Unraveling viscosity effects on the hysteresis losses of magnetic nanocubes.

Hysteresis losses in magnetic nanoparticles constitute the basis of magnetic hyperthermia for delivering a local thermal stress. Nevertheless, this therapeutic modality is only to be realised through a careful appraisal of the best possible intrinsic and extrinsic conditions to the nanoparticles for which they maximise and preserve their heating capabilities. Low frequency (100 kHz) hysteresis loops accurately probe the dynamical magnetic response of magnetic nanoparticles in a more reliable manner than calorimetry measurements, providing conclusive quantitative data under different experimental conditions. We consider here a set of iron oxide or cobalt ferrite nanocubes of different sizes, through which we experimentally and theoretically study the influence of the viscosity of the medium on the low frequency hysteresis loops of magnetic colloids, and hence their ability to produce and dissipate heat to the surroundings. We analyse the role of nanoparticle size, size distribution, chemical composition, and field intensity in making the magnetisation dynamics sensitive to viscosity. Numerical simulations using the stochastic Landau-Lifshitz-Gilbert equation model the experimental observations in excellent agreement. These results represent an important contribution towards predicting viscosity effects and hence to maximise heat dissipation from magnetic nanoparticles regardless of the environment.


Introduction
Heat dissipation due to hysteresis losses in magnetic materials generally constitutes a difficult issue to circumvent. But in the case of magnetic nanoparticles (MNPs), heat losses are exploited for treating tumors in the so called magnetic hyperthermia theraphy. 1 Under alternating magnetic fields (AMF), MNPs release heat to their surroundings leading to a local temperature elevation -typically up to 43ºC-that destroy malignant tissues. [2][3] A handful of clinical trials have employed MNPs as magnetic heating mediators with partial success depending on the cancer type [2][3] . Learnt lessons evidence that the current understanding of the physical parameters controlling magnetic heating at the nanoscale still needs to be improved, thus precluding the development of suitable instrumentation and clinical protocols. In this regard, viscosity () is one of the less studied physical parameters that affect nanoparticle heating efficiency. [4][5] Fortin et al. presented one of the few comprehensive experimental studies considering the influence of the medium viscosity, and found a strong decrease of heat dissipation while increasing the viscosity of the medium. 6 This effect, as expected, is far more pronounced in highly anisotropic systems, like cobalt ferrites. The underlying reason is related to the magnetic relaxation processes (i.e. Néel or Brownian mechanisms) causing heat dissipation. [6][7] Whereas Néel relaxation is related to the reorientation of the particle magnetic moment, Brownian relaxation is related to the particle rotation for reorienting its magnetic moment. These relaxation processes are ruled by different parameters such as MNP size, 7 or magnetic anisotropy constant, 6 that significantly determine the relaxation mechanisms responsible of magnetic heating. In addition, magnetic dipolar interactions 8 or magnetic field intensity [9][10][11] have shown to significantly influence the relaxation mechanisms responsible of heat losses. In absence of magnetic field, the relaxation times are given by the following expressions 9 for Néel and Brownian processes, respectively: where  0 is the inverse attempt frequency, K is the magnetic anisotropy constant, V is the nanoparticle volume, k B is the Boltzman constant, T is the nanoparticle temperature, V H is the hydrodynamic volume. Although the faster mechanism dominates, both mechanisms coexist, are coupled through a torque, 10 and contribute to a different extent to the magnetic heat dissipation. While the Brownian relaxation renders heat dissipation of MNPs sensitive to , Néel relaxation is not influenced by at all. Considering the interaction of MNP and biological entities, recent studies [12][13][14][15] have shown that the magnetic heating efficiency is significantly reduced when MNPs are located inside cells or tissues. The soundest explanation of such behaviour is related to either the increase of viscosity [16][17] and/or nanoparticle aggregation [18][19][20] imposed by the host biological matrices. Whereas aggregation effects have been explored and understood in terms of magnetic dipolar effects, [21][22] little is known about the parameters ruling viscosity effects. Hence, it is crucial to unveil what determines those viscosity effects to predict any derived underperformance and choose the right MNP design strategy in order to avoid variations of magnetic heat losses related to their surroundings. Successive computational models describing hysteresis losses -with emphasis on their application to magnetic hyperthermia-have been developed in parallel to the experimental research on viscosity effects during last years. 1 Many of them derive from the seminal Rosensweig's model, 7 which constituted a first attempt to calculate the heating losses of MNPs. Mamiya et al considered different scenarios upon the ratio between the externally applied AMF and the anisotropy field (H K ) of the MNPs 10 and performed numerical simulations based on the coupling of the mechanical and magnetic torques behind the Brownian and Néel relaxation mechanisms. Using a kinetic Monte-Carlo method, several works proposed a unified model for heating mediated by MNPs to improve Rosensweig's theory that included interparticle interactions, 23-25 a topic of increasing interest in nanoparticle-mediated magnetic hyperthermia. One of the most remarkable aspects of this model is that it offers a common "umbrella" to the underlying nanoparticle heating mechanisms across the transition from the superparamagnetic regime to the metastable hysteresis one. So far, Usov and Liubimov presented -perhaps the only one-a model with a rigorous treatment of viscosity effects on nanoparticle heating. 26 They first coined the terms viscous and magnetic modes, established according to the ratio between field intensity (H MAX ) and H K . Besides, calorimetry measurements are currently employed to quantify the MNP heat dissipation under changing conditions of concentration, or aggregation, 10,21,23,25,[27][28] in spite of the potential error sources in measuring and determining SAR values, which are difficult to keep under control. 29-30 However, hysteresis loop measurements under AMF are the most direct and accurate tool to probe heat losses as a function of different extrinsic or intrinsic parameters such as particle aggregation, 21 or concentration, 21, 31 especially for the sake of comparison with theoretical models. 21 For instance, how medium viscosity influences the AC hysteresis loops of MNPs is still an open question, which requires to be properly addressed. The present study merges the choice of bespoke nanoparticle systems, consisting in a set of cobalt ferrite and iron oxide nanocubes with increasing magnetic anisotropy, with a suitable theoretical model for hysteresis losses featuring distinct considerations to best explain the actual role of viscosity on the dynamical magnetic response of MNPs. The role of nanoparticle size, size distribution, chemical composition, and field intensity in rendering the dynamical magnetic response sensitive to viscosity is studied.

Results and discussion
In order to probe the role of the above mentioned intrinsic and extrinsic parameters on the viscosity effects, iron oxide nanocubes (IONCs) of edge size (l C ) 14±2 nm and 24±4 nm, and cobalt ferrite nanocubes (CoFeNCs) of 21 ± 2nm were studied (see ESI for more details related to MNP synthesis and characterisation). This set of MNPs allows one to probe the viscosity effects while progressively increasing KV by changing either size (i.e. V ranging from 1.4x10 -24 to 2.7x10 -23 m 3 ) or MNP chemical composition 32 (cobalt ferrite K=290 kJ/m 3 , magnetite K=-13 kJ/m 3 ). This is thanks to the recently developed chemical routes 33-34 that supply iron oxide and cobalt ferrite nanocubes with precise control of size and morphology, then resulting in outstanding SAR values. According to the Stoner-Wohlfarth model 35 , heat losses in magnetic materials are proportional to K. The magnetisation cycles of the studied MNPs at 260 K show different hysteretic behaviour ( Figure S2), more pronounced for CoFeNCs, negligible for 14 nm IONCs, and intermediate for 24 nm IONCs. Such different magnetic properties of MNPs under quasi-static conditions agree with the expected trend of the KV values for the studied MNPs. Furthermore, similar tendency will be also observed in the dynamical magnetisation response, including magnetic heating efficiency that clearly varies with chemical composition (i.e. K) and size (i.e. V). 6 Additionally, MNPs shape 36 , surface defects, 37 coating 38 or core/shell nanostructure 39 also influence the K value, and therefore, modulate the magnetic heating efficiency. The studied magnetic colloids present a homogenous cubic shape, and do not aggregate in the studied viscous media (see Figure S1 in ESI). SAR values are commonly determined to appraise viscosity effects on the dynamical magnetic response of MNPs. Considering that SAR = Af, one may expect that any variation of SAR values with will be translated into variations of shape and area of the corresponding AC hysteresis loops. In order to verify this point, we have measured hysteresis loops at different viscosity values ranging from 0.9 to 153.5 mPas and under AMF conditions (100 kHz and H MAX =24 kA/m), which are very close to the ones currently employed in magnetic hyperthermia treatments. 2 Figures 1A-C shows the representative viscosity dependence of the hysteresis loops for the studied MNPs. As expected, the opening of the AC hysteresis loops under the same conditions significantly depends on MNPs, and hence the KV value. Indeed, the AC hysteresis loops are less opened in case of IONCs (independently of size) than in case of CoFeNCs. The latter are also highly sensitive to in comparison to IONCs, whose viscosity effects diminishes with particle size. Such viscosity behaviour is similar to the trend observed for the SAR under similar experimental conditions (see Figure 2 Further analysis of the AC hysteresis loops evidence the related viscosity effects. On the one hand, AC hysteresis loops of CoFeNCs feature a reduction of all characteristic hysteresis parameters (i.e. coercive field (H C ), remanence (M R ), and maximum magnetisation (M MAX )) when increasing viscosity as shown in Figure 3. On the other hand, the viscosity changes of shape and the area of AC hysteresis loops for IONCs depend on particle size. Whereas M R and M MAX decrease for both IONC sizes, but in a more pronounced manner for the larger size, H C does not vary with viscosity for any size (see Figure 3D). Hence, the shape and area of the AC hysteresis loops for the larger IONCs behave differently with  similarly to those of CoFeNCs. Furthermore, the variation of A with  ( Figure 3B  respective SAR dependence (Figures 1 and 2). As revealed in Figures 3A and C 6 . In order to confirm this hypothesis, we have analysed the magnetic relaxation processes by AC susceptibility (ACS) measurements on the most and least sensitive MNPs to . At a first glance, Figure 4 shows that the Brownian mechanism dominates the relaxation processes for those MNPs whose dynamical magnetic response is sensitive to viscosity (i.e. CoFeNCs) under the applied AMF conditions (i.e. 100 kHz, see dotted line in Figure 4) (Figure 5E,G).
Hence, AC hysteresis loops of CoFeNCs are highly sensitive to viscosity ( Figure S6B) because  Beff is much smaller than  Neff Figure S6A). Unlikely, 14 nm IONCs present no viscosity effects on the dynamical magnetic response when  Neff are shorter partially explains why AC hysteresis loops of 14 nm IONCs are so sensitive to  at 4kA/m. However, the key parameter required to explain such observations is the particle size distribution. Figures 5A-D show that l C -which defines V and consequently the anisotropy energy barrier-leads to the raise of the  Neff values. Hence, the crossover between  Neff and  Beff regimes is favored by the size distribution (i.e. large values of l C ). Therefore, when the dynamical magnetic contribution of MNPs of a given l C is weighted according to the related size histogram ( Figures S1D-F), AC hysteresis loops are differently sensitive to viscosity depending on H MAX (as shown in Figures  5B, D, F, H). Finally, the concomitant influence of H MAX and size distribution on  Neff and  Beff is required to simulate the experimental results at low field intensities. The balance between several parameters determining  Neff and  Beff , namely size distribution, , H MAX and K eventually define the effective relaxation mechanism. Thus, AC hysteresis loops of 14 nm IONCs at H MAX = 24 kA/m present no viscosity effects ( Figure  1C) because the Néel relaxation is dominant for l C values due to the stronger field-dependence of  Neff in comparison with  Beff¸1 1 as opposed to the observations at 4 kA/m ( Figures 5E-H). Moreover, the implicit consideration of both  Neff and  Beff in the stochastic LLG equation results in an overall good agreement between experimental and theoretical hysteresis loops under different experimental conditions. This model stresses the fact that not only MNP size and K favor Brownian relaxation process, but also low H MAX and large size distribution. In a first approach, MNP size and K define the crossover between Néel and Brownian mechanisms, resulting in magnetic dynamics that are sensitive to viscosity for large MNP sizes and K values. Secondly, H MAX and size distribution play also a critical role in the dynamic magnetic response, as both parameters strongly influence relaxation mechanisms. The so-produced results by feeding these parameters into the LLG equations provide a valuable guidance to predict the influence of viscosity on the magnetic heating efficiency under given MNP intrinsic (i.e. MNP size, size distribution, and K) and extrinsic conditions (H MAX ).

Conclusions
We have demonstrated the convenience of AC hysteresis loop measurements to determine the influence of viscosity on the dynamical magnetic response of MNPs through modelling experimental results. AC hysteresis loops from selected MNPs of different anisotropy constants dispersed in increasingly viscous media provide conclusive evidences on: i) the balance between those parameters determining the dominant magnetic relaxation process -namely size, size distribution, H MAX and K-eventually define the viscosity sensitivity of the dynamical magnetic response of MNPs, ii) the viscosity dependence of the AC hysteresis area is correlated to that of the SAR, due to demagnetisation effects related to changes in the magnetisation dynamics when viscosity increases, and iii) size distributions and field-effective relaxation times in the stochastic LLG equation precisely reproduce the hysteresis loops in most of the considered cases. Our reported results encourage further exploration of the viscosity influence on the

Magnetic nanoparticles
The MNPs studied in this work were iron oxide nanocubes with edge sizes of 14 ± 2 and 24 ± 4 nm and Co x Fe 3−x O 4 nanocubes with edge sizes of 21 ± 2 nm and cobalt fraction of x = 0.7, synthesised by thermal decomposition method as described elsewhere. [33][34]43 More details related to the MNP synthesis, surface modification and physico-chemical characterisation are described in ESI.

Magnetic characterisation AC Magnetometry
Quasi-static conditions: field dependent magnetic measurements under quasi-static conditions were carried out at 260 K in an ever-cooled Magnetic Property Measurement System (MPMS-XL, Quantum Design) on MNP dispersion volumes of 100L at concentration of 2 g Fe /L ( Figure S2). MNPs were physically blocked by slowly cooling the sample from room temperature to 260 K. The magnetisation signal was normalized to the mass of magnetic material (i.e. magnetite or cobalt ferrite). Dynamical conditions: field dependent magnetic measurements under dynamical conditions (100 kHz and 24 kA/m) were performed using a home-made inductive magnetometer set up similar to the one described by Connord et al. 44 AMF was generated by an air-cooled Litz wire coil. Two contrariwise-wound compensated pick-up coils connected in series were set inside the AC magnetic field generator coil. The system quantifies the magnetic signal from MNP dispersions, and it is calibrated by comparing magnetisation values in a similar field intensity range obtained under AC and quasi-static magnetic field conditions. The AC magnetisation signal was normalized to the mass of magnetic material (i.e. magnetite or cobalt ferrite, respectively).

Calorimetry measurements
SAR values were determined for the studied MNPs dispersed in different glycerol dilutions by employing a non-adiabatic procedure described elsewhere. 45 The AMF employed is a home-made frequency and intensity adjustable field generator with frequencies up to 500 kHz and field intensities up to 56 kA/m. Temperature variations after AMF application were monitored via a commercial optical fibre probe TS2/2 connected to a FOTEMP2-16 two-channel signal conditioner from Optocon AG with an experimental error of ± 0.2 ºC. The temperature vs time curves of IONC and CoFeNCs dispersions were recorded three times under similar dynamical conditions than AC hysteresis loops (100 kHz and 24 kA/m). Then, the maximum slope values at initial times after switching on AMF (i.e. dT/dt| max ) were extracted. Average dT/dt slope value and standard deviation were determined for calculating SAR values by using the following expression: where C d is the mass specific heat of the dispersion medium, m d is the dispersion mass and m mag is the mass of the magnetic material (i.e. magnetite or cobalt ferrite MNPs, respectively) and dT/dt| max is the average value of the maximum slope at initial times after switching on AMF. More details on Cd values of the employed glycerol dispersions are described in ESI.

Complex ac-susceptibility measurements
The complex AC-susceptibility (ACS) measurements were carried out using two setups 46 operating from 10 Hz to 10 kHz and 1 kHz to 1 MHz at magnetic field amplitudes of 0.46 kA/m and 0.07 kA/m, respectively. The ACS measurements at lower frequencies (2 Hz to 9 kHz) were performed using a fluxgatebased rotating magnetic field setup at 0.16 kA/m magnetic field amplitude. 47 The ACS measurements were carried out at 295 K in a volume of 150 μL of MNP dispersions with different glycerol fractions (0%, 36%, 81%, and 86%) but at fixed iron concentration of 2g Fe /L.

Numerical simulations
The curve-fitting of ACS measurements was performed using a least-squares fit routine to solve Debye equations and written in MATLAB. The routine uses the Levenberg-Marquardt algorithm which is a powerful scheme for non-linear problems. where 0 , which is given by 0 = /2 , is the attempt frequency, is the magnitude of the magnetic moment, γ is the gyromagnetic ratio, λ is a dimensionless damping coefficient, is the magnetic anisotropy constant, and is the individual MNP volume. The effective magnetic field eff ⃗⃗⃗⃗⃗⃗⃗⃗ has been calculated assuming a uniaxial magnetic anisotropy and neglecting magnetic dipolar interactions; thus: where K ⃗⃗⃗⃗⃗ = 2 0 ⃗⃗⃗⃗ •⃗ ⃗ ⃗ ⃗ , and ⃗ ⃗ is unit vector along the easy axis.
Thermal fluctuations have been taken into account in the form of a random torque (as described in ESI). The influence of the applied field on the Néel and Brownian relaxation times as well as their relative contribution to their relaxation processes have been computed through the following set of equations: (1−ℎ 2 )(cosh −ℎ sinh ) Please do not adjust margins Please do not adjust margins where = , = 0 , ℎ = 2 ,  N and  B are Néel and Brown relaxation times in the absence of an external field (Equations 1 and 2). For more realistic results, the experimental nanoparticle size distributions were inserted in the relevant equations. The algorithm was developed in C++ with CUDA running on a graphics processing unit (NVIDIA Tesla C2075). More details on numerical simulations of the magnetic properties under AMF are described in ESI.