Tuning the dynamics in Fe₃O₄ nanoparticles for hyperthermia optimization

Iron oxide nanoparticles are under intensive investigation for biomedical applications,^1–4 including magnetic hyperthermia cancer treatment, which use AC magnetic fields for excitation. Despite this, many questions remain about how to optimize the heat delivery.^5–10 Clinical hyperthermia uses a single excitation frequency, most commonly between 100 and 400 kHz, and the efficiency of heating depends on the degree of overlap with the spectrum of relaxation frequencies. The Brownian rotation frequency f_B is on the order of hundreds of kHz or lower, depending on the hydrodynamic diameter, but when particles are taken up by cells, the rotation could be hindered. For an isolated 10 nm particle at 300 K, the Néel relaxation frequency f_N is over 10 MHz.¹¹ However, when clustered, the resulting magnetostatic interactions can shift f_N and broaden the range of Néel relaxation rates. Here, we describe how magnetostatic interactions impact the real and imaginary parts of the AC magnetic susceptibility, measured over a wide range of excitation frequencies (10 kHz–4 MHz) at different temperatures (100 K–150 K), using a custom-built high frequency AC magnetic susceptibility (ACS) insert for a Quantum Design PPMS,¹² and how the addition of a small DC magnetic field can enhance the response and shift the relaxation peak frequency. While we use a model system of immobilized monodisperse nanoparticles, this approach is general and could be readily adapted for use in clinical applications.

Monodisperse Fe₃O₄ nanoparticles were synthesized by thermal decomposition in a high boiling point organic solvent and coated with a mixture of oleic acid and oleyl amine surfactants. Further synthetic details have been reported previously^13,14 and show a high specific saturation magnetization of ∼82 J/Tkg.¹⁵ The particle diameter was found to be (10.7 ± 1.2) nm, based on transmission electron microscopy measurements. The particles were dispersed in melted eicosane (0.15 vol. %) and then cooled to solidify the sample. Electron micrographs, the size distribution analysis, and the zero field cooled (ZFC) and field cooled (FC) magnetization curves are shown in the supplementary material.

In the DC magnetization measurements, the maximum in the ZFC occurs at approximately (105 ± 10) K, and there is a slow drop at higher temperatures. Below the peak, the FC curve is almost constant and the ZFC curve decreases. The gradual decay of the ZFC curve is characteristic of systems with interparticle interactions.^16–18 In a non-interacting model where the peak is at the Blocking temperature, T_B, the anisotropy energy density K estimated from the relation $KV=25kBTB$ would be 5.6 × 10⁴ J/m³, where V is the average particle volume and k_B is the Boltzmann constant. This is higher than that found for ultra-dilute samples of magnetite nanoparticles, ∼2 × 10⁴ J/m³,⁹ where the curve drops rapidly above the peak.

The real (χ′) and imaginary (χ″) parts of the AC susceptibility reveal the distribution of relaxation frequencies. For these samples in our measurement range, the greatest response was found between 110 and 150 K, at and slightly above the ZFC peak of the DC measurements. Figure 1 shows the AC magnetic susceptibility components as a function of frequency, for H_DC = 0, at different temperatures, T. The real part shows a linear decay with frequency, for a logarithmic scale, at 110 K. At higher T, the real part has a more typical roll-off curvature and higher amplitude as fewer spins remain frozen. At lower T, some of the particles are still frozen, while well above it, the particles cannot keep up with the phase of the driving field. At 110 K, the imaginary part shows a very broad peak with a peak frequency at about 100 kHz. At higher temperatures, the dispersive peak sharpens and the peak shifts to higher frequencies. AC hysteresis loops and specific absorption rate (SAR) measurements made with different fixed frequency excitations have also indicated a distribution of relaxation times.¹⁹ 

The real and imaginary parts of the magnetic susceptibility can be described by

and

respectively, where $χ∞$ is the infinite frequency susceptibility, $χ0$ is the DC susceptibility, and $τ$ is a characteristic relaxation time. $χ′f$ is flat at frequencies where the sample response is in phase with the AC driving field and drops off at higher frequencies as a phase lag develops. $χ″f$ represents the response that is 90° out of phase with the driving field. It rises and peaks at a frequency of f_c = 1/2πτ. For non-interacting immobilized monodomain particles, $τ$ is often written as

where $τ0$ is the Larmor precession time, ∼1 ns, H is the DC field applied along the particle easy axis, H_K is the anisotropy field, and k_B is the Boltzmann constant. The data of Fig. 1 show qualitatively that f_c increases at higher temperature, as expected. While it is possible to fit χ′ and χ″ data at 150 K with the TEM size distribution, there are deviations at lower temperatures. Parameters extracted from fits with a non-interacting model for a single temperature are, therefore, unreliable; further details are provided in the supplementary material.

Interactions of magnetic nanoparticles have been empirically modeled in terms of a Vogel–Fulcher relation^20–22 or a Cole–Cole or Cole–Davidson distribution of relaxation times.^23–25 In the most common form of the Vogel–Fulcher equation, the temperature in Eq. (3) is replaced by the quantity (T-T₀), where T₀ is interpreted as a spin freezing temperature. The origin of this factor was explained in spin glasses by Shtrikman and Wohlfarth,²⁶ who assumed a statistical distribution of interaction energies E_int and a Langevin function with an average proportional to tanh(E_int/k_BT) ∼ 1/T. This extra term in the exponential of the Néel–Brown–Arrhenius law leads to the apparent 1/(T− T₀) in the Vogel–Fulcher law and k_BT₀ = E_int²/E, where E is the Néel–Brown–Arrhenius law energy. Because a Langevin function has an H/T dependence, lowering the temperature or increasing the magnetic field slows down the fluctuation rate. The $χ″$ data for different temperatures can be scaled with the correct choice of T₀, as shown in Fig. 2, with T₀ = 30 K. If k_BT₀ is interpreted in terms of the magnetostatic energy of a particle in an average interaction field, M_sVH_int, where is the saturation magnetization, then H_int ∼ 1.5 mT. This indicates that the energy barrier height KV and the magnetostatic energy M_sHV are of comparable magnitude. At large fields, the latter dominates, and the spins are mainly in the ground state. Ambiguity in the peak frequency of $χ″$ for 140 and 150 K and the flatness of the 110 K data lead to some uncertainty in estimating T₀. While the peaks of the $χ″(f)$ data can be scaled with a spin freezing temperature or average interaction energy, real interactions can either increase or decrease the energy barrier. Previous work found T₀ = 3 K for 9 nm Fe₃O₄ nanoparticles measured between 0.01 Hz and 1 kHz.²⁰ Just as the Blocking temperature depends on the measurement time, the use of higher frequencies, probing fluctuations over shorter time scales, leads to a higher spin freezing temperature.

Depending on H, f, and T, different subsets of spins respond to an AC driving field and contribute to the dissipation $χ″$⁠. The Blocking temperature measured with a slowly varying field depends on the time scale of the measurement. At high temperature, the spins equilibrate within the measurement time, and a superparamagnetic magnetization curve is measured. Below T_B, hysteresis is seen because the response of the spins lags changes in the driving field. $χ″(f)$ can be viewed as a distribution of Blocking frequencies f_c or the related energy barriers.

The AC susceptibility as a function of a DC magnetic field provides insight into the Blocking frequency distributions. Large DC magnetic fields lock the spins in place, so that they respond minimally to a low amplitude AC driving field, and both $χ′$ and $χ″$ are very small across the entire frequency range. However, for smaller DC fields, surprising non-monotonic behavior is observed, as shown in Fig. 3. At 140 K, both the amplitude of the real part and the relaxation peak frequency changed non-monotonically with H_dc. Below 1.25 mT, the peak frequency decreases and the $χ″$ amplitude increases with increasing DC fields. At and above 1.25 mT, these trends reverse. The dissipation at 100 kHz increases by a factor of 3.8 between 0 and 1.25 mT. We note that 1.25 mT is very close to H_int estimated for with Vogel–Fulcher scaling (about 1.5 mT).

Cole–Cole analysis of χ″ vs χ′ data from Fig. 3 was used to get a more accurate estimate of the average relaxation time τ and the relaxation time distribution. For non-interacting particles, a Debye model such as that described in Eqs. (1) and (2) applies, and the Cole–Cole plot will be a semicircle with a peak at τ. In cases where the center of the circle is depressed below χ″ = 0, the data can be fit with an empirical parameter α associated with a distribution of relaxation times,

Representative fits of the Cole–Cole plots are shown in the supplementary material, and Fig. 4 summarizes the results as a function of H_dc. The average τ is sharply peaked and is nearly 20 times larger at 1.25 mT than at H = 0. α was found to vary between 0.46 and 0.67. While care should be taken in the quantitative interpretation of α, its non-zero value shows that magnetostatic interactions impact the relaxation time. τ tracks the peak frequency χ″, and the relative width of the distribution associated with α is the greatest at 1.25 mT, consistent with the data of Fig. 3(b).

In the uniaxial case with the DC field applied along the easy axis, the application of a small field would increase some energy barriers and decrease others, broadening the distribution of relaxation rates. However, the particles are randomly oriented relative to each other and to the applied field. Even if a small amount of shape anisotropy made them effectively uniaxial when isolated,²⁷ with magnetostatic interactions, the energy landscape is complex, with many local minima. Dipolar interactions among nanoparticles can lead to DC hysteresis characteristic of a soft ferromagnet, with a steeper rise in M(H) and the development of a small coercivity.²⁸ In the assembly of iron oxide nanoparticles studied here, the effect of a small H_dc is to orient more of the moments preferentially, and so they respond to the AC field and dissipate energy.

We have shown that magnetostatic interactions of nanoparticles affect the AC susceptibility even for low concentrations and the limitations of fitting temperature-dependent data with a consistent set of parameters when interactions are ignored (see the supplementary material). While interactions broaden the range of frequencies, the magnitude and peak frequency of χ″ can be tuned using a small DC field comparable to the average interaction field. While we demonstrated this effect for a relatively dilute sample of 10 nm particles at low temperature, it should also be present for more concentrated samples at clinical hyperthermia temperatures, using size and/or shape anisotropy to tune the zero field thermal stability parameter, Δ = KV/k_BT. Our approach combines AC susceptibility measurements with a variable H_dc to quantify the effect of magnetostatic interactions, enabling the peak frequency and amplitude to be tuned. In the future, this concept could be used to maximize the SAR for a single excitation frequency. Magnetic particle imaging (MPI), which uses low frequency AC excitation together with spatial gradients of DC magnetic fields,²⁹ shows that incorporating H_dc in a clinical environment is feasible, and it could similarly be combined with existing hyperthermia equipment.^30,31

See the supplementary material for further information about the particle size distribution, zero field-cooled and field-cooled magnetization, fitting to a non-interacting particle model, field-dependent AC susceptibility, and Cole–Cole model fitting.

S.A.M. acknowledges support from U.S. NSF Grant No. ECCS-1709845 and DOE Grant No. DE-SC0019237. S.R.G. acknowledges support from EPSRC Grant No. EP/S016465/1.