When increasing the sigma parameter of a Lennard-Jones potential, e.g. to make a single particle represent multiple particles (that can be represented with a Lennard-Jones potential), then you have to keep in mind that this process will also make the resulting coarse-grain particle “softer”. Thus this kind of coarse-graining is usually only used for representing particles at a 1:3 or 1:4 ratio not more. For larger ratios, you usually want to use a custom functional form that will be more representative, i.e. make a single particle have interactions more similar to a cluster of individual atoms, e.g. represented via fix rigid in LAMMPS. This could be something like colloid-colloid interaction in pair_style colloid command — LAMMPS documentation or a custom table for pair_style table command — LAMMPS documentation
That is a very specific question about your research and less so about LAMMPS. Thus the best place to look for information is the published literature. Please keep in mind that a lot of research and literature about coarse-graining focuses on molecular systems and there the coarse-grain ratio is rather small, since they still want to represent molecules, a very popular model in that domain is the MARTINI model (http://cgmartini.nl/), but I would be concerned that that is not an approach for your system (it uses a fixed 1:4 ratio).
In general, you have - as far as I can tell - two options:
- you pick an empirical functional form that is suitable for the kind of objects and coarse-grain ratio you want to implement and then parameterize it like one would a normal force field, i.e. adjust the available parameters to match data from experiments and all-atom or quantum calculations.
- follow one of the available “bottom-up” coarse-graining schemes, where you start with an all-atom model simulation and then pick a coarse-graining ratio and then use available tools to determine parameters automatically to represent structural properties of the all-atom calculation (e.g. the g(r)). In fact the inversion of the g(r) would be the simplest approach, but that is technically only reliable for very diluted systems.
From what you describe, following the first approach with the functional form from the colloid pair style (which is effectively an integrated form of a Lennard-Jones potential of the aggregate atoms of the nano-particles) seems like the simplest approach. However, only careful testing and comparing to other methods and experimental data, as well as knowing what kind of accuracy and predictive quality you expect from your coarse-grain model, can inform you whether that is a good approach and suitable.