Supplementary materials: Hyperthermic effects of dissipative structures of magnetic nanoparticles in large alternating magnetic fields

Transcription

1 Supplementary materials: Hyperthermic effects of dissipative structures of magnetic nanoparticles in large alternating magnetic fields Hiroaki Mamiya and Balachandran Jeyadevan Appendix A: Verification of our methods Test simulations were performed to check the validity of the used method by setting conditions similar to already published reports 1,2. The results agree with the behaviour reported in earlier papers 1,2. First, the method for the simulation of the reversal in the non-rotatable nanoparticles was verified. Carrey et al. 1 recently performed numerical simulations of hysteresis loops for the reversals, and we used their parameters in the test simulation. As shown in Supplementary Figs. S1 and S2, the results obtained from our simulation of the reversal were consistent with the results reported by Carrey et al. Next, the method for the simulation of the reversal and rotation in the rotatable nanoparticles was verified. There have been no prior theoretical studies on systems where both reversal and rotation occur simultaneously in a large AC magnetic field. Consequently, the comparisons with prior studies were performed at the two extreme conditions. First, high viscosities were assumed, and because reversals are predominant over rotation under these conditions, the results were compared with those reported by Carrey et al 1 (see Supplementary Figs. S1, S2). Second, a high anisotropic field was assumed. Because reversals are predominant over rotation, the results were compared with the numerical simulations of Yoshida et al. for nonlinear Brownian rotational relaxation of magnetic fluids in a large excitation field 2 (see Supplementary Figs. S3, S4). The results obtained from our simulation of the reversal and rotation were consistent with these studies as shown in Supplementary Figs. S1 S4. Figure S1 Hysteresis areas obtained from the simulations plotted as a function of = H ac /k B T for various uniaxial anisotropy constants, K eff, where H ac = 0.8 ka/m, T = 300 K, f = 100 khz, and M s = 10 6 A/m. The small dots are the results of simulations reported by Carrey et al.*, and the open symbols are the results of our test simulations for the reversals. The large solid symbols are for the results of our test simulations for the reversals and rotation, where the viscosity was set at 1000 mpa s and the hydrodynamic volume was assumed to be the same as that of the magnetic core. *Reproduced from Carrey et al. with permission (Copyright 2011 American Institute of Physics)..

2 Figure S2 Normalized hysteresis area as a function of the normalized magnetic field for various nanoparticle radii, where K eff = J/m 3, M s = 10 6 A/m, f = 100 khz, H ac = 16 ka/m, T = 300 K, and f 0 = s 1. The solid lines are for the results of the simulations reported by Carrey et al.*, and the open symbols are for the results of our test simulations for the reversals. The large solid symbols are for the results of our test simulations for the reversals and rotation, where the viscosity was set at mpa s and the hydrodynamic volume was assumed to be the same as that of the magnetic core. *Reproduced from Carrey et al. with permission (Copyright 2011 American Institute of Physics). Figure S3 Relationship between and magnetisation <M> st in steady state. The open circles are the results of the simulations reported by Yoshida et al.* The solid curves are the results of our test simulations for the reversals and rotation, where the anisotropy field is set at 400 ka/m. *Reproduced from Yoshida et al. with permission (Copyright 2009 The Japan Society of Applied Physics).

3 Figure S4 Frequency dependency of the real and imaginary parts of the susceptibility of (a) the fundamental component and (b) the third harmonic when sinusoidal fields with of 10 were applied. The open symbols are the results of the simulations reported by Yoshida et al.* The solid symbols are the results of our test simulations for the reversals and rotation, where the anisotropy field was set at 400 ka/m. The solid curves are the theoretical values.* *Reproduced from Yoshida et al. with permission (Copyright 2009 The Japan Society of Applied Physics). Appendix B: Other results of the simulations. Other supporting results obtained for nanoparticles with various sizes and shapes are detailed in Figs. S5 and S6. Figure S5 presents the contour plots of P H /(H ac f) for elongated spheroidal nanoparticles with the same shape ( = 1.4) and various sizes, as a function of the amplitude H ac and frequency f. For the smaller nanoparticles with 2R M = 12 nm, which are considered SPIONs, an evident rise of the P H /(H ac f) due to the rotatability was observed at 100 khz and 32 ka/m where H (H, ) was comparable to B. For the slightly larger nanoparticles with 2R M of 18 nm, the rotatability shifted the peak maxima of P H /(H ac f) towards higher H ac at higher frequencies. Because the slightly larger nanoparticles with N (H = 0) of 0.02 s is considered as ferromagnetic nanoparticles in the frequency range of hyperthermia treatments, this shift can be also attributed to the formation of dissipative structures. The behaviour for the larger nanoparticles with 2R M of 24 nm has been already discussed in the main text. Figure S6 is the contour plot of P H /(H ac f) of the spheroidal nanoparticles (2R M = 18 nm) with various aspect ratios. For the nearly spherical nanoparticles with = 1.1, there is the secondary maximum due to the rotatability as discussed in the main text. On the other hand, the shift in the peak maximum of P H /(H ac f) towards higher H ac at higher frequencies can be found for the rotatable elongated spheroidal nanoparticles with of 1.4, as discussed above. For the nanoparticles with an intermediate shape ( = 1.25), N (H = 0) of 8 s is as long as B of 9 s, and it is comparable to the period 1/f of the AC field. In this borderline case, P H /(H ac f) for the rotatable nanoparticles seems to show both the features mentioned above (to be discussed elsewhere.) Apart from such a borderline case, we can say that superparamagnetic nanoparticles have the features common to the typical SPION described in the main text, while ferromagnetic nanoparticles have the features equivalent to the typical ferromagnetic nanoparticles described in the main text.

4 Figure S5 Efficiency of heat dissipation of the elongated spheroidal nanoparticles with various sizes, in non-rotatable case ((a), (c) and (e)) and in rotatable case ((b), (d) and (f)). Dashed lines represent the Néel relaxation time (2 N ) 1 ; dotted lines show the Brownian relaxation time (2 B ) 1, solid lines indicate a typical angular velocity, H (H = H ac, = /4)/2, of the rotation due to magnetic torque. and dot-dash-line lines indicate f p calculated by the equation (5). White lines show the thresholds for biomedical safety.

5 Figure S6 Efficiency of heat dissipation of the spheroidal nanoparticles with the same equatorial diameter and various shapes, in non-rotatable case ((a), (c) and (e)) and in rotatable case ((b), (d) and (f)). Dashed lines represent the Néel relaxation time (2 N ) 1 ; dotted lines show the Brownian relaxation time (2 B ) 1, solid lines indicate a typical angular velocity, H (H = H ac, = /4)/2, of the rotation due to magnetic torque, and dot-dash-line lines indicate f p calculated by the equation (5). White lines show the thresholds for biomedical safety.

6 1. Carrey, J., Mehdaoui, B., & Respaud, M. Simple models for dynamic hysteresis loop calculations of magnetic single-domain nanoparticles: Application to magnetic hyperthermia optimization, J. Appl. Phys. 109, (2011). 2. Yoshida, T., & Enpuku, K. Simulation and Quantitative Clarification of AC Susceptibility of Magnetic Fluid in Nonlinear Brownian Relaxation Region, Jpn. J. Appl. Phys. 48, (2009).