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Anisometric and anisotropic magnetic colloids: How to tune the response
Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Anisometric and anisotropic magnetic colloids: How to tune the response J.G. Donaldson b, E.S. Pyanzina a, E.V. Novak a, S.S. Kantorovich a,b,n a b
Ural Federal University, Lenin Av. 51, Ekaterinburg, Russia University of Vienna, Sensengasse 8, Vienna, Austria
art ic l e i nf o
a b s t r a c t
Article history: Received 22 June 2014 Received in revised form 6 November 2014 Accepted 6 November 2014
We present a comparative study of the anisometric and anisotropic magnetic colloids at low temperatures. As examples we choose the ellipsoidal and cubic magnetic colloids to illustrate the influence of the shape (particle anisometry) on the ground state structures. To scrutinise the influence of the internal particle anisotropy we address particles with dipoles shifted out from the centre of mass. Of the latter, we distinguish between two types: the first type has a dipole moment pointing radially outwards; the other has a dipole pointing perpendicular to the radius along which it is shifted. & 2014 Elsevier B.V. All rights reserved.
Keywords: Magnetic colloid Ground state Anisometry Anisotropy
1. Introduction Nowadays, materials whose properties can be fine-tuned by external fields of moderate and low strengths are widely studied. “Green” environmentally friendly systems and biologically compatible materials are of particular interest, especially for medical applications. In order to obtain a system with desired properties, for example, with a controllable response to electric/magnetic fields, one needs to design this system on the level of its microstructure. When it comes to dipolar systems, the characteristic sizes of this microstructure can vary from several Ångström to tens of microns. Experimental studies of these new dipolar “smart materials” often involve rather expensive equipment and compounds, and can turn out to be very time consuming. Thus, the first step would be to predict theoretically (analytically and/or in computer simulations) the relationship between a certain microstructure and the macroscopical behaviour of the system. The self-assembly of nano and micron sized particles play a crucial part in the microstructure formation. Understanding and predicting the processes by which these particles assemble is allowing taylor-made materials to be built from the bottom up. Usual model to study self-assembly in dipolar soft matter employs the system of dipolar hard spheres (DHS)—monodisperse hard spheres of diameter d, possessing a point dipole moment m in their centres [1–10]. The main conclusion of the latter works is n
Correspondence to: University of Vienna, Sensengasse 8, Vienna, Austria. E-mail address: [email protected] (J.G. Donaldson).
that DHS at room temperatures form a gas phase, then, on cooling, the chain formation starts, which at low temperatures are replaced by that of rings or branched structures depending on the particle concentration. There are two possible ways to make the DHS model more complex. The first one is to change the properties of the carrier, and to introduce a certain magneto-elastic coupling into the system, like, for example, in magnetic gels [11] or magnetoelastomers [12]. Another avenue often exploited in self-assembling systems is to keep the carrier matrix simple, but to modify the particles themselves [15–20]. This can come in a number of forms: the shape of the particle can be anisotropic e.g. spheroids, rods and spherocylinders [21,22], or you can manipulate the positioning of the dipole within the spherical particle, like with sd-particles or magnetic Janus ones [23]. In the present manuscript we decided to study one pair of anisometric magnetic colloids, namely magnetic ellipsoids [24] and magnetic cubes; and one pair of anisotropic particles, namely sd-partcles [14,25–27] and magnetic Janus particles. We focus here on the ground states of these systems, i.e. we investigate the mostly energetically advantageous configurations of particles at 0 K. This way, one can deeper understand the influence of various parameters on the energy landscapes for various anisotropic and anisometric systems. The manuscript is organised as follows. The first section is dedicated to shape anisotropy. In the first part of Section 1.1 we present the comparative analysis of the two-particle ground states for ellipsoids and cubes with various orientations of the dipole moment. In the second part of this section we discuss the topology
http://dx.doi.org/10.1016/j.jmmm.2014.11.013 0304-8853/& 2014 Elsevier B.V. All rights reserved.
Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
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J.G. Donaldson et al. / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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in order to make a comparison easier, for ellipsoids the shortest axes (a) is 1, the next one (b) is 2 , and the longest axes (c) is 3 . We employ computer simulations and analytical calculations to find the ground states of the systems. The details of the method could be found in [7,28]. Computer simulations are performed in ESPResSo [29]. 1.2. Two particles Let us consider two-particle interaction as the one between two hard bodies with point dipoles in their centres of mass. In this case the interaction potentials have the following form:
Ud (ij) = −
(m i · m j ) ⎤ μ 0 ⎡ (mi ·rij )(m j ·rij ) ⎥, ⎢3 − 5 ⎥⎦ 4π ⎢⎣ rij rij3
(1)
rij = |rij | = |ri − r j | , where rij is the displacement vector of the two particles and μ0 is the vacuum permeability. The steric potential in a general form can be written
Fig. 1. Sketch of the dipole orientations in theory and simulations for anisometric particles. Cubes in simulations are made of spheres, whereas the ellipsoids are modified Gay–Berne gaussians [13]. In simulations the dipole moment is in the virtual-site particle [14]; the violet particle is just to show the orientation of the dipole. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
of small clusters and study the dependence of the clusters’ total dipole moment on the cluster size. Section 2 contains the study of the spherical particles with internal anisotropy, namely, we compare a model for so-called sd-particles to a model for magnetic Janus particles. Here, we mainly focus on the two-particle ground states, and then discuss shortly the larger clusters. The manuscript ends with a brief summary and outlook.
⎧∞ if ovelap; Us = ⎨ ⎩ 0 otherwise.
(2)
Thus, for two ellipsoids only two ground states are possible. If the dipole is aligned along the shorter axes, the head-to-tail configuration is the most energetically advantageous, whereas if the dipole is aligned along the long axes, and the minimal semiaxis ratio is smaller than a critical one (21/3), the antiparallel pair becomes the state with the lowest energy. For cubes we obtain the head-to-tail or zig-zag structures to provide the ground states. It is important to underline that already at this stage we see an important difference between the cubes and the ellipsoids: two ellipsoids do have the ground state with a zero total dipole moment, whereas two cubes do not. 1.3. Small clusters
1.1. Ellipsoids versus cubes Let us first consider an ellipsoidal and a cubic particle. For an ellipsoid we choose three possible orientations of the dipole moment (along each of the axis), see Fig. 1. As one can see, for a cube, one can also choose three different directions of a dipole (crystallographic axes (001), (110) and (111)). In the present manuscript
In this subsection we investigate small systems of ellipsoids and cubes in quasi two dimensions (q2D) with the total number of particles N less than 16. From earlier studies we know that for ellipsoids with the dipole moment along the short axis, both for two and three dimensions, that a chain or a daisy (a ring of ellipsoids) are the ground state topologies. The number of particles
Fig. 2. The energy per particle as a function of N: (left) cubes, two different orientations are used as shown in the inset; (right) ellipsoids and three different orientations of the dipole. Note that all chain-forming ellipsoids will exhibit a closing chain-ring transition, whereas the cubes never do, and the chain remains.
Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
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Fig. 3. The total dipole moment of the cluster calculated per particle as a function of N: (left) cubes, two different orientations are used as shown in the inset; (right) ellipsoids and three different orientations of the dipole. Note that all clusters of ellipsoids have a vanishing dipole moment, the same holds true for the (111) cubes. The cubes with (001) orientation form however clusters with a constant dipole moment per particle.
number in a q2D case is presented in Fig. 2, where the insets show the actual configurations. If one calculates the total dipole moments of these clusters, an interesting fact can be observed: the total dipole moment for all clusters, except for those made of cubes with (001) orientation of dipoles, is a vanishing quantity with an asymptotic behaviour of const/N (see Fig. 3). This remarkable difference between the cubes with (001) orientation of the dipole and the other anisometric particles, studied here, will undoubtably influence the thermodynamic behaviour of these systems when the thermal fluctuations become pronounced. This is the investigation that we are currently working on, and it surely deserves a separate manuscript.
2. SD-partcles versus Janus-like ones
Fig. 4. Sketch of the dipole orientations in theory and simulations for particles with internal anisotropy. Both sd-particles and Janus-like ones are modelled with soft spheres [30]. In simulations the dipole moment is in the virtual-site particle [14]; the violet particle is just to show the orientation of the dipole. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
for which the transition between a chain and a daisy occurs depends not only on the number of particles in the structure (as for the system of magnetic spheres), but also on the particle shape anisotropy. The more elongated the ellipsoids are, the more particles are needed for a daisy to become the ground state. For ellipsoids or cylinders, whose point dipole is coaligned with the long axis, in two dimensional case, the ground state might be a chain or a ring if the elongation of the particles is small, or a carpet made of side-by-side ellipsoids (cylinders) with neighbouring dipoles oriented antiparallel. In 3D, similar to 2D, the ground state of strongly elongated particles with dipole moment along the main axis is either a carpet or a bracelet of side-by-side particles with antiparallel orientation of dipoles [24]. For cubes we have two possibilities: chains when the dipole moment point in (001) direction, and some parquet-like structures, whose building blocks are four-cubes rings with the zig-zag orientation of the moments. The total energy for ground states as a function of the particle
In this section we analyse the ground states of systems whose building blocks are perfect spheres, however the dipole moment is shifted outwards from the centre of mass towards the particle surface. We focus on two possible cases. The first one is inspired by the study of Baraban et al. [19], where they presented the analysis of capped colloids: silica colloidal particles with a thin cap. This leads to a dipole moment pointing radially outside the spherical colloid. We call these particles sd-particles [14]. The second model is a toy model of a magnetic Janus particle, in which, one half is made of a nonmagnetic material, whereas the other hemisphere is magnetic. This usually leads to a dipolar alignment parallel to the division plane, i.e. perpendicular to the case of capped colloids. Earlier, we actively investigated the model of capped colloids [14,25–27], and inspired some other research in this area [31,32]. As for the theory of magnetic Janus particles, to our knowledge there is only one work [33], addressing the subject, however some of the ground states are missing in the latter investigation. 2.1. Two particles To describe the interaction between two particles, one needs to take into account that the interparticle distance now depends on the dipolar orientation. We use the following variables x = sin ((ψ − ϕ)/2), y = sin ((ψ + ϕ)/2), where, as it is shown in Fig. 4, ψ is the angle between the radius vector and the magnetic moment of the first particle, ϕ is the angle between the radius vector and the magnetic moment of the second particle, and s is the shift of the magnetic moment. The pair potential for the
Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
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Fig. 5. The energy of the particle pair as a function of the shift s: (left) sd-particles [14]; (right) Janus-like particles. For the sd-particles we see three possible configurations, with the head-to-tail orientation remaining a ground state up to s ∼ 4 . For the Janus-like particles, the head-to-tail orientation is the ground state only for s ¼ 0. For high shifts in both cases antiparallel pair of dipoles is formed and the energy diverges fast to −∞.
sd-particles can be written as shift (s , x , y ) = Udd
4x 2
(
x2
− 2)
s2
+ 4xy (
5x 2
− 4) s
5/2
(4s2x2 − 4sxy + 1) +
1 − 5x 2 − 3y 2 5/2
(4s2x2 − 4sxy + 1)
(3)
For the toy model of magnetic Janus particles we can write the potential in the following form: Janus Udd (s , x , y ) =
(
) 5/2 (s2x2 − 2sxy + 1)
s 2x 4 − 2sx 3y + s 2 + 1 x2 − 2sxy
+
3y 2 − 2 5/2
(s2x2 − 2sxy + 1)
. (4)
It is worth mentioning that both potentials are divided by m2 /d 3. Besides that, if for sd-particles the head-to-tail orientation corresponds to ϕ = ψ = 0, for the Janus-type particle, the head-totail configuration is described by ϕ = ψ = π /2. Minimising the total energy we obtain that the orientation of the dipole brings drastic changes in the energetically advantageous configurations. In Fig. 5 we present the plot for the ground state energy of the sd-particles and of the Janus-type particles as a function of the shift together with sketches of the configurations. The main similarity in these models can be seen at high shifts, where both sf-particle and Janus-like particles form antiparallel pairs at high values of the shift s. Note, however, that for low shifts the head-to-tail orientation provides the ground state only for the sd-particles, whereas for the Janus-type particles immediately assume such a mutual orientation that the dipoles are aligned trapezoidal, and keep rotating towards each other with the growing shift until the antiparallel orientation is reached.
3. Conclusion We presented a study of anisotropic and anisometric magnetic colloids. We scrutinised the two particle ground states and the behaviour of small clusters at low temperature. We made a critical comparison between two systems of anisometric magnetic colloids, namely, cubes and ellipsoids. They were chosen for their three possible orientations of dipoles: the short, the middle and the long axis for the ellipsoid; and the edge,
the face diagonal and the cube diagonal for the cube. The analysis of the two particle ground states showed that only for the systems in which the dipole is oriented along the shortest axis the “classical” head-to-tail orientation is the most energetically advantageous. However, if we look at larger systems, only cubes with (001) orientation keep the chain structure in the ground state, and as such exhibit a nonvanishing dipole moment. All other anisometric magnetic colloids at zero K form structures, whose dipole moment tends to be closed. Our study of internally anisotropic particles was based on two possible orientations of the dipole moment within the spherical hard body. One was so-called sd-particle with the dipole moment shifted out of the particle centre of mass radially outwards and another one was a prototype of a magnetic Janus particle, with a dipole moment shifted out of the centre of mass of a sphere but being perpendicular to the radius on which it is fixed. We showed that the two particle ground states are very different depending on the dipolar orientation and not only on the shift. Thus, for example, two sd-particles exhibit the lowest energy in a head-to-tail orientation even if the dipole moment is shifted by more than 30 percent. For a toy model of magnetic Janus colloids the head-totail orientation stops being a minimum of the energy as soon as the shift is larger than zero. We are currently working on the detailed analysis of larger clusters and infinite systems of anisometric and anisotropic particles at low temperatures in order to make the next step in our understanding of the shape and internal particle anisotropy influence on the thermodynamic and magnetic properties of various magnetic soft matter systems.
Acknowledgments The research has been partially supported by Austrian Science Fund (FWF): START-Project Y 627-N27, RFBR grant mol-a-ved 1202-33106, the Ural Federal University stimulating programme and by the Ministry of Education and Science of the Russian Federation (Contract 02.A03.21.000, Project 3.12.2014/K).
References [1] P. de Gennes, P. Pincus, Pair correlations in a ferromagnetic colloid, Zeitschrift für Phys. B Condens. Matter 11 (1970) 189–198. [2] J.J. Weis, D. Levesque, Chain formation in low density dipolar hard spheres: a Monte Carlo study, Phys. Rev. Lett. 71 (17) (1993) 2729–2732.
Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
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[3] J.M. Tavares, J.J. Weis, M.M. Telo da Gama, Quasi-two-dimensional dipolar fluid at low densities: Monte Carlo simulations and theory, Phys. Rev. E 65 (6) (2002) 061201. http://dx.doi.org/10.1103/PhysRevE.65.061201. [4] P.J. Camp, J.C. Shelley, G.N. Patey, Isotropic fluid phases of dipolar hard spheres, Phys. Rev. Lett. 84 (1) (2000) 115–118. [5] C. Holm, A. Ivanov, S. Kantorovich, E. Pyanzina, E. Reznikov, Equilibrium properties of a bidisperse ferrofluid with chain aggregates: theory and computer simulations, J. Phys.: Cond. Matter 18 (2006) S2737–S2756. [6] Z. Wang, C. Holm, Structure and magnetization properties of polydispersed ferrofluids: a molecular dynamics study, Phys. Rev. E 68 (2003) 041401. [7] T. Prokopieva, V. Danilov, S. Kantorovich, C. Holm, Ground state structures in ferrofluid monolayers, Phys. Rev. E. 80 (2009) 031404. [8] L. Rovigatti, J. Russo, F. Sciortino, No evidence of gas-liquid coexistence in dipolar hard spheres, Phys. Rev. Lett. 107 (2011) 237801. http://dx.doi.org/ 10.1103/PhysRevLett.107.237801. [9] L. Rovigatti, J. Russo, F. Sciortino, Structural properties of the dipolar hardsphere fluid at low temperatures and densities, Soft Matter 8 (2012) 6310–6319. http://dx.doi.org/10.1039/C2SM25192B. [10] S. Kantorovich, E. Pyanzina, C. De Michele, F. Sciortino, How to calculate structure factors of self-assembling anisotropic particles, Soft Matter 9 (2013) 4412–4427. http://dx.doi.org/10.1039/C3SM27895F. [11] R. Weeber, S. Kantorovich, C. Holm, Deformation mechanisms in 2d magnetic gels studied by computer simulations, Soft Matter 8 (2012) 9923–9932. http: //dx.doi.org/10.1039/C2SM26097B. [12] J.J. Cerda, P.A. Sanchez, C. Holm, T. Sintes, Phase diagram for a single flexible stockmayer polymer at zero field, Soft Matter 9 (2013) 7185–7195. http://dx. doi.org/10.1039/C3SM50278C. [13] J.G. Gay, B.J. Berne, Modification of the overlap potential to mimic a linear site potential, J. Chem. Phys. 74 (1981) 3316–3319. http://dx.doi.org/10.1063/ 1.441483. [14] S. Kantorovich, R. Weeber, J.J. Cerda, C. Holm, Ferrofluids with shifted dipoles: ground state structures, Soft Matter 7 (11) (2011) 5217–5227. [15] S. Varga, I. Szalai, J. Liszi, G. Jackson, A study of orientational ordering in a fluid of dipolar gay–berne molecules using density-functional theory, J. Chem. Phys. 116 (20) (2002) 9107–9119. http://dx.doi.org/10.1063/1.1469607, URL 〈http:// link.aip.org/link/?JCP/116/9107/1〉. [16] S. Sacanna, L. Rossi, B.W.M. Kuipers, A.P. Philipse, Fluorescent monodisperse silica ellipsoids for optical rotational diffusion studies, Langmuir 22 (4) (2006) 1822–1827. http://dx.doi.org/10.1021/la052484o, pMID: 16460113. arXiv: http://pubs.acs.org/doi/pdf/10.1021/la052484o. [17] M. Nakade, T. Ikeda, M. Ogawa, Synthesis and properties of ellipsoidal hematite/silicone core–shell particles, J. Mater. Sci. 42 (2007) 4815–4823. http: //dx.doi.org/10.1007/s10853-006-0761-9. [18] A. Malijevsky, G. Jackson, S. Varga, Many-fluid onsager density functional theories for orientational ordering in mixtures of anisotropic hard-body fluids, J. Chem. Phys. 129 (14) (2008) 144504. http://dx.doi.org/10.1063/1.2982501, URL 〈http://link.aip.org/link/?JCP/129/144504/1〉. [19] L. Baraban, D. Makarov, M. Albrecht, N. Rivier, P. Leiderer, A. Erbe, Frustrationinduced magic number clusters of colloidal magnetic particles, Phys. Rev. E 77 (2008) 031407.
5
[20] J. Sánchez, C. Rinaldi, Magnetoviscosity of dilute suspensions of magnetic ellipsoids obtained through rotational Brownian dynamics simulations, J. Colloid Interface Sci. 331 (2) (2009) 500–506. http://dx.doi.org/10.1016/j. jcis.2008.11.061, URL 〈http://www.sciencedirect.com/science/article/pii/ S0021979708015695〉. [21] M. Yan, J. Fresnais, J.-F. Berret, Growth mechanism of nanostructured superparamagnetic rods obtained by electrostatic co-assembly, Soft. Matter 6 (2010) 1997–2005. http://dx.doi.org/10.1039/B919992F. [22] A. Günther, P. Bender, A. Tschöpe, R. Birringer, Rotational diffusion of magnetic nickel nanorods in colloidal dispersions, J. Phys.: Cond. Matter 23 (2011) F5103. http://dx.doi.org/10.1088/0953-8984/23/32/325103. [23] S.K. Smoukov, S. Gangwal, M. Marquez, O.D. Velev, Reconfigurable responsive structures assembled from magnetic Janus particles, Soft Matter 5 (2009) 1285–1292. http://dx.doi.org/10.1039/B814304H. [24] S. Kantorovich, A.O. Ivanov, L. Rovigatti, J.M. Tavares, F. Sciortino, Nonmonotonic magnetic susceptibility of dipolar hard-spheres at low temperature and density, Phys. Rev. Lett. 110 (2013) 148306. http://dx.doi.org/10.1103/ PhysRevLett.110.148306. [25] M. Klinkigt, R. Weeber, S. Kantorovich, C. Holm, Systems of particles of particles with shifted magnetic dipoles, Magnetohydrodynamics 47 (2011) 143– 148. [26] M. Klinkigt, R. Weeber, S. Kantorovich, C. Holm, Cluster formation in systems of shifted-dipole particles, Soft Matter 9 (2013) 3535–3546. http://dx.doi.org/ 10.1039/C2SM27290C. [27] R. Weeber, M. Klinkigt, S. Kantorovich, C. Holm, Microstructure and magnetic properties of magnetic fluids consisting of shifted dipole particles under the influence of an external magnetic field, J. Chem. Phys. 139 (21) (2013) 214901. [28] V.A. Danilov, T.A. Prokopieva, S. Kantorovich, Ground state structures and structural transitions in a monolayer of magnetic dipolar particles in the presence of an external magnetic field, Phys. Rev. E 86 (2012) 061408. [29] A. Arnold, B.A. Mann, H. Limbach, C. Holm, ESPResSo—an extensible simulation package for research on soft matter systems, in: K. Kremer, V. Macho (Eds.), Forschung und wissenschaftliches Rechnen 2003, GWDG-Bericht, vol. 63, Gesellschaft für wissenschaftliche Datenverarbeitung mbh, Göttingen, Germany, 2004, pp. 43–59. [30] J.D. Weeks, D. Chandler, H.C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, J. Chem. Phys. 54 (1971) 5237. [31] M. Piastra, E.G. Virga, Phase polarity in a ferrofluid monolayer of shifted-dipole spheres, Soft Matter 8 (2012) 10969–10981. http://dx.doi.org/10.1039/ C2SM25984B. [32] E.C. Gartland, E.G. Virga, An analytic mean-field model for the magnetic response of a ferrofluid monolayer, Soft Matter 9 (2013) 5991–6008. http://dx. doi.org/10.1039/C3SM27939A. [33] A.I. Abrikosov, S. Sacanna, A.P. Philipse, P. Linse, Self-assembly of spherical colloidal particles with off-centered magnetic dipoles, Soft Matter 9 (2013) 8904–8913. http://dx.doi.org/10.1039/C3SM27128E.
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Anisometric and anisotropic magnetic colloids: How to tune the response J.G. Donaldson b, E.S. Pyanzina a, E.V. Novak a, S.S. Kantorovich a,b,n a b
Ural Federal University, Lenin Av. 51, Ekaterinburg, Russia University of Vienna, Sensengasse 8, Vienna, Austria
art ic l e i nf o
a b s t r a c t
Article history: Received 22 June 2014 Received in revised form 6 November 2014 Accepted 6 November 2014
We present a comparative study of the anisometric and anisotropic magnetic colloids at low temperatures. As examples we choose the ellipsoidal and cubic magnetic colloids to illustrate the influence of the shape (particle anisometry) on the ground state structures. To scrutinise the influence of the internal particle anisotropy we address particles with dipoles shifted out from the centre of mass. Of the latter, we distinguish between two types: the first type has a dipole moment pointing radially outwards; the other has a dipole pointing perpendicular to the radius along which it is shifted. & 2014 Elsevier B.V. All rights reserved.
Keywords: Magnetic colloid Ground state Anisometry Anisotropy
1. Introduction Nowadays, materials whose properties can be fine-tuned by external fields of moderate and low strengths are widely studied. “Green” environmentally friendly systems and biologically compatible materials are of particular interest, especially for medical applications. In order to obtain a system with desired properties, for example, with a controllable response to electric/magnetic fields, one needs to design this system on the level of its microstructure. When it comes to dipolar systems, the characteristic sizes of this microstructure can vary from several Ångström to tens of microns. Experimental studies of these new dipolar “smart materials” often involve rather expensive equipment and compounds, and can turn out to be very time consuming. Thus, the first step would be to predict theoretically (analytically and/or in computer simulations) the relationship between a certain microstructure and the macroscopical behaviour of the system. The self-assembly of nano and micron sized particles play a crucial part in the microstructure formation. Understanding and predicting the processes by which these particles assemble is allowing taylor-made materials to be built from the bottom up. Usual model to study self-assembly in dipolar soft matter employs the system of dipolar hard spheres (DHS)—monodisperse hard spheres of diameter d, possessing a point dipole moment m in their centres [1–10]. The main conclusion of the latter works is n
Correspondence to: University of Vienna, Sensengasse 8, Vienna, Austria. E-mail address: [email protected] (J.G. Donaldson).
that DHS at room temperatures form a gas phase, then, on cooling, the chain formation starts, which at low temperatures are replaced by that of rings or branched structures depending on the particle concentration. There are two possible ways to make the DHS model more complex. The first one is to change the properties of the carrier, and to introduce a certain magneto-elastic coupling into the system, like, for example, in magnetic gels [11] or magnetoelastomers [12]. Another avenue often exploited in self-assembling systems is to keep the carrier matrix simple, but to modify the particles themselves [15–20]. This can come in a number of forms: the shape of the particle can be anisotropic e.g. spheroids, rods and spherocylinders [21,22], or you can manipulate the positioning of the dipole within the spherical particle, like with sd-particles or magnetic Janus ones [23]. In the present manuscript we decided to study one pair of anisometric magnetic colloids, namely magnetic ellipsoids [24] and magnetic cubes; and one pair of anisotropic particles, namely sd-partcles [14,25–27] and magnetic Janus particles. We focus here on the ground states of these systems, i.e. we investigate the mostly energetically advantageous configurations of particles at 0 K. This way, one can deeper understand the influence of various parameters on the energy landscapes for various anisotropic and anisometric systems. The manuscript is organised as follows. The first section is dedicated to shape anisotropy. In the first part of Section 1.1 we present the comparative analysis of the two-particle ground states for ellipsoids and cubes with various orientations of the dipole moment. In the second part of this section we discuss the topology
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Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
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in order to make a comparison easier, for ellipsoids the shortest axes (a) is 1, the next one (b) is 2 , and the longest axes (c) is 3 . We employ computer simulations and analytical calculations to find the ground states of the systems. The details of the method could be found in [7,28]. Computer simulations are performed in ESPResSo [29]. 1.2. Two particles Let us consider two-particle interaction as the one between two hard bodies with point dipoles in their centres of mass. In this case the interaction potentials have the following form:
Ud (ij) = −
(m i · m j ) ⎤ μ 0 ⎡ (mi ·rij )(m j ·rij ) ⎥, ⎢3 − 5 ⎥⎦ 4π ⎢⎣ rij rij3
(1)
rij = |rij | = |ri − r j | , where rij is the displacement vector of the two particles and μ0 is the vacuum permeability. The steric potential in a general form can be written
Fig. 1. Sketch of the dipole orientations in theory and simulations for anisometric particles. Cubes in simulations are made of spheres, whereas the ellipsoids are modified Gay–Berne gaussians [13]. In simulations the dipole moment is in the virtual-site particle [14]; the violet particle is just to show the orientation of the dipole. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
of small clusters and study the dependence of the clusters’ total dipole moment on the cluster size. Section 2 contains the study of the spherical particles with internal anisotropy, namely, we compare a model for so-called sd-particles to a model for magnetic Janus particles. Here, we mainly focus on the two-particle ground states, and then discuss shortly the larger clusters. The manuscript ends with a brief summary and outlook.
⎧∞ if ovelap; Us = ⎨ ⎩ 0 otherwise.
(2)
Thus, for two ellipsoids only two ground states are possible. If the dipole is aligned along the shorter axes, the head-to-tail configuration is the most energetically advantageous, whereas if the dipole is aligned along the long axes, and the minimal semiaxis ratio is smaller than a critical one (21/3), the antiparallel pair becomes the state with the lowest energy. For cubes we obtain the head-to-tail or zig-zag structures to provide the ground states. It is important to underline that already at this stage we see an important difference between the cubes and the ellipsoids: two ellipsoids do have the ground state with a zero total dipole moment, whereas two cubes do not. 1.3. Small clusters
1.1. Ellipsoids versus cubes Let us first consider an ellipsoidal and a cubic particle. For an ellipsoid we choose three possible orientations of the dipole moment (along each of the axis), see Fig. 1. As one can see, for a cube, one can also choose three different directions of a dipole (crystallographic axes (001), (110) and (111)). In the present manuscript
In this subsection we investigate small systems of ellipsoids and cubes in quasi two dimensions (q2D) with the total number of particles N less than 16. From earlier studies we know that for ellipsoids with the dipole moment along the short axis, both for two and three dimensions, that a chain or a daisy (a ring of ellipsoids) are the ground state topologies. The number of particles
Fig. 2. The energy per particle as a function of N: (left) cubes, two different orientations are used as shown in the inset; (right) ellipsoids and three different orientations of the dipole. Note that all chain-forming ellipsoids will exhibit a closing chain-ring transition, whereas the cubes never do, and the chain remains.
Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
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Fig. 3. The total dipole moment of the cluster calculated per particle as a function of N: (left) cubes, two different orientations are used as shown in the inset; (right) ellipsoids and three different orientations of the dipole. Note that all clusters of ellipsoids have a vanishing dipole moment, the same holds true for the (111) cubes. The cubes with (001) orientation form however clusters with a constant dipole moment per particle.
number in a q2D case is presented in Fig. 2, where the insets show the actual configurations. If one calculates the total dipole moments of these clusters, an interesting fact can be observed: the total dipole moment for all clusters, except for those made of cubes with (001) orientation of dipoles, is a vanishing quantity with an asymptotic behaviour of const/N (see Fig. 3). This remarkable difference between the cubes with (001) orientation of the dipole and the other anisometric particles, studied here, will undoubtably influence the thermodynamic behaviour of these systems when the thermal fluctuations become pronounced. This is the investigation that we are currently working on, and it surely deserves a separate manuscript.
2. SD-partcles versus Janus-like ones
Fig. 4. Sketch of the dipole orientations in theory and simulations for particles with internal anisotropy. Both sd-particles and Janus-like ones are modelled with soft spheres [30]. In simulations the dipole moment is in the virtual-site particle [14]; the violet particle is just to show the orientation of the dipole. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
for which the transition between a chain and a daisy occurs depends not only on the number of particles in the structure (as for the system of magnetic spheres), but also on the particle shape anisotropy. The more elongated the ellipsoids are, the more particles are needed for a daisy to become the ground state. For ellipsoids or cylinders, whose point dipole is coaligned with the long axis, in two dimensional case, the ground state might be a chain or a ring if the elongation of the particles is small, or a carpet made of side-by-side ellipsoids (cylinders) with neighbouring dipoles oriented antiparallel. In 3D, similar to 2D, the ground state of strongly elongated particles with dipole moment along the main axis is either a carpet or a bracelet of side-by-side particles with antiparallel orientation of dipoles [24]. For cubes we have two possibilities: chains when the dipole moment point in (001) direction, and some parquet-like structures, whose building blocks are four-cubes rings with the zig-zag orientation of the moments. The total energy for ground states as a function of the particle
In this section we analyse the ground states of systems whose building blocks are perfect spheres, however the dipole moment is shifted outwards from the centre of mass towards the particle surface. We focus on two possible cases. The first one is inspired by the study of Baraban et al. [19], where they presented the analysis of capped colloids: silica colloidal particles with a thin cap. This leads to a dipole moment pointing radially outside the spherical colloid. We call these particles sd-particles [14]. The second model is a toy model of a magnetic Janus particle, in which, one half is made of a nonmagnetic material, whereas the other hemisphere is magnetic. This usually leads to a dipolar alignment parallel to the division plane, i.e. perpendicular to the case of capped colloids. Earlier, we actively investigated the model of capped colloids [14,25–27], and inspired some other research in this area [31,32]. As for the theory of magnetic Janus particles, to our knowledge there is only one work [33], addressing the subject, however some of the ground states are missing in the latter investigation. 2.1. Two particles To describe the interaction between two particles, one needs to take into account that the interparticle distance now depends on the dipolar orientation. We use the following variables x = sin ((ψ − ϕ)/2), y = sin ((ψ + ϕ)/2), where, as it is shown in Fig. 4, ψ is the angle between the radius vector and the magnetic moment of the first particle, ϕ is the angle between the radius vector and the magnetic moment of the second particle, and s is the shift of the magnetic moment. The pair potential for the
Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
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Fig. 5. The energy of the particle pair as a function of the shift s: (left) sd-particles [14]; (right) Janus-like particles. For the sd-particles we see three possible configurations, with the head-to-tail orientation remaining a ground state up to s ∼ 4 . For the Janus-like particles, the head-to-tail orientation is the ground state only for s ¼ 0. For high shifts in both cases antiparallel pair of dipoles is formed and the energy diverges fast to −∞.
sd-particles can be written as shift (s , x , y ) = Udd
4x 2
(
x2
− 2)
s2
+ 4xy (
5x 2
− 4) s
5/2
(4s2x2 − 4sxy + 1) +
1 − 5x 2 − 3y 2 5/2
(4s2x2 − 4sxy + 1)
(3)
For the toy model of magnetic Janus particles we can write the potential in the following form: Janus Udd (s , x , y ) =
(
) 5/2 (s2x2 − 2sxy + 1)
s 2x 4 − 2sx 3y + s 2 + 1 x2 − 2sxy
+
3y 2 − 2 5/2
(s2x2 − 2sxy + 1)
. (4)
It is worth mentioning that both potentials are divided by m2 /d 3. Besides that, if for sd-particles the head-to-tail orientation corresponds to ϕ = ψ = 0, for the Janus-type particle, the head-totail configuration is described by ϕ = ψ = π /2. Minimising the total energy we obtain that the orientation of the dipole brings drastic changes in the energetically advantageous configurations. In Fig. 5 we present the plot for the ground state energy of the sd-particles and of the Janus-type particles as a function of the shift together with sketches of the configurations. The main similarity in these models can be seen at high shifts, where both sf-particle and Janus-like particles form antiparallel pairs at high values of the shift s. Note, however, that for low shifts the head-to-tail orientation provides the ground state only for the sd-particles, whereas for the Janus-type particles immediately assume such a mutual orientation that the dipoles are aligned trapezoidal, and keep rotating towards each other with the growing shift until the antiparallel orientation is reached.
3. Conclusion We presented a study of anisotropic and anisometric magnetic colloids. We scrutinised the two particle ground states and the behaviour of small clusters at low temperature. We made a critical comparison between two systems of anisometric magnetic colloids, namely, cubes and ellipsoids. They were chosen for their three possible orientations of dipoles: the short, the middle and the long axis for the ellipsoid; and the edge,
the face diagonal and the cube diagonal for the cube. The analysis of the two particle ground states showed that only for the systems in which the dipole is oriented along the shortest axis the “classical” head-to-tail orientation is the most energetically advantageous. However, if we look at larger systems, only cubes with (001) orientation keep the chain structure in the ground state, and as such exhibit a nonvanishing dipole moment. All other anisometric magnetic colloids at zero K form structures, whose dipole moment tends to be closed. Our study of internally anisotropic particles was based on two possible orientations of the dipole moment within the spherical hard body. One was so-called sd-particle with the dipole moment shifted out of the particle centre of mass radially outwards and another one was a prototype of a magnetic Janus particle, with a dipole moment shifted out of the centre of mass of a sphere but being perpendicular to the radius on which it is fixed. We showed that the two particle ground states are very different depending on the dipolar orientation and not only on the shift. Thus, for example, two sd-particles exhibit the lowest energy in a head-to-tail orientation even if the dipole moment is shifted by more than 30 percent. For a toy model of magnetic Janus colloids the head-totail orientation stops being a minimum of the energy as soon as the shift is larger than zero. We are currently working on the detailed analysis of larger clusters and infinite systems of anisometric and anisotropic particles at low temperatures in order to make the next step in our understanding of the shape and internal particle anisotropy influence on the thermodynamic and magnetic properties of various magnetic soft matter systems.
Acknowledgments The research has been partially supported by Austrian Science Fund (FWF): START-Project Y 627-N27, RFBR grant mol-a-ved 1202-33106, the Ural Federal University stimulating programme and by the Ministry of Education and Science of the Russian Federation (Contract 02.A03.21.000, Project 3.12.2014/K).
References [1] P. de Gennes, P. Pincus, Pair correlations in a ferromagnetic colloid, Zeitschrift für Phys. B Condens. Matter 11 (1970) 189–198. [2] J.J. Weis, D. Levesque, Chain formation in low density dipolar hard spheres: a Monte Carlo study, Phys. Rev. Lett. 71 (17) (1993) 2729–2732.
Please cite this article as: J.G. Donaldson, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.11.013i
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[3] J.M. Tavares, J.J. Weis, M.M. Telo da Gama, Quasi-two-dimensional dipolar fluid at low densities: Monte Carlo simulations and theory, Phys. Rev. E 65 (6) (2002) 061201. http://dx.doi.org/10.1103/PhysRevE.65.061201. [4] P.J. Camp, J.C. Shelley, G.N. Patey, Isotropic fluid phases of dipolar hard spheres, Phys. Rev. Lett. 84 (1) (2000) 115–118. [5] C. Holm, A. Ivanov, S. Kantorovich, E. Pyanzina, E. Reznikov, Equilibrium properties of a bidisperse ferrofluid with chain aggregates: theory and computer simulations, J. Phys.: Cond. Matter 18 (2006) S2737–S2756. [6] Z. Wang, C. Holm, Structure and magnetization properties of polydispersed ferrofluids: a molecular dynamics study, Phys. Rev. E 68 (2003) 041401. [7] T. Prokopieva, V. Danilov, S. Kantorovich, C. Holm, Ground state structures in ferrofluid monolayers, Phys. Rev. E. 80 (2009) 031404. [8] L. Rovigatti, J. Russo, F. Sciortino, No evidence of gas-liquid coexistence in dipolar hard spheres, Phys. Rev. Lett. 107 (2011) 237801. http://dx.doi.org/ 10.1103/PhysRevLett.107.237801. [9] L. Rovigatti, J. Russo, F. Sciortino, Structural properties of the dipolar hardsphere fluid at low temperatures and densities, Soft Matter 8 (2012) 6310–6319. http://dx.doi.org/10.1039/C2SM25192B. [10] S. Kantorovich, E. Pyanzina, C. De Michele, F. Sciortino, How to calculate structure factors of self-assembling anisotropic particles, Soft Matter 9 (2013) 4412–4427. http://dx.doi.org/10.1039/C3SM27895F. [11] R. Weeber, S. Kantorovich, C. Holm, Deformation mechanisms in 2d magnetic gels studied by computer simulations, Soft Matter 8 (2012) 9923–9932. http: //dx.doi.org/10.1039/C2SM26097B. [12] J.J. Cerda, P.A. Sanchez, C. Holm, T. Sintes, Phase diagram for a single flexible stockmayer polymer at zero field, Soft Matter 9 (2013) 7185–7195. http://dx. doi.org/10.1039/C3SM50278C. [13] J.G. Gay, B.J. Berne, Modification of the overlap potential to mimic a linear site potential, J. Chem. Phys. 74 (1981) 3316–3319. http://dx.doi.org/10.1063/ 1.441483. [14] S. Kantorovich, R. Weeber, J.J. Cerda, C. Holm, Ferrofluids with shifted dipoles: ground state structures, Soft Matter 7 (11) (2011) 5217–5227. [15] S. Varga, I. Szalai, J. Liszi, G. Jackson, A study of orientational ordering in a fluid of dipolar gay–berne molecules using density-functional theory, J. Chem. Phys. 116 (20) (2002) 9107–9119. http://dx.doi.org/10.1063/1.1469607, URL 〈http:// link.aip.org/link/?JCP/116/9107/1〉. [16] S. Sacanna, L. Rossi, B.W.M. Kuipers, A.P. Philipse, Fluorescent monodisperse silica ellipsoids for optical rotational diffusion studies, Langmuir 22 (4) (2006) 1822–1827. http://dx.doi.org/10.1021/la052484o, pMID: 16460113. arXiv: http://pubs.acs.org/doi/pdf/10.1021/la052484o. [17] M. Nakade, T. Ikeda, M. Ogawa, Synthesis and properties of ellipsoidal hematite/silicone core–shell particles, J. Mater. Sci. 42 (2007) 4815–4823. http: //dx.doi.org/10.1007/s10853-006-0761-9. [18] A. Malijevsky, G. Jackson, S. Varga, Many-fluid onsager density functional theories for orientational ordering in mixtures of anisotropic hard-body fluids, J. Chem. Phys. 129 (14) (2008) 144504. http://dx.doi.org/10.1063/1.2982501, URL 〈http://link.aip.org/link/?JCP/129/144504/1〉. [19] L. Baraban, D. Makarov, M. Albrecht, N. Rivier, P. Leiderer, A. Erbe, Frustrationinduced magic number clusters of colloidal magnetic particles, Phys. Rev. E 77 (2008) 031407.
5
[20] J. Sánchez, C. Rinaldi, Magnetoviscosity of dilute suspensions of magnetic ellipsoids obtained through rotational Brownian dynamics simulations, J. Colloid Interface Sci. 331 (2) (2009) 500–506. http://dx.doi.org/10.1016/j. jcis.2008.11.061, URL 〈http://www.sciencedirect.com/science/article/pii/ S0021979708015695〉. [21] M. Yan, J. Fresnais, J.-F. Berret, Growth mechanism of nanostructured superparamagnetic rods obtained by electrostatic co-assembly, Soft. Matter 6 (2010) 1997–2005. http://dx.doi.org/10.1039/B919992F. [22] A. Günther, P. Bender, A. Tschöpe, R. Birringer, Rotational diffusion of magnetic nickel nanorods in colloidal dispersions, J. Phys.: Cond. Matter 23 (2011) F5103. http://dx.doi.org/10.1088/0953-8984/23/32/325103. [23] S.K. Smoukov, S. Gangwal, M. Marquez, O.D. Velev, Reconfigurable responsive structures assembled from magnetic Janus particles, Soft Matter 5 (2009) 1285–1292. http://dx.doi.org/10.1039/B814304H. [24] S. Kantorovich, A.O. Ivanov, L. Rovigatti, J.M. Tavares, F. Sciortino, Nonmonotonic magnetic susceptibility of dipolar hard-spheres at low temperature and density, Phys. Rev. Lett. 110 (2013) 148306. http://dx.doi.org/10.1103/ PhysRevLett.110.148306. [25] M. Klinkigt, R. Weeber, S. Kantorovich, C. Holm, Systems of particles of particles with shifted magnetic dipoles, Magnetohydrodynamics 47 (2011) 143– 148. [26] M. Klinkigt, R. Weeber, S. Kantorovich, C. Holm, Cluster formation in systems of shifted-dipole particles, Soft Matter 9 (2013) 3535–3546. http://dx.doi.org/ 10.1039/C2SM27290C. [27] R. Weeber, M. Klinkigt, S. Kantorovich, C. Holm, Microstructure and magnetic properties of magnetic fluids consisting of shifted dipole particles under the influence of an external magnetic field, J. Chem. Phys. 139 (21) (2013) 214901. [28] V.A. Danilov, T.A. Prokopieva, S. Kantorovich, Ground state structures and structural transitions in a monolayer of magnetic dipolar particles in the presence of an external magnetic field, Phys. Rev. E 86 (2012) 061408. [29] A. Arnold, B.A. Mann, H. Limbach, C. Holm, ESPResSo—an extensible simulation package for research on soft matter systems, in: K. Kremer, V. Macho (Eds.), Forschung und wissenschaftliches Rechnen 2003, GWDG-Bericht, vol. 63, Gesellschaft für wissenschaftliche Datenverarbeitung mbh, Göttingen, Germany, 2004, pp. 43–59. [30] J.D. Weeks, D. Chandler, H.C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, J. Chem. Phys. 54 (1971) 5237. [31] M. Piastra, E.G. Virga, Phase polarity in a ferrofluid monolayer of shifted-dipole spheres, Soft Matter 8 (2012) 10969–10981. http://dx.doi.org/10.1039/ C2SM25984B. [32] E.C. Gartland, E.G. Virga, An analytic mean-field model for the magnetic response of a ferrofluid monolayer, Soft Matter 9 (2013) 5991–6008. http://dx. doi.org/10.1039/C3SM27939A. [33] A.I. Abrikosov, S. Sacanna, A.P. Philipse, P. Linse, Self-assembly of spherical colloidal particles with off-centered magnetic dipoles, Soft Matter 9 (2013) 8904–8913. http://dx.doi.org/10.1039/C3SM27128E.
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