## Macromolecular Orientation of Rigid Dumbbells in Shear Flow

##### Abstract

We use molecular modeling to study the non-equilibrium behaviour of polymeric fluids. Specifically, we explore the macromolecular orientation induced by shear flow fields. We choose the simplest molecular model, the rigid dumbbell model, which describes the molecular structure of the polymer with two identical spherical beads joined by a massless fixed rod. For rigid dumbbells suspended in a Newtonian solvent, the viscoelastic response depends exclusively on the dynamics of dumbbell orientation. The orientation distribution function ψ (θ,φ,t) represents the probability of finding dumbbells within the range (θ,θ+dθ) and (φ,φ+dφ). This function is expressed in terms of a partial differential equation called the diffusion equation, which, for any simple shear flow, is solved by postulating a series expansion in the shear rate magnitude. Each order of this expansion yields a new partial differential equation, for which one must postulate a form for its solution. This work finds a simple and direct pattern to these solutions. The use of this pattern reduces the amount of work required to determine the coefficients of the power series expansion of the orientation distribution function,ψi. To demonstrate the usefulness of this new pattern, we arrive at new expressions for these coefficients up to and including the sixth power of the shear rate magnitude. This work also completes previous findings that ended at the fourth power of the shear rate magnitude. We then use the general results found for any simple shear flow to derive the solution for the special case of large-amplitude oscillatory shear (LAOS). We extend the orientation distribution function to the 6th power of the shear rate amplitude. We arrive at the Fourier solution for each harmonic contribution to the total orientation distribution function, separating each harmonic into its coefficients in and out-of-phase with cosnωt, ψ'n and ψ" n, respectively. We plot, for the first time, the evolving normalized alternant macromolecular orientation, in the nonlinear viscoelastic regime. Moreover, to deepen our understanding of the macromolecular motions, we distinguish and study two types of possible rotations, tumbling and wobbling

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