Open Access Article

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Arzu B.
Yener
and
Sabine H. L.
Klapp
*

Institute of Theoretical Physics, Technical University Berlin, Hardenbergstr. 36, 10625 Berlin, Germany. E-mail: klapp@physik.tu-berlin.de

Received
26th October 2015
, Accepted 8th January 2016

First published on 8th January 2016

We consider a model of colloidal spherical particles carrying a permanent dipole moment which is laterally shifted out of the particles' geometrical centres, i.e. the dipole vector is oriented perpendicular to the radius of the particles. Varying the shift δ from the centre, we analyse ground state structures for two, three and four hard spheres, using a simulated annealing procedure. We also compare earlier ground state results. We then consider a bulk system at finite temperatures and different densities. Using molecular dynamics simulations, we examine the equilibrium self-assembly properties for several shifts. Our results show that the shift of the dipole moment has a crucial impact on both the ground state configurations as well as the self-assembled structures at finite temperatures.

Yet, not only the application of functionalized particles is of interest but also their capability to serve as model systems to study fundamental concepts of physics such as self-organization,^{8,9} chirality,^{2,10–12} synchronisation,^{8,13–15} critical phenomena,^{16,17} entropic effects^{18,19} and active motion,^{20,21} to name a few. A paradigm example is the model of dipolar hard and soft spheres which is a well-established model to examine and understand the properties of magnetic colloidal particles immersed in a solvent, also called ferrofluids. From numerous studies of the phase behavior of dipolar liquids (e.g.ref. 22 and 23), and especially of their structural properties (e.g.ref. 24 and 25), it is known that the dipoles assemble into chains, rings and branched structures at sufficiently large dipolar strengths and low densities.

Here, we focus on (permanent) ferromagnetic colloids with anisotropic symmetry, i.e. the magnetic moment within the colloidal particle is not located at the geometric centre of the particle. A first theoretical description for decentrally located dipoles was introduced by Holm et al.^{26–28} In their model, spherical particles carry a dipole moment which is shifted out of the particle centre and is oriented parallel to the radius vector of the particle. The model describes very well the cluster formation of particles carrying a magnetic cap.^{29} Yet, it is insufficient to mimic the self-assembly of so-called patchy colloids,^{9} that is, silicon balls carrying magnetic cubes beneath their surfaces. Furthermore, the model does not reproduce the zig-zag chained structures formed by magnetic Janus particles in an external field, i.e. particles in which one hemisphere of silica spheres is covered with a magnetic coat.^{8,30} The concept of shifting the dipole was later extended by fixing the amount of shift and varying the orientation of the dipole moment vector within the particle^{31} which was proven to be more convenient for patchy collids.

In the present contribution we consider a model in which the dipole moment is laterally shifted such that the radius vector and the dipole moment vector are oriented perpendicular. The same model was also proposed in ref. 31. However, here we fix the orientation of the dipole moment and vary the amount of shift. Thereby, we do not only aim at modeling synthesized particles mentioned above. Rather, we are interested in understanding the impact of successively shifting the dipole on the self-assembly of such particles. To this end, we perform ground state calculations for a small number of dipolar hard spheres and conduct Molecular Dynamics (MD) simulations to study the bulk at finite temperature in three dimensions (3D). Very recently, Novak et al. have considered the same model;^{32} however, they restricted their study to systems where the particles are fixed in a plane with freely rotating dipoles. Thus, they considered a quasi-two-dimensional (q2D) system. Besides, the authors examined the system at one fixed density. Here, we examine a three dimensional system of such particles at zero temperature and conduct MD simulations of the bulk at several thermodynamic state points. Thereby, we aim at quantitive characterization in which the shift is the state parameter in the system.

The remainder of the paper is structured as follows. In Section 2, we present the model and the equations of motion, Section 3 refers to the computational methods and Sections 4 and 5 include the results for the ground state calculations and for the structural analysis of the bulk systems, respectively. We close the paper by a summary and outlook.

Fig. 1 Sketch of a dipolar sphere with a laterally shifted dipole moment. Also shown are the axes of the body-fixed coordinate system. |

In the laboratory reference frame, r_{i} is the position vector of the particle centre while the position vector of μ_{i} is given by r_{i}′ = r_{i} + d_{i}, where d_{i} now denotes the shift vector in the laboratory frame. For d = 0, r_{i}′ coincides with r_{i} yielding conventional dipolar systems with centered dipoles. A discussion of typical values of the relative shift δ = |d|/2|R| (where |R| is the particle radius) is given at the end of Section 5.1. The total pair potential between two particles i and j consists of a short-range repulsive potential, u_{short}(r_{ij}), and the dipole–dipole potential,

(1) |

(2) |

(3) |

Second, for the ground-state calculations presented in Section 4, we set u_{short}(r_{ij}) equal to the hard sphere (HS) potential defined as

(4) |

(5) |

(6) |

(7) |

m_{i} = F_{i} | (8) |

(9) |

(10) |

(11) |

The case of three particles was also considered in ref. 32. However, for very small and very high shifts, we find slightly different ground state energies and thus structures than those in ref. 32. Here, we discuss the main results and differences.

For three particles, shifting the dipole has the same effect on the ground state energies and configurations as for two particles. Thus, the former also rapidly decreases with increasing shift qualitatively in the same way as shown in Fig. 10 in the Appendix (and the same holds for the case N = 4). In terms of the ground state configurations, the dipoles first organize into slightly curved chainlike geometries [Fig. 2(b)] for very small shifts (see Table 1) and pass to triangular orientations with increasing shift, as shown in Fig. 2. For very high shifts, the particles keep their triangular arrangement while two of the dipoles within the particles form an antiparallel pair which is joined by the third one in a perpendicular manner. In this way, the order of the dipoles is still head-to-tail [see Fig. 2(d)].

δ |
E
_{gs} in a.u. |
---|---|

0.0125 | −4.2556 |

0.01875 | −4.2667 |

0.02 | −4.2722 |

0.025 | −4.2911 |

Our simulation results show that the ground state energies for curved chain configuration [Fig. 2(b)] are slightly lower (see Table 1) than those for the corresponding structure proposed in ref. 32 which the authors call a “zipper”. In a “zipper” configuration the dipoles have a head-to-tail orientation and are organized in a staggered manner.

Furthermore, at shifts higher than δ ≈ 0.4, we find again a difference to the results in ref. 32. The authors propose a configuration containing an antiparallel pair which is joined by the third particle via a head to tail orientation with one of the dipoles of the antiparallel pair. To clarify this issue, we have derived an analytical expression for the rectangular configuration in Fig. 2(d). It is given by

Evaluating this energy, we find that the rectangular configuration is energetically slightly more favourable than that of ref. 32. Fig. 3 shows the results for the absolute values of u_{rect}(δ), the results for the absolute values of eqn (7) of ref. 32, u_{ap+p}(δ), and the difference |u_{rect}(δ)| − |u_{ap+p}(δ)|, which is positive for all values considered.

Fig. 3 Absolute values for u_{rect}(δ), u_{ap+p}(δ)^{32} and for the difference |u_{rect}(δ)| − |u_{ap+p}(δ)|, in a.u., respectively. |

In conclusion, the four-particle system is the smallest system for which the dimensionality of the system is crucial for the resulting ground state structure at high shifts. While two and three particles always lie in a plane, four particles can arrange in a 3D structure.

Due to the fact that the magnitude of the ground state energy E_{G}(δ) is an increasing function of the shift (see previous discussions), also the dipolar coupling strength λ, which is defined as the ratio of the half ground state energy and the thermal energy, λ(δ) = |E_{G}(δ)|/2k_{B}T, becomes an increasing function of the shift. This yields an irreversible agglomeration of the particles, which cannot be counteracted by the soft-core potential. For the present choices for ε and n this situation occurs if the shift exceeds the value of δ = 0.33. We examined higher shifts than δ = 0.33 by appropiate choices for ε and n but did not gain any new insights of the system beyond those already observed for smaller shifts. Therefore, instead of adjusting λ(δ), e.g. by the appropriate reduction of μ* with increasing shifts, or instead of enhancing the soft-sphere potential values ε and n, we limit the shift at δ_{limit} = 0.33 in order to prevent agglomeration. In this way the structural properties of the system can be directly related to the amount of shift which hence is the parameter of interest in our examinations.

In order to roughly estimate which value for δ corresponds to real magnetic Janus particles, such as studied in ref. 36, we conducted test simulations of particles with various shifts in a static magnetic field (corresponding to the experimental situation considered in Fig. 1c of ref. 36). There, the particles organize in a staggered chain configuration with a head-to-tail orientation of the dipoles. The angle θ between neighbouring centre-to-centre distance vectors in the staggered chain configuration was measured as a function of the thickness of the magnetic coating of the particles.^{36} Evaluating snapshots of our test simulations, we found, by comparing with the measurements of ref. 36, that values in the range between δ ≈ 0.25 and δ ≈ 0.3 qualitatively describe Janus particles. In a perfect staggered-chain-configuration, one has cos(θ/2) = 2δ. For θ ≈ 110°,^{36} corresponding to the smallest thickness^{36} of nickel coating, this expression yields the value δ ≈ 0.287. This confirms that considering shifts larger than our limiting value δ > 0.33, does not have experimental relevance.

We consider a strongly coupled system with μ* = 3 with the densities ρ* = 0.07, ρ* = 0.1 and ρ* = 0.2 and at the two temperatures T* = 1.0 and T* = 1.35, respectively. This yields coupling strengths ranging from λ(δ = 0) = μ^{2}/(k_{B}Tσ^{3}) = 9 to λ(δ = 0.33) = 72 for T* = 1.0, and λ(δ = 0) ≈ 6.67 to λ(δ = 0.33) ≈ 53.33 for T* = 1.35. For a thorough investigation of the equilibrium properties of the shifted system, we performed MD simulations and calculated various structural properties, as described in the next section.

Qualitatively, the structures appearing for the considered values of δ can be divided into four groups. These are chains (A), staggered chains (B), rings built by staggered chains (C) and small clusters (D) of the types presented in Fig. 10(d), 2(c) and 4(c). Structures of type (A) can consist of a few (e.g. 2–5) as well as of many (e.g. more than 10) particles, i.e., the chains can be short or long. Structures of types (B) and (C) always consist of more than 10 particles [Fig. 5(d) and (e)]. In accordance with the ground state configurations (see Fig. 2 and 4), the structures found in the finite temperature systems for different shifts pass from chainlike geometries to circular close-packed clusters upon the increase of δ. Accordingly, structures of the first group are formed for zero and small shifts in the range δ = 0.01–δ ≈ 0.1 [Fig. 5(a) and (d)]. In this shift region, the overall chainlike structure with a head-to-tail orientation as formed by nonshifted dipoles is maintained. Yet, the shift causes more and more curved structures compared to the nonshifted particles. As is generally known for dipolar systems, the chain length, i.e. the number of particles within a chain, has a polydisperse distribution.^{37} This holds also for the shifted system (see also the discussion of the cluster analysis in Section 5.2.2).

For intermediate shifts, e.g. δ = 0.24, Fig. 5(b) and (e), the particles within the chains become staggered and we observe the coexistence of structures of the types (B), (C) and (D). Structures of group (D) are consistent with ground state configurations of this and higher shifts. Although groups (B) and (C) are not observed for zero temperature, they can be understood as a modification of chains, as they appear for small δ, and of rings which occur at zero temperature.

If δ takes values near 0.33, all large aggregates (B) and (C) vanish and only small clusters (D) remain, as shown in Fig. 5(c) and (f).

The same structural behaviour at the different shift regions is observed for the other state points considered. Thus we conclude that the described self-assembly of the particles at different shifts is a quite general behaviour which results from the increasing dipolar coupling strength for increasing shifts. The latter causes more and more close-packed structures as we already confirmed in the case of hard spheres.

5.2.1 Radial distribution function.
As a first quantitative measure of the structure formation, we consider the radial distribution function

for several shifts.

for several shifts.

The plots in Fig. 6 show g(r) for δ = 0 and δ = 0.33 for T* = 1.0 and T* = 1.35. The g(r) at zero shift is dominated by first and second neighbour correlations. This is a typical feature of strongly coupled dipolar systems^{38,39} and reflects the formation of chain-like structures. When we successively increase the shift, the second peak exists up to a value of δ ≈ 0.25. Beyond this value, only nearest neighbour correlations at r/σ = 1 are present in the system signifying the presence of only small and close-packed clusters (D), as seen in the snap shots of Fig. 5(c).

Fig. 6 Radial distribution functions g(r) for densities ρ* = 0.07 (turquoise), ρ* = 0.1 (black) and ρ* = 0.2 (red) at two temperatures T* = 1.0 in (a) and (b), and T* = 1.35 in (c) and (d). |

Noticeably, the results for the higher temperature T* = 1.35 completely coincide with those of T* = 1.0 in the high shift region [Fig. 6(b) and (d)]. This is because for sufficiently high shifts, the increase of the dipolar coupling strength is already enhanced, and thus, the increase of temperature does not affect the self-assembly.

5.2.2 Cluster analysis.
To further characterize the aggregates, we perform a cluster analysis. In particular, we are interested in the cluster size distribution for several shifts, the mean cluster size and the mean cluster magnetization as a function of δ. The bases of this analysis are distance and energy criteria. Specifically, all particles with a distance lower than r_{c} = 1.3σ and binding energy are regarded as being clustered. Here, denotes the dipolar energy [see eqn (1)] between all pairs i, i′ within the critical distance r_{c}.
_{c}(S) is a measure of the parallel alignment of the dipole vectors within the individual clusters. Specifically, values of M_{c}(S) near to one reflect a high degree of head to tail orientation, while vanishing values of this quantity indicate an antiparallel or a triangular orientation. Therefore, mean cluser magnetization gives insights into the organization of the dipoles within the formed structures and thus allows us to evaluate if a given assembly is chainlike [types (A) and (B)] or closed [types (C) and (D)]. Note that the total magnetization, which is usually calculated by summing over all particles, has vanishing values as the system is globally isotropic at the state points considered here.

The detected clusters were collected in a histogram in which the number of clusters with size S, n(S), is counted and normalized by the total number of clusters, , such that

Based on the function n(S), the mean cluster magnetization is calculated by

Finally, the mean cluster size is obtained from

(a) Normalised cluster size distribution. The results for h(S) for different characteristic shifts, namely for δ = 0.1 (small shift), δ = 0.16 (intermediate shift) and δ = 0.27 (high shift), are presented in Fig. 7. Fig. 7(a) and (d) show that mostly large aggregates, that can contain up to 25–30 particles, are formed. On the other hand, Fig. 7(c) and (f) indicate the formation of only small assemblies with 3–4 particles.

Fig. 7 Normalized cluster size distribution for the same densities and colors as in Fig. 6: (a–c) T* = 1.0; (d–f) T* = 1.35. |

However, in Fig. 7(b) and (e), although there is a preferential emergence of small assemblies, large aggregates of up to 20 particles are present in a non-negligible number and secondary peaks at, e.g. S = 15 (for T* = 1.0) and S = 13 (for T* = 1.35) are visible. Evidently, for this and comparable shifts, small and large assemblies can coexist.

One also finds that for higher temperature, large aggregates are less often formed than for the smaller temperature. This is indicated by the fact that the peaks in Fig. 7(e) and (f) are enhanced compared to those in Fig. 7(b) and (c).

(c) Mean cluster magnetization. In order to evaluate the types of the occurring structures for a given shift, we determine 〈M〉 as a function of the shift and plot the results in Fig. 8(b) and (d).

Fig. 8 Mean cluster size 〈S〉 and mean cluster magnetization 〈M〉 as a function of the shift at two temperatures T* = 1.0 ((a), (b)) and T* = 1.35 ((c), (d)). Colors are the same as in Fig. 6. |

For zero and initial shifts, 〈M〉 takes the value ≈0.7, reflecting predominantly the parallel orientation of the dipoles within their aggregates. From this and from the cluster size distribution [Fig. 7(a) and (d)] we conclude that for small shifts (up to δ ≈ 0.1), mainly short and long polar chains of type (A) or (B) are formed.

If the shift is further increased, 〈M〉 decreases, indicating that polar chains occur less often. Instead, the aggregates become more and more closed structures of the types (C) or (D) with increasing shifts. Hence, the decrease of 〈M〉 implies the coexistence of types (B), (C) and (D) [see Fig. 5(b) and (e)]. At the high shift end, 〈M〉 drops down to vanishing values indicating only pairwise antiparallel or triangular arrangements of the dipoles within the clusters, which is also consisent with the results shown in Fig. 7(c) and (f). The fact that the mean cluster magnetization has vanishing values at large δ also suggests that the clusters poorly interact.

Note that for all values of δ, the corresponding aggregates are isotropically oriented such that the total magnetization is zero for all shifts (not shown here).

(b) Mean cluster size. Finally, we examine the influence of the shift on the mean cluster size and plot 〈S〉 as a function of the shift in Fig. 8(a) and (c).

Starting at δ = 0, the mean cluster size grows to its maximum with about 17 particles for T* = 1.0 and about 13 particles for T* = 1.35. The maximum is reached at δ ≈ 0.05, respectively. This increase can be understood by the effective increase of the dipolar coupling strength λ (see preceding discussion) such that initial shifts result in the formation of longer chains of type (A). If δ exceeds this value, 〈S〉 starts to gradually decrease because with increasing shift, smaller aggregates are formed more frequently (see Fig. 7). Finally, 〈S〉 attains the value of about 3 particles in the high shift end, which is a highly representative value for both temperatures considered [Fig. 7(c) and (f)]. Significant differences between the results of the two temperatures can be seen only for shifts smaller than δ ≈ 0.1 where mainly chainlike aggregates are formed. Here, the increase of temperature, which involves the decrease of the coupling strength from λ = 9 to λ ≈ 6.67, causes the formation of chains with less particles. Moreover, for these values of δ, shifting the dipoles does not impose fundamentally different self-assembly patterns compared to nonshifted dipoles. Therefore, small shifts can be regarded as perturbation of the nonshifted system.

On the other hand, high shifts impose significantly different structures: the particles exclusively form structures of type (D) that correspond to ground state configurations of two, three and four hard spheres [see Fig. 10(d), 2(c) and 4(c)]. This is possible due to the large values of λ = 72 for T* = 1 or λ ≈ 53.33 for T* = 1.35.

Finally, for intermediate shifts, where large aggregates as well as small clusters are formed, the decrease of 〈S〉 (and at the same time of 〈M〉) can be interpreted as a transition region in which large aggregates gradually dissolve into small clusters until no large structures appear at all. Within this region, the competition between energy minimization and entropy maximization results in the coexistence of both, small and large aggregates. With increasing shift (i.e., effectively increasing λ(δ)), the particles accomplish to form structures equivalent to ground state configurations.

To summarize, in the bulk systems at the finite temperatures and densities considered here, we can qualitatively distinguish between three shift regions (small, intermediate and high) each of which is characterized by its own structural characteristics. By contrast, in the ground states of two particles, we determined only a small (with head-to-tail dipolar order) and a high (with antiparallel dipolar orientation) shift region. The intermediate shift region, observed for the bulk systems, is not detected for zero temperature. This is consistent with the fact that the corresponding structures of types (B) and (C) are not observed in the ground state calculations.

Shifting the dipole does not only fundamentally affect the ground state configurations but also the self-assembly patterns in finite temperature systems. For these, we could determine three regions of shift. In each region, the self-assembly of the particles is fundamentally different. The system passes from a state which is similar to that of nonshifted dipoles to a clustered structural state. By test simulations and by comparisson with experiments,^{36} we estimated that shift between δ ≈ 0.25 and δ ≈ 0.3 reproduces the qualitative behaviour of Janus particles.^{36} Moreover, the comparison also showed that shifts larger than our choice of the limiting value δ = 0.33 is not experimentally relevant.

Further, it is an interesting observation that the asymmetry of the particles, caused by the off-centred location of the dipole moment, is overcome for small shifts insofar as the behaviour of the small shift region can be recognized as a perturbation of the nonshifted system. On the other hand, if the shift is too high, the system compensates the off-centred location of the dipole by building symmetric aggregates.

So far, we focused on the self-assembled structures at low densities. One topic of further investigations should be the interaction between the aggregates in the different shift regions. In the present study, the cluster magnetization showed a vanishing value in the high shift region which hints to a negligible interaction between the clusters. For smaller shifts, we expect similar behaviour to that for the centred system. Moreover, it would be clearly desirable to have a full phase diagram as it is known for centred dipolar soft spheres.^{40} A particularly interesting aspect is the impact of the shift on long-range orientational ordering. Such an investigation is clearly beyond the scope of the present paper, but will be considered in future.

In view of the severe effects of the shift on the equilibrium properties, one expects new types of pattern formation if the system is out of equilibrium. An interesting case are systems of shifted dipoles exposed to several types of external magnetic fields.^{41} The case of a constant field was examined in ref. 32 demonstrating that shifted dipoles form staggered chains for appropriate values for the field strength and the shift. Even more exciting phenomena are expected if the field is time-dependent, e.g. precessing or rotating. In particular, it is interesting to explore whether the model of laterally shifted dipoles is capable of describing phenomena such as the formation of a tubular^{8} or a crystalline structure^{42} accompanied by a synchronization effect. Computer simulations in these directions are on the way.

To this end, we first derive an analytical expression for the pair energies as a function of the relative shift δ = |d|/(2|R|), where |R| = σ/2 is the particle radius. A similar derivation (leading to the same result) was very recently presented in ref. 32. Here, we include the derivation as a background for investigations of N > 2 particles (see Section 4). The basis of the derivation is the coordinate system shown in Fig. 9. Note that this is a two-dimensional system (x–z-plane) where the orientations of the dipoles along the y-axis, i.e. out-of-plane orientations, are neglected. This assumption is confirmed by simulation studies of q2D dipolar systems showing that out-of-plane fluctuations vanish for decreasing temperatures.^{43} In Fig. 9, the angles α and β describe the orientations of the shift vectors d_{i} and d_{j} with respect to the z-axis. As the shift vector and the dipole vector have a fixed orthogonal orientation to each other, the orientations of μ_{i} and μ_{j} with respect to the z-axis follow as α + π/2 and β + π/2. With these definitions of the angles, the results for our lateral shift can be directly compared to those for the radial shift given in ref. 27. Clearly, the distance |r_{ij}′| varies with α, β and δ. Finally, we obtain

(12) |

Fig. 9 Sketch of two dipolar hard spheres i and j and the orientations of their shift and dipole vectors in the x–z-plane. |

Fig. 10 (a) Ground state pair energy E_{G} normalized by the corresponding ground state energy E^{0}_{G} = −2μ^{2}/σ^{3} of centred dipoles. The results are obtained by simulated annealing (circles) and by minimization of eqn (12) (solid line). (b)–(d) Ground state configurations of two dipoles. |

When the results shown in Fig. 10(a) are compared to the corresponding results of radially shifted dipoles of ref. 26, a qualitative agreement of the function E_{G}(δ) can be seen. Yet, in the case of lateral shifts, the reduction of energy sets in earlier, i.e. for smaller shifts δ than those of radial shifts for which the energy starts to decrease only at δ ≈ 0.25. This can be understood by inspecting the ground state configurations corresponding to a given shift, which are shown in the inset of Fig. 10. These are determined by those values for the angles α and β that minimize eqn (12). Shifting the dipoles out of the centres, the parallel orientation of the dipoles for zero shift is gradually abandoned in favour of reducing the dipolar distance. In other words, the upper particle in Fig. 10(b) rotates clockwise while the lower one rotates counterclockwise upon increasing the shift. At the value δ ≈ 0.13, the dipoles attain a perpendicular orientation [Fig. 10(c)]. Finally, at δ = 0.2, the dipoles arrange in an antiparallel configuration of μ_{1} and μ_{2} where each of the dipoles has a perpendicular orientation relative to the connecting line between the particle centres. For all higher shifts, the antiparallel orientation is kept and only the dipolar distance is further reduced [Fig. 10(d)].

Compared to radially shifted dipoles,^{26} the main difference in the ground state structures is that radially shifted dipoles keep their parallel head-to-tail orientation for small shifts. Unlike laterally shifted dipoles, the antiparallel orientation is never energetically favourable. This demonstrates that not only the location but also the orientation of the dipole vector within the particle play a crucial role in the ground states of the particles.^{44}

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