Abstract
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 nanoscale mechanical resonator. The energy transfer performance is also discussed for a newly proposed bionanoresonator composed of magnetosomes coated with a net of protein fibers.
Keywords:
Nanoresonators; Magnetic nanoparticles; Magnetosomes; Energy transfer; Nanotube radioBackground
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. Nanoscale mechanical resonators coupled with electromagnetic fields have been receiving significant attention recently [13]. The ability to interact with electromagnetic fields allow such resonators to be essential parts of nanoscale systems. Imaging, sensing, and targeted actuation in nanoscale 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 nearfield range [4]. 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 nontargeted 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. [7] and focus on lowdissipation 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.
Methods
In general, the mechanical structure of a nanoresonator consists of an elastic cantilever
beam equipped with a specialized tip, which is responsible for electromagnetic interaction,
vibrating in a lowviscosity fluid such as lowpressure air. The viscoelastic model
of the nanoresonator includes the coefficient of mechanical elasticity, k, and the dissipation coefficient, D. For a cylindrical beam with a spherical tip, k ≅ EI_{c}/L^{3}, where E, I_{c}, and L are the Young’s modulus, second moment of crosssection, and the length of the beam,
respectively. Moreover, as shown in [9], the combination of intrinsic (e.g., plastic deformation and surface effects) and
extrinsic (e.g., viscous forces of the surrounding fluid) dissipation mechanisms determines
the value of D. Because the size of the nanoresonator is much smaller than the wavelength of the
external field, the energy transfer is in the form of interactions between the incoming
field and dipole moment of the nanoresonator’s tip. As shown in Figure 1, we consider two nanoresonators that have identical mechanical structures, yet interact
with electromagnetic fields via different coupling mechanisms: electric coupling (
Figure 1. An overview of nanoresonators with electric (right) and magnetic (left) coupling. The viscoelastic properties of the resonators are identical.
The dynamics of the system can be expressed by a Langevin equation for the resonator tip. After linearization for small deflections, we have
where x is the displacement at the tip of the beam; m, the effective mass of the system; and D, the dissipation coefficient. F = qE for electric coupling, while
where x_{m} and φ are the maximum deflection of the tip and the phase shift, respectively, given by
A resonance can be achieved if
The dynamics of the system can also be expressed by the following Langevin equation for rotational oscillation [10]:
Here, θ = x/L is the angular displacement, I ≅ mL^{2} is the system’s second moment of inertia, κ ≅ kL^{2} is the rotational spring constant of the cantilever, and ψ is the stochastic torque caused by the thermal noise. For the magnetic coupling,
Discussion
Energy analysis
We now consider the total energy of the oscillator
When ω=ω_{0}, this quantity is timeindependent and is given by
One can think of U as the energy capacity of the resonator. An important observation is that U scales with Q^{2}.
Next, we consider the energy absorbed by the nanoresonator during the relaxation time τ=Q/ω_{r}. This quantity can be calculated by averaging the instantaneous power absorbed by the nanoresonator, P, over τ. For our system, P can be written as inner product of incident force and velocity of the resonator
After some algebra, the average absorbed power,
At the resonance frequency,
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 dipoledipole interactions. In the next section, we will use this more convenient method to study the resonant energy transfer.
Resonant scattering analysis
The coupling between external fields and the nanoresonator consists of an absorption and a scattering process. According to the scattering theory, the power absorbed by the resonant system equals to P_{a} = Φσ_{a}, where Φ is the incident electromagnetic power flux, and σ_{a} is the absorption crosssection given by [7,11]
Here, c is the speed of light; Γ_{a}, the absorption width; and Γ_{s}, the scattering width. The widths are the ratio of the power loss to the characteristic energy of the corresponding process. For process i, Γ_{i} = 1/τ_{i} = ω/Q_{i}, where τ_{i} and Q_{i} represent the relaxation time and the quality factor, respectively. The total energy absorbed by the resonant system during the resonant process is given by the following:
where Γ = Γ_{a} + Γ_{s} is the total width of the system. For nanoscale systems of interest, Γ ≈ Γ_{a} because Γ_{s} ≪ Γ_{a}. The maximal energy transfer occurs at the resonant frequency and can be written as
Q_{a} is obtained from the steady state solution.
By definition, the width of the scattering process is equal to the inverse of the decay time of radiating dipole given by the following:
where U is the energy of the resonator, and P_{r} = dU/dt is the radiative power of the resonator’s dipole. For the electric dipole model (
where p_{0} = qx_{m} is the maximal amplitude for the electric moment of the resonator. Thus, one obtains the following scattering width for the system:
Replacing (17) in (13) results in
which confirms that the energy deposited scales as Q^{2}/k.
Using classical electrodynamics [12], one can show that the the radiative power of an oscillating magnetic dipole of moment
μ_{eff} is given by replacing p_{0} by μ_{eff}/c in 15. In the case of the spherical MNP shown in Figure 1, if θ_{m } is the maximum angular deflection, then the oscillating part of the magnetic dipole
of moment is
Therefore, the corresponding radiation width and deposited energy are
Note that Q_{a} and k in Equations (18) and (22) only depend on the viscoelastic structure of the resonator and are independent from the coupling type (magnetic or electric). Given a similar viscoelastic structure, the energy absorption value for electric and magnetic coupling will be comparable if μ/Lc ≈ q. For nanoscale systems of interest, the condition Lc < 1,000 m^{2}/s normally holds. By comparing 22 to 10, it is possible to derive an expression for the average magnetic force experienced by the nanoresonator over the resonance relaxation time, which is given by
Applications
Having discussed the mechanical dynamics of the nanoresonator as well as the theoretical formulation for energy transfer performance of different coupling mechanism, we apply our analysis to a possible nanoresonator sketched in Figure 2, and we also discuss the feasibility of a bionanoresonator composed of protein coated Fe_{3}O_{4} nanoparticles.
Figure 2. Magnetosome arrangement in magnetotactic bacteria. The magnified part shows how elastic protein fibers embed magnetite (Fe_{3}O_{4}) crystals in the cytoskeleton. Interaction of the magnetic dipole of the crystal with external fields within its viscoelastic environment can be analyzed by our presented theoretical model as a torsional nanoresonator shown on the right hand side. Magnetic torque rotates the MNP around its center of mass. The rotational spring constant is given by κ = kR, where k is the aggregate rigidity of the connecting protein fibers, and R is the radius of the MNP. Since the Reynold number of the MNP is very small, the drag forces are given by Stokes’ law. Therefore, the rotational damping coefficient is C = 6π ηR^{3}, where η is the viscosity of the surrounding fluid.
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 [1]. 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 μm 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} Am^{2}, which can be obtained by placing a magnetite nanoparticle of radius R approximately 160 nm.
Another interesting application is the possibility of transmitting energy to magnetic nanoparticles in the biological setting. Biogenic magnetite nanoparticles called magnetosomes, first discovered in magnetotactic bacteria [13], are also found in the brain of many animals and are believed to participate in determining the orientation in several species such as migratory birds [14]. Interestingly, magnetosomes consist of magnetite particles of radius 50 to 100 nm and are embedded in the cytoskeleton bound to a viscoelastic system formed by a net of protein fibers. Because magnetic nanoparticles of such size are single domain with high coercivity [15,16], the magnetosome can be represented as a torsional nanoresonator with magnetic coupling (see Figure 2). According to Winklhofer and Kirschvink [17], the rigidity of the cytoskeleton can be estimated by κ = 100k_{B}T/ Rad per connecting filament. Thus, for magnetite particles with a density of ρ = 5,200 Kg/m^{3}, a radius of R = 100nm, and the number of the connecting filaments ranging from 1 to 1,000, the natural frequency of the oscillator will fall between 2 and 66 MHz. A resonance is in principle not possible if we adopt the standard viscosity of the cytoplasm [18]. However, in a carefully engineered synthetic system, one could lower the drag forces in order to achieve resonance and higher quality factor up to Q = s100. Figure 3 shows possible quality factor values for a nanoparticle of radius 100 nm, assuming that the viscosity and elastic constants could be controlled.
Figure 3. Quality factor of the resonance for reasonable range of values (color online). For the environment viscosity and the rotational spring constant of elastic environment (in terms of k_{B}T). We assume the design includes a magnetite nanoparticle of radius 100 nm. Note that resonance is possible in the region above Q = 0.5line. It is shown how resonance of a given quality can be achieved in lower frequency by reducing the viscosity experienced by the resonator.
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 multilayer shell of hydrophobic proteins may be used to engulf the nanoresonator and repel water molecules [21]. 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. [22] 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/m^{2}, deposits a significant amount of energy ΔU = 2,500k_{B}T 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 = 0.1μs. As shown in Figure 3, Q = 10corresponds to ω_{0} = 66MHz, in air (η = 10^{−5}Pa.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 [23]; however, these applications have not yet benefited from the resonant energy transfer since their quality factors are well below one.
Conclusion
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 nanoscale device or organism. Characterizing the transmission of information and the channel capacity [24] will be discussed in future studies.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
HJ designed the theoretical model, carried out the analysis, and wrote the manuscript. BB participated in developing the model and helped in conceptualizing the ideas. GN defined the research theme and participated in writing the manuscript. All authors read and approved the final manuscript.
Acknowledgements
We would like to thank Professor Mark Dykman for his important suggestions and enlightening comments. We also thank the participants of NSF Workshop on Biologicallyenabled Wireless Networks for stimulating discussions. This work is supported in part by the National Science Foundation under grant number NSF CNS1051240 and the US Department of Energy, Office of Science, Basic Energy Sciences with contract numbers DEFG0207ER46352 and DEFG0208ER46540 (CMSN), and benefited from allocation of computer time at the NERSC and NUASCC computation centers.
References

Jensen K, Weldon J, Garcia H, Zettl A: Nanotube radio.
Nano lett 2007, 7(11):35083511. PubMed Abstract  Publisher Full Text

Degen CL, Poggio M, Mamin HJ, Rettner CT, Rugar D: Nanoscale magnetic resonance imaging.
P Natl Acad Sci Usa 2009, 106(5):13131317. Publisher Full Text

Dykman MI, Khasin M, Portman J, Shaw SW: Spectrum of an oscillator with jumping frequency and the interference of partial susceptibilities.
Phys Rev Lett 2010, 105(23):230601. PubMed Abstract  Publisher Full Text

Karalis A, Joannopoulos J, Soljacic M: Efficient wireless nonradiative midrange energy transfer.
Ann Phys 2008, 323:3448. Publisher Full Text

Kirschvink JL, KobayashiKirschvink A, DiazRicci JC, Kirschvink SJ: Magnetite in human tissues: a mechanism for the biological effects of weak ELF magnetic fields.
Bioelectromagnetics 1992,, Suppl 1:10113. PubMed Abstract

Poon A, O’Driscoll S, Meng T: Optimal frequency for wireless power transmission into dispersive tissue.

Hamam R, Karalis A, Joannopoulos J, Soljačić M: Coupledmode theory for general freespace resonant scattering of waves.

Stipe B, Mamin H, Stowe T, Kenny T, Rugar D: Magnetic dissipation and fluctuations in individual nanomagnets measured by ultrasensitive cantilever magnetometry.
Phys Rev Lett 2001, 86(13):28742877. PubMed Abstract  Publisher Full Text

Sazonova V: A tunable carbon nanotube resonator.
Dissertation
Cornell University, Physics Department; 2006

Javaheri H, Barbiellini B, Noubir G: Efficient magnetic torque transduction in biological environments using tunable nanomechanical resonators.
Conf Proc IEEE Eng Med Biol Soc 2011, 2011:18631866. PubMed Abstract  Publisher Full Text

Blatt JM, Weisskopf VF: Theoretical Nuclear Physics. New York: Wiley; 1952.

Jackson J: Classical Electrodynamics. New York: Wiley; 1967.

Blakemore R: Magnetotactic bacteria.
Science 1975, 190(4212):377379. PubMed Abstract  Publisher Full Text

Kirschvink JL, Winklhofer M, Walker MM: Biophysics of magnetic orientation: strengthening the interface between theory and experimental design.
J R Soc Interface 2010, 7:S179S191. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Muxworthy AR, Williams W: Critical superparamagnetic/singledomain grain sizes in interacting magnetite particles: implications for magnetosome crystals.
J R Soc Interface 2009, 6(41):12071212. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Hergt R, Dutz S, Müller R, Zeisberger M: Magnetic particle hyperthermia: nanoparticle magnetism and materials development for cancer therapy.
J Phys: Condens Matter 2006, 18:S2919. Publisher Full Text

Winklhofer M, Kirschvink JL: A quantitative assessment of torquetransducer models for magnetoreception.
J R Soc Interface 2010, 7:S273S289. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Adair RK: Vibrational resonances in biological systems at microwave frequencies.
Biophys J 2002, 82(3):11471152. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Phizicky EM, Fields S: Proteinprotein interactions: methods for detection and analysis.
Microbiol Rev 1995, 59:94123. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Samanta B, Yan H, Fischer NO, Shi J, Jerry DJ, Rotello VM: Proteinpassivated Fe3O4 nanoparticles: low toxicity and rapid heating for thermal therapy.
J Mater Chem 2008, 18(11):1204. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Hu X, Cebe P, Weiss AS, Omenetto F, Kaplan DL: Proteinbased composite materials.
Mater Today 2012, 15(5):208215. Publisher Full Text

del Barco ED, Asenjo J, Zhang X, Pieczynski R, Julia A, Tejada J, Ziolo R, Fiorani D, Testa A: Free rotation of magnetic nanoparticles in a solid matrix.
Chem mater 2001, 13(5):14871490. Publisher Full Text

Alphandéry E, Faure S, Raison L, Duguet E, Howse PA, Bazylinski DA: Heat production by bacterial magnetosomes exposed to an oscillating magnetic field.
J Phys Chem C 2011, 115:1822. Publisher Full Text

Sidles JA: Spin microscopy’s heritage, achievements, and prospects.
P Natl Acad Sci Usa 2009, 106(8):24772478. Publisher Full Text