Testing orientational closure approximations in dilute and non-dilute suspensions with Rheo-SANS

Orientational closure approximations play a crucial role in the development and use of theories for complex fluids with orientable microstructures. In such theories, the fluid rheology is completely determined by moments of the orientation distribution function (OPDF). Closure approximations enable one to solve directly for moments of the OPDF, rather than the more computationally intense procedure of first solving for the OPDF and then calculating the moments. Numerous orien-tational closure approximations have been published over the past half century. These closures are typically tested in the context of a proposed theory for calculating the OPDF, or in some cases the stress, where the proposed theory itself may introduce more physical assumptions than the closure approximation. In the absence of an exact theory, the ideal would be to test closures based upon experimental measurement of the full 3D OPDF. Recently, a new method was proposed, called MAPSI, which enables inference of the full 3D structure of an orientable rodlike nanoparticle suspension determined from small angle scattering measurements. In this work, we directly test the accuracy of a number of orientational closure approximations that relate the fourth moment of the OPDF to the second moment in steady simple shear flow. We use two approaches. First, we follow some previous studies and use the OPDF from the exact solution to a Smoluchowski equation for a dilute system to compare the fourth moment calculated directly from the OPDF, with that obtained via the second moment and various closure approximations. Second, we follow the more novel idea of measuring the OPDF of a model rodlike system via rheo-SANS, and testing the closure via fourth moments calculated both directly from the measured OPDF and via a closure approximation using the second moment obtained from the OPDF. Out of the closures tested, we find that there does not exist a single closure that clearly outperforms others, but that there are several closures that provide good approximations, despite being derived from very different starting assumptions. Furthermore, differences between closures are not reflected in rheological quantities derived from moments of the OPDFs, highlighting the necessity for direct testing of closures, rather than testing through rheological predictions.