Interaction Effects in Assembly of Magnetic Nanoparticles

Interaction Effects in Assembly of Magnetic Nanoparticles A specific absorption rate of a dilute assembly of various random clusters of iron oxide nanoparticles in alternating magnetic field has been calculated using Landau–Lifshitz stochastic equation. This approach simultaneously takes into account both the presence of thermal fluctuations of the nanoparticle magnetic moments and magneto-dipole interaction between the nanoparticles of the clusters. It is shown that for usual 3D clusters, the intensity of the magneto-dipole interaction is determined mainly by the cluster packing density η = N V/V ,where N is p cl p the average number of the particles in the cluster, V is the nanoparticle volume, and V is the cluster volume. cl The area of the low frequency hysteresis loop and the assembly-specific absorption rate have been found to be considerably reduced when the packing density of the clusters increases in the range of 0.005 ≤ η <0.4. The dependence of the specific absorption rate on the mean nanoparticle diameter is retained with an increase of η, but becomes less pronounced. For fractal clusters of nanoparticles, which arise in biological media, in addition to a considerable reduction of the absorption rate, the absorption maximum is shifted to smaller particle diameters. It is found also that the specific absorption rate of fractal clusters increases appreciably with an increase of the thickness of nonmagnetic shells at the nanoparticle surfaces. Keywords: Iron oxide nanoparticles, Magneto-dipole interaction, Specific absorption rate, Numerical simulation PACS: 75.20.-g, 75.50.Tt, 75.40.Mg Background because they are biocompatible and biodegradable and Magnetic hyperthermia [1–4] is one of the most promis- can be detected in the human body using clinical MRI. In ing directions in contemporary biomedical research this study, we consider assemblies of nanoparticles with related with cancer treatment. The performance of mag- magnetic parameters typical of iron oxide nanoparticles. It netic nanoparticles to generate heat in alternating external has been found recently [4, 6] that being embedded in a magnetic field is affected by various factors, such as their biological environment, for example, into a tumor, mag- geometrical and material parameters, the concentration of netic nanoparticles turn out to be tightly bound to the sur- nanoparticles in the media, as well as the frequency and rounding tissues. Therefore, the rotation of magnetic amplitude of the alternating magnetic field. In this paper, nanoparticles as a whole under the influence of alternating the effect of mutual magneto-dipole interaction on the external magnetic field is greatly hindered. In such a case, specific absorption rate (SAR) of an assembly of magnetic the Brownian relaxation is unimportant [4]. Therefore, nanoparticles on an alternating magnetic field is studied only the motion of the particle magnetic moments under theoretically. The nanoparticles of iron oxides seem most the influence of an alternating magnetic field and thermal promising for use in magnetic hyperthermia [2–5], fluctuations has to be considered. In addition, one must take into account the influence of the magnetic-dipole interaction between particles. The latter effect is especially * Correspondence: usov@obninsk.ru important since magnetic nanoparticles in biological National University of Science and Technology “MISIS”, 119049 Moscow, Russia media tend to agglomerate [2, 4, 7] forming dense aggre- Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave gates of nanoparticles having fractal [8, 9] geometrical Propagation, Russian Academy of Sciences, IZMIRAN, 108480, Troitsk, structure. Moscow, Russia © The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Usov et al. Nanoscale Research Letters (2017) 12:489 Page 2 of 8 The effect of thermal fluctuations on the heat dissipa- shown to be determined mainly by the nanoparticle tion in a dilute assembly of magnetic nanoparticles in packing density η = N V/V , where N is the average p cl p alternating magnetic field has been studied in detail in number of particles in the cluster, V is the nanoparticle Refs. [10–13]. In particular, it has been shown [10] that volume, and V is the cluster volume. The area of the cl the SAR of such assembly depends substantially on the hysteresis loop and the assembly SAR are found to be mean nanoparticle diameter, among other factors. For a considerably reduced when the packing density of the dilute nanoparticle assembly detailed calculations [10] 3D clusters increases in the range of packing densities allow one to determine the optimum diameter of the studied, 0.005 ≤ η < 0.4. For fractal clusters of magnetic nanoparticles at the given particle magnetic parameters nanoparticles, in addition to considerable reduction of and given amplitude and frequency of the alternating SAR, the maximum absorption rate is shifted to smaller magnetic field. With optimal choice of geometric and particle diameters, as a rule. It is found also that the magnetic parameters of the nanoparticles very high SAR SAR of fractal clusters increases appreciably with an in- values, of the order of 1000 kW/kg, have been predicted crease of the thickness of nonmagnetic shells at the [10, 11]. It is notable that the SAR values reported in a nanoparticle surfaces. This effect may be important for number of experiments [14–17] are really close to the application of magnetic nanoparticle assemblies in mag- above theoretical estimates. At the same time, in many netic hyperthermia. experiments [5, 18–21] significantly lower values of SAR ~ 20–50 kW/kg were measured. This fact can be Numerical Simulation explained, most likely, by the influence of strong magneto- Non-Interacting Nanoparticles dipole interaction in dense assemblies of magnetic It is instructive to remind first the behavior of an assem- nanoparticles. bly of non-interacting superparamagnetic nanoparticles Indeed, it has been experimentally shown [22, 23] that in an alternating magnetic field. It enables one to see the SAR in the dense assembly of magnetic nanoparti- clearly the influence of the magneto-dipole interaction cles essentially depends on the aspect ratio of the test on the assembly properties. Based on the Fokker–Planck sample, i.e., the ratio of sample length to width. This is equation derived by W.F. Brown [39], one can get an indirect evidence of the influence of magneto-dipole approximate kinetic equation [10] for the population interaction on the response of an assembly of nanoparti- numbers n (t) and n (t) of two potential wells of uniaxial 1 2 cles on an alternating external magnetic field. The effect superparamagnetic nanoparticle of magneto-dipole interaction on the energy absorption ∂n n n 1 2 1 rate by the assembly of magnetic nanoparticles has been ¼ − ; n ðÞ t þ n ðÞ t ¼ 1: ð1Þ 1 2 ∂t τ ðÞ T τ ðÞ T 2 1 studied in a number of recent theoretical and experi- mental investigations [7, 24–38]. However, further inves- Here, τ (T) and τ (T) are the corresponding relaxation 1 2 tigations seem necessary to take into account the fractal times at a given temperature T for the first and second nature [8, 9] of the nanoparticle distribution in biological potential wells, respectively. The relaxation times τ (T) media. and τ (T) depend essentially on the amplitude and direc- To see clearly the effect of magneto-dipole interaction, tion of the applied magnetic field with respect to particle in this paper we first calculate the SAR of an assembly of easy anisotropy axis (see Appendix in Ref. [10]). The it- non-interacting iron oxide nanoparticles. To study the eration procedure can be used to calculate the well effect of magneto-dipole interaction, we solve numerically population numbers n (t) and n (t) for several periods of 1 2 the Landau–Lifshitz stochastic equation [13, 39–41], the alternating magnetic field. It is sufficient to obtain a which simultaneously takes into account both the pres- stationary hysteresis loop of a particle in an alternating ence of thermal fluctuations of the particle magnetic magnetic field. To do so, one can use an approximate moments and magneto-dipole interaction between the relation for the component of the reduced particle nanoparticles of the clusters. Two types of magnetic clus- magnetization along the magnetic field direction ters are considered, the usual random 3D clusters of nanoparticles distributed in a rigid media and the fractal M ¼ m ðÞ t ¼ n ðÞ t cos θ −θ ðÞ h ðÞ t h 2 0 min;2 e clusters of nanoparticles which usually arise within the M V intracellular space. Note that within the cluster, the nano- þn ðÞ t cos θ −θ ðÞ h ðÞ t 1 0 min;1 e particles are coupled by a strong magneto-dipole inter- ð2Þ action. At the same time, for a dilute assembly of clusters, the magnetic interaction between the clusters can be Here, θ is the angle of the external magnetic field neglected in a first approximation. with respect to the particle easy anisotropy axis, θ min,1 The influence of magneto-dipole interaction on the and θ are the locations of the potential well minima min,2 properties of a dilute assembly of random 3D clusters is as the functions of the reduced applied magnetic field, Usov et al. Nanoscale Research Letters (2017) 12:489 Page 3 of 8 h (t)= H sin(ωt)/H , where ω =2πf is the angular fre- For an assembly of completely random 3D clusters, the e 0 a quency, H being the particle anisotropy field. To get a orientations of the easy anisotropy axes of nanoparticles hysteresis loop of an assembly of randomly oriented in- {e }, i =1, 2, .. N , are chosen randomly and independently i p dependent nanoparticles, it is necessary to average the on the unit sphere. Alternatively, one can assume that reduced magnetization m (t) over the magnetic field during the formation of clusters in a solution under the directions. It is worth noting that the accuracy of an influence of magneto-static interaction, certain correlation approximate analytical solution, Eq. (1), (2), of the Fokker- occurs in the distribution of the nanoparticle easy anisot- Planck equation has been validated [10] by means of ropy axis directions. One possibility to describe such direct comparison with the numerical solutions of the sto- partially ordered clusters is to assume that the easy anisot- chastic Landau–Lifshitz equation for non-interacting ropy axes of the nanoparticles are uniformly distributed in magnetic nanoparticles. a solid angle, θ < θ , in the spherical coordinates. max Random 3D clusters with a given number of particles Nanoparticle Clusters N of diameter D were created in this study as follows. To investigate the effect of magneto-dipole interaction First, we generated dense enough and approximately on the specific absorption rate of an assembly of inter- uniform set of N random points {ρ } within a spherical acting magnetic nanoparticles in an alternating magnetic volume of the radius R , so that |ρ | ≤ R for all gener- cl i cl field, in this paper, we study the behavior of a dilute ated points, i = 1, 2... N, N >> N . The center of the first assembly of usual 3D clusters of superparamagnetic nanoparticle was placed in the first random point, nanoparticles and that of fractal clusters [8, 9] which r = ρ . Then, all random points with coordinates |ρ – 1 1 i arise usually in biological media loaded with fine mag- r | ≤ D were removed from the initial set of the random netic nanoparticles. points. After this operation, any point in the remaining A quasi-spherical 3D cluster of nanoparticles shown set of random points could be used as a center of the schematically in Fig. 1a can be characterized by its radius second nanoparticle. For example, one can put simply R , and the number of nanoparticles, N >> 1, within its r = ρ . At the next step, one removes all random points cl p 2 2 volume. It is assumed that the nanoparticles have nearly whose coordinates satisfy the inequality |ρ –r | ≤ D. This i 2 the same diameter D, and their centers, {r }, i =1, 2,.. N , procedure is repeated until all N nanoparticle centers i p p are randomly distributed in the cluster volume. We also are placed within the cluster volume. As a result, all ran- assume that the particles are coated with thin non- dom nanoparticle centers lie within a sphere of radius magnetic shells, so that the exchange interaction between R , so that |r | ≤ R , i = 1, 2,... N . Furthermore, none of cl i cl p the neighboring nanoparticles of the cluster is absent. As the nanoparticles is in a direct contact with the neigh- we mentioned above, such 3D cluster is characterized by boring nanoparticles. This algorithm enables one to con- the nanoparticle packing density η = N V/V .Thisis a struct random quasi-spherical 3D clusters of magnetic p cl total volume of the magnetic material distributed in the nanoparticles for moderate values of the nanoparticle volume of the cluster. One can define the average distance volume fraction η < 0.5. between the nanoparticles of the cluster by means of the For a given set of initial parameters, i.e., D, R ,and N , cl p 1/3 relation D =(6V /πN ) . Then, the nanoparticle pack- various random 3D clusters differ by the sets of the coor- av cl p ing density is given by η =(D/D ) . dinates of the nanoparticle centers {r } and orientations av i ab Fig. 1 Geometry of quasi-spherical random 3D cluster of single-domain nanoparticles (a) and fractal cluster (b) with fractal descriptors D = 2.1 and k = 1.3 f Usov et al. Nanoscale Research Letters (2017) 12:489 Page 4 of 8 {e } of the particle easy anisotropy axes. However, the cal- For nanoparticles of nearly spherical shape with uniaxial culations show that in the limit N >> 1, the hysteresis type of magnetic anisotropy the magneto-crystalline loops obtained for different realizations of random vari- anisotropy energy is given by ables {r}and {e } differ only slightly from each other. To Np i i →→ 2 characterize the behavior of a dilute assembly of random W ¼ KV 1− α e Þ Þ; ð5Þ i i i¼1 nanoparticle clusters, it is necessary to calculate assembly hysteresis loop averaged over a sufficiently large number where e is the orientation of the easy anisotropy axis of of random cluster realizations. It is found that in the limit i-th particle of the cluster. Zeeman energy W of the N >> 1, the averaged hysteresis loop of cluster assembly cluster in applied magnetic field is given by has a rather small dispersion even being averaged over Np 20–30 independent realizations of random clusters with → W ¼ −M V α H sinðÞ ωt Þ: ð6Þ Z s i 0 the fixed values of the initial parameters D, R ,and N . cl p i¼1 The geometry of fractal clusters of single-domain Next, for spherical uniformly magnetized nanoparti- nanoparticles is characterized [42, 43] by the fractal de- cles, the magneto-static energy of the cluster can be rep- scriptors D and k . By definition, the total number of f f resented as the energy of the point interacting dipoles nanoparticle N in the fractal cluster is given by the rela- located at the particle centers r within the cluster. Then, D i tion N ¼ k 2R =D , where D is the fractal dimen- p f g f the magneto-dipole interacting energy is sion, k is the fractal prefactor, and R being the radius of f g →→ →→ →→ 2 2 gyration. It is defined [43] via the mean square of the α α −3 α n α n i ij j ij M V i W ¼ ; ð7Þ distances between the particle centers and the geomet- 3 → → r −r i j i≠j rical center of mass of the aggregate. In this paper, the fractal clusters with various fractal descriptors were cre- where n is the unit vector along the line connecting the ij ated using the well-known Filippov et al.’s algorithm centers of i-th and j-th particles, respectively. [43]. As an example, Fig. 1b shows the geometrical struc- Thus, the effective magnetic field acting on the i-th ture of fractal cluster with fractal descriptors D = 2.1 and nanoparticle of the cluster is given by k = 1.3 consisting of N = 90 single-domain nanoparticles. f p → → →→ → H ¼ H α e e þ H sinðÞ ωt Geometrically, it seems that the main difference between a ef ;i i i i 0 →→ → 3D and fractal clusters is that in the latter case, every α −3 α n n j ij ij þM V : nanoparticle has at least one neighbor located at the clos- s → → r −r j≠i i j est possible distance between nanoparticle centers equal to the nanoparticle diameter D. ð8Þ Dynamics of unit magnetization vector α of i-th where H =2 K/M is the particle anisotropy field. a s. single-domain nanoparticle of the cluster is determined The thermal fields, H , i = 1, 2...N , acting on various by the stochastic Landau–Lifshitz (LL) equation th;i nanoparticles of the cluster are statistically independent, ∂α → → i → → with the following statistical properties [39] of their ¼ −γ α  H þ H Þ−κγ α ef ;i i 1 th;i 1 ∂t components for every nanoparticle → → α  H þ H ; i ef ;i th;i DE DE ðÞ α ðÞ α ðÞ β H ðÞ t ¼ 0; H ðÞ t H ðÞ t ð3Þ th th th 2k Tκ ¼ δ δðÞ t−t ; α; β ¼ðÞ x; y; z : ð9Þ αβ 1 where γ is the gyromagnetic ratio, κ is phenomenological γM V damping parameter, γ = γ/(1+κ ), H is the effective ef ;i Here, k is the Boltzmann constant, δ is the Kroneker magnetic field and H is the thermal field. The effect- B αβ th;i symbol, and δ(t) is the delta function. ive magnetic field acting on a separate nanoparticle can The procedure for solving stochastic differential Eqs. be calculated as a derivative of the total cluster energy (3), (8), and (9) is described in detail in Refs. [13, 40, 41]. ∂W H ¼ − : ð4Þ ef ;i Results and Discussion VM ∂α Non-Interacting Iron Oxide Nanoparticles The totalmagneticenergy ofthe clusterW=W +W +W a Z m Consider a dilute assembly of superparamagnetic nano- is a sum of the magneto-crystalline anisotropy energy W , a particles with an average diameter D. The particles are Zeeman energy W of the particles in applied magnetic Z assumed to be tightly packed in a surrounding media, field H sinðÞ ωt , and the energy of mutual magneto-dipole and their easy anisotropy axes are randomly oriented in interaction of the particles W . space. The hysteresis loop of such an assembly in an m Usov et al. Nanoscale Research Letters (2017) 12:489 Page 5 of 8 alternating magnetic field H = H sin(ωt) can be calcu- Fig. 2 shows, for the assembly of non-interacting iron lated [10] using Eqs. (1) and (2). This approach, due to oxide nanoparticles, the peak of the energy absorption in its simplicity, allows one to carry out detailed calcula- an alternating magnetic field corresponds to particles tions of the assembly hysteresis loops for various particle with diameter D = 20 nm. Therefore, first we calculated sizes depending on frequency and amplitude of the alter- the hysteresis loops of an assembly of 3D clusters with nating magnetic field. In the calculations performed, in particle diameter D = 20 nm. accordance with the experimental data [2–6], the satur- Figure 3a shows the evolution of the assembly hyster- ation magnetization of iron oxide nanoparticles is as- esis loops depending on the average distance between sumed to be M =70Am /kg, the magnetic anisotropy the nanoparticle centers D at the fixed value of the s. av 4 3 constant being K =10 J/m . The assembly temperature particle diameter D. The frequency and amplitude of is T = 300 K, and the nanoparticle diameters are in the alternating magnetic field are fixed at f = 400 kHz and range D =10–30 nm. These parameters seem typical for H = 8 kA/m, respectively. The number of particles in experiments carried out on iron oxide nanoparticles. the clusters equals N = 40. The calculations are carried Figure 2 shows the SAR of non-interacting assemblies of out at T = 300 K, and magnetic damping constant is iron oxide nanoparticles at various frequencies at a fixed taken to be κ = 0.5. amplitude of alternating magnetic field, H = 8 kA/m. As Evidently, the decrease of the average distance between can be seen, for the range of frequencies that are charac- the nanoparticles of the cluster leads to an increase of teristic for magnetic hyperthermia, f =200–500 kHz, SAR intensity of the magneto-dipole interaction within the has a maximum for the assembly of iron oxide nanoparti- cluster. Note that for N = 40, the ratios D /D specified p av cles with diameters D =20–21 nm. It is notable that even in Fig. 3a correspond to the cluster packing densities at relatively moderate amplitude of an alternating mag- η = 0.005, 0.04, and 0.32, correspondingly. One can see netic field, the assembly SAR reaches sufficiently high in Fig. 3a that the hysteresis loop area rapidly decreases values, 350–450 kW/kg, if the nanoparticle diameters are as a function of the parameter η. For comparison, Fig. 3a chosen properly. also shows the hysteresis loop 4, calculated for an However, the experimentally measured SAR values for assembly of non-interacting particles, i.e., in the limit assemblies of iron oxides nanoparticles are, as a rule, sig- D /D → ∞, N = const, using Eqs. (1) and (2). av p nificantly below [18–21] these theoretical values. As we One can see that the hysteresis loop 3 (η = 0.005) in shall see in the next section, this fact can be explained Fig. 3a turns out to be close to the hysteresis loop of the [22–38] by the influence of strong magneto-dipole inter- assembly of non-interacting nanoparticles. Therefore, in action in dense assemblies of magnetic nanoparticles. the case η ≤ 0.005 the magneto-dipole interaction of the nanoparticles within the cluster can be neglected. How- Assembly of 3D Clusters ever, for η ≥ 0.04 the magneto-dipole interaction has a Consider now the hysteresis loops of a dilute assembly significant influence on the properties of an assembly of of 3D random clusters having easy anisotropy axes of in- random 3D clusters. A similar evolution of the assembly dividual nanoparticles randomly oriented in space. As hysteresis loops has been obtained also for the frequen- cies f = 300 and 500 kHz, respectively. The hysteresis loops shown in Fig. 3a are calculated for different ratios D /D, but for the fixed number of f = 200 kHz av H = 8 kA/m nanoparticles in the cluster N = 40. However, the de- f = 300 kHz 400 p T = 300 K f = 400 kHz tailed computer simulations show that the shape of the f = 500 kHz hysteresis loop of a dilute assembly of random 3D clus- ters is practically unchanged, if the number of particles, N >> 1, and the radius of the cluster R are changed so p cl that the nanoparticle packing density η remains con- stant. Therefore, the hysteresis loop of dilute assembly of random 3D clusters depends mainly on the cluster packing density η. Figure 3b shows the SAR of assemblies of random 16 18 20 22 24 clusters of iron oxide nanoparticles for different η D (nm) values. The SAR of the assembly is calculated [10] as Fig. 2 The specific absorption rate of non-interacting assembly of SAR = M fA/ρ,where A is the hysteresis loop area in the iron oxides nanoparticles, obtained by means of Eqs. (1) and (2), as a variables (M/M , H), ρ being the density of iron oxide s. function of average particle diameter at different frequencies of the 3 3 nanoparticles which is assumed to be ρ =5× 10 kg/m . alternating magnetic field As Fig. 3b shows, the SAR decreases as a function of η SAR (kW/kg) Usov et al. Nanoscale Research Letters (2017) 12:489 Page 6 of 8 ab Fig. 3 (a) Evolution of the hysteresis loops of dilute assembly of clusters of iron oxide nanoparticles with diameter D = 20 nm for various ratios D /D:(1) D /D = 1.46; (2) D /D = 2.92; (3) D /D = 5.84. Hysteresis loop 4 corresponds to assembly of non-interacting nanoparticles of the same av av av av diameter. It is calculated by means of Eqs. (1) and (2). (b) SAR as a function of the average nanoparticle diameter D for dilute assemblies of clusters of nanoparticles with different packing density η due to an increase of the intensity of magneto-dipole nanoparticles, the SAR as a function of the particle interaction within the clusters. At the same time, the diameter also decreases considerably with respect to that dependence of the assembly SAR on the average par- of the assembly of non-interacting nanoparticles. How- ticle diameter still remains, though it becomes less ever, in contrast to the assembly of 3D clusters, the peak pronounced. values of SAR are shifted systematically to smaller par- For small values of η ≤ 0.005, the SAR of random as- ticle diameters, except for the case of fractal dimension sembly of 3D clusters actually coincides with that one for D = 2.7, which is close to the case of 3D clusters with an assembly of non-interacting nanoparticles, shown in D = 3.0. It is interesting to note also that for non- Fig. 2. On the other hand, SAR falls about six times when optimal nanoparticle diameters, for example, for nano- the cluster packing density increases up to η =0.32. Then, particles with diameters D ≤ 17 nm, the influence of it becomes close to typical SAR values ~ 50–100 kW/kg, magneto-dipole interaction leads to increase of the SAR which are obtained in a number of experiments [5, 18–21] with respect to the case of assembly of non-interacting with iron oxide nanoparticle assemblies. nanoparticles, as the SAR of the assembly of non- interacting nanoparticles is very small for nanoparticles Assembly of Fractal Clusters with diameters D ≤ 17 nm. Similar calculations were carried out for dilute assem- The calculations shown in Fig. 4 were carried out blies of fractal clusters of nanoparticles with various assuming the existence of thin non-magnetic shells with fractal descriptors. As Fig. 4 shows, for fractal clusters of thickness t = 1 nm at the surface of magnetic nanoparti- Sh cles. This prevents the nanoparticles of the fractal cluster from direct exchange interaction. Evidently, the increase D = 1.8, k = 1.5 H = 8 kA/m f f 0 of the non-magnetic shell thickness reduces the intensity D = 2.1, k = 1.3 f = 300 kHz f f of the magneto- dipole interaction of closest nanoparti- D = 2.7, k = 0.5 Np = 90 cles, as the average distance between the magnetic cores f f on the nanoparticles increases. Figure 5 shows that the in- Non interacting crease of the non-magnetic shell thickness is a proper way nanoparticles to raise the SAR of the assembly of fractal clusters of nanoparticles. Namely, for sufficiently large thickness of non-magnetic shells the dependence of the SAR on the particle diameter resembles that for weakly interacting magnetic nanoparticles. This fact may be important for the application of magnetic nanoparticle assemblies in 14 16 18 20 22 24 magnetic hyperthermia. D (nm) Fig. 4 SAR as a function of the average nanoparticle diameter D for Conclusions dilute assemblies of fractal clusters of nanoparticles with various fractal The main conclusion of this study is that the SAR of a descriptors. The SAR of the assembly of non-interacting nanoparticles dilute assembly of clusters of magnetic nanoparticles in is calculated by means of Eqs. (1) and (2) alternating magnetic field is significantly reduced with SAR (kW/kg) Usov et al. Nanoscale Research Letters (2017) 12:489 Page 7 of 8 fluctuations. The cluster model studied allows obvious generalization that can make it more practical. First, it is Non interacting D = 2.1 f necessary to take into account the size distribution of t_Sh = 1 nm k = 1.3 t_Sh = 5 nm magnetic nanoparticles in the assembly. Second, in some t_Sh = 10 nm cases exchange interaction may exist between neighbor- ing nanoparticles of the cluster if they are in direct H = 8 kA/m atomic contact. The theoretical results obtained in this study seem to f = 300 kHz be in a satisfactory agreement with recent experimental data [35] for iron oxide nanoparticles of optimal diame- ters. Indeed, according to Ref. [35], the SAR of the iron oxide nanoparticles increases with the average diameter of the nanoparticles and peaks for nanoparticles with 14 16 18 20 22 24 mean diameter D =20–21 nm. In addition, the SAR D (nm) decreases [35] with a decrease in the average distance Fig. 5 The dependence of the SAR of dilute assembly of fractal clusters between the nanoparticles due to increasing intensity of on the thickness t of the non-magnetic shells at the surface of the Sh the magneto-dipole interaction. nanoparticles. The SAR of the assembly of non-interacting nanoparticles is calculated by means of Eqs. (1) and (2) Unfortunately, in some experimental studies [5, 21] carried out to optimize the properties of magnetic nano- particles for use in magnetic hyperthermia, often do not increasing of the intensity of magneto-dipole interaction take into account the theoretical predictions [10, 11] in the clusters. For usual 3D clusters of nanoparticles, about significant dependence of the assembly SAR on the intensity of the magneto-dipole interaction can be the characteristic size of the magnetic nanoparticles. As characterized by dimensionless packing density, η = shown in this paper, this dependence can be substantial N V/V =(D/D ) . The latter determines the average even for rather dense nanoparticle assemblies. From a p cl av distance between the nanoparticles of the cluster. The theoretical point of view, it is obvious [10] that the as- calculations show that for the assembly of random 3D sembly of iron oxide nanoparticles with very small, clusters, the energy absorption peak, which for iron D ≤ 10 nm, or too big, D ≥ 30 nm diameters can hardly oxide nanoparticles corresponds to particles with aver- provide a sufficiently high SAR values for typical for age diameter D = 20 nm, is reduced about six times magnetic hyperthermia frequencies, f = 200–600 kHz, when the packing density increases from η = 0.005 up to and magnetic field amplitudes H ~ 8 kA/m. The cre- η = 0.32. The dependence of the assembly SAR on the ation of mono-crystalline iron oxide nanoparticles with mean nanoparticle diameter is retained with increase of sharp size distribution near the optimal diameter has to η, but becomes less pronounced. be promising for application in magnetic hyperthermia. For dilute assemblies of fractal clusters of magnetic Acknowledgements nanoparticles, the SAR values also decrease several times The authors gratefully acknowledge the financial support of the Ministry of irrespective on the fractal descriptors of the assembly. In Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISIS”,contract № K2-2015-018. addition, the peak values of SAR are shifted systematically to smaller particle diameters, as a rule. It is important to Funding note, however, that the increase of the non-magnetic shell The funding of this study is from Contract № K2–2015-018 of the Ministry of thickness at the nanoparticle surfaces restores the SAR Education and Science of the Russian Federation. values close to that of the assembly of weakly interacting Availability of Data and Materials nanoparticles. This fact can be important for various bio- Not applicable medical applications of magnetic nanoparticle assemblies. The model considered in this paper takes into account Authors’ Contributions the geometrical structure of nanoparticle assemblies The manuscript was written through the contributions of all authors. All authors have read and approved the final version of the manuscript. observed experimentally in biological media [4, 8, 9] (in particular in tumors), i.e., the agglomeration of nanopar- Ethics Approval and Consent to Participate ticles in a sufficiently dense fractal clusters of different Not applicable sizes, with different numbers of nanoparticles in the Consent for Publication clusters. The stochastic LL Eq. (3) accurately describes Not applicable the real dynamics of the magnetic moments of nanopar- ticles taking into account both the magneto-dipole inter- Competing Interests action between the particles and the effect of thermal The authors declare that they have no competing interests. SAR (kW/kg) Usov et al. Nanoscale Research Letters (2017) 12:489 Page 8 of 8 Publisher’sNote absorption in the dispersion of iron oxide nanoparticles in a viscous Springer Nature remains neutral with regard to jurisdictional claims in medium. J Magn Magn Mater 374:508–515 published maps and institutional affiliations. 22. Gudoshnikov SA, Liubimov BY, Usov NA (2012) Hysteresis losses in a dense superparamagnetic nanoparticle assembly. AIP Advances 2:012143 Received: 24 May 2017 Accepted: 2 August 2017 23. Gudoshnikov SA, Liubimov BY, Popova AV, Usov NA (2012) The influence of a demagnetizing field on hysteresis losses in a dense assembly of superparamagnetic nanoparticles. 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Interaction Effects in Assembly of Magnetic Nanoparticles

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Abstract

A specific absorption rate of a dilute assembly of various random clusters of iron oxide nanoparticles in alternating magnetic field has been calculated using Landau–Lifshitz stochastic equation. This approach simultaneously takes into account both the presence of thermal fluctuations of the nanoparticle magnetic moments and magneto-dipole interaction between the nanoparticles of the clusters. It is shown that for usual 3D clusters, the intensity of the magneto-dipole interaction is determined mainly by the cluster packing density η = N V/V ,where N is p cl p the average number of the particles in the cluster, V is the nanoparticle volume, and V is the cluster volume. cl The area of the low frequency hysteresis loop and the assembly-specific absorption rate have been found to be considerably reduced when the packing density of the clusters increases in the range of 0.005 ≤ η <0.4. The dependence of the specific absorption rate on the mean nanoparticle diameter is retained with an increase of η, but becomes less pronounced. For fractal clusters of nanoparticles, which arise in biological media, in addition to a considerable reduction of the absorption rate, the absorption maximum is shifted to smaller particle diameters. It is found also that the specific absorption rate of fractal clusters increases appreciably with an increase of the thickness of nonmagnetic shells at the nanoparticle surfaces. Keywords: Iron oxide nanoparticles, Magneto-dipole interaction, Specific absorption rate, Numerical simulation PACS: 75.20.-g, 75.50.Tt, 75.40.Mg Background because they are biocompatible and biodegradable and Magnetic hyperthermia [1–4] is one of the most promis- can be detected in the human body using clinical MRI. In ing directions in contemporary biomedical research this study, we consider assemblies of nanoparticles with related with cancer treatment. The performance of mag- magnetic parameters typical of iron oxide nanoparticles. It netic nanoparticles to generate heat in alternating external has been found recently [4, 6] that being embedded in a magnetic field is affected by various factors, such as their biological environment, for example, into a tumor, mag- geometrical and material parameters, the concentration of netic nanoparticles turn out to be tightly bound to the sur- nanoparticles in the media, as well as the frequency and rounding tissues. Therefore, the rotation of magnetic amplitude of the alternating magnetic field. In this paper, nanoparticles as a whole under the influence of alternating the effect of mutual magneto-dipole interaction on the external magnetic field is greatly hindered. In such a case, specific absorption rate (SAR) of an assembly of magnetic the Brownian relaxation is unimportant [4]. Therefore, nanoparticles on an alternating magnetic field is studied only the motion of the particle magnetic moments under theoretically. The nanoparticles of iron oxides seem most the influence of an alternating magnetic field and thermal promising for use in magnetic hyperthermia [2–5], fluctuations has to be considered. In addition, one must take into account the influence of the magnetic-dipole interaction between particles. The latter effect is especially * Correspondence: usov@obninsk.ru important since magnetic nanoparticles in biological National University of Science and Technology “MISIS”, 119049 Moscow, Russia media tend to agglomerate [2, 4, 7] forming dense aggre- Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave gates of nanoparticles having fractal [8, 9] geometrical Propagation, Russian Academy of Sciences, IZMIRAN, 108480, Troitsk, structure. Moscow, Russia © The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Usov et al. Nanoscale Research Letters (2017) 12:489 Page 2 of 8 The effect of thermal fluctuations on the heat dissipa- shown to be determined mainly by the nanoparticle tion in a dilute assembly of magnetic nanoparticles in packing density η = N V/V , where N is the average p cl p alternating magnetic field has been studied in detail in number of particles in the cluster, V is the nanoparticle Refs. [10–13]. In particular, it has been shown [10] that volume, and V is the cluster volume. The area of the cl the SAR of such assembly depends substantially on the hysteresis loop and the assembly SAR are found to be mean nanoparticle diameter, among other factors. For a considerably reduced when the packing density of the dilute nanoparticle assembly detailed calculations [10] 3D clusters increases in the range of packing densities allow one to determine the optimum diameter of the studied, 0.005 ≤ η < 0.4. For fractal clusters of magnetic nanoparticles at the given particle magnetic parameters nanoparticles, in addition to considerable reduction of and given amplitude and frequency of the alternating SAR, the maximum absorption rate is shifted to smaller magnetic field. With optimal choice of geometric and particle diameters, as a rule. It is found also that the magnetic parameters of the nanoparticles very high SAR SAR of fractal clusters increases appreciably with an in- values, of the order of 1000 kW/kg, have been predicted crease of the thickness of nonmagnetic shells at the [10, 11]. It is notable that the SAR values reported in a nanoparticle surfaces. This effect may be important for number of experiments [14–17] are really close to the application of magnetic nanoparticle assemblies in mag- above theoretical estimates. At the same time, in many netic hyperthermia. experiments [5, 18–21] significantly lower values of SAR ~ 20–50 kW/kg were measured. This fact can be Numerical Simulation explained, most likely, by the influence of strong magneto- Non-Interacting Nanoparticles dipole interaction in dense assemblies of magnetic It is instructive to remind first the behavior of an assem- nanoparticles. bly of non-interacting superparamagnetic nanoparticles Indeed, it has been experimentally shown [22, 23] that in an alternating magnetic field. It enables one to see the SAR in the dense assembly of magnetic nanoparti- clearly the influence of the magneto-dipole interaction cles essentially depends on the aspect ratio of the test on the assembly properties. Based on the Fokker–Planck sample, i.e., the ratio of sample length to width. This is equation derived by W.F. Brown [39], one can get an indirect evidence of the influence of magneto-dipole approximate kinetic equation [10] for the population interaction on the response of an assembly of nanoparti- numbers n (t) and n (t) of two potential wells of uniaxial 1 2 cles on an alternating external magnetic field. The effect superparamagnetic nanoparticle of magneto-dipole interaction on the energy absorption ∂n n n 1 2 1 rate by the assembly of magnetic nanoparticles has been ¼ − ; n ðÞ t þ n ðÞ t ¼ 1: ð1Þ 1 2 ∂t τ ðÞ T τ ðÞ T 2 1 studied in a number of recent theoretical and experi- mental investigations [7, 24–38]. However, further inves- Here, τ (T) and τ (T) are the corresponding relaxation 1 2 tigations seem necessary to take into account the fractal times at a given temperature T for the first and second nature [8, 9] of the nanoparticle distribution in biological potential wells, respectively. The relaxation times τ (T) media. and τ (T) depend essentially on the amplitude and direc- To see clearly the effect of magneto-dipole interaction, tion of the applied magnetic field with respect to particle in this paper we first calculate the SAR of an assembly of easy anisotropy axis (see Appendix in Ref. [10]). The it- non-interacting iron oxide nanoparticles. To study the eration procedure can be used to calculate the well effect of magneto-dipole interaction, we solve numerically population numbers n (t) and n (t) for several periods of 1 2 the Landau–Lifshitz stochastic equation [13, 39–41], the alternating magnetic field. It is sufficient to obtain a which simultaneously takes into account both the pres- stationary hysteresis loop of a particle in an alternating ence of thermal fluctuations of the particle magnetic magnetic field. To do so, one can use an approximate moments and magneto-dipole interaction between the relation for the component of the reduced particle nanoparticles of the clusters. Two types of magnetic clus- magnetization along the magnetic field direction ters are considered, the usual random 3D clusters of nanoparticles distributed in a rigid media and the fractal M ¼ m ðÞ t ¼ n ðÞ t cos θ −θ ðÞ h ðÞ t h 2 0 min;2 e clusters of nanoparticles which usually arise within the M V intracellular space. Note that within the cluster, the nano- þn ðÞ t cos θ −θ ðÞ h ðÞ t 1 0 min;1 e particles are coupled by a strong magneto-dipole inter- ð2Þ action. At the same time, for a dilute assembly of clusters, the magnetic interaction between the clusters can be Here, θ is the angle of the external magnetic field neglected in a first approximation. with respect to the particle easy anisotropy axis, θ min,1 The influence of magneto-dipole interaction on the and θ are the locations of the potential well minima min,2 properties of a dilute assembly of random 3D clusters is as the functions of the reduced applied magnetic field, Usov et al. Nanoscale Research Letters (2017) 12:489 Page 3 of 8 h (t)= H sin(ωt)/H , where ω =2πf is the angular fre- For an assembly of completely random 3D clusters, the e 0 a quency, H being the particle anisotropy field. To get a orientations of the easy anisotropy axes of nanoparticles hysteresis loop of an assembly of randomly oriented in- {e }, i =1, 2, .. N , are chosen randomly and independently i p dependent nanoparticles, it is necessary to average the on the unit sphere. Alternatively, one can assume that reduced magnetization m (t) over the magnetic field during the formation of clusters in a solution under the directions. It is worth noting that the accuracy of an influence of magneto-static interaction, certain correlation approximate analytical solution, Eq. (1), (2), of the Fokker- occurs in the distribution of the nanoparticle easy anisot- Planck equation has been validated [10] by means of ropy axis directions. One possibility to describe such direct comparison with the numerical solutions of the sto- partially ordered clusters is to assume that the easy anisot- chastic Landau–Lifshitz equation for non-interacting ropy axes of the nanoparticles are uniformly distributed in magnetic nanoparticles. a solid angle, θ < θ , in the spherical coordinates. max Random 3D clusters with a given number of particles Nanoparticle Clusters N of diameter D were created in this study as follows. To investigate the effect of magneto-dipole interaction First, we generated dense enough and approximately on the specific absorption rate of an assembly of inter- uniform set of N random points {ρ } within a spherical acting magnetic nanoparticles in an alternating magnetic volume of the radius R , so that |ρ | ≤ R for all gener- cl i cl field, in this paper, we study the behavior of a dilute ated points, i = 1, 2... N, N >> N . The center of the first assembly of usual 3D clusters of superparamagnetic nanoparticle was placed in the first random point, nanoparticles and that of fractal clusters [8, 9] which r = ρ . Then, all random points with coordinates |ρ – 1 1 i arise usually in biological media loaded with fine mag- r | ≤ D were removed from the initial set of the random netic nanoparticles. points. After this operation, any point in the remaining A quasi-spherical 3D cluster of nanoparticles shown set of random points could be used as a center of the schematically in Fig. 1a can be characterized by its radius second nanoparticle. For example, one can put simply R , and the number of nanoparticles, N >> 1, within its r = ρ . At the next step, one removes all random points cl p 2 2 volume. It is assumed that the nanoparticles have nearly whose coordinates satisfy the inequality |ρ –r | ≤ D. This i 2 the same diameter D, and their centers, {r }, i =1, 2,.. N , procedure is repeated until all N nanoparticle centers i p p are randomly distributed in the cluster volume. We also are placed within the cluster volume. As a result, all ran- assume that the particles are coated with thin non- dom nanoparticle centers lie within a sphere of radius magnetic shells, so that the exchange interaction between R , so that |r | ≤ R , i = 1, 2,... N . Furthermore, none of cl i cl p the neighboring nanoparticles of the cluster is absent. As the nanoparticles is in a direct contact with the neigh- we mentioned above, such 3D cluster is characterized by boring nanoparticles. This algorithm enables one to con- the nanoparticle packing density η = N V/V .Thisis a struct random quasi-spherical 3D clusters of magnetic p cl total volume of the magnetic material distributed in the nanoparticles for moderate values of the nanoparticle volume of the cluster. One can define the average distance volume fraction η < 0.5. between the nanoparticles of the cluster by means of the For a given set of initial parameters, i.e., D, R ,and N , cl p 1/3 relation D =(6V /πN ) . Then, the nanoparticle pack- various random 3D clusters differ by the sets of the coor- av cl p ing density is given by η =(D/D ) . dinates of the nanoparticle centers {r } and orientations av i ab Fig. 1 Geometry of quasi-spherical random 3D cluster of single-domain nanoparticles (a) and fractal cluster (b) with fractal descriptors D = 2.1 and k = 1.3 f Usov et al. Nanoscale Research Letters (2017) 12:489 Page 4 of 8 {e } of the particle easy anisotropy axes. However, the cal- For nanoparticles of nearly spherical shape with uniaxial culations show that in the limit N >> 1, the hysteresis type of magnetic anisotropy the magneto-crystalline loops obtained for different realizations of random vari- anisotropy energy is given by ables {r}and {e } differ only slightly from each other. To Np i i →→ 2 characterize the behavior of a dilute assembly of random W ¼ KV 1− α e Þ Þ; ð5Þ i i i¼1 nanoparticle clusters, it is necessary to calculate assembly hysteresis loop averaged over a sufficiently large number where e is the orientation of the easy anisotropy axis of of random cluster realizations. It is found that in the limit i-th particle of the cluster. Zeeman energy W of the N >> 1, the averaged hysteresis loop of cluster assembly cluster in applied magnetic field is given by has a rather small dispersion even being averaged over Np 20–30 independent realizations of random clusters with → W ¼ −M V α H sinðÞ ωt Þ: ð6Þ Z s i 0 the fixed values of the initial parameters D, R ,and N . cl p i¼1 The geometry of fractal clusters of single-domain Next, for spherical uniformly magnetized nanoparti- nanoparticles is characterized [42, 43] by the fractal de- cles, the magneto-static energy of the cluster can be rep- scriptors D and k . By definition, the total number of f f resented as the energy of the point interacting dipoles nanoparticle N in the fractal cluster is given by the rela- located at the particle centers r within the cluster. Then, D i tion N ¼ k 2R =D , where D is the fractal dimen- p f g f the magneto-dipole interacting energy is sion, k is the fractal prefactor, and R being the radius of f g →→ →→ →→ 2 2 gyration. It is defined [43] via the mean square of the α α −3 α n α n i ij j ij M V i W ¼ ; ð7Þ distances between the particle centers and the geomet- 3 → → r −r i j i≠j rical center of mass of the aggregate. In this paper, the fractal clusters with various fractal descriptors were cre- where n is the unit vector along the line connecting the ij ated using the well-known Filippov et al.’s algorithm centers of i-th and j-th particles, respectively. [43]. As an example, Fig. 1b shows the geometrical struc- Thus, the effective magnetic field acting on the i-th ture of fractal cluster with fractal descriptors D = 2.1 and nanoparticle of the cluster is given by k = 1.3 consisting of N = 90 single-domain nanoparticles. f p → → →→ → H ¼ H α e e þ H sinðÞ ωt Geometrically, it seems that the main difference between a ef ;i i i i 0 →→ → 3D and fractal clusters is that in the latter case, every α −3 α n n j ij ij þM V : nanoparticle has at least one neighbor located at the clos- s → → r −r j≠i i j est possible distance between nanoparticle centers equal to the nanoparticle diameter D. ð8Þ Dynamics of unit magnetization vector α of i-th where H =2 K/M is the particle anisotropy field. a s. single-domain nanoparticle of the cluster is determined The thermal fields, H , i = 1, 2...N , acting on various by the stochastic Landau–Lifshitz (LL) equation th;i nanoparticles of the cluster are statistically independent, ∂α → → i → → with the following statistical properties [39] of their ¼ −γ α  H þ H Þ−κγ α ef ;i i 1 th;i 1 ∂t components for every nanoparticle → → α  H þ H ; i ef ;i th;i DE DE ðÞ α ðÞ α ðÞ β H ðÞ t ¼ 0; H ðÞ t H ðÞ t ð3Þ th th th 2k Tκ ¼ δ δðÞ t−t ; α; β ¼ðÞ x; y; z : ð9Þ αβ 1 where γ is the gyromagnetic ratio, κ is phenomenological γM V damping parameter, γ = γ/(1+κ ), H is the effective ef ;i Here, k is the Boltzmann constant, δ is the Kroneker magnetic field and H is the thermal field. The effect- B αβ th;i symbol, and δ(t) is the delta function. ive magnetic field acting on a separate nanoparticle can The procedure for solving stochastic differential Eqs. be calculated as a derivative of the total cluster energy (3), (8), and (9) is described in detail in Refs. [13, 40, 41]. ∂W H ¼ − : ð4Þ ef ;i Results and Discussion VM ∂α Non-Interacting Iron Oxide Nanoparticles The totalmagneticenergy ofthe clusterW=W +W +W a Z m Consider a dilute assembly of superparamagnetic nano- is a sum of the magneto-crystalline anisotropy energy W , a particles with an average diameter D. The particles are Zeeman energy W of the particles in applied magnetic Z assumed to be tightly packed in a surrounding media, field H sinðÞ ωt , and the energy of mutual magneto-dipole and their easy anisotropy axes are randomly oriented in interaction of the particles W . space. The hysteresis loop of such an assembly in an m Usov et al. Nanoscale Research Letters (2017) 12:489 Page 5 of 8 alternating magnetic field H = H sin(ωt) can be calcu- Fig. 2 shows, for the assembly of non-interacting iron lated [10] using Eqs. (1) and (2). This approach, due to oxide nanoparticles, the peak of the energy absorption in its simplicity, allows one to carry out detailed calcula- an alternating magnetic field corresponds to particles tions of the assembly hysteresis loops for various particle with diameter D = 20 nm. Therefore, first we calculated sizes depending on frequency and amplitude of the alter- the hysteresis loops of an assembly of 3D clusters with nating magnetic field. In the calculations performed, in particle diameter D = 20 nm. accordance with the experimental data [2–6], the satur- Figure 3a shows the evolution of the assembly hyster- ation magnetization of iron oxide nanoparticles is as- esis loops depending on the average distance between sumed to be M =70Am /kg, the magnetic anisotropy the nanoparticle centers D at the fixed value of the s. av 4 3 constant being K =10 J/m . The assembly temperature particle diameter D. The frequency and amplitude of is T = 300 K, and the nanoparticle diameters are in the alternating magnetic field are fixed at f = 400 kHz and range D =10–30 nm. These parameters seem typical for H = 8 kA/m, respectively. The number of particles in experiments carried out on iron oxide nanoparticles. the clusters equals N = 40. The calculations are carried Figure 2 shows the SAR of non-interacting assemblies of out at T = 300 K, and magnetic damping constant is iron oxide nanoparticles at various frequencies at a fixed taken to be κ = 0.5. amplitude of alternating magnetic field, H = 8 kA/m. As Evidently, the decrease of the average distance between can be seen, for the range of frequencies that are charac- the nanoparticles of the cluster leads to an increase of teristic for magnetic hyperthermia, f =200–500 kHz, SAR intensity of the magneto-dipole interaction within the has a maximum for the assembly of iron oxide nanoparti- cluster. Note that for N = 40, the ratios D /D specified p av cles with diameters D =20–21 nm. It is notable that even in Fig. 3a correspond to the cluster packing densities at relatively moderate amplitude of an alternating mag- η = 0.005, 0.04, and 0.32, correspondingly. One can see netic field, the assembly SAR reaches sufficiently high in Fig. 3a that the hysteresis loop area rapidly decreases values, 350–450 kW/kg, if the nanoparticle diameters are as a function of the parameter η. For comparison, Fig. 3a chosen properly. also shows the hysteresis loop 4, calculated for an However, the experimentally measured SAR values for assembly of non-interacting particles, i.e., in the limit assemblies of iron oxides nanoparticles are, as a rule, sig- D /D → ∞, N = const, using Eqs. (1) and (2). av p nificantly below [18–21] these theoretical values. As we One can see that the hysteresis loop 3 (η = 0.005) in shall see in the next section, this fact can be explained Fig. 3a turns out to be close to the hysteresis loop of the [22–38] by the influence of strong magneto-dipole inter- assembly of non-interacting nanoparticles. Therefore, in action in dense assemblies of magnetic nanoparticles. the case η ≤ 0.005 the magneto-dipole interaction of the nanoparticles within the cluster can be neglected. How- Assembly of 3D Clusters ever, for η ≥ 0.04 the magneto-dipole interaction has a Consider now the hysteresis loops of a dilute assembly significant influence on the properties of an assembly of of 3D random clusters having easy anisotropy axes of in- random 3D clusters. A similar evolution of the assembly dividual nanoparticles randomly oriented in space. As hysteresis loops has been obtained also for the frequen- cies f = 300 and 500 kHz, respectively. The hysteresis loops shown in Fig. 3a are calculated for different ratios D /D, but for the fixed number of f = 200 kHz av H = 8 kA/m nanoparticles in the cluster N = 40. However, the de- f = 300 kHz 400 p T = 300 K f = 400 kHz tailed computer simulations show that the shape of the f = 500 kHz hysteresis loop of a dilute assembly of random 3D clus- ters is practically unchanged, if the number of particles, N >> 1, and the radius of the cluster R are changed so p cl that the nanoparticle packing density η remains con- stant. Therefore, the hysteresis loop of dilute assembly of random 3D clusters depends mainly on the cluster packing density η. Figure 3b shows the SAR of assemblies of random 16 18 20 22 24 clusters of iron oxide nanoparticles for different η D (nm) values. The SAR of the assembly is calculated [10] as Fig. 2 The specific absorption rate of non-interacting assembly of SAR = M fA/ρ,where A is the hysteresis loop area in the iron oxides nanoparticles, obtained by means of Eqs. (1) and (2), as a variables (M/M , H), ρ being the density of iron oxide s. function of average particle diameter at different frequencies of the 3 3 nanoparticles which is assumed to be ρ =5× 10 kg/m . alternating magnetic field As Fig. 3b shows, the SAR decreases as a function of η SAR (kW/kg) Usov et al. Nanoscale Research Letters (2017) 12:489 Page 6 of 8 ab Fig. 3 (a) Evolution of the hysteresis loops of dilute assembly of clusters of iron oxide nanoparticles with diameter D = 20 nm for various ratios D /D:(1) D /D = 1.46; (2) D /D = 2.92; (3) D /D = 5.84. Hysteresis loop 4 corresponds to assembly of non-interacting nanoparticles of the same av av av av diameter. It is calculated by means of Eqs. (1) and (2). (b) SAR as a function of the average nanoparticle diameter D for dilute assemblies of clusters of nanoparticles with different packing density η due to an increase of the intensity of magneto-dipole nanoparticles, the SAR as a function of the particle interaction within the clusters. At the same time, the diameter also decreases considerably with respect to that dependence of the assembly SAR on the average par- of the assembly of non-interacting nanoparticles. How- ticle diameter still remains, though it becomes less ever, in contrast to the assembly of 3D clusters, the peak pronounced. values of SAR are shifted systematically to smaller par- For small values of η ≤ 0.005, the SAR of random as- ticle diameters, except for the case of fractal dimension sembly of 3D clusters actually coincides with that one for D = 2.7, which is close to the case of 3D clusters with an assembly of non-interacting nanoparticles, shown in D = 3.0. It is interesting to note also that for non- Fig. 2. On the other hand, SAR falls about six times when optimal nanoparticle diameters, for example, for nano- the cluster packing density increases up to η =0.32. Then, particles with diameters D ≤ 17 nm, the influence of it becomes close to typical SAR values ~ 50–100 kW/kg, magneto-dipole interaction leads to increase of the SAR which are obtained in a number of experiments [5, 18–21] with respect to the case of assembly of non-interacting with iron oxide nanoparticle assemblies. nanoparticles, as the SAR of the assembly of non- interacting nanoparticles is very small for nanoparticles Assembly of Fractal Clusters with diameters D ≤ 17 nm. Similar calculations were carried out for dilute assem- The calculations shown in Fig. 4 were carried out blies of fractal clusters of nanoparticles with various assuming the existence of thin non-magnetic shells with fractal descriptors. As Fig. 4 shows, for fractal clusters of thickness t = 1 nm at the surface of magnetic nanoparti- Sh cles. This prevents the nanoparticles of the fractal cluster from direct exchange interaction. Evidently, the increase D = 1.8, k = 1.5 H = 8 kA/m f f 0 of the non-magnetic shell thickness reduces the intensity D = 2.1, k = 1.3 f = 300 kHz f f of the magneto- dipole interaction of closest nanoparti- D = 2.7, k = 0.5 Np = 90 cles, as the average distance between the magnetic cores f f on the nanoparticles increases. Figure 5 shows that the in- Non interacting crease of the non-magnetic shell thickness is a proper way nanoparticles to raise the SAR of the assembly of fractal clusters of nanoparticles. Namely, for sufficiently large thickness of non-magnetic shells the dependence of the SAR on the particle diameter resembles that for weakly interacting magnetic nanoparticles. This fact may be important for the application of magnetic nanoparticle assemblies in 14 16 18 20 22 24 magnetic hyperthermia. D (nm) Fig. 4 SAR as a function of the average nanoparticle diameter D for Conclusions dilute assemblies of fractal clusters of nanoparticles with various fractal The main conclusion of this study is that the SAR of a descriptors. The SAR of the assembly of non-interacting nanoparticles dilute assembly of clusters of magnetic nanoparticles in is calculated by means of Eqs. (1) and (2) alternating magnetic field is significantly reduced with SAR (kW/kg) Usov et al. Nanoscale Research Letters (2017) 12:489 Page 7 of 8 fluctuations. The cluster model studied allows obvious generalization that can make it more practical. First, it is Non interacting D = 2.1 f necessary to take into account the size distribution of t_Sh = 1 nm k = 1.3 t_Sh = 5 nm magnetic nanoparticles in the assembly. Second, in some t_Sh = 10 nm cases exchange interaction may exist between neighbor- ing nanoparticles of the cluster if they are in direct H = 8 kA/m atomic contact. The theoretical results obtained in this study seem to f = 300 kHz be in a satisfactory agreement with recent experimental data [35] for iron oxide nanoparticles of optimal diame- ters. Indeed, according to Ref. [35], the SAR of the iron oxide nanoparticles increases with the average diameter of the nanoparticles and peaks for nanoparticles with 14 16 18 20 22 24 mean diameter D =20–21 nm. In addition, the SAR D (nm) decreases [35] with a decrease in the average distance Fig. 5 The dependence of the SAR of dilute assembly of fractal clusters between the nanoparticles due to increasing intensity of on the thickness t of the non-magnetic shells at the surface of the Sh the magneto-dipole interaction. nanoparticles. The SAR of the assembly of non-interacting nanoparticles is calculated by means of Eqs. (1) and (2) Unfortunately, in some experimental studies [5, 21] carried out to optimize the properties of magnetic nano- particles for use in magnetic hyperthermia, often do not increasing of the intensity of magneto-dipole interaction take into account the theoretical predictions [10, 11] in the clusters. For usual 3D clusters of nanoparticles, about significant dependence of the assembly SAR on the intensity of the magneto-dipole interaction can be the characteristic size of the magnetic nanoparticles. As characterized by dimensionless packing density, η = shown in this paper, this dependence can be substantial N V/V =(D/D ) . The latter determines the average even for rather dense nanoparticle assemblies. From a p cl av distance between the nanoparticles of the cluster. The theoretical point of view, it is obvious [10] that the as- calculations show that for the assembly of random 3D sembly of iron oxide nanoparticles with very small, clusters, the energy absorption peak, which for iron D ≤ 10 nm, or too big, D ≥ 30 nm diameters can hardly oxide nanoparticles corresponds to particles with aver- provide a sufficiently high SAR values for typical for age diameter D = 20 nm, is reduced about six times magnetic hyperthermia frequencies, f = 200–600 kHz, when the packing density increases from η = 0.005 up to and magnetic field amplitudes H ~ 8 kA/m. The cre- η = 0.32. The dependence of the assembly SAR on the ation of mono-crystalline iron oxide nanoparticles with mean nanoparticle diameter is retained with increase of sharp size distribution near the optimal diameter has to η, but becomes less pronounced. be promising for application in magnetic hyperthermia. For dilute assemblies of fractal clusters of magnetic Acknowledgements nanoparticles, the SAR values also decrease several times The authors gratefully acknowledge the financial support of the Ministry of irrespective on the fractal descriptors of the assembly. In Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISIS”,contract № K2-2015-018. addition, the peak values of SAR are shifted systematically to smaller particle diameters, as a rule. It is important to Funding note, however, that the increase of the non-magnetic shell The funding of this study is from Contract № K2–2015-018 of the Ministry of thickness at the nanoparticle surfaces restores the SAR Education and Science of the Russian Federation. values close to that of the assembly of weakly interacting Availability of Data and Materials nanoparticles. This fact can be important for various bio- Not applicable medical applications of magnetic nanoparticle assemblies. The model considered in this paper takes into account Authors’ Contributions the geometrical structure of nanoparticle assemblies The manuscript was written through the contributions of all authors. All authors have read and approved the final version of the manuscript. observed experimentally in biological media [4, 8, 9] (in particular in tumors), i.e., the agglomeration of nanopar- Ethics Approval and Consent to Participate ticles in a sufficiently dense fractal clusters of different Not applicable sizes, with different numbers of nanoparticles in the Consent for Publication clusters. The stochastic LL Eq. (3) accurately describes Not applicable the real dynamics of the magnetic moments of nanopar- ticles taking into account both the magneto-dipole inter- Competing Interests action between the particles and the effect of thermal The authors declare that they have no competing interests. SAR (kW/kg) Usov et al. Nanoscale Research Letters (2017) 12:489 Page 8 of 8 Publisher’sNote absorption in the dispersion of iron oxide nanoparticles in a viscous Springer Nature remains neutral with regard to jurisdictional claims in medium. J Magn Magn Mater 374:508–515 published maps and institutional affiliations. 22. Gudoshnikov SA, Liubimov BY, Usov NA (2012) Hysteresis losses in a dense superparamagnetic nanoparticle assembly. AIP Advances 2:012143 Received: 24 May 2017 Accepted: 2 August 2017 23. Gudoshnikov SA, Liubimov BY, Popova AV, Usov NA (2012) The influence of a demagnetizing field on hysteresis losses in a dense assembly of superparamagnetic nanoparticles. 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Nanoscale Research LettersSpringer Journals

Published: Aug 14, 2017

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