Empirically, we find that the parametric plots of the imaginary vs real parts of the complex viscosity may depend neither on temperature nor on average molecular weight. Moreover, for a fixed polydispersity, these viscosity Cole–Cole curves amplify both rightward and upward with long-chain branching content. In this paper, we find that general rigid bead–rod theory [O. Hassager, “Kinetic theory and rheology of bead–rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60(10), 4001–4008 (1974)] can explain these rightward and upward amplifications. We explore the effects of branching along a straight chain in small-amplitude oscillatory shear flow. Specifically, we explore the number of branches, branch length, branch position, and branch distribution.

Atomically thin flat sheets of carbon, called graphene, afford interesting opportunities to study the role of orientation in suspensions. In this work, we use general rigid bead-rod theory to arrive at general expressions from first principles for the complex viscosity of graphene suspensions. General rigid bead-rod theory relies entirely on suspension orientation to explain the elasticity of the liquid. We obtain analytical expressions for the complex viscosity of triangular and hexagonal graphene sheets of arbitrary size. We find good agreement with new complex viscosity measurements.

Whereas much is known about the complex viscosity of polymeric liquids, far less is understood about the behavior of this material function when macromolecules are confined. By confined, we mean that the gap along the velocity gradient is small enough to reorient the polymers. We examine classical analytical solutions [O. O. Park and G. G. Fuller, “Dynamics of rigid and flexible polymer chains in confined geometries. II. Time-dependent shear flow,” J. Non-Newtonian Fluid Mech. 18, 111–122 (1985)] for a confined rigid dumbbell suspension in small-amplitude oscillatory shear flow. We test these analytical solutions against the measured effects of confinement on both parts of the complex viscosity of a carbopol suspension and three polystyrene solutions. From these comparisons, we find that both parts of the complex viscosity decrease with confinement and that macromolecular orientation explains this. We find the persistence length of macromolecular confinement, Lp, to be independent of both λω and λγ̇0.

One good way to explain the elasticity of a polymeric liquid is to just consider the orientation distribution of the macromolecules. When exploring how macromolecular architecture affects the elasticity of a polymeric liquid, we find the general rigid bead–rod theory to be both versatile and accurate. This theory sculpts macromolecules using beads and rods. Whereas beads represent points of Stokes flow resistances, the rods represent rigid separations. In this way, how the shape of the macromolecule affects its rheological behavior in suspension is determined. Until recently, general rigid bead–rod theory has neglected interferences of the Stokes flow velocity profiles between nearby beads. We call these hydrodynamic interactions, and we here employ our new method for exploring how these interactions affect the complex viscosity of suspensions of multi-bead rods. These multi-bead rods are also called shish-kebabs. We use the center-to-center distance between adjacent beads as the characteristic length. We proceed analytically, beginning with a geometric expression for the shish-kebab bead positions. Our analytical solution for the complex viscosity presents as one for [Formula: see text], one for [Formula: see text], and another for the rigid dumbbell, [Formula: see text]. We find that for shish-kebabs, hydrodynamic interactions (i) increase zero-shear viscosity, (ii) increase zero-shear first normal stress coefficient, (iii) decrease the real part of the dimensionless complex viscosity, and (iv) increase minus the dimensionless imaginary part. We find that the combination of (iii) and (iv) explains crossovers of the parts of the complex viscosity. We further find that for a monodisperse polystyrene solution, the general rigid bead–rod theory with hydrodynamic interaction, for both parts of the complex viscosity, provides stunning improvement over without.

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