Accepted Manuscript

Finite-Difference Time-Domain (FDTD) Design of Gold Nanoparticle Chains

with Specific Surface Plasmon Resonance

Cristian Tira, Daniela Tira, Timea Simon, Simion Astilean

PII: S0022-2860(14)00453-0

DOI: http://dx.doi.org/10.1016/j.molstruc.2014.04.086

Reference: MOLSTR 20593

To appear in: Journal of Molecular Structure

Received Date: 17 February 2014

Revised Date: 28 April 2014

Accepted Date: 29 April 2014

Please cite this article as: C. Tira, D. Tira, T. Simon, S. Astilean, Finite-Difference Time-Domain (FDTD) Design

of Gold Nanoparticle Chains with Specific Surface Plasmon Resonance, Journal of Molecular Structure (2014),

doi: http://dx.doi.org/10.1016/j.molstruc.2014.04.086

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Finite-Difference Time-Domain (FDTD) Design of Gold Nanoparticle Chains with Specific Surface Plasmon

Resonance

Cristian Tira, Daniela Tira, Timea Simon and Simion Astilean* Nanobiophotonics and Laser Microspectroscopy Center, Interdisciplinary Research Institute in Bio-Nano-Sciences and Faculty of Physics, Babes-Bolyai University, 1 M. Kogalniceanu Str., 400084, Cluj-Napoca, Romania

*Corresponding author: simion.astilean@phys.ubbcluj.ro

Abstract

We employ Finite-Difference Time-Domain (FDTD) simulations to analyze the electromagnetic far- and near-field response of gold nanoparticles (NPs) organized in chain-like structures as function of the number of particles and inter-particle distance in structures. As a result an empirical formula to predict the position of collective localized surface plasmon resonance (LSPR) as function of number of particles in the chain is devised. On the other hand the experimental LSPR spectrum recorded from a colloidal solution exhibiting a certain degree of aggregation has been effectively reconstructed by linear combination of individual LSPR contribution as calculated for NP ensembles of different size (monomers, dimers, trimers, etc.). Notably, we find that the maximum of electric field intensity (E2) in between adjacent NPs increases from dimeric to trimeric and tetrameric ensembles, followed by a steady state decrease as the number of NPs per chain further increases. The central gap in a long chain of NPs accommodate the highest field enhancement (‘hot-spots’). Our findings are relevant for designing effective substrates for Surface-Enhanced Raman Scattering (SERS) and plasmonic waveguides.

Keywords gold nanoparticles, FDTD, SERS, LSPR 1. Introduction The interaction between light and noble-metal nanoparticles (NPs) enables a unique optical

response of NPs known as localized surface plasmon resonance (LSPR) which can be exploited in various applications from sensing, diagnostics and enhanced light harvesting to optical nano-antenna and waveguides [1-6]. The LSPR of assemblies of NPs can be engineered by controlling the size and shape of the individual NPs in structure as well as their spatial arrangement and mutual distance. In recent years several groups have studied theoretically and experimentally the plasmonic properties of various assemblies of metal nanoparticles either in solution or deposited onto solid substrates [7-16]. The linear chain of nanoparticles has emerged as one of reasonably simple nanostructures to be fabricated with the aim of producing intense electromagnetic fields in the interstices between NPs and confine electromagnetic energy on nanoscale [17, 18]. For instance, previous research works have demonstrated the potential of small aggregates made of few NPs as effective SERS substrates due to the high electric field that reside in the inter-particle gaps called ‘hot-spots’ [19-24]. However, there is still an insufficient understanding as regards to the optimum number of coupled NPs and the inter-particle gaps which can ensure a maximum SERS activity. Primarily, it is important to understand if dimeric or trimeric assemblies would be substantially more effective as SERS substrates than linear chains containing several NPs (>4). Such an answer is relevant from the experimental point of view as longer chains would not be easily fabricated, stabilized or directed in biosensing applications. Secondly, a predictable spectral position for LSPR band of NPs arrangement would be helpful in selecting the right laser line needed for excitation to optimize the electromagnetic enhancement in SERS. Therefore, better understanding and rational optimization of collective plasmonic properties of linear chains as function of number and distance between NPs would be highly desirable.

Here, we report a theoretical analysis on the far- and near-field plasmonic properties of linearly assembled gold NPs. The analysis is based on comprehensive electromagnetic FDTD simulations, i.e. numerically solving Maxwell’s equations by iteration over time. A theoretical

formula for LSPR position as function of number of NPs in the chain is devised and experimental spectra recorded from colloidal solutions with various degrees of NP aggregation analyzed. Near-field

examination reveals that the maximum of electric field intensity ( 2E ) in between adjacent NPs increases from dimers to trimers and tetramers followed by a steady state decrease as the number of NPs per chain increases. We explain this behavior by a “trade-off” between electromagnetic losses and localization of electromagnetic energy over the whole structure in collective plasmonic modes.

2. Theory and Experiment

2.1. FDTD Method In order to predict the near- and far–field optical properties of metallic nanoparticles, it is

necessary to solve a set of four partial differential equations known as Maxwell’s equations which describe the electric and magnetic components of the electromagnetic wave in interaction with quasi-free charges existing in metallic NPs. Gustav Mie was the first to solve analytically the Maxwell’s equations in the case of light interaction with a small metallic sphere and to calculate the scattering and absorption coefficients [25]. While accurate analytical solutions still exist for a few symmetrical nanoparticles (elliptical and rod-like shape) [26-29], for two or more assembled spheres the Maxwell’s equations should be solved numerically. The FDTD is one of the most popular numerical methods to solve Maxwell’s equations for complex arrangement of nanoparticles that cannot be approached analytically. In present work, a commercial program from Lumerical [30] is used to investigate the plasmon resonances properties of linear chains of gold NPs. The approach of the FDTD method is that of dividing the physical space into small cubic cells, Yee cells (Figure 1A). Starting with some guess values, the electric fields are calculated on the edges of the cube and the magnetic fields crossing the faces using Maxwell’s equations. After these first order approximation values are found, they are introduced in Maxwell’s equations to obtain the second order approximation and continue this process until convergent results are obtained. The convergence limit corresponds to the real physical value of the fields. Using this approach, one of the most important parameters that we have to take into consideration is the mesh size of the simulation region (the length of the Yee cell edges). This parameter must be decreased until the results using meshes of different sizes converge. In all our simulations we have used a mesh of 0.5 nm because we found out that this is small enough to meet the convergence criteria, the far-field response being identical to simulations with mesh 0.2 nm or even 0.1 nm. The only problem with this greater mesh was a small numerical artifact which produced a small intensity peak between 800 and 850 nm but we did not consider this a problem for our research purposes. To model the complex permittivity of gold, we used the experimental data of Johnson and Christy [31]. The source was a Total Field Scatter Field (TFSF), one that uses plane waves, suitable for problems concerning the scattering of small particles. We used power monitors arranged in such a way that we formed small boxes, one inside the TFSF source for the calculation of the absorption coefficients and one that contained the source for the scattering coefficient (see Figure 1B).

Figure 1. (A) Yee cell representation; (B) Representation of the monitors, source and boundary

conditions used in our simulations.

To probe both the longitudinal and transverse plasmonic response of NPs chain, the incident plane wave was considered either polarized parallel to the interparticle axis or perpendicular to this axis (see Figure 2).

Figure 2. The two polarizations of the incident wave: (A) electric field parallel to the interparticle axis and (B) electric field perpendicular to the interparticle axis.

Numerical simulations were performed on chains containing different numbers of spheres for each polarization presented above and results combined to depict the “unpolarized” LSPR spectrum. The extinction coefficient was calculated using the data provided by the total and scattered field power monitors. Summing up the contributions from all the planes of the total field box we obtain the absorbed power (Pabs) of the structure. The second monitor gives the particles’ scattered power (Pscatt) because at the TFSF boundaries the power from the incident waves (Iincident) is subtracted. Thus, for a certain wavelength, the absorption, scattering and extinction coefficients are given by:

, ,

Finally, to compare chains of different number of spheres, the absorption, scattering or extinction coefficients were normalized to the total scattering area of chains:

, ,

where R is the sphere radius and n represents the number of spheres in the chain. 2.2. Synthesis and characterization of gold NPs

In this work spherical gold nanoparticles were synthesized according to the Turkevich-Frens method [32] by the aqueous reduction of HAuCl4 with trisodium citrate. Briefly, 100 mL of 1 mM HAuCl4

.3H2O was boiled and a solution of 38.8 mM sodium citrate (10 mL) was added under continuous magnetic stirring. When the solution changed its color to intense burgundy-red, it was removed from the heat and the stirring process continued for another 10-15 min. To induce assembly of gold nanoparticles, 1 ml of aqueous colloidal solution was mixed with 500 µl of ethanol. 0.2 mM Pluronic F127 polymer was added at different periods of time (0, 15, 30, 50 s) to as prepared colloidal solution in order to stabilize the formed aggregates in solution. Extinction spectra of colloidal solutions with different degree of aggregation were recorded using a Jasco V-670 UV-Vis-NIR spectrometer with a slit width of 2 nm and 1 nm spectral resolution. The NPs assemblies were examined using a JEOL 100 U type transmission electron microscope operated at 100 kV accelerating voltage.

3. Results and Discussion Figure 3A shows a sequence of simulated LSPR spectra corresponding to increasing number

of NPs in chains with constant inter-particle distance of 0.5 nm. While important red-shifts of LSPR band assigned to longitudinal coupling can be noticed only minor red-shifts occur for LSPR bands assigned to transversal polarization as compared to spectrum of individual NPs.

More interestingly, we found that the spectral position of LSPR band assigned to longitudinally coupled NPs can be described by a sigmoid function y(x):

where y and x represents the spectral position of band and number of NPs in the chain, respectively. In mathematical statistics the logistic sigmoid function is employed to model the evolution of a certain quantity over time when the quantity increases initially through an exponential and finally follows an asymptotic regime to reach some upper limit. This saturation of the LSPR shift as a function of NPs could be explained in terms of a trade-off between the energy accumulation in the system through the addition of a sphere and the dissipation of the whole chain. In the above equation A1, A2, x0 and p are fitting parameters. The value A2=739nm is obtained from a simulation considering an infinite chain while other parameters can be obtained by fitting considering y=616nm for x=2 (A1=525nm,

x0=2.601 and p=1.04 and the correlation coefficient R=0.982). Based on this equation it is possible to predict the wavelength at which a chain composed of a given number of spheres exhibits the maximum LSPR.

500 550 600 650 7000.0

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.u.)

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eA

0 2 4 6 8 10 12 14 16 18 20500

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eak

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B

Figure 3. (A) Simulated LSPR spectra of chains of gold NPs of 20 nm diameter and interparticle distance of 0.5 nm surrounded by water (n=1.33) and composed of 1 (a-black), 2 (b-red), 3 (c-green), 4 (d-blue) and 5 (e-cyan) NPs. (B) LSPR spectral position as function of the number of particles in

the chain.

400 500 600 700 8000,0

0,2

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1,2

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inct

ion

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.)

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A

a

b

c

d

e

Figure 4 (A) Normalized extinction spectra recorded from colloidal solutions with various degree of aggregation. The measurements proceeded at various time intervals after adding of polymer to solution (0 s – blue, 15 s – black, 30 s – red and respectively 50 s – green). For reference the spectrum of non-aggregated solution is given in cyan. (B) Representative TEM pictures of chains formed in solution. (C) Histogram of chain lengths distribution in solution after immediate adding of Pluronic.

The analysis of TEM pictures recorded after stabilization of colloidal solution is the usual

way to provide the histogram of distribution of monomers, dimers, trimers, etc. existing in colloidal solution (Figure 4C). However, alternative methods to provide some similar information about the size of chains and their distribution in a colloidal solution extracted from their LSPR spectra would be highly desirable. Here we demonstrate that the experimental, composite LSPR spectrum recorded from a colloidal solution of a certain degree of aggregation can be analyzed with a “trial and error” method based on linear combination of theoretical spectra computed above for chain of different lengths. Indeed, the analysis of the experimental spectrum (Figure 4A, spectrum b) provides the percentage of monomers, dimers, trimers, etc. (see data in Table 2) which are consistent with the histogram provided by TEM pictures. Therefore we can use the results provided by a “trial and error” method as first indicator for the “composition” of aggregated samples c, d and e (corresponding spectra are presented in Figure 4A) for which we do not provide TEM data.

NPs / chain

Sample /method

1 2 3 4 5 6 7 > 7

d

TEM 32.5 23.6 14.6 7 5.6 - - 16.5

Fit 28.2 25 9.1 4.5 3.4 0.9 0.8 28

c Fit 2.4 39.8 12.1 12.3 6 0.6 0.6 26

d Fit 0.6 22.6 14.2 14.9 16.2 3.4 2.6 25

e Fit 0.4 17.5 11 11.7 12.5 7 18.6 21.2

Table 1. Percentage of chains of different lengths existing in the colloidal solutions as provided by TEM (sample b) and “trial and error” method (samples b, c, d, and e).

Figure 5A shows the experimental spectra together with spectra reconstructed by linear

combination of above simulated spectra for chains of different lengths (see Figure 3A). As can be seen the reconstructed spectra feature much better spectral resolution of longitudinal and transversal bands relative to the experimental spectra which show bands much broader. At this point we suggest that additional effects have to be taken in consideration to better reproduce the real case of aggregated solution. It is conceivable to take in consideration for instance the effect of interparticle distance and angle. Therefore in the next step we discuss how the interparticle distance between NPs in the chain could play on the appearance of LSPR spectra.

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efgh

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Figure 5 (A) Experimental spectra (continuous line) together with theoretical spectra reconstructed by

linear combination of above simulated spectra for chains of different lengths. (B) Simulated LSPR spectra of chains composed of three gold nanoparticles of 20 nm diameter for different interparticle distances 0.5 nm (a-black), 1 nm (b-red), 2 nm (c-green), 5 nm (d-blue), 10 nm (e-cyan), 15 nm (f-

magenta), 20 nm (g-yellow) and, as reference, single isolated sphere (h-olive).

As shown in Figure 5B, the LSPR band blue shifts and gradually reduces its intensity as the interparticle distance increases from 0.5 nm to 20 nm. Moreover, we can notice that for distances larger than 15 nm, the profile of extinction spectrum is almost identical with that of isolated NP. The limit of 0.5 nm between NPs is the lowest possible one that we could simulate due to computational restrictions. However, for a smaller separation distance the contribution of the quantum tunneling effect increases and the classical electrodynamics theory is no longer an accurate approximation [33,34].

In the above simulations the chains were considered to be perfectly aligned. However, as can be seen in the TEM image, some deviations from the linear chain-like structure could

exist. Therefore a series of simulations were performed to reveal the optical response of non-perfectly aligned chains and estimate their contribution to overall spectra recorded from the colloidal solution. An example of geometry to illustrate a non-perfectly aligned chain comprising 5 nanoparticles is depicted in Figure 6A. For simulation we consider a gradual variation of the deviation angle within the range 0-30 degrees, while keeping the inter-particle distance fixed at 0.5nm. The computed spectra for deviation angles of 0, 10, 20 and 30 degrees are plotted in Figure 6B. As function of increasing angle, the main longitudinal surface plasmon resonance decreases in intensity and shifts toward blue revealing a second multipolar resonance located in the red side. It is conceivable that such effects can explain at least partially the broad feature of experimental spectra when compared to reconstructed spectra generated only by linear chain-like structure.

500 600 700 8000

1

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5

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7

Ext

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(a.u

.)

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B a

b

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Figure 6. (A) Top-view schematic of the geometry of the chains with different angles between

adjacent NPs. (B) Simulated extinction spectra obtained from 5 sphere chains with different angles (0 degrees - a, 10 degrees - b, 20 degrees - c and 30 degrees - d) between the NPs in water (n=1.33)

with an interparticle distance of 0.5nm.

Apart from the far-field optical properties of the gold NPs chains we were interested to

analyze the electrical field intensity ( 2E ) at the surface of NPs in a plane perpendicular to the propagation direction of the waves that contains the NPs centers. For simulation we considered light polarized parallel to the interparticle axis. The maps represent the distribution of logarithmic values of

electrical field intensity normalized to the incident field 20E considered equal to unity ( ( )2

02log EE ).

Figure 7. Maps of electric field enhancement ( ( )2

02log EE ) in the X0Y plane for chains composed

of different number of NPs. A cross-section representation of field enhancement is given for each map.

As we can see in Figure 7, the highest field intensity is always located in the middle of the

chain, either in the central gap in the case of chains composed of even number of spheres or at the interstices between the central sphere and its two neighbors for chains composed of odd number of spheres. This result can be explained taking into consideration the differences between the electrical field superposition of the central gap and its left neighbor for example for an even number of spheres in the chain. The distances between the central gap and all the gaps on one side of the chain are (d+D), 2(d+D) … (n/2)(d+D) where d is the interparticle distance, D is the diameter of the sphere and n is the number of spheres in the chain. Due to the symmetry conditions the same calculation is valid for the other side of the chain as well. In contrast, the first gap to the left is separated by the following distances (d+D), 2(d+D) … (n/2-1)(d+D) to the gaps on the left side of the chain and respectively (d+D), 2(d+D), … (n/2)(d+D), (n/2+1)(d+D) to the gaps on the right side of the chain. Comparing these two gaps we find that the difference between them is that when we calculate the total superposition of the electric field at their position, the central gap will have a contribution of an electrical dipole formed at a distance of (n/2)(d+D) while the left gap neighbor will have a contribution of a dipole formed at a distance of (n/2+1)(d+D). Due to this distance difference the total superposition will have a maximum at the central gap. This reasoning can be applied to any chain length of odd or even number of spheres. The electric field of the source induces oscillations of the free electric charges (electrons) in the material, thus the sign of charges alternates over the structure and the interaction increases when NPs get close to each other. The electrical dipole oscillates in the opposite direction relative to electrical field of the source. We find in each case a total electrical field several orders of magnitude higher at these particular locations relative to the rest of the structure. That is why these points are known in literature as “hot spots” [19]. It is supposed that “hot spots” exhibit the highest field intensity for a particular gap size at the excitation wavelength which corresponds to the maximum of the extinction spectrum [35].

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)

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Figure 8 (A) Plot of logarithmic intensity maximum of the normalized electric field as function of the

number of spheres in the chain. The spheres are of 20 nm diameter and separated by 0.5 nm. The black squares are for water (n=1.33) and the red triangles are for air (n=1) as surrounding media. (B) The imaginary part of the electric permittivity of gold (black line – Johnson and Christy) and

fitted by the FDTD program (red line). Also depicted are the LSPR positions for chains of different number of NPs in water (black vertical dotted lines) and in air (blue vertical dotted lines).

Figure 8A presents a plot of the maximum enhancement factor ( ( )20

2log EE ) for the intensity of the electric field as function of different number of spheres. We can observe that the maximum enhancement is reached for a chain composed of 4 NPs. This can be explained taking into consideration that the minimum of damping coefficient for gold coincides with LSPR wavelength for the chain of 4 sphere chain (655nm) as can be seen in figure 8B. For comparison we considered the case of chains in a surrounding medium with refractive index n=1 (air). Unlike chains in water, in the case of air the field enhancement increases toward a saturation level. For an infinite number of spheres the saturation level in water is the smallest of all chain lengths simulated whereas for air it is the highest. This observation can be attributed to the fact that the extinction peak for the infinite chain of spheres in air is at 620 nm, where the damping function is still decreasing at this wavelength.

Figure 9. Maps of logarithmic distribution of the electric field intensity surrounding chains of 3 gold

nanospheres of 20 nm diameter in a medium with n=1.33 for different interparticle distances: 0.5, 1, 2, 5, 10, 15, 20 nm. All plots are calculated at the LSPR wavelength.

A similar study for somewhat larger particles (40 nm diameter) has been done in literature using a Finite Integration Technique (FIT) method and authors found that the maximum enhancement is obtained for a chain of 8 particles [36]. They also found that in the case of silver chains the highest enhancement is obtanied for dimers because the LSPR wavelength of the dimeric structure corresponds to the smallest value of the damping coefficient.

Figure 8 depicts several examples of logarithmic distributions of the electric field intensity in the case of trimers with different separation distances between NPs. We observe that for d=0.5 nm a

value of 6.420

2 10≅EE is obtained which provides an electromagnetic amplification of about 109 as SERS substrate, which is promising for applications.

4. Conclusions The present paper analyzes the far- and near field optical properties of chain-like structures as function of the number of NPs and the interparticle distance. FDTD simulations indicate a strong coupling effect in chains for a polarization parallel to the axis uniting the spheres centers. This effect is characterized by a red shift of the LSPR peak with the increase of the number of spheres in the chain, which is in very good agreement with the experimental UV-Vis extinction data of linear assemblies of citrate capped gold nanoparticles. An empirical formula for direct calculation of the LSPR position as function of number of particles in the chain has been demonstrated. The highest field enhancement position was found at the center of the chain (for even number of spheres) or surrounding the center sphere (for odd number of spheres). Furthermore, the highest field enhancement factor was obtained for a chain composed by 4 NPs and explained by the coincidence between the wavelength of LSPR and the minimum of damping coefficient for gold. With the increase of particle distance the field enhancement decreases and the extinction peaks blue shift and decrease in intensity until reaching the limit of the isolated sphere case.

Understanding the dependence of collective plasmonic properties on the number and distance between NPs in linear chains offers useful design rules for the development of SERS-based sensing applications.

5. Acknowledgments This study was supported by CNCSIS–UEFISCDI (Romania) under the project number PNII-ID PCCE 129/2008. C.A. Tira acknowledges the performance scholarship from Babes-Bolyai University, Cluj-Napoca, Romania. T. Simon gratefully acknowledges the financial support from Babes-Bolyai University, project number GTC_34015/2013.

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Graphical abstract

Highlights

• FDTD analysis of plasmonic properties of chain like assembly of GNPs;

• Empirical formula for LSPR band of chain like assembly of GNPs; • Useful rules for designing highly effective SERS-active substrates;