Role of viscoelasticity in non-equilibrium steady states of sheared entangled polymer mixtures

Introduction

Since almost all systems found in nature are not in thermodynamic equilibrium, the non-equilibrium dynamics play a significant role in modern statistical physics. There are two significant classes of non-equilibrium systems:¹ those undergo phase separation after temperature quench and those are continuously driven by applied shear force. In this article, we consider systems that contain both features—fluids undergoing phase separation in a shear flow. However, it is challenging to understand the morphological and rheological properties of these systems with both features. Early experimental^2–6 and numerical^7–12 studies revealed the basic features of non-equilibrium phase separation in binary fluids and many intriguing effects were observed, such as the highly elongated domains in very weak shear and a string phase under strong shear. There are still some discrepancies between theoretical and experimental opinions. For example, the relation $L_x~\stackrel{·}{\gamma }t{L}_y$ was predicted by theories in zero-shear systems, where the velocity field does not fluctuate.^13–15 In real fluids, however, the velocity fluctuates strongly and the hydrodynamic scaling arguments balance either interfacial and viscous or interfacial and inertial forces,¹ leading to the power law of the form $L(t)~t^{-\alpha }$.

Despite these previous efforts, it is still an open question whether the shear effect interrupts domain coarsening and leads to a non-equilibrium steady state independent of the system size, until Stansell’s group^1,16 provided convincing evidence for the existence of non-equilibrium steady states with finite domain size. Stansell et al.¹ found that the sheared binary fluid mixture could attain a dynamic steady state with finite domain lengths. Besides, they also predicted a power law dependence of characteristic length $L_{\parallel }$ and $L_{\perp }$ on shear rate as $L_{\parallel }~{\stackrel{·}{\gamma }}^{-2/3}$ and $L_{\perp }~{\stackrel{·}{\gamma }}^{-3/4}$ using the Lattice Boltzmann approach. After that, Fielding¹⁷ took a further research and found no evidence for non-equilibrium steady states in inertialess systems, which confirmed the irreplaceable role of inertia in non-equilibrium steady states.

All aforementioned studies of non-equilibrium steady states focused on binary viscous fluids with symmetric composition and ignored the role of viscoelasticity. However, most materials in the natural world exhibit some viscoelastic...