Cross-stream migration in dilute solutions of rigid polymers undergoing rectilinear flow near a wall Joontaek Park, Jonathan M. Bricker, and Jason E. Butler * Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, USA Received 27 July 2007; published 22 October 2007 Kinetic theory is used to investigate cross-stream migration of a rigid polymer undergoing rectilinear flow in the vicinity of a wall. Hydrodynamic interactions between the polymers and the boundary result in a cross- stream migration. In simple shear flow, polymers migrate away from the wall, creating a depletion layer in the vicinity of the wall which thickens as the flow strength increases relative to the Brownian force. In pressure- driven flow, an off-center maximum in the center-of-mass distribution occurs due to a competition between hydrodynamic interactions with the wall and the anisotropic diffusivity induced by the inhomogeneous flow field. DOI: 10.1103/PhysRevE.76.040801 PACS numbers: 83.80.Rs, 47.15.G, 47.27.nd, 47.57.ef Flexible polymers in dilute solution migrate across streamlines in simple shear and pressure-driven flows. Though the origin and direction of the migration were con- troversial 1, recent work 2–5 has clarified that flexible polymers primarily migrate away from bounding walls due to a hydrodynamic lift force. The local shear flow extends the polymer, generating tension in the chain and an addi- tional flow field around the polymer. The flow field becomes asymmetric near a no-slip boundary and results in a net drift away from the wall for both simple shear and pressure- driven flow. In inhomogeneous flows, the variation in the local shear rate alters the position-dependent conformation and consequent diffusivity transverse to the flow. This addi- tional mechanism results in a weak displacement of the poly- mers away from the centerline in pressure-driven flow, though the net migration still occurs away from the wall unless hydrodynamic interactions with the wall are screened, as occurs for highly confined polymers. The mechanism and direction of migration remain unclear for rigid polymers in dilute solution. Measurements on semi- rigid xanthan molecules in pressure-driven flow indicate mi- gration away from the wall, resulting in a depletion layer 6. When considering only steric interactions with the walls, simulations of a rigid dumbbell predicted a limited increase of the depletion layer at sufficiently high shear 7. However, mechanisms based on the anisotropic diffusivity predict a net migration of rigid polymers toward the wall in pressure- driven flow 8,9, similar to the mechanism for flexible poly- mers. Most recently, simulations of rigid polymers in pressure-driven flow predicted migration away from the wall 10, though the depletion layer is larger than predicted by steric interactions alone. The authors proposed that a subtle combination of orientation effects and hydrodynamic inter- actions with the walls produces the overall migration. We develop a kinetic theory for the migration of a dilute solution of rigid polymers undergoing rectilinear flows and include hydrodynamic interactions with the bounding walls to confirm the results of Ref. 10 and clarify the origins of the observed migration. The theory contains approximations similar to those made for flexible polymers 2: the polymer distribution function is factorized into a product of a center- of-mass and orientation distribution and a far-field approxi- mation for the hydrodynamic interaction with the bounding wall is made. We first derive and present results for a rigid polymer undergoing simple shear flow near a single wall and show that rigid polymers migrate away from the wall due to hydrodynamic interactions with the wall. The theory is ex- tended to pressure-driven flow between two bounding walls; in this case, we observe an off-center maximum in the center-of-mass distribution due to a competition between hy- drodynamic interactions with the wall and the anisotropic diffusivity induced by the inhomogeneous flow field. The evolution of a rigid polymer in solution see Fig. 1 is governed by a continuity equation for the distribution func- tion, r c , p , t, of the center-of-mass, r c , and orientation, p,  t =-  · r ˙ c  -  p · p ˙  . 1 The probability distribution function is separated into a center-of-mass, n, and orientation distribution function , r c , p, t = nr c , tr c , p, t , 2 where nr c , t = r c , p , tdp. Integrating Eq. 1 over p and solving for the steady result gives *butler@che.ufl.edu FIG. 1. A rigid polymer of length L and aspect ratio A sus- pended in a shear flow of strength  ˙ . The coordinate s describes positions along the polymer axis, and a no-slip boundary is located at y =0. PHYSICAL REVIEW E 76, 040801R2007 RAPID COMMUNICATIONS 1539-3755/2007/764/0408014 ©2007 The American Physical Society 040801-1