Hydrodynamic interaction within canonical macromolecular structures

In general rigid bead-rod theory, we deduce the rheological properties of a suspension of macromolecules from the orientation distribution that arises during flow. The most important feature governing this orientation is macromolecular architecture, and right behind this, enters hydrodynamic interaction. Until now, general rigid bead-rod theory has neglected hydrodynamic interactions, namely, the interferences of Stokes flow velocity profiles between nearby beads. The lopsidedness of the architecture affects orientability, and so do these heretofore unexplored interferences within the macromolecule. We here employ a new method for exploring how such hydrodynamic interactions affect the complex viscosity. This method has, with great effort, been used to examine hydrodynamic interactions in complex architectures, namely, multi-bead rods and backbone-branched polymers. However, it has yet to be applied to canonical forms. In this paper, we focus on the simplest of rigid architectures: (i) rigid dumbbell, (ii) tridumbbell, (iii) rigid rings, and (iv) planar stars. We call these forms canonical. We arrive at beautiful algebraic expressions for the complex viscosity for each canonical form. We find that for the dimensionless complex viscosity, for all canonical forms, hydrodynamic interactions just depend on the ratio of the bead diameter to the nearest bead separation, $d/2L≡A$⁠. Furthermore, we find that for the dimensionless complex viscosity, for canonical forms (i) and (iii), hydrodynamic interactions shift the real part upward and minus the imaginary part downward. For canonical forms (ii), both parts are unaffected. For canonical forms (iv), the story depends interestingly on the number of beads. We advance the mathematics of fluids by establishing, for intramolecular hydrodynamic interactions, the foundational equations which future work must recover.