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A. Akouibaâ, M. Benhamou, A. Derouiche, F. Benzouine, H. Ridouane, A.R. Senoudi, A. Boussaid.

Numerical study of the plasmonic resonance of multi-phases nanoparticles. International Journal of Academic Research Part A; 2012; 4(6), 213-223. DOI: 10.7813/2075-4124.2012/4-6/A.29

NUMERICAL STUDY OF THE PLASMONIC RESONANCE OF MULTI-PHASES NANOPARTICLES 1

1,2

1

1

A. Akouibaâ , M. Benhamou , A. Derouiche , F. Benzouine , 1 3 3 H. Ridouane , A.R. Senoudi , A. Boussaid 1

Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences Ben M’sik, 2 3 Casablanca, Maroc, ENSAM, Université Moulay Ismaïl, Meknès, Maroc, Laboratoire de Recherches sur les Macromolécules Faculté des Sciences, Tlemcen, Algérie (MOROCCO) benhamou.mabrouk@gmail.com DOI: 10.7813/2075-4124.2012/4-6/A.29 ABSTRACT In this study, use is made of finite element method (FEM) for the calculation of the complex effective permittivity of the silica-core nanoparticles coated with gold or gold/polymer shells. Such a permittivity allows the determination of the absorption spectrum of these clothed nanoparticles. Several plasmonics structures have dimensions much smaller than the wavelength of the incident light. Under these conditions, the retardation effects are negligible and the field distribution problem then reduces to solve the Laplace's equation. A simple scheme based on FEM is developed, which enables us to compute the optical properties, such as the effective dielectric function and absorption cross-section of silica-nanoparticles coated with gold or gold/polymer layers, which are embedded in a dielectric matrix. In particular, calculations reveal the capital role played by the polymer-shell that shifts the Surface Plasmon Resonance peak towards the infrared domain and increases the absorption amplitude. Finally, our numerical results are compared to those obtained using the so-called Maxwell Garnett theoretical model.

Key words: Silica-core nanoparticles, Gold-shell, Polymer-shell, Optical properties, Finite element method 1. INTRODUCTION The gold nanoparticles [1,2] have attracted significant interest as a novel platform for the nanobiotechnology and nanomedicine [3,4], because of the convenient surface bioconjugation with molecular probes and remarkable optical properties related with the Surface Plasmonic Resonance (SPR). The mechanism is due to the collective oscillations of their conductive electrons in response to an optical excitation [5-8]. SPR frequency of the metal nanoparticles depends on their size [9-11] and shape [12,13], dielectric properties [14], aggregate morphology [15], surface modification [16], and refractive index of the surrounding medium [17].

Recently, various spherical core-shell nanostructures, in which silica is used as a core, shell or medium for nanoparticles encapsulation, are synthesized [18]. In practice, there are many more possible solutions of the coreshell nanostructures based on the use of different shapes (nanorings, nanorods, and nanostars), and on the arrangement of the nanostructure components.

The core-shell nanospheres composed of a nanosphere-core and a noble metal-shell, have been intensively studied in the field of nanooptics, especially in regard to their SPR characteristics [19,20]. The SPR of core-shell nanoparticles (CSNs) can be tuned from visible to infrared by varying the ratio of the outer and inner radii of the metal shell [21]. The optical properties of CSNs have been investigated, for various synthesized cores and shells [22-25]. In particular, CSNs with finely tunable near-infrared absorption can be utilized in biomedical and bioimaging devices, for cancer therapy [26-29]. Silica, whose chemical formula is SiO₂, is an insulating dielectric material and transparent in the visible (about 400nm to 800nm). Its high transmittance, ease of fabrication by different techniques, the thermal stability and environmental as well as its availability, and therefore its low cost, make of the silica a material of choice for a wide range of applications. SiO₂ is widely used in the optical applications and is, in most cases, the basic material employed in the manufacture of optical systems.

Generally, particles like rods or spheres are coated by a polymeric shell, typically a thiolated PEG (poly ethylene glycol) or a PPG (poly propylene glycol). These last renders the particles biocompatible and stable. In addition, these polymers may be used as an intermediate for attaching active substances to the surface of these particles.

In order to simulate the plasmon behaviors, several methods were developed including the finite-difference time-domain method [30], Fourier pseudo-spectral time-domain method [31], multiple multipole method [32], volume integral equations with dyadic Green's function [33], boundary-element method [34-36], and surface integral method [37], etc. The effective medium theory (EMT) is a powerful way to handle the optical properties [38] of the composite materials, and the most popular EMT is the Maxwell Garnett theory (MGT) [39] (mixing rule). However, this theory often claimed that a weak particle interaction is a condition for the validity of MGT.

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Numerical techniques are designed to solve the relevant filed equation in the computational domain, subject to the boundary constraints imposed by the geometry. Without making a priori assumption about which the field interactions are most significant, the numerical techniques analyze the entire geometry provided as input. FEM [40] that is a powerful numerical tool, has been widely used for modeling the electromagnetic wave interaction with complex materials. Recently, Senoudi and coworkers [41] have used this method to study the effects of the polymeric shell on the plasmonic resonance of nanogolds (of arbitrary geometry).

The main purpose of this paper is to present various numerical results (in the quasi-static limit) of the effective permittivity and absorption cross-section computed by FEM, for a periodic composite structure in which the inclusions are nanospherical silica-core covered with a gold shell. The calculated data are compared with MGT rule.

The effects of the silica-core radius and the gold-shell thickness (interphase) are then observed. We also present numerical results in the case where the silica-core/gold-shell nanoparticles are coated with a polymeric layer (interphase). The effects of the polymeric layer thickness and its dielectric permittivity are also studied.

The remainder of the paper is organized as follows. Useful backgrounds are presented in sections 2 and 3.

In the first, we give a brief review of some important properties of the effective medium approaches, some classical mixing rules and gold dispersive function, and present a geometric description of the studied structure. Formulae for calculating the optical properties like effective permittivity and absorption cross-section for CSNs are recalled in section 3. We describe, in section 4, the used finite element methodology and computational aspects. Discussion of results is the aim of section 5. Finally, some concluding remarks are drawn in the last section.

2. EFFECTIVE MEDIUM THEORY 2.1. Effective dielectric constant Several theory have been developed to describe the optical properties of the metal inclusions in a dielectric matrix, among these, we can quote the popular one that is MGT [39]. Such a theory allows the calculation of the effective dielectric function of a dielectric medium, in which several metal spheres are dispersed, provided that the volume fraction of the nanoparticles is not very high, in order to neglect the interaction between inclusions.

In some cases, the volume concentration of the nanoparticles is too high. For their description, the used approach is the so-called effective medium approximation. The latter models the dielectric constant of a nanocomposite in quasi-static limit, where the diameter of the incorporated particles, , is very small compared to the lightwavelength, , that is ≪ . In this limit, the light scattered by the particles can be ignored. The effective dielectric permittivity, , must satisfy the equation of the electric displacement [42] ⃗=

⃗

=

⃗

+ ⃗.

(1)

Here, ⃗ is the macroscopic polarization vector, which can be related to the polarizability of the inclusions by the following relationship ⃗=

⃗,

(2)

where is the dielectric permittivity of the medium free from inclusions. There, ⃗ and ⃗ are the internal and local electric fields, respectively. These are not independent each form other and satisfy ⃗ = ⃗

⃗.

+

(3)

The second part on the right-side of the above relation accounts for the depolarization term, where is the depolarization tensor that depends on the geometric shape of inclusions, and is their number density.

Combining relations (1), (2) and (3) yields the expression of the effective dielectric permittivity. In the case of an anisotropic medium with space randomly oriented nanoparticles, the effective permittivity is given by =

1+

, = ( , , ),

(4)

where and are the polarizability and depolarization factor of a nanoparticle in the space-direction , respectively. In the case of spherical nanoparticles, the polarizabilities are equal along the three axes, and we have: = = ≡ and = = = 1/3.

Now, to calculate the effective permittivity defined in relation (4), one must know the expression of the polarizability of each inclusion, by solving the Laplace's equation in the quasi-static limit. Analytical expressions were obtained with simple geometric shapes (sphere or ellipsoid) in the case of low-density inclusions (dispersed phase) [40]. In this work, however, we choose to model the optical properties of spherical nanoparticles with two and three phases.

2.1.2. Silica-core/Gold-shell nanoparticles Consider now a low-density assembly of spherical nanoparticles that are silica-cores of radius and permittivity , coated each with a gold-layer of thickness and permittivity . We suppose that these particles are immersed in a medium of dielectric permittivity . Then, the total radius of the nanoparticle is = + (figure 1). The volume and the volume fraction of inclusions are denoted by and , respectively.

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In this case, the polarizability of inclusions writes [43] (

=3

(

)(

) (

)(

)

)(

)

(

)(

)

,

(5)

with the volume fraction of the shell = / . In the monodisperse case, that is when the particles have the same mass and the same volume, the volume fraction of inclusions is given by: = / , where represents the total number of inclusions and is the total volume of the medium. Therefore, the volume fraction of inclusions is given by = , (6) where

is their number density.

Formulae (4), (5) and (6), together, lead to the desired expression of the effective permittivity (nanoparticles with two phases) =

1+3

,

)(

+2 )+ (

(7)

with the notations =( − =

(1 − )[

+2 +2 (

− )] +

− )(

+ 2 ),

(2 + )[

(7a)

+2 − (

− )],

(7b)

Notice that the above expression is valid only in the case of monodisperse inclusions of low volume fraction.

In the case of nanoparticles without shell, that is = 1 and = , we recover that formula obtained in the framework of MGT [39].

2.1.3. Silica-core/Gold and Polymer-shells Start with spherical nanoparticles, of silica-core radius and permittivity , which are coated each with a first layer formed by a gold-layer of thickness and permittivity , and a second one that is a polymer-layer of thickness and permittivity . These particles are assumed to be immersed in a dielectric medium of permittivity . In this case, the total radius of a nanoparticle is: = + + (figure 2). The volume and the volume fraction of inclusions are denoted by and , respectively.

In this case, the polarizability of each nanoparticle is given by the following formula [44] =3

,

(8)

with the notations =

−

=

+ 2

+2

(

+

+2

) ( (

−

(

/

)

(

)( (

)

)(

/

(

)

(

) )

)( )(

/

) )

,

(9)

/

/

.

(10)

/

We have used the notations /

=

,

/

=

(

)

,

/

=(

)

.

(11)

Formula (5) is then recovered for a single coating (in the absence of polymer-shell). Combining Eqs. (8) to (11) gives the polarizability of nanoparticles coated with two phases (god/polymer-shells). We also recover those formulas obtained by MGT corresponding to the situation of nanoparticles without coating. In this sense, the obtained expression for the nanoparticle polarizability is more general.

2.2. Dielectric function of nanogolds The optical response of a metal is quantified by its dielectric constant, which is a function of frequency of an electromagnetic wave that interacts with the material. Indeed, the interaction of an electromagnetic wave (with a pulse ) with a metal will lead to the polarization of the medium. Then, such a polarization shall generate a change in the complex refractive index, ( ), which can be related to the dielectric constant by the following relationship ( ) = ( ).

(12)

In the case of metal nanoparticles, the effects of a size reduction on the electronic properties appear when their diameter are small-enough compared to the mean free path of electrons. From a classical point of view, this corresponds to the fact that the electron-surface collisions are not negligible compared to the other interactions (electron-electron, electron-phonon) processes, and must be taken into account in the rate of the optical collision of electrons.

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To study the interaction of an electromagnetic wave with gold nanoparticles, several models were developed in order to express the dependence of the dielectric constant on the wave-frequency. Among these, we can cite the phenomenological Drude [45-49] and Drude-Lorentz [50] models. Each has its limit validity. For instance, the first ignores the inter-band transitions, although these same transitions occur when the incident photon energy exceeds a certain threshold. The latter model describes (in reasonable way) the dielectric function of gold, only in a frequency-band between 0.5 and 3.5 . We recall that the use of the so-called two critical points Drude (TCPD) model [51], for the description of the permittivity of gold in a wide-frequency-band, is reliable only when the frequency-band is in the interval 0.5 to 6.5 corresponding to wavelengths between 200 and 2400 . In this model, the permittivity expression of the gold nanoparticles is =

∞

+∑

−

Ω

,

Ω

Γ

+

Ω

Γ

,

(13)

with = −1 ( is the pure imaginary number). Here, ∞ is the permittivity at infinite frequency, being the frequency of the volume plasmon, is the collision rate of the electrons, and the third part in the right-side of the above equality represents the Lorentz term.

3. SURFACE PLASMON RESONANCE OF NANOSTRUCTURES From the evolution of the effective complex dielectric function depending on the wavelength (or frequency) of the incident field, the resonance modes that may occur in nanoparticles are identified. For this, it would be interesting to calculate the scattering cross-sections and absorption. The sum of these two quantities defines the extinction cross-section = + .

(14) In the case where the dimensions are very small compared to the wavelength, the light scattering can be ignored, and we have: ≈ .

The cross-section of extinction (absorption) can be determined from the imaginary part of the effective dielectric function of the composite using the following equation [51] ′′

=

.

(15)

Here, is the volume of a nanoparticle, is the volume fraction of nanoparticles, is the amplitude of the wave-vector of the electromagnetic wave, and represents the refractive index that can be related to the real and imaginary parts, ′ and ′′ , of the effective permittivity by [52] / ′

′′

′

=

.

(16)

Therefore, the position of the optical resonance peak of these nanostructures is predicted from the effective medium theory through the imaginary part of the effective permittivity, which can be calculated from FEM. This will be the aim of the next section.

4. FINITE ELEMENT METHODOLOGY Given the progress achieved in the power of computer processing, understanding the dielectric behavior of heterogeneous structures has stimulated many theoretical studies based on numerical calculations, especially in the case where the inclusions are randomly dispersed in the matrix [53-56]. For this particular case, the laws governing the dielectric behavior of the composite materials are deduced [53]. In order to study some details of the dielectric properties of the periodic composites, in the quasi-static limit, FEM is used in an objective to determine the effective permittivity. A detailed description of the method can be found in the literature [55]. We consider a parallel plate capacitor, as shown in figure 3, which is formed by two metal plates of area = × and separated by the height . The two plates are submitted to a potential difference − .

Solving the problem at hand means finding the local potential distribution inside the unit cell volume by solving the first principle of electrostatics, namely the Laplace's equation ∇⃗.

( ⃗)∇⃗ ( ⃗) = 0,

(17)

where ∇⃗ is the first derivative with respect to the position-vector, ⃗, of the representative point of the medium, ( ⃗) and are the local relative permittivity and the potential distribution inside the spatial domain (with zero charge density). There, = 8.85 × 10⁻¹² ⁻¹ is the permittivity of vacuum. The electrostatic energy, , and losses, , can be expressed in terms of the potential derivatives for each tetrahedral element by =

∭

′

( , , )

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+

+

,

(18)

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH =

∭

′′ ( , , )

Vol. 4. No. 6. November, 2012 +

+

,

(19)

where ( ) and represent the permittivity and the volume of the tetrahedron element . The phases of the composite are described through their complex permittivity, that is ( ) = ′ + ′′ .

Periodic (Neumann) boundary conditions / = 0 are enforced on the faces perpendicular to and directions. Consequently, the edge fringing effects can be eliminated, and the effective permittivity of the composite can be determined from the energy and losses stored in the capacitor as = =

′

( ′′

− (

−

) , ) .

(20) (21)

All results are obtained by the commercial finite element solver COMSOL-MULTIPHYSICS installed on a personal computer.

Once we have established the mathematical equation that defines the physics of the system, i.e. Eq. (17), we must point-out how to solve this equation for the interested domain, with the appropriate boundary conditions. In FEM, the domain is decomposed into a number of uniform or non-uniform finite elements that are connected via nodes. For each subspace element, the function modeling the potential is defined by a polynomial interpolation function. Then, the boundary values are replaced by an equivalent integral formulation. The interpolation polynomials are then replaced into the integral equation and integrated in the interested domain. Finally, the results of this step are converted into matrix equation, which is afterward solved for [56,57].

5. RESULTS AND DISCUSSION Two structures of the core-shelled nanoparticles will be studied and analyzed quantitatively by FEM: (1) spherical silica-cores covered with a gold-shell, and (2) spherical silica-cores covered with two successive shells (gold and polymer). In the two cases, the clothed nanoparticle is embedded in a surrounding medium. The permittivity of gold (function of frequency) will be described within the framework of TCPD model. In the following, the dependence of SPR on the silica-core radius, thicknesses of gold and polymer layers, polymer nature, will be discussed systematically.

To verify the precision of the used FEM, we report, in figure 4, the evolution of the absorption cross-section upon the photon energy, for two different nanostructures, which are a spherical silica-core of radius = 10 , covered with a gold-shell of thickness = 4 , and a gold nanosphere (without shell) of radius = 14 . For these curves, the dielectric function of the host medium, , of the surrounding medium and the inclusion volume fraction, , are fixed to the respective values: = 2.25 and = 0.1. The curves show that the peak of SPR appears at the photon energy = 2.28 ( = 544 ), for the gold nanosphere, and = 1.86 ( = 668 ), for SGNs. For the latter, we remark a sharp increase in the amplitude of the resonance peak and a decrease in its width. This curve also shows that, there is a good agreement between numerical results and analytical ones.

Validation of FEM allows the application to the detection of the optical resonance via the absorption crosssection. The fundamental aim of the next step is to present a numerical study on the influences of geometrical and physical parameters of the core-shells nanoparticles on the location, amplitude and width of the optical resonance peak.

Now, consider homogeneous spherical silica-cores that are uniformly coated with a gold-shell and placed into some dielectric medium. The dielectrics functions, and , of the surrounding medium and silica-core, and the inclusion volume fraction, , are fixed to the values: = 2.25, = 3.9 and = 0.1 (see figure 2).

In figure 5, we depict the variation of the absorption cross-section calculated by FEM upon photon energy, for different values of the silica-core radius by fixing the total radius of the nanoparticle to the value = 18 . This curve shows that, when the radius of the silica-core increases from = 10 to = 14 , together with the thickness of the gold-shell that decreases from = 8 to = 4 , the position of the energy resonance-peak moves from = 2.1 ( = 591 ) to = 1.7 ( = 730 ). The conclusion is that, an increase of the silicacore radius and a decay of the gold-shell thickness indicate that the peak-position is shifted towards the infrared photon energy. Also, we remark a considerable increase in the resonance-peak height.

To examine the effect of the thickness of the metal-shell, we fix the silica-core radius to the value = 10 , and we report, in figures 6a, 6b and 6c, the real and imaginary parts of the effective permittivity and the absorption cross-section versus the photon energy, for different values of the thickness of the gold-shell. These curves shows that, when the thickness of the metal-shell increases, the optical resonance peak moves to the visible wavelengths, while its amplitude increases considerably. We note a decrease in the real and imaginary parts of , when the gold-shell thickness is augmented.

Gold nanoparticles are very useful in biology and medicine; they are used in many applications: therapeutic, biomedical, biological imagining, etc. Indeed, it is the optical properties and biocompatibility of the gold that count.

To improve these properties, these nanoparticles are often coated with a polymer-layer. The latter can also increase the stability of these nanoparticles and to fix active substances on their surfaces. In what follows, we will study the effects of coating of the nanoparticles with polymers.

For this, we consider a nanoparticle formed by a silica-core of radius = 10 and dielectric permittivity = 3.9, coated with a gold-layer of thickness ₁, and a polymer-layer of thickness ₂, and (polymer) permittivity = 3.2. This particle is emerged in a dielectric medium of permittivity = 2.25, with a volume fraction = 0.1.

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In figure 7, we report the evolution of the absorption cross-section upon the photon energy, choosing higher values of the polymer-layer thickness, which are ₂ = 0 , ₂ = 4 and ₂ = 6 . With these values, the positions of SPR peak appear at the respective photon energies: = 2.02 , = 1.98 and = 1.96 corresponding to the wavelengths = 615 , = 627 and = 639 . The conclusion is that, when the polymer-shell thickness is increased, the peak position is shifted towards the infrared photon energy. Also, we remark that, as the polymer-shell thickness is augmented, the intensity of the absorbed light increases. The same effect can be observed, when the permittivity of the polymer-shell increases, as can be seen in figure 8.

Finally, figure 9 shows the absorption spectra of a core-shell nanoparticle (Silica/Gold/Polymer) for three values of the gold-shell thickness: ₁ = 4 , ₁ = 6 and ₁ = 8 . For theses curves, the silica-core radius and the polymer-shell thickness are fixed to the respective values: = 10 and ₂ = 4 . Theses curves show that, when the thickness of the gold-shell increases, the peak of the SRP is shifted towards the visible wavelengths, whereas the amplitude of the absorption cross-section substantially increases.

Fig. 1. Nanoparticle with two phases

Fig. 2. Nanoparticle with three phases

Fig. 3. Schematic representation of a three-dimensional composite medium by a cubic cell with length .

This cell contains a single inclusion. On the boundary of each cell, the potential is on the top face, and on the bottom one, and its normal derivative, / , is equal to zero along the vertical walls

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Fig. 4. Absorption cross-section versus the photon energy, for two spherical nanostructures: gold (curve with open circles) and silica-core/gold-shell, (curve with solid circles).

These curves are drawn choosing: = 14 , = 0.1, = 2.25.

Points indicate numerical results and the solid line describes the analytical ones from MGT

Fig. 5. Numerical results of the absorption cross-section for several values of the silica-core radius, , and gold-shell thickness, . The total radius of the nanoparticle is fixed to the value = 18 .

For these curves, the chosen parameters are: = 2.25, = 3.9, = 0.1

Fig. 6a. Evolution of the real part of the effective permittivity for different gold-shell thicknesses, , with fixed parameters: = 10 , = 2.25, = 3.9, = 0.1

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Fig. 6b. Evolution of the imaginary part of the effective permittivity for different gold-shell thicknesses, , with fixed parameters: = 10 , = 2.25, = 3.9, = 0.1

Fig. 6c. Evolution of the absorption cross-section for different gold-shell thicknesses, , with fixed parameters: = 10 , = 2.25, = 3.9, = 0.1

Fig. 7. Absorption cross-section for different values of the polymeric-shell thickness, with fixed parameters: = 2.25, = 3.9, = 3.2, = 6 , = 10 , = 0.1

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Fig. 8. Absorption cross-section for three values of the polymeric-shell permittivity , with fixed parameters: = 2.25, = 3.9, = 6 , = 4 , = 10 , = 0.1

Fig. 9. Absorption cross-section for different values of the gold-shell thickness, , with fixed parameters: = 2.25, = 3.9, = 3.2, = 4 , = 10 , = 0.1

6. CONCLUDING REMARKS In conclusion, we have established a series of 3D-simulations by finite element method for determining the optical properties of the core/shelled nanoparticles with two and three phases that are embedded in a host dielectric medium. For the silica-core/gold-shell nanoparticles, we saw that the optical properties strongly depend on the silica-core radius and gold-shell thickness. Confrontation with analytical models showed that the results agree very well for small particle fractions, for which the analytical models are recognized as valid, and that, MEF provides a powerful tool for the study of the composite systems with large particle fractions.

The presence of silica-core has many consequences. As a matter of fact, this silica-core increases strongly the light adsorption and leads to a diminution of the effective index of the medium, in comparison with gold nanoparticles.

For the silica-core/gold and polymer-shells nanoparticles, we numerically studied the influence of each parameter (taken separately). In particular, we have shown the crucial role played by the polymer and gold-shells and the dielectric permittivity of the polymer-layer. Indeed, the variation of these parameters gives arise to drastic changes of the position and amplitude of the optical resonance peak.

As last word, we emphasize that the numerical method developed in this paper, for the investigation of the optical properties of the clothed nanogolds, can be extended to other types of gold nanoparticles, with different sizes and shapes, such as nanorods, nanoshells, nanocages and nanotips, in order to improve the optical properties suitable for the biomedical applications. Such considerations are in progress.

ACKNOWLEDGMENTS Authors M.B., A.D. and F.B. would like to thank the Tlemcen University of Algeria, where a part of this work has been done, for their kind hospitality during their regular visits.

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