Since the scattering amplitude scales with size21 as \(\tilde{d}^{6}\), it is an optical property that scales strongly at any nanoparticle size level, allowing accurate measurement of nanoparticle sizes even smaller than 50 nm. The only limitation of this novel approach is the instrumental capability of the equipment used to detect and measure nanoparticles, i.e. it is necessary to improve the signal-to-background ratio as much as possible. As the nanoparticles analyzed here are beyond the Rayleigh regime (i.e. sizes above ~ 10 nm), they exhibit anisotropic emission, with the forward scattering signal being significantly larger than the backward scattering contribution.22,23 For this reason, the use of a transmission detection system is more advantageous than the episcopic system used in the previous work.
A schematic drawing of the experimental setup is shown in the Fig. 1. The white light coming from a halogen lamp is directed to an electro-optical filter, which separates the light into its constituent wavelengths. The filtered light is directed to a dark-field condenser that focuses it onto the surface of the transparent sample where the nanoparticles are located. The scattered light is collected by a dark-field objective, and redirected to a monochromatic CMOS camera placed at the image plane of the experimental setup, which collects the images at each wavelength of illumination (see "Methods" for more technical details). The experimental setup is also equipped with an episcopic illuminator capable of imaging the same sample region.
The measurement of the scattering amplitude is more complex than the spectral measurement of the wavelength, because it depends not only on the morphological and optical characteristics of the particle, but also on the measurement system used to obtain the signal (such as the numerical aperture of the objective,24 the angle of illumination,25 the polarization of the light, etc.). Therefore, it is essential to use a batch of nanoparticles measured under the same conditions as a calibration system in order to eliminate the influence of any instrumental dependencies coming from the AR-SPS setup.
Each plasmonic nanostructure has a specific relationship between its scattering amplitude and its size, which can be determined by either analytical or semi-numerical methods26; for example, in the case of spherical geometries, a well-defined relationship between the size \(d\) of spherical nanoparticles and their scattering amplitude \(A\) can be established thanks to the Mie theory20;
where α, 𝛽, b and c are constants depending on the optical properties of the medium and the nanoparticles themselves (for more information on the theoretical derivation, see Supplementary Information)19,27,28, and \({d}_{ref}\) is the diameter of the reference batch. The scattering amplitude \(A\) in Eq. (1) has been normalized by \({A}_{ref}\), which corresponds to the scattering amplitude of the nanoparticles of the reference lot. Normalization is a key aspect of this method as it removes the influence of measurement system dependencies.
The AR-SPS includes a simple and rapid sample preparation based on the drop casting method17,29. A schematic of the sample preparation is shown in Fig. 2a; a drop of 50 µL was placed on a microscope slide and dried at room temperature. A drop of 2 µL of glycerol28 was added to the dried GNPs before the glass coverslip was placed on the sample substrate. The addition of glycerol reduces the optical mismatch between the refractive index of the glass substrate and that of the surrounding environment, thus avoiding the splitting of substrate-mediated plasmonic modes and allowing the optical response to be well predicted by standard Mie theory (see "Methods" for more technical details on sample preparation).
çOnce the sample is prepared, dark field microspectrophotometry is performed by sequentially illuminating the sample at different wavelengths, as shown in Fig. 2b. Cropped images of a spectral measurement of 100 nm GNPs are shown, while the normalized spectra of approximately 1000 nanoparticles are shown in Fig. 2c. For the sake of clarity, normalized scattering spectra have always been obtained in this paper by dividing the scattering signal of each nanoparticle by the scattering signal coming from the substrate; this type of normalization allows obtaining a direct estimate of the signal-to-noise ratio for each nanoparticle measured (for more technical information on data processing, see Supplementary Information).
In good agreement with Mie theory, we can observe how the resonance peak reaches its maximum at about 575 nm. Although the size variability of the studied nanoparticles is about 4%, the brightness variability is as high as 22%. This increased relative uncertainty in the amplitude is expected because, as mentioned above, the amplitude scales with the diameter by about a power of six (i.e. ∂A⁄A ≈ 6 ∂d⁄d)21. Therefore, the scattering amplitude of each individual nanoparticle can be calculated by performing a Lorentzian fit to each individual particle spectrum (for more details on data analysis, see the Supplementary Information).
We have demonstrated the effectiveness of the AR-SPS technique in a proof-of-concept experiment using spherical nanoparticles with diameters ranging from 40 to 150 nm (Nanopartz, Inc.). In total, we used nine different batches of nanoparticles for these experiments. Although the experiments were performed with spherical nanoparticles, it is important to note that it is possible to extend this method also to other shapes, such as nanorods30, shells31 and other geometries 32, as long as a relationship between particle size and its plasmonic emission can be calculated, either analytically or numerically26.
Figure 3a shows the normalized scattering spectra of individual particles for each of the characterized batches. The data show that while the plasmonic peak undergoes minimal spectral changes with decreasing nanoparticle size, the amplitude signal consistently increases in magnitude for all measured sizes.
In Fig. 3b, the experimental data of each batch were directly compared with the theoretical Eq. (1), using the 90 nm batch as the reference sample. Since we considered spherical nanoparticles immersed in glycerol in the following experiments, the constants used in Eq. (1) are as follows: \(\alpha =3.21·{10}^{7}\), \(\beta =-3.52\), \(b=-37.85 nm\), and \(c=14.58\). These numerical values are valid as long as we express \({d}_{ref}\) in nm in Eq. (1).
To improve the comparison between experiment and theory, the TEM nanoparticle size was used instead of the nominal size in Fig. 3b (see the Supplementary Information for more technical details). The amplitude data shown here are from approximately 5000 individual nanoparticle spectra for each batch analyzed, with the error bars representing the corresponding standard deviations. The black dashed line in Fig. 3b represents the Mie theory, obtained by using Eq. (1); the excellent agreement observed between the experimental data and the theoretical predictions is a strong indication of the effectiveness of the AR-SPS method.
The relationship between diameter and scattering amplitude observed in Fig. 3b shows that the main limitation of the method is not theoretical, but due to the limitations of the experimental setup. Although we have measured particles down to 40 nm in the following experiments, we could go even lower by improving the signal-to-noise ratio of the instrument; this could be achieved, for example, by using objectives with a larger numerical aperture, or by amplifying the scattering signal using multi-dielectric substrates33,34, or a detection scheme based on total internal reflection35,36.
To evaluate the accuracy of the proposed method in estimating nanoparticle diameters, a direct comparison with TEM is necessary. Figure 4 shows the correlation between the diameter obtained by TEM and the diameter estimated by AR-SPS (for more technical details, see the Supplementary Information). The error bars in both X and Y axes represent the standard deviations derived from the TEM and amplitude data, respectively; at least 500 particles per batch were characterized by TEM and more than 5000 individual nanoparticle spectra were obtained for the amplitude data.
It is noteworthy that the experimental data closely follow the ideal correlation curve (black dashed line in Fig. 4), indicating excellent agreement between the diameter estimated from the AR-SPS and the true diameter obtained from the TEM, with an average discrepancy of about 2.6%.