Non-Uniform Hysteresis in Small Clusters of Magnetic Nanoparticles
NNon-Uniform Hysteresis in Small Clusters of MagneticNanoparticles
Manish Anand ∗ Department of Physics, Bihar National College,Patna University, Patna-800004, India. (Dated: March 1, 2021)
Abstract
Using first-principle calculations and kinetic Monte Carlo simulation, we study the local andaveraged hysteresis in tiny clusters of k magnetic nanoparticles (MNPs) or k -mers. We also analyzethe variation of local dipolar field acting on the constituent nanoparticles as a function of theexternal magnetic field. The dipolar interaction is found to promote chain-like arrangement insuch a cluster. Irrespective of cluster size, the local hysteresis response depends strongly on thecorresponding dipolar field acted on a nanoparticle. In a small k -mer, there is a wide variation inlocal hysteresis as a function of nanoparticle position. On the other hand, the local hysteresis ismore uniform for larger k -mer, except for MNPs at the boundary. In the case of superparamagneticnanoparticle and weak dipolar interaction, the local hysteresis loop area A i is minimal and dependsweakly on the k -mer size. While for ferromagnetic counterpart, A i is considerably large even forweakly interacting MNPs. The value of A i is found to be directly proportional to the dipolar fieldacting on the nanoparticle. The dipolar interaction and k -mer size also enhances the coercivity andremanence. There is always an increase in A i with clutser size and dipolar interaction strength.Similarly, the averaged hysteresis loop area A also depends strongly on the k -mer size, particle sizeand dipolar interaction strength. A and A i always increase with k -mer size and dipolar interactionstrength. Interestingly, the value of A saturates for k ≥
20 and considerable dipolar interactionirrespective of particle size. We believe that the present work would help understand the intricaterole of dipolar interaction on hysteresis and organizational structure of MNPs and their usage indrug delivery and hyperthermia applications. ∗ [email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b . INTRODUCTION Magnetic nanoparticles (MNPs) have tremendous potential in diagnostic and therapeuticapplications [1–6]. The reasons behind their applicability are many [7, 8]. For instance, theycan be bound to drugs, dyes and biological entities, thereby providing opportunities for site-specific drug delivery, improved quality of magnetic resonance imaging, manipulation ofcell membranes, etc. [9–11]. Most importantly, they can be made to dissipate heat whensubjected to an oscillating magnetic field. Therefore, when targeted on malignant tumours,MNPs can kill the cancerous cells or introduce a modest rise in temperature to increase theefficacy of drug reactions with the body. This phenomenon results in selective warming ofthe local area and is usually referred to as hyperthermia in the medical literature [12–14].Several procedures, often ad-hoc, are being used for hyperthermia. Usually, dilute so-lutions of MNPs (coated, functionalized and often bound to chemotherapeutic drugs) areinjected into the bloodstream and magnetically targeted to affected tissues or organs [15, 16].An advantage here is their rapid clearance from the body by way of renal and biliary excre-tions. A significant disadvantage, however, is the danger of damage to healthy cells. Thereis also the possibility of insufficient heating and inefficient drug delivery due to particle dis-persion. There has been a surge of activity in recent years to design temperature-sensitiveliposomes to overcome this shortcoming [17–21]. The latter are lipid sacs encapsulatingfluids containing chemotherapeutic drugs and MNPs. The membranes are chosen to be bio-compatible, with a melting temperature slightly above the body temperature. On applyingan oscillating magnetic field, the contents are released when the heat generated by the en-closed MNPs is sufficient to melt the lipid membrane. Therefore, these micron-sized ferrieshold tremendous promise, as they combine the advantages of targeted drug delivery withlocalized hyperthermia treatment.Many theoretical descriptions have emerged to understand heat dissipation in such asystem. These are generally based on single-particle models which ignore the inter-particledipole-dipole interactions [22–24]. However, wide discrepancies between experimental mea-surements of heat dissipation and corresponding theoretical calculations have been reportedin the literature [25, 26]. Investigations using sophisticated experimental techniques suchas electron magnetic resonance (EMR), transmission electron microscopy (TEM), zero-fieldbirefringence, etc. have indicated small agglomerates of MNPs in these carriers [27, 28]. It2s not unexpected, recalling the presence of the ubiquitous dipolar interaction [29–31]. Thelatter is long-ranged and anisotropic, which has a diverse effect on the morphologies andmagnetic properties of an assembly of MNPs [32–34]. Further, the constituting MNPs mayform clusters in many different ways. Therefore, there is a plethora of geometric configura-tions and magnetic moments orientations. Consequently, it is crucial to understand the roleof dipolar interactions, as a function of cluster’s geometry and the orientation of magneticmoments, to identify the efficient heat generators from the inefficient ones to optimise heatdissipation for their efficient usage in hyperthermia applications.In MNPs assembly, the impact of dipolar interaction should not be overlooked as clus-tering of nanoparticles is also caused by the cellular environment and other factors suchas the spatial variation in the applied magnetic field, nanoparticle-synthesis procedures,etc. [35, 36]. As a consequence, the dipolar interaction drastically affects the amount of heatdissipation in such assembly. Therefore, the study of magnetic hysteresis as a function ofdipolar interaction is equally important for better usage of MNPs in hyperthermia applica-tions. Generally, such a system’s heating efficiency is measured by the specific absorptionratio (SAR), which is related to the hysteresis loop area and the frequency of the appliedalternating magnetic field [37]. On the other hand, some recent works stress the importanceof local heat dissipation by the constituent nanoparticles [38, 39]. Recent studies also sug-gest that the nanoparticle’s local heat is way more useful than the entire system’s averagedheating [40–43]. Therefore, it is crucial to understand the role of dipolar interaction onthe local (individual nanoparticle) and the averaged hysteresis response of the underlyingsystem. To probe it more microscopically, it is also valuable to understand the variation oflocal dipolar field acted on each nanoparticle as a function of the applied magnetic field.This paper aims to understand the consequences of dipolar interactions and their im-plications on the organizational structure of small cluster of nanoparticles; local (due tothe individual nanoparticle) and averaged hysteresis response of such a system. We alsostudy the detailed mechanism of the variation of local dipolar field acting on the constituentnanoparticle as a function of the external magnetic field. We first identify the low energyconfigurations of clusters of k MNPs or k -mers ( k = 2, 3, 4, 5, etc.) by a first principlecalculation. We then use the kinetic Monte Carlo (kMC) simulation technique to study thelocal and the global hysteresis behaviour as a function of dipolar interaction strength, sizeof k -mer and particle size. To probe further, we also perform a detailed analysis of the3ocal dipolar field acted on each nanoparticle by all other MNPs present in the system as afunction of the external magnetic field.The rest of the paper is organized as follows. In Section II, we discuss the methodologiesrequired to evaluate the low-energy configurations. kMC algorithm is also discussed in brief.In Section III, we present the numerical results. Finally, in Section IV, we provide a summaryand the conclusion of our work. II. THEORETICAL FRAMEWORK
The energy associated with a single nanoparticle due to magnetocrystalline anisotropy isgiven by [37, 44] E K = K eff V sin θ (1)Here K eff is the uniaxial anisotropy constant, V is the volume of the nanoparticle, and θ isthe angle between anisotropy axis and magnetic moment. Clustering is relatively commonin suspensions of MNPs or magnetic fluids, as observed in electron microscopy or lightscattering experiments [45, 46]. As the particles are magnetised, they adhere and formagglomerates. We can calculate the dipolar interaction energy E dip of i th nanoparticle in anassembly of MNPs as [30, 31, 47] E dip = µ o π (cid:88) j, j (cid:54) = i (cid:20) (cid:126)µ i · (cid:126)µ j s − (cid:126)µ i · (cid:126)s ) ( (cid:126)µ j · (cid:126)s ) s (cid:21) . (2)Here µ o is the permeability of free space; (cid:126)µ i and (cid:126)µ j are the magnetic moment vectors of i th and j th nanoparticle respectively, and s is the center-to-center separation between µ i and µ j . The particle has a magnetic moment µ = M s V , M s is the saturation magnetization.The corresponding dipolar field µ o (cid:126)H dip is given by [31, 48] µ o (cid:126)H dip = µ o π (cid:88) j, j (cid:54) = i (cid:20) (cid:126)µ j · (cid:126)s ) (cid:126)ss − (cid:126)µ j s (cid:21) , (3)When the particles are in contact and the moments are aligned, (cid:126)µ i · (cid:126)µ j = µ and( (cid:126)µ i · (cid:126)s ) ( (cid:126)µ j · (cid:126)s ) = µ s = µ D . In the latter case, D is the diameter of the nanoparticle.In such a case, the dipole-dipole contact energy E dd can be evaluated using the followingexpression [31]: E dd = 112 µ o M s V. (4)4s it is directly proportional to the magnetic volume and M s , smaller particles are less likelyto aggregate. However, the aggregation can be disrupted by the available thermal energy k B T r , where k B is the Boltzmann constant and T r is the temperature. The effectiveness ofdisruption is governed by the ratio of the thermal and dipole-dipole contact energy [31]: E R = k B T r E dd = 12 k B T r µ o M s V . (5)To escape agglomeration, E R must be greater than unity [30, 31] yielding: D ∗ c ≤ (cid:0) k B T r /πµ o M s (cid:1) / . (6)Thus particles with diameter D ∗ c are on the agglomerating threshold, but those with diame-ters less than D ∗ c manage to escape this fate. They at the most form small (nano) clusters,e.g., dimers, trimers, tetramers, etc. The mean cluster size is expected to be governed by abalance between the energies responsible for the complementary mechanisms of aggregationand fragmentation. These can be further tailored by choice of surfactant coating and itsthickness [30, 49–51].We apply an oscillating magnetic field to probe the hysteresis behaviour of minimumenergy geometrical configurations of k -mer. It is given by [29] µ o H = µ o H max cos ωt, (7)where µ o H max and ω = 2 πν are the amplitude and angular frequency of the applied magneticfield respectively, ν is the linear frequency, and t is the time. The total energy of the i th nanoparticle under the influence of dipolar and external magnetic field is given by [48, 52] E i = K eff V sin θ i + E dip − (cid:126)µ i · µ o (cid:126)H (8)Here θ i is the angle between the anisotropy axis and the i th magnetic moment of the system.Using first-principle calculation, we first identify low energy geometric configurations ofthe tiny cluster in the absence of an external magnetic field. We then implement kMCsimulations to probe the local and the averaged hysteresis response as a function of dipolarinteraction strength, particle size and size of k -mer. To see the effect of dipolar interaction atthe particle level, we also probe the variation of local dipolar field acting on each nanoparticleas a function of the applied magnetic field. In the kMC algorithm, dynamics is captured moreaccurately, which is essential to study the dynamic hysteresis response of dipolar interacting5NPs [53]. Using this technique, we can accurately describe the dynamical properties ofMNPs in the superparamagnetic or ferromagnetic regime without any artificial or abruptseparation between them [48]. We have used the same algorithm, which is described ingreater detail in the work of Tan et al. and Anand et al. [48, 52]. Therefore, we do notreiterate it here to avoid repetition. The local hysteresis loop area A i (due to individualnanoparticle) of the i th nanoparticle can be calculated as [6] A i = (cid:73) M i ( H ) dH, (9)The above intergral is evaluated over the entire period of the external magnetic field change. M i ( H ) is the magnetization of i th magnetic nanoparticle at magnetic field H . III. SIMULATIONS RESULTS
We can expect clusters of fewer particles due to the ubiquitous dipole-dipole interactionsin MNPs carriers. These tiny clusters are expected to have significantly different hysteresisbehaviour from the monomers and to see this; it is essential to identify the cluster geometriesand their spin configurations. Although this task is humongous, the following simplificationsmake our analysis tractable. We choose representative geometric arrangements for dimers,trimers and tetramers, pentamers, etc. in two dimensions. We do not expect a significant lossof information on this account as the clusters are tiny. These prototypical arrangements aredepicted in Fig. 1(a). Each cluster geometry is then decorated by magnetic moments. Theyare allowed to assume nine orientations, specified by angles nπ/ n = 0 , , , ..,
8. Thus, if G is the number of geometric configurations for a k -mer, the total configurations are G × k . Forinstance, a trimer could be formed in 4 × = 2916 ways and a tetramer in 12 × = 78732.So the number of configurations in our simplified model is large even for tiny clusters andgrows exponentially with cluster size. The particle is assumed to have a diameter D = 8nm to perform model calculations. The other material parameters are: K eff = 13 × Jm − and M s = 4 . × Am − . These parameters correspond to magnetite (Fe O ),one of the best candidates for biomedical applications due to its biocompatibility [54]. Asdiscussed above, we have used Eq. (2) to evaluate all the possible configurations’s dipolarinteraction energy. At body temperature T b ≈
310 K, the thermal energy k B T b ≈ . × − Joule. So, clusters with energies ∼ O ( k B T r ) will be unstable. Therefore, we consider only6he minimum energy configurations, which are not destroyed by the thermal energy. Wehave shown typical morphologies of the tiny cluster with their dipolar interaction energyin Fig. 1(b) and Fig. 1(c). It is evident that magnetic moments tend to form head to thetail arrangement as the corresponding structure’s dipolar interaction energy is the minimumcompared to the other arrangements. One of the most interesting facts to note is that thechain arrangement of MNPs is the most stable of all the possible geometrical structure. Thisobservation is in perfect agreement with the earlier works [55–57]. Researchers have alsofound the chain-like arrangement of k -mers in liposome [58, 59].From the above observations, it is evident that the chain arrangement of k -mer is anideal candidate to probe the fundamental effects of dipolar interactions on local and aver-aged hysteresis response in the presence of an external alternating magnetic field. Therefore,in the rest of the paper, we study the physical processes involved in local (due to individualnanoparticle) and global hysteresis mechanism under the influence of dipolar interactionin the chain-like structure of k -mer. Our study is primarily focused on gathering and un-derstanding the essential effects caused by dipolar interactions in magnetic hyperthermiaexperiments. The anisotropy axes of the MNPs are assumed to have random orientationsto remain closer to the real experiments. To vary the dipolar interaction, we define a scaleparameter λ = D/d , where d is the centre to centre distance between two consecutive MNPsin a k -mer as shown in the schematic Fig. 1(d). So, λ = 0 . λ = 1 . T r = 300 K and ν = 10 Hz, and the externalmagnetic field is applied along the z -direction [please see Fig. 1(d)]. The applied magneticfield’s amplitude is taken as µ o H max = 0 .
06 T, which is slightly larger than single-particleanisotropy field H K = 2 K eff /M s [37].In Fig. (2) we plot the local (individual nanoparticle), and the averaged magnetic hys-teresis for four values of k -mers ( k = 2, 4, 6 and 10). To probe the effect of dipolarinteraction at the particle level, we also analyze the variation of local dipolar field acted oneach nanoparticle as a function of the external magnetic field. The particle size is taken as D = 8 nm and dipolar interaction strength λ = 0 .
2. The local hysteresis curve and dipolarfield variation are shown with different colour depending on the nanoparticle’s position in aparticular k -mer. The averaged hysteresis curve is shown with black colour. As the particlesize lies in the superparamagnetism regime, and dipolar interaction is negligibly small, the7ocal and averaged hysteresis loop show negligibly small value of the coercive field and re-manent magnetization. The local hysteresis depends strongly on the corresponding dipolarfield acting on a nanoparticle. For instance, the dipolar field acted on each nanoparticle isthe same in the dimer ( k = 2). As a result, the local hysteresis curve is exactly the samefor both particles. Similarly, in the case of tetramer ( k = 4), the dipolar field acted onnanoparticles at both the ends ( i = 1 and i = 4) is the same. In contrast, the remainingtwo nanoparticles experience same dipolar field. Consequently, the local hysteresis curve fornanoparticles positioned at i = 1 and i = 4 is exactly similar to each other; the remainingMNPs have identical hysteresis curves. We can infer similar observation for hexamer ( k = 6)and decamer ( k = 10). These observations are in qualitative agreement with the theoreticalworks of Torche et al. [39]. It is also in qualitative agreement with the works of Vald´es etal. [60]. The averaged hysteresis curve is found to be the mean of local hysteresis curves ofconstituent MNPs. Interestingly, the variation of the local dipolar field acted on individualnanoparticle is the same as that of corresponding magnetic hysteresis response. As the sizeof k -mer increases, there is more uniformity in constituent nanoparticles’s local hysteresisbehaviour except for MNPs at the both ends.Next, we study the hysteresis behaviour for strongly dipolar interacting MNPs. InFig. (3), we plot the local and averaged hysteresis curves for λ = 1 .
0. All the other pa-rameters are the same as that of Fig. (2). Even in the case of large dipolar interactionstrength, the shape of the local hysteresis curve for both the constituent particles is iden-tical for dimer structure. It is due to the fact that the same dipolar field is acting on eachparticle in the dimer ( k = 2). The value of the dipolar field experienced by the nanoparti-cle is way larger than the previous case. The variation of local dipolar field acted on eachnanoparticle is the same as that of corresponding local hysteresis response irrespective of thesize of k -mer. For all the k -mer, the local hysteresis response of the constituting nanoparti-cles is dictated by the dipolar field acted on it. For k >
10, the local hysteresis curves areuniform except for MNPs at both the ends (curve are not shown). The value of the coercivefield and remanent magnetization are also very large irrespective of the size of the k -mer.It can be attributed to the enhanced ferromagnetic coupling between the MNPs. The areaunder the hysteresis curve is also huge as compared to the weakly interacting case.What happens to the averaged and local hysteresis response in the case of the ferromag-netic nanoparticle? To analyze it, we study the magnetic hysteresis and variation of local8ipolar field acted on each nanoparticle in a k -mer as a function of an external magneticfield for D = 24 nm with weak dipolar interaction λ = 0 . λ = 1 .
0) [Fig. (5)]. Even with ferromagnetic nanoparticles, the local and globalhysteresis response is dictated by the dipolar field acted on the constituent MNPs. Thevariation of the local dipolar field acted on each nanoparticle is exactly similar to that of thecorresponding hysteresis curve, irrespective of the size of the k -mer and dipolar interactionstrength. Interestingly, the shape of the hysteresis curve is like a perfect square for stronglyinteracting MNPs even with randomly oriented anisotropy. It means that dipolar interac-tion creates an additional anisotropy so-called shape anisotropy which weakens the impactof randomly oriented uniaxial anisotropy. Consequently, remanence also gets enhanced withan increase in the size of k -mer and dipolar interaction strength. The coercive field andremanent magnetization have larger values even in the case of weak dipolar interaction.The hysteresis has less dependence on the size of k -mer for weak dipolar interacting MNPs.The hysteresis loop area is very large compared to superparamagnetic nanoparticle even inthe case of the weak interacting case [please see Fig. (4)]. On the other hand, there is anincrease in the area under the hysteresis curve with an increase in the size of the k -mer forstrongly interacting MNPs. These results could help choose precise values of particle size,and interaction strength to optimize the heat dissipation for drug delivery and hyperthermiaapplications.To quantify the above observations, we analyze the local hysteresis loop area A i of con-stituent MNPs for four values of k -mer ( k = 2 , ,
6, and 10) with particle size D = 8 and 24nm. We have considered two representative values of dipolar interaction strength ( λ = 0 . λ = 1 . A i has an intimate relationship withlocal dipolar field acting on a nanoparticle. For instance, as the dipolar field acted on bothparticles in the dimer ( k = 2) is equal, the corresponding local hysteresis loop area A i isalso equal to each other. We can draw similar observations for other values of k -mer, ir-respective of dipolar interaction strength and particle size. The value of A i is larger forstrongly interacting MNPs as compared to the weakly dipolar interacting case. The sameis true for ferromagnetic nanoparticle D = 24 nm even for weak interacting MNPs. It isclear that A i of the nanoparticles at the boundary is smaller than the central MNPs. It canbe explained from the fact that the boundary MNPs experience a smaller dipolar field than9he MNPs at the centre. For smaller k -mer, there is a large variation in A i as local dipolarfield acted on the nanoparticle in such a case varies rapidly as a function of the positionof MNPs. On the other hand, the local dipolar field acted on the particle in larger k -merhas less variation except for MNPs at the boundary. Therefore, there is more uniformityin A i for larger k -mer. A i is directly proportional to the dipolar field, acting on the indi-vidual particle. The larger the dipolar field, the larger the hysteresis loop area. The localhysteresis loop area A i increases with an increase in the size of k -mer. It also increases withan increase in dipolar interaction strength and particle size. These observations can helptune the dipolar interaction to optimize the local and averaged hysteresis response, whichis essential for hyperthermia applications.Finally, we study the dependence of the averaged hysteresis loop area A on the size of k -mer and dipolar interaction strength. In Fig. (7), we plot A as a function of k and λ for D = 8 and 24 nm. k is changed from 2 to 30, and λ is varied from 0 to 1.0. In thecase of superparamagnetic nanoparticle ( D = 8 nm) and small dipolar interaction strength( λ ≤ . A is very small and depends weakly on the size of k -mer. On the other hand,there is an increase in A with k for strongly interacting MNPs ( λ > . A is more considerable for ferromagnetic nanoparticle ( D = 24 nm) as comparedto superparamagnetic counterpart ( D = 8 nm) even for weakly dipolar interacting MNPs( λ ≤ . A is significantly large for D = 24 nm in comparison with D = 8 nm, irrespectiveof λ and k . Interestingly, the value of A saturates for k >
20, irrespective of D . It is inperfect agreement with the recent work of Vald´es et al. [60]. These findings could be usefulfor optimizing the size of k -mer and interaction strength to control the heat dissipation,which is essential for hyperthermia applications. IV. SUMMARY AND CONCLUSION
Now, we summarize and discuss the main results presented in this work. Using first-principles calculations, we first identified minimum energy configurations of clusters of k MNPs or k -mers (k = 2, 3, 4, 5, etc.). After that, we have used kinetic Monte Carlo simula-tions technique to probe the local (individual nanoparticle) and averaged hysteresis responseas a function of cluster size, dipolar interaction strength and particle size. To investigatethe effect of dipolar interaction at the particle level, we also analyze the variation of the10ocal dipolar field acted on each nanoparticle as a function of the applied magnetic field.Dipolar interaction is found to be the primary factor which helps in clustering of MNPs.One can confine the agglomeration process to fewer MNPs by particle size selection. Evenfor these tiny clusters, there is a possibility of a huge number of configurational arrangments.Of all the possible configurations, chains of k -mers are most stable as they have the lowestinteraction energy. The local hysteresis behaviour is primarily dictated by the dipolar fieldacting on a particle. The variation of the local dipolar field acting on a nanoparticle isprecisely similar to that of the corresponding hysteresis curve. For smaller k -mer, there isa considerable variation in the local dipolar field as a function of nanoparticle’s position inthe k -mer. Consequently, the local hysteresis response also varies rapidly. For superparam-agnetic nanoparticle and weak dipolar interaction, the hysteresis shows zero coercivity andremanence. While for sizeable dipolar interaction strength, the coercive field and remanentmagnetization have significant values. The same is true for ferromagnetic nanoparticle withnegligible dipolar interaction. Interestingly, the shape of the hysteresis curve is like a perfectsquare with large dipolar interaction strength.The local hysteresis loop area A i of a nanoparticle is directly proportional to the corre-sponding local dipolar field. In the case of superparamagnetic nanoparticle and weak dipolarinteraction, A i is small, irrespective of the size of k -mer. On the hand, A i is very large forstrongly interacting nanoparticles because of enhanced ferromagnetic coupling. The sameis true for ferromagnetic nanoparticle with small dipolar interaction strength. In a k -mer( k > A i is the maximum for centrally positioned nanoparticle and its value decrease as wemove towards both ends of the k -mer. There is more uniformity in A i for larger k -mer exceptfor MNPs at both the ends. Similarly, there is a strong dependence of averaged hysteresisloop area A on the dipolar interaction strength. A is very small for small values of dipolarinteraction strength, irrespective of k -mer. There is an increase in A with the size of k -merand λ . In the case of ferromagnetic nanoparticle, A is very large even with small dipolarinteraction strength. Therefore, it is evident that the local and averaged hysteresis loop areaincreases with an increase in dipolar interaction strength. Irrespective of particle size anddipolar interaction strength, the value of A saturates for large size of k -mer ( k > k -mer. We emphasize thatour methodologies are generic and applicable to a various tiny cluster in distinct physicalsettings. Our methodologies could provide a theoretical basis to the often used ad-hocprocedures in therapeutic applications. We believe the similar observations can be drawn forthe diverse clusters as long as the magnetic behaviour is dictated by the dipolar interaction. DATA AVAILABILITY
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2. The local dipolar field’svariation is exactly similar as that of local hysteresis response. The coercive field and remanencemagnetization have negligible values, indicating the dominance of superparamagnetic character. (a)(c)(d) (e)(g)(h) k=4 (b) (f) k=2k=6k=10 D = λ= FIG. 3. The study of local and the averaged hysteresis curve for large dipolar interaction strength λ = 1 . k -mer [(e)-(f)]. The particle size is taken as D = 8 nm. Even in the case ofsizeable dipolar interaction, the local hysteresis curve is intimately related to the amount of dipolarfield acted on it. There is an increase in averaged and local hysteresis loop area with an increasein the size of k -mer. (a)(c)(d) (e)(g)(h) k=2 (b) (f) k=4k=6k=10 D =
24 nm λ= FIG. 4. The local and averaged hysteresis response variation for ferromagnetic nanoparticle ( D =24 nm) and weak dipolar interaction ( λ = 0 . k -mer: k = 2[(a) and (e)], k = 4 [(b) and (f)], k = 6 [(c) and (g)], and k = 10 [(d) and (h)]. It is clearly seen thatthe local and the averaged hysteresis loop area is huge compared to D = 8 nm, even in the caseof weak dipolar interaction. The local hysteresis is also directly proportional to the correspondingdipolar field acted on it. (a)(c)(d) (e)(g)(h) k=2k=6k=10k=4 (b) (f) D =
24 nm λ= FIG. 5. The local and averaged hysteresis curve are shown for strongly interacting MNPs ( λ = 1 . D = 24 nm, and four values of k -mer ( k = 2, 4, 6, and 10) [(a)-(d)]. The correspondingvariation of the local dipolar field as a function of an applied magnetic field is shown in (e)-(h).There is an increase in the hysteresis loop area and dipolar field with the size of k -mer. Irrespectiveof the size of k -mer, the variation of the local dipolar field is the same as that of the correspondinghysteresis response. (a)D=8 nm, λ D=8 nm, D=24 nm, D=24 nm,(b)(c)(d) (e)(f)(g)(h) (i)(j)(k)(l) (m)(n)(o)(p) λ= λ= λ= λ= FIG. 6. We have depicted the value of the local hysteresis loop area A i (due to individual nanopar-ticle) for four values of k -mer: k = 2, 4, 6 and 10. The other parameters are: D = 8 nm, λ = 0 . D = 8 nm, λ = 1 . D = 24 nm, λ = 0 . D = 24 nm, λ = 1 . A i in smaller k -mer, as the local dipolar field variedrapidly with position of nanoparticle in a k -mer. On the other hand, there is more uniformity in A i for the larger size of k -mer. There is an increase in A i with the size of k -mer, particle size anddipolar interaction strength. (a)(b) FIG. 7. The variation of averaged hysteresis loop area A (due to entire system) as a function ofdipolar interaction strength λ and size of k -mer for D = 8 nm [(a)] and D = 24 nm [(b)]. Forsuperparamagnetic nanoparticle ( D = 8 nm) and weak dipolar interaction strength ( λ ≤ . A is significantly small and depends weakly on k . While for ferromagnetic nanoparticle, A is moresignificant even with weakly interacting MNPs. There is an increase in A with k and D . Irrespectiveof particle size, the value of A saturates for k ≥
20 and considerable dipolar interaction strength.20 and considerable dipolar interaction strength.