Application of the Eckart frame to soft matter: rotation of star polymers under shear flow

The Eckart co-rotating frame is used to analyze the dynamics of star polymers under shear flow, either in melt or solution and with different types of bonds. This formalism is compared with the standard approach used in many previous studies on polymer dynamics, where an apparent angular velocity $\omega$ is obtained from relation between the tensor of inertia and angular momentum. A common mistake is to interpret $\omega$ as the molecular rotation frequency, which is only valid for rigid-body rotation. The Eckart frame, originally formulated to analyze the infrared spectra of small molecules, dissects different kinds of displacements: vibrations without angular momentum, pure rotation, and vibrational angular momentum (leading to a Coriolis cross-term). The Eckart frame co-rotates with the molecule with an angular frequency $\Omega$ obtained from the Eckart condition for minimal coupling between rotation and vibration. The standard and Eckart approaches are compared with a straight description of the star's dynamics taken from the time autocorrelation of the monomers positions moving around the molecule's center of mass. This is an underdamped oscillatory signal, which can be described by a rotation frequency $\omega_R$ and a decorrelation rate $\Gamma$. We consistently find that $\Omega$ coincides with $\omega_R$, which determines the characteristic tank-treading rotation of the star. By contrast, the apparent angular velocity $\omega < \Omega$ does not discern between pure rotation and molecular vibrations. We believe that the Eckart frame will be useful to unveil the dynamics of semiflexible molecules where rotation and deformations are entangled, including tumbling, tank-treading motions and breathing modes.