Propagation of Collective Surface Plasmons in Linear Periodic Ionic

Article pubs.acs.org/JPCC

Propagation of Collective Surface Plasmons in Linear Periodic Ionic Structures: Plasmon Polariton Mechanism of Saltatory Conduction in Axons Witold A. Jacak* Institute of Physics, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland ABSTRACT: The propagation of collective surface plasmons, called plasmon polaritons, in infinite equidistant electrolyte sphere chains is analyzed. The quenching of radiative losses of these ionic excitations in the chain is demonstrated by the inclusion of the retarded near-, medium-, and far-field components of the dipole interaction between the spheres in the chain in analogy to metallic nanochains. It is shown that the damping of the plasmon polariton self-modes in the wide region of the domain is reduced to only residual Ohmic losses that are much lower than the radiation losses for the separate electrolyte sphere. The self-frequencies and group velocities of plasmon polaritons for longitudinal and transversal (with respect to the chain orientation) polarization are determined and assessed for various ion and electrolyte parameters. It is proved that there exist weakly damped self-modes of plasmon polaritons in the chain for which the propagation range is limited only by relatively small Ohmic losses. Completely undamped collective waves are also described in the presence of persistent external excitations of some fragments of the chain. The possibility of applying the plasmon polariton model to describe the so-called saltatory conduction in periodically myelinated nerve axons is discussed. The accommodation of the plasmon polariton parameters to requirements of explanation of the saltatory conduction in neurons of the central and peripheral nervous system is presented.

■

INTRODUCTION The periodic linear structures of metallic nanoparticles serve as plasmon waveguides with low damping.1,2 The wavelengths of propagating in such structure plasmon polaritons are considerably shorter in comparison to the light with the same frequency. This allows for avoiding diffraction limits for light circuits when one transforms the light signal into plasmon polaritons.3,4 This is regarded as prospective for forthcoming construction of plasmon optoelectronic nanodevices not available when using only ordinary light waveguides because of diffraction constraints for the latter. The relatively long-range practically undamped propagation of plasmon polaritons in various metallic nanostructures has been experimentally demonstrated.5 In a previous paper,6 we developed a description of plasmon oscillations in ionic spheres in analogy to plasmons in metallic nanospheres. We demonstrated the existence of the surface and volume modes of plasmons in electrolyte spheres and analyzed their attenuation, including scattering and radiation losses, where the latter is expressed in terms of the Lorentz friction of oscillating charge fluctuations. The majority of the details of ionic systems remains analogous to those of electrons in metals, albeit with an overall scale shifted by several orders of magnitude (depending on ion parameters and concentration) toward lower frequencies, larger system sizes, and larger wavelengths of resonance. In the present paper, we will utilize the plasmonic properties of ionic electrolyte spheres to discuss the kinetics of surface plasmon wave-type excitations in a chain of electrolyte spheres, © 2015 American Chemical Society

in analogy to plasmon polariton propagation in metallic nanochains. Issues that we address in the present paper include the possibility that ionic carriers may exhibit plasmon polariton properties similar to those of electrons and the verification of the efficiency of electrolyte plasmon waveguides. The electrolyte spheres of the chain are assumed to be closed by sphere-shaped membranes similar to those frequently found in biological structures. The use of ions in cell signaling, membrane transfer, and nerve cell conductivity are examples of such functional ionic structures, and the question of the usefulness of soft plasmonics in these biological ionic systems arises. For the theoretical model, we will consider an infinite chain of equidistant spheres of radius a, though a generalization to oblate or prolate spheroids or even to other chain-element shapes is natural. We will describe the ionic system in each electrolyte sphere using the model developed in the previous paper,6 i.e., the simplified effective jellium approach for electrolytes. Bearing in mind that metallic nanochains create very effective lownoise waveguides for electromagnetic signals in the form of the wave-like plasmon polaritons, we will model the analogous phenomenon in chains of ionic spheres (Figure 1). In the next paragraph of the present paper, we will derive the basic equations for the dynamics of collective surface plasmons in Received: March 12, 2015 Revised: April 6, 2015 Published: April 7, 2015 10015

DOI: 10.1021/acs.jpcc.5b02418 J. Phys. Chem. C 2015, 119, 10015−10030

Article

The Journal of Physical Chemistry C

attenuation, which enables long-range, nearly undamped propagation. The second important property is the reduction of the signal group velocity (in metallic chains) by at least 1 order of magnitude compared with the velocity of light, thus allowing for the subdiffraction arrangement of metallic plasmon polariton circuits. The collective plasmon excitations in the chain, which have a shorter wavelength compared with photons of the same energy, can efficiently transport information signals over large distances, with transfer parameters that can be controlled over a relatively wide range by modifying the chain size and geometry. If these properties could be reproduced in electrolyte periodic structures (with the scale of the corresponding quantities appropriately shifted), they would offer important insight into signal transfer in biological systems. The periodicity of the chain makes the system similar to a 1D crystal. The interaction between the chain elements when their surface dipole plasmons are excited can be regarded as a dipoletype coupling. For chains of ionic spheres, one can adopt the results of the corresponding analysis for metallic chains, which support a dipole model of interactions between spheres when the distance between neighboring spheres is larger than the sphere radius, i.e., when d > 3a (otherwise, multipole channels may contribute to the intersphere interaction).7−9 Note also that the model of interacting dipoles10,11 was originally developed for the description of stellar matter12,13 but was then adopted for metal particle systems.14,15 The dipole interaction resolves itself into the electric and magnetic fields created by an oscillating dipole D(r,t) at any distant point. If this point is represented by the vector r0 (with one end fixed at the end of r, where the dipole is placed), then the electric field produced by the dipole D(r,t) has the following form, including the relativistic retardation16,17

Figure 1. Schematic representation of a chain of ionic spheres with propagating plasmon polaritons (upper); creation of a dipole D in a single spheroid and the wave-type dipole excitation in a chain of spheroids (the longitudinal mode is depicted in the prolate geometry) traversing the chain with group velocity v (lower).

a chain of ionic spheres, including the self-oscillations of the surface plasmons in each chain component, the Joule heat damping of the plasmons due to irreversible energy dissipation (Ohmic losses) induced by the microscopic scattering of ions, the radiation losses described by the Lorentz friction hampering the movement of charges, and the mutual energy supplementation in the chain via the radiation of all elements of the chain. The energy balance will be analyzed to detect low-noise windows for ionic plasmon polariton propagation. The solutions of the dynamic equation in the homogeneous and inhomogeneous cases are next presented. In the final two paragraphs, several commonalities with the electric signaling in nerve axons are preliminarily discussed in view of the possible application of the developed description of ionic plasmon polaritons in ionic periodic systems. An explanation of the so-called saltatory conduction in myelinated nerve axons is suggested by the propagation of plasmon polaritons in periodic linear ionic structures.

E(r, r0 , t ) =

+

■

1 ⎛ ∂2 1 1⎞ ∂ 1 ⎜− 2 2 − − 3 ⎟D(r, t − r0/v) 2 v∂t r0 ε1 ⎝ v ∂t r0 r0 ⎠

1 ⎛ ∂2 1 3⎞ ∂ 3 ⎜ 2 2 + + 3 ⎟n 0(n 0· D(r, t − r0/v)) 2 v∂t r0 ε1 ⎝ v ∂t r0 r0 ⎠ (1)

DYNAMIC EQUATION FOR COLLECTIVE SURFACE PLASMONS IN A CHAIN OF SPHERICAL ELECTROLYTE SYSTEMS Let us consider an infinite chain of equidistant spherical electrolyte systems of radius a, separated by a distance d between the sphere centers aligned along the z axis. We will attempt to describe the collective surface dipole plasmons in such systems in analogy to the propagation of plasmon polaritons in metallic nanochains.2,7 Because of the similarity of ionic plasmon excitations in finite electrolyte systems to plasmons in metallic nanospheres, one can expect a corresponding similarity in the collective plasmon behavior of complex ionic systems and of metallic ordered nanostructures. Although plasmon oscillations have a much lower energy in finite electrolyte systems compared with plasmons in metals, their radiation properties, with their typical size dependence, may allow for the effective transfer of plasmon energy along the electrolyte sphere chain, analogous to that in metallic nanochains. Such electrolyte sphere chains would serve as waveguides for plasmon signals along ionic structures, with a potential relation to certain biological ionic information channels. The most important property of metallic plasmonic waveguides is their perfect transmittance, i.e., the absence of radiation losses and the reduction of signal damping to only Ohmic

(c/ε11/2),

with n0 = (r0/r0) and v = where c is the light velocity in vacuum and ε1 is the dielectric susceptibility of the medium surrounding the chain. The terms with denominators of r03, r02, and r0 usually are called the near-field, medium-field, and far-field components of the interaction, respectively. The above formula allows for the description of the mutual interaction of the plasmon dipoles with each sphere in the chain. The spheres in the chain are numbered by integers l, and the equation for the surface plasmon oscillation of the lth sphere can be written as follows ⎡ ∂2 ⎤ 2 ∂ + ω12 ⎥Dα (ld , t ) ⎢ 2 + τ0 ∂t ⎣ ∂t ⎦ m =∞

= ε1ω12a3

∑ m =−∞ , m ≠ l

+

ε1ω12a3ELα(ld ,

⎛ | l − m| d ⎞ ⎟ Eα⎜md , t − ⎝ ⎠ v

t ) + ε1ω12a3Eα(ld , t )

(2)

The first term on the right-hand side of eq 2 describes the dipole coupling between spheres, and the other two terms correspond to the plasmon attenuation due to Lorentz friction (as described in the previous paper6 with regard to a single electrolyte sphere) and the forcing field arising from an external electric field, 10016


Article

The Journal of Physical Chemistry C respectively; ω1 = ωp/(3ε1)1/2 is the frequency of the dipole surface plasmons6 (note that this frequency derived upon random phase approximation18 agrees with the classical Mie ωp frequency19 and ωMie = 2ε + 1 for ε1 = 1, whereas for ε1 > 1

one thus obtains the Fourier representation (a discrete Fourier transform for the sphere positions and a continuous Fourier transform for time) of eq 2 in the following form ⎛ 2 ⎞ 2 ⎜ −ω − i ω + ω12⎟Dα (k , ω) τ0 ⎝ ⎠

1

even better than classical formula approximates the more accurate frequency found by the Kohn−Sham equation type numerical calculus.18) Ohmic losses are included via the term20 1 v Cv = + τ0 2λB 2a

= ω12

(3)

Fz(k , ω) = 4

+ ωd / v

=

ω)

⎛

m=1

⎝

m

⎞⎤ cos(mkd) cos(mkd) cos(mωd /v) − (ωd /v)2 sin(mωd /v)⎟⎥ 2 ⎠⎥⎦ m m

■

CALCULATION OF THE RADIATIVE DAMPING OF PLASMON POLARITONS IN AN IONIC ELECTROLYTE CHAIN The problem of propagation of the plasmon polaritons along the metallic nanochans was formulated and discussed originally by Markel21 and then developed among others by Citrin.22 The appropriate mathematical apparatus for the metallic nanosphere chain was applied by Citrin23 in order to assess the radiative losses of plasmon polariton in agreement with former observations related to radiation losses in one-dimensional and twodimensional crystals.24 Similarly to matallic nanochains, in the case of the electrolyte sphere chain, one can analytically calculate the sums in the functions ImFz(k,ω) and ImFx(y)(k,ω) (in eq 8), which correspond to the radiative damping of the longitudinal and transverse plasmon polariton modes. Both functions precisely vanish when 0 < kd ± ωd/v < 0. To show it one can utilize the following formulas25

(5)

⎧ ∞ ⎪∑ ⎪m=1 ⎪ ∞ ⎪ ⎪∑ ⎪m=1 ⎨ ⎪ ∞ ⎪∑ ⎪m=1 ⎪ ∞ ⎪ ⎪∑ ⎩m=1

⎛ md ⎞ −i(kld − ωt ) ⎟e Dα ⎜ ±md + ld ; t − ⎝ v ⎠

md ei(∓kmd + ω v )Dα (k ,

∞

) ∑ ⎜⎜ cos(mkd cos(mωd /v) 3

(8)

In the Fourier picture of eq 2 (the discrete Fourier transform (DFT) with respect to the positions and the continuous Fourier transform (CFT) with respect to time), this solution takes a form similar to that of the solution for the case of phonons in 1D crystals. Let us note that the DFT is defined for a finite set of numbers; therefore, we consider a chain with 2N + 1 spheres, i.e., a chain of finite length L = 2Nd. Then, for any discrete characteristic f(l), l = −N, ..., 0, ..., N of the chain, such as a selected polarization of the dipole distribution, one must consider the DFT picture f(k) = ∑Nl=−N f(l)eikld where k = (2π/2Nd)n and n = 0, ..., 2N. This means that kd ∈ [0,2π) because of the periodicity of the equidistant chain. The Born−Karman boundary condition is imposed on the entire system, resulting in the form of k given above. For a chain of infinite length, one can take the limit N → ∞, which causes the variable k to become quasicontinuous, although kd ∈ [0,2π) still holds. Let us multiply both sides of eq 2 by (ei(kld−ωt))/(2π) and then perform a summation with respect to the positions of the spheres and an integration over t. Taking into account that

l =−N

m

⎞ cos(mkd) cos(mkd) sin(mωd /v) − (ωd /v)2 cos(mωd /v)⎟⎟ 2 m m ⎠ ∞ ⎛ ⎡ cos( ) mkd 2 3 sin(mωd /v) − i⎢ − (ωd /v) + 2 ∑ ⎜⎜ ⎢⎣ 3 m3 m=1 ⎝

Because of the periodicity of the chain (analogous to a 1D crystal), one can assume a wave-type collective solution of the dynamical eq 2

∫∞ ∑

⎝

+ ωd / v

(4)

N

m=1

Fx(y)(k , ω) = − 2

1 ⎛ 1 d ∂ + ⎜ ε1d3 ⎝ |m − l|3 v|m − l|2 ∂t d2 ∂2 ⎞ + 2 ⎟ × Dx(y)(md , t − |m − l|d /v) v |l − d| ∂t 2 ⎠

∞

⎛

⎡ ⎞ cos(mkd) ⎟ + 2i⎢ 1 (ωd /v)3 ω m d v sin( / ) ⎟ ⎢⎣ 3 m2 ⎠ ∞ ⎛ ⎞⎤ cos(mkd) cos(mkd) ⎟⎥ , ω ω ω m d v d v m d v + 2 ∑ ⎜⎜ − sin( / ) / cos( / ) ⎠⎥⎦ m3 m2 m=1 ⎝

Ex(y)(md , t ) = −

1 2π

∞

) ∑ ⎜⎜ cos(mkd cos(mωd /v) 3

+ ωd / v

2 ⎛ 1 d ∂⎞ ⎜ ⎟ Ez(md , t ) = + 3⎝ 3 2 ε1d |m − l| v|m − l| ∂t ⎠ × Dz (md , t − |m − l|d /v),

2π d

(7)

with

where λB is the mean free path of the carriers in the bulk electrolyte; v is the mean velocity of the carriers at a temperature T; v = ((3kT)/m)1/2; m is the mass of the ion; k is the Boltzmann constant; C is a constant of order unity; and a is the radius of a sphere. The first term in the expression for (1/τ0) approximates ion scattering losses such as those occurring in the bulk electrolyte, whereas the second term describes the losses due to the scattering of ions on the boundary of a sphere of radius a. α = z is the longitudinal polarization, and α = x(y) is the transversal polarization (the chain orientation is assumed to be along the z direction). According to eq 1, we can write the following quantities that appear in eq 2

Dα (ldt ) = Dα (k , t )e−ikld , 0 ≤ k ≤

a3 Fα(k , ω)Dα (k , ω) + ε1a3ω12E0α(k , ω) d3

(6) 10017

sin(mz) π−z , for 0 < z < 2π , = m 2 ⎞ cos(mz) 1 ⎛ 1 = ln⎜ ⎟, m 2 ⎝ 2 − 2 cos(z) ⎠ cos(mz) 1 π2 π = − z + z 2 , for 0 < z < 2π , 2 6 2 4 m sin(mz) 1 3 π2 π z − z2 + z , for 0 < z < 2π = 3 6 4 12 m

(9)


Article

The Journal of Physical Chemistry C Using the above formulas, one can show that if 0 < kd ± ωd/ v < 2π, then

A similar analysis can be performed for the transversal polarization case, i.e., for ImFx(y)(k,ω). This function is exactly zero only in the region corresponding to arguments of 0 < kd − ωd/v < 2π and 0 < kd + ωd/v < 2π, where one can write

⎡ sin(m(kd + ωd /v)) − sin(m(kd − ωd /v)) ⎢ m3 m=1 ⎣ ∞

ImFz(k , ω) = 2

∑⎢

∞ ⎡ sin(m(kd + ωd /v)) − sin(m(kd − ωd /v)) ImFx(y)(k , ω) = − ∑ ⎢ ⎢ m3 m=1 ⎣

cos(m(kd + ωd /v)) + cos(m(kd − ωd /v)) ⎤ 2 3 − (ωd /v) ⎥ + (ωd /v) ⎦ 3 m2 ⎡ π2 ⎤ 1 π (kd + ωd /v)3 ⎥ = 2⎢ (kd + ωd /v) − (kd + ωd /v)2 + 4 12 ⎣6 ⎦

− (ωd /v)

⎡ π2 ⎤ 1 π (kd − ωd /v)3 ⎥ − 2⎢ (kd − ωd /v) − (kd − ωd /v)2 + 4 12 ⎣6 ⎦

cos(m(kd + ωd /v)) + cos(m(kd − ωd /v)) m2

− (ωd /v)2

sin(m(kd + ωd /v)) − sin(m(kd − ωd /v)) ⎤ 2 3 ⎥ + (ωd /v) ⎦ 3 m

⎡ π2 ⎤ 1 π (kd + ωd /v)3 ⎥ = − ⎢ (kd + ωd /v) − (kd + ωd /v)2 + 4 12 ⎣6 ⎦

⎡ π2 ⎤ 1 π − 2(ωd /v)⎢ − (kd + ωd /v) + (kd + ωd /v)2 ⎥ 2 4 ⎣6 ⎦

⎡ π2 ⎤ 1 π (kd − ωd /v)3 ⎥ + ⎢ (kd − ωd /v) − (kd − ωd /v)2 + 4 12 ⎣6 ⎦

⎡ π2 ⎤ 2 1 π − 2(ωd /v)⎢ − (kd − ωd /v) + (kd − ωd /v)2 ⎥ + (ωd /v)3 2 4 3 ⎣6 ⎦ ≡0

⎡ π2 ⎤ 1 π + (ωd /v)⎢ − (kd + ωd /v) + (kd + ωd /v)2 ⎥ 2 4 ⎣6 ⎦

(10)

However, if kd − ωd/v < 0 or kd + ωd/v > 2π for certain values of the wave vector k, then a more general formula must be used (by applying the Heaviside step function, one can extend the formulas in eq 9 to the second period of their left-hand sides). This extended form of ImFz(k,ω) is as follows (here, we use the dimensionless variables x = kd and y = d/a and n = x + ωya/v, u = x − ωya/v)

⎡ π2 ⎤1 1 π + (ωd /v)⎢ − (kd − ωd /v) + (kd − ωd /v)2 ⎥ (ωd /v)2 2 4 ⎣6 ⎦2 1 2 [π − kd − ωd /v] − [π − kd + ωd /v] + (ωd /v)3 ≡ 0 2 3

(12)

Nevertheless, outside the region 0 < kd ± ωd/v < 2π, the value of ImFx(y) is not zero, as demonstrated in Figure 4, and can be accounted for by the formula (x = kd, y = d/a, n = x + ωay/v, u = x − ωay/v)

⎡ π2 1 3⎤ π ImFz(k , ω) = Θ(2π − n)2⎢ n − n2 + n⎥ 4 12 ⎦ ⎣6 ⎤ ⎡ π2 1 π (n − 2π )3 ⎥ Θ(−2π + n)2⎢ (n − 2π ) − (n − 2π )2 + 4 12 ⎦ ⎣6

⎡ π2 1 3⎤ π ImFx(y)(k , ω) = −Θ(2π − n)⎢ n − n2 + n⎥ 4 12 ⎦ ⎣6

⎡ π2 1 3⎤ π u⎥ − Θ(u)2⎢ u − u 2 + 4 12 ⎦ ⎣6

⎡ π2 ⎤ 1 π (n − 2π )3 ⎥ − Θ(− 2π + n)⎢ (n − 2π ) − (n − 2π )2 + 4 12 ⎣6 ⎦

⎤ ⎡ π2 1 π − Θ(−u)2⎢ (u + 2π ) − (u + 2π )2 + (u + 2π )3 ⎥ 4 12 ⎦ ⎣6

⎡ π2 1 3⎤ π u⎥ + Θ(u)⎢ u − u 2 + 4 12 ⎦ ⎣6 ⎡ π2 ⎤ 1 π + Θ(− u)⎢ (u + 2π ) − (u + 2π )2 + (u + 2π )3 ⎥ 4 12 ⎣6 ⎦

⎡ π2 1 ⎤ π − Θ(2π − n)2(ωay/v)⎢ − n + n2 ⎥ 2 4 ⎦ ⎣6

⎡ π2 1 ⎤ π + Θ(2π − n)(ωay/v)⎢ − n + n2 ⎥ 2 4 ⎦ ⎣6

⎤ ⎡ π2 1 π − Θ(−2π + n)2(ωay/v)⎢ − (n − 2π ) + (n − 2π )2 ⎥ 2 4 ⎦ ⎣6

⎡ π2 1 ⎤ π + Θ(u)(ωay/v)⎢ − u + u2⎥ 2 4 ⎦ ⎣6

⎤ ⎡ π2 π 1 − Θ(−u)2(ωay/v)⎢ − (u + 2π ) + (u + 2π )2 ⎥ 2 4 ⎦ ⎣6

⎡ π2 ⎤ 1 π + Θ(− 2π + n)(ωay/v)⎢ − (n − 2π ) + (n − 2π )2 ⎥ 2 4 ⎣6 ⎦

⎡ π2 π 1 ⎤ 2 − Θ(u)2(ωay/v)⎢ − u + u 2 ⎥ + (ωay/v)3 2 4 ⎦ 3 ⎣6

⎡ π2 ⎤ 1 π + Θ(− u)(ωay/v)⎢ − (u + 2π ) + (u + 2π )2 ⎥ 2 4 ⎣6 ⎦

(11)

The function expressed in eq 11 is depicted in Figure 3. The expression 11 allows one to account for the inconsistency of periodic functions written in terms of the sums of sine and cosine functions with the nonperiodic Lorentz friction term as well as the disagreement of the arguments kd ± ωd/v of the trigonometric functions outside of the first period. In Figure 2, we plot the solution of the equation (kd − ωd/v)(kd + ωd/ v − 2π) = 0, which determines the region for kd (denoted by x) versus d/a (denoted by y) inside which the exact cancellation of the Lorentz friction by the radiative energy originating from the other electrolyte spheres occurs. In Figure 3, a comparison of this cancellation for various sphere diameters is presented for longitudinally polarized collective plasmon excitations.

1 1 + Θ(2π − n) (ωay/v)2 [π − n] + Θ(− 2π + n) (ωay/v)2 [3π − n] 2 2 1 1 2 − Θ(u) (ωay/v) (ωay/v) [π − u] 2 2 1 2 − Θ(− u) (ωay/v)2 [− π − u] + (ωay/v)3 2 3

(13)

This function is plotted in Figure 4. The discontinuity at the border between the region with vanishing radiative damping and that with nonzero radiative attenuation is caused by the fact that the discontinuous function ∑∞ n=1 [(sin(nz))/n] (cf. Figure 5) appears in ImFx(y) but not in ImFz (cf. eq 8). 10018


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Figure 2. Region (gray, 0 < kd ± ω1d/v < 2π) in which the radiation losses vanish for infinite chains of electrolyte spheres with various radii and separations such that d/a ∈ [3,6], for ω = ω1 = 1.2 × 1012 (3.8 × 1013) 1/s, v = c, and an ionic concentration of n = 10−3N0 (left) or n = 10−2N0 (right), where N0 is a one-molar concentration.

Figure 3. Function ImFz(k;ω = ω1) for infinite chains of electrolyte spheres of radius a at separations of d = 3a, 4a, and 6a for an ion of mass m = 104me and charge q = 3e, where the dielectric susceptibility of the surroundings is ε = 2 and T = 300 K and the ionic concentration is n = 10−3N0 (upper) or n = 10−2N0 (lower) (N0 is a one-molar electrolyte concentration).

forms using the expressions for Fα(k,ω1) given by eq 8. The real parts of the functions Fα(k,ω1) renormalize the corresponding frequencies for both types of polarization, and they take the following forms for longitudinal and transversal polarization, respectively (according to eq 20)

To compare the magnitude of the various contributions to the damping of the collective plasmons in the chain, let us plot the dimensionless values for the longitudinal polarization, (1/ω1τ) = (1/ω1τ0) + (a3/2d3)ImFz(k) (in red in Figure 6), for comparison with the Lorentz friction contribution (1/3)(ω1a/v)3 (the blue line in Figure 6, cf. eq 19). In this figure, it is evident that the Lorentz term is much larger than the Ohmic attenuation for the exemplary spheres (the bottom plateau of the red line, cf. also eq 3). For large spheres, the Ohmic damping is governed by the ratio (v/λB) in eq 3, which is small because of the extremely low mean velocity of the carriers in the electrolyte despite the small mean free path. The same can be demonstrated for transversal polarization, as shown in Figure 6.

⎡ 4a3 ωz2(k) = ω12⎢1 − 3 ⎢⎣ d +

■

ω1d cos(mkd) ⎛ mω1d ⎞⎞⎤ ⎟⎟⎥ sin⎜ ⎝ v ⎠⎠⎥⎦ v m2

⎡ 2a3 = ω12⎢1 − 3 ⎢⎣ d

CALCULATION OF THE SELF-FREQUENCIES AND GROUP VELOCITIES OF PLASMON POLARITONS IN IONIC ELECTROLYTE CHAINS According to eqs 20 and 21 for the self-frequencies of plasmon polaritons in an electrolyte chain, one can write their explicit

+

10019

⎛ cos(mkd) ⎛ mω d ⎞ cos⎜ 1 ⎟ 3 ⎝ v ⎠ ⎝ m m=1 ∞

∑⎜

⎛ cos(mn1) + cos(mu1) ⎝ m3 m=1 ∞

∑⎜

ω1ya sin(mn1) − sin(mu1) ⎞⎤ ⎟⎥ ⎠⎥⎦ v m2

(14)


Article


Figure 4. Function ImFx(y)(k;ω = ω1) for infinite chains of ionic spheres of radius a at separations of d = 3a, 4a, and 6a for an ion of mass m = 104me and charge q = 3e, where the dielectric susceptibility of the surroundings is ε = 2 and T = 300 K and the ionic concentration is n = 10−3N0 (upper) or n = 10−2N0 (lower) (N0 is a one-molar electrolyte concentration).

∞ ∞ 2 3 Figure 5. Sums ∑∞ n=1[(sin(nx))/n] (left), ∑n=1[(cos(nx))/n ] (center), and ∑n=1[(sin(nx))/n ] (right) for x ∈ (−1, 15).

⎡ 2a3 ωx2(y)(k) = ω12⎢1 + 3 ⎢⎣ d

∞

⎛ cos(mkd)

∑ ⎜⎜ m=1

⎝

m3

⎛ mω d ⎞ ω d cos(mkd) cos⎜ 1 ⎟ + 1 ⎝ v ⎠ v m2

⎛ mω d ⎞⎞⎤ ⎛ mω d ⎞ ⎛ ω d ⎞ cos(mkd) sin⎜ 1 ⎟− ⎜ 1 ⎟ cos⎜ 1 ⎟⎟⎟⎥ ⎝ v ⎠⎠⎥⎦ ⎝ v ⎠ ⎝ v ⎠ m 2

⎡ a3 = ω12⎢1 + 3 ⎢⎣ d

⎛ cos(mn + cos(mu ) ω ay sin(mn1) − sin(mu1) 1 1 + 1 3 v m m2 ⎝ m=1 ∞

∑ ⎜⎜

⎛ ω ay ⎞2 cos(mn1) + cos(mu1) ⎞⎤ ⎟⎟⎥ −⎜ 1 ⎟ ⎝ v ⎠ m ⎠⎥⎦

(15)

where x = kd and y = d/a and n1 = x + ω1ya/v, u1 = x − ω1ya/v. The shift in the self-frequencies of the plasmon polaritons that is caused by their attenuation is addressed by the formula 21, where ωα(k) is given by eqs 14 and 15 and τα(k) has the form given by eq 19. Using the explicit form of ImFα(k,ω1), i.e., expressions 11 and 13, one can easily calculate the selffrequencies ω′α(k); these functions are depicted in Figures 7 and 8 for the longitudinal and transversal polarizations, respectively. Note that for the transversal polarization in eq 15, the sum

Figure 6. Comparison of various contributions to the damping of collective plasmons in the chainthe red line corresponds to the total damping ratio (in dimensionless units) (1/ω1τz) = (1/ω1τ0) + (a3/2d3) ImFz(k) and the bottom plateau of this line represents only the Ohmic contribution in the segment of the period of k in which all radiation losses vanish, whereas the blue line represents the Lorentz friction (in dimensionless units) (1/3)(ω1a/v)3for an ion of mass m = 104me and charge q = 3e, where the dielectric susceptibility of the surroundings is ε = 2 and T = 300 K; the ionic concentration is n = 10−3N0 (N0 is a one-molar electrolyte concentration); and the ion mean free path is λB = 5 × 10−9 m (left) or λB = 10−8 m (right).

∞

∑ m=1

cos(m(x + ω1ya/v)) + cos(m(x − ω1ya/v)) m

(16)

can be performed analytically using formula 9, resulting in the following contribution 10020


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Figure 7. Self-frequency ω′z(k) (in units of ω1) and the group velocity (in m/s) of longitudinally polarized plasmon polaritons in an infinite chain of electrolyte spheres of radius a = 20 μm at a separation of d = 3a, 4a, or 5a, where the ion concentration is n = 10−3N0, T = 300 K, and ω1 = 1.2 × 1012 1/s; the right panel depicts the effective removal by the chain finite length of the logarithmic singularity in the group velocity (occurring upon the perturbative approach) due to the far-field contribution of a finite chain of 15 spheres; the logarithmic singularity for the infinite chain in the perturbative formula for vz (though not in the formula for ω′z) leads to a local excess of v = c/(ε1)1/2, and this artifact of the perturbative approach is removed by the exact solution even for an infinite chain (cf. Appendix).

Figure 8. Self-frequency ωx(y) ′ (k) (in units of ω1) and the group velocity (in m/s) of transversally polarized plasmon polaritons in an infinite chain of electrolyte spheres of radius a = 20 μm at a separation of d = 3a, 4a, or 5a, where the ion concentration is n = 10−3N0, T = 300 K, and ω1 = 1.2 × 1012 1/s; ′ (k) due to the constructive interference of the far-field contribution of the dipole interaction and the related hyperbolic the logarithmic singularity in ωx(y) singularity in the transverse group velocity are artifacts of the perturbative solution of eq 7, and in the exact solution of this equation, the singularities are quenched to merely small local minima (cf. Appendix). −

1 ln[(2 − 2 cos(x + ω1ya/v))(2 − 2 cos(x − ω1ya/v))] 2

local enhancement of the transversal group velocity in close proximity to singularity points. The same result can be applied to ionic electrolyte sphere chains. To determine the group velocities of particular self-modes of plasmon polaritons in the chain, the derivative of ω′α(k) with respect to k must be performed, which, according to the expressions 14, 15, 21, and 19, is a straightforward though extended calculation. The sums that appear in the formulas for ωα(k) still cannot be performed analytically, with the exception of the sum with the linear denominator in eq 15. The resulting group velocities calculated for both types of polarization and for exemplary chain parameters (kd ∈ [0,2π] and d/a ∈ [3,10]) are presented in Figures 7 and 8.

(17)

(the other sums in eqs 14 and 15 must be performed numerically). This logarithmic singularity in the self-frequencies for transversally polarized plasmon polaritons on the edge of the region 0 < x − ω1ya/v < 2π (inside which the radiative damping vanishes) is depicted in the left-hand plot in Figure 8. This singularity gives rise to a hyperbolic discontinuity in the transverse group velocity (cf. Figure 8 right). Nevertheless, the logarithmic singularity in the self-energy and the related hyperbolic discontinuity of the transversal group velocity are artifacts of the perturbative solution of eq 7. The exact numerical solution of this equation yields an effective quenching of the logarithmic singularity to a small local minimum, resulting in a finite group-velocity discontinuity, as shown in the Appendix addressed to the exact solution of eq 7. This property of the transversally polarized propagation of plasmon polaritons in an electrolyte sphere chain, which is caused by the constructive interference of the far-field component of the dipole interactions between spheres, was numerically analyzed for a metallic chain in ref 26 and discussed in refs 27 and 28. The numerical studies of plasmon polariton propagation in metallic nanochains26 indicated a very narrow and weak but long-range mode in addition to the wide spectrum of quickly damped modes. This long-range “fainting” mode may be associated with the interference of the far-field component of the dipole−dipole interaction of the nanospheres in the chain, which results in a

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PLASMON POLARITON SELF-MODES IN THE CHAIN AND THE PROPAGATION OF THE ELECTROMAGNETIC SIGNAL ALONG THE PERIODIC IONIC STRUCTURE The real parts of the functions Fα renormalize the self-frequency of the plasmon polaritons in the chain, whereas the imaginary parts renormalize the damping rates of these modes. ReFα(k,ω) and ImFα(k,ω) are both functions of k and ω. In the first-order approximation, one can substitute ω = ω1 into ReFα as well as into the residual nonzero ImFα outside the region 0 < kd ± ω1d/v < 2π. Let us emphasize, however, that the vanishing of ImFα(k,ω) inside the region 0 < kd ± ωd/v < 2π holds for any value of ω27 (and thus not only for the ω = ω1 approximation but also for the exact frequency solution). 10021


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The Journal of Physical Chemistry C Hence, in the perturbative approach, one can rewrite the dynamic eq 7 for plasmon polariton modes in the chain in the following form

Dα″(ld , t ) =

Dα″(ld , t ) =

(19)

⎛ ⎞ a3 ωα2(k) = ω12⎜1 − 3 ReFα(k , ω1)⎟ ⎝ ⎠ d

γ = ωα(k*) 1 −

(25)

2 (τα(k*)ωα(k*))2

(26)

In a chain that is subjected to a persistent time-dependent electric-field excitation applied to some number (even a small number) of spheres, one must consider the undamped wave packet propagating along the entire chain. Such modes, depending on the particular shape of the wave packet arising from a specific chain excitation, may be responsible for the experimentally observed long-range, nearly undamped plasmon polariton propagation in metallic nanochains.5,20,29,30 Similar behavior is also expected in electrolyte microchains, although with a shift of all size parameters toward the microscale, for plasmon polaritons in ionic systems. The self-modes described by eq 22 are damped, and their propagation depends on appropriately prepared initial conditions that permit nonzero values of Aα,k. These initial conditions might be prepared by switching off the time-dependent external electric field that initially excited certain fragments of the chain. The resulting wave packet may adopt wavenumbers k from some subset of the [0,2π) region. If only the wavenumbers k for which 0 < kd ± ω1d/v < 2π contribute to the wave packet, then its damping is only of the Ohmic type (as shown in the previous paragraphs). The corresponding value of (1/τ0) decreases with increasing a (cf. eq 3)thus, larger spheres are favorable for the formation of longer-range excitations in the chain. The limiting value of the rate of Ohmic losses for large spheres is (1/τ0) → (v/2λB) 1/s, which determines the maximal range of propagation for irradiation-free plasmon polariton modes. This value depends on both the mean velocity of the ions, v, and their mean free path length, λB; for the exemplary values v ∼ 1100 m/s and λB ∼ 50 × 10−8 m, one obtains an ionic plasmon polariton range of ∼3 × 10−3 m (with the group velocity of the wave packet assumed to be ∼0.01c m/s). Estimations of the group velocities of plasmon polaritons for both types of polarization were presented in the previous paragraph.

(21)

i.e., for each k and α D′α (k , t ) = Aα , k ei(ωα ′ t + ϕα ,k)e−t / τα(k)

4γ 2 τα2(k*)

The above expression describes undamped wave motion with a frequency γ and a velocity, an amplitude, and a phase shift determined by k*. The amplitude attains its maximal value at resonance, when

Equation 18 can be easily solved for both the inhomogeneous and homogeneous (when E0α = 0) cases. The general solution of eq 18 has the form of the sum of the general solution of the homogeneous equation and a single particular solution of the inhomogeneous equation. The former includes the initial conditions and describes damped self-oscillations with the frequency 1 τα2(k)

(24)

2π cos(k*ld − γt − ηα , k )ε1a3ω12E0α (k*) * d 1

(20)

(22)

where the constants Aα,k and ϕα,k are adjusted to reflect the initial conditions. For the inhomogeneous case, the particular solution is as follows 1 (ωα2(k) − γ 2)2 +

4γ 2 τα2(k)

(ωα(k*)2 − γ 2)2 +

and the renormalized self-frequency

Dα″(k , t ) = εa3ω12E0α(k)ei(γt + ηα ,k)

− γ 2)2 +

This integral can be rewritten in the following form using the mean value theorem

with the renormalized attenuation rate

ωα2(k) −

dk cos(kld − γt − ηα , k )ε1a3ω12E0α (k)

(ωα2(k)

(18)

ω′α =

2π / d

1

⎛ 2 ⎞ 2 ω + ωα(k)2 ⎟Dα(k , ω) = ε1a3ω12E0α(k , ω) ⎜−ω − i τα(k) ⎝ ⎠

⎧1 ⎪ , for 0 < kd ± ω1d /v < 2π , ⎪ τ0 ⎪ 1 =⎨1 a3ω1 τα(k) ⎪ + ImFα(k , ω1), 2d3 ⎪ τ0 ⎪ for kd − ω d /v < 0 or kd + ω d /a > 2π ⎩ 1 1

∫0

4γ 2 τα2(k)

(23)

where one assumes a single Fourier time component of E0α(k,t) = 2γ / τ (k) E0α(k)eiγt and tg(ηαk ) = ω 2(k)α– γ 2 as is typical for a forced α

oscillator. Let us emphasize that E0α(k) is a real function such that E0α(ld)* = E0α(ld) = E0α(−ld). An appropriate choice of the latter function allows for the modeling of its Fourier picture E0α(k). In practice, this choice reflects the number of externally excited spheres in the chain, corresponding to, e.g., a suitably focused external excitation. This function yields the envelope of the wave packet if one takes the inverse of the Fourier transform of the solution given by eq 23 to recover the position variables. In the case of the external excitation of only a single sphere, the wave packet envelope homogeneously includes all wave vectors kd ∈ [0,2π). When a larger number of spheres is simultaneously excited, a wave packet envelope that is narrower in k can be selected. For E0α(ld)* = E0α(ld) = E0α(−ld), the Fourier transform has the same properties, i.e., E0α(k)* = E0α(k) = E0α(−k), and the latter equality can be rewritten as E0α(−k) = E0α((2π/d) − k) =E0α(k) by virtue of the (2π/d) period of k. The inverse Fourier picture with respect to eq 23 (the real part) has the form

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PLASMON POLARITON MODEL OF THE SALTATORY CONDUCTION OF ACTION POTENTIAL ALONG MYELINATED AXONS In the context of the rich physics of plasmon polaritons in ionic systems, which mirrors the properties of plasmon polaritons in metallic nanostructures, there arises the question of how to confirm these properties experimentally. Ionic systems with 10022


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myelinated axon does not form a chain of electrolyte spheres but rather is a thin electrolyte cord wrapped in myelin sheath on periodically distributed sectors separated by very short unmyelinated intervalsRanvier nodes. The periodic structure of the dielectric isolation allows, however, for consideration of collective plasmon wave-type oscillations propagating along the cord which resolve themselves to synchronic of wave-type dipole polarizations of consecutive Ranvier intervals. Taking into account that Ranvier nodes are short, these polarization dipoles are equivalent with similarly synchronically oscillating surface dipoles of ends of the myelinated sectors. The latter can be thus mimicked by dipole surface plasmon oscillations propagating along the chain of prolate spheroids with small separations and with the longitudinal dipole polarization, despite continuous character of the cordthis is sketched in Figure 12 (right panel). To estimate characteristics of such collective dipole wave-type excitation along the cord sectors one can use the model of the chain of spherical electrolyte systems with proportionally diluted ion concentration (resulting in the same total ion number as in the cord fragments) and also suitably decreased resonance frequency as for the longitudinal polarization in the case of a highly prolate spheroid (or elongated cylinder rather). This makes the proposed chain of auxiliary electrolyte spheres the effective model for the true initial system with the advantage that it allows for the tentative quantitative estimations of relevant propagation characteristics and the verification of whether plasmon polariton dynamics might fit to the observed features of the saltatory conduction of the myelinated axon, being somewhat mysterious and obscure as of yet. The consistency of this model requires highly exceeding the size of a single myelinated segment by the plasmon polariton wavelength, and this condition is satisfactorily fulfilled as will be shown in the next paragraph (for realistic parameters the plasmon polariton wavelength occurs more than 50 times longer than the myelinated segment length). If the signal conduction along the axon is driven by a plasmonic mechanism, then it must also contribute with triggering role to the electrochemical mechanism of the polarization/depolarization of the fragments of the axon membrane that are unmyelinated in the Ranvier nodes. The signal-dependent reversible release of ions through the lipid membrane first sharply enhances and subsequently reduces this polarization following the cyclic scheme of neuronal activity signaling, called the action potential. The arrival of a relatively low signal at a Ranvier node triggers the opening of the Na+ and K+ intermembrane ion channels, which results in a characteristically larger activation signal due to the transfer of ions through the opened gates caused by the difference in their concentrations on either side of the membrane. The entire cycle requires a few milliseconds, but the initial increase in polarization caused by the rapid opening of the Na+ channel occurs on the scale of a single millisecond. Because the myelin layer surrounded by the Schwann cell prohibits crossmembrane ion transfer, the local polarization/depolarization of the internal cytoplasm of the axon occurs only in the Ranvier nodes, thus strengthening the dipole formation in the axon fragment surrounded by the Schwann cell resulting in wave-type collective dipole oscillations in the electrolyte segments. The plasmon polariton mode that traverses the axon structure consisting of periodically polarized electrolyte segments wrapped in Schwann cells can cause the one-by-one individual ignition of the Ranvier node sequence along the axon. The triggering role of the plasmon polaritons might thus elucidate how the action signal jumps between neighboring active Ranvier nodes, in which

sufficiently high ionic concentrations are known as liquid electrolytes and are not amenable to simple fragmentation into spheres and chains. Nevertheless, finite electrolyte systems confined by appropriately formed membranes might serve for the practical realization of such model ionic systems. Such organizations of matter are frequent in biological structures on the cellular level and, moreover, at dimensions that are well consistent with the typical size scale of ionic plasmon effects, i.e., the micrometer scale. Cells and their internal components are frequently spheroid-shaped structures, and ionic effects are of primary significance to the functionality of such cells and in cell signaling. The search for certain commonalities with the collective plasmonic effects of the ions involved in these systems is a novel aspect of this approach. The high efficiency, low damping, and relatively fast nature of the kinetics of plasmon polaritons in chains of ionic systems are attractive features, especially the frequencies of the plasmon oscillations of the ions, which are lower than the high frequencies of plasmons in metals by several orders of magnitude and, moreover, are extensively tunable by means of modifying the ionic concentration. The possibility of modeling ionic plasmon polariton signaling in nerve cells is of particular interest, considering the structure of the long axons associated with the prolate-spheroid shape of Schwann cells, which are attached to the axon core with a myelin sheath and are separated by Ranvier nodes (cf. Figure 9). Although the core of

Figure 9. Schematic illustration of a long myelinated axon with a chain of Schwann cells with periodically repeated sectors of approximately 100 μm in length separated by unmyelinated Ranvier nodes, corresponding to a number of segments of order 10 000 per 1 m of axon length.

the axon is a long, thin, continuous insert of the nerve cell, its electrical conductivity is relatively poor, and it is commonly agreed that the Schwann cell structure (resembling a chain of spheroids) is crucial to the acceleration and maintenance of signals over long distances. The myelin sheath must be sufficiently thick to ensure the proper functionality of the neuron linkage, and a myelin deficiency results in the slowing of signal transmission, as observed in multiple sclerosis. This slowed transmission may be caused not only by the complete removal of the myelin cover but also even merely by a diminution of its thickness. Note, however, that a myelin sheath of reduced thickness still provides efficient electrical isolation of the core of the axon from its intercell surroundings, although it is most likely insufficient for the dielectric separation that is required for plasmon polariton formation in the linear periodic electrolyte/ dielectric structure. This observation may suggest that myelin serves some distinct role in addition to simple electrical isolation, namely, the creation of appropriately large dielectric/electrolyte periodic segments to optimize the plasmon polariton kinetics. The main idea of the plasmon polariton model for the saltatory conduction of axons is as follows. The periodic structure of the 10023


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required to maintain the same long-range propagation of plasmon polaritons at constant amplitude. This property seems to be also consistent with experimental observations.

the dipole oscillation amplitude is increased by the functioning of the nonlinear block of two ion channels for Na+ and K+. Because of the nonlinearity (the mutually dependent feedback of both ion channels) of this block, the polarization amplitude is saturated at a constant level (it remains the same for all nodes). These active elements of the chain play also the role of an external energy supply, which compensates for the Ohmic losses of the ions and ensures the long-distance transduction of the electric signal with a constant amplitude along the axon. The velocity of the plasmon polariton along the axon cord combined with the ion-gate blocks in the Ranvier nodes must be accommodated to the plasmon frequency of dipole oscillation of electrolyte segments, as is illustrated in a simplified manner in the next paragraph. The direct calculation of the density of the ions participating in plasmon polariton formation with a submillisecond frequency implies a plasmon polariton velocity on the order of 100 m/s (cf. the next paragraph), which is consistent with the observed signal velocity in long myelinated axons. The plasmon polariton scenario in an ionic chain, although simplified in comparison with the real structure of neurons, might share certain features in common with the ability of nerve systems to achieve very efficient and energetically economical electric signaling despite the rather poor ordinary conductivity of axons. The plasmon polariton kinetics in the periodic structure of the axon may be a convenient possible explanation for high performance nerve signaling by providing quick signal propagation despite very low ordinary conductivity and with a practically unlimited range when energy is persistently supplied to cover only relatively small Joule losses because the plasmon polariton does not irradiate any energy. This energy supply is provided by the ATP/ADP mechanism in cells, which energetically contributes to the signal-dependent opening and closing of Na+ and K+ channels in the Ranvier nodes and then to the restoration of the steady-state conditions, i.e., the active transport of ions across the membrane against the ionic concentration gradient. Moreover, the coincidence of the micrometer scale of the axon’s periodic structure of Schwann cells (of approximately 100 μm in length) with the typical requirements for the size of ionic chains supports the suggested plasmon polariton concept as the explanation of the transduction of the action potential along the axon, as illustrated by the tentative quantitative fitting presented in the next paragraph. It must also be emphasized that plasmon polaritons do not interact with electromagnetic waves or, equivalently, with photons (even with adjusted energy), which is a consequence of the large difference in group velocity between the plasmon polaritons and the photon velocity c/(ε)1/2. The resulting large discrepancy in wavelength between a photon and a plasmon polariton of the same energy prohibits the mutual transformation of the two types of excitation because of momentum conservation constraints. Therefore, plasmon polariton signaling by means of dipole oscillations along a chain can be neither detected nor perturbed by external electromagnetic radiation. This also well fits with neural signaling properties in the peripheral nervous system and in white myelinated matter in the central system. It is also worth noting that surface plasmon frequencies are independent of temperature, although not independent for volume plasmons, as shown in our previous paper.6 However, the temperature influences the mean velocity of the ions, v = ((3kT)/m)1/2, thereby causing an enhancement in Ohmic losses with increasing temperature (cf. eq 3), which, in turn, strengthens the plasmon polariton damping. Hence, at higher temperatures, higher external energy supplementation is

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FITTING OF PLASMON POLARITON FEATURES TO THE PARAMETERS OF AN AXON IN THE PERIPHERAL NERVOUS SYSTEM The bulk plasmon frequency is related to the ionic parameters via the formula ωp = ((q2n)/(ε0m))1/2 (in Gaussian units, ωp = ((q2n4π)/(m))1/2), where q is the charge of the ion, n the ionic concentration, ε0 the dielectric susceptibility of the vacuum, and m the mass of the ion. In our model, we assume the ion charge to be q = 1.6 × 10−19 C (the electron charge) and the ion mass to be m = 104me, where me = 9.1 × 10−31 kg is the mass of an electron; furthermore, we assume n = 2.1 × 1014 1/m3, and thus we obtain the Mie-type frequency for ionic dipole oscillations, ω1 ≃ 0.1 (ωp/(3ε1)1/2) ≃ 4 × 106 1/s, where the relative permittivity of water is ε1 ≃ 80 for frequencies in the MHz range31 (although for light frequencies, beginning at approximately 10 GHz, this value decreases to approximately 1.7, corresponding to the familiar refractive index of water, η ≃ ε11/2 = 1.33). For the thin and strongly prolate inner ionic cord segment, we reduce the longitudinal Mie-type frequency by a factor of 0.132,33 compared with the isotropic spherical case, in which it is ω1 = ωp/(3ε1)1/2. Let us emphasize that the axon consists of a cord with a small diameter of 2r, and this thin cord is wrapped with a myelin sheath with a length of 2a per segment; however, for the effective model, we consider electrolyte spheres of radius a. Thus, the auxiliary concentration n of ions in the fictitious spheres in fact corresponds to a concentration of ions in the cord of n′ = ((n4/ 3πa3)/(2a πr2)), which yields a typical concentration of ions in a nerve cell of n′ ∼ 10 mM (i.e., approximately 6 × 1024 1/m3). This is because all ions in a segment of length 2a that participate in the dipole oscillation that are represented in the fictitious spherical model correspond in the sphere model to a much smaller volume in the real system, that of the thin cord portion (the insulating myelinated sheath consists of a lipid substance without any ions). This insulating, relatively thick myelin coverage creates the periodically broken channel required for plasmon polariton formation and propagation. To reduce the coupling with the surrounding intercellular electrolyte and protect against any leakage of the plasmon polaritons, the myelin sheath must be sufficiently thick, much thicker than what is required merely for electrical insulation. Moreover, to accommodate the conductivity parameters that we calculated for a spherical geometry to the highly prolate geometry of the real oscillating ionic system, we must account for the fact that the Mie-type frequency of the longitudinal oscillations must be significantly lower than that for a sphere with a diameter equal to the elongation axis. As a rough estimate, we assume a correction factor of 0.1. For the resulting Mie-type frequency, ω1 ≃ 4 × 106 1/s, one can determine the plasmon polariton self-frequencies in a chain of spheres of radius 50 μm (for a Schwann cell length of 2a) and for small chain separations of d/a = 2.01, 2.1, and 2.2 (corresponding to Ranvier node lengths of 0.5, 5, and 10 μm, respectively) using the approach presented in the previous subsections. The derivative of the self-frequency with respect to the wave vector k determines the group velocity of the plasmon polariton modes. The results are presented in Figure 10. We observe that for the ionic system parameters listed above the group velocity of the plasmon polaritons reaches 100 m/s for longitudinal modes and 40 m/s for transversal modes. 10024


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Figure 10. Solutions for the self-frequencies and group velocities of the longitudinal (upper) and transversal (middle, lower) modes of a plasmon polariton in the model ionic chain; ω is presented in units of ω1, and here ω1 = 4 × 106 1/s, for a chain of spheres of radius a = 50 μm and a separation of d/a = 2.01, 2.1, or 2.2 for an equivalent ion concentration in the inner ionic cord of the axon of n′ ∼ 6 × 1024 1/m3; the upper panels represent a group velocity amplitude for the longitudinal mode of ≃100 m/s, whereas for the transversal mode, a similar velocity amplitude is achieved when the resonance transversal dipole oscillation frequency in the prolate geometry is assumed to be twice as large (2ω1);32,33 in the lower panels, the transversal mode characteristics corresponding to no frequency increase (ω1) are presented.

occurs throughout practically the entire kd ∈ [0,2π) region. Additionally, the above-mentioned singularities are characteristic of infinite chains and therefore cannot fully develop because the nerve model electrolyte chains are of a finite length. Although the ionic system chain model for a myelinated axon appears to be a crude approximation of the real axon structure, it can serve for the comparison of the energy and time scales of plasmon polariton propagation implied by the model with the kinetic parameters of nerve signals. In the model, the propagation through the axon chain of a plasmon polariton that is excited by an initial action potential on the first Ranvier node (after the synapse or, for the reverse signal direction, in the synapse hillock) (Figure 11) sequentially ignites consecutive Ranvier node blocks of Na+ and K+ ion gates, and the resulting firing of the action potential traverses the axon with an observed velocity of approximately 100 m/s, consistent with the velocity actually observed in myelinated axons. Each new ignition releases an excess of energy at a particular Ranvier node block; because of its nonlinearity, the signal growth saturates at a constant level, and the overall timing of each action potential spike has the stable shape of the local polarization/depolarization scheme in the

Nevertheless, if one assumes a larger transversal plasmon oscillation frequency,32,33 such as that for a more realistic model for prolate spheroids (i.e., for the inner Schwann cell cord that contains the ions), one also obtains a value of 100 m/s for the transversal group velocity of the plasmon polaritons for an increase in frequency of only twice the initial value (cf. Figure 10). Such transversal ion oscillations are, however, inconvenient in the considered structure, and moreover, the initial postsynaptic action potential or that from the synapse hillock excites predominantly longitudinal oscillations. Note that for ω1 = 4 × 106 1/s and a = 50 μm, the interference conditions kd − ω1d/c = 0 and kd + ω1d/c = 2π are fulfilled for extremely small values of kd and 2π − kd, respectively (of the order of 10−6 for d/a ∈ [2,2.5]), and thus are negligible with regard to the plasmon polariton kinetics under these conditions. Hence, the singularities induced by the far- and medium-field contributions to the dipole interaction are effectively removed near borders of the k wave vector domain. Moreover, the quenching of the radiative losses (i.e., the perfect balance of the Lorentz friction in each sphere by the radiation income from other spheres in the chain) for the plasmon polariton modes 10025


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Figure 11. Schematic illustration of the periodic fragments of a myelinated long axon (myelin is a multilamellar lipid membrane that wraps around the axon cord in segments separated by Ranvier nodes; the myelin sheath is produced by specialized cells (Schwann cells) in the peripheral nervous system and oligodendrocytes in the central nervous system); a fictitious periodic chain of spherical ionic systems is proposed as an effective model. In right panel the equivalence of polarized Ranvier nodes with longitudinal surface plasmons on myelinated sectors is sketched; effective concentration n of ions in the auxiliary sphere corresponding to the actual ion concentration n′ in the axon cord are indicated.

Figure 12. Comparison of the group velocities, in units of m/s, of the longitudinal plasmon polariton mode with respect to the wave vector k ∈[0,2π/d) in axon models for a Schwann cell myelinated sector with a length of 100 μm; Ranvier separations of 0.5 μm, 5 μm, and 10 μm (represented by d/a = 2.01, 2.1, and 2.2 in the figure, respectively) and axon cord radii of r = 20, 50, and 100 nm.

concentration in the cord of 10 mM, i.e., ∼6 × 1024 1/m3, which is typical for cell ion concentrations), a length of 100 μm for each myelinated sector wrapped with Schwann cells, and Ranvier intervals of 0.5, 5, and 10 μm. The dependence on the length of the Ranvier interval is weak (i.e., negligible), but the increase in the velocity with increasing thickness of the internal cord is significant, similarly as in the real axons with growing diameter. To comment on the appropriateness of the chain model to the axon case let us note that even though the thin core of the axon is a continuous ion conducting fiber the surface electromagnetic field can be closely pinned to the linear conductor similarly as for the Goubau line,34,35 and if periodically wrapped by dielectric shells, the plasmon resonantly coupled with the field propagates as the plasmon polariton in the chain of periodic segments despite continuity of the fiber. The fragments of the thin axon cord wrapped with the myelin thick shells with typical for neurons ion concentration n′ ∼ 10 mM ≃ 6 × 1024 1/m3 inside

short fragment of the cell membrane that corresponds to the Ranvier node. This permanent supply of energy contributes to the plasmon polariton dynamics assuring the same amplitude of each Ranvier block dipole excitation and in this way compensates for the Ohmic thermal losses of the plasmon polariton and ensures an undamped propagation over an unlimited range. Although the entire signal cycle of the action potential on a single Ranvier node block requires several milliseconds (or even longer when one includes the time required to restore the steady-state conditions, which, on the other hand, conveniently blocks the reversal of the signal), subsequent nodes are ignited more rapidly than this, corresponding to the velocity of the triggering plasmon polariton wave packet, as illustrated in Figure 14. The direction of the velocity of the plasmon polariton wave packet is adjusted to the semi-infinite geometry of the model chain (in fact, the chain is finite and is excited at one end). The firing of the action potential that is triggered by the plasmon polariton travels along the axon in only one direction because the nodes that have already fired have had their Na+ gates inactivated and will require a relatively long time to restore these gates to their original status (the entire sodium/potassium block requires time of the order of even a second as well as a sufficient energy supply to bring the concentrations to their normal values via cross-membrane active ion pumps). The described above plasmon polariton scheme for the ignition of the action potential spikes in the chain of Ranvier nodes along the axon is thus well consistent with the saltatory conduction observed in myelinated axons. In Figure 12, the group velocity for an action potential traversing a firing myelinated axon is plotted for various diameters of the internal cord of the axon (cf. Figure 12) (for an ionic

Figure 13. Examples of chains of elongate cylindrical rods and of prolate three-axis spheroids. 10026


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Figure 14. Schematic presentation of the firing of a nerve cell triggered by the propagation of the plasmon polariton wave packet along the periodic structure of the myelinated axon; for group velocity of the plasmon polariton wave packet of order of 100 m/s within 1 ms the packet traverses 10 cm distance and initiates one by one the forming of the action potential on 1000 in sequence Ranvier nodes being thus in various time phases as indicated by red dots for exemplary nodes.

α = x(y) is the transversal polarization (the chain orientation is assumed to be along the z direction). The dipole coupling is independent of the shape of chain elements. The described above independence of dipole oscillations with distinct polarization follows from the linearity of the dynamical equation with respect to the dipole, regardless of the metal or electrolyte-conducting elements. Because the structure of the dynamics equation is not affected by the anisotropy, its solutions for each polarization have the same form as in the spherical case with the exception of modification of the related frequency of dipole oscillation in each direction and the small correction to the orientation-dependent contribution to the scattering ratio (this part is related to the boundary scattering of carriers). One can thus renormalize the equation for dipole oscillations independently for each polarization direction introducing in a phenomenological manner the resonance oscillation frequency for each direction ωα1 (they can be estimated numerically, whereas for a sphere ω1 = ωp/ (3ε1)1/2; in general, the longer semiaxis, the lower the related dipole oscillation frequency is). Moreover, the size correction of the boundary scattering term is of order, as described above. Such a renormalization was done for a considered model of axon segment chain.

the cell can be equivalently modeled by a sphere with diameter equal to the length of the cord fragment and with the ion concentration n ≃ 2 × 1014 1/m3 (for the axon cord diameter assumed 100 nm) conserving the number of ions in the segment taking part in dipole oscillations. Such a model is justified by the same structure of the dynamics equation for dipole plasmon fluctuations in the chain of spheres, i.e., of eq 2, and of its modification for the prolate spheroid or elongated cylindrical rod chain (as depicted in Figure 13). This modification resolves to the following equation instead of eq 2 ⎡ ∂2 ⎤ 2 ∂ + ωα21⎥Dα (ld , t ) ⎢ 2 + τα 0 ∂t ⎣ ∂t ⎦ m =∞

=(

∑ m =−∞ , m ≠ l

⎛ |l − m|d ⎞ ⎟ + (E (ld , t ) Eα ⎜md , t − Lα ⎝ ⎠ v

+ (Eα(ld , t )

(27)

2

where ( = V((nq )/m is a shape-independent factor proportional to the number of ion carriers with concentration n in the volume of the spheroid with semiaxes a, b, c, V = ((4π)/3)abc = ((4π)/3)a3 (the latter for a sphere). Taking into account that the plasmon frequency in bulk electrolyte with ion concentration n equals to ωp = ((nq24π)/m)1/2, one can rewrite ( as follows: ( = (abcω2p)/3 = ε1a3ω21 the latter for a sphere, for which ω1 = ωp/(3ε1)1/2. Ohmic losses are included via the term 1 v Cv = + α 2λB 2a τα 0

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CONCLUSIONS The analysis presented above, in combination with the quantitative estimations, demonstrates that collective wave-type surface plasmon modes, similar to plasmon polaritons in metallic nanochains, may be excited in chains consisting of finite electrolyte systems (e.g., the considered model of spheres of radius a at the micrometer scale). These ionic plasmon polaritons can efficiently transfer information and energy over relatively large distances, and chains of ionic systems can act as low-loss ionic plasmon waveguides. These features of electrolyte chains, which

(28)

where aα is the dimension (semiaxis) of the spheroid in the direction α (equaled to a,b,c as in Figure 13). The first term in the expression for (1/τα0) approximates ion scattering losses such as those occurring in the bulk electrolyte, whereas the second term describes the losses due to the scattering of ions on the boundary of the spheroid. α = z is the longitudinal polarization, and 10027


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the divergence of the group velocities, although they result in a local maximum of truncated singularities, sharply cut at the level of the velocity of light (as presented by an exact solution of eq 7 in the Appendix). This local enhancement of the group velocity may lead to the occurrence of fast-moving but narrow wave packets of plasmon polaritons in ionic chains, taking advantage of the higher velocity of plasmon polariton modes near singular points. However, it must be emphasized that this property is efficiently quenched in chains of finite length because the contribution of an infinite number (in practice, a large number) of spheres in the chain is required for the full development of the mentioned singularities. By contrast, the other properties of plasmon polariton dynamics in such chains prove to be much more robust against the shortening of the chain because of the very rapid convergence of the series representing the influence of the mutual radiation of the chain elements, except in the case of a singular element (as many elements are required to develop the divergence). The utilization of the plasmon polariton propagation irradiatively undamped in the chain of electrolyte subsystems may lead to an explanation of the efficient and long-range socalled saltatory conduction in myelinated axons in peripheral neural system and in white matter of brain and spinal cord. The effective plasmon polariton model of the triggering of action potential fires along the axon in the peripheral nervous system myelinated by Schwann cells separated with Ranvier nodes fit well with the observed conduction velocity and its temperature and size dependence. This coincidence together with the immunity of plasmon polariton to the external e−m perturbation or detection support the propriety of the proposed new model for saltatory conduction of action potential, which is very efficient and energy frugal despite the poor ordinary conductivity of axons.

are convenient for plasmon polariton kinetics, are related to the following properties: •Ideal cancellation of the radiation losses of the collective dipole-type surface plasmons in the chain for both polarizations of the oscillations. This occurs because of the ideal compensation of the radiative energy losses caused by the Lorentz friction on the oscillating charges in each sphere of the chain by the radiative income from all neighboring spheres. The contributions of the mutual radiation of all spheres in the chain, including the near-, medium-, and far-field components of the dipole interaction, when retardation effects are accounted for, perfectly cancel all radiation losses and compress the entire electromagnetic field into the interior of the chain, resulting in ideal plasmon polariton guidance along the ionic chain. The extent of this plasmon polariton propagation in the chain is limited only by the residual Ohmic losses caused by the microscopic scattering of ions, including collisions with other ions, solvent particles, and admixtures, and with the boundaries of the electrolyte spheres. It must be emphasized that the ionic plasmon damping caused by these irreversible scattering processes is much smaller than the radiation losses in a single individual electrolyte sphere with a large radius (on the order of several micrometers, depending on the electrolyte parameters). Note that the residual heat losses (Ohmic losses) of plasmon polaritons in the chain may be compensated for by relatively low external energetic support in the form of a persistent excitation or a coupling to an active medium (in metal nanochains, such an active medium may be a system of quantum dots coupled with the metallic nanospheres in the chain and may operate in the spaser regime.36) The free kinetics of ionic plasmon polaritons in the chain can reach the millimeter scale in terms of the propagation range and may be practically unlimited in the case of the external compensation of heat dissipation. •The group velocity for ionic plasmon polaritons may vary in a wide range depending on the electrolyte parameters (primarily the ion concentration) and on the shape of the signal wavepacket (due to the selection of an appropriate envelope for the wave vectors included in the packet of plasmon polariton modes). Typically, the ionic plasmon polariton mode velocities occur at least 2 orders of magnitude lower than the velocity of light for an ion concentration of ∼10−2N0 (where N0 is a one-molar concentration), despite the very low mean value of the velocity of ions in electrolytes due to their large mass. The velocities of ionic plasmon polaritons in such chains are also correlated with their frequency and, for high ionic concentrations, are lower than the plasmon frequency in metals by 3−4 orders of magnitude (i.e., ∼1011−12 1/s compared with 1015−16 1/s in metals). For lower ionic concentrations and for special selection of the electrolyte subsystem chain, the group velocity can be, however, much lower, as in the case of the plasmon polariton model for the saltatory conduction in axons. The primary characteristics of ionic plasmon polaritons, as listed above, are also associated with many specific particularities, for instance with a local increase in the group velocities for both types of polarization for oscillations in the near vicinity of wave vectors that satisfy the interference condition 0 = kd ∓ ωd/v = 2π (where k is the wave vector, d is the spacing in the chain, ω is the plasmon polariton frequency, and v =c/(ε1)1/2 is the velocity of light in the surrounding medium), which occurs as a result of the constructive interference of the radiation of the chain elements in the far- and medium-field zones. Remarkably, the singular contributions to the dipole interaction are effectively quenched by the nonlinearity of the dynamics and do not lead to

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APPENDIX

Exact Solution of Equation 7: The Problem of the Logarithmic Divergence of the Far-Field Contribution to the Self-Frequency for the Transversal Polarization of Plasmon Polaritons and That of the Medium-Field Contribution to the Group Velocities for Both Types of Polarization

The solution of eq 7 in the homogeneous case yields the Fourier exponent ωα(k), which is, in general, a complex function. Its imaginary part defines the plasmon polariton damping rate, whereas the real part of the ω function defines the self-frequency of a particular k mode with α polarization. One can then determine the group velocities for particular modes by taking the derivative of the real part of ωα(k) with respect to the wave vector k. As noted above, the far-field contribution to the dipole interaction in the chain produces a logarithmically singular term in eq 7, of the form given by eqs 16 and 17. Because of the infinite divergence of this term, one cannot apply the perturbative method to solve eq 7 in the vicinity of singular points in the k domain. The perturbative method of solving eq 7 involves transforming this logarithmic singularity into the solution for the self-frequency at the singular point k. The derivative of this frequency, i.e., the group velocity of the singular mode k, thus acquires a hyperbolic singularity. This occurs in the perturbative formula for the transversal group velocity. These infinite divergences are visible in Figure 8 (left), near the edges of the domain kd ∈ [0, 2π). When applying the perturbative approach to the longitudinal polarization case, one also encounters an infinite singularity of the group velocity at the same k as for the transversal polarization 10028


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Figure 15. Exact solutions for the self-frequencies for the longitudinally and transversally polarized modes of plasmon polaritons in an ionic chain (ω in units of ω1) obtained by exactly solving eq 7 at 1000 points in the region kd ∈ [0,2π) (left) and the corresponding group velocities for both types of polarization (right).

perturbative formula for the group velocities. The exact solution of the Lorentz-invariant dynamical equation also inherently possesses this property. The exact self-energies are suitably regularized with respect to k such that their derivatives do not exceed v = c/(ε1)1/2. For the sake of completeness, it is worth emphasizing that the inclusion of the magnetic field of the dipoles does not modify this scenario because the contribution of the magnetic field to the self-energies is a few orders of magnitude lower than the contribution of the electric field. This difference in magnitude between the two contributions arises because the velocity of the ions is similarly lower than the velocity of light, which significantly reduces the force acting on ions induced by the magnetic field. Therefore, the magnetic field of the dipoles16,17

but no singularity in the longitudinal self-frequency. This astonishing singularity in the group velocity is of the logarithmic type and actually arises for both types of polarization when one takes the derivative of eqs 14 and 15 with respect to k. After taking the derivative with respect to x (x = kd), the contribution to the self-energies for both types of polarization originating from a medium-field irradiation term with a factor ∑((cos(nx))/n2) is divergent. For the longitudinal polarization, this divergence is visible in Figure 7. For the transversal polarization, this logarithmic singularity interferes with the hyperbolic singularity described above. All these singularities occur at isolated points for which kd ± ω1d/v = lπ(linteger). These singularities are apparent artifacts of the perturbative approach because they lead to a local velocity exceeding the velocity of light. To resolve and clarify this problem of unphysical divergence, one must solve eq 7 exactly, without applying the perturbative technique. It is clear that because of the divergence of expression 16 the corresponding contributions cannot be treated as a small perturbation. The exact solution of eq 7 can be obtained numerically. The numerical solutions (obtained using a Newtontype algorithm) for the self-frequencies for both types of polarization and for the entire k domain are plotted in Figure 15. Upon comparing the corresponding plots obtained using the perturbative method with those presented in Figure 15, we observe that the exact solutions for the self-frequencies do not differ significantly from those obtained using the perturbative approach. Nevertheless, the exact solutions for the group velocities do not contain any singularities. A thorough analysis28 reveals that all divergences in the exact solutions for the group velocities exhibit a cutoff such that they do not exceed the velocity of light. This quenching of the singularities to prevent exceeding the velocity of light is a manifestation of the Lorentz relativistic invariance of the plasmon polariton dynamics. The retardation of the electric signals prohibits the collective excitation group velocity from exceeding the velocity of light. This quenching regularizes the infinite singularities that occur in the perturbative expressions for the self-energy and then in the

⎛ ik 1⎞ Bω = ik(Dω × n)⎜ − 2 ⎟eikr0 r0 ⎠ ⎝ r0

(29)

although it contributes to the far-field and medium-field components of the plasmon polariton self-energies, does not significantly affect the relevant terms related to the electric field; it gives rise only to essentially negligible corrections.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +048 71 320 20 27. Fax: +048 71 328 36 96. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

The present work was supported by the NCN project no. 2011/ 03/D/ST3/02643 and the NCN project no. 2011/02/A/ST3/ 00116. 10029


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(28) Jacak, W. Exact solution for velocity of plasmon-polariton in metallic nano-chain. Opt. Express 2014, 22, 18958. (29) Krenn, J. R.; Dereux, A.; Weeber, J. C.; Bourillot, E.; Lacroute, Y.; Goudonnet, J. P.; Schider, G.; Gotschy, W.; Leitner, A.; Aussenegg, F. R.; Girard, C. Squeezing the Optical Near-Field Zone by Plasmon Coupling of Metallic Nanoparticles. Phys. Rev. Lett. 1999, 82, 2590. (30) Maier, S. A.; Kik, P. G.; Atwater, H. A. Optical pulse propagation in metal nanoparticle chain waveguides. Phys. Rev. B 2003, 67, 205402. (31) Meissner, T.; Wentz, F. The Complex Dielectric Constant of Pure and Sea Water from Microwave Satellite Observations. IEEE THRS 2004, 42, 1836. (32) El-Brolossy, T. A.; Abdallah, T.; Mohamed, M. B.; Abdallah, S.; Easawi, K.; Negm, S.; Talaat, H. Shape and size dependence of the surface plasmon resonance of gold nanoparticles studied by Photoacoustic technique. Eur. Phys. J. Special Top. 2008, 153, 361. (33) Liz-Marzan, L. M. Tuning nanorod surface plasmon resonances. SPIE Newsroom 2007, DOI: 10.1117/2.1200707.0798. (34) Goubau, G. Surface Waves and Their Application to Transmission Lines. J. Appl. Phys. 1950, 21, 119. (35) Sommerfeld, A. Ü ber die Fortpflanzung Elektrodynamischer Wellen Langs Eines Drahts. Ann. Phys. Chem. 1899, 67, 233. (36) Andrianov, E. S.; Pukhov, A. A.; Dorofeenko, A. V.; Vinogradov, A. P.; Lisyansky, A. A. Stationary behavior of a chain of interacting spasers. Phys. Rev. B 2012, 85, 165419.

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Propagation of Collective Surface Plasmons in Linear Periodic Ionic

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