Single Silver Nanoparticles: Local Refractive Index Response to Localized Surface Plasmon Resonance and Molar Attenuation Coefficient

Abstract

In this paper, we analyze the optical properties and sensitivities of spherical silver nanoparticles (AgNPs) with varying diameters (10–60 nm) in different surrounding medium, including water, ethanol, and chloroform. The investigation focuses on analytically assessing scattering efficiencies (${{Q}_{{{\text{sca}}}}}$), absorption efficiencies (${{Q}_{{{\text{abs}}}}}$), total extinction efficiencies (${{Q}_{{{\text{ext}}}}}$), shedding light on how the properties of AgNPs change with the size and surrounding medium of AgNPs. The results show that smaller AgNPs exhibit sharper plasmon resonance peaks at shorter wavelengths, while larger AgNPs display broader peaks in the visible spectrum, showcasing the size-dependent behavior of AgNPs. Additionally, the study calculates the molar attenuation coefficient (ε) and extinction cross section (${{\sigma }_{{{\text{ext}}}}}$) for AgNPs, highlighting the differences in absorption properties between small and large particles and their sensitivity to the surrounding medium. The research also discusses the implications of these properties for applications such as plasmonic sensing and sensor design, emphasizing the importance of particle size and surrounding medium in optimizing sensor performance. Moreover, the refractive index sensitivity (S) and the figure of merit (FOM) are introduced as critical parameters for assessing sensor performance, revealing their relationship with particle size, and providing valuable insights into sensor design and optimization.

APPENDIX A

MODELING

Electromagnetic scattering by a homogeneous sphere was referred to as Mie scattering theory. The distribution by spherical NPs was solved exactly regardless of the size of the NPs as long as the resonance of NPs near the frequency of the incident wavelength by Gustava Mie [14]. According to the Mie scattering solution, scattering S₁(θ) and extinction S₂(θ) coefficients were expanded using spherical functions for a homogeneous plane wave [28–32].

$${{S}_{1}}\left( \theta \right) = \mathop \sum \limits_{n = 1}^\infty \frac{{2n + 1}}{{n\left( {n + 1} \right)}}\left( {{{a}_{n}}{{\pi }_{n}}(\cos {\kern 1pt} \theta ) + {{b}_{n}}{{\tau }_{n}}(\cos {\kern 1pt} \theta )} \right),$$

(1A)

$${{S}_{2}}\left( \theta \right) = \mathop \sum \limits_{n = 1}^\infty \frac{{2n + 1}}{{n\left( {n + 1} \right)}}\left( {{{a}_{n}}{{\tau }_{n}}(\cos {\kern 1pt} \theta ) + {{b}_{n}}{{\pi }_{n}}(\cos {\kern 1pt} \theta )} \right).$$

(2A)

Here S₁(θ) and S₂(θ) are related to scattering and extinction coefficients (Q_sca & Q_ext), respectively. The functions ${{\tau }_{n}}$ and ${{\pi }_{n}}$ are related to the Legendre polynomials ${{P}_{n}}$ as follows:

$${{\pi }_{n}}(\cos {\kern 1pt} \theta ) = \frac{{P_{n}^{1}\cos {\kern 1pt} \theta }}{{\sin {\kern 1pt} \theta }} = \frac{{d{{P}_{n}}\cos {\kern 1pt} \theta }}{{d{\kern 1pt} \cos {\kern 1pt} \theta }},$$

(3A)

$${{\tau }_{n}}(\cos {\kern 1pt} \theta ) = \frac{{P_{n}^{1}\cos {\kern 1pt} \theta }}{{d\theta }} = \cos {\kern 1pt} \theta {{\pi }_{n}} - {{\sin }^{2}}\theta \frac{{d{{\pi }_{n}}\cos (\theta )}}{{d{\kern 1pt} \cos (\theta )}}.$$

(4A)

The standard evaluation of these representations can be found elsewhere (2,3). ${{a}_{n}}$ and ${{b}_{n}}$ are the expressions of Mie coefficients. The assumptions were made that the refractive index of the nanoparticles is ${{n}_{p}}$, and the refractive index of the surrounding medium is ${{n}_{m}}$. Then the coefficients embedded in the refractive indexes are given by the following [27–29]:

$${{a}_{n}} = \frac{{{{n}_{m}}\psi _{n}^{'}\left( y \right){{\psi }_{n}}\left( x \right) - {{n}_{p}}\psi _{n}^{'}\left( x \right){{\psi }_{n}}\left( y \right)}}{{{{n}_{m}}\psi _{n}^{'}\left( y \right){{\zeta }_{n}}\left( x \right) - {{n}_{p}}{{\psi }_{n}}\left( y \right)\zeta _{n}^{'}\left( x \right)}},$$

(5A)

$${{b}_{n}} = \frac{{{{n}_{p}}\psi _{n}^{'}\left( y \right){{\psi }_{n}}\left( x \right) - {{n}_{m}}{{\psi }_{n}}\left( y \right)\psi _{n}^{'}\left( x \right)}}{{{{n}_{p}}\psi _{n}^{'}\left( y \right){{\zeta }_{n}}\left( x \right) - {{n}_{m}}{{\psi }_{n}}\left( y \right)\zeta _{n}^{'}\left( x \right)}}.$$

(6A)

The primes “'” indicated to the derivatives. ${{\psi }_{n}}$ and $\zeta _{n}^{'}\left( x \right)$ are spherical functions known as Riccati–Bessel functions, which are related to the spherical Bessel functions $z{{j}_{n}}\left( z \right)$ and ${{\zeta }_{n}}\left( z \right)$ as follows:

$${{\psi }_{n}}\left( z \right) = z{{j}_{n}}\left( z \right),$$

(7A)

$${{\zeta }_{n}}\left( z \right) = z\left( {{{j}_{z}}\left( z \right) - i{{y}_{n}}\left( z \right)} \right).$$

(8A)

The argument z indicates that it is an arbitrary complex number, and the arguments x and y are used for Mie coefficients related to the particle size (x) and surrounding medium (y). They defined as follow:

$$X = \frac{{2\pi r}}{\lambda }{\text{ and }}Y = \frac{{2\pi r}}{\lambda }.$$

(9A)

Here, λ is the wavelength in vacuum, and r is the radius of the NPs. In the case of the surrounding medium is absorbed, ${{n}_{m}}$ has a real and imaginary part of the refracctive index. Therefore, x is a complex function. Consequencelly, the spherical functions (Riccati–Bessel functions) become unstable because these functions entered the exponential domain and passed out the limit of boundaries. The Eqs. (5A) and (6A) were rewritten to fit with these boundary conditions as follows:

$${{a}_{n}} = \frac{{{{n}_{m}}{{\psi }_{n}}\left( x \right){{A}_{n}}\left( y \right) - {{n}_{p}}{{A}_{n}}\left( x \right)}}{{{{n}_{m}}{{\zeta }_{n}}\left( x \right){{A}_{n}}\left( y \right) - {{n}_{p}}{{B}_{n}}\left( x \right)}},$$

(10A)

$${{b}_{n}} = \frac{{{{n}_{p}}{{\psi }_{n}}\left( x \right){{A}_{n}}\left( y \right) - {{n}_{m}}{{A}_{n}}\left( x \right)}}{{{{n}_{p}}{{\zeta }_{n}}\left( x \right){{A}_{n}}\left( y \right) - {{n}_{m}}{{B}_{n}}\left( x \right)}},$$

(11A)

$${{A}_{n}} = \frac{{\psi _{n}^{'}}}{{{{\psi }_{n}}}}{\text{ and }}{{B}_{n}} = \frac{{\zeta _{n}^{'}}}{{{{\zeta }_{n}}}}.$$

(12A)

The optical properties of spherical AgNPs were presented in terms of their calculated scattering, extinction, and scattering efficiencies by Mie scattering theory represented in the calculated method section for the homogenous sphere. The incident plane wavelength was set to be in the visible range at 390 nm, where most spherical AgNPs have LSPR around 390 nm. The required parameters for the calculations are the radius (r), the refractive index of the surrounding medium of the NPs, which is n_p = 1.33, 1.36, and 1.45 for water, ethanol, and chloroform, respectively, as well the refractive index of the vacuum, which is n_m = 1. Equations (1A) and (2A) can be simplified as follow:

$${{Q}_{{{\text{sca}}}}} = \frac{2}{{{{x}^{2}}}}\mathop \sum \limits_{n = 1}^\infty \left( {2n + 1} \right)\left( {a_{n}^{2} + b_{n}^{2}} \right),$$

(13A)

$${{Q}_{{{\text{ext}}}}} = \frac{2}{{{{x}^{2}}}}\mathop \sum \limits_{n = 1}^\infty \left( {2n + 1} \right)\mathcal{R}\left( {{{a}_{n}} + {{b}_{n}}} \right),$$

(14A)

$${{Q}_{{{\text{ext}}}}} = {{Q}_{{{\text{sca}}}}} + {{Q}_{{{\text{abs}}}}}.$$

(15A)

The efficiences described above can be translated to the corresponding cross sections by multiplying Q_{abs,sca,and ext} with cross sectional areas (πr²) of spherical AgNPs. Hence, the corresponding cross section of absorption, scattering, and extinction can be written as follows:

Fig. A1.

The calculated FWHM with respect to the diameter of AgNPs the exponential growth fitting of FWHM.

Fig. A2.

The calculated FOM with respect to diameter of AgNPs and the linear decreased fitting of FOM.

$${{\sigma }_{{{\text{abs}}}}} = \pi {{r}^{2}}{{Q}_{{{\text{abs}}}}},$$

(16A)

$${{\sigma }_{{{\text{sca}}}}} = \pi {{r}^{2}}{{Q}_{{{\text{sca}}}}},$$

(17A)

$${{\sigma }_{{{\text{ext}}}}} = \pi {{r}^{2}}{{Q}_{{{\text{ext}}}}}.$$

(18A)

In brief, the calculations pertaining to Q_sca and Q_ext were executed with the aid of the user-friendly software known as MiePlot [34], which played a vital role in validating the precision of the computational results and ensuring their alignment with established findings. Furthermore, the results obtained from the MiePlot software underwent rigorous verification through the application of a modified version of the FORTRAN 77 code, as initially outlined in the research by Haiss et al. [32] and originally introduced by Bohr and Huffman [29]. Succinctly, this methodology encompassed the incorporation of silver dielectric data as outlined by Palik [33] and the meticulous selection of optical constants for the purpose of generating simulated extinction data. Essential parameters, including the plasma frequency (9.5 × 10¹⁴ Hz for silver), collision frequency (0.4 × 10¹⁴ Hz for silver), and Fermi velocity (1.4 × 10⁸ for silver), were finely tuned within the code to faithfully represent the unique characteristics of silver.