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.
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