Energy transfer performance of mechanical nanoresonators coupled with electromagnetic fields
© Javaheri et al.; licensee Springer. 2012
Received: 23 July 2012
Accepted: 6 September 2012
Published: 17 October 2012
We study the energy transfer performance in electrically and magnetically coupled mechanical nanoresonators. Using the resonant scattering theory, we show that magnetically coupled resonators can achieve the same energy transfer performance as for their electrically coupled counterparts or even outperform them within the scale of interest. Magnetic and electric coupling are compared in the nanotube radio, a realistic example of a nano-scale mechanical resonator. The energy transfer performance is also discussed for a newly proposed bio-nanoresonator composed of magnetosomes coated with a net of protein fibers.
KeywordsNanoresonators Magnetic nanoparticles Magnetosomes Energy transfer Nanotube radio
Mechanical nanoresonators exhibit resonance behavior involving the mechanical vibrations of the system elements. The natural frequencies of such resonances will, generally, be in the radio frequency range. Nano-scale mechanical resonators coupled with electromagnetic fields have been receiving significant attention recently [1–3]. The ability to interact with electromagnetic fields allow such resonators to be essential parts of nano-scale systems. Imaging, sensing, and targeted actuation in nano-scale are among several emerging technologies that rely on efficient energy and information transfer.
In principle, nanoresonators may couple to electromagnetic fields by the charge distributions (electric coupling) or by the magnetic moment they carry (magnetic coupling). Traditionally, the energy transfer via electric coupling has received more attention since materials are mostly transparent to the magnetic field. Also, magnetic field intensity in electromagnetic radiations is significantly smaller than the electric field. Consequently, magnetic coupling of mechanical resonators with electromagnetic radiations becomes impractical unless the size of the system significantly decreases. A desirable magnetic coupling, however, can be achieved if the coupling occurs within the near-field range . Take the example of a mechanical nanoresonator operating in a biological environment. In this case, magnetic coupling holds important advantages over electric coupling. First, magnetically coupled systems can provide more selective and localized energy transfer that is due to the fact that magnetic fields, unlike electric fields, couple weakly with non-targeted surrounding media, which are often not magnetic [5, 6]. Therefore, magnetic signals suffer from considerably less attenuations while propagating in the surrounding biological media and can drive a targeted resonator inaccessible to electric signals with the same level of energy. In addition, magnetic dipoles are normally more stable than electric dipoles and do not require significant energy from outside to maintain their state.
This work revisits the interactions of radiofrequency electromagnetic fields with mechanical nanoresonators. In particular, we are interested in the quantitative assessment of the energy transfer in such nanoresonators. We use the same methodology presented by Hamam et al.  and focus on low-dissipation conditions that permit resonance. The feasibility of achieving such conditions has been demonstrated in the literature [1, 8]. The outline of this paper is as follows. We first present a general model for mechanical nanoresonators including electric and magnetic coupling mechanisms and describe the dynamics of the model. Then, we compare the resonant energy transfer performance of the resonator for electric and magnetic coupling using resonant scattering theory. Finally, we sketch a roadmap for a new nanoresonator composed of a magnetite nanoparticle embedded in a net of protein fibers.
A resonance can be achieved if.
Here, θ = x/L is the angular displacement, I ≅ m L2 is the system’s second moment of inertia, κ ≅ k L2 is the rotational spring constant of the cantilever, and ψ is the stochastic torque caused by the thermal noise. For the magnetic coupling, , while in the case of the electric coupling.
One can think of U as the energy capacity of the resonator. An important observation is that U scales with Q2.
Note that the energy absorbed by the resonator over the relaxation time matches the resonator energy capacity. In general, the calculation of the force (or torque) exerted on the nanoresonator through electromagnetic coupling is not straightforward. As an alternative approach, one can use scattering theory [7, 11], which allows to work with fluxes instead of forces, to estimate the energy deposited on the resonant system. The two approaches are equivalent since our theoretical model is solely based on dipole-dipole interactions. In the next section, we will use this more convenient method to study the resonant energy transfer.
Resonant scattering analysis
Qa is obtained from the steady state solution.
which confirms that the energy deposited scales as Q2/k.
In our first example, we compare the energy transfer performance of the magnetic and electric coupling in the nanotube radio, a realistic example of a mechanical nanoresonator . We replace the electric dipole of the nanotube tip with a magnetic dipole in the form of spherical magnetite nanoparticle. According to the original study, a nanotube radio built from a cylindrical carbon nanotube of length L ≈ 1 μ m holding a net charge of q = 200 e− absorbs an amount of energy enough to detect radio signals from the electromagnetic radiation. To achieve the same amount of energy deposit, the magnetic moment of the replacement tip should be in the order of μ ≈ qLc = 9.6 × 10−15 Am2, which can be obtained by placing a magnetite nanoparticle of radius R approximately 160 nm.
In order to achieve higher quality factor, the nanoresonator should experience smaller viscous resistance. For instance, one can reduce drag forces in the system by coating the magnetite nanoparticle with hydrophobic proteins or lipids. In this case, the hydrophobic coat acts as a lubricant [19, 20]. In a more sophisticated design, a multi-layer shell of hydrophobic proteins may be used to engulf the nanoresonator and repel water molecules . In order to aggressively reduce the rotational friction, the nanoresonator could be packed in an inorganic shell that completely excludes the system from the cytoplasm. The elastic protein fibers may be replaced by synthetic nanowires or nanotubes with carefully designed rigidity. For example, del Barco et al.  have demonstrated the possibility to have free rotation of magnetic nanoparticle embedded in a solid matrix.
Assuming that high quality factor, Q = 100, can be achieved, one finds that an AC magnetic field of intensity B = 3.5mT, generating an electromagnetic flux of 10 W/m2, deposits a significant amount of energy ΔU = 2,500kBT into the system over the resonance relaxation time. Since this field intensity is well below the coercivity field of the nanoparticle [15, 16], we neglect the energy losses via magnetic reversal. If this energy was entirely manifested as heat, the temperature of the magnetosome would be increased by 0.5°C during the relaxation time τ = 0.1μ s. As shown in Figure 3, Q = 10corresponds to ω0 = 66MHz, in air (η = 10−5Pa.s), while the same quality factor can be achieved at frequencies as low as about 1 MHz if the viscosity can be reduced by a factor of 100 compared to air. Magnetically coupled mechanical nanoresonators with high quality factor show good energy transfer performance while being tunable and may be useful in frequency selective heat production in the biological environment. The important contribution of mechanical motions in magnetic hypothermia has been experimentally shown in ; however, these applications have not yet benefited from the resonant energy transfer since their quality factors are well below one.
In conclusion, we have shown that carefully engineered magnetically coupled nanoresonators can match the energy transfer performance of its electrically coupled counterpart, while providing a more selective and robust interaction in biological environments. We have used a unifying framework of resonant energy transfer for electrically coupled and magnetically coupled mechanical nanoresonators and compared the performance for the two couplings. Our analysis suggests that if the interacting electric dipole of a small electrically coupled resonator is replaced by a magnetic dipole, a comparable amount of energy can still be deposited on the system. We have considered the example of nanotube radio, and we have shown that the strength of electromagnetic coupling remains the same using a magnetite nanoparticle of radius 160 nm instead of the charged tip. We have proposed a new resonator composed of magnetosomes embedded in a net of protein fibers and analyzed its energy transfer performance. We have discussed possible pathways to further improve the quality factor of the resonator. While this article focuses on quantitative aspect of energy transfer, our work also opens up new interesting questions on how to use efficient energy channels to transmit information to a nano-scale device or organism. Characterizing the transmission of information and the channel capacity  will be discussed in future studies.
We would like to thank Professor Mark Dykman for his important suggestions and enlightening comments. We also thank the participants of NSF Workshop on Biologically-enabled Wireless Networks for stimulating discussions. This work is supported in part by the National Science Foundation under grant number NSF CNS-1051240 and the US Department of Energy, Office of Science, Basic Energy Sciences with contract numbers DE-FG02-07ER46352 and DE-FG02-08ER46540 (CMSN), and benefited from allocation of computer time at the NERSC and NU-ASCC computation centers.
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