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Validity limits of the Néel relaxation model of magnetic nanoparticles for hyperthermia
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2010 Nanotechnology 21 015706 (http://iopscience.iop.org/0957-4484/21/1/015706) View the table of contents for this issue, or go to the journal homepage for more
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IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 21 (2010) 015706 (5pp)
doi:10.1088/0957-4484/21/1/015706
Validity limits of the N´eel relaxation model of magnetic nanoparticles for hyperthermia Rudolf Hergt, Silvio Dutz and Matthias Zeisberger Institute of Photonic Technology, Albert-Einstein-Strasse 9, D-07745 Jena, Germany E-mail:
[email protected]
Received 28 August 2009 Published 30 November 2009 Online at stacks.iop.org/Nano/21/015706 Abstract The derivation of the optimum mean diameter of magnetic nanoparticles (MNP) for hyperthermia as a tumour therapy in the literature is commonly reduced to application of the N´eel relaxation model. Serious restrictions of this model for MNP for hyperthermia are discussed and a way is outlined to a more comprehensive model including hysteresis.
magnetic spinel compounds Brown relaxation is prevalent in aqueous suspensions, which may be well studied by polarization–optical relaxation measurements. There are many experimental studies showing excellent agreement with the Brown relaxation model (e.g. Delaunay et al 1995, Bacri et al 1997, Payet et al 1998). In comparison, N´eel relaxation is less satisfyingly verified by experiment. Due to the exponential size dependence of the relaxation time given below, the commonly observed log-normal size distribution leads to flat spectra, a typical example of which was reported by Raikher and Pshenichnikov (1985) for magnetite in kerosene for frequencies up to 106 Hz. For a similar ferrofluid type having a smaller distribution width Hanson (1991) found a well-pronounced N´eel relaxation peak at about 107 Hz. Wide band measurements up to 6 GHz were reported by Fannin et al (1999) for magnetite in a hydrocarbon (Isopar M) carrier liquid. For ferrofluids with particles being immobilized by freezing at −100 ◦ C the measured spectra show an increase of χ above 107 Hz—presumably due to N´eel relaxation—which passes to a large maximum at 1 GHz related to ferromagnetic resonance (FMR).
1. Introduction In the last few years an increasing interest has evolved in the application of magnetic nanoparticles for hyperthermia as a tumour therapy (for a survey see, e.g., Andr¨a et al 2007). Several papers deal with models of relaxation processes in nanoparticle systems in order to find ways of increasing the specific loss power of magnetic nanoparticles, which is necessary for a successful implementation of the therapy (Hergt et al 2006). However, most authors restrict their considerations to N´eel and Brown relaxation (e.g. Rosensweig 2002). Fortin et al (2007, 2008) discuss the specific loss power of a magnetic nanoparticle system in the framework of the relaxation model of N´eel (1949). They claim to be ‘able to derive the best nanoparticle design’ (Fortin et al 2008). Those authors overlook that the simple relaxation model has a limited range of validity due to the restricting presuppositions it is based on. It is shown in the present paper that, to attain a high output of heating power in hyperthermia, one has to go beyond the narrow framework of the common theory of superparamagnetism and the N´eel relaxation model. There are many experimental investigations in the literature on relaxation in magnetic particle systems. Mainly aqueous suspensions of nanoparticles of magnetite and maghemite were investigated (e.g. Fannin and Charles 1989; for further references see Hergt and Andr¨a 2007). But also non-aqueous systems like magnetite in kerosene, oil or hydrocarbons (e.g. Hanson 1991, Dormann et al 1997, Fannin et al 1999), as well as other types of ferrite nanoparticles like Co-ferrite, were intensively studied. Since Co-ferrite exhibits large anisotropy in comparison to magnetite and other 0957-4484/10/015706+05$30.00
After giving a short overview on theory, in the present paper limiting conditions of the N´eel model will be discussed which lead to the conclusion that nanoparticles with optimum magnetic properties for hyperthermia may be found beyond the superparamagnetic size range. The condition for eddy current limitation in healthy tissue is discussed with respect to the optimum choice of frequency and amplitude of the external magnetic ac field for hyperthermia apart from the N´eel model. 1
© 2010 IOP Publishing Ltd Printed in the UK
Nanotechnology 21 (2010) 015706
R Hergt et al
2. Theory
Table 1. Parameters separating Brown and N´eel spectral regions.
In general, a magnetic sample being characterized by the magnetic moment m responds to a variation of an external magnetic field H in a retarded nonlinear way which is known as hysteresis (Bertotti 1998). While hysteresis in general is a quite nonlinear phenomenon, in special cases one may assume for sufficiently small field amplitudes that the magnetic susceptibility is approximately independent of the external magnetic field. Irreversibility is taken into account by assuming the susceptibility to be complex. Its imaginary part characterizing magnetic losses may be experimentally determined from the out-of-phase response of the sample for a sinusoidal excitation at the frequency f . The specific loss power (SLP) may be expressed by the imaginary part of the susceptibility χ (see textbooks on electromagnetism, e.g. Landau and Lifshitz 1960): SLP(H, f ) = μo πχ ( f )H 2 f /ρ
(1)
χo = const. (2)
The linear approximation represented by equations (1) and (2) may be useful for the description of certain types of materials only in special cases. To our knowledge, it was used for the first time by Debye (1929) to describe the dielectric response of polar molecules in liquid media. For ferrofluids, equation (2) was derived from an equation of motion by Shliomis (1974) with χo = μo MS2 V /(3kT ), where MS is saturation magnetization and V is particle volume. A more rigorous theory of the dynamic susceptibility of magnetic fluids was presented by Shliomis and Stepanov (1994) for a wide frequency range including ferromagnetic resonance (FMR). However, those models, which are based on the Landau–Lifshitz equation, are restricted to the small-amplitude precession regime with low damping. Thus, they are not suitable for describing hysteresis losses at large amplitudes being of interest for magnetic particle heating. For ferrofluids, the relaxation time being widely used in the literature may be derived in the framework of a simple magnetic two-level model. Following N´eel (1949) it is given in terms of thermal activation across the energy barrier K V ( K is anisotropy energy density) separating the two levels:
τN = τo exp[K V /(kT )]
(τo ∼ 10−9 s).
fC (kHz)
Maghemite/water Maghemite/ester oil Ba-hexaferrite/water Co-ferrite/water Co-ferrite/glycerin Co, hexag./water
25 24 11 7 9 6
8 0.1 50 100 00 0.1 2500
3. Limitations of the N´eel theory There are two conditions which are supposed in the derivation of the relaxation model of N´eel (1949). First, anisotropy energy has to be larger than thermal energy K V kT . Second, in the presence of a magnetic field applied along the axis of anisotropy H < 3kT /(μo MS V ) should hold. This means that the field amplitude is within the linear range of the Langevin curve. In particular the latter condition is of relevance for magnetic particle hyperthermia. As a consequence, the validity range of the linear relaxation theory is restricted to small values of field amplitude and particle diameter, which are given in figure 1 by the area below the dashed line. Clearly, the particle diameter being acceptable for linear theory decreases with increasing field amplitude. For instance, for maghemite ( MS ∼ 400 kA m−1 ) a value of the field amplitude of 10 kA m−1 being appropriate for
(3)
Besides N´eel relaxation, so-called Brown relaxation due to re-orientation of particles may occur in a fluid suspension having a viscosity η. It is characterized by the relaxation time
τB = πηdh3 /(2kT )
dC (nm)
(dh is the hydrodynamic diameter which, due to, for example, particle coating, may be essentially larger than the diameter of the magnetic particle core). Generally, the faster of both relaxation mechanisms is governing the absorption of the particle system. Due to the very different size dependence given by equations (3) and (4) for the two relaxation regimes, there is a separation of the spectral regions of N´eel and Brown relaxation. The boundary between both regions is defined by τN = τB . In general, N´eel relaxation prevails at higher frequency combined with smaller particle size and vice versa for Brown relaxation. The boundary frequency f C and the corresponding particle diameter dC are given in table 1 for examples of typical ferrofluid systems. Besides the very anisotropic Co-ferrite and metallic Co particles, Brown relaxation is restricted to frequencies being of minor practical interest for hyperthermia. Moreover, though viscous frictional losses may persist up to large particle diameters (Hergt and Andr¨a 2007) they are of minor relevance for hyperthermia since mobility of magnetic particles—located either interor intracellular—is highly suppressed in biological tissue. Insofar as it is not appropriate to consider Brown relaxation playing any essential role in magnetic particle hyperthermia as discussed by Fortin et al (2008). It should be pointed out that magnetic losses—apart from Brown heating—are independent of the intra- or intercellular position of the particles in tumour tissue. Astonishingly, the term ‘intracellular heating’ is still used by some authors (e.g. Fortin et al 2008) though it was clearly shown by Rabin (2002) that it is not reasonable to expect any special heating effect of magnetic nanoparticles on the intracellular level.
where ρ is the mass density and μo = 4π × 10−7 V s A−1 m−1 . Obviously, this formula is restricted to small amplitudes of the magnetic field since the typical magnetic saturation is not comprised. In simple relaxation models, equation (1) is combined with the assumption of an exponential decay of the magnetization with a relaxation time τR . For a linear system this is equivalent to a frequency spectrum χ ( f ) of the type (e.g. Bertotti 1998)
χ ( f ) = χo 2π f τR /[1 + (2π f τR )2 ],
Ferrofluid type
(4) 2
Nanotechnology 21 (2010) 015706
R Hergt et al
Figure 1. Critical diameter above which linear theory ceases dependent on magnetic field amplitude (for magnetite particles with an effective anisotropy of 10 kJ m−3 at a frequency of 400 kHz).
Figure 2. Dependence of the specific hysteresis loss per cycle on mean particle diameter and magnetic field amplitude for monodisperse maghemite nanoparticles (Hergt et al 2008, by permission of IOP).
hyperthermia (e.g. Andr¨a et al 2007) restricts the validity of the relaxation model to d < 16 nm. In comparison, Fortin et al (2008) consider within the framework of the N´eel model particle diameters up to 30 nm at a field amplitude of 25 kA m−1 for which a critical particle diameter d < 12 nm may be derived from figure 1. Moreover, in addition to the just-discussed restriction there is a general constraint in superparamagnetism, which implies a limitation of the particle diameter. A negligible coercivity will be found for a magnetic particle system only if the characteristic time of measurement τM is much larger than the relaxation time τN of the particle ensemble. Only in this case may the system be termed to be superparamagnetic. For instance, for a magnetic field frequency of 400 kHz—which is appropriate for hyperthermia—and a magnetic anisotropy energy density of 104 J m−3 (for magnetite particles of ellipsoidal shape with an aspect ratio of 1.4) the critical diameter following from the condition τN ∼ τM is about 17 nm. Above this critical size nonlinear hysteresis losses have to be taken into account. Accordingly, the validity range for the superparamagnetic relaxation model is further reduced to the dashed area shown in figure 1. Equations (1)–(3) of the linear relaxation model may be applied only within the limits of this area. The N´eel relaxation model predicts a maximum of specific loss power (SLP) for the value of particle diameter corresponding to the relaxation time τN (Hergt et al 1998). This maximum is interpreted by some authors (e.g. Fortin et al 2007, 2008) to represent an optimal mean particle size for hyperthermia, which is the wrong conclusion for the following reason. While in the range below the maximum the N´eel model is confirmed by an experimentally found steep increase of SLP with increasing diameter (Hergt and Andr¨a 2007); it fails for diameters above the maximum. There, equations (1)–(3) predict N´eel relaxation losses to approach zero. Instead, ferromagnetic hysteresis losses arise for sufficient field amplitude. Accordingly, in the transition region
from superparamagnetic to stable ferromagnetic behaviour a superposition of N´eel relaxation and ferromagnetic hysteresis losses has to be considered (Hergt et al 1998), in particular if the particle size distribution is broad. Recently, a phenomenological model was presented (Hergt et al 2008) which allows us to describe the dependence of magnetic losses on mean particle size and magnetic field amplitude in the whole size range from superparamagnetic up to multidomain particles including the effect of size distribution. As a result, figure 2 shows the dependence of specific hysteresis loss (SHL) per cycle on mean particle size and field amplitude for monodisperse maghemite nanoparticles. The specific loss power results by multiplying the hysteresis loss per cycle by frequency. Using the formalism developed previously (Hergt et al 2008) the dependence of losses per cycle on the mean size of the particle ensemble was calculated for a distribution of particle size with a distribution width parameter D0 = 0.7 nm. (There, the phenomenological parameter d1 of the model presented previously was changed from 15 to 18 nm.) Results are shown in figure 3 for different values of the field amplitude as a parameter. For comparison, a typical curve resulting from the N´eel model is shown as a dashed line. This curve was derived from the work of Fortin et al (2008) for a field amplitude of 25 kA m−1 and comparable distribution width parameter. Note that frequency is eliminated by giving the loss per cycle in figure 3. The figure shows that losses are fairly well described by the N´eel model for small mean diameter and small field amplitude (
