# Solvent-Dispersible Nanostructured MIMI: An Experimental and Computational Study
## 2 Materials and Methods
### Theoretical Calculations  
Simulations regarding the optical characteristics of the nanorods-derived nano-matryoshka were conducted using the commercially available finite element analysis software COMSOL Multiphysics (COMSOL, Inc., Burlington, MA, USA) version 5.4, module 'Electromagnetic Waves, Frequency Domain' (EWFD) [54; 55; 56; 57; 58]. With this module, it is possible to study electromagnetic wave propagation, resonance effects at the nanoscale, electromagnetic field distributions through a scattered field formulation. The finite element method operates by discretizing the modeling domains into smaller, simpler domains, named elements. The solution is computed by assembling and solving a set of equations over all the elements of the model. The 3D simulation domain contained the nanoparticle immersed in the surrounding medium, which was in turn surrounded by a Perfectly Matched Layer (PML) to mimic an open and nonreflecting infinite domain. The amplitude of the plane wave \(|\mathbf{E_{0}}|\) was calculated for the incoming illumination intensity \(\mathbf{I_{0}}\) according to \(\mathbf{I_{0}}=n\mathbf{c}\mathbf{\epsilon_{0}}|\mathbf{E_{0}}|^{2}/2\), where \(\mathbf{n}\) is the refractive index of the surrounding medium, \(\mathbf{c}\) is the speed of light, and \(\mathbf{\epsilon_{0}}\) is the vacuum permittivity. [PERSON]'s equations, intended for the scattering of electromagnetic radiation with nanoparticles, were solved in the model [59]. In particular, the wavelength-dependent optical characteristics, including the absorption cross section and the scattering cross section, were calculated. Parametric Sweep was introduced from 400 nm to 1000 nm, with a step size of 5 nm. The simulation computed the local electromagnetic field on each mesh point, accompanied by an incident plane wave propagatingin the z-direction by the electric field polarized along the \(x\)-axis and \(y\)-axis to opportunely excite the transversal and longitudinal resonance of the gold nanorods. The nanoparticle was modeled in all its constituent parts, starting from (i) AuNR@CTAB, (ii) AuNR@SiO\({}_{2}\), (iii) AuNR@SiO\({}_{2}\)@Au and (iv) AuNR@SiO\({}_{2}\)@Au@SiO\({}_{2}\). An \"extra-fine\" mesh, consisting of a free triangular and tetrahedral geometry with about 25,000-80,000 elements (from the AuNR@CTAB to the AuNR@SiO\({}_{2}\)@Au@SiO\({}_{2}\) structure) was considered. For the outer boundary of the spherical domain, the scattering boundary condition was used to make the boundary transparent for the scattered wave. All the parameters used in the numerical modelling are listed in Table S1. The absorption cross section and the scattering cross section were derived by the calculation of the rate of the electromagnetic energy that was absorbed (\(W_{abs}\)) and scattered (\(W_{sca}\)) by the nanoparticle with respect to the incident irradiance, defined as the energy flux of the incident wave (\(P_{inc}\)): \(\sigma_{abs}=W_{abs}/P_{inc}\) and \(\sigma_{sca}=W_{sca}/P_{inc}\), respectively. The total absorbed energy was derived by integrating the energy loss (\(Q_{loss}\)) over the volume of the particle (\(W_{abs}=\big{\|}\big{\|}V_{p}\,Q_{loss}dV\)), while the scattered energy was derived by integrating the Poynting vector (\(P_{sca}\)) over an imaginary sphere around the particle (\(W_{sca}=\iint_{S}P_{sca}\,\,\textit{n}\,\,dS\)). The extinction cross section (\(\sigma_{ext}\)) represents the amount of energy removed from the incident field due to absorption and scattering (\(\sigma_{ext}=\sigma_{abs}+\sigma_{sca}\)).