We've successfully computed the reflectance from a multilayered system of glass, organosilanes and gold nanoparticle films (SEE ATTACHED FIGURE) using the transfer matrix method, and treating the gold nanoparticles as an effective medium theory defined in this paper:
http://www.sciencedirect.com/science/article/pii/S0009261499012063
The results were actually quite good. It's cool that the film of gold can be approximated using an effective medium theory, such that the heterogenous gold/water mixtures can be parametrized by a single complex index of refraction, n.
I'm trying to extend this process to model films that contain gold AND silver nanoparticles (maybe of different sizes), and complex morphologies such as nanorods. This is where my understanding is lost.
There's a great deal of research out there on numerical analysis on exotic nanoparticles, but most of it involves the computation of single-particle properties: for example, the extinction spectrum of a nanorod. Where I get lost is what implications does this have for the film? I can model a film of nanospheres just fine, and I can compute the optical properties of a disperse solution of nanorods using DDA or FDTD, but how do I model a FILM of nanorods? How could I compute the reflectance from an array of nanorods on a glass slide? We'd expect the reflectance from nanorods to be different than nanospheres; however, how would this manifest in an effective medium theory? In other words, can I still represent this film of nanorods with a single parameter, n? Presumably n for nanorods would be similar to gold, but not identical.
What is the correspondence between near-field features like particle shape, and the effect they have on the far-field properties of the film? In other words, is it possible to predict a single index of refraction, n, that would parametrize an entire film of complex morphologies (eg rods), or multiple particle types (eg gold and silver nanospheres), similiar to how this was done for a layer of just gold sphers? And if so, is there a theoretical prescription? IE a cooking recipe from start to finish that would start with individual particle properties and then spit out an index of refraction approximation of the film? Is there a way to take near-field features like aspect ratio of a nanorod, and say how this would contribute to the effective index of refraction of the film, n?
Adam,
Take a look at the book by Michael Quinten, "Optical properties of nanoparticle systems. Mie and Beyond" (Wiley, 2011). He considers densely packed systems of NPs. This is what comes to mind first.
Thanks Dr. Stepanov. This is fairly similar to our results as they are now. I noticed your paper said:
"Consequently, a closer coincidence of experimental and theoretical spectra would require a calculation of structures with multiple layers, and also taking into account additional terms of effective media theories [20] by considering the dipole±dipole interactions between particles and their size distribution."
So would you say the approach to go and do this would be kind of a trial-and-error or basically messing around with mixing model parameters until they fit the experimental data? Essentially, we'd looking for some concrete guidelines to pursue exactly what you've stated.
I followed up on the book by Dr. Quinten, and it was supremely helpful. It looks like there are many avenues to approach this for those interested:
1) Generalized mie-theory, which is a coherent superposition of spheres of arbitrary topology. The downsides are that it is computationally intensive, so can only be done for arrangements of a hundred or so spheres (as of 2011 when the book was written). It's also, AFAIK, only available for spheres.
2) Incoherent superposition, as described in the book. The book offers a hybrid treatment to account for aggregates without. Also, I believe, only works for spheres.
3) Rigorous coupled wave analysis. This is an extension of Fresnel's calculations to include periodic dielectric structures directly in transfer matrix method calculations. I've found at least one attempt to utilize it in nanoparticle films, but more most is require.
4) I've found a plethora of effective medium theories that can handle inclusions of different sizes, shapes and material compositions. Spectral density theory, and more generalized versions involving tensor descriptions (https://www.researchgate.net/publication/258392346_Comment_on_Generalized_effective-medium_theory_of_induced_polarization%27%27_%28Michael_Zhdanov_2008_GEOPHYSICS_73_F197-F211%29) are more and more likely to reproduce the myriad of topologies and heterogenous materials (eg gold and silver nanoparticles on the same surface), but medium theories fail in some cases. First, finding the correct theory to best describe one's system is often a matter of trial and error and tuning. Secondly, according to Quiten's book, highly scattering particles (eg big metal nanoparticles, and some other edge cases) can't be described with effective medium theories. Not sure if this holds true still.
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