Field-driven dynamics of dilute gases, viscous liquids and polymer chains

Abstract

This thesis is concerned with the exploration of field-induced dynamical phenomena arising in dilute gases, viscous liquids and polymer chains. The problems considered herein pertain to the slip-induced motion of a rigid, spherical or nonspherical particle in a fluid in the presence of an inhomogeneous temperature or concentration field or an electric field, and the dynamics of charged polymers animated by the application of an electric field. The problems studied in this thesis are unified by the existence of a separation of length scales between the macroscopic phenomena of interest and their microscopic underpinnings, and are treated by means of coarse-graining principles that exploit this scale separation. Specifically, the first part of this thesis investigates the dynamics caused by the existence of a slip velocity at a fluid-solid interface. The macroscopic slip boundary condition obtains from the asymptotic matching of the velocity within the microscale layer of fluid adjoining the solid surface, and the velocity in the bulk fluid. In the case of a gas, the microscopic length scale is constituted by the mean free path, and the layer of gas adjoining the solid boundary having a thickness of the order of the mean free path is referred to as the Knudsen layer. The parameter representing the ratio of the mean free path to the macroscopic length scale is the Knudsen number, denoted Kn. The widely-used Navier-Stokes and Fourier equations are valid away from the solid boundary at distances large compared to the mean free path in the limit Kn < 1, and necessitate the imposition of continuum boundary conditions on the gas velocity and temperature at the outer limit of the Knudsen layer. These macroscopic equations are typically solved subject to the no-slip of velocity and the equality of the gas and solid temperatures at the solid boundary.

(cont) However, as first pointed out by Maxwell, the no-slip boundary condition fails to explain experimentally observed phenomena when imposed at the surface of a nonuniformly heated solid, and must be replaced by the thermal slip condition obtained via the asymptotic matching of the velocity within the Knudsen layer with that in the bulk gas. Slip has also been proposed to occur at liquid-solid boundaries under conditions of inhomogeneous temperature or concentration. In this thesis, we extend Faxen's laws for the force and torque acting on a spherical particle in a fluid with a prescribed undisturbed flow field to account for the existence of fluid slip at the particle surface. Additionally, we investigate the effect of particle asymmetry by studying the motion of a slightly deformed sphere in a fluid having a uniform unperturbed flow field, and demonstrate that the velocity of a force- and torque-free particle is independent of its size or shape. While the slip-induced motions studied in this thesis are presented in the context of thermally-induced slip arising from the existence of a temperature gradient, the results are equally applicable to more general phoretic transport, encompassing the electrokinetic slip condition employed in the treatment of charged particle dynamics in an electrolytic liquid. Analogous to the thermal slip condition imposed on a gas at the outer limit of the Knudsen layer, the electrokinetic slip condition is imposed at the outer limit of the layer of counterions surrounding a charged surface in an electrolytic liquid. The studies presented in this thesis have potential applications in aerosol and colloid technology, in the nonisothermal transport of particulates in porous media and MEMS devices, and in the electrophoresis of charged bodies. The behavior of a charged polymer molecule in an electric field constitutes the subject of the second part of this thesis.

(cont) Motivated by the medical and technological necessity to effect the size-separation of DNA chains in applications ranging from the Human Genome Project to DNA-based criminology, we consider specifically the dynamics of electric-field driven DNA chains in size-based separation devices. The conventional technique of constant-field gel electrophoresis is ineffective in achieving the separation of long DNA chains whose sizes exceed a few tens of kilobase pairs, owing to the fact that the velocity becomes independent of chain size for long chains in a gel. This limitation of gel electrophoresis has spurred the development of alternative separation devices, such as obstacle courses confined to microchannels wherein the obstacles may be either microfabricated or formed from the self-assembly of paramagnetic beads into columns upon the imposition of a magnetic field transverse to the channel plane. Size separation in the latter devices arises from the fact that longer chains, when driven through the channel by an applied electric field, are more likely to collide with the obstacles and take longer to disentangle from the obstacle once a collision has occurred, relative to shorter chains. Consequently, a longer chain requires more time to traverse the array compared to a shorter chain. As a model for the transient chain stretching occurring subsequent to the collision of an electrophoresing DNA molecule with an obstacle, we study the unraveling of a single, tethered polymer molecule in a uniform solvent flow field. In the context of a polymer, the microscopic length scale is associated with the size of a monomer. We, however, employ a coarse-grained representation wherein the polymer is modeled by a chain of entropic springs connected by beads, with each bead representing several monomers, thereby enabling a continuum description of the solvent. We adopt the method of Brownian dynamics applied to the bead-spring model of the polymer chain.

(cont) We consider both linear force-extension behavior, representative of chain stretching in a weak field, and the finitely-extensible wormlike chain model of DNA elasticity, which dominates chain stretching under strong fields. The results yield insight into the mechanism of tension propagation during chain unraveling, and are more generally applicable to situations involving transient stretching, such as chain interactions arising in entangled polymer solutions. We next conduct investigations of chain dynamics in obstacle-array based separation devices by means of coarse-grained stochastic modeling and Brownian dynamics simulation of a chain in a self-assembled array of magnetic beads, and predict the separation achievable among different chain sizes. We examine the influence of key parameters, namely, the applied electric field strength and the spacing between obstacles, on the separation resolution effected by the device. Our results elucidate the mechanisms of DNA dynamics in microfluidic separation devices, and are expected to aid in the design of DNA separation devices and the selection of parameters for their optimal operation.