This paper summarizes and elaborates on some new results on the optimal absorption in small spherical suspensions based on electrophoretic (plasmonic) resonances and lossy surrounding media. The main application here is to study the physical limitations for radio frequency absorption in gold nanoparticle (GNP) suspensions and its potential to achieve GNP targeted hyperthermia in cancer therapy. Numerical parameter studies are included to demonstrate the analysis approach

Bayford, R.

Dalarsson, M.

Ivanenko, Y.

Nordebo, S.

Sjöberg, D.

English

Middlesex University Research Repository

32nd URSI GASS, Montreal, 19–26 August 2017Parameter studies on optimal absorption and electrophoretic resonances in lossy mediaSven Nordebo*(1), Mariana Dalarsson(1), Yevhen Ivanenko(1), Daniel Sjöberg(2) and Richard Bayford(3)(1) Department of Physics and Electrical Engineering, Linnæus University, 351 95 Växjö, Sweden.E-mail: {sven.nordebo,mariana.dalarsson,yevhen.ivanenko}@lnu.se.(2) Department of Electrical and Information Technology, Lund University, Box 118, 221 00 Lund, Sweden. E-mail:daniel.sjoberg@eit.lth.se.(3) The Biophysics and Bioengineering Research Group, Middlesex University, Hendon campus, The Burroughs, London,NW4 4BT, United Kingdom. E-mail: R.Bayford@mdx.ac.uk.AbstractThis paper summarizes and elaborates on some new resultson the optimal absorption in small spherical suspensionsbased on electrophoretic (plasmonic) resonances and lossysurrounding media. The main application here is to studythe physical limitations for radio frequency absorption ingold nanoparticle (GNP) suspensions and its potential toachieve GNP targeted hyperthermia in cancer therapy. Nu-merical parameter studies are included to demonstrate theanalysis approach.1 IntroductionThe study of optimal absorption based on electrophoretic(or plasmonic) resonances in lossy materials provides anumber of interesting scientific questions with a potentialapplication in radiotherapeutic hyperthermia based meth-ods to treat cancer. In particular, gold nanoparticles (GNPs)can be coated with ligands that target specific cancer cellsas well as providing a net electronic charge of the GNPs.One hypothesis is then that the charged GNPs will facilitatean electrophoretic current that can destroy the cancer un-der electromagnetic radiation [1, 4, 5]. However, it is alsoimportant to observe the complexity of the clinical appli-cation with many possible physical and biophysical phe-nomena to consider, as well as the fact that several authorshave questioned whether metal nanoparticles can be heatedin radio frequency at all, see e.g., [1, 3]. We have recentlyperformed a study on the physical limitations for radio fre-quency absorption in a spherical suspension where the maintheoretical result is the derivation of an optimal conjugatematch with respect to the surrounding medium [4]. It isdemonstrated in [4] that for a surrounding medium consist-ing of a weak electrolyte solution, a significant RF-heatingcan be achieved inside a small GNP suspension, providedthat an electrophoretic particle acceleration (Drude) mecha-nism is valid and can be tuned into resonance at the desiredfrequency. In this paper, we summarize briefly the new the-ory, we expand the asymptotic analysis and we demonstratesome new parameter studies related to this application.2 Optimal absorption for the sphereConsider a spherical region of radius r1 consisting of adielectric material with permittivity ε1 and which is sus-pended inside a lossy dielectric background medium hav-ing permittivity ε . A relative heating coefficient is definedas F = Ploc(r1)/Pb(r) where Ploc(r1) is the mean local heat-ing (in W/m3) generated inside the sphere of radius r1, andPb(r) the mean background heating (in W/m3) at some ob-servation radius r > r1. In a close vicinity to a small sphere(both r and r1 are small), an asymptotic analysis, similaras presented in [4], reveals that the optimal excitation thatmaximizes F is an electric dipole field yielding the relativeheating coefficientFa(ε1) = 9|ε|2ℑ{ε}ℑ{ε1}|ε1 +2ε|2. (1)An alternative analysis is given by the general Mie theoryfor lossy media where the absorption cross section Cabs forplane wave incidence is given byCabs = 12πk0r31|ε|2ℜ{√ε}ℑ{ε1}|ε1 +2ε|2, (2)and where k0 is the wave number of vacuum, see [4, 5]. Ithas recently been shown in [4] that (2), and hence also (1)are locally concave functions in the complex parameter ε1and which are maximized by the conjugate matchε1o =−2ε∗. (3)It is noted that the conjugate match ε1o yields a desiredfunction with the metamaterial characteristics ℜε1o < 0 fora “normal” surrounding medium with ℜε > 0. It is alsonoted that the parameters of a Drude model can always betuned to the conjugate match at a single frequency, but justlike the metamaterials with real valued and negative permit-tivity [2], the conjugate match (with small imaginary part)is not realizable as a passive material over a finite band-width, cf., [4].3 Numerical examplesConsider a surrounding medium consisting of a weak elec-trolyte solution with relative permittivityε(ω) = ε∞ +εs− ε∞1− iωτ+ iσωε0, (4)where ε∞, εs and τ are the high frequency permittivity, thestatic permittivity and the dipole relaxation time in the cor-responding Debye model for water, respectively, and σ theconductivity of the saline solution. The effective permittiv-ity of the spherical GNP suspension is modeled asε1(ω) = ε(ω)+ iσ1ωε011− iωτ1, (5)where σ1 = N q2/β and τ1 = m/β , where N is the num-ber of particles per unit volume, q the particle charge, β thefriction constant of the host medium and m the mass of theparticle, cf., [5]. The Drude parameters can furthermore betuned to the conjugate match by choosingτ1 =1ωdℜε(ωd)−ℜε1o(ωd)ℑε1o(ωd)−ℑε(ωd), (6)σ1 = ε0 (ℜε(ωd)−ℜε1o(ωd))1+ω2d τ21τ1, (7)where ωd = 2π fd is the desired resonance frequency.In the numerical examples below, the surrounding salinewater is modeled by using ε∞ = 5.27, εs = 80, τ = 10−11 sand σ = 0.1S/m. For the tuned Drude model we have usedfd = 1GHz. For the realistic Drude models below, the ra-dius of the GNP with ligands is a = 2.5nm, the radius ofthe gold core is aAu = 0.75nm and the mass density of goldis ρAu = 19300kg/m3. The ligands are assumed to have amass density similar to water. The friction constant β ismodelled as β = 6πµfa, where µf = 10−3 Nsm−2 is the dy-namic shear viscosity of the host medium (water at roomtemperature) as in [5]. The number of particles per unitvolume is given by N = φ/(4πa3/3), where φ is the vol-ume fraction of GNPs in the spherical suspension. Here,we have chosen φ = 2 ·10−3 as in [5]. The net charge of theGNP is modeled as q = (3aAu + 0.5a2Au)e0 + nLe0, wherethe charge of the gold core is modeled as in [5], nL is theassumed net electron count in the ligand shell and e0 de-notes the electron charge e0 = 1.6 ·10−19 C.In Figure 1 is shown the absorption cross section Cabs givenby (2), plotted as a function of frequency and where the ra-dius of the spherical suspension is r1 = 1 µm. In Figure2 is shown the relative heating coefficient Fa(ε1) given by(1) representing the mean power dissipation inside a smallsphere relative the mean power dissipation in a close vicin-ity of the sphere. In both figures, ε1 is either the optimalconjugate match, the tuned Drude model or the realisticDrude models with qL = nLe0 ∈ {10,102,103}e0.7 8 9 10 11−8−6−4−2Frequency log f (Hz)logCabs/πr2 1Absorption cross section Cabs/πr21 Optimal bound: ε1oTuned Drude: ε1DDrude: qL = 103e0Drude: qL = 102e0Drude: qL = 10e0fd = 1GHzFigure 1. Absorption cross section Cabs for a small spheri-cal GNP suspension of radius r1 = 1 µm. The surroundingmedium is a saline solution with σ = 0.1S/m.7 8 9 10 110123Frequency log f (Hz)logFaRelative heating FaOptimal bound: ε1oTuned Drude: ε1DDrude: qL = 103e0Drude: qL = 102e0Drude: qL = 10e0fd = 1GHzFigure 2. Relative heating coefficient Fa in a close vicin-ity of a small spherical GNP suspension. The surroundingmedium is a saline solution with σ = 0.1S/m.4 AcknowledgementsThis work was supported by the Swedish Foundation forStrategic Research (SSF).References[1] C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J.Collins, and C. J. Ackerson. Radiofrequency heatingpathways for gold nanoparticles. Nanoscale, 6, 8459–8472, 2014.[2] M. Gustafsson and D. Sjöberg. Sum rules and physi-cal bounds on passive metamaterials. New Journal ofPhysics, 12, 043046, 2010.[3] G. W. Hanson, R. C. Monreal, and S. P. Apell.Electromagnetic absorption mechanisms inmetal nanospheres: bulk and surface effects inradiofrequency-terahertz heating of nanoparticles. J.Appl. Phys., 109, 2011. 124306.[4] S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg,and R. Bayford. On the physical limitations for ra-dio frequency absorption in gold nanoparticle suspen-sions. J. Phys. D: Appl. Phys, 2017. In Press.arXiv:1608.06512 [physics.class-ph].[5] E. Sassaroli, K. C. P. Li, and B. E. O’Neil. Radio fre-quency absorption in gold nanoparticle suspensions: aphenomenological study. J. Phys. D: Appl. Phys., 45,1–15, 2012. 075303.

Parameter studies on optimal absorption and electrophoretic resonances in lossy media

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Parameter studies on optimal absorption and electrophoretic resonances in lossy media

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