Balasoiu, M.

Melenev, P. V.

Raikher, Yu L.

Ryzhkov, A. V.

Journal of Physics Conference Series

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

1Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd1234567890 ‘’“”CMSMS17 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 994 (2018) 012004  doi :10.1088/1742-6596/994/1/012004The eﬀect of magnetic nanoparticle concentrationon the structure organisation of a microferrogelA V Ryzhkov1,2, P V Melenev2,3, M Balasoiu4,5, Yu L Raikher1,21 Perm National Research Polytechnic University, Perm, 614990, Russia2 Institute of Continuous Media Mechanics – the Division of Perm Federal Research Center,Russian Academy of Sciences, Ural Branch, Perm, 614013, Russia3 Ural Federal University, Ekaterinburg, 620083, Russia4 Laboratory of Neutron Physics, Joint Institute of Nuclear Research, Dubna, Moscow region141980, Russia5 Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele,RomaniaE-mail: ryzhkov.a@icmm.ruAbstract. Coarse-grained molecular dynamics simulation is applied to study the structuralresponse of micro-sized magnetopolymer objects – microferrogels (MFG). The results for MFGswith diﬀerent magnetic properties and concentrations of magnetic ﬁller nanoparticles areanalysed to detect the transition between non-aggregated conﬁgurations and the states withpronounced chains. The nanoparticles are assumed to be either magnetically isotropic or topossess inﬁnite magnetic anisotropy. It is shown that, depending on the type of the particleanisotropy, an applied ﬁeld in rather diﬀerent ways aﬀects the MFG structure and shape.Diagrams describing the degree of aggregation as a function of the parameter of the interparticlemagnetodipolar interaction and concentration are presented. In particular, it is found that inthe case of inﬁnitely anisotropic nanoparticles the aggregation transitions undergoes via a non-trivial scenario. The eﬀect of the structure transformations on the volume change of the MFGobjects is studied as well.1. IntroductionMicroferrogels (magnetic microgels, microscopic ferrogels, MFGs) are isolated soft polymersobjects with typical sizes in the range of 0.1-10 μm containing embedded magnetic nanoparticles(NPs); typical NP materials are magnetite and cobalt ferrite [1, 2]. Strong coupling betweenthe magnetic NPs and the polymer mesh, in which the NPs take on the role of cross-linkers,ensures the dependence of mechanical and structural properties of a MFG object on the externalmagnetic ﬁeld. Therefore, an inherent property of MFGs is the remote control over theirstructure, volume and shape. Such a possibility of ﬁeld-aided manipulations is very important,for example, for drug delivery, drug release and other biomedical applications [3, 4, 5, 6].As it was shown in Refs. [7, 8], structure (i.e., chain formation), magnetisation and volumechanges due to application of an external uniform magnetic ﬁeld depend on the type ofuniaxial magnetic anisotropy of NPs. However, the above-cited works were focused exclusivelyon highly-ﬁlled samples with strong dipolar magnetic interaction between NPs. Hereby wecontinue molecular dynamics study of magneto-structural and magneto-mechanical responses of21234567890 ‘’“”CMSMS17 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 994 (2018) 012004  doi :10.1088/1742-6596/994/1/012004individual MFGs and analyse the dependence of structural properties of MFGs on concentrationof the magnetic ﬁller particles as well as on the parameter of interparticle interaction.2. ModelWe use two models of nanoparticles which correspond to the limiting cases of their internalmagnetic anisotropy. In the ﬁrst case, it is a particle whose magnetic moment is entirely free torotate inside its body, i.e., the particle has zero anisotropy and, thus, is perfectly magneticallysoft. The other model implies that the magnetic moment is tightly “frozen” in the particlebody, i.e., the particle is absolutely magnetically hard, its magnetic anisotropy is inﬁnite. Todistinguish between those two situations, we deﬁne parameter σ = EA/kBT that relates themagnetic anisotropy energy to thermal one. Therefore, the considered particles have eitherσ = 0 or σ = ∞.The model MFG samples generated for simulations contain the particles of the same σ,which make the nodes of a simple cubic lattice constructed of bead-spring polymer strands. Inthe framework of coarse-grained molecular dynamics, each strand is constructed of a chain ofstructure units (polymer blobs) which, but conditionally, are termed as “monomers”. Eachmonomer has two neighbours, to which it is linked by central-bonded harmonic potentialsequivalent to simple springs with equal rigidity and equilibrium lengths. In an MFG, the endmonomers of some strands are attached in the same way to the NPs. Under such linking,the MFG subsystems turn out to be always coupled, while, as shown below, the parameter σdetermines the extent of that coupling.The concentration of magnetic ﬁller in the samples is deﬁned on the basis of the number ofthe lattice (mesh) nodes occupied by the particles. The residual fraction of the nodes are thesites where the polymer chains are connected only with each other; in simple cubic lattice thismeans a “meeting point” of six chains.The samples with diﬀerent fractions of magnetic ﬁller were generated in the following way.First a “pre-initial” conﬁguration, being a simple cubic lattice with 103 nodes, is used, wherethe lattice period is equal to the contour length of the polymer strand. In our simulations all thestrands were equal and comprised 12 identical blobs (monomer). To make the polymer objectsuﬃciently soft and to provide mobility for the magnetic particles, the size of the monomer istaken three times smaller that that of the NP., and no angular rigidity is introduced.As soon as the magnetic phase concentration c for a given simulation is deﬁned, then thenumbers of the occupied and particle-free nodes are calculated as Np = cNtot, where Ntot is thetotal number of nodes in the sample. Thus determined amount of NPs is distributed at randomover all the nodes.After this “pre-initial” conﬁguration of the sample is assembled, it is subjected tothermalisation by setting it in the thermostat of given temperature and letting the polymerstrands to ﬂuctuate. This thermalisation stage is also preliminary, as during it the NPs aretreated as non-magnetic, and when performing simulation, only steric potentials for the structureunits (blobs and NPs altogether) and the bonding potentials of the blob strands are taken intoaccount. To allow for the stochastic nature of the thermostat action as well as for randomdistribution of NPs over the mesh nodes, the averaging over several implementations of the samesample is carried out. The snapshots of MFG samples with diﬀerent particle concentrations areshown in Fig. 1.Upon having attained the thermalised state, the magnetodipole interaction between theNPs, all of which are assumed to bear equal permanent magnetic moments μ and be of thesame diameter dnp, is turned on. The intensity of this interaction is characterised by thenondimensional parameter λ = μ2/(d3npkBT ) that relates the magnetodipole energy of twoparticles in tight contact to thermal energy. Then the ﬁnal stage of simulation is commenced:the equation of motion of all the elements of the MFG sample are integrated (with all the31234567890 ‘’“”CMSMS17 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 994 (2018) 012004  doi :10.1088/1742-6596/994/1/012004a b c d eFigure 1. (Colour online) Thermalised conﬁgurations with diﬀerent number of nodes occupiedby NPs; 20% of nodes (a), 35% (b), 50% (c), 65% (d), 80% (e).constraints and random forces and torques) until the energetically stationary state is attained.3. Results and discussionIn this work we focus on the structures induced in the two afore-mentioned types of MFGs (withσ = 0 or ∞) in the absence of external magnetic ﬁeld. For that, we use an indicator functionq(j, d˜ ) introduced in Ref. [7]. It renders the fraction of NPs possessing at least j neighbourswhose center-to-center distances are less or equal the value d˜ that is chosen quite close to dnp;in our calculations we set d˜ = 1.1dnp. Therefore, the values q(1, d˜), q(2, d˜), and q(3, d˜)yieldthe number of particles which have at least one, two or three nearest neighbours, respectively.As a general fact, we remark that in all the MFGs studied, the value of q(3, d˜)turns outto be virtual zero that points out the absence of multi-particle clusters and/or branched NPaggregates. In other words, practically all the forming aggregates are chains.In all the simulations we ﬁnd that for a given concentration, the value of q(1, d˜)that reﬂectsthe presence of any aggregates (including pair clusters) grows with the increase of the dipolarinteraction λ. As shown in Fig. 2, indicator q(1, d˜)changes mostly fast (from 0.2 to 0.8) ina relatively narrow interval of λ. Because of that, one may interpret the occurring change ofthe MFG as a structure transition from a nearly homogeneous (the majority of the particles aresingle) to the aggregated state where the majority of the particles are united in chains. It seemsreasonable to associate the ﬁnite width of this interval (at c =const) with the distributions of(i) the local interparticle distances and (ii) the local eﬀective elasticity of the mesh in a givenrealisation of the MFG sample; ﬂuctuations of those distributions in diﬀerent realisations of thesample should also contribute to smearing of the transition region. We remark that in a givenMFG the only way to initiate the above-discussed transition is to move parameter λ by varyingtemperature. However, temperature is a factor that inﬂuences all the properties of a gel thathas to be taken into account for real systems.From the diagrams of Fig. 2 it is seen that the transition zones are bow-shaped (leftwardinﬂected) independently of the type of the particle anisotropy. Therefore, in any of MFGsunder consideration the aggregation transition requires minimal intensity of the interparticleinteraction at moderate concentrations of the magnetic phase, while to induce the same transitionat either low or high concentration one needs to enhance the value of λ.Note that the bow-shaped conﬁguration of the transition zones in Fig. 2 is due to two diﬀerenteﬀects. At the low-concentration end it is entailed by the fact that mean interparticle distances insuch MFGs are large. As the magnetodipole coupling decreases rather fast with the distance, toprovide a suﬃcient attraction force at given temperature it is necessary to enhance the referencevalue of the interaction, i.e., to increase λ in fact is. In a system with high NP concentration,the factor impeding aggregation is most probably the self-averaging of the local ﬁeld. Mutual41234567890 ‘’“”CMSMS17 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 994 (2018) 012004  doi :10.1088/1742-6596/994/1/012004a bFigure 2. (Colour online) Dependence of the aggregation indicator q(1, d˜)on concentrationc and parameter of magnetodipolar interaction λ for: (a) magnetically isotropic particles σ = 0and (b) magnetically anisotropic particles σ = ∞. Solid black lines are isolines of q(1, d˜)= 0.2and 0.8, dashed black lines correspond to intermediate values of q(1, d˜).attraction of a given pair of particles is slackened by the ﬁelds induced by the surrounding oneswhich are positioned quite close to the pair. To overcome this screening, the enhanced pairwiseinteraction is required.Comparison of the diagrams for MFGs with σ = 0 and σ = ∞ in Fig. 2 reveals that foranisotropic particles the aggregation is notably hampered. To understand the origin of thiseﬀect, it suﬃces to recall that in a system with σ = ∞ the orientation of the NP magneticmoment is strongly coupled to the positions of the blob chains grafted to the NP surface.Therefore, to point its magnetic moment along the direction optimising the magnetodipoleinteraction with other particles, the given one has to move its mesh environment. Evidently,this displacement requires some extra energy in comparison with the case of magnetically softNP where the internal rotations of the magnetic moment do not cost any energy. This causesapparent diﬀerence between Figs. 2 a and b: the enhanced width of the transition “band” anda notable enhancement of the reference values of λ corresponding to the aggregation transitionfor any given concentration. On this basis, one concludes that, if to take two MFG samplespossessing the same concentration and same magnitude of parameter of λ, then under zeroexternal ﬁeld the sample with anisotropic particles would display a considerably lower degree ofaggregation.The above-described character of the transition from non-aggregated to chained state inMFGs is conﬁrmed by the plots of indicator q(2, d˜), see Fig. 3. This function characterise thefraction of NPs which are incorporated in chains comprising at least three-particle clusters.Aggregation of the particles resulting in formation of “coil” structures, see Fig. 3, aﬀects thepolymer conformations in the MFGs. By that, it entails the change of the sample densityand, thus, its volume V . For MFGs with magnetically isotropic ﬁller the volume changebehaves expectably: the greater the magnetodipolar interaction, i.e., the higher the degreeof chaining, the larger is the relative ”volume defect” ΔV = (V − V0)/V0 where V0 is thevolume of a thermalised sample with the switched oﬀ magnetic interactions. Under growth ofthe interparticles magnetic forces, i.e., of parameter λ, the volume defect ΔV of such a samplesaturates due to pure sterical reasons. This is illustrated by Fig. 5a, see the dashed lines there.51234567890 ‘’“”CMSMS17 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 994 (2018) 012004  doi :10.1088/1742-6596/994/1/012004a bFigure 3. (Color online) Dependence of the aggregation indicator q(2, d˜)on concentration cand parameter of magnetodipolar interaction λ for: (a) magnetically isotropic particles σ = 0and (b) magnetically hard particles σ = ∞. Solid black lines are isolines of q(2, d˜)= 0.2 and0.8, dashed black lines correspond to intermediate values of the degree of aggregation.Note that in this case the magnitude of ΔV increases monotonously with the NP concentration.We remark also that the behaviour reﬂected by Fig. 5a to a good extent correlates with thediagram of indicator q(2, d˜)in Fig. 3a.Figure 4. (Colour online)“Coil”-like aggregated conﬁguration of an MFG that is densely ﬁlled(c = 0.8) with magnetically soft particles (σ = 0) possessing high dipolar moments (λ = 11.5)For the model MFG with inﬁnitely hard particles, the reference interval of the volume defectto the smaller value side, cf. colour scales in Figs. 5a and 5b. The cause of that is the necessity forthe anisotropic particles to overcome rotational constraints imposed by the polymer strands inaddition to the translational ones. Indeed, for the interaction parameters from λ = 2.5 to aboutλ = 7.0, the magnitude of ΔV grows rather slowly for low and moderate concentrations thatimplies that the distortion of the polymer in the vicinity of the “magnetic” nodes are yet small.61234567890 ‘’“”CMSMS17 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 994 (2018) 012004  doi :10.1088/1742-6596/994/1/012004In the range of parameters where signiﬁcant aggregation occurs (λ > 7.0) the volume defectchanges faster but the saturation levels are attained at much larger values of the magnetodipolarparameter than in the case of an MFG with σ = 0.Notably, at large NP concentrations for all the studied values of λ, the volume defect changesnon-monotonically: its magnitude displays maximum about c = 0.65. In our view, for the weakerdipoles (say, λ ≤ 7) such a feature is result of mutual competition of dipolar forces of closelypositioned particles, i.e., the self-averaging of the local ﬁelds combined with lower complianceof anisotropic NPs to those forces. Under strong dipole interaction, the most probable reasonof ΔV decrease is formation of rather extended NP chains which go through the whole samplethus making it less dense.a bFigure 5. (Colour online) Volume defect of an MFG sample as a function of concentration c andmagnetodipolar parameter λ: (a) magnetically isotropic particles σ = 0 and (b) magneticallyanisotropic particles σ = ∞. Solid black lines are isolines of volume, dot-dashed black linerenders the average volume between the minimal and the maximal value at the diagram4. ConclusionsCoarse-grained molecular dynamics simulations are performed of micro-size ferrogel objectswith the reference number of magnetic nanoparticles being several hundreds. The structurestate of the system might be non-aggregated as well as aggregated (chained) depending onthe concentration of the magnetic phase and the intensity of interparticle interaction. Withrespect to the magnetodipolar parameter, the structure changes might be treated as a transitionwhich is quite sharp for the microferrogels comprising magnetically soft particles and rathersmeared for those comprising magnetically hard ones. The samples with the particles of diﬀerentanisotropies possess substantially diﬀerent magnitudes of the volume defect, i.e., eﬀective density,as well. The results obtained show that the simulated MFGs can be used, in principle, to studythe locking ability of microferrogels to carry microscale cargo since, under a ﬁxed amount ofincorporated particles, this ability depends on the structure of spontaneous conﬁgurations of thesample.AcknowledgmentsThis work was supported by RFBR grants #17-42-590504 and #17-41-590160.71234567890 ‘’“”CMSMS17 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 994 (2018) 012004  doi :10.1088/1742-6596/994/1/012004References[1] Backes S, Witt M U, Roeben E, Kuhrts L, Aleed S, Schmidt A M and von Klitzing R 2015 Journal of PhysicalChemistry B 119 12129–12137[2] Turcu R, Socoliuc V, Craciunescu I, Petran A, Paulus A, Franzreb M, Vasile E and Vekas L 2015 Soft Matter11 1008–1018[3] Campbell S, Maitland D and Hoare T 2015 ACS Macro Letters 4 312–316[4] Li Y, Huang G, Zhang X, Li B, Chen Y, Lu T, Lu T J and Xu F 2013 Advanced Functional Materials 23660–672[5] Thevenot J, Oliveira H, Sandre O and Lecommandoux S 2013 Chemical Society Review 42 7099–7116[6] Ha¨ring M, Schiller J, Mayr J, Grijalvo S, Eritja R and Daz D D 2015 Gels 1 135–161[7] Ryzhkov A V, Melenev P V, Balasoiu M and Raikher Yu L 2016 Journal of Chemical Physics 145 074905[8] Ryzhkov A and Raikher Yu L 2017 Journal of Magnetism and Magnetic Materials 431 192–195

The effect of magnetic nanoparticle concentration on the structure organisation of a microferrogel

http://elar.urfu.ru/bitstream/10995/73770/1/10.1088-1742-6596-994-1-012004.pdf

The effect of magnetic nanoparticle concentration on the structure organisation of a microferrogel

Abstract

Similar works

Full text

Available Versions

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin