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Localized SPR Phenomenon and Theoretical Background for Its Application in SensingTable of Contents:
Theoretical Background of Localized SPR MethodSensitive element model for LSPR sensorLocalized SPR is one of the peculiar optical properties of noble metal nanoparticles that occurs when the light falling on them is resonant with the collective oscillations of free electrons in the nanoparticle (Fig. 2.12). This electronicresponse causes unusual optical properties of nanosized metal—a peak appears in the light extinction spectrum of noble metal nanoparticles (Fig. 2.13), which is not observed in the corresponding spectrum for bulk material. The shape of the spectrum and the position of the LSPR peak depend on the shape of the nanoparticle [41, 42], its size [43, 44], the interparticle distance [45], and the dielectric properties of the material of the nanoparticles [43, 46], as well as the dielectric properties of the environment [47— 50] and the charge of the molecules contained therein [51], which is important for sensor applications. Figure 2.12 Excitation of a dipolar localized surface plasmon by an electric field of an incident light wave. Reprinted with permission from Ref. [52], Copyright 2003, American Chemical Society. Figure 2.13 Light extinction spectra of random gold and silver nanoparticle arrays on glass substrates (nanochips). Since the operating principle of LSPR sensors is based on the change in the optical properties of absorption and scattering of light by highconductive metal nanoparticles when the adsorption of molecules or molecular process occurs on their surface, for the development of a theoretical background for describing the operation of the LSPR sensor, an important stage is the development of a model for the "nanoparticlemolecule” system, which forms the sensitive element of the sensor. To do this, one needs to specify the optical and geometric parameters of the nanoparticle, the molecular component, and the ambient environment. Optical constants of gold nanoparticlesIt is known that the optical properties of nanostructured and bulk materials differ, because the socalled size effects exist [53] due to the dependence of the dielectric permittivity e[co, R) on the size of the nanoparticle. Reducing the size of the metallic nanoparticle leads to an increase in the influence of the classical (a reduction in the mean free path of electrons [53], a decrease in the concentration of free electrons in a nanoparticle due to the spillout effect [53]) and quantum mechanical (Landau damping [54], the interaction of plasmon oscillations with individual discrete electronic states [55]) size effects. The typical size of a nanoparticle, which the size effects begin to affect the dielectric constant at, varies: for the reduction in the mean free path of electrons, this size is smaller or comparable to the free path of electrons in a bulk material; for the spillout effect, it is less than 10 nm, and quantummechanical size effects are important when considering only very small metal clusters [53]. Since the typical nanoparticulate materials for use in LSPR sensors are gold and silver, for which the electron mean free path is 42 and 52 nm [53], respectively, the most significant effect on the optical properties of nanoparticles produces an effect of reduction in the electron mean free path. Thus, this model, which is most commonly used to specify the optical constants of nanoparticles, is described below. Optical constants of bulk gold were taken from Table II (pp. 290295), Part II in Ref. [56] and approximated using 79 order polynomials with wavelength steps in the range of 0.1 to 1 nm (depending on the system under consideration). The dielectric function of gold has been modified in accordance with the size of the nanoparticle according to the model of reducing the mean free path of electrons. This modification was carried out by introducing an effective electron relaxation time where r_{bulk} = 9.3 x 10'^{15} s [57] is the electron relaxation time for bulk gold, = 1.4 x 10^{6} m/s [58] is the Fermi velocity, R is the spherical nanoparticle radius, and A is a constant, which can be set to 1 for spherical nanoparticles studied and isotropic surface electron scattering [53,59]. The sizedependent electron relaxation time was further used to modify the values of the dielectric function in the DrudeLorentz model [60, 61]: where £_{x} and e_{2} stand for the real and imaginary parts of the dielectric permittivity, mis the angular frequency of light, and co_{p} = 1.37 x 10^{16 }rad/s [61] is the plasma frequency for bulk gold. Optical constants adjusted for the size of the gold nanoparticles were calculated according to the following equations [62]: For nonspherical gold nanoparticles, which were also considered a sensitive element of LSPR sensor, the optical constants of bulk gold were used without modification. This approach was used due to the fact that the geometric size of the studied nonspherical nanoparticles was predominantly greater than the mean free path of electrons in gold, so that the influence of dimensional effects on the optical constants of nanoparticles can be neglected. Optical constants of the molecular component and environmentIntroduction to the model of the sensitive element of the LSPR sensor of the molecular component was carried out by adding one or two dielectric layers of a certain thickness, which formed a coating on the surface of the gold nanoparticle. Two approaches were used to specify the optical properties of molecular layers. The first approach supposes the specification of the optical constants of the layer irrespective of the geometric dimensions of the molecules and the nanoparticle, i.e., the molecular layer was considered a uniform dielectric layer with a certain refractive index. In the framework of this approximation, the refractive index of the layer was considered both real and complex, as well as both dispersive and nondispersive. The refractive indexes of the environment and the substrate (in the case of an array of nanostructures on the solid surface) were chosen as real and independent of the light wavelength. In the second approach, which was realized only for spherical gold nanoparticles, the molecular layer was interpreted as a saturated monolayer of globular molecules, which are approximated with solid spheres (Fig. 2.14). To describe the shell consisting of a densely packed monolayer of globular molecules, a symmetric Bruggeman effective medium theory (EMT) [59] was used, which specified the effective value of the refractive index of the shell n_{2} as a solution to the equation where / is the shell fill factor by molecules, n_{m} is the molecules' refractive index, which was chosen to be equal to 1.46 as for biomolecular species [64, 65] regardless of the wavelength of light, n_{0} is the refractive index of the environment (water), which was calculated by the formula (A in nanometers) [66] Figure 2.14 Schematic representation of a sensitive element model for LSPR sensor with a molecular layer in the form of a dense monolayer of globular molecules. Reprinted with permission from Ref. [63], Copyright 2011, IEEE. Obviously, the fill factor for a saturated monolayer of spherical molecules depends on the ratio between the diameter of the nanoparticle and the shell thickness. This dependence for a spherical nanoparticle was obtained with the following considerations. Let us imagine the surface of the sphere of the radius R + r (where r is the radius of the molecule), where the centers of the molecules are located on, and calculate the number of cross sections of the molecules with this surface. If one approximates the arrangement of these cross sections with a dense square grid of circles of radius r on a plane, one can obtain an expression for the number of molecules on the surface of a nano particle: The fill factor is calculated as a ratio between the volume occupied by molecules and the total volume of the shell: Method of optical properties calculation based on the Mie scattering theory for LSPR sensor sensitive elementTo simulate the optical properties of light absorption, scattering, and extinction for individual gold nano particles, the Mie theory of light scattering on a spherical nonmagnetic particle with a shell [59] was used. Within this approach, cross sections of extinction, scattering, and absorption of light (the rates of energy totally lost, scattered, and absorbed, respectively, divided by the incident light intensity) are expressed in the following form: where к is the wave vector of light in the environment; L is the number of considered multipole modes; m_{1} = (n_{1} + /7c_{1})/n_{0} and m_{2 }= n_{2}/n_{0} are complex refractive indices of core and shell relative to the environment; x =k a , у =/c b,a= R, and b = R + 2rare the core and shell radii; w[z), ^,(z), and xi^{z}) are the RiccatiBessel functions. The parameter L value was calculated according to the relationship indicated in Ref. [67]: where the square brackets [ ] mean rounding to the nearest integer. To precisely determine the wavelength positions of maxima of the simulated cross sections spectra, parabolic approximation in the vicinity of absolute maximum position of the spectrum was used. Influence of “NanoparticleMolecular Layer” System Parameters on the Optical Response of LSPR SensorComparison of LSPR and SPR sensors responseThe main method used in LSPRbased sensing in noble metal nanoparticles is the measurement of light extinction. Therefore, the response (specifically, the shift in the wavelength of the LSPR peak position in the extinction spectrum) was calculated for a single spherical gold nanoparticle immersed in water upon a thickening molecular monolayer using the equations given in Subsection 2.3.1. Figure 2.15 shows a simulated LSPR response of gold nanoparticles Figure 2.15 Response of LSPR sensor on the basis of a gold nanoparticle with a radius of 25 nm and SPR sensor based on a gold film with a thickness of 50 nm, depending on the thickness of the molecular layer. Calculations of the response of SPR sensor were carried out according to the theoretical model developed in Subsection 2.2.1 for the wavelength of 650 nm. The refractive index of a homogeneous layer for SPR and LSPR simulations without EMT was 1.398, corresponding to a flat square grid (inset) of molecules (refractive index 1.46) with a fill factor of n/6, and the refractive index of the environment (water) was 1.331. Reprinted with permission from Ref. [63], Copyright 2011, IEEE. with a radius of 25 nm with increasing thickness of the coating according to the models, which treated a shell as homogeneous as well as composed of globular molecules (according to Eqs. (2.37), (2.38), (2.40)), compared with the response of a usual SPR sensor based on a 50 nm thick gold film. Usually, a small range of shell thicknesses (up to 30 nm [68,69]) is studied, in which the shift in the LSPR peak rapidly increases with an increase in the layer thickness on the nanoparticle surface. A further increase in the thickness of the shell results in a slower change in LSPR response and is usually not studied. Indeed, an LSPR sensor is theoretically able to sense much thicker molecular coatings; in particular, for a nanoparticle with a radius of 25 nm, the LSPR response is saturated when the thickness of the molecular layer reaches 130 nm (as can be seen from Fig. 2.15). Further increase in the thickness of the shell does not show significant growth of response. This behavior of the LSPR response with saturation at thick molecular layers is similar to the response in the slowly decaying electric field of the nanopyramidal particle, which was considered in Ref. [41]. This is expected, since the spherical nanoparticle with excited surface plasmons does not have sharp edges typical for pyramids, in which the electric field is the most intense, but also rapidly decaying (socalled "hot spots”). This value of the sensitivity "upper limit,” which is equal to 130 nm, is comparable to the value for the SPR sensor (250 nm, see Fig. 2.15). Figure 2.15 also shows that the application of the effective medium theory reduces the response of the sensor due to a decrease in the effective refractive index of the shell with increasing diameter of the molecule. 
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