- Email: [email protected]
Structural changes in microferrogels cross-linked by magnetically anisotropic particles
Journal of Magnetism and Magnetic Materials 431 (2017) 192–195
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Structural changes in microferrogels cross-linked by magnetically anisotropic particles
MARK
⁎
A.V. Ryzhkova,b, , Yu. L. Raikherb a b
Perm National Research Polytechnic University, Perm 614990, Russia Institute of Continuous Media Mechanics, Russian Academy of Sciences, Ural Branch, Perm 614013, Russia
A R T I C L E I N F O
A BS T RAC T
Keywords: Ferrogel Molecular dynamics Magnetic nanoparticles Magnetic anisotropy Radial distribution function
Chaining of magnetic nanoparticles in a microscopic ferrogel (MFG) due to interparticle interaction and external field is analyzed by the coarse-grained molecular dynamics. The embedded nanoparticles, unlike existing conventional models, are assumed to possess uniaxial magnetic anisotropy. By that, the consideration is brought closer to reality. Evolution of particle chains, both in length and straightness, is handled with the aid of “axial” radial distribution function that is sensitive to orientation of the aggregates. The effect of the particle magnetic anisotropy on the structural alterations as well as on volume changes of MFGs is demonstrated.
1. Introduction The goal of this work is to analyze the structural changes occurring in a ferrogel object in response to an applied magnetic field. We consider a gel sample of microscopic size where the role of cross-linkers is given to properly functionalized magnetic nanoparticles, such entities are known as microferrogels (MFGs) [1,2]. MFGs are considered and tested as remotely controlled containers for drug delivery and release [3–5] and as devices for hypethermia cancer treatment [6,7]. The MFGs have dimensions from hundreds to thousands nanometers and contain several hundreds of magnetic nanoparticles. When an MFG sample is magnetized, its structure changes drastically under the action of interparticle magnetic forces. Cluster and chain formation along with the overall volume changes take place. Evidently, those effects, strongly affecting the magnetomechanics of MFGs, are important to be accounted considering the potential applications. The absolute majority of theoretical studies of MFGs, when describing the magnetic behavior of nanoparticles, employ the two most simple schemes. The particles are assumed to be either perfectly magnetically soft [8,9] or infinitely magnetically hard [10,11]. The first approximation implies that the magnetic moment is completely free to rotate inside the particle, while the second one treats the magnetic moment as “frozen-in”, so that it rotates only together with the particle body. However, to the typical MFG fillers, viz. nanodisperse magnetite [12] and cobalt ferrite [13,14], both assumptions apply quite poorly. In order to overcome this drawback, here we study an MFG filled with the particles which possess uniaxial magnetic anisotropy of finite intensity. ⁎
This means that the particle magnetic moment is able to turn inside the particle but this rotation is hindered substantially by the presence of the magnetic anisotropy barrier. We show that depending on the parameter of magnetic anisotropy, an MFG might considerably vary its response to the applied field. In what follows we, first, describe in brief the numerical model and the method of structure analysis. The results of simulations obtained in that way and their analysis are presented followed by some conclusions. 2. Model 2.1. Coarse-grained simulation We use many-particle coarse-grained molecular dynamics (MD) model that imitates a micro-object whose initial structure is that of a simple cubic lattice. A certain fraction of the junctions (nodes) of this lattice are made of magnetic nanoparticles of identical size, the other part of the nodes is nonmagnetic. In the given lattice topology (see Fig. 1A), to each internal magnetic particle six polymer strands are attached. The attachment points are fixed at the particle surface, and due to that the translational and rotational displacements of the particle and the polymer strands affect each other. All the internode polymer strands of the model MFG comprise equal number of building blocks (blobs) which in the parlance of coarse-grained molecular dynamics are called “monomers”. Any monomer (except for those which occupy the lattice nodes) is linked to two of its nearest neighbors by a center-to-center harmonic potential and also is subjected to purely repulsive interaction – Weeks–Chandler–
Corresponding author at: Institute of Continuous Media Mechanics, Russian Academy of Sciences, Ural Branch, Perm 614013, Russia. E-mail address: [email protected] (A.V. Ryzhkov).
http://dx.doi.org/10.1016/j.jmmm.2016.09.056 Received 1 July 2016; Accepted 11 September 2016 Available online 19 September 2016 0304-8853/ © 2016 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 431 (2017) 192–195
A.V. Ryzhkov, Y.L. Raikher
Fig. 1. (A) Pre-initial configuration of a simple cubic lattice network with embedded magnetic particles. The insets show an internal node made of a nanoparticle and the sketch of the internal particle structure, here μ is the magnetic moment and n is the easy magnetization axis. (B) Thermalized configuration of an MFG put inside Langevin thermostat for the case where the Zeeman and dipole–dipole magnetic interactions are absent. (C) The same but the interparticle magnetic interaction is “switched on”. (D) The configuration that evolves from that of (C) under external magnetic field H of the magnitude H = 10kB T /μ .
Andersen (WCA) potential [15] – that prevents their overlapping. The energy of a magnetic particle that possesses the dipolar magnetic moment μ , along with the Zeeman term and the all-with-all dipole–dipole interaction, includes also the WCA potential. Besides that, vector μ is coupled inside the particle with the anisotropy (easy magnetization) axis n. All the coarse-grained MD simulations were carried out with the aid of ESPResSo software [16]. First the pre-initial state of fully extended simple cubic lattice is generated. Then the sample is put inside Langevin thermostat for thermalization (Fig. 1B), at the next step the interparticle magnetic interaction is introduced (Fig. 1C), and finally an external uniform magnetic field is applied to the sample (Fig. 1D).
perpendicular to the cone axis. Then one defines RDFA as
RDFA(ν) (r ) =
1 N magnρ
N magn
∑ i
Nisect (rj , rj +1, ν) V sect (rj , rj +1, φ)
(1)
Nisect (rj ,
rj +1, ν) renders the number of particles in the portion of where spherical sector of the angle 2φ that has inner radius rj and outer radius rj+1. The sector axis points along ν and r = (rj + rj+1)/2 . In Eq. (1) N magn is the total number of magnetic nanoparticles and ρ = N magn / Vbox the net number density of nanoparticles in the simulation box Vbox . 3. Results
2.2. Structural analysis The considered MFG sample (Fig. 1) is an isolated cubic lattice (no periodic boundary conditions are applied) with 1000 nodes about 80% of which are occupied by magnetic nanoparticles of diameter dnp. Their magnetic moments have the absolute value μ and randomly distributed directions. Each internode segment of the lattice is a strand 1 of 12 monomers of diameter d m = 3 dnp . The parameter defining the strength of dipolar magnetic interaction of the particles in contact is 3 λ = μ2 /(dnp kB T ) = 16 ; here kB is the Boltzmann constant and T the absolute temperature. The effect of magnetic anisotropy is characterized by the ratio of anisotropy energy barrier EA to thermal energy: σ = EA / kB T . After step-by-step process of MD calculation (from A to D in Fig. 1),
As seen from Figs. 1C and D, upon magnetization, in an MFG the particles self-organize into chain-like structures. To monitor the formation of such anisotropic aggregates, we use an angle-resolved radial distribution function (RDF). Unlike the standard definition of RDF [17], this modification renders the distribution of the particles in a narrow cone whose axis points the preferred direction. The idea of this “axial” radial distribution function (RDFA) is as follows. Two symmetrical spherical sectors with vertex angles 2φ are built (Fig. 2). Their common vertex is positioned on the i-th magnetic particle, while the axes are directed along the anisotropy axis ν . The obtained spherical sectors are divided into thin layers which are 193
Journal of Magnetism and Magnetic Materials 431 (2017) 192–195
A.V. Ryzhkov, Y.L. Raikher
Fig. 2. Scheme of the space “probed” by RDFA.
the set of final configurations is obtained. To obtain statistical relevance, for a given set of governing parameters, we simulate 10 replicas of the sample. Assuming three values of the magnetic anisotropy parameter, σ = 0, 10, ∞, we simulate quasi-static magnetization of the MFG under the applied field normalized as ξ = μH / kB T . Occurrence of nanoparticle aggregates (chains) and their orientation are analyzed with the aid of RDFA. Chain-like structures manifest themselves as high peaks of RDFA (directed along the field) at a distance of about one diameter of nanoparticle. The higher this peak, the more straight the chains are. The examples of RDFA patterns corresponding to sequential steps of magnetization are shown in Fig. 3. These plots clearly demonstrate evolution of the particle aggregates in MFG. Under the given value of λ, the chains do exist yet in the absence of the field. In this state they are inflected in all the directions, and the sample structure is in general isotropic. When the field is applied, it changes the force balance formerly established between the interparticle magnetic forces and the elastic ones induced in the polymer mesh by aggregation of the particles at H = 0 . The field affects the magnetic moments which, in turn, entrain the particles themselves into mechanical displacements. As the field grows, the peak of RDFA in its direction becomes higher while the value of RDFA along the perpendicular direction diminishes. Comparing RDFA for different magnetic anisotropy energies, one observes that with the increase of anisotropy the peak becomes lower, the polar plot becomes less sharp. Evidently, the sample with the maximal (infinite) magnetic anisotropy has the most curved chain structures. The same RDFA-analysis applied to MFGs with the magnetic interparticle interaction as low as λ = 4 (not shown here), the values of peaks would be an order of magnitude lower. Furthermore, in the absence of field the amount of clusterized particles is next to zero. To characterize the degree of the particle aggregation, we use indicator q (m, d ′) that renders the fraction of nanoparticles each of which has at least m neighbors whose centers are located in the spherical layer of radius d′ around the center of the given one. To make q (m, d ′) sensitive to tight chains, we set d ′ = 1.1dnp . Calculating this indicator for the MFG with λ = 16 and σ = 0 under the field ξ = 20 we find q (1, 1.1dnp ) ∼ 1.0 , q (2, 1.1dnp ) ∼ 0.8 and q (3, 1.1dnp ) ∼ 0.0 . This proves
Fig. 3. Polar plots of RDFA at distance r ≈ dnp ; the field is directed along the Y-axis; the interaction parameter is λ = 16 .
194
Journal of Magnetism and Magnetic Materials 431 (2017) 192–195
A.V. Ryzhkov, Y.L. Raikher
volume changes, etc., need a thorough study. Especially this concerns possible biomedical applications. Taking a microsample of a ferrogel as a test object, with a number of examples we show that internal magnetic anisotropy of the particles matters considerably for the type and intensity of the MFG magneto-structural and magneto-configurational behavior. Acknowledgments The work was supported by Grants RFBR-DFG #16-51-12001 and RFBR #14-02-96003. This corresponds to joint project of Russian Foundation for Basic Research (RFBR) and Deutsche Forschungsgemeinschaft (DFG). Fig. 4. Volume change of the MFG under the increase of applied magnetic field ξ for different magnetic anisotropy parameters σ, here for all samples: λ = 16
References [1] Y. Li, G. Huang, X. Zhang, B. Li, Y. Chen, T. Lu, T.J. Lu, F. Xu, Magnetic hydrogels and their potential biomedical applications, Adv. Funct. Mater. 23 (2012) 660–672. http://dx.doi.org/10.1002/adfm.201201708. [2] J. Thévenot, H. Oliveira, O. Sandre, S. Lecommandoux, Magnetic responsive polymer composite materials, Chem. Soc. Rev. 42 (2013) 7099–7116. http:// dx.doi.org/10.1039/c3cs60058k. [3] P.M. Mendoza Zelis, D. Muraca, J.S. Gonzalez, G.A. Pasquevich, V.A. Alvarez, K.R. Pirota, F.H. Sánchez, Magnetic properties study of iron-oxide nanoparticles/ pva ferrogels with potential biomedical applications, J. Nanopart. Res. 15 (2013). http://dx.doi.org/10.1007/s11051-013-1613-6 Art. no. 1613. [4] C.A. Cezar, S.M. Kennedy, M. Mehta, J.C. Weaver, L. Gu, H. Vanderburgh, D.J. Mooney, Biphasic ferrogels for triggered drug and cell delivery, Adv. Healthc. Mater. 3 (11) (2014) 1869–1876. http://dx.doi.org/10.1002/adhm.201400095. [5] S. Campbell, D. Maitland, T. Hoare, Enhanced pulsatile drug release from injectable magnetic hydrogels with embedded thermosensitive microgels, ACS Macro Lett. 4 (3) (2015) 312–316. http://dx.doi.org/10.1021/acsmacrolett.5b00057. [6] J.K. Oh, J.M. Park, Iron oxide-based superparamagnetic polymeric nanomaterials: design, preparation, and biomedical application, Prog. Polym. Sci. 36 (8) (2011) 168–189. http://dx.doi.org/10.1016/j.progpolymsci.2010.08.005. [7] M. Häring, J. Schiller, J. Mayr, S. Grijalvo, R. Eritja, R.D. Díaz, Magnetic gel composites for hyperthermia cancer therapy, Gels 1 (2015) 135–161. http:// dx.doi.org/10.3390/gels1020135. [8] D.S. Wood, P. Camp, Modeling the properties of ferrogels in uniform magnetic fields, Phys. Rev. E 83 (1) (2011). http://dx.doi.org/10.1103/PhysRevE.83.011402 Art. no. 011402. [9] D. Ivaneyko, V. Toshchevikov, M. Saphiannikova, Dynamic moduli of magnetosensitive elastomers: a coarse-grained network model, Soft Matter 11 (2015) 7627–7638. http://dx.doi.org/10.1039/C5SM01761K. [10] P. Cremer, H. Löwen, A.M. Menzel, Tailoring superelasticity of soft magnetic materials, Appl. Phys. Lett. 107 (2015). http://dx.doi.org/10.1063/1.4934698 Art. no. 171903. [11] R. Weeber, S. Kantorovich, C. Holm, Ferrogels cross-linked by magnetic particles: field-driven deformation and elasticity studied using computer simulations, J. Chem. Phys. 143 (2015). http://dx.doi.org/10.1063/1.4932371 Art. no. 154901. [12] S. van Berkum, J.T. Dee, A.P. Philipse, B.H. Erné, Frequency-dependent magnetic susceptibility of magnetite and cobalt ferrite nanoparticles embedded in paa hydrogel, Int. J. Mol. Sci. 14 (2013) 10162–10177. http://dx.doi.org/10.3390/ ijms140510162. [13] R. Barbucci, D. Pasqui, G. Giani, M. De Cagna, M. Fini, R. Giardino, A. Atrei, A novel strategy for engineering hydrogels with ferromagnetic nanoparticles as crosslinkers of the polymer chains. potential applications as a targeted drug delivery system, Soft Matter 7 (2011) 5558–5565. http://dx.doi.org/10.1039/ C1SM05174A. [14] R. Messing, N. Frickel, L. Belkoura, R. Strey, H. Rahn, S. Odenbach, A.M. Schmidt, Cobalt ferrite nanoparticles as multifunctional cross-linkers in paa ferrohydrogels, Macromolecules 44 (8) (2011) 2990–2999. http://dx.doi.org/10.1021/ ma102708b. [15] J.D. Weeks, D. Chandler, H.C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, J. Chem. Phys. 54 (12) (1971) 5237–5247. http://dx.doi.org/10.1063/1.1674820. [16] 〈http://espressomd.org〉. [17] A.V. Ryzhkov, P.V. Melenev, C. Holm, Yu.L. Raikher, Coarse-grained molecular dynamics simulation of small ferrogel objects, J. Magn. Magn. Mater. 383 (2015) 277–280. http://dx.doi.org/10.1016/j.jmmm.2014.11.008.
that almost all the particles are aggregated, and these clusters are not clots but chains. For the MFG with λ = 4 under the same conditions we got q (1, 1.1dnp ) ∼ 0.2 , q (2, 1.1dnp ) ∼ 0.0 and q (3, 1.1dnp ) ∼ 0.0 . This means the rate of aggregation is about 20%, and these aggregates are mostly pairs, no long chains turn up. We note that with the increase of anisotropy the respective values of q decrease whatever λ and ξ. Structure rearrangements, moving the particles jointly with polymer strands, entail the changes of the volume occupied by the MFG sample. The corresponding results of our simulations are presented in Fig. 4 for the MFGs differing by the magnetic anisotropy of the particles. As seen, the smaller the initial (ξ = 0 ) volume of the sample, the greater the particle anisotropy. We infer that, as the chains at greater σ are more curved, see Fig. 3, their neighboring fragments experience lower repulsion. This makes the sample more dense. On turning on the field, in all the cases the MFG volume enhances. The evident explanation is that under magnetization the initially “coiled” structures (Fig. 1C) unwind. The emerging chains point approximately along the field and repel each other in the plane perpendicular to that direction. The more the particles are anisotropic, the greater the field is necessary to straighten the chains. That is why, in Fig. 4, the curves for smaller σ's almost saturate while the curves with σ ≳ ξ keep growing. As the relative volume increment at ξ = 20 is about the same for all the samples (9–12%), this implies that in the long run an MFG with more anisotropic particles would enhance its volume to a larger extent than that with the particles which are more magnetically soft. A possible explanation is as follows. The chains in an MFG with magnetically hard particles are substantially shorter than those in a magnetically softer ferrogel. Shorter chains are more free to move away from each other adjusting the structure in response to the mutual repulsion induced by the chain orientation. In this way we can conclude that uniaxial magnetic anisotropy has considerable influence on the process of structural rearrangements in MFG under the magnetization. It determines the fraction of aggregated particles (and length of induced chain-like aggregates as well), the curvilinearity of chains and as a result it affects on the mechanical response. 4. Conclusions Magnetoactive behavior (the magnetomechanical response to applied magnetic field) of ferrogels is the effect that makes them so interesting functional materials. The multiple manifestations of this effect, viz. magnetization, field-induced structuring and birefringence,
195
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Structural changes in microferrogels cross-linked by magnetically anisotropic particles
MARK
⁎
A.V. Ryzhkova,b, , Yu. L. Raikherb a b
Perm National Research Polytechnic University, Perm 614990, Russia Institute of Continuous Media Mechanics, Russian Academy of Sciences, Ural Branch, Perm 614013, Russia
A R T I C L E I N F O
A BS T RAC T
Keywords: Ferrogel Molecular dynamics Magnetic nanoparticles Magnetic anisotropy Radial distribution function
Chaining of magnetic nanoparticles in a microscopic ferrogel (MFG) due to interparticle interaction and external field is analyzed by the coarse-grained molecular dynamics. The embedded nanoparticles, unlike existing conventional models, are assumed to possess uniaxial magnetic anisotropy. By that, the consideration is brought closer to reality. Evolution of particle chains, both in length and straightness, is handled with the aid of “axial” radial distribution function that is sensitive to orientation of the aggregates. The effect of the particle magnetic anisotropy on the structural alterations as well as on volume changes of MFGs is demonstrated.
1. Introduction The goal of this work is to analyze the structural changes occurring in a ferrogel object in response to an applied magnetic field. We consider a gel sample of microscopic size where the role of cross-linkers is given to properly functionalized magnetic nanoparticles, such entities are known as microferrogels (MFGs) [1,2]. MFGs are considered and tested as remotely controlled containers for drug delivery and release [3–5] and as devices for hypethermia cancer treatment [6,7]. The MFGs have dimensions from hundreds to thousands nanometers and contain several hundreds of magnetic nanoparticles. When an MFG sample is magnetized, its structure changes drastically under the action of interparticle magnetic forces. Cluster and chain formation along with the overall volume changes take place. Evidently, those effects, strongly affecting the magnetomechanics of MFGs, are important to be accounted considering the potential applications. The absolute majority of theoretical studies of MFGs, when describing the magnetic behavior of nanoparticles, employ the two most simple schemes. The particles are assumed to be either perfectly magnetically soft [8,9] or infinitely magnetically hard [10,11]. The first approximation implies that the magnetic moment is completely free to rotate inside the particle, while the second one treats the magnetic moment as “frozen-in”, so that it rotates only together with the particle body. However, to the typical MFG fillers, viz. nanodisperse magnetite [12] and cobalt ferrite [13,14], both assumptions apply quite poorly. In order to overcome this drawback, here we study an MFG filled with the particles which possess uniaxial magnetic anisotropy of finite intensity. ⁎
This means that the particle magnetic moment is able to turn inside the particle but this rotation is hindered substantially by the presence of the magnetic anisotropy barrier. We show that depending on the parameter of magnetic anisotropy, an MFG might considerably vary its response to the applied field. In what follows we, first, describe in brief the numerical model and the method of structure analysis. The results of simulations obtained in that way and their analysis are presented followed by some conclusions. 2. Model 2.1. Coarse-grained simulation We use many-particle coarse-grained molecular dynamics (MD) model that imitates a micro-object whose initial structure is that of a simple cubic lattice. A certain fraction of the junctions (nodes) of this lattice are made of magnetic nanoparticles of identical size, the other part of the nodes is nonmagnetic. In the given lattice topology (see Fig. 1A), to each internal magnetic particle six polymer strands are attached. The attachment points are fixed at the particle surface, and due to that the translational and rotational displacements of the particle and the polymer strands affect each other. All the internode polymer strands of the model MFG comprise equal number of building blocks (blobs) which in the parlance of coarse-grained molecular dynamics are called “monomers”. Any monomer (except for those which occupy the lattice nodes) is linked to two of its nearest neighbors by a center-to-center harmonic potential and also is subjected to purely repulsive interaction – Weeks–Chandler–
Corresponding author at: Institute of Continuous Media Mechanics, Russian Academy of Sciences, Ural Branch, Perm 614013, Russia. E-mail address: [email protected] (A.V. Ryzhkov).
http://dx.doi.org/10.1016/j.jmmm.2016.09.056 Received 1 July 2016; Accepted 11 September 2016 Available online 19 September 2016 0304-8853/ © 2016 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 431 (2017) 192–195
A.V. Ryzhkov, Y.L. Raikher
Fig. 1. (A) Pre-initial configuration of a simple cubic lattice network with embedded magnetic particles. The insets show an internal node made of a nanoparticle and the sketch of the internal particle structure, here μ is the magnetic moment and n is the easy magnetization axis. (B) Thermalized configuration of an MFG put inside Langevin thermostat for the case where the Zeeman and dipole–dipole magnetic interactions are absent. (C) The same but the interparticle magnetic interaction is “switched on”. (D) The configuration that evolves from that of (C) under external magnetic field H of the magnitude H = 10kB T /μ .
Andersen (WCA) potential [15] – that prevents their overlapping. The energy of a magnetic particle that possesses the dipolar magnetic moment μ , along with the Zeeman term and the all-with-all dipole–dipole interaction, includes also the WCA potential. Besides that, vector μ is coupled inside the particle with the anisotropy (easy magnetization) axis n. All the coarse-grained MD simulations were carried out with the aid of ESPResSo software [16]. First the pre-initial state of fully extended simple cubic lattice is generated. Then the sample is put inside Langevin thermostat for thermalization (Fig. 1B), at the next step the interparticle magnetic interaction is introduced (Fig. 1C), and finally an external uniform magnetic field is applied to the sample (Fig. 1D).
perpendicular to the cone axis. Then one defines RDFA as
RDFA(ν) (r ) =
1 N magnρ
N magn
∑ i
Nisect (rj , rj +1, ν) V sect (rj , rj +1, φ)
(1)
Nisect (rj ,
rj +1, ν) renders the number of particles in the portion of where spherical sector of the angle 2φ that has inner radius rj and outer radius rj+1. The sector axis points along ν and r = (rj + rj+1)/2 . In Eq. (1) N magn is the total number of magnetic nanoparticles and ρ = N magn / Vbox the net number density of nanoparticles in the simulation box Vbox . 3. Results
2.2. Structural analysis The considered MFG sample (Fig. 1) is an isolated cubic lattice (no periodic boundary conditions are applied) with 1000 nodes about 80% of which are occupied by magnetic nanoparticles of diameter dnp. Their magnetic moments have the absolute value μ and randomly distributed directions. Each internode segment of the lattice is a strand 1 of 12 monomers of diameter d m = 3 dnp . The parameter defining the strength of dipolar magnetic interaction of the particles in contact is 3 λ = μ2 /(dnp kB T ) = 16 ; here kB is the Boltzmann constant and T the absolute temperature. The effect of magnetic anisotropy is characterized by the ratio of anisotropy energy barrier EA to thermal energy: σ = EA / kB T . After step-by-step process of MD calculation (from A to D in Fig. 1),
As seen from Figs. 1C and D, upon magnetization, in an MFG the particles self-organize into chain-like structures. To monitor the formation of such anisotropic aggregates, we use an angle-resolved radial distribution function (RDF). Unlike the standard definition of RDF [17], this modification renders the distribution of the particles in a narrow cone whose axis points the preferred direction. The idea of this “axial” radial distribution function (RDFA) is as follows. Two symmetrical spherical sectors with vertex angles 2φ are built (Fig. 2). Their common vertex is positioned on the i-th magnetic particle, while the axes are directed along the anisotropy axis ν . The obtained spherical sectors are divided into thin layers which are 193
Journal of Magnetism and Magnetic Materials 431 (2017) 192–195
A.V. Ryzhkov, Y.L. Raikher
Fig. 2. Scheme of the space “probed” by RDFA.
the set of final configurations is obtained. To obtain statistical relevance, for a given set of governing parameters, we simulate 10 replicas of the sample. Assuming three values of the magnetic anisotropy parameter, σ = 0, 10, ∞, we simulate quasi-static magnetization of the MFG under the applied field normalized as ξ = μH / kB T . Occurrence of nanoparticle aggregates (chains) and their orientation are analyzed with the aid of RDFA. Chain-like structures manifest themselves as high peaks of RDFA (directed along the field) at a distance of about one diameter of nanoparticle. The higher this peak, the more straight the chains are. The examples of RDFA patterns corresponding to sequential steps of magnetization are shown in Fig. 3. These plots clearly demonstrate evolution of the particle aggregates in MFG. Under the given value of λ, the chains do exist yet in the absence of the field. In this state they are inflected in all the directions, and the sample structure is in general isotropic. When the field is applied, it changes the force balance formerly established between the interparticle magnetic forces and the elastic ones induced in the polymer mesh by aggregation of the particles at H = 0 . The field affects the magnetic moments which, in turn, entrain the particles themselves into mechanical displacements. As the field grows, the peak of RDFA in its direction becomes higher while the value of RDFA along the perpendicular direction diminishes. Comparing RDFA for different magnetic anisotropy energies, one observes that with the increase of anisotropy the peak becomes lower, the polar plot becomes less sharp. Evidently, the sample with the maximal (infinite) magnetic anisotropy has the most curved chain structures. The same RDFA-analysis applied to MFGs with the magnetic interparticle interaction as low as λ = 4 (not shown here), the values of peaks would be an order of magnitude lower. Furthermore, in the absence of field the amount of clusterized particles is next to zero. To characterize the degree of the particle aggregation, we use indicator q (m, d ′) that renders the fraction of nanoparticles each of which has at least m neighbors whose centers are located in the spherical layer of radius d′ around the center of the given one. To make q (m, d ′) sensitive to tight chains, we set d ′ = 1.1dnp . Calculating this indicator for the MFG with λ = 16 and σ = 0 under the field ξ = 20 we find q (1, 1.1dnp ) ∼ 1.0 , q (2, 1.1dnp ) ∼ 0.8 and q (3, 1.1dnp ) ∼ 0.0 . This proves
Fig. 3. Polar plots of RDFA at distance r ≈ dnp ; the field is directed along the Y-axis; the interaction parameter is λ = 16 .
194
Journal of Magnetism and Magnetic Materials 431 (2017) 192–195
A.V. Ryzhkov, Y.L. Raikher
volume changes, etc., need a thorough study. Especially this concerns possible biomedical applications. Taking a microsample of a ferrogel as a test object, with a number of examples we show that internal magnetic anisotropy of the particles matters considerably for the type and intensity of the MFG magneto-structural and magneto-configurational behavior. Acknowledgments The work was supported by Grants RFBR-DFG #16-51-12001 and RFBR #14-02-96003. This corresponds to joint project of Russian Foundation for Basic Research (RFBR) and Deutsche Forschungsgemeinschaft (DFG). Fig. 4. Volume change of the MFG under the increase of applied magnetic field ξ for different magnetic anisotropy parameters σ, here for all samples: λ = 16
References [1] Y. Li, G. Huang, X. Zhang, B. Li, Y. Chen, T. Lu, T.J. Lu, F. Xu, Magnetic hydrogels and their potential biomedical applications, Adv. Funct. Mater. 23 (2012) 660–672. http://dx.doi.org/10.1002/adfm.201201708. [2] J. Thévenot, H. Oliveira, O. Sandre, S. Lecommandoux, Magnetic responsive polymer composite materials, Chem. Soc. Rev. 42 (2013) 7099–7116. http:// dx.doi.org/10.1039/c3cs60058k. [3] P.M. Mendoza Zelis, D. Muraca, J.S. Gonzalez, G.A. Pasquevich, V.A. Alvarez, K.R. Pirota, F.H. Sánchez, Magnetic properties study of iron-oxide nanoparticles/ pva ferrogels with potential biomedical applications, J. Nanopart. Res. 15 (2013). http://dx.doi.org/10.1007/s11051-013-1613-6 Art. no. 1613. [4] C.A. Cezar, S.M. Kennedy, M. Mehta, J.C. Weaver, L. Gu, H. Vanderburgh, D.J. Mooney, Biphasic ferrogels for triggered drug and cell delivery, Adv. Healthc. Mater. 3 (11) (2014) 1869–1876. http://dx.doi.org/10.1002/adhm.201400095. [5] S. Campbell, D. Maitland, T. Hoare, Enhanced pulsatile drug release from injectable magnetic hydrogels with embedded thermosensitive microgels, ACS Macro Lett. 4 (3) (2015) 312–316. http://dx.doi.org/10.1021/acsmacrolett.5b00057. [6] J.K. Oh, J.M. Park, Iron oxide-based superparamagnetic polymeric nanomaterials: design, preparation, and biomedical application, Prog. Polym. Sci. 36 (8) (2011) 168–189. http://dx.doi.org/10.1016/j.progpolymsci.2010.08.005. [7] M. Häring, J. Schiller, J. Mayr, S. Grijalvo, R. Eritja, R.D. Díaz, Magnetic gel composites for hyperthermia cancer therapy, Gels 1 (2015) 135–161. http:// dx.doi.org/10.3390/gels1020135. [8] D.S. Wood, P. Camp, Modeling the properties of ferrogels in uniform magnetic fields, Phys. Rev. E 83 (1) (2011). http://dx.doi.org/10.1103/PhysRevE.83.011402 Art. no. 011402. [9] D. Ivaneyko, V. Toshchevikov, M. Saphiannikova, Dynamic moduli of magnetosensitive elastomers: a coarse-grained network model, Soft Matter 11 (2015) 7627–7638. http://dx.doi.org/10.1039/C5SM01761K. [10] P. Cremer, H. Löwen, A.M. Menzel, Tailoring superelasticity of soft magnetic materials, Appl. Phys. Lett. 107 (2015). http://dx.doi.org/10.1063/1.4934698 Art. no. 171903. [11] R. Weeber, S. Kantorovich, C. Holm, Ferrogels cross-linked by magnetic particles: field-driven deformation and elasticity studied using computer simulations, J. Chem. Phys. 143 (2015). http://dx.doi.org/10.1063/1.4932371 Art. no. 154901. [12] S. van Berkum, J.T. Dee, A.P. Philipse, B.H. Erné, Frequency-dependent magnetic susceptibility of magnetite and cobalt ferrite nanoparticles embedded in paa hydrogel, Int. J. Mol. Sci. 14 (2013) 10162–10177. http://dx.doi.org/10.3390/ ijms140510162. [13] R. Barbucci, D. Pasqui, G. Giani, M. De Cagna, M. Fini, R. Giardino, A. Atrei, A novel strategy for engineering hydrogels with ferromagnetic nanoparticles as crosslinkers of the polymer chains. potential applications as a targeted drug delivery system, Soft Matter 7 (2011) 5558–5565. http://dx.doi.org/10.1039/ C1SM05174A. [14] R. Messing, N. Frickel, L. Belkoura, R. Strey, H. Rahn, S. Odenbach, A.M. Schmidt, Cobalt ferrite nanoparticles as multifunctional cross-linkers in paa ferrohydrogels, Macromolecules 44 (8) (2011) 2990–2999. http://dx.doi.org/10.1021/ ma102708b. [15] J.D. Weeks, D. Chandler, H.C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, J. Chem. Phys. 54 (12) (1971) 5237–5247. http://dx.doi.org/10.1063/1.1674820. [16] 〈http://espressomd.org〉. [17] A.V. Ryzhkov, P.V. Melenev, C. Holm, Yu.L. Raikher, Coarse-grained molecular dynamics simulation of small ferrogel objects, J. Magn. Magn. Mater. 383 (2015) 277–280. http://dx.doi.org/10.1016/j.jmmm.2014.11.008.
that almost all the particles are aggregated, and these clusters are not clots but chains. For the MFG with λ = 4 under the same conditions we got q (1, 1.1dnp ) ∼ 0.2 , q (2, 1.1dnp ) ∼ 0.0 and q (3, 1.1dnp ) ∼ 0.0 . This means the rate of aggregation is about 20%, and these aggregates are mostly pairs, no long chains turn up. We note that with the increase of anisotropy the respective values of q decrease whatever λ and ξ. Structure rearrangements, moving the particles jointly with polymer strands, entail the changes of the volume occupied by the MFG sample. The corresponding results of our simulations are presented in Fig. 4 for the MFGs differing by the magnetic anisotropy of the particles. As seen, the smaller the initial (ξ = 0 ) volume of the sample, the greater the particle anisotropy. We infer that, as the chains at greater σ are more curved, see Fig. 3, their neighboring fragments experience lower repulsion. This makes the sample more dense. On turning on the field, in all the cases the MFG volume enhances. The evident explanation is that under magnetization the initially “coiled” structures (Fig. 1C) unwind. The emerging chains point approximately along the field and repel each other in the plane perpendicular to that direction. The more the particles are anisotropic, the greater the field is necessary to straighten the chains. That is why, in Fig. 4, the curves for smaller σ's almost saturate while the curves with σ ≳ ξ keep growing. As the relative volume increment at ξ = 20 is about the same for all the samples (9–12%), this implies that in the long run an MFG with more anisotropic particles would enhance its volume to a larger extent than that with the particles which are more magnetically soft. A possible explanation is as follows. The chains in an MFG with magnetically hard particles are substantially shorter than those in a magnetically softer ferrogel. Shorter chains are more free to move away from each other adjusting the structure in response to the mutual repulsion induced by the chain orientation. In this way we can conclude that uniaxial magnetic anisotropy has considerable influence on the process of structural rearrangements in MFG under the magnetization. It determines the fraction of aggregated particles (and length of induced chain-like aggregates as well), the curvilinearity of chains and as a result it affects on the mechanical response. 4. Conclusions Magnetoactive behavior (the magnetomechanical response to applied magnetic field) of ferrogels is the effect that makes them so interesting functional materials. The multiple manifestations of this effect, viz. magnetization, field-induced structuring and birefringence,
195