Hydrodynamic interactions are a key component of the dynamics of polymer solutions; for example, the diffusion coefficient of a linear polymer with N segments scales as N -0.6 with hydrodynamic interactions but as N -1 in the free-draining limit. Hydrodynamic interactions arise from the thermal motion of individual monomers, which causes flow in the surrounding solvent that is propagated by viscous diffusion of momentum to distant monomers. We have adapted the lattice-Boltzmann model to solutions of flexible polymers [1], so as to investigate hydrodynamic effects in confined geometries. The results are similar to Brownian dynamics, but at reduced computational cost for large numbers of segments. While the maximum chain length that can be studied using Brownian dynamics is of the order of 100 beads [2], we have simulated chains with up to 1024 beads per polymer. We have used this method to study the effects of flow fields and external forces on the migration of a confined polymer chain.
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Center-of-mass distribution of a confined polymer in an external force field. Results with a force-field alone (left) are compared with a combined force field and flow field acting in opposition to each other (right). The flow field was characterized by a mean shear rate γ, with Pef = γR 2/D = 12.5 |
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Recently, we have calculated the center-of-mass distribution of a confined polymer that is subjected to a combination of external force and pressure-driven flow [3]. Previous studies [2,4] have focused on migration in a pressure driven flow only. The new simulations show that a polymer driven by an external force parallel to the channel walls migrates cross-stream, towards the channel center (left). The strength of the applied force is characterized by the polymer Peclet number Pe = UR/D, where U is the mean polymer velocity, R is its radius of gyration and D is its diffusivity. At Peclet numbers in excess of 100, this transverse migration results in nonuniform center of mass distribution with the polymer concentrated in the middle of the channel, as in the case of a pressure-driven flow [2,4]. However, if hydrodynamic interactions are neglected there is no migration, and the distribution is uniform at all Peclet numbers.
An external force and pressure driven flow working in conjunction lead to enhanced migration towards the center, when compared to either force or flow acting alone. However, an external force acting in opposition to a weak pressure-driven flow leads to more complex migration, with the polymer moving towards the channel walls when the external force is small, and towards the channel center when the force is large. Our results are in good agreement with experimental measurements on DNA driven through microchannels by combinations of electric fields and pressure-driven flow [5]. The mechanisms explaining these migrations are illustrated in the figure below.
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Illustration of the different migration mechanisms. In each figure the solid lines represent the forces while the dashed lines represent the velocities generated by these forces. A) The lift due to shear: the tension in the polymer near a solid boundary generates a net velocity away from the boundary. B) Rotation due to the external field: the velocity field due to the external force on each bead results in a rotation about the center of mass of the polymer. C) Drift of a rotated polymer: two beads oriented at an angle to the external force drift due to the hydrodynamic interactions between the beads. |
In a pressure-driven flow, the local shear rate near a boundary generates a tension in the polymer. The flow field generated by a bead interacts with the boundary to produce an upward force on the other bead, leading to a net lift away from the boundary [6]. In an external force the wall mediated hydrodynamic produces a rotation of the polymer, with the leading end pointing towards the center; the action of the force on the oriented polymer again leads to migration towards the center of the channel. There is a third mechanism, which operates in the bulk of the channel, and requires the combined action of flow and force. The shear flow rotates the polymer and the applied force causes it to migrate, either towards the center or towards the wall depending on the direction of the force. In combination these mechanisms explain both the numerical and experimental observations [3].