  Bottle brushes are polymeric macromolecules made of a linear polymeric
backbone grafted with side chains. The choice of the grafting density
{\sigma}g, the length ns the grafted side chains and their chemical nature
fully determines the properties of each macromolecule, such as its elasticity
and its folding behaviour. Typically, experimental bottle brushes are systems
made of tens of thousands of monomeric units, rendering a computational
approach extremely expensive, especially in the case of bottle brush solutions.
A proper coarse graining description of these macromolecules thus appears
essential. We present here a theoretical approach able to develop a general,
transferable and analytical multi-scale coarse graining of homopolymeric bottle
brush polymers under good solvent conditions. Starting from scaling theories,
each macromolecule is mapped onto a chain of tethered star polymers, whose
effective potential is known from scaling predictions, computational and
experimental validations and can be expressed as a function of the number of
arms f, and the length na of each arm. Stars are then tethered to one another
and the effective potential between them is shown to only depend on the key
parameters of the original bottle brush polymer ({\sigma}g, ns). The
generalised form of the effective potential is then used to reproduce
properties of the macromolecules obtained both with scaling theories and with
simulations. The general form of the effective potentials derived in the
current study allows a theoretical and computational description of the
properties of homopolymeric bottle brush polymers for all grafting densities
and all lengths of both backbone and grafted arms, opening the path for a
manifold of applications.
