For a certain class of branched structures of arbitrary but non-interacting sub-units, we have derived a Feynmann like diagrammatic formalism for deriving form factors for these structures in dilute solution. Form factors are required to analyze small-angle X-ray and neutron scattering spectra for such systems.
Feynman rules for the Fourier transform of the intra-intra pair correlation function (form factor), intra-reference point pair correlation function (form factor amplitude) and reference-reference point pair correlation function (phase factor) for a generic sub-unit I.
Example of a structure composed of many sub-units where certain the phase factor of paths through the structure have been illustrated.
The definitions of the form factor (Fs), form factor amplitude (As), and phase factor (Phis) for a generic structure expressed in terms of the corresponding terms expressed for the sub-units.
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 allow 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 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.
Paper: "A formalism for scattering of complex composite structures. I. Applications to branched structures of asymmetric sub-units" Journal of Chemical Physics 136, 104105 (2012). Authors: Carsten Svaneborg and Jan Skov Pedersen
Recently, we developed a formalism for the scattering from linear and acyclic branched structures build of mutually non-interacting sub-units. [C. Svaneborg and J. S. Pedersen, J. Chem. Phys. 136, 104105 (2012)] We assumed each sub-unit has reference points associated with it. These are well defined positions where sub-units can be linked together. In the present paper, we generalize the formalism to the case where each reference point can represent a distribution of potential link positions. We also present a generalized diagrammatic representation of the formalism. Scattering expressions required to model rods, polymers, loops, flat circular disks, rigid spheres, and cylinders are derived, and we use them to illustrate the formalism by deriving the generic scattering expression for micelles and bottle-brush structures and show how the scattering is affected by different choices of potential link positions and sub-unit choices.
"A formalism for scattering of complex composite structures. II. Distributed reference points" Journal of Chemical Physics 136, 154907 (2012). Authors: Carsten Svaneborg and Jan Skov Pedersen