Combining the Modal Expansion and Dipole Equivalence Methods for Coated Plasmonic Particles of Various Shapes

From a structural point of view, plasmonic nanoparticles are always at least two-layer structures with a dielectric layer of stabilizing, targeting, fluorescent, Raman, or other functional molecules, often embedded in a matrix such as porous silica. To optimize the optical properties of such bioconjugates, one needs efficient analytical models based on simple physical ideas and with reasonable accuracy comparable to rigorous numerical methods requiring significant computer resources. Recently, we suggested an analytical approach ( J. Phys. Chem. C 2024, 128, 15029–15040) based on a combination of the modal expansion method (MEM) and the dipole equivalence method (DEM). The MEM+DEM method was exemplified by coated gold nanorods excited along the symmetry axis. Here, we extend the MEM+DEM approach for particles of various shapes and orientations, including random ones. To illustrate the possibilities of our method, we calculate extinction and scattering spectra of gold and silver nanorods, nanodiscs, triangle nanoprisms, bicones, and bipyramids with a dielectric coating thickness from 0 to 30 nm. For the main plasmonic peaks, we found excellent agreement between our analytical method and numerical simulations performed with COMSOL, except for some disagreement between MEM and COMSOL solutions for bicones with small rounding radii. This drawback of MEM for coated particles is explained by the unusual local field distribution with strong localization near the particle tips. To fix the point, we propose a factorized version of MEM that brings analytical spectra of coated bicones and bipyramids in good agreement with numerical counterparts. To illustrate the application of our method to real experimental colloids, we prepared 12 AuNR samples using a chemical etching with fine-tuning of the LPR over the 1020–630 nm range. By including the CTAB shell in the simulation model, we achieved excellent agreement between the calculated and measured dependence of the LPR peak position on the AuNR aspect ratio. In summary, our method makes it possible to use the MEM parameters of the original metal particles. It does not require additional numerical calculations to build analytical models for bilayered or multilayered conjugates. The presented analytical solutions depend on the particle shape, the shell thickness, and the shell refractive index, regardless of the particle size and the metal’s nature. Due to the simplicity and reasonable accuracy of the MEM+DEM analytical models, they can help apply machine learning to predict various plasmonic responses such as light absorption, scattering, surface enhanced Raman scattering (SERS), and metal-enhanced fluorescence (MEF).