry (Fig. 1e). Here, particles in contact exhibit an antiparallel cap orientation. Such a cluster has a net magnetic moment that points perpendicular to the particle chain. Thus, the staggered cluster aligns in an external magnetic field, provided that the field strength is relatively weak and does not distort the cluster structure. If stronger external fields are applied, the caps align with the field and eventually the cluster structure is altered. This can be used to transform one structure into another, as is described below and depicted in the image sequence in Fig. 1f. A magnetic field, which is applied perpendicular to the layer of colloids, forces the individual particles to rotate such that the caps point perpendicular to the image plane. The dominating dipole interaction between coaligned side-by-side dipoles leads to a strong repulsion and, thus, separation of the particles. Releasing the field results in reorientation and subsequent recombination of the particles. During the application of successive magnetic field pulses, any of the two cluster types (6th and 9th image of Fig. 1e) can form (see Movie, ESI†), provided that sufficient time between the pulses is allowed for cluster recombination. This suggests that both clusters are metastable and their energies are of a similar order of magnitude. After the field is released, the formation of one of the states instead of the other one depends on the relative orientations and positions of the particles before recombination and is, therefore, strongly influenced by thermal fluctuations and external factors such as magnetic fields. For example, if weak in-plane fields are applied during the recombination process of three particles, we primarily observed staggered configuration.

It is important to note that the discrepancy between the rotational symmetry, which is attributed to the capped particles, and the mirror symmetry of the proposed model has no qualitative relevance for the present study. It has been previously shown^{28} that for a monolayer of strongly interacting dipolar particles, the magnetic flux does not, on average, fluctuate out of the plane of the particle assembly. Since we restrict our study to two-dimensional particle assemblies, all three dipoles of each particle will lie in this plane. As the model breaks rotational symmetry in the direction perpendicular to this plane, we do not expect qualitative differences between the results of our analytical investigations and the experimental observations.

In comparison with the 1sd-particle model, where the dipole shift s_{0} is the only parameter, the increased number of parameters in the 3sd-particle model results in a more complex interaction energy landscape and potentially causes richer assembly behavior. In the following, we show step by step how the individual parameters impact the three-particle configurations, and extract suitable values for the parameters that match the capped particles. As a starting point, we made several assumptions listed below. First, the previously reported comparison between simulations of the 1sd-model^{24,25,27} and artificially grown compact clusters of the capped particles^{23} suggests a dipole shift of s_{0} = 0.6, where s_{0/1} is measured in units of the particle radius. We can use this value here since the side dipoles only cause a perturbation of the energy landscape. Second, analytic investigations revealed that the configurations of three-particle clusters of 3sd-particles qualitatively do not depend on the absolute values of the dipoles, but only on their ratio m_{1}/m_{0}. We can assume that m_{1}/m_{0} = 0.5 since the thickness t of the cap at intermediate angles β of the side dipoles is approximately half the thickness t_{max} at the top of the cap, which varies as t(γ) = t_{max}cosγ with the angle γ measured with respect to its symmetry axis. With these starting values, it is possible to determine suitable values for s_{1} and β in the model by comparing our experimental observations with analytical calculations and numerical simulations using the 3sd-particle model.

By energy minimization of the analytical interaction potential between 3sd-particles (Fig. 2b), we identify all stable configurations of three-particle clusters for a range of the parameters s_{1} and β. The configuration space is depicted in Fig. 3a as polar plot, where the central dipole is drawn as an orange arrow. Below a critical value of s_{1}, which is around 0.76 for the examined range of β, only a triangle with the central dipoles forming a flux-closure ring (Fig. 2c) is found to be stable. Larger values of s_{1} result in two energy minima, which correspond to the ring and the staggered chain with antiparallel orientation of the central dipoles (Fig. 2d). The critical value of s_{1} increases only slightly with decreasing β (Fig. 3a). The impact of the magnitudes of the dipoles has been examined by recalculating the configuration space with m_{1}/m_{0} = 1 (Fig. 3b). From the comparison between configuration spaces, we can derive that by increasing or decreasing m_{1}/m_{0} moderately, the critical value of s_{1} (above which coexistence occurs) slightly decreases or increases, respectively. In any case, the side dipoles must always be shifted further off-center than the central dipole (s_{1} > s_{0}) to ensure the coexistence of two stable three-particle configurations. From the configuration space it can be concluded that, in this 3sd-particle model, one can observe the coexistence of the two the stable configurations if the parameters are chosen appropriately.

The influence of s_{1} and β within the coexistence space is scrutinized in the following for the case m_{1}/m_{0} = 0.5. With increasing s_{1} the global energy minimum switches from the ring (Fig. 3a, blue regime) to the staggered chain (black regime). More specifically, the relative difference of the energy between both states can be tuned gradually by s_{1} (Fig. 3c). Which of both states is the global minimum depends on the sign of the energy difference. This implies that the stability of the ring can be reduced in favor of the staggered chain primarily by increasing s_{1}. Fig. 3c shows that the variation of β results in only a small and consistent vertical shift of the energy difference. Therefore, s_{1} acts as major tuning parameter for the energy landscape. Based on Fig. 3a, we can conclude that the modeling of the presented capped particles requires a value of s_{1} > 0.76.

In the next step, we investigate the influence of s_{1} and β on the structure of the three-particle clusters. While the ring has a unique structure, the staggered chain can assume different shapes by varying the stagger angle θ (Fig. 2d). A systematic understanding is gained by calculating the energy of the staggered chain versus the stagger angle θ for various values of the shift s_{1} and the angle β, which is presented in Fig. 4. It can be seen that for s_{1} < 0.76 no energy minimum exists for a staggered configuration. For larger values of s_{1} a minimum is found which increases in depth with increasing shift. This is consistent with the conclusions that we have drawn from Fig. 3. Further, it can be seen that for a fixed value of β (in each plot), the stagger angle, at which the energy exhibits a minimum, decreases only slightly with increasing s_{1}. In contrast, increasing β (among the four plots) results in a significant variation of the stagger angle.

The linear dependence of θ on β in the equilibrium configuration is visualized in Fig. 5 for suitable values of s_{1}. Here, varying s_{1} causes only small vertical shifts of the linear function. Thus, in a staggered configuration, θ is predominantly affected by β. The comparison of Fig. 5 and the experimentally obtained staggered chains allows for the extraction of suitable values of β. For the capped particles, we find experimental values in the range of 21π/36 < θ < 23π/36. In Fig. 5, this range is marked by a semi-transparent grey vertical bar. Taking into account that s_{1} > 0.76, we can derive suitable values of β = [6π/25…8π/25]. Based on this range, an upper limit of s_{1} can be determined from Fig. 3c. The relative energy difference should be small since both three-particle structures are of comparable energy as derived from their field-induced interconversion (Fig. 1f). Assuming a maximum relative energy difference of 10% and β = [6π/25…8π/25] gives an upper limit of s_{1} < 0.86.

So far, we have demonstrated by analytical calculations that the proposed 3sd-particle is a suitable model to explain the coexistence of two distinct self-assembly patterns exhibited by three capped colloids. Clearly, the model is not limited to the presented capped particles, but can also be transferred to other variants with a fixed anisotropic magnetization distribution. One alternative is given by Janus-like magnetic hydrogel particles^{29} with anisotropic distribution of magnetic nanoparticles.^{30–34} We anticipate that, by using this model, general design recommendations can be derived on how to tune the magnetization distribution in order to provide a bistability of three-particle configurations. Moreover, the results obtained here evidence that a magnetic particle with anisotropic magnetization distribution can be sectioned into regions with broadly similar net magnetic anisotropy direction, and that the regions can be effectively approximated by a single point dipole. The finite number of dipole interactions between particles enables an analytic description. The magnetic center of mass of these regions is determined by the shift parameters s_{0} and s_{1} and the net magnetization in these regions is given by the parameters m_{0} and m_{1}. The introduced angle β corresponds to the spreading of the magnetic material in the particle.

The coated particles are re-suspended in distilled water via sonication. A droplet of 1.5 μl of this particle suspension is enclosed between two glass slides and sealed with UV glue to prevent evaporation of the solvent. Due to the density mismatch the particles sediment. To test the stability of the suspension, we have first examined demagnetized particles. They form a stable dispersion in water without coagulation and, thus, the particles do not require any further surface preparation for stabilization. This further proves that the presented agglomeration of magnetically saturated particles primarily results from magnetostatic interactions and other sources such as attractive electrostatic interactions can be neglected. The sample cell is studied via video microscopy (Leica DMI3000B transmission light microscope). The cap orientation is resolved via digital image analysis with an uncertainty of less than 3°. An electromagnetic coil for the alignment of the capped particles is mounted parallel to the sample cell.

(1) |

Z_{ij} = cos(φ − ψ − β(i−j)); |

r_{ij} = d^{2} + s_{i}^{2} + s_{j}^{2} − 2s_{i}s_{j}Z_{ij} − 2d[s_{i}cos(βi + ψ) − s_{j}cos(βj + φ)] |

For three particles, the situation is more complex and we used the following approach for finding a ground state based on ref. 16, 25 and 28: we first performed molecular dynamics simulations to define the ground-state candidates, which turned out to be a linear chain, a staggered chain and a ring; then, using the reduced phase space, we could analytically obtain the energies for these candidates and pinpoint the lowest one. These analytic expressions for the energies are provided in the ESI.†

Beyond this flexibility, the instantaneous control over the ratio between both existing assembly patterns might be desirable. Here, one possibility to interconvert between both assembly patterns of the three-particle clusters has been presented. We are currently investigating the possibilities to control the relative occurrence of both assembly patterns. This can be realized by imprinting a certain spatial and directional distribution of the single particle suspension by, e.g., external magnetic fields or geometric confinements.

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## Footnote |

† Electronic supplementary information (ESI) available: Analytic expression for the energies of three 3sd-particles. Movie: field-induced transformation between staggered and compact three-particle cluster. See DOI: 10.1039/c5sm02899j |

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