^{1}, Hongbo Ma

^{1}, Juan J. de Pablo

^{1,a)}and Michael D. Graham

^{1,b)}

### Abstract

Theory and Brownian dynamics (BD) simulations are used to study cross-stream migration in confined dilute flowing polymer solutions, using bead-spring chain and dumbbell models for the polymer molecules. Different degrees of confinement are explored, from a chain above a single wall to slits whose widths are much bigger than the polymer contour length and radius of gyration , much bigger than the radius of gyration but comparable with the contour length , and comparable with the polymer radius of gyration . The results show that except in the latter case, polymer chains migrate in shear flow away from the confining surfaces due to the hydrodynamic interactions between chains and walls. In contrast, when , the chain migration in flow is toward the walls. This is a steric effect, caused by extension of the chain in the flow direction and corresponding shrinkage of the chains in the confined direction; here the hydrodynamic effects of each wall cancel one another out. Considering the polymer chain as a Stokeslet-doublet (point-force-dipole) as in a previously developed kinetic theory captures the correct far-field (relative to the walls) behavior. Once a finite-size dipole is used, the theory improves its near-wall predictions. In the regime , the results are significantly affected by the level of discretization of the polymer chain, i.e., number of springs, because the spatial distribution of the forces exerted by the chain on the fluid acts on the scale of the channel geometry.

This work was supported by the National Science Foundation, Grants No. EES/BES/CTS-0085560 and No. DMR-0425880 (Nanoscale Science and Engineering Center).

I. INTRODUCTION

II. POINT-DIPOLE THEORY OF POLYMER MIGRATION

III. POLYMER MODEL AND SIMULATION METHOD

IV. RESULTS AND DISCUSSION

A. Single wall migration in simple shear

B. Slit confinement: shear flow

C. Highly confined polymer chains

D. General flux expression for dumbbells

V. CONCLUSIONS

### Key Topics

- Polymers
- 30.0
- Hydrodynamics
- 27.0
- Tensor methods
- 15.0
- DNA
- 12.0
- Microscale flows
- 12.0

## Figures

Schematic of different regimes of confinement: (a) Single wall confinement; (b) weak confinement: ; (c) strong confinement: ; and (d) extreme confinement: .

Schematic of different regimes of confinement: (a) Single wall confinement; (b) weak confinement: ; (c) strong confinement: ; and (d) extreme confinement: .

Time evolution of axially averaged fluorescence intensity of fluorescent labeled T2-DNA solution as a function of cross-sectional position. The channel walls are at . The solution is undergoing oscillatory pressure-driven flow at a maximum strain rate of and a frequency of in a microchannel (Ref. 12). The bright band at the center indicates higher concentration of T2-DNA molecule and the dark region represents the depletion layer near the channel walls.

Time evolution of axially averaged fluorescence intensity of fluorescent labeled T2-DNA solution as a function of cross-sectional position. The channel walls are at . The solution is undergoing oscillatory pressure-driven flow at a maximum strain rate of and a frequency of in a microchannel (Ref. 12). The bright band at the center indicates higher concentration of T2-DNA molecule and the dark region represents the depletion layer near the channel walls.

Steady-state center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation at , 5, 10, and 20 in simple shear flow. The concentration is normalized by its value at .

Steady-state center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation at , 5, 10, and 20 in simple shear flow. The concentration is normalized by its value at .

Migration velocity scaled with the point-dipole value for different dumbbell (force-dipole) sizes, as a function of distance from the wall.

Migration velocity scaled with the point-dipole value for different dumbbell (force-dipole) sizes, as a function of distance from the wall.

Near-field center-of-mass steady-state concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation and finite-size dumbbells, and the BD simulation at in simple shear flow.

Near-field center-of-mass steady-state concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation and finite-size dumbbells, and the BD simulation at in simple shear flow.

Steady-state center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation of 10 springs chains, at and 10 in simple shear flow.

Steady-state center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation of 10 springs chains, at and 10 in simple shear flow.

Steady-state center-of-mass concentration profiles predicted by theory, using far-field and single-reflection approximations, and the BD simulation at , 5, and 20 in shear flow.

Steady-state center-of-mass concentration profiles predicted by theory, using far-field and single-reflection approximations, and the BD simulation at , 5, and 20 in shear flow.

Steady-state center-of-mass concentration profiles predicted by the BD simulation at in shear flow, for different polymer discretizations: , 5, and 10.

Steady-state center-of-mass concentration profiles predicted by the BD simulation at in shear flow, for different polymer discretizations: , 5, and 10.

Schematic of two different discretization levels of the same molecule (a) dumbbell: the effect of the molecule on the solvent is approximated as two point forces with large separation; (b) chain: the effect of the molecule on the solvent is approximated as several point forces with smaller separation.

Schematic of two different discretization levels of the same molecule (a) dumbbell: the effect of the molecule on the solvent is approximated as two point forces with large separation; (b) chain: the effect of the molecule on the solvent is approximated as several point forces with smaller separation.

Steady-state center-of-mass concentration profiles predicted by the theory, using far-field and single-reflection approximations, and the BD simulation at in shear flow. The steady-state center-of-mass concentration profile at equilibrium and the bead-distribution from the simulation at are also shown.

Steady-state center-of-mass concentration profiles predicted by the theory, using far-field and single-reflection approximations, and the BD simulation at in shear flow. The steady-state center-of-mass concentration profile at equilibrium and the bead-distribution from the simulation at are also shown.

Steady-state center-of-mass concentration profiles predicted by the BD simulation of chains for a highly confined polymer solution, .

Steady-state center-of-mass concentration profiles predicted by the BD simulation of chains for a highly confined polymer solution, .

Steady-state bead-concentration profiles predicted by the BD simulation of chains for a highly confined polymer solution, .

Steady-state bead-concentration profiles predicted by the BD simulation of chains for a highly confined polymer solution, .

Polymer stretch as a function of the wall-normal direction, , for (no flow); .

Polymer stretch as a function of the wall-normal direction, , for (no flow); .

Polymer stretch in the flow direction, , as a function of the wall normal direction, ; .

Polymer stretch in the flow direction, , as a function of the wall normal direction, ; .

Polymer stretch in the confined direction, , as a function of the wall normal direction, ; .

Polymer stretch in the confined direction, , as a function of the wall normal direction, ; .

Schematic of the hydrodynamic migration mechanism (a) : wall-induced migration—momentum diffusion to the wall and back to the particle is fast; (b) : No wall-induced migration—the shear flow distorts the velocity perturbation due to the particle so that the particle is not affected by the presence of the wall.

Schematic of the hydrodynamic migration mechanism (a) : wall-induced migration—momentum diffusion to the wall and back to the particle is fast; (b) : No wall-induced migration—the shear flow distorts the velocity perturbation due to the particle so that the particle is not affected by the presence of the wall.

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