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Low temperature structural transitions in dipolar hard spheres: The influence on magnetic properties
Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Low temperature structural transitions in dipolar hard spheres: The influence on magnetic properties A.O. Ivanov a , S.S. Kantorovich a,b, L. Rovigatti b, J.M. Tavares c, F. Sciortino d a
Ural Federal University, Lenin Av. 51, Ekaterinburg, Russia University of Vienna, Sensengasse 8, Vienna, Austria c Instituto Superior de Engenharia de Lisboa-ISEL, Rua Conselheiro Em'dio Navarro 1, Lisbon, Portugal d Universit di Roma La Sapienza, Piazzale Aldo Moro 2, Roma, Italy b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 May 2014 Received in revised form 24 September 2014
We investigate the structural chain-to-ring transition at low temperature in a gas of dipolar hard spheres (DHS). Due to the weakening of entropic contribution, ring formation becomes noticeable when the effective dipole–dipole magnetic interaction increases. It results in the redistribution of particles from usually observed flexible chains into flexible rings. The concentration (ρ) of DHS plays a crucial part in this transition: at a very low ρ only chains and rings are observed, whereas even a slight increase of the volume fraction leads to the formation of branched or defect structures. As a result, the fraction of DHS aggregated in defect-free rings turns out to be a non-monotonic function of ρ. The average ring size is found to be a slower increasing function of ρ when compared to that of chains. Both theory and computer simulations confirm the dramatic influence of the ring formation on the ρ-dependence of the initial magnetic susceptibility (χ) when the temperature decreases. The rings due to their zero total dipole moment are irresponsive to a weak magnetic field and drive to the strong decrease of the initial magnetic susceptibility. & 2014 Published by Elsevier B.V.
Keywords: Initial magnetic susceptibility Low temperature Structural transition Dipolar hard spheres Chain Ring
1. Introduction Nowadays magnetic soft materials form an emerging area of soft matter research, as they provide a possibility to efficiently control their behaviour by an external magnetic field. A lot of promising applications of magnetic soft materials in medicine, robotics and industry [1–9] rely on the detailed understanding of the relationship between the nano-scale structure of these materials and their microscopic responses. Magnetic nanoparticles tend to form various structures due to the anisotropic interparticle dipole–dipole interaction. In order to focus on the magnetic interactions and get rid of often non-crucial steric (chemical) details of the interactions, theoreticians and computer scientists usually employ the model of DHSs to investigate and predict structural and phase transitions in dipolar soft matter. One of the well-known and actively studied examples of dipolar soft matter is magnetic fluids (ferrofluids). Synthesised in the 1960s [10], these systems are suspensions of magnetic singledomain nanoparticles in magnetopassive carriers. Ferrofluids prove to have a complex microstructure. Chain aggregates composed by magnetic particles are responsible for magnetooptical E-mail address: [email protected] (A.O. Ivanov).
[11–13], rheological [14–16], scattering [17–20] and many other anomalies usually observed in magnetic fluids when an external field is applied. As such, ferroparticle chains became a subject of numerous theoretical [21–26] and simulation [27–40] studies. The main conclusion of these works is that the chains are very flexible, and their average length, as well as the concentration, rapidly grow with decreasing temperatures and increasing concentration. In addition, ferroparticle chains, being highly correlated objects, inevitably contribute to a sharp increase of the initial magnetic susceptibility [26]. Lately, the investigation of DHS at low temperatures revealed the existence of a different cluster type, namely ferroparticle rings [28,41–43,37,38]. Their formation is to be expected, as the long flexible chains close the magnetic flux when collapsing into rings. Despite the loss of entropy, a ferroparticle ring is energetically more advantageous, that is why the ideal ring of DHS was proven to be a ground state in quasi 2D [44]. Unlike chains, ferroparticle rings have a very weak response to an applied magnetic field, and do not contribute to the increase of χ [45]. In Ref. [40] the structural transition at low temperature, earlier obtained in simulations, was accurately quantified theoretically and was shown to be directly responsible for the nonmonotonic temperature dependence of the DHS gas initial magnetic susceptibility. However, in the work [40] only very low concentrations were
http://dx.doi.org/10.1016/j.jmmm.2014.10.013 0304-8853/& 2014 Published by Elsevier B.V.
Please cite this article as: A.O. Ivanov, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.10.013i
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addressed to guarantee that all clusters were well defined rings and chains, and the intercluster interactions, as well as the deviations of cluster topologies from the “ideal” ones, were irrelevant. In the present contribution we continue scrutinising the structural peculiarities of the DHS gas at low temperatures, but for much broader range of particle concentrations. As such, the present study allows to find the concentration dependence of cluster topology and to clarify the limitations of the previously developed approach [40]. For the first time (to our knowledge), we pinpoint the concentrations at which the branched structures start emerging. The paper is organised as follows. In Section 2 we discuss Monte Carlo simulation results and theoretical predictions on the chain and ring concentrations and their average sizes. Here, we show that the non-monotonic ρ-dependence of the fraction of DHSs aggregated in defect-free rings is a signature of a more complex structure forming already at very low concentrations. One of the possible structures, suggested in Ref. [46], is a “Y”shaped structure obtained via amalgamation of three chains. Recently, we systemised defect structures in Ref. [47], where we showed the importance of branched rings as well. Our detailed investigation of ring structures provided in this section brings out the concentration range in which the “ideal” structures start being replaced by more sophisticated topologies. In Section 3 we show two distinct temperature regimes for the ρ-dependence of χ. At rather high and intermediate temperatures, where the majority of particles either remain non-aggregated, or become members of chains, χ monotonically increases with concentration and inverse temperature. In contrast, for low temperatures, at which rings start dominating in the system, χ increases with concentration, but decreases with inverse temperature. The summary and a short outlook are provided in the Conclusion.
2. Chain-to-ring transition To investigate the structural transition in DHS gas, we use a combination of Monte Carlo simulations and Density Functional Theory. Monte Carlo simulations are performed in the canonical ensemble of N ¼ 5000 particles. Implementing the advanced volumebias techniques [48,37] allows us to equilibrate the systems at rather low temperatures. Metallic periodic boundary conditions in combination with Ewald summation are used. In simulations the following criterion was applied: two particles are considered as bonded if their interaction energy is negative and if their relative distance is smaller than 1.3 of particle diameter [38]. For further details on the simulation approach, see Ref. [37]. Here, we distinguish between three types of clusters, namely chains, rings and branched structures. Branched structures are composed by chains and rings connected to each other via the so-called “defects” (particles that have more than two bonded neighbours) [47], which is why chains and rings will be addressed as defectfree clusters in the present paper. In theory, instead, only defect-free clusters are taken into account. The free energy (F) of a DHS gas can be written as a functional of n-particle chain (gn) and ring (fn) volume fractions [49–52]:
F [{gn }, {fn }] Vk B T
∞
=
∑ gn ln n= 1
gn v eQ n
∞
+
∑ fn ln n= 5
fn v eWn
,
numerical results [53,37,38], we assume that rings smaller than five particles do not form. In our calculations we assume chains to be flexible and rings to have an ideal circular shape. The minimisation of the free-energy functional (Eq. (1)) with respect to the distributions {gn } and {fn } preserving the concentration ρ = 6Nv/πV : ∞
∞
∑ gn n + ∑ fn n = n= 1
n= 5
π ρ , 6v
(2)
leads to the following solution:
gn =
1 Q n pn , v
fn =
1 Wn pn . v
(3)
Here, p, the Lagrange multiplier to be found from Eq. (2), has the meaning of activity. In the present paper we use the dimensionless temperature T ⁎ = 6vk B T /πm2, where m is the particle magnetic moment. This temperature is the inverse dipolar coupling parameter, characterising the energy per particle of two touching DHS with coaligned dipolar moments in units of k B T . We start describing the low-Tn behaviour of DHS gas by plotting the distributions of defect-free chains and rings over their sizes. In this case, it is convenient to use normalised fractions of nparticle clusters:
Cn =
6 nvgn , π ρ
Rn =
6 nvfn . π ρ
(4)
These functions show the fraction of particles belonging to a corresponding cluster of a fixed length. The fraction of single particles is given by C1. Note that for Rn the value of n is larger than five. In Fig. 1 the characteristic cluster distributions are plotted for ρ = 5 × 10−4 at various low temperatures. Here, one can see that for the highest T ⁎ = 0.155 the majority of particles is aggregated in chains. The amount of particle in rings is rather low. This difference levels out with decreasing temperature, and for the lowest T ⁎ = 0.125 we observe an inversion: the population of rings evidently dominates. This crossover shows the presence of a structural transition from the system with mainly chain clusters to that of rings. It is worth mentioning that the position of Cn maximum almost does not depend on Tn and is noticeably closer to unity, than the one for Rn. The latter strongly depends on temperature and shifts towards the larger rings with decreasing Tn. Both distributions become wider on cooling, which is the
(1)
where V stands for the system volume, kBT is the thermal energy, v has a meaning of the particle volume. Qn and Wn denote the chain and ring partition functions respectively, whose calculations have been performed in Ref. [40]. Following the results of Ref. [44] and
Fig. 1. Distributions Cn and Rn from Eq. (4) at fixed ρ = 5 × 10−4 . Solid lines show the fraction of particles in chains, dashed lines describe those in rings. Dimensionless temperatures are given in the legend. For the sake of clarity we plot here only theoretical predictions that are in a good agreement with the simulation data.
Please cite this article as: A.O. Ivanov, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.10.013i
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Fig. 2. Concentration dependence of the ring portion νr , see Eq. (5). Theoretical predictions are shown with solid lines; simulation data are plotted with symbols. Theoretical lines are not extended to the whole concentration range due to the reasons explained in the text. Dimensionless temperatures are given in the legend.
signature of the growing average cluster size. It is important that such a crossover is observed in a broad range of ρ ∼ 10−5 − 10−3 for 0.125 ≤ T ⁎ ≤ 0.155. For lower densities, the crossover is observed at temperatures lower than T ⁎ = 0.125, whereas for higher densities, the structure of the DHS becomes more complex, as it will be shown below. The characteristics of chain-like aggregates have been exhaustively studied in the last decades ([21–24,26–40], and references therein), which is why, here, we will focus on the description of rings. In Fig. 2 we show the portion of rings νr defined as follows:
νr =
∑∞ n = 5 fn ∞ ∑∞ n = 1 gn + ∑n = 5 fn
.
Fig. 3. Concentration dependence of the branched cluster portion (not defect-free chains and rings). Only simulation data are presented. Dimensionless temperatures are given in the legend.
[54,55]: χL = 4πρ /3T ⁎. Once the temperature decreases T ⁎ ∼ 0.5 − 1, the interparticle magnetic correlations begin to play an important part in the system collective response to a weak applied magnetic field. Several approaches to deal with this phenomena have been put forward [56–60,31,61,62]. For diluted DHS gas, all of these models give the same asymptotic behaviour: χd = χL (1 + χL /3). With increasing interparticle interaction T ⁎ ≤ 0.3 the chain formation occurs and contributes noticeably to the initial susceptibility, since the flexible chains exhibit a correlated reaction to an external magnetic field. In this case χ is derived in the following form [63]:
χc = (5)
In other words, the total portion of clusters is a sum of defect-free chains and rings only. At the same time, in Monte Carlo simulations this portion is calculated explicitly by summing up all possible aggregates. Theory and simulations agree well only up to ρ ∼ 10−3. For larger concentrations our “idealised” theoretical approach (Eqs. (1)–(3)) predicts further growth of νr , whereas in simulations a clear maximum is observed. The reason for the maximum of the ring portion can be easily understood when looking at Fig. 3, where the analogous portion but of branched structures (νb ) obtained in simulations is plotted versus ρ. The point is that starting from concentration of ∼10−3 almost independently from temperature the portion of branched structures starts growing rapidly and basically reaches unity for ρ ≳ 0.1. In other words, at low Tn, being rather seldom in very diluted DHS gas, branched structures become basically the only possible aggregates when ρ exceeds one per cent, and create a percolating network for concentrations ten percent and higher [38,47]. Since our theory in its present form is able to capture only one of the two low-T structural transitions in DHS gas, namely that from defect-free chains to defect-free rings and not the one to branched structures, below we focus on magnetic properties of the systems with ρ not exceeding 1 per cent.
3. Initial magnetic susceptibility The initial susceptibility of DHSs has been studied for more than 100 years. At very high temperatures, when the interparticle interactions are negligible, χ is described by the Langevin law
3
χd ρ
∞
∑ gn 〈mn2 〉, n= 1
〈mn2 〉 = n + 2
(6)
⎛ ⎞ K ⎜n − 1 + K n − nK ⎟. 2 ⎠ (1 − K ) ⎝
Here, the final sum contains the chain contribution, with 〈mn2 〉 having the meaning of a dimensionless mean-squared dipole moment of a chain made of n DHS particles, and the expression for 〈mn2 〉 has been derived in Ref. [63]. The dipolar correlation coefficient K assumes the simple form: K = coth(1/2T ⁎) − 2T ⁎. In the present paper, we use Eq. (6) to describe the initial magnetic susceptibility of the DHS gas at low temperatures. One can do that because the total dipole moment of a ring is zero, and a weak magnetic field is not able to perturb such an energetically advantageous structure. In other words, we assume that the particles aggregated in rings are simply excluded from the contribution to χ. To do it in theory, we minimise the free energy functional from Eq. (1), Section 2 for fixed ρ and Tn. As a result, we obtain the concentration ρc of particles aggregated in chains, and the chain distribution gn. We plug ρc in Eq. (6) instead of the total concentration ρ and evaluate the susceptibility χ. In simulations, the initial magnetic susceptibility is calculated using fluctuationdissipation theorem [40], and as a consequence, the contribution of each particle, regardless of its membership of any aggregate class, is taken into account. In Fig. 4 the concentration dependence of χ is shown for different Tn. We distinguish between two temperature ranges. In Fig. 4(a) the initial susceptibility monotonically grows with both increasing concentration and decreasing temperature. For the highest values of T ⁎ = 1 and 0.5 this effect is simply described by the functional ρ- and Tn-dependences of χd. The formation of
Please cite this article as: A.O. Ivanov, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.10.013i
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4. Conclusion
Fig. 4. Concentration dependence of the DHS initial magnetic susceptibility. (a) High temperature regime. (b) Low temperature regime. In both figures simulation results are plotted with symbols; solid lines represent theoretical predictions. Dimensionless temperatures are given in the legends.
chains, which takes place at T ⁎ = 0.25 and is stimulated by the growth of ρ, does not physically change the matter. The following decrease of temperature, however, leads to a drastic change of χ behaviour, as it is shown in Fig. 4(b). Even though, in the temperature range, corresponding to the chain to ring structural transition (see, Fig. 1, Section 2), the initial susceptibility keeps growing with increasing total concentration ρ, but the inversion of Tn dependence is observed. As shown in Fig. 3, Section 2, the amount of rings grows with density and inverse temperature; it leads to an effective exclusion of growing amount of particles (ring members) from the collective magnetic response to a weak external magnetic field. Hence, for T ⁎ = 0.125 one finds the curve for χ versus ρ to be the lowest. This effect, equally observed in simulations and confirmed by our simple theoretical approach, is rather unexpected, as the magnetic correlations only keep increasing with the increase of concentration and decrease of temperature. On the other hand, for higher densities at low Tn one observes another structural transition (see Fig. 3, Section 2) accompanied by the formation of branched structures and even percolating network, which at the moment cannot be described theoretically.
In this contribution we analyse in detail the behaviour of diluted DHS gas at low temperature. We focus on the influence of concentration on the possible structural transitions in the system and their influence on the collective magnetic response to a weak applied external magnetic field. We show that at low temperature and very low concentration, the chain to ring structural transition takes place. This transition can be stimulated by growing concentration only up to the values of ρ in the order of one per cent. For higher concentrations, the chain to ring transition is first smeared out and then completely replaced by the appearance of new type of aggregates, i.e. branched structures. For the first low-concentration transition we develop a theoretical model that appears to be in a good agreement with simulation data. Our theory is based on the assumption that chains are flexible, whereas the rings are ideally circular, and considers the entropy loss for the ring formation. From this we conclude that the energy of a flexible “breathing” ring does not differ much from that of an ideal ring in the range of parameters addressed here. In terms of magnetic response, both an ideal and a flexible ring have a zero total dipole moment. This allows our theory to describe accurately the initial susceptibility of the diluted DHS gas in a broad range of temperature and concentrations, by using an assumption that particles aggregated in rings simply do not react to a weak applied magnetic field. We show that an increasing fraction of rings result in a significant decrease of the initial susceptibility. The structural transition to the branched structures deserves a special separate study, as it, on one hand, leads to a formation of the percolating network, thus, promising the existence of a new magnetic “polymeric material” kept together by exclusively magnetic forces. On the other hand, this transition sheds light on the long-lasting debate concerning the possibility of the gas–liquid transition in DHSs. In addition, the presence of branched structures is also expected in real magnetic fluids at low temperatures, and will necessarily change their viscous, optic and magnetic properties.
Acknowledgments The research has been partially supported by Austrian Science Fund (FWF): START-Projekt Y 627-N27. J.M.T. acknowledges financial support from the Portuguese Foundation for Science and Technology under Contract no. EXCL/FIS-NAN/0083/2012. A.O.I. and S.S.K. are supported by the Ministry of Education and Science of the Russian Federation (Contract 02.A03.21.000, Project 3.2.2014/K).
References [1] A. Lubbe, C. Alexiou, C. Bergemann, Clinical applications of magnetic drug targeting, J. Surg. Res. 95 (2001) 200–206(7). [2] J. Dobson, Magnetic micro- and nano-particle-based targeting for drug and gene delivery, Nanomedicine 1 (1) (2006) 31–37. [3] M.E. Hayden, U.O. Häfeli, Magnetic bandages for targeted delivery of therapeutic agents, J. Phys.: Condens. Matter 18 (38) (2006) S2877. [4] J. Durán, J. Arias, V. Gallardo, A. Delgado, Magnetic colloids as drug vehicles, J. Pharm. Sci. 97 (8) (2008) 2948–2983. [5] S. Hughes, S. McBain, J. Dobson, A.J. El Haj, Selective activation of mechanosensitive ion channels using magnetic particles, J. R. Soc. Interface 5 (25) (2008) 855–863. [6] Q.A. Pankhurst, N.T.K. Thanh, S.K. Jones, J. Dobson, Progress in applications of magnetic nanoparticles in biomedicine, J. Phys. D: Appl. Phys. 42 (22) (2009) 224001.
Please cite this article as: A.O. Ivanov, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.10.013i
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[7] C.S. Brazel, Magnetothermally-responsive nanomaterials: Combining magnetic nanostructures and thermally-sensitive polymers for triggered drug release, Pharm. Res. 26 (3) (2009) 644–656. [8] E. Pollert, G. Goglio, S. Mornet, E. Duguet, Magnetic nanoparticles for magnetic resonance imaging and hyperthermia applications, in: R. Brayner, F. Fiévet, T. Coradin (Eds.), Nanomaterials: A Danger or a Promise? Springer, London, 2013, pp. 99–129. [9] D. Ortega, N. Perez, J.L. Vilas, J.S. Garitaonandia, K. Suzuki, J.R. Marin, M. Rodriguez, Nonylphenol polyethoxylate coated body-center-cubic iron nanocrystals for ferrofluids with technical applications, J. Appl. Phys. 113 (17) (2013) 17B505. [10] L.R. Jr., R.E. Rosensweig, Magnetocaloric power, J. AIAA 2 (1964) 1418–1422. [11] Y.N. Skibin, V.V. Chekanov, Y.L. Raikher, Birefringence in a ferromagnetic liquid, J. Exp. Theor. Phys. 45 (3) (1977) 496–499. [12] P.C. Scholten, The origin of magnetic birefringence and dichroism in magnetic fluids, IEEE Trans. Magn. 16 (1980) 221–225. [13] S. Taketomi, Magnetic fluids anomalous pseudo-cotton mouton effects about 107 larger that that of nitrobenzene, Jpn. J. Appl. Phys. 22 (1983) 1137–1143. [14] V. Buzmakov, A. Pshenichnikov, On the structure of microaggregates in magnetite colloids, J. Colloid Interface Sci. 182 (1996) 63–70. [15] S. Odenbach, H. Gilly, Taylor vortex flow of magnetic fluids under the influence of an azimuthal magnetic field, J. Magn. Magn. Mater. 152 (1996) 123. [16] S. Odenbach, Magnetoviscous Effects in Ferrofluids, Lecture Notes in Physics, vol. 71, Springer, Berlin, Heidelberg, 2002. [17] E. Dubois, R. Perzynski, F. BouÃ, V. Cabuil, Liquid–gas transitions in charged colloidal dispersions: a small-angle neutron scattering coupled with phase diagrams of magnetic fluids, Langmuir 16 (13) (2000) 5617–5625. [18] L. Pop, S. Odenbach, Investigation of the microscopic reason for the magnetoviscous effect in ferrofluids studied by small angle neutron scatting, J. Phys.: Condens. Matter 18 (2006) 2758. [19] M. Klokkenburg, B.H. Erné, A. Wiedenmann, A.V. Petukhov, A.P. Philipse, Dipolar structures in magnetite ferrofluids studied with small-angle neutron scattering with and without applied magnetic field, Phys. Rev. E 75 (2007) 051408. [20] D. Bica, L. Vekas, M. Avdeev, O. Marinica, V. Socoliuc, M. Balasoiu, V. Garamus, Sterically stabilized water based magnetic fluids: synthesis, structure and properties, J. Magn. Magn. Mater. 311 (2007) 17. [21] P. de Gennes, P. Pincus, Pair correlations in a ferromagnetic colloid, Z. Phys. B Condens. Matter 11 (1970) 189–198. [22] A.Y. Zubarev, L.Y. Iskakova, Theory of physical properties of magnetic liquids with chain aggregates, J. Exp. Theor. Phys. 80 (1995) 857. [23] M.A. Osipov, P.I.C. Teixeira, M.M. TelodaGama, Structure of strongly dipolar fluids at low densities, Phys. Rev. E 54 (3) (1996) 2597. [24] S. Kantorovich, A. Ivanov, Formation of chain aggregates in magnetic fluids: an influence of polydispersity, J. Magn. Magn. Mater. 252 (2002) 244–246. [25] K. Morozov, M. Shliomis, Magnetic fluid as an assembly of flexible chains, in: S. Odenbach (Ed.), Ferrofluids: Magnetically Controllable Fluids and their Applications, Lecture Notes in Physics, vol. 594, 2002, pp. 162–184. [26] A.O. Ivanov, S.S. Kantorovich, V.S. Mendelev, E.S. Pyanzina, Ferrofluid aggregation in chains under the influence of a magnetic field, J. Magn. Magn. Mater. 300 (2006) 206–209. [27] G.N. Patey, D. Levesque, J.J. Weis, On the theory and computer simulations of dipolar fluids, Mol. Phys. 38 (1979) 219. [28] J.J. Weis, D. Levesque, Chain formation in low density dipolar hard shperes: a Monte Carlo study, Phys. Rev. Lett. 71 (17) (1993) 2729–2732. [29] P.J. Camp, G.N. Patey, Structure and scattering in colloidal ferrofluids, Phys. Rev. E 62 (4) (2000) 5403–5408. [30] Z. Wang, C. Holm, Structure and magnetization properties of polydispersed ferrofluids: a molecular dynamics study, Phys. Rev. E 68 (2003) 041401. [31] T. Kristóf, I. Szalai, Magnetic properties and structure of polydisperse ferrofluid models, Phys. Rev. E 68 (4) (2003) 041109. [32] P.D. Duncan, P.J. Camp, Structure and dynamics in a monolayer of dipolar spheres, J. Chem. Phys. 121 (22) (2004) 11322–11331. [33] P. Ilg, E. Coquelle, S. Hess, Structure and rheology of ferrofluids: simulation results and kinetic models, J. Phys.: Condens. Matter 18 (38) (2006) S2757. [34] L. Luo, S.H.L. Klapp, Fluctuations in a ferrofluid monolayer: an integral equation study, J. Chem. Phys. 131 (3) (2009) 034709.
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[35] H. Schmidle, S.H.L. Klapp, Phase transitions of two-dimensional dipolar fluids in external fields, J. Chem. Phys. 134 (11) (2011) 114903. [36] J. Richardi, J.-J. Weis, Low density mesostructures of confined dipolar particles in an external field, J. Chem. Phys. 135 (12) (2011) 124502. [37] L. Rovigatti, J. Russo, F. Sciortino, No evidence of gas–liquid coexistence in dipolar hard spheres, Phys. Rev. Lett. 107 (2011) 237801. [38] L. Rovigatti, J. Russo, F. Sciortino, Structural properties of the dipolar hardsphere fluid at low temperatures and densities, Soft Matter 8 (2012) 6310–6319. [39] J. Richardi, J.-J. Weis, Influence of short range potential on field induced chain aggregation in low density dipolar particles, J. Chem. Phys. 138 (24) (2013) 244704. [40] 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. [41] P.J. Camp, J.C. Shelley, G.N. Patey, Isotropic fluid phases of dipolar hard spheres, Phys. Rev. Lett. 84 (1) (2000) 115–118. [42] P.D. Duncan, P.J. Camp, Aggregation kinetics and the nature of phase separation in two-dimensional dipolar fluids, Phys. Rev. Lett. 97 (2006) 107202. [43] G. Ganzenmuller, P.J. Camp, Vapor–liquid coexistence in fluids of charged hard dumbbells, J. Chem. Phys. 126 (19) (2007) 191104. [44] T. Prokopieva, V. Danilov, S. Kantorovich, C. Holm, Ground state structures in ferrofluid monolayers, Phys. Rev. E 80 (2009) 031404. [45] 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. [46] T. Tlusty, S.A. Safran, Defect-induced phase separation in dipolar fluids, Science 290 (5495) (2000) 1328–1331. [47] L. Rovigatti, S. Kantorovich, A.O. Ivanov, J.M. Tavares, F. Sciortino, Branching points in the low-temperature dipolar hard sphere fluid, J. Chem. Phys. 139 (13) (2013) 134901. [48] B. Chen, J.I. Siepmann, Improving the efficiency of the aggregation volume biasing Monte Carlo algorithm, J. Phys. Chem. B 105 (45) (2001) 11275–11282. [49] R.P. Sear, G. Jackson, Thermodynamic perturbation theory for association into chains and rings, Phys. Rev. E 50 (1994) 386–394. [50] A. Avlund, G. Kontogeorgis, W. Chapman, Mol. Phys. 109 (2011) 1759. [51] A. Galindo, S. Burton, G. Jackson, D. Visco, D.A. Kofke, Mol. Phys. 100 (2002) 2241. [52] J.M. Tavares, L. Rovigatti, F. Sciortino, Quantitative description of the selfassembly of patchy particles into chains and rings, J. Chem. Phys. 137 (4) (2012) 044901. [53] S. Kantorovich, J.J. Cerdà, C. Holm, Microstructure analysis of monodisperse ferrofluid monolayers: theory and simulation, Phys. Chem. Chem. Phys. 10 (14) (2008) 1883–1895. [54] P. Langevin, Sur la theory du magnetism, J. Phys. 4 (1905) 678. [55] P. Langevin, Magnetism et theory des electrons, Ann. Chim. Phys. 5 (1905) 70. [56] K.I. Morozov, A.F. Pshenichnikov, Y.L. Raikher, M.I. Shliomis, Magnetic properties of ferrocolloids: the effect of interparticle interactions, J. Magn. Magn. Mater. 65 (1987) 269–272. [57] M. Shliomis, A. Pshenichnikov, K. Morozov, I. Shurubor, Magnetic properties of ferrocolloids, J. Magn. Magn. Mater. 85 (1–3) (1990) 40–46. [58] A.O. Ivanov, O.B. Kuznetsova, Magnetic properties of dense ferrofluids: an influence of interparticle correlations, Phys. Rev. E 64 (2001) 23, pp. 041405–1. [59] B. Huke, M. Lücke, Magnetization of ferrofluids with dipolar interactions: a Born–Mayer expansion, Phys. Rev. E 62 (2000) 6875–6890. [60] B. Huke, M. Lücke, Magnetization of concentrated polydisperse ferrofluids: cluster expansion, Phys. Rev. E 67 (2003) 051403. [61] K.I. Morozov, The dielectric virial expansion and the models of dipolar hardsphere fluid, J. Chem. Phys. 19 (2007) 126. [62] I. Szalai, S. Nagy, S. Dietrich, Comparison between theory and simulations for the magnetization and the susceptibility of polydisperse ferrofluids, J. Phys.: Condens. Matter 25 (46) (2013) 465108. [63] V.S. Mendelev, A.O. Ivanov, Ferrofluid aggregation in chains under the influence of a magnetic field, Phys. Rev. E 70 (2004) 051502–051510.
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Low temperature structural transitions in dipolar hard spheres: The influence on magnetic properties A.O. Ivanov a , S.S. Kantorovich a,b, L. Rovigatti b, J.M. Tavares c, F. Sciortino d a
Ural Federal University, Lenin Av. 51, Ekaterinburg, Russia University of Vienna, Sensengasse 8, Vienna, Austria c Instituto Superior de Engenharia de Lisboa-ISEL, Rua Conselheiro Em'dio Navarro 1, Lisbon, Portugal d Universit di Roma La Sapienza, Piazzale Aldo Moro 2, Roma, Italy b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 May 2014 Received in revised form 24 September 2014
We investigate the structural chain-to-ring transition at low temperature in a gas of dipolar hard spheres (DHS). Due to the weakening of entropic contribution, ring formation becomes noticeable when the effective dipole–dipole magnetic interaction increases. It results in the redistribution of particles from usually observed flexible chains into flexible rings. The concentration (ρ) of DHS plays a crucial part in this transition: at a very low ρ only chains and rings are observed, whereas even a slight increase of the volume fraction leads to the formation of branched or defect structures. As a result, the fraction of DHS aggregated in defect-free rings turns out to be a non-monotonic function of ρ. The average ring size is found to be a slower increasing function of ρ when compared to that of chains. Both theory and computer simulations confirm the dramatic influence of the ring formation on the ρ-dependence of the initial magnetic susceptibility (χ) when the temperature decreases. The rings due to their zero total dipole moment are irresponsive to a weak magnetic field and drive to the strong decrease of the initial magnetic susceptibility. & 2014 Published by Elsevier B.V.
Keywords: Initial magnetic susceptibility Low temperature Structural transition Dipolar hard spheres Chain Ring
1. Introduction Nowadays magnetic soft materials form an emerging area of soft matter research, as they provide a possibility to efficiently control their behaviour by an external magnetic field. A lot of promising applications of magnetic soft materials in medicine, robotics and industry [1–9] rely on the detailed understanding of the relationship between the nano-scale structure of these materials and their microscopic responses. Magnetic nanoparticles tend to form various structures due to the anisotropic interparticle dipole–dipole interaction. In order to focus on the magnetic interactions and get rid of often non-crucial steric (chemical) details of the interactions, theoreticians and computer scientists usually employ the model of DHSs to investigate and predict structural and phase transitions in dipolar soft matter. One of the well-known and actively studied examples of dipolar soft matter is magnetic fluids (ferrofluids). Synthesised in the 1960s [10], these systems are suspensions of magnetic singledomain nanoparticles in magnetopassive carriers. Ferrofluids prove to have a complex microstructure. Chain aggregates composed by magnetic particles are responsible for magnetooptical E-mail address: [email protected] (A.O. Ivanov).
[11–13], rheological [14–16], scattering [17–20] and many other anomalies usually observed in magnetic fluids when an external field is applied. As such, ferroparticle chains became a subject of numerous theoretical [21–26] and simulation [27–40] studies. The main conclusion of these works is that the chains are very flexible, and their average length, as well as the concentration, rapidly grow with decreasing temperatures and increasing concentration. In addition, ferroparticle chains, being highly correlated objects, inevitably contribute to a sharp increase of the initial magnetic susceptibility [26]. Lately, the investigation of DHS at low temperatures revealed the existence of a different cluster type, namely ferroparticle rings [28,41–43,37,38]. Their formation is to be expected, as the long flexible chains close the magnetic flux when collapsing into rings. Despite the loss of entropy, a ferroparticle ring is energetically more advantageous, that is why the ideal ring of DHS was proven to be a ground state in quasi 2D [44]. Unlike chains, ferroparticle rings have a very weak response to an applied magnetic field, and do not contribute to the increase of χ [45]. In Ref. [40] the structural transition at low temperature, earlier obtained in simulations, was accurately quantified theoretically and was shown to be directly responsible for the nonmonotonic temperature dependence of the DHS gas initial magnetic susceptibility. However, in the work [40] only very low concentrations were
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addressed to guarantee that all clusters were well defined rings and chains, and the intercluster interactions, as well as the deviations of cluster topologies from the “ideal” ones, were irrelevant. In the present contribution we continue scrutinising the structural peculiarities of the DHS gas at low temperatures, but for much broader range of particle concentrations. As such, the present study allows to find the concentration dependence of cluster topology and to clarify the limitations of the previously developed approach [40]. For the first time (to our knowledge), we pinpoint the concentrations at which the branched structures start emerging. The paper is organised as follows. In Section 2 we discuss Monte Carlo simulation results and theoretical predictions on the chain and ring concentrations and their average sizes. Here, we show that the non-monotonic ρ-dependence of the fraction of DHSs aggregated in defect-free rings is a signature of a more complex structure forming already at very low concentrations. One of the possible structures, suggested in Ref. [46], is a “Y”shaped structure obtained via amalgamation of three chains. Recently, we systemised defect structures in Ref. [47], where we showed the importance of branched rings as well. Our detailed investigation of ring structures provided in this section brings out the concentration range in which the “ideal” structures start being replaced by more sophisticated topologies. In Section 3 we show two distinct temperature regimes for the ρ-dependence of χ. At rather high and intermediate temperatures, where the majority of particles either remain non-aggregated, or become members of chains, χ monotonically increases with concentration and inverse temperature. In contrast, for low temperatures, at which rings start dominating in the system, χ increases with concentration, but decreases with inverse temperature. The summary and a short outlook are provided in the Conclusion.
2. Chain-to-ring transition To investigate the structural transition in DHS gas, we use a combination of Monte Carlo simulations and Density Functional Theory. Monte Carlo simulations are performed in the canonical ensemble of N ¼ 5000 particles. Implementing the advanced volumebias techniques [48,37] allows us to equilibrate the systems at rather low temperatures. Metallic periodic boundary conditions in combination with Ewald summation are used. In simulations the following criterion was applied: two particles are considered as bonded if their interaction energy is negative and if their relative distance is smaller than 1.3 of particle diameter [38]. For further details on the simulation approach, see Ref. [37]. Here, we distinguish between three types of clusters, namely chains, rings and branched structures. Branched structures are composed by chains and rings connected to each other via the so-called “defects” (particles that have more than two bonded neighbours) [47], which is why chains and rings will be addressed as defectfree clusters in the present paper. In theory, instead, only defect-free clusters are taken into account. The free energy (F) of a DHS gas can be written as a functional of n-particle chain (gn) and ring (fn) volume fractions [49–52]:
F [{gn }, {fn }] Vk B T
∞
=
∑ gn ln n= 1
gn v eQ n
∞
+
∑ fn ln n= 5
fn v eWn
,
numerical results [53,37,38], we assume that rings smaller than five particles do not form. In our calculations we assume chains to be flexible and rings to have an ideal circular shape. The minimisation of the free-energy functional (Eq. (1)) with respect to the distributions {gn } and {fn } preserving the concentration ρ = 6Nv/πV : ∞
∞
∑ gn n + ∑ fn n = n= 1
n= 5
π ρ , 6v
(2)
leads to the following solution:
gn =
1 Q n pn , v
fn =
1 Wn pn . v
(3)
Here, p, the Lagrange multiplier to be found from Eq. (2), has the meaning of activity. In the present paper we use the dimensionless temperature T ⁎ = 6vk B T /πm2, where m is the particle magnetic moment. This temperature is the inverse dipolar coupling parameter, characterising the energy per particle of two touching DHS with coaligned dipolar moments in units of k B T . We start describing the low-Tn behaviour of DHS gas by plotting the distributions of defect-free chains and rings over their sizes. In this case, it is convenient to use normalised fractions of nparticle clusters:
Cn =
6 nvgn , π ρ
Rn =
6 nvfn . π ρ
(4)
These functions show the fraction of particles belonging to a corresponding cluster of a fixed length. The fraction of single particles is given by C1. Note that for Rn the value of n is larger than five. In Fig. 1 the characteristic cluster distributions are plotted for ρ = 5 × 10−4 at various low temperatures. Here, one can see that for the highest T ⁎ = 0.155 the majority of particles is aggregated in chains. The amount of particle in rings is rather low. This difference levels out with decreasing temperature, and for the lowest T ⁎ = 0.125 we observe an inversion: the population of rings evidently dominates. This crossover shows the presence of a structural transition from the system with mainly chain clusters to that of rings. It is worth mentioning that the position of Cn maximum almost does not depend on Tn and is noticeably closer to unity, than the one for Rn. The latter strongly depends on temperature and shifts towards the larger rings with decreasing Tn. Both distributions become wider on cooling, which is the
(1)
where V stands for the system volume, kBT is the thermal energy, v has a meaning of the particle volume. Qn and Wn denote the chain and ring partition functions respectively, whose calculations have been performed in Ref. [40]. Following the results of Ref. [44] and
Fig. 1. Distributions Cn and Rn from Eq. (4) at fixed ρ = 5 × 10−4 . Solid lines show the fraction of particles in chains, dashed lines describe those in rings. Dimensionless temperatures are given in the legend. For the sake of clarity we plot here only theoretical predictions that are in a good agreement with the simulation data.
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Fig. 2. Concentration dependence of the ring portion νr , see Eq. (5). Theoretical predictions are shown with solid lines; simulation data are plotted with symbols. Theoretical lines are not extended to the whole concentration range due to the reasons explained in the text. Dimensionless temperatures are given in the legend.
signature of the growing average cluster size. It is important that such a crossover is observed in a broad range of ρ ∼ 10−5 − 10−3 for 0.125 ≤ T ⁎ ≤ 0.155. For lower densities, the crossover is observed at temperatures lower than T ⁎ = 0.125, whereas for higher densities, the structure of the DHS becomes more complex, as it will be shown below. The characteristics of chain-like aggregates have been exhaustively studied in the last decades ([21–24,26–40], and references therein), which is why, here, we will focus on the description of rings. In Fig. 2 we show the portion of rings νr defined as follows:
νr =
∑∞ n = 5 fn ∞ ∑∞ n = 1 gn + ∑n = 5 fn
.
Fig. 3. Concentration dependence of the branched cluster portion (not defect-free chains and rings). Only simulation data are presented. Dimensionless temperatures are given in the legend.
[54,55]: χL = 4πρ /3T ⁎. Once the temperature decreases T ⁎ ∼ 0.5 − 1, the interparticle magnetic correlations begin to play an important part in the system collective response to a weak applied magnetic field. Several approaches to deal with this phenomena have been put forward [56–60,31,61,62]. For diluted DHS gas, all of these models give the same asymptotic behaviour: χd = χL (1 + χL /3). With increasing interparticle interaction T ⁎ ≤ 0.3 the chain formation occurs and contributes noticeably to the initial susceptibility, since the flexible chains exhibit a correlated reaction to an external magnetic field. In this case χ is derived in the following form [63]:
χc = (5)
In other words, the total portion of clusters is a sum of defect-free chains and rings only. At the same time, in Monte Carlo simulations this portion is calculated explicitly by summing up all possible aggregates. Theory and simulations agree well only up to ρ ∼ 10−3. For larger concentrations our “idealised” theoretical approach (Eqs. (1)–(3)) predicts further growth of νr , whereas in simulations a clear maximum is observed. The reason for the maximum of the ring portion can be easily understood when looking at Fig. 3, where the analogous portion but of branched structures (νb ) obtained in simulations is plotted versus ρ. The point is that starting from concentration of ∼10−3 almost independently from temperature the portion of branched structures starts growing rapidly and basically reaches unity for ρ ≳ 0.1. In other words, at low Tn, being rather seldom in very diluted DHS gas, branched structures become basically the only possible aggregates when ρ exceeds one per cent, and create a percolating network for concentrations ten percent and higher [38,47]. Since our theory in its present form is able to capture only one of the two low-T structural transitions in DHS gas, namely that from defect-free chains to defect-free rings and not the one to branched structures, below we focus on magnetic properties of the systems with ρ not exceeding 1 per cent.
3. Initial magnetic susceptibility The initial susceptibility of DHSs has been studied for more than 100 years. At very high temperatures, when the interparticle interactions are negligible, χ is described by the Langevin law
3
χd ρ
∞
∑ gn 〈mn2 〉, n= 1
〈mn2 〉 = n + 2
(6)
⎛ ⎞ K ⎜n − 1 + K n − nK ⎟. 2 ⎠ (1 − K ) ⎝
Here, the final sum contains the chain contribution, with 〈mn2 〉 having the meaning of a dimensionless mean-squared dipole moment of a chain made of n DHS particles, and the expression for 〈mn2 〉 has been derived in Ref. [63]. The dipolar correlation coefficient K assumes the simple form: K = coth(1/2T ⁎) − 2T ⁎. In the present paper, we use Eq. (6) to describe the initial magnetic susceptibility of the DHS gas at low temperatures. One can do that because the total dipole moment of a ring is zero, and a weak magnetic field is not able to perturb such an energetically advantageous structure. In other words, we assume that the particles aggregated in rings are simply excluded from the contribution to χ. To do it in theory, we minimise the free energy functional from Eq. (1), Section 2 for fixed ρ and Tn. As a result, we obtain the concentration ρc of particles aggregated in chains, and the chain distribution gn. We plug ρc in Eq. (6) instead of the total concentration ρ and evaluate the susceptibility χ. In simulations, the initial magnetic susceptibility is calculated using fluctuationdissipation theorem [40], and as a consequence, the contribution of each particle, regardless of its membership of any aggregate class, is taken into account. In Fig. 4 the concentration dependence of χ is shown for different Tn. We distinguish between two temperature ranges. In Fig. 4(a) the initial susceptibility monotonically grows with both increasing concentration and decreasing temperature. For the highest values of T ⁎ = 1 and 0.5 this effect is simply described by the functional ρ- and Tn-dependences of χd. The formation of
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4. Conclusion
Fig. 4. Concentration dependence of the DHS initial magnetic susceptibility. (a) High temperature regime. (b) Low temperature regime. In both figures simulation results are plotted with symbols; solid lines represent theoretical predictions. Dimensionless temperatures are given in the legends.
chains, which takes place at T ⁎ = 0.25 and is stimulated by the growth of ρ, does not physically change the matter. The following decrease of temperature, however, leads to a drastic change of χ behaviour, as it is shown in Fig. 4(b). Even though, in the temperature range, corresponding to the chain to ring structural transition (see, Fig. 1, Section 2), the initial susceptibility keeps growing with increasing total concentration ρ, but the inversion of Tn dependence is observed. As shown in Fig. 3, Section 2, the amount of rings grows with density and inverse temperature; it leads to an effective exclusion of growing amount of particles (ring members) from the collective magnetic response to a weak external magnetic field. Hence, for T ⁎ = 0.125 one finds the curve for χ versus ρ to be the lowest. This effect, equally observed in simulations and confirmed by our simple theoretical approach, is rather unexpected, as the magnetic correlations only keep increasing with the increase of concentration and decrease of temperature. On the other hand, for higher densities at low Tn one observes another structural transition (see Fig. 3, Section 2) accompanied by the formation of branched structures and even percolating network, which at the moment cannot be described theoretically.
In this contribution we analyse in detail the behaviour of diluted DHS gas at low temperature. We focus on the influence of concentration on the possible structural transitions in the system and their influence on the collective magnetic response to a weak applied external magnetic field. We show that at low temperature and very low concentration, the chain to ring structural transition takes place. This transition can be stimulated by growing concentration only up to the values of ρ in the order of one per cent. For higher concentrations, the chain to ring transition is first smeared out and then completely replaced by the appearance of new type of aggregates, i.e. branched structures. For the first low-concentration transition we develop a theoretical model that appears to be in a good agreement with simulation data. Our theory is based on the assumption that chains are flexible, whereas the rings are ideally circular, and considers the entropy loss for the ring formation. From this we conclude that the energy of a flexible “breathing” ring does not differ much from that of an ideal ring in the range of parameters addressed here. In terms of magnetic response, both an ideal and a flexible ring have a zero total dipole moment. This allows our theory to describe accurately the initial susceptibility of the diluted DHS gas in a broad range of temperature and concentrations, by using an assumption that particles aggregated in rings simply do not react to a weak applied magnetic field. We show that an increasing fraction of rings result in a significant decrease of the initial susceptibility. The structural transition to the branched structures deserves a special separate study, as it, on one hand, leads to a formation of the percolating network, thus, promising the existence of a new magnetic “polymeric material” kept together by exclusively magnetic forces. On the other hand, this transition sheds light on the long-lasting debate concerning the possibility of the gas–liquid transition in DHSs. In addition, the presence of branched structures is also expected in real magnetic fluids at low temperatures, and will necessarily change their viscous, optic and magnetic properties.
Acknowledgments The research has been partially supported by Austrian Science Fund (FWF): START-Projekt Y 627-N27. J.M.T. acknowledges financial support from the Portuguese Foundation for Science and Technology under Contract no. EXCL/FIS-NAN/0083/2012. A.O.I. and S.S.K. are supported by the Ministry of Education and Science of the Russian Federation (Contract 02.A03.21.000, Project 3.2.2014/K).
References [1] A. Lubbe, C. Alexiou, C. Bergemann, Clinical applications of magnetic drug targeting, J. Surg. Res. 95 (2001) 200–206(7). [2] J. Dobson, Magnetic micro- and nano-particle-based targeting for drug and gene delivery, Nanomedicine 1 (1) (2006) 31–37. [3] M.E. Hayden, U.O. Häfeli, Magnetic bandages for targeted delivery of therapeutic agents, J. Phys.: Condens. Matter 18 (38) (2006) S2877. [4] J. Durán, J. Arias, V. Gallardo, A. Delgado, Magnetic colloids as drug vehicles, J. Pharm. Sci. 97 (8) (2008) 2948–2983. [5] S. Hughes, S. McBain, J. Dobson, A.J. El Haj, Selective activation of mechanosensitive ion channels using magnetic particles, J. R. Soc. Interface 5 (25) (2008) 855–863. [6] Q.A. Pankhurst, N.T.K. Thanh, S.K. Jones, J. Dobson, Progress in applications of magnetic nanoparticles in biomedicine, J. Phys. D: Appl. Phys. 42 (22) (2009) 224001.
Please cite this article as: A.O. Ivanov, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.10.013i
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1 2 3 4 5 6 7 8 9 Q5 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
[7] C.S. Brazel, Magnetothermally-responsive nanomaterials: Combining magnetic nanostructures and thermally-sensitive polymers for triggered drug release, Pharm. Res. 26 (3) (2009) 644–656. [8] E. Pollert, G. Goglio, S. Mornet, E. Duguet, Magnetic nanoparticles for magnetic resonance imaging and hyperthermia applications, in: R. Brayner, F. Fiévet, T. Coradin (Eds.), Nanomaterials: A Danger or a Promise? Springer, London, 2013, pp. 99–129. [9] D. Ortega, N. Perez, J.L. Vilas, J.S. Garitaonandia, K. Suzuki, J.R. Marin, M. Rodriguez, Nonylphenol polyethoxylate coated body-center-cubic iron nanocrystals for ferrofluids with technical applications, J. Appl. Phys. 113 (17) (2013) 17B505. [10] L.R. Jr., R.E. Rosensweig, Magnetocaloric power, J. AIAA 2 (1964) 1418–1422. [11] Y.N. Skibin, V.V. Chekanov, Y.L. Raikher, Birefringence in a ferromagnetic liquid, J. Exp. Theor. Phys. 45 (3) (1977) 496–499. [12] P.C. Scholten, The origin of magnetic birefringence and dichroism in magnetic fluids, IEEE Trans. Magn. 16 (1980) 221–225. [13] S. Taketomi, Magnetic fluids anomalous pseudo-cotton mouton effects about 107 larger that that of nitrobenzene, Jpn. J. Appl. Phys. 22 (1983) 1137–1143. [14] V. Buzmakov, A. Pshenichnikov, On the structure of microaggregates in magnetite colloids, J. Colloid Interface Sci. 182 (1996) 63–70. [15] S. Odenbach, H. Gilly, Taylor vortex flow of magnetic fluids under the influence of an azimuthal magnetic field, J. Magn. Magn. Mater. 152 (1996) 123. [16] S. Odenbach, Magnetoviscous Effects in Ferrofluids, Lecture Notes in Physics, vol. 71, Springer, Berlin, Heidelberg, 2002. [17] E. Dubois, R. Perzynski, F. BouÃ, V. Cabuil, Liquid–gas transitions in charged colloidal dispersions: a small-angle neutron scattering coupled with phase diagrams of magnetic fluids, Langmuir 16 (13) (2000) 5617–5625. [18] L. Pop, S. Odenbach, Investigation of the microscopic reason for the magnetoviscous effect in ferrofluids studied by small angle neutron scatting, J. Phys.: Condens. Matter 18 (2006) 2758. [19] M. Klokkenburg, B.H. Erné, A. Wiedenmann, A.V. Petukhov, A.P. Philipse, Dipolar structures in magnetite ferrofluids studied with small-angle neutron scattering with and without applied magnetic field, Phys. Rev. E 75 (2007) 051408. [20] D. Bica, L. Vekas, M. Avdeev, O. Marinica, V. Socoliuc, M. Balasoiu, V. Garamus, Sterically stabilized water based magnetic fluids: synthesis, structure and properties, J. Magn. Magn. Mater. 311 (2007) 17. [21] P. de Gennes, P. Pincus, Pair correlations in a ferromagnetic colloid, Z. Phys. B Condens. Matter 11 (1970) 189–198. [22] A.Y. Zubarev, L.Y. Iskakova, Theory of physical properties of magnetic liquids with chain aggregates, J. Exp. Theor. Phys. 80 (1995) 857. [23] M.A. Osipov, P.I.C. Teixeira, M.M. TelodaGama, Structure of strongly dipolar fluids at low densities, Phys. Rev. E 54 (3) (1996) 2597. [24] S. Kantorovich, A. Ivanov, Formation of chain aggregates in magnetic fluids: an influence of polydispersity, J. Magn. Magn. Mater. 252 (2002) 244–246. [25] K. Morozov, M. Shliomis, Magnetic fluid as an assembly of flexible chains, in: S. Odenbach (Ed.), Ferrofluids: Magnetically Controllable Fluids and their Applications, Lecture Notes in Physics, vol. 594, 2002, pp. 162–184. [26] A.O. Ivanov, S.S. Kantorovich, V.S. Mendelev, E.S. Pyanzina, Ferrofluid aggregation in chains under the influence of a magnetic field, J. Magn. Magn. Mater. 300 (2006) 206–209. [27] G.N. Patey, D. Levesque, J.J. Weis, On the theory and computer simulations of dipolar fluids, Mol. Phys. 38 (1979) 219. [28] J.J. Weis, D. Levesque, Chain formation in low density dipolar hard shperes: a Monte Carlo study, Phys. Rev. Lett. 71 (17) (1993) 2729–2732. [29] P.J. Camp, G.N. Patey, Structure and scattering in colloidal ferrofluids, Phys. Rev. E 62 (4) (2000) 5403–5408. [30] Z. Wang, C. Holm, Structure and magnetization properties of polydispersed ferrofluids: a molecular dynamics study, Phys. Rev. E 68 (2003) 041401. [31] T. Kristóf, I. Szalai, Magnetic properties and structure of polydisperse ferrofluid models, Phys. Rev. E 68 (4) (2003) 041109. [32] P.D. Duncan, P.J. Camp, Structure and dynamics in a monolayer of dipolar spheres, J. Chem. Phys. 121 (22) (2004) 11322–11331. [33] P. Ilg, E. Coquelle, S. Hess, Structure and rheology of ferrofluids: simulation results and kinetic models, J. Phys.: Condens. Matter 18 (38) (2006) S2757. [34] L. Luo, S.H.L. Klapp, Fluctuations in a ferrofluid monolayer: an integral equation study, J. Chem. Phys. 131 (3) (2009) 034709.
5
[35] H. Schmidle, S.H.L. Klapp, Phase transitions of two-dimensional dipolar fluids in external fields, J. Chem. Phys. 134 (11) (2011) 114903. [36] J. Richardi, J.-J. Weis, Low density mesostructures of confined dipolar particles in an external field, J. Chem. Phys. 135 (12) (2011) 124502. [37] L. Rovigatti, J. Russo, F. Sciortino, No evidence of gas–liquid coexistence in dipolar hard spheres, Phys. Rev. Lett. 107 (2011) 237801. [38] L. Rovigatti, J. Russo, F. Sciortino, Structural properties of the dipolar hardsphere fluid at low temperatures and densities, Soft Matter 8 (2012) 6310–6319. [39] J. Richardi, J.-J. Weis, Influence of short range potential on field induced chain aggregation in low density dipolar particles, J. Chem. Phys. 138 (24) (2013) 244704. [40] 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. [41] P.J. Camp, J.C. Shelley, G.N. Patey, Isotropic fluid phases of dipolar hard spheres, Phys. Rev. Lett. 84 (1) (2000) 115–118. [42] P.D. Duncan, P.J. Camp, Aggregation kinetics and the nature of phase separation in two-dimensional dipolar fluids, Phys. Rev. Lett. 97 (2006) 107202. [43] G. Ganzenmuller, P.J. Camp, Vapor–liquid coexistence in fluids of charged hard dumbbells, J. Chem. Phys. 126 (19) (2007) 191104. [44] T. Prokopieva, V. Danilov, S. Kantorovich, C. Holm, Ground state structures in ferrofluid monolayers, Phys. Rev. E 80 (2009) 031404. [45] 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. [46] T. Tlusty, S.A. Safran, Defect-induced phase separation in dipolar fluids, Science 290 (5495) (2000) 1328–1331. [47] L. Rovigatti, S. Kantorovich, A.O. Ivanov, J.M. Tavares, F. Sciortino, Branching points in the low-temperature dipolar hard sphere fluid, J. Chem. Phys. 139 (13) (2013) 134901. [48] B. Chen, J.I. Siepmann, Improving the efficiency of the aggregation volume biasing Monte Carlo algorithm, J. Phys. Chem. B 105 (45) (2001) 11275–11282. [49] R.P. Sear, G. Jackson, Thermodynamic perturbation theory for association into chains and rings, Phys. Rev. E 50 (1994) 386–394. [50] A. Avlund, G. Kontogeorgis, W. Chapman, Mol. Phys. 109 (2011) 1759. [51] A. Galindo, S. Burton, G. Jackson, D. Visco, D.A. Kofke, Mol. Phys. 100 (2002) 2241. [52] J.M. Tavares, L. Rovigatti, F. Sciortino, Quantitative description of the selfassembly of patchy particles into chains and rings, J. Chem. Phys. 137 (4) (2012) 044901. [53] S. Kantorovich, J.J. Cerdà, C. Holm, Microstructure analysis of monodisperse ferrofluid monolayers: theory and simulation, Phys. Chem. Chem. Phys. 10 (14) (2008) 1883–1895. [54] P. Langevin, Sur la theory du magnetism, J. Phys. 4 (1905) 678. [55] P. Langevin, Magnetism et theory des electrons, Ann. Chim. Phys. 5 (1905) 70. [56] K.I. Morozov, A.F. Pshenichnikov, Y.L. Raikher, M.I. Shliomis, Magnetic properties of ferrocolloids: the effect of interparticle interactions, J. Magn. Magn. Mater. 65 (1987) 269–272. [57] M. Shliomis, A. Pshenichnikov, K. Morozov, I. Shurubor, Magnetic properties of ferrocolloids, J. Magn. Magn. Mater. 85 (1–3) (1990) 40–46. [58] A.O. Ivanov, O.B. Kuznetsova, Magnetic properties of dense ferrofluids: an influence of interparticle correlations, Phys. Rev. E 64 (2001) 23, pp. 041405–1. [59] B. Huke, M. Lücke, Magnetization of ferrofluids with dipolar interactions: a Born–Mayer expansion, Phys. Rev. E 62 (2000) 6875–6890. [60] B. Huke, M. Lücke, Magnetization of concentrated polydisperse ferrofluids: cluster expansion, Phys. Rev. E 67 (2003) 051403. [61] K.I. Morozov, The dielectric virial expansion and the models of dipolar hardsphere fluid, J. Chem. Phys. 19 (2007) 126. [62] I. Szalai, S. Nagy, S. Dietrich, Comparison between theory and simulations for the magnetization and the susceptibility of polydisperse ferrofluids, J. Phys.: Condens. Matter 25 (46) (2013) 465108. [63] V.S. Mendelev, A.O. Ivanov, Ferrofluid aggregation in chains under the influence of a magnetic field, Phys. Rev. E 70 (2004) 051502–051510.
Please cite this article as: A.O. Ivanov, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j. jmmm.2014.10.013i
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