Power Losses

If a magnetic material is composed of few-nanometer particles, domains are no longer able to form and the external field will produce (depending on its strength) the coherent rotation of all magnetic moments in the material.

In this situation, the process for their magnetization reversal is the thermally activated jump over an energy barrier, the height of which depends on the anisotropy K, on the volume V of the particle, and the applied magnetic field. The exact shape of the hysteresis in the general case must be calculated numerically and depends on f, H_max, K, V, and on the temperature T. The hysteresis loop evolves progressively from a Langevin function for a zero frequency – or equivalently an infinite temperature - to a Stoner-Wohlfarth (SW) hysteresis at infinite frequency or null temperature.

The power absorbed by a magnetic nanoparticle is given by P= A.f, where A is the area of the hysteresis loop at the frequency and magnetic field at which the experiment is conducted. SAR amplitude and its dependence on external parameters strongly depend on the NP magnetic properties.

To calculate A and interpret hyperthermia experiments, two models valid in different domains can be used. First, when the applied magnetic field is small compared to the saturation field of the NPs, the linear response theory can be used. In this case the hysteresis loop is an ellipse of area

with

M_s: saturation magnetization,

ρ the density of the material,

τ the Néel relaxation time,

 

where η is the viscosity of the liquid, V_hyd is the hydrodynamic volume of the nanoparticles, and k_BT is the thermal energy. Néel relaxation time depends on the magnetic properties and it is given by

In fact, the SPA values measured for a ferrofluid containing magnetic nanoparticles depends strongly of the magnetic, morphological and rheological properties of the system, specifically to M_S, K_eff , V_mag, η and V_hyd [5]. From theoretical point of view, these mechanisms are analyzed from different ways. Rosensweig [6] proposed a model where both magnetic and mechanical relaxations are taken into account as independent and occur in parallel, and the effective relaxation time is τ_eff^-1 = τ_B^-1 + τ_N^-1, but the use of τ_eff is not completely justified. Recently, Usov and Liubimov [1] calculated numerically the hysteresis loop of the system taking into account simultaneously both, magnetic and mechanical relaxation, and they define the viscous and magnetic mode, where mechanical or magnetic mechanism dominates according to the value of H₀ in comparison to the anisotropy field H_K. However it is important to focus on the relaxation times τ_B and τ_N. For several experiments, in agglomerated nanoparticles for example, it is interesting to consider only the magnetic mechanism. For this case, Carrey et al [7] describe the area of minor hysteresis loop of the system as an ellipse, making easy to estimate theoretically the SPA value of the system.

 

Despite these theoretical analyses on the SPA values of magnetic monodomains in the presence of ac magnetic field, we detected the need of a tool that easily compiles the theoretical information on the heat generation in MFH, allowing that the interdisciplinary community working on MFH determines the main mechanism acting in this process. At the same time, this tool must allow the estimation of the SPA value of a ferrofluid for in vivo and in vitro MFH experiments.

 

 

[1] N. A. Usov and B. Ya. Liubimov, J. Appl. Phys. 112 (2012) 023901.

[2] S. Laurent, S. Dutz, U. O. Häfeli, M. Mahmoudi, Adv. Colloid Interface Sci. 166 (2011) 8 – 23.

[3] R. E. Rosensweig, J. Magn. Magn. Mater. 252 (2002) 370-374.

[4] G. F. Goya, V. Grazu, M. Ricardo Ibarra, Current Nanoscience 4 (2008) 1-16.

[5] Seongtae Bae, Sang Won Lee, Atsuo Hirukawa, Yasushi Takemura, Youn Haeng Jo, Sang Geun Lee, IEEE Trans Nanotechnol 8 (2009) 86 - 94.

[6] R. E. Rosensweig, J. Magn. Magn. Mater. 252 (2002) 370-374.

[7] J. Carrey, B. Mehdaoui, M. Respaud, J. Appl. Phys. 109 (2011) 083901.

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For single-domain MNPs dispersed in liquid medium (the usual case for biomedical applications), the magnetic losses under ac magnetic fields are originated in the delay between applied field and magnetic moment of the system.[1] The relaxation of magnetic moment is related with two mechanisms [2], the mechanical mechanism associated with the physical rotation of the whole nanoparticle in the fluid (Brown relaxation process), and the magnetic relaxation process related to the fluctuation of magnetization through magnetic energy barrier (Néel relaxation time). Brown relaxation time depends on the rheological properties of the system and can be calculated by

where V_mag is the magnetic volume, K_eff is the effective anisotropy of the monodomain and τ₀ is the characteristic relaxation time (= 10^-9 s – 10^-11 s). Both mechanisms happen simultaneously in the system, however, both depends on different parameters: τ_B is proportional to the hydrodynamic volume and viscosity, while τ_N depends exponentially of the anisotropy and the magnetic volume of the particle. It is important to clarify that in real experiments of MFH involving magnetic nanoparticles the hydrodynamic radius is generally greater than the magnetic one as consequence of the functional layer attached to the surface of magnetic material, generally an organic layer, which influences the relation between τ_B and τ_N.