Increase of magnetic hyperthermia efficiency due to dipolar interactions in low-anisotropy magnetic nanoparticles: Theoretical and experimental results

Increase of magnetic hyperthermia efficiency due to dipolar interactions in low anisotropy magnetic nanoparticles : theoretical and experimental results. B. Mehdaoui, R. P. Tan, A. M effre, J. Carrey *, S. Lachaize, B. Chaudret and M. Resp aud Université de Toulouse; INSA; UPS; LPCNO (Laboratoire de Phy sique et Chimie d es NanoObjets), 135 avenue de Rangueil, F-31077 Toulouse, France and CNRS; UMR 5215; LPCNO, F-31077 Toulouse, France

Abstract: When magnetic n anop articles (MNPs) are single-domain and magnetically indep endent, their magnetic p rop erties and the conditions to optimize their efficiency in magnetic hyp erthermia app lications are now well-understood. However, the influence of magn etic interactions on magnetic hy p erthermia prop erties is still unclear. Here, we rep ort hyp erthermia and highfrequency hy steresis loop measurements on a model system consisting of M NPs with the same size but a vary ing anisotropy , which is an interesting way to tune the relative strength of magnetic interactions. A clear correlation between the M NP anisotropy and the squareness of their hy steresis loop in colloidal solution is observed : the larger the anisotropy, the smaller the squareness. Sin ce low anisotropy MNPs disp lay a squareness high er than the one of magnetically indep endent nanoparticles, magnetic interactions enhance their heatin g p ower in this case. Hysteresis loop calculations of indep endent and coup led M NPs are comp ared to exp erimental results. It is shown that the observed features are a natural consequence of the formation of chains and colu mns of MNPs during hyp erthermia exp eriments: in these structures, when the MNP magnetocristallin e anisotropy is small enough to be dominated by magnetic interactions, the hy steresis loop shap e tends to be rectan gular, which enhan ce their efficien cy. On the contrary , when MNPs do not form chains and columns, magn etic interactions reduces the hy steresis loop squareness and the efficiency of M NPs comp ared to indep endent ones. Our finding can thus exp lain contradictory results in the literature on the influence of magnetic interactions on magnetic hyp erthermia. It also p rovides an alternative exp lanation to some experiments where an enhanced specific absorp tion rate for M NPs in liqu ids has b een found comp ared to the one of MNPs in gels, usually interp reted with some contribution of the brownian motion. The p resent work should improve the understanding and interp retation of magnetic hyp erthermia exp eriments.

Main Text: 1. Introduction M aximizin g the sp ecific absop tion rate (SAR) of nanoparticles in magnetic hyp erthermia and interp reting exp erimental results is a comp lex task since the magn etic p rop erties of assemblies of magnetic nanop articles (MNPs) dep end of a large numb er of p arameters. When MNPs can be considered as sin gle-do main and magnetically indep endent, underlying mechan isms are now well-und erstood [1, 2]. However, several experimental and theoretical

1

studies have shown that the p resence of magnetic interactions between MNPs dramatically influence their SAR [3, 4, 5, 6, 7, 8, 9, 10]. When MNPs are small enou gh to clearly have a sup erp aramagn etic behaviour, the influence of magnetic interactions is relatively simple: magnetic interactions increase their effective anisotropy [3]; in most cases, this is exp ected to enhance their susceptibility and SAR by bringin g them closer to the sup erp aramagn etic/ferro magnetic transition. However, such NPs are not the most suitable for magnetic hyp erthermia and larger NPs close to the sup erp aramagnetic-ferromagn etic transition or well into the ferromagn etic regime have been shown to be more efficient [1]. Influence of magnetic interactions in these latter is more comp lex and ap p arently contradictory results have been found. Exp erimentally , an increase [5], a decrease [7, 8] or a non-monotonic [10] variation of SAR with interactions have been rep orted. In our group, quantitative analy sis of SAR measurements on two different typ es of single-do main M NPs (FeCo and Fe) had led us to conclude that magnetic interactions decrease SAR [11, 12]. From a theoretical p oint of view, most theoretical works agree that the general trend is that SAR decreases with interactions [3, 8, 9, 10] although a limited increase in a restricted range of NP concentration has also been rep orted [9, 10]. Excep t Ref. [9], theoretical studies have been p erformed on isotrop ic assemblies of NPs, despite that it is well-known that app ly ing a magnetic field to a ferrofluid induces the formation of chains or colu mns [13, 14]. The p resent work aims at p resenting a set of both theoretical and exp erimental results related to the influence of magnetic interactions in magnetic hy p erthermia efficiency . The measurements were p erformed on an ensemble of samples in which the size of the MNPs was mainly kept constant but their anisotropy could be varied by changing their co mp osition (Fe, FeCo, FexCy ) and their intimate structure. This is a unique way to tune the relative strength of magnetic interactions and study their influence on hy p erthermia p rop erties. The strength of this work lies in the fo llowin g features: i) high frequ ency hysteresis loop s were measured in the same conditions as hyp erthermia. The hysteresis loop s contain more information on the samp le magnetic p rop erties than basic SAR measurements and are thus a p recious tool to understand the p hy sics of magnetic hy p erthermia. ii) Exp erimental results were compared with hy steresis loop calculations of indep endent and coup led MNPs. In p articular, the fact that MNPs form chains or columns has b een taken into account and is shown to be a k ey element to understand exp erimental results. iii) These M NPs display optimized characteristics for magnetic hyp erthermia and very large SARs, so these results are of real interest for eventual ap plications. Our different app roaches converge to the same conclusion that increasin g the strength of magnetic interactions can imp rove the efficiency of MNPs in colloidal solution, i.e. when the nanop articles are free to arrange in chains and columns. However, this imp rovement only occurs when dip olar interactions are sufficiently strong to overcome the magn etocristalline anisotropy of NPs and increase the squareness of their hy steresis loop . This is p recisely the case for low anisotropy NPs, which are the p referred cand idates for magnetic hy perthermia. However, our numerical simu lations eviden ce that the conclusion is opp osite if the nanop articles are in isotrop ic conditions, i.e. if they are not able to move because they are blocked in a gel or in a cell for instance. Thus our results are not contradictory with p revious theoretical works but only more gen eral; they p rovide an alternative exp lanation for the increase of SAR of nanop articles in liquid comp ared to a gel, which has been often attributed in the literature to the contribution of Brownian motion. Finally, they should imp rove the understanding of both in vivo and in vitro hyp erthermia exp eriments and of the mechanisms of magnetic hyp erthermia. 2

2. Methods Samples under study are collo idal solutions comp osed of Fe, FexCy , Fe@FexCy (an iron core surrounded by a FexCy shell) and FeCo nanop articles. Their sy nthesis methods and detailed structural characteristizations have been described in Refs. [11] [15] and [16]. Mean diameter and size distribution of the different samp les were measured by transmission electron microscop y (TEM). Standard magnetic characterizations were p erformed using a SQUID magnetometer on a p owder. For SAR and high-frequency hy steresis loop measurements, 10 mg of p owder were diluted in 0.5 ml of mesity lene inside a schlen ck tube filled with Ar to protect the samples against oxidation. SAR and high-frequ ency hy steresis loop measurements were p erformed on the same sample. SAR measurements were p erformed on a home-mad e electromagn et sp ecially designed for hyp erthermia exp eriments [17]. High-frequency hysteresis loop s were performed using a setup described elsewhere [18]. Hysteresis loop s measured on this setup have p reviously shown to be consistent with SAR measurements [18]. Hysteresis loop calculations of magnetically indep endent MNPs have been described in details in Ref. [1] and fully tested; they are able to calculate the hy steresis loop area A for an assembly of magnetically indep endent sp herical uniaxial NPs with their anisotropy axis randomly distributed in space or aligned with the magnetic field. Hysteresis loop calculations of coup led nanop articles are based on the solving of the Landau-Lifshitz-Gilbert (LLG) equation. Details on the mod el used and results on the influence of magnetic interactions on the magnetic and magnetotransp ort p rop erties of 2D isotrop ic assemblies of M NPs can be found in Ref. [19].

3. Results Sample names and characterization results are summarized in Table 1. TEM micrograp hs are shown in Fig. 1. All the samp les are comp osed of M NPs with a similar diameter, which varies slightly around 13.5 nm. The saturation magnetization p er unit mass σS of samp le 1 is 140±8 2 -1 2 -1 Am .kg , well b elow the bulk value (240 Am kg ), due to an imp erfect alloyin g of Fe and Co atoms and an amorphous structure of the NPs [20, 21]. Their amorp hous structure also exp lains their very low anisotropy value, p reviously deduced from hy p erthermia exp eriments [11]. Extensive structural characterization by X-Ray diffraction, high-resolution TEM and M össbauer sp ectroscopy on Fe, FexCy, and Fe@FexCy samples (samp les 2-6) have been p ublished in Ref [16]. In this series, ch an ges in the sy nthesis p arameters leads to chan ge in the carbon content and p hases of the M NPs. Sample 2 contains pure iron monocry stalline NPs. Sample 3 contains NPs comp osed of a crystalline Fe2.2C core and a thin amorp hous Fe2.5C shell. In samp les 4-6, the carbon content and p hases vary . Since magn etocristalline an isotropy is very sensitive to exact comp osition, its value is strongly modulated in the series of Fe, FexCy, and Fe@FexCy samples. M agnetization saturation is close to the one of bulk Fe for samp le 2 (Fe) and samp les 4-6 (Fe@FexCy ) and is reduced in sample 3 (FexCy ). SAR and high-frequency hy steresis loop measurements have been p erformed at the same frequency f = 54 kHz. In SAR measurements, the maximum app lied magnetic field µ0H m ax was varied between 0 and 60 mT. When a large magnetic field is ap plied during hyp erthermia 3

exp eriments, it is seen with the naked ey e that MNPs self organ ize into needles, similarly to what was rep orted in Ref [13]. To evidence it, we hav e dep osited a drop of colloidal solution of Fe MNPs on a TEM grid and let it dry under the application of a 40 mT alternating magnetic field at 54 kHz (see Fig. 2). Although the result obtained cannot be fully rep resentative of what happens during hy p erthermia exp eriments, it still gives an idea of the magnetic field influence on the MNP organization during hy p erthermia. In Fig. 3(a), SAR values as a function of the magnetic field measured on the different samp les are shown. High-frequency hy steresis loops were measured at µ 0H m ax= 42 mT. Raw and normalized hy steresis loop s are disp lay ed in Figs. 3(b) and 3(c), resp ectively . From the area A of these hy steresis loop s, it is also possible to calculate a SAR value at 42 mT, using the equation SAR = Af . The calculated values are shown in Table 1 along with SAR values deduced from temp erature measurements. A reasonable agreement is found between the two methods, similarly to what was found in Ref. [18]. The shap e of the hysteresis loops combined with the SAR measurements for each samp le lead to very useful information on the magnetic p rop erties of each samp le. Excep t samp le 3, all the samp les disp lays a behavior typ ical of the ferromagnetic regime [1, 11, 12], as evidenced by i) the measurement of widely opened saturated hy steresis loop s and ii) an abrup t increase in SAR(µ 0Hm ax) curves followed by a saturation. For samp le 3, which is mainly comp osed of cry stalline Fe2.2C, the different behavior is p robably due to an anisotropy value much higher than the other samp les – and then a coercive field higher than our maximum ap p lied field. This is what will be assumed in the remainin g of this article. From the hysteresis loop s shown in Figs. 3(a) and 3(b), a second observation can b e made : samp les 1, 4 and 5 disp lays a hysteresis loop typ ical of M NPs with an anisotropy axis oriented with the magnetic field, characterized by both large squaren ess and remnant magnetization MR. On the contrary , samp le 2 disp lays a shap e typ ical of MNPs with randomly oriented anisotropy axis, characterized by lower squareness and remanence MR. Samp le 6 is intermediate between these two behaviors. Sample 3, as discussed above, is not saturated. In the following, we will p erform a detailed an aly sis and interp retation of these various hysteresis loop s. The first step of this analy sis is to try to provide an estimation of the effective anisotropy Keff of the MNPs, which will be usefu l for the rest of the an aly sis. Keff is derived by two different ways, one based on the hyp erthermia experiments and the other one on the hy steresis loop s. For the first one, we use the highest slop e of the SAR(µ 0Hm ax) function to estimate the anisotropy , similarly to Ref. [12]. For samp le 2 and 6 the equation valid for randomly oriented M NPs was used:

µ 0 H CHyp

µ 0 K eff   k B T  kBT = 0.926 1−  ln   M S   K eff V  4µ 0 H CHyp M S Vfτ 0 

   

0 .8

  ,  

(1)

where µ 0 H CHyp is the point of highest slope in SAR(µ 0H max) functions, V the MNP volume, τ0 the frequency factor of the Néel-Brown relaxation time, M S the magnetization per unit of volume and T the temperature. The experimental M S value was used. For samples 1, 4 and 5, the following equation was used, which was derived similarly to Equ. (1) using numerical simulations [12]

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and is valid for M NPs aligned with the magnetic field. Appendix A describes the origin of this equation: 0 .5    µ0 K eff   k B T  k BT  . µ 0 H CHyp = 1.85 1−  ln (2) M S   Keff V  4µ 0 H CHyp M S Vfτ 0      The second method to deduce Keff uses the coercive f ield v alues fro m the hy steresis loop s µ 0H C. The following equation was used for samp les 2 and 6 [1]:

µ 0H C

µ 0 Keff   k BT  kBT = 0.96 1−  ln   MS  K V  4 µ0 H max M S Vfτ 0   eff

   

0. 8

   

(3)

This one was used for samples 1, 4 and 5:

µ0 K eff   k B T  k BT µ 0H C = 2 1−  ln  MS K V  4µ 0 H max M S Vfτ 0   eff

    

0 .5

   

(4)

For sample 3, since both µ0H C and µ 0 H CHyp are above our maximum available magnetic field, a lower limit for K eff was provided using Equs. (1) and (3). The values found for K eff using these two methods are summarized in Table 1. Please note that these Keff value should not be taken as the one of individual M NPs, but the one of MNPs inside the assembly. In other words, K eff is the value that individual MNPs would have to display a coercive field similar to the one measured. A similar approach was for instance used in Ref. [3] to analyse the influence of magnetic interactions on anisotropy fields and barriers. Except for sample 2, the methods based on hyperthermia measurements and the one based on hysteresis loops converge to K eff values close to each other. For sample 2, the discrepancy probably comes from the large error bar on the µ 0 H CHyp value due to the smooth shape of the SAR(µ 0H max) function. We come now to the description of the most interesting features of our set of experimental data. In Figs. 4(a) and 4(b), the values of remnant magnetization M R normalized by the magnetization at 42 mT Msat and of the slope of the hysteresis loop at the coercive field are plotted as a function of K eff . For Keff , the value found using magnetic measurements is taken since the error on µ 0H C is lower than on µ 0 H CHyp . It is clear from these figures that a decrease of the MNP anisotropy is correlated with an increase of both normalized remnant magnetization and slope at the coercive field. Another way to characterize this fact is to calculate the squareness of the hysteresis loop, which is directly related to the efficiency of M NPs for their application in hyperthermia [1]. Here, we define the squareness S as S=

A , 4µ 0 H satM sat

(5)

5

where Msat is the magnetization of the samp le when it is saturated by a magnetic field µ 0H sat. For instance, for sample 1, Msat = 33 Am2kg-1 and µ 0H sat = 12 mT [see Fig. 3(b)] . With this definition, S = 1 for a perfectly square hy steresis loop . In Fig. 4(c), the evolution of the squareness with Keff is plotted for the different samples : an increase of squareness with a decreasin g anisotropy is evidenced. The p revious findings can in no case be exp lained in the framework of magnetically indep endent single-do main M NPs. To illustrate it, we have p erformed numerical simu lations of -11 6 -1 hy steresis loop s using the mod el described in Ref. [1], with τ0 = 5 ×10 s, MS = 2 ×10 A m , T = 300 K, f = 54 kHz and a vary in g an isotropy Kind. Illustration of the hy steresis loop s can be found elsewhere (see Figs. 3(a) and 3(b) in Ref [1]). In Fig. 5, the evolution of the slop e at the coerciv e field, of MR/MS and of the squareness are p lotted as a function of Kind in the case of MNPs with anisotropy axes oriented with the magn etic field and in the case of anisotropy axes randomly oriented in sp ace. Although the slop e at the coercive field varies with anisotropy similarly to our exp erimental results- this is not the case for the MR/M S ratio and for the squareness. These latter are rather independent of Kind in a large range and drop sharp ly only when the low-anisotropy nanop articles are in the sup erp aramagnetic regime. The exp erimental results are op p osite to this tendency since low anisotropy nanoparticles show a strong increase of their MR/Msat ratio and squaren ess. On one other side, exp erimental results are very well exp lain ed by the presence of magnetic interactions. To illustrate it, we have p erformed simu lations based on the solvin g of the LLG equation at T = 0 K. Sin gle-domain NPs are p laced on a square lattice and carry macrospins considered as magnetic p oint dip oles. M ore details on the resolution can b e found in Ref [19]. In 6 -1 the present study , the following p arameters have been consid ered: MS = 2 ×10 A m , D = 13 nm, the center-to-center interparticle distance was 15 nm, and the individual anisotropy Kind of the nanop articles was varied. Hy steresis loop s of 3D isotrop ic assemblies of M NPs as well as anisotrop ic ones have been studied, with array s ranging from 6 x6 x6 to 6x6 x60. In Fig. 6, the evolution of the slop e at the coercive field, of MR/M S and of the squaren ess are plotted as a function of the nanop article individu al anisotropy Kind. Disp lay ing these data as a function of Keff (i.e. the anisotropy that one would have deduced from the coercive field of the hy steresis loop ) does not change the trend of these curves and the conclusion of this p aragrap h. In Fig. 7 examples of calcu lated hy steresis loop s are shown. In Fig. 8, the magnetic configuration of the MNPs in the remanent state in four typical cases are shown (large/low anisotropy , and isotrop ic/anisotrop ic array). As exp ected, for large anisotropy MNPs, the magnetic beh aviour is indep endent of the array configuration: hy steresis loops display the typ ical features of non-coupled assemblies of MNPs at T = 0 K [1] : the MR/MS ratio is 0.5 [see Figs 6(a), 7(a) and 7(b)], squareness equals 0.25 2K ind [see Figs 6(c)], the co ercive field is ap p roximately half of the anisotropy field µ 0 H K = MS [see Figs. 7(e) and 7(f)] and the magnetic moment of the M NPs are randomly oriented in sp ace in the remanent state [see Figs. 8(b) and 8(d)]. All these features indicate that any influence of dip olar interactions is dominated by the strong anisotropy of individual M NPs.

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On the opp osite, for low anisotropy M NPs, magnetic p rop erties are extremely sensitive to the configuration of the array . In isotropic arrays, the squareness [see Fig. 6(c)] as well as the MR/MS ratio [see Figs. 6(a), 7(c) and 7(e)] both decrease. This is a logical consequ ence of the fact that when a null magnetic field is ap plied, a demagnetized state of the assemb ly is favoured, leadin g to a wasp-waisted hysteresis loop [see Fig. 7(c)]. Elon gated arrays display a very different behaviour : decreasin g the anisotropy leads to an increase of the MR/MS ratio [see Figs. 6(a), 7(b) and 7(d)] and of the squaren ess [see Fig. 6(c)]. Fig. 8(c) eviden ces visually that this originates from the stabilization of the MNP magnetization along the lon ger axis of the assembly by dip olar interactions (in other words, there is a global easy axis in the assembly resulting from its shap e anisotropy ). When comparing the various graphs of Fig. 6 with the exp erimental data shown in Fig. 4, it is clear that the p rop erties observed in hy p erthermia exp eriments are typ ical of the last case described. We conclude that the formation of chains and /or colu mns of M NPs during hyp erthermia exp eriments leads to an increase of the squareness of the hy steresis loops – and so to their efficiency - when their anisotropy is small enough. 4. Discussion In the remainin g of this article, we discuss in details our results and make comp arisons with the literature. The agreement between our simulations and our experiments is qualitative and not quantitative. Indeed, the anisotropy value for which there is a transition between the coupled and the non-coup led regime is much high er in simulations than experimentally , indicating a weaker influen ce of magn etic interactions in experiments. This could be due to the fact that our simulations take into account n either the temp erature nor the presence of disorder/vo ids in the assembly and are performed on assemblies of reduced size. A quantitative analy sis would require more realistic simulations as well as a much detailed characterization of the chains and columns formed durin g the exp eriments. There is a tricky and unsolved issue emerging from our exp erimental results. The hy steresis loop s measured on all the samples of the p resent studies display very large squareness, with values up to 0.74 (see Table 1). With such hysteresis loop shap e, it should be exp ected that the measured SAR are mu ch higher than the one measured, getting close to the maximum p ossible SAR [1]: SARmax ≈ 4µ 0 M S H max f .

(6)

In several previous articles we had attributed reduced SAR comp ared to what could be exp ected theoretically to the p resence of magn etic interactions, which would have reduced the squareness of hysteresis loop s [11, 12]. The present study indicates that this hyp othesis was p robably wrong since, in our sy stems, the squareness is on the contrary enhanced by the p resence of interactions. This p oint is clearly demonstrated by the shap e of the hysteresis loop s. A careful look to Fig. 3(b) and Table 1 indicates a surp rising second exp lanation to the reduction of SAR: the saturation magnetization M sat measured from the high frequency hysteresis loop s is for every samp le much lower than the M S measured in SQUID, desp ite that the hy steresis loop shap e seems to indicate that the MNPs are saturated or nearly saturated. One could think that the M sat value provided by our setup is not correct. We cannot comp letely exclude it; however, the fact that temp erature measurem ents and high-fr equency hy steresis loops are coherent and give ap p roximately the same

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SAR value do es not supp ort this hyp othesis. It is as if only a part (in the range 20-50%) of the MNPs contributed to the magnetic r esp onse and then the heatin g of the sample. In the case of sample 1, we attributed this discrep ancy to the p resence of many sup erparamagnetic MNPs which would not be saturated by the magnetic field [ 22]. However, in the other samp les under study here, there is no trace of such small M NPs. It could also be possible that a p art of the MNPs adopt a configuration with their easy axis p erp endicular to the magnetic field, where they would not contribute much to the magnetic signal and to the SAR. It has b een p redicted theoretically that such a configuration occurs when magnetic fields much below the saturating field of the MNPs are app lied [23]. Finally , it is also p ossible that, in the samp les, there still have individual MNPs, which would have very different p rop erties from the ones in assemblies. In any case, more work is required to elu cidate this issue. Ap art the p resent exp erimental results, a few other exp erimental studies can be discussed in view of our findin gs. In Refs [24] and [25], Gudoshnikov et al. eviden ced that increasin g the asp ect ratio of dense assemb lies of M NPs increase their SAR, due to effect of the demagnetizing field. This is in agreement with our theoretical finding that anisotrop ic assemblies of M NPs displays an enhanced SAR co mp ared to the isotrop ic ones. In Ref. [26], Alp handery et al. have shown that chains of MNPs are dominated by dip olar interactions, irresp ective of the exact orientation of the MNP individual easy axis inside them, and that the squareness of their quasistatic hysteresis loop is enhanced by the p resence of magnetic interactions. We show here that this finding is certain ly true for very low anisotropy MNPs (the ones studied by the authors belon g to this category ) but cannot be generalized since lar ge anisotropy MNPs maintain the characteristics of non-interactin g MNPs. In Ref [27], Müller et al. observed very different SAR values between M NPs which were gelled with or without app ly ing a magnetic field. Although no interp retation was p rovided by the authors, we attribute the increase of SAR in the samples textured by a large magn etic field to the formation of MNP chains or columns. Some authors have attributed the increase of SAR values when p articles are in a liquid comp ared to the ones found in a gel to Brownian motion [28]. This exp lanation should be p roposed by the authors with caution because there is no specific contribution of brownian motion to magnetic hy p erthermia, which could be separated from another contribution. SAR value is the result of the global M NP hy steresis loop . This p oint is clearly explained and illustrated in Ref. [23]. Our p resent study p roposes an alternative p lausible exp lanation to the increase of SAR in liqu ids, resulting from the formation of chains of M NPs. This hyp othesis should be seriously envisaged by authors when such experimental result is found. Finally , our theoretical results are consistent with theoretical works of Refs [3] and [9] since we similarly show that, in isotrop ic assemblies of M NPs, magnetic interactions decrease SAR comp ared to indep endent MNPs. We show an additional feature which is that, due to the formation of chains and columns in magnetic hyp erthermia experiments, low anisotropy M NPs p resent enhanced SAR comp ared to indep endent NPs. This p oint was missed in p revious theoretical works: in Refs [3], [8] and [10], on ly isotrop ic assemblies were studied; in Ref [9], in sp ite that anisotrop ic assemblies of low anisotropy M NPs near the sup erp aramagn etic/ferro magnetic transition were studied, only a very limited increase of SAR was detected. This may be due to the fact that the effect of magnetic interactions in these assemblies also leads to an increase of coercive field (see Fig. 6) and that simulations in Ref [9]

8

were p erformed at a small magnetic field which d id not saturate the hysteresis loops of interacting NPs and masked the effect. 5. Conclusion Our main f indin g can be sum marized this way : the magnetic interactions in the chains of MNPs formed during hy p erthermia exp eriments have a tendency to induce a uniaxial anisotropy , which incr eases the squareness of the hy steresis loop and thus MNP efficiency . This process is in comp etition with the M NP magn etocristalline anisotropy, which has the tendency to decrease the squareness and efficiency of the MNPs toward the one of magnetically indep endent MNPs. This effect is then visible on ly for low-anisotropy M NPs saturated by the applied magn etic field. Since this configur ation is p recisely the one where M NPs are op timized for magnetic hy p erthermia and display the highest SARs [1], this finding is imp ortant to interp ret exp eriments. Chains of MNPs with a uniaxial anisotropy are the only way to reach the m aximu m p ossible SAR with a given magnetic material. These results evidence the imp ortance of consider in g the chains and co lumns of M NPs formed during the ap p lication of the magnetic field to interp ret correctly magnetic hyperthermia. Future theoretical work on magnetic interactions should not miss this p oint. It would be very interesting to know to which extent such chains are for med durin g in vivo and in vitro exp erim ents : in sp ite of their low overall concentration in tumors, M NPs are in some case group ed in intracellular comp artments of cells and locally reach high concentration. These are conditions where magnetic interactions could be non-negligible and could then induce the formation of chains that lead to large SAR valu es.

Acknowledgements : This work was supported by the Fondation InNaBioSanté and the Région M idi-Py renées.

Appendix Determination of Equ. (2) follows the same p rincip le as the determination of Equ. (1), which was p reviously described in [12]. In the ferromagnetic r egim e, the magnetic field dep endence of SAR in hy perthermia m easurem ents shows an abrupt increase, which occurs when the app lied magn etic field exceeds the coercive field. Sin ce the coercive field is not an intrinsic p arameter of the MNPs but dep ends on the amp litude of the applied magnetic field, the determination of the exact valu e of this abrupt increase r equires a dedicated study . For that p urpose, SAR(µ 0Hmax) functions were calculated for different NP diam eter in the case where their anisotropy axis is aligned with the magn etic field. The maximum slop e of the SAR(µ 0H max) H CHyp function occurs when µ 0 H max = µ 0 H CHyp . In Fig.9, is plotted as a function of κ , where HK

 k BT  . It has been shown in ln MS K ind V  4µ 0 H max M S Vf τ 0  p revious studies that κ is a good dimensionless p arameter to describ e the coercive field in the ferromagnetic regime [1, 12, 29]. Data in Fig. 9 were then fitted using variations of Equ. (2) with different p re-factors. The best fit occurs when the p refactor equals 1.85 (see Fig. 9), which p ermits to determine the final form of Equ. (2). µ 0H K =

2K ind

is the anisotropy field and κ =

k BT

9

References : *

Electronic mail: julian. carrey @insa-toulouse.fr

[1] J. Carrey, B. Mehdaoui and M. Resp aud, J. App l. Phys. 109, 033901 (2011) [2] N. A. Usov, J. App l. Phy s. 107, 123909 (2010) [3] F. Burrows, C. Parker, R. F. L. Evans, Y. Hancock, O. Hovorka and R. W. Chantrell, J. Phy s. D : App l. Phys. 43, 474010 (2010) [4] C. L. Dennis, A. J. Jackson, J. A. Borchers, P. J. Hoop es, R. Strawbridge, A. R. Foreman, J. van Lierop , G. Grüttner and R. Ivkov, Nanotechnology 20, 395103 (2009) [5] C. L. Dennis, A. J. Jackson, J. A. Borchers, P. J. Hoop es, R. Ivkov, A. R. Foreman, J. W. Lau, E. Goernitz, G. Gruettner, J. App l. Phy s. 103, 07A319 (2008) [6] S. A. Gudoshnikov, B. Ya. Liubimov and N. A. Usov, AIP Advances 2, 012143 (2012) [7] A. Urtizberea, E. Natividad, A. Arizaga, M . Castro and A. M ediano, J. Phys. Chem C 114, 4916 (2010) [8] D. Serantes et al., J. App l. Phys. 108, 073918 (2010) [9] C. Haase and U. Nowak, Phys. Rev. B 85, 045435 (2012) [10] C. M artinez-Boubeta et al. Adv. Func. M ater., DOI: 10.1002/adfm.201200307 (2012) [11] L.-M . Lacroix, R. Bel-M alaki, J. Carrey , S. Lach aize, G. F. Goy a, B. Chaudret, M. Resp aud, J. App l. Phys. 105, 023911 (2009) [12] B. M ehdaoui, A. M effre, J. Carrey , S. Lachaize, L. M . Lacroix, M. Gougeon, B. Chaudret, and M . Respaud, Adv. Func. M ater. 21, 4573 (2011) [13] B. M ehdaoui, A. M effre, L.-M . Lacroix, J. Carrey , S. Lachaize, M. Goujeon, M. Resp aud, and B. Chaudret, J. M agn. M agn. Mat. 322, L49 (2010) [14] M . Klokkenburg, B. H. Erné, J. D. M eeldijk, A. Wiedenm ann, A. V. Petukhov, R. P. A. Dullens, and A. P. Philip se, Phy s. Rev. Lett. 97, 185702 (2006) [15] A. M effre, S. Lachaize, C. Gatel, M. Resp aud and B. Chaudret, J. M ater. Chem. 21, 13464 (2011) [16] A. M effre, B. M ehdaoui, V. Kelsen, P. F. Fazzini, R.P. Tan, J. Dugay , J. Carrey , S. Lachaize, M. Resp aud and B. Chaudret, Nanoletters 12, 4722 (2012) [17] L.-M . Lacroix, J. Carrey , M . Resp aud, Rev. Sci. Instrum. 79, 093909 (2008) [18] B. M ehdaoui, J. Carr ey , M . Stadler, A. Cornejo, C. Nay ral, F. Delpech, B. Ch audret and M . Respaud, App l. Phys. Lett. 100, 052403 (2012) [19] R.P. Tan et al., Journal of Physics D: App lied Physics 43, 165002 (2010). [20] C. Desvau x, C. Amiens, P. Fejes, P. R enaud, M . Resp aud, P. Lecantes, E. Snoeck and B. Chaudret, Nature Materials 4, 750 (2005) [21] C. Desvaux, F. Dumestre, C. Amiens, M . Resp aud, P. Lecantes, E. Snoeck, P. Fejes, P. Renaud and B. Chaudr et, J. M ater. Chem 19, 3268 (2009) [22] B. M ehdaoui, J. Carr ey , M . Stadler, A. Cornejo, C. Nay ral, F. Delp ech, B. Ch audret and M . Respaud, App l. Phys. Lett. 100, 052403 (2012) [23] H. M amiy a and B. Jeyadevan, Scientific Reports 1, 157 (2011) [24] S. A. Gudoshnikov, B. Ya. Liubimov and N. A. Usov, AIP Advances 2, 012143 (2012)

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[25] S. A. Gudoshnikov, B. Ya. Liub imov, A. V. Pop ova and N. A. Usov, J. Magn. M agn. M ater, in p ress [26] E. Alphandéry, Y. Ding, A. T. Ngo, Z. L. Wang, L. F. Wu and M. P. Pileni, ACS Nano 3, 1539 (2009) [27] R. M üller, S. Dutz, R. Hergt, C. Schmidt, H. Steinmetz, M . Zeisberger, W. Gawalek, J. M agn. M agn. M ater 310, 2399 (2007) [28] E. Alp handéry, S. Faur e , L. Raison, E. Du guet, P. A. Howse and D. A. Bazy linski, J. Phy s. Chem. C 115, 18 (2011). [29] N. A. Usov and and Y. B. Grebenshchikov, J. App l. Phy s. 106, 023917 (2009)

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Tables Samples

d0 %Fe2.2 C Ms µ 0 HCHyp µ 0 HC 2 (nm) (Am /kg) (mT ) (mT ) T EM XRD SQUID hyperthermia loops

Keff K eff A A Msat MR/Msat Squareness Slope 3 3 2 (kJ /m ) (kJ /m ) (mJ/g) (mJ/g) (Am /kg) (a. u.) hyperthermia loops hyperthermia loops loops loops loops loops

Sample 1 FeCo Sample 2 Fe(0) Sample 3 Fex Cy Sample 4 Fe@Fex Cy Sample 5 Fe@Fex Cy Sample 6 Fe@Fex Cy

12.8

0

140

8

8

13.7

0

232

38

20

12.1

100

146

> 60

> 42

13.1

22

202

17

18

13.1

16

191

32

30

14.6

59

203

25

25

45.7 1D 103 3D >117 3D 54.3 1D 68.2 1D 67.4. 3D

45.7 1D 69 3D >95 3D 55.3 1D 66.2 1D 67.4 3D

1.05

1.1

33.0

0.92

0.69

544

3.80

2.7

47.0

0.63

0.37

66

1.57

1.08

20.2

0.45

0.32

36

4.66

6.00

89.4

0.94

0.74

292

6.35

8.2

75.1

0.92

0.71

217

7.9

8.6

97.8

0.83

0.56

116

Table 1 : Summary of magnetic and hyperthermia properties for different samples: Columns labeled with “hyperthermia” refers to data deduced from temperature measurements. Columns labeled with “loops” refers to data extracted from hysteresis loops measured at f = 54 kHz and µ 0H max = 42 mT. “d0”: mean diameter determined by TEM . “%Fe2.2C”: phase fraction of Fe2.2C deduced from XRD. “MS”: saturation magnetization for µ 0H m ax = 5 T.deduced from SQUID measurements on powder and from microanalysis. “µ0HCHy p” coercive field deduced from the highest slope of the SAR(µ 0Hm ax) function. “µ0H C“: coercive field deduced from hysteresis loops. “Keff “: effective anisotropy deduced from hyperthermia measurements or hysteresis loops. In the first (second) column, µ 0H CHy p (µ 0HC) have been used to calculate Keff using the analytical equations presented in text (“1D” refers to axis aligned with the magnetic field and “3D” to randomly oriented axis). “A”: losses per cycle deduced from hyperthermia and hysteresis loop measurements. “Msat” : magnetization at saturation (see text). “MR/Msat” ratio between remnent and saturation magnetization. “Squareness” is calculated from Equ.(5). “Slope” is the slope of the hysteresis loop at the coercive field.

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Figures :

Figure 1: TEM micrographs of the samples. The length of the scale bar is 100 nm.

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Figure 2: (a) and (b) TEM micrographs of experiments where a drop of colloidal solution containing MNPs of Fe was deposited on a grid of microscopy and let dry under the application of a magnetic field of 40 mT at 54 kHz. The length of the bar is (a) 2 µm (b) 200 nm.

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Figure 3 (color online): (a) SAR as a function of the magnetic field deduced from temperature measurements; f = 54 kHz (b) and (c) Hysteresis loop measurements performed at f = 54 kHz and µ 0H max = 42 mT. In (c) the hysteresis loops are normalized by the magnetization value at 42 mT.

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Figure 4 : Data extracted from the hysteresis loops shown in Fig. 3 and plotted as a function of the estimated Keff value of the different samples. The parameters plotted are (a) the normalized remanent magnetization MR/Msat, (b) the slope at the coercive field, (c) the squareness S.

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Figure 5 : Data extracted from numerical simulations of non-coupled single-domain M NPs. MNPs have either their anisotropy axis oriented with the magnetic fied or randomly oriented in -11 6 -1 space. Parameters are τ0 = 5 ×10 s, MS = 2×10 A m , T = 300 K, f = 54 kHz and a varying Kind. The graphs are (a) the normalized remanent magnetization MR/MS, (b) the slope at the coercive field and (c) the squareness. Examples of the corresponding hysteresis loops can be found in Ref [1].

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Figure 6: (color online) Data extracted from numerical simulations of magnetically coupled MNPs. 3D arrays of 6x6xZ MNPs have been simulated. Parameters are MS = 2×106 A m-1, D = 13 nm, the center-to-center interparticle distance was 15 nm, Kind was varied. The data plotted are (a) the normalized remnant magnetization MR/MS, (b) the slope at the coercive field and (c) the squareness. Examples of the corresponding hysteresis loops are shown in Fig. 7.

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Figure 7: (color online) Numerical simulations of magnetically coupled MNPs. Simulation parameters are given in Fig. 6. The array size is (a) (b) (c) 6x6x6 and (d) (e) (f) 6x6x42. In (e) 2K ind and (f) the magnetic field µ0H is normalized by the anisotropy field µ 0 H K = . MS

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Figure 8: Magnetic configuration of the nanoparticles in the remanent state for two different values of aspect ratio and anisotropies. Each arrow indicates the magnetization direction of a 3 3 6 3 3 3 MNP. (a) 6x6x6, Kind = 10 J/m . (b) 6x6x6, Kind = 10 J/m . (c) 6x6x18, Kind = 10 J/m . (d) 6x6x18, Kind = 106J/m3.

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Figure 9: Normalized µ0HCHy p values plotted as a function of κ. Each µ0H CHy p value has been determined from the maximum of the derivative of SAR(µ0H m ax) functions. The numerical 4 3 -11 simulations were run with Kind = 1x10 J/m , T = 300 K, f = 100 kHz, τ 0 = 5x10 s, µ 0Hmax in the range 0-70 mT, and with a varying NP diameter ranging from 2 to 40 nm. Dots correspond to the values extracted from the numerical simulations and the solid line to the numerical solving of Equ. (2).

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