Heating ability of cobalt ferrite nanoparticles showing dynamic and interaction effects

Abstract

The heating ability of magnetic nanoparticles (MNPs) intended for magnetic hyperthermia is quantified by means of the specific absorption rate (SAR), also referred to as specific loss power. This quantification is mainly performed on ferrofluids, even though the SAR values so obtained are often not representative of the nanoparticle performance inside tissues (solid matrices). In this study, the SAR of cobalt ferrite MNPs with mean crystallite diameters of 5.5 nm and 7.4 nm, functionalized or not, and dispersed in liquid or solid media, was determined in the 180–300 K temperature range. Higher SAR values were systematically obtained for samples in liquid media. On the one hand, heat capacity data together with zero-field-cooled and field-cooled magnetization curves allowed correlation of these results with ferrofluid dynamics originating from viscosity changes in samples dispersed in diethylene glycol. On the other hand, the higher degree of agglomeration attained by the functionalized MNPs after immobilization in paraffin wax seemed responsible for the decrease in SAR values and the shift in blocking temperature. In sum, MNP's spatial arrangements acquired after ferrofluid injection in magnetic hyperthermia should be taken into account to predict SAR values during therapies.

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Introduction

Magnetic nanoparticles (MNPs) have turned out to be promising materials for biomedical applications.^1–4 In addition to the useful properties related to their magnetic nature, their small size allows them to flow through blood vessels, cross cell membranes or prevent phagocytic response. Among other functionalities, MNPs can release heat when subjected to an alternating magnetic field, being the amount of released heat a relevant parameter for biomedical applications such as magnetic fluid hyperthermia (MFH) for cancer treatment or thermally assisted drug release.^5,6 In MFH, the achievement of local temperature increments for a certain period of time has proven to hinder tumor growth, reinforce the effect of radiotherapy or induce direct necrosis of cancerous cells while preserving healthy tissue. Several in vitro, in vivo and clinical trials have widely corroborated these statements.^7,8 The heat released by MNPs generally increases with the frequency, f, and the amplitude, H₀, of the alternating magnetic field. However, these parameters should be limited in order to avoid side effects in patients like additional tissue heating by eddy currents and/or undesirable muscular stimulations due to electromagnetic radiation. A widespread limit for the biological range of field application is H₀f < 485 kA kHz m⁻¹, with 50 kHz < f < 1.2 MHz and H₀ < 15 kA m⁻¹,⁹ although this limit can be overcome for certain body areas.¹⁰

In the last years, great efforts have been made to synthesize nanomaterials with improved heating ability to be used in MFH. This ability is currently evaluated by several methods (magnetic, calorimetric),¹¹ over samples with different MNP concentration, distribution and dispersive medium,^12,13 and in different thermal (adiabatic, non-adiabatic, in vitro, in vivo) and magnetic (H₀, f) conditions. This spectrum of variables often makes difficult comparing the results. In particular, with respect to concentration, distribution and dispersive medium, most authors measure the heating efficiency in liquid,^12–19 where distances between particles are expected to be large and homogeneous throughout the sample. However, MNPs suspended in a liquid carrier are also free to move, form structures and orientate with the alternating magnetic field.^20–22 The results obtained in these conditions can strictly only be compared with samples of the same concentration and similar solvent, since inter-particle distances affect interaction effects, and the properties of the dispersive medium also play a role in heat dissipation mechanisms.^19,23 Furthermore, given that in real applications the MNP suspension is previously injected into the tissue, results obtained in liquid cannot be directly extrapolated to in vivo experiments. After injection into a solid matrix, MNP distribution varies, interparticle distances turn non-constant, interparticle interactions change, and even some heating mechanisms are no longer active (orientation in liquid, Brown relaxation).^24,25 In addition, results of in vivo heating experiments are not usually accompanied by a description of the MNP distribution and concentration acquired upon injection of MNPs. Studies considering all these variables would certainly help to compare heating efficiencies between samples synthesized by different authors.

In this work, we evaluate the heating ability of cobalt ferrite MNPs dispersed in liquid, considering the influence of size, functionalization, dispersive media and particle distribution. We also study and justify the change in their performance upon immobilization in solid matrices on the basis of ferrofluid dynamics and interparticle interactions. Eventually, the importance of measuring the dissipated heating power as a function of temperature to infer the effect of these variables is highlighted.

Experimental methods

Sample preparation

The samples under study were synthetized, starting from iron and cobalt acetates, via the polyol method reported elsewhere.²⁶ This route allows obtaining stable and narrow-sized cobalt ferrite MNPs in diethylenglycol (DEG) without surfactants. Two different synthetic approaches were used: batch CF1 was obtained by a one-step growth (see synthesis of sample 1),²⁶ while batch CF2 was synthesized starting from an initial seed (see synthesis of samples 2–6)²⁶ through 4 growth steps. The magnetic fluids obtained were named CF1L and CF2L, respectively. Afterwards, the DEG of a part of both fluids was replaced by a polyethylenglycol matrix (PEG, average MW of 10 000 Da). For this purpose, the dried MNPs were mixed with PEG; the mix was then heated at 80 °C, stirred for 1 h, and finally cooled in LN₂ to provide fast entrapping of the MNPs in the solidifying PEG. This process gave rise to samples CF1S and CF2S. In addition, another part of fluid CF1L was functionalized with cetyl phosphate in ethanol, according to the method previously reported.²⁶ The functionalized MNPs, hydrophobic, were separated with hexane, precipitated and named CF1f. Eventually, one part of these MNPs was re-suspended in n-dodecane, to provide sample CF1fL, while another part was embedded in paraffin wax in a similar way to samples CF1S and CF2S in PEG, giving rise to sample CF1fS. Table 1 summarizes all prepared samples.

Table 1 Batch, size, functionalization, dispersive media and mass concentration of samples

Batch	Size (nm)	Sample	f.^a	Disp. medium	c^b (gMNP/g)
a f. is functionalization with cetyl phosphate.b c is the sample concentration in grams of nanoparticles per grams of sample.
CF1	5.2 (TEM), 5.5 (XRD)	CF1L	No	DEG	0.0256
		CF1S	No	PEG	0.0089
		CF1fL	Yes	n-dodecane	0.0596
		CF1fS	Yes	Paraffin wax	0.0236
CF2	7.4 (XRD)	CF2L	No	DEG	0.0262
		CF2S	No	PEG	0.0142

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Composition and microstructure

Cobalt concentration of ferrofluids CF1L and CF2L was measured using Inductively Coupled Plasma – Optical Emission Spectrometry (ICP-OES, Jobin Ybon 2000). For this purpose, samples were digested in acidic media, and the concentration of each fluid was then obtained assuming stoichiometric composition of the CoFe₂O₄ nanoparticles.

X-ray diffraction (XRD) experiments were performed on a Phillips X'per Pro diffractometer (Cu Kα radiation). Diffraction patterns were recorded in the 2θ range 10–70° with a scan step of 0.05° (2θ) for 5 s, from which the crystalline structure of the samples was identified. Also, the nanoparticle crystallite size was estimated from the peak broadening using the Scherrer method.

Particle size and arrangement were studied by Transmission Electron Microscopy (TEM) with a JEOL 2000 FXII instrument working at an acceleration voltage of 200 kV. Drops of ferrofluids were put onto copper grids with a carbon membrane film. The solid samples were prepared by ultramicrotomy, using epoxy resin (Epofix) as embedding medium. Blocks were sectioned with a diamond knife and slices were also laid on similar copper grids.

Determination of heating ability

The heating ability of MNPs was characterized through the specific absorption rate (SAR) which is defined as the heating power released per unit mass of magnetic material during ac-field exposure. Given that this heating ability is related to magnetization reversal processes, SAR values are, like magnetic properties, temperature dependent. In order to obtain SAR(T) trends, SAR was measured by a calorimetric method using a unique special-purpose magnetothermal setup²⁷ working in adiabatic conditions, and following the procedure described elsewhere.²⁸ In the present case, the temperature of the specimens was first decreased down to 180 K, and a gentle positive sample temperature slope was established by means of the temperature control, so that measurements can take place during heating ramps. SAR values were determined up to 300 K by applying successive ac-field pulses. During each ac-field pulse of duration Δt, the specimen releases heat and undergoes self-heating. The temperature increment, ΔT, of the sample due to each ac-field pulse, is measured taking into account the sample temperature drifts in absence of applied magnetic field. ΔT is calculated as T₂ − T₁, where T₂ and T₁ are, respectively, the backward and forward extrapolation of the temperature drifts after and before the ac-field application, at the midpoint of Δt.²⁷ From these data, SAR is calculated as:where C (J K⁻¹) is the heat capacity of the specimen (sample + container), and m_MNP is the mass of magnetic nanoparticles. Each SAR value is assigned to the midpoint temperature T = (T₁ + T₂)/2.

Liquid samples were measured inside quartz containers sealed with adhesive to prevent liquid leakage in the vacuum environment required for adiabatic measuring conditions. Solid samples were prepared either in pellets (CF1fS) or embedded in epoxy resin (CF1S, CF2S). Eventually, measurements were performed with ac-field frequency and amplitude of 111 kHz and 3 kA m⁻¹, respectively, in order not to overtake the biological range of ac-field application and study the low-field heating of the nanoparticles.

Heat capacity measurements

The determination of SAR by calorimetric methods requires precise heat capacity data of the samples, especially when solvents or matrices with possible thermal anomalies are used. The specific heat of the epoxy resin, DEG, quartz and samples (except CF1fL) was measured by Differential Scanning Calorimetry (DSC) in the 100–300 K temperature range. The specimen heat capacity necessary for SAR calculation was estimated using the mass and the specific heat of each component at the adequate temperature. The specific heat of sample CF1fL was assumed to be that of n-dodecane,²⁹ as the contribution of MNPs to the heat capacity is negligible.

Magnetic measurements

Static magnetic measurements were performed with a SQUID magnetometer (Quantum Design, MPMS-XL) on the same specimens used for the magnetothermal characterization. Magnetization vs. field, M(H), curves were recorded at 300 K with fields up to 5 × 10⁴ Oe (3980 kA m⁻¹). Also, zero-field-cooled (ZFC) and field-cooled (FC) magnetization vs. temperature, M(T), curves were acquired during heating ramps with a static field of 38 Oe (3 kA m⁻¹, the same value than H₀ in SAR determination) in the 10–350 K temperature range. Diamagnetic corrections regarding sample holders and matrices were applied in all magnetic measurements.

Results and discussion

Composition and morphology

Fig. 1 displays the XRD patterns of MNP powder from batches CF1 and CF2, whose peak positions match well with the inverse spinel structure expected for the CoFe₂O₄ phase. The (311) peak corresponding to batch CF2 appears slightly narrower than that of batch CF1, indicating a larger mean crystallite size for the former. Best fits of these peaks to Lorentzian distributions derive mean crystallite diameters of 5.5 and 7.4 nm for MNPs of batches CF1 and CF2, respectively, using the Scherrer equation.³⁰

Fig. 1 Above: XRD patterns of CoFe₂O₄ database reference and CF1 (dashed line) and CF2 (continuous line) powder. Below: TEM image and diameter histogram for batch CF1.

Fig. 1 also shows a TEM image of batch CF1. According to the fit of the histogram to a normal distribution, these nanoparticles have an average diameter of 5.2 ± 1.0 nm. This result indicates that the size of the MNPs synthesized in batch CF1 has a standard deviation less than 20%. Given the small size of these MNPs, we can consider that they present a narrow size distribution, even if they do not fulfill strictly the standard criterion for monodispersity (10%).

Sample CF1L (Fig. 2) presents quasi-2D arrangements onto the copper grid indicative of a slight fractal aggregation. This arrangement is expectable in absence of functionalization since MNPs, due to their magnetic nature, tend to interact and form some structures in liquid.²¹ The image of sample CF1fL shows a 2D distribution of MNP islands formed upon solvent evaporation. This indicates that the synthesis of batch CF1 has produced well-dispersed MNPs, and that functionalization has enabled maintaining the absence of aggregation. Eventually, sample CF2L displays a raspberry-like arrangement, with visible edge particles similar in size and shape to those of sample CF1L. The successive growth steps of the synthetic method together with an eventual agglomeration of the seeds have induced formation of raspberry-like nanoparticles with an overall mean diameter of 40 nm. Even though these particles show considerable polydispersity, they have proven useful to explain the behavior of samples coming from batch CF1, as it is described below. According to the previous results, crystallite size as obtained by XRD agrees well with TEM particle size for batch CF1, while the results for batch CF2 would indicate that raspberry nanoparticles consist in several nanocrystallites with a certain size distribution, fact that should be reflected in the physical properties.

Fig. 2 Left: TEM images of dried drops of the three liquid samples. Right: TEM images of ultrathin slices of epoxy resin containing small fragments of the three solid samples. The arrangement of the MNPs inside their matrix can be observed.

Fig. 2 also illustrates the MNP distribution within solid samples. The images of samples CF1S and CF1fS show regions of very dense nanoparticle packaging, while the raspberry-like structure of sample CF2S is preserved upon solid matrix transfer.

Concentration and saturation magnetization

A precise determination of the sample mass concentration, that is, the mass fraction of magnetic material present in each sample, is of great importance for obtaining correct specific values. The nanoparticle concentration of ferrofluids CF1L and CF2L, determined by ICP are, respectively, 0.0256 and 0.0262 g_MNP/g (Table 1). Specific magnetization data (Fig. 3) for both samples were calculated using these concentrations. Assuming saturation at H = 5 × 10⁴ Oe, M_S values of 67.2 and 78.0 emu g_MNP⁻¹ are obtained respectively, that, considering a mass density, ρ, of 4.907 g cm⁻³ for CoFe₂O₄, turn into 329.8 and 382.8 kA m⁻¹, respectively, in SI units. These M_S values are lower than those expected for bulk samples (425 kA m⁻¹), and support the hypothesis that sample CF2L, with a higher M_S value, presents larger crystallites than sample CF1L.³¹

Fig. 3 M(H) curves at 300 K.

The concentration of the other samples (Table 1) was calculated with the help of M(H) curves, assuming that M_S values are similar for samples of the same synthetic batch (CF1 or CF2). The resulting specific magnetization data are displayed in Fig. 3.

Heat capacity

Fig. 4 displays the specific heat of samples CF1L, CF1S, CF2L, CF2S and CF1fS, together with that of DEG and n-dodecane in the temperature range of the SAR(T) measurements. According to these data, DEG undergoes on heating a glass transition at 174 K with a heat capacity step of 0.96 J g⁻¹ K⁻¹. At higher temperature, a cold crystallization starts at around 230 K, although the formed crystalline phase is readily melted at 260 K. The enthalpy content of the cold crystallization and melting peaks is similar (around 5.7 J g⁻¹). This fact points the absence of a crystalline phase at low temperature or, in other words, reveals the amorphous nature of the state reached after cooling from room temperature. According to the curves corresponding to samples CF1L and CF2L, the presence of MNPs slightly modifies the behavior of DEG. The glass transition of these samples happens, respectively, at 174 and 171 K, with heat capacity steps of 0.91 J g⁻¹ K⁻¹ and 0.93 J g⁻¹ K⁻¹ after correction taking into account the mass of MNPs. These values are very close to those of pure DEG. Also, an amorphous low temperature phase is still reached. However, the cold crystallization seems to be blocked by the presence of the MNPs. Heating thermograms performed at 1 K min⁻¹ (not shown) in the same samples did not reflect the cold crystallization and subsequent melting either. Eventually, the heat capacity of n-dodecane (from literature) shows the presence of solid crystalline and liquid phases, with a melting point at about 260 K (melting peak not shown).

Fig. 4 Specific heat of some samples and solvents.

The specific heat of samples CF1fS and CF2S grows monotonically from 150 K to around 300 K, where the melting peak of the solid matrices (paraffin wax and PEG, not shown) starts to contribute. Sample CF1S follows the same trend but, in addition, presents a glass transition at around 172 K. This transition is assigned to the existence of some DEG trapped with the NPs during solidification of PEG. The quantification of this step (heat capacity of 0.27 J g⁻¹ K⁻¹) has allowed estimating the DEG content in the sample between 25–30% of the total weight.

From previous data, the heat capacity of the specimens used in SAR measurements was determined as C = Σc_P,im_i, where c_P,i and m_i, are, respectively, the specific heat and mass of each contribution to C. Table 2 collects the mass of these contributions in each case.

Table 2 Sample and container mass contributions to the heat capacity of the measured specimens

Sample	msample (g)	Container	mcontainer (g)
CF1L	0.1146	Quartz	0.3799
CF1S	0.1204	Epoxy	0.1331
CF1fL	0.0826	Quartz	0.3998
CF1fS	0.1545	—	—
CF2L	0.1032	Quartz	0.3539
CF2S	0.1233	Epoxy	0.1222

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ZFC/FC magnetization

It is widely known that MNPs can behave as superparamagnets provided that the thermal energy is high enough to overcome the energy barrier for magnetization reversal, E_b. For non-interacting MNPs, this barrier is E_b = KV, where K and V are, respectively, the magnetic anisotropy constant and the magnetic volume of each MNP. CoFe₂O₄ is a compound that, displaying almost the same saturation magnetization as magnetite, presents a magnetocrystalline anisotropy one order of magnitude larger.³² This implies that the ferro/ferrimagnetic to superparamagnetic (FM/SPM) transition is achieved at room temperature for much smaller cobalt ferrite particles (<10 nm) than iron oxide ones (∼25 nm).² Given the size of the MNPs involved in the present work, their FM/SPM transition should lie below or next to room temperature. Although E_b and, consequently, the transition temperature could be modified in case of interacting MNPs, this variation is expected to be less pronounced for high-anisotropy materials.³³

ZFC-FC magnetization vs. temperature measurements at low field are a useful tool for determining FM/SPM transitions. Three characteristic temperatures can be distinguished within ZFC-FC curves as temperature increases:³⁴ T_sat, temperature above which the FC magnetization starts to decrease, T_max, temperature corresponding to the maximum of the ZFC curve, and T_bra, temperature at which the ZFC and FC curves join together. T_bra is representative of the temperature at which the majority of the MNPs is unblocked and behaves as a superparamagnet. In this situation, the magnetization of each MNP flips direction due to thermal energy, with a characteristic time for magnetization reversal higher than the measuring time. T_max determines the onset of the superparamagnetic behavior assuming non-interacting and monodisperse MNPs, and its value is related to E_b. Accordingly, these curves must also reflect differences related to the particle size.

Fig. 5 shows the ZFC-FC M(T) curves of all the samples. From this data, values of T_max and T_bra were obtained and collected in Table 3. T_max and T_bra data of samples coming from batch CF1 are lower than those of samples coming from CF2. This result is indicative of a larger average particle volume of the later, as previously deduced from XRD and M(H) measurements. In effect, in our MNPs, main contributions to K are magnetocrystalline, K_M, and surface, K_S, ones, so K can be expressed as K ≅ K_M + K_S = K_M + λ(S/V), where λ is a positive constant,³¹ V is the magnetic volume of the MNP and S, its surface. K_M is the same for batches CF1 and CF2, but K_s increases for smaller particles due to their higher S/V ratio. This implies that, if K (CF1) > K (CF2) and T_max, T_bra (CF1) < T_max, T_bra (CF2), then V (CF1) < V (CF2).

Fig. 5 ZFC-FC M(T) trends for ferrofluids (black lines + small circles) and solid-matrix samples (purple lines + crosses) at 3 kA m⁻¹. Inset: temporal evolution of magnetization at 200 K. Blue dotted lines: specific heat of DEG (CF1 and CF2) and n-dodecane (CF1f).

Table 3 Parameters related to the FM/SPM transition of the samples

Sample	Tmax^a (K)	Tbra^a (K)	Tb^b (K)
a Obtained from FC/ZFC magnetization data (see text).b Obtained from SAR data (see text).
CF1L	—	206	244
CF1S	194	223	245
CF1fL	190	—	238
CF1fS	196	220	249
CF2L	—	273	—
CF2S	301	301	—

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Apart from this difference, ZFC-FC curves reflect other remarkable features. Let us pay attention to the differences between the ZFC-FC curves of samples CF2L and CF2S. While the latter shows a typical ZFC-FC trend for FM/SPM transitions, sample CF2L present several untypical features, related to the temperature dependence of the DEG viscosity. In the ZFC curve, taken upon heating, once DEG undergoes the glass transition, M(T) grows faster than it does in PEG, and at 194 K a great step is observed. This sharp magnetization rise is a consequence of the decrease of the DEG viscosity to a value low enough to allow physical orientation and/or reorganization of the MNPs by the action of the static magnetic field, thus increasing χ₀. Such behavior has been also observed in cobalt MNPs in several organic solvents.^35,36 Afterwards, at 243 K, the magnetization reaches a local minimum that coincides with the recrystallization phenomenon reflected in the c_P of DEG, and afterwards recovers the decreasing trend upon re-melting of the formed crystallites. In FC curves, magnetization of sample CF2L is considerably higher than that of sample CF2S, given that, in this case, the sample is solidified under the static magnetic field that generates the ordered high-χ₀ state.

Similar effects are also present in sample CF1L. At 194 K, when the DEG viscosity is low enough to allow physical orientation and/or reorganization of the MNPs, magnetization starts to increase, but the magnetization rise is counteracted by the transition of the sample to the superparamagnetic state, and only a slight bump is observed. In order to highlight this rise, we performed new ZFC measurements over samples CF1L and CF1S (with 25–30% of DEG) stopping at 200 K to record the temporal evolution of M. These results are shown in the inset of Fig. 5. In both cases the magnetization increases when the system is allowed to orientate within the applied magnetic field, although it is sample CF1L which undergoes the higher rise.

Finally, sample CF1fL does not present an increase in χ₀ upon melting of n-dodecane. This is probably related to the fact that this sample is already in the superparamagnetic state when the phase transition of the dispersive media takes place.

Evaluation of SAR

For non-interacting monodisperse MNPs, the out-of-phase component of the ac magnetic susceptibility, χ′′, can be expressed as:³²where χ₀ is the equilibrium susceptibility and τ, the effective relaxation time of the thermally induced magnetic-moment relaxation mechanisms, Néel and Brown, which depend on temperature. The function χ′′(T) presents a maximum when 2πfτ = 1. This maximum defines the blocking temperature, T_b, which determines the onset of the superparamagnetic behavior and, accordingly, is related to previously described T_max. Although the presence of interparticle magnetic interactions is known to modify eqn (2), the temperature at which χ′′(T) is maximum is still referred to as T_b. Consequently, χ′′(T) measurements are also a useful method for determining FM/SPM transitions although, in contrast to T_max and T_bra values from ZFC-FC M(T) curves, T_b increases with the frequency of the alternating magnetic field. Most widespread commercial devices for magnetic measurements allow performing χ′′ measurements as a function of T, f and H₀, although the f and H₀ available range is some lower than that used in MFH.

According to the Linear Response Theory (LRT), valid for low H₀ values, SAR depends on the out-of-phase component of the ac magnetic susceptibility as:

(in SI units),³² where μ₀ is the permeability of free space. Considering the relationship between SAR and χ′′, the maximum of SAR(T) defines as well T_b. Then, SAR(T) measurements of the samples, collected in Fig. 6, give simultaneous information of heating efficiency and FM/SPM transition with f and H₀ values more useful for MFH.

Fig. 6 SAR(T) of the samples measured at 111 kHz and 3 kA m⁻¹.

Samples CF2L and CF2S show the highest SAR values at room temperature, 1.2 W g⁻¹ and 0.6 W g⁻¹, respectively, measured under 111 kHz frequency and 3 kA m⁻¹ amplitude, while samples obtained from batch CF1 display lower SAR values at room temperature. This fact makes batch CF2 more interesting for MFH applications. However, the SAR(T) trend of these samples does not allow full understanding of the effects studied in the present work, given that their FM/SPM transition is beyond the temperature measuring limits. For this purpose, samples obtained from batch CF1 are of greater interest.

Certainly, the SAR values mentioned above may seem dramatically low compared to other values reported in the literature like, for example, that of bacterial magnetosomes, which show a breathtaking SAR of 960 W g⁻¹ at 410 kHz and 10 kA m⁻¹,³⁷ or the unprecedented SAR of 415 W g⁻¹ displayed by synthetic iron/iron carbide core–shell MNPs³⁸ at 96 kHz and 20 mT (16 kA m⁻¹). However, these values are not directly comparable with our data, as SAR depends strongly on the ac-field parameters, and much greater SAR is obtained for the same MNPs when using higher f or, especially, H₀ values, often beyond the physiological limits of the therapy. Comparison then implies the use of extrapolations.

A linear dependence of SAR on f is usually considered, although recent studies contradict this consideration, especially in presence of inter-particle magnetic interactions³⁹ or when high frequencies are used.⁴⁰ The dependence of SAR with H₀ usually shows a H₀ⁿ relationship, with n ranging between 0 and 3, depending on the characteristics of the MNP assembly and on the value of H₀. In the case of magnetosomes an average dependence of H^2.5 was found³⁷ using static hysteresis cycles with H₀ ranging from 0.5 to 10 kA m⁻¹. Using this fit and the typical lineal relation with f, the SAR of magnetosomes at 111 kHz and 3 kA m⁻¹ is estimated to be a still high but more modest 12.8 W g⁻¹. In the case of synthetic iron/iron carbide core–shell MNPs, measurements at 3.8 mT (≅ 3 kA m⁻¹) derive negligible SAR values.

Due to the variable and often unknown dependency of SAR on H₀ and f, fair comparisons between MNP assemblies can be made when SAR values are obtained with more similar parameters of the ac-field. For example, the highest SAR of assemblies of CoFe₂O₄ powder with different sizes was found to be 0.07 W g⁻¹ at 500 kHz and 5.4 kA m⁻¹ for samples with a mean diameter of 8.4 nm,⁴¹ close to the size of the crystallites of samples CF2. Also, highly concentrated Y₃Fe₅O₁₂ powder in Epoxy resin showed a SAR of 0.2 W g⁻¹ at 200 kHz and 1.77 kA m⁻¹.⁴² The SAR of assemblies of different ferrite MNPs, measured at 110 kHz and 4.8 kA m⁻¹, was found to be highest for MgFe₂O₄ particles, 33 W g⁻¹, and more modest for CoFe₂O₄ MNPs, 1 W g⁻¹.⁴³ Extrapolating these values to 111 kHz and 3 kA m⁻¹, using the Hⁿ dependence obtained in their work (n = 2.7 for MgFe₂O₄ and n = 2.8 for CoFe₂O₄), SAR turns into 9.4 W g⁻¹ and 0.3 W g⁻¹ for MgFe₂O₄ and CoFe₂O₄, respectively. Eventually, Fe₃O₄ assemblies with different coatings showed SAR values of 6.7 W g⁻¹ under 750 kHz and 2 kA m⁻¹.⁴⁴ The extrapolation of these values to 111 kHz and 3 kA m⁻¹, with n = 1.9, gives a SAR of 2.1 W g⁻¹. In sum, samples from batch CF2 show quite high SAR values, compared to samples of akin composition and measured with similar ac-field parameters.

Furthermore, although not among the highest ever found, SAR values of samples CF2 are high enough to achieve therapeutic temperatures in MFH treatments for certain tumor volumes⁴⁵ and MNP delivery methods.⁴⁶ For example, therapeutic temperatures were recently achieved in animal bladders by using injected magnetite MNPs,⁴⁷ whose SAR, measured at 40 kHz and in liquid, was about 0.8 and 5.6 W g⁻¹ at 2 and 5.5 kA m⁻¹, respectively.

Going back to Fig. 6, and just considering SAR values at 300 K, one could conclude that samples obtained from batch CF1, functionalized or not, have similar heating behavior. However, the study of the SAR(T) trends over the whole temperature range reveal interesting differences among the samples. Note that data acquired during the melting of n-dodecane (sample CF1fL) were not considered, since the first-order character of the transition prevents a correct evaluation of the heat capacity at this temperature range.

T_b data obtained from SAR(T) trends are collected in Table 3, for comparison with the characteristic temperatures obtained from ZFC-FC M(T) curves. Given that SAR was measured at high frequency (111 kHz), T_b values are systematically higher than T_max and T_bra data. However, T_b and T_max are linearly proportional, confirming the correlation between both magnitudes.

During measurements, taking place on heating, samples CF1L and CF2L, dispersed in DEG, go gradually from a high-viscosity to a liquid state, and the SAR(T) trends of both samples are continuous. The T_b of sample CF1L is 244 K, while sample CF2L is in the FM state within the whole temperature range. Sample CF1fL is in the solid state up to 263 K. In this state, the trend shows a maximum at T_b = 238 K that corresponds to the unblocking of the Néel relaxation mechanism, since the physical movement of the MNPs within the dispersive medium is inhibited. Above 263 K, in the liquid state, the trend is similar to that of sample CF1L. Measurements of samples CF1S, CF2S and CF1fS take place entirely in the solid state. The most outstanding difference between liquid and solid samples is the decrease of SAR values in all cases. Samples CF1S and CF2S present qualitatively the same trend than samples CF1L and CF2L, respectively, but lower values. However, sample CF1fS shows an increase of 11 K in T_b and the strongest reduction in SAR compared to CF1fL. As both CF1fL and CF1fS were solid at T_b, the observed shift cannot be assigned to the suppression of Brown relaxation in sample CF1fS. For the same reason, the SAR reduction undergone by all solid samples cannot be justified on the basis of different relaxation mechanisms.

Effects of ferrofluid dynamics and interparticle interactions in SAR

The study of the SAR(T) trends has pointed out the influence of diverse phenomena. According to eqn (2) and (3), SAR depends on the equilibrium susceptibility of the MNPs. Then, in order to explain the differences between the SAR of solid and liquid samples, changes in χ₀(T) values or, likewise, in M(T) = χ₀(T)H, must be considered.

As explained in the previous section, a great step is observed in the ZFC M(T) curve of the ferrofluid with uncoated raspberry-like MNPs (sample CF2L) once DEG undergoes the glass transition, and its viscosity becomes low enough to allow physical orientation and/or reorganization of the MNPs by the action of the magnetic field. As a consequence, magnetization of sample CF2L gets considerably higher than that of sample CF2S, in which field-induced reorganizations are inhibited due to the solid matrix. This greater static magnetic susceptibility of sample CF2L at 3 kA m⁻¹ with respect to that of sample CF2S would explain the SAR differences between them, although the increase in χ₀ may not be fully reflected in ac-measurements due to frequency effects.

The ferrofluid with uncoated and slightly-agglomerated MNPs (sample CF1L) undergoes the same effect, although the M(T) rise is much smaller, since such ferrofluid dynamics are readily inhibited by the transition of the sample to the superparamagnetic state. This effect should be reflected in SAR(T) measurements. Given that, during SAR measurements, the sample is subjected to an alternating magnetic field, and that T_b, unlike T_max, shifts to higher values as the ac-field frequency increases, the counteraction of the FM/SPM transition in SAR(T) measurements occurs at higher temperature. This fact should allow, according to the results observed in sample CF2L, a more pronounced increase in χ₀ for sample CF1L compared to sample CF1S than that observed in static measurements. This could be the reason why the SAR of sample CF1L appears considerably higher than that of sample CF1S, although χ₀(T) is similar in both samples in ZFC-FC curves.

The ferrofluid with well-dispersed functionalized MNPs (sample CF1fL) does not show an increase in χ₀ upon melting of the solvent, due to the higher melting temperature of n-dodecane, above the blocking temperature. Accordingly, the SAR differences between samples CF1fL, which displays the highest SAR values among the samples of the same particle size, and CF1fS, which is the worst performing sample, cannot be assigned to ferrofluid dynamics. However, the increase of 11 K in T_b of sample CF1fS with respect to that of sample CF1fL points an appreciable modification of E_b, presumably due to the effect of interparticle interactions. Indeed, the most outstanding change in particle distribution upon embedding in a solid matrix was clearly undergone by functionalized samples of batch CF1 (Fig. 2). While sample CF1fL presents well dispersed MNPs in n-dodecane thanks to functionalization, sample CF1fS shows very dense nanoparticle packing in paraffin wax.

In order to justify this change in T_b, the Berkov and Gorn (BG) model³³ was considered. This is a numerical simulation of χ′′(T) for various excitation frequencies, and for samples with different homogeneous particle concentration, damping parameter (λ) and magnetic anisotropy. It predicts variations both in the position and in the value of the maximum of χ′′(T). Depending on the value of the adimensional parameter β = 20K(J m⁻³)/M_S(kA m⁻¹)² two different regimes can be distinguished: high- and moderate-anisotropy regime (β > 1) and low-anisotropy case (β < 1), for which the behavior of χ′′(T) is quite different.

To roughly estimate the regime in which sample CF1fL lies, β can be calculated using the M_S value obtained from magnetic measurements (330 kA m⁻¹), and the K value derived from the Néel relaxation time, τ_N = τ₀ exp(E_b/k_BT), assuming E_b = KV, that is, overlooking the effect of magnetic interactions. At T_b, τ₀ exp(E_b/k_BT_b) = 1/2πf, and, considering τ₀ = 10⁻⁹ s, the value of K for sample CF1fL at 238 K is 274.025 kJ m⁻³ and then β = 50 ≫ 1 at this temperature. Even if the calculation is just an estimate, we can conclude that sample CF1fL lies in the high- and moderate anisotropy regime. For this case, the BG model predicts that inter-particle interactions do not fully govern the energy barriers, but such barriers are already created by the relatively high single-particle anisotropy. Interactions then decrease E_b, shift the χ′′(T) peak to lower temperatures and decrease its value for concentrations c_v > 0.08 (volume concentration). This is not consistent with the evolution of the functionalized samples coming from batch CF1f upon embedding in paraffin wax, since the SAR(T) trend of sample CF1fS is lower than that of sample CF1fL, but shifted to higher temperatures.

The BG model attributes this disagreement to the presence of MNP agglomeration within the samples. Such an agglomeration would lead to correlations between the anisotropy axis directions of the neighboring particles, increasing the energy barriers in the system, thus leading to the shift of the χ′′(T) peak towards higher temperatures with increasing particle concentration. In effect, the MNPs in sample CF1fS can be considered aggregated rather than concentrated, so a shift of the χ′′(T) peak towards higher temperatures together with a decrease in its value is predictable.

When comparing samples CF1L and CF1S, no appreciable shift in the position of the maximum of SAR(T) is observed. One possible reason is that, given that sample CF1L presents already some aggregation due to the lack of coating, the embedding in PEG has not substantially changed its agglomeration. This is also the case of samples CF2L and CF2S, for which TEM images reveal similar MNPs arrangement, although the high position of their SAR(T) maximum, out of the measuring range, prevents further discussion.

Conclusions

In summary, several are the factors to be taken into account when evaluating the heating efficiency of MNPs. The most relevant among them is the size, which affects the saturation magnetization and the temperature of the FM/SPM transition, especially in materials with high magnetic anisotropy like cobalt ferrites. But the influence of both the dispersive media and the MNP arrangement cannot be neglected. The SAR drop of up to 50% observed in MNPs unable to be oriented or form structures upon application of the magnetic field suggest that, at evaluating a sample in form of ferrofluid, we are most probably overestimating its heating ability in real application conditions. Also, the fact that a similar SAR degradation can be undergone by a sample that, initially well dispersed in liquid, acquires a high degree of agglomeration when transferred to a solid matrix, arises the question whether the use of initially isolated and well-dispersed nanoparticles is really the best strategy to obtain high SAR values in magnetic fluid hyperthermia applications, where nanoparticles are susceptible of aggregation inside a tissue. Eventually, the evaluation of SAR at just one temperature is shown to provide a partial view of the heating ability that may lead to incorrect interpretations, while the study of SAR(T) trends derives useful and complementary information.

Acknowledgements

This work has been funded by the Spanish MICINN and FEDER, projects, Grant no. MAT2007-61621 and Grant no. CSD 2007-00010, and by the University of Zaragoza and Banco Santander Central Hispano S.A., project UZ2012-CIE-10. I. A. thanks the Spanish CSIC for her JAE-Predoc contract.

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