Tumbling and Deformation of Isolated Polymer Chains in Shearing Flow Indranil Saha Dalal, Alex Albaugh, Nazish Hoda, and Ronald G. Larson* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States ABSTRACT: Using Brownian dynamics simulations of polymer chains over a wide range of resolution, we find universal scaling laws for polymer coil dimensions and tumbling time. At high shear rates γ̇ , the coil thickness in the gradient direction becomes independent of chain length, scaling as N K 0 γ̇ -1/4 , where N K denotes the numbers of Kuhn steps, and the tumbling time scales as N K γ̇ -3/4 , correcting scaling laws presented in prior studies. We find this to be a consequence of the formation of loops whose length is limited by the time required to stretch them and derive scaling laws from a balance of convection and diffusion of monomers. We find that, in the absence of hydrodynamic interaction (HI) and excluded volume (EV), for wormlike chains, the shrinkage in chain stretch observed at ultrahigh shear rates is pushed out to arbitrarily high shear rates if the chain is resolved increasingly finely below the persistence length. Finally, scaling laws in the presence of excluded volume and hydrodynamic interactions are derived that are expected to be valid for long chains at high shear rates. ■ INTRODUCTION The dynamics of isolated flexible and semiflexible polymer chains in shear flow has attracted considerable attention and is thought to be well understood as a result of single-molecule imaging of DNA molecules 1-3 and Brownian dynamics simulations. 4,5 These studies have shown that the average molecular extension of a polymer chain in the flow direction (x) increases with shear rate (γ̇ ) and reaches a plateau at high shear rates. The failure to reach full extension at high shear rates is a consequence of repeated end-over-end tumbling of the polymer molecule, due to the equal strength of the extensional and rotational components in a shear flow. The plateau of polymer extension at high shear rate was assumed to be the asymptotic response at high shear rates, until some recent simulations, 6,9,12 performed over a wide range of shear rates, revealed a highly nonmonotonic response in chain stretch. Quite surprisingly, without excluded volume (EV) and hydrodynamic interactions (HI), the average chain stretch was shown to reach a maximum and then decrease at the highest shear rates. (This effect almost vanished in the presence of EV but became more pronounced again when both HI and EV were included. 6 ) We show in what follows that this non- monotonic behavior disappears in the absence of EV and HI when a bending potential is included in the chain model to represent more accurately the behavior of a semiflexible wormlike chain, such as double-stranded DNA or other biopolymers. In addition, coarse-grained simulations 7,8 have indicated that the tumbling time decreases with shear rate as γ̇ 2/3 at high shear rates, which was further confirmed by a theoretical analysis. 11 For a DNA chain, the measured tumbling time approximately followed this power law for shear flows with Wi > 10, 7,8 where Wi is the Weissenberg number, which is the product of shear rate γ̇ and the relaxation time of the polymer chain. It seems doubtful, however, that results using coarse-grained chains will be accurate at high shear rates, where small subsections of the chain are excited by the flow and perturbed away from equilibrium. Using a fine-grained bead-rod model, we obtain a more accurate scaling law for the tumbling time, and derive this law theoretically, in what follows. At high shear rates, in the absence of EV and HI, the bead-rod model shows that radius of gyration of the chain in the flow gradient direction (R gy ) scales with shear rate as R gy ∼ γ̇ -1/4 . 5 A Graetz-Le ̀ vê que analysis in that study was consistent with this shear-rate scaling but also predicted a scaling with chain length, as R gy ∼ N K 1/4 , where N K is the number of Kuhn steps in the chain. We show below, however, that this latter prediction is incorrect and derive the correct result from a new physical picture of the chain dynamics under fast shearing flow. In this article, we resolve the physics for both the deformations and tumbling dynamics of chains over a wide range of chain lengths in steady shear flow using both fine- grained and coarse-grained models, encompassing both slow and fast shearing. The results are explained using simple balances of diffusion and convection of chain segments. In the absence of HI and EV, at high shear rates, chain deformations depend on the discretization level of the chain model; for insufficiently discretized models there is a decrease in chain Received: July 11, 2012 Revised: November 8, 2012 Published: November 27, 2012 Article pubs.acs.org/Macromolecules © 2012 American Chemical Society 9493 dx.doi.org/10.1021/ma3014349 | Macromolecules 2012, 45, 9493-9499