A Formalism for Scattering of Complex Composite Structures. 1 Applications to Branched Structures of Asymmetric Sub-Units

TL;DR

This paper introduces a versatile formalism for calculating the scattering from complex branched structures built from arbitrary sub-units, enabling easier analysis of hierarchical and asymmetric assemblies in materials science.

Contribution

The formalism allows hierarchical structures to be analyzed by decoupling connectivity from internal sub-unit structure, and it can incorporate complex sub-units within the same framework.

Findings

Derived scattering expressions for branched structures like stars and dendrimers.

Demonstrated the formalism's ability to handle asymmetric two-functional sub-units.

Applied the method to obtain pair distributions of flexible polymer sub-units.

Abstract

We present a formalism for the scattering of an arbitrary linear or acyclic branched structure build by joining mutually non-interacting arbitrary functional sub-units. The formalism consists of three equations expressing the structural scattering in terms of three equations expressing the sub-unit scattering. The structural scattering expressions allows a composite structures to be used as sub-units within the formalism itself. This allows the scattering expressions for complex hierarchical structures to be derived with great ease. The formalism is furthermore generic in the sense that the scattering due to structural connectivity is completely decoupled from internal structure of the sub-units. This allows sub-units to be replaced by more complex structures. We illustrate the physical interpretation of the formalism diagrammatically. By applying a self-consistency requirement we derive the pair distributions of an ideal flexible polymer sub-unit. We illustrate the formalism by deriving generic scattering expressions for branched structures such as stars, pom-poms, bottle-brushes, and dendrimers build out of asymmetric two-functional sub-units.