Michael D. Graham

Stochastic simulations of DNA in flow: Dynamics and the effects of hydrodynamic interactions

We present a fully parametrized bead–spring chain model for stained λ-phage DNA. The model accounts for the finite extensibility of the molecule, excluded volume effects, and fluctuating hydrodynamic interactions (HI). Parameters are determined from equilibrium experimental data for 21 μm stained λ-phage DNA, and are shown to quantitatively predict the non-equilibrium behavior of the molecule. The model is then used to predict the equilibrium and nonequilibrium behavior of DNA molecules up to 126 μm. In particular, the HI model gives results that are in quantitative agreement with experimental diffusivity data over a wide range of molecular weights. When the bead friction coefficient is fit to the experimental relaxation time at a particular molecular weight, the stretch in shear and extensional flows is adequately predicted by either a free-draining or HI model at that molecular weight, although the fitted bead friction coefficients for the two models differ significantly. In shear flow, we find two regi...

Shear-induced migration in flowing polymer solutions: Simulation of long-chain DNA in microchannels

We simulate dilute solution dynamics of long flexible polymer molecules in pressure driven flow in channels with widths of roughly 0.1-10 times the polymer bulk radius of gyration. This is done using a self-consistent coarse-grained Langevin description of the polymer dynamics and a numerical simulation of the flow in the confined geometry that is generated by the motions of polymer segments. Results are presented for a model of DNA molecules of approximately 10-100 microm contour length in micron-scale channels. During flow, the chains migrate toward the channel centerline, in agreement with well-known experimental observations. The thickness of the resulting hydrodynamic depletion layer increases with molecular weight at constant flow strength; higher molecular weight chains therefore move with a higher average axial velocity than lower molecular weight chains. In contrast, if the hydrodynamic effects of the confining geometry are neglected, depletion of concentration is observed in the center of the channel rather than at the walls, contradicting experimental observations. The mechanisms for migration are illustrated using a simple kinetic theory dumbbell model of a confined flexible polymer. The simple theory correctly predicts the trends observed in the detailed simulations. We also examine the steady-state stretch of DNA chains as a function of channel width and flow strength. The flow strength needed to stretch a highly confined chain away from its equilibrium length is shown to increase with decreasing channel width, independent of molecular weight; this is fairly well explained using a simple blob picture.

Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis

Abstract The Karhunen-Loeve (KL) decomposition is a statistical pattern analysis technique for finding the dominant structures in an ensemble of spatially distributed data. Recently the technique has been used to analyze and perform model reduction on both experimental and simulated spatiotemporal patterns from reactive and fluid-dynamical systems. We propose alternative ensembles for the KL decomposition that address some of the shortcomings of the usual procedure. Two examples are presented. In the first, the question of optimal low-dimensional bases for a reaction-diffusion model is addressed. We consider an ensemble constructed from short time integrations of a large set of initial conditions. This ensemble contains information about the global dynamics that is not contained in an ensemble comprised only of snapshots close to a particular attractor. A low-dimensional KL basis for this alternative ensemble is found to represent the dynamics better than a KL basis obtained only from points on the attractor. The second example shows how different ensemble averages give different results for the representation of “intermittent” attractors. An average based on arclength in phase space stresses the intermittent components of an attractor, features that are de-emphasized in the usual time-average based procedure.

Hydrodynamic interactions in long chain polymers: Application of the Chebyshev polynomial approximation in stochastic simulations

We have simulated Brownian bead-spring chains of up to 125 units with fluctuating hydrodynamic and excluded volume interactions using the Chebyshev polynomial approximation proposed by Fixman [Macromolecules 19, 1204 (1986)] for the square root of the diffusion tensor. We have developed a fast method to continuously determine the validity of the eigenvalue range used in the polynomial approximation, and demonstrated how this range may be quickly updated when necessary. We have also developed a weak first order semiimplicit time integration scheme which offers increased stability in the presence of steep excluded volume potentials. The full algorithm scales roughly as O(N2.25) and offers substantial computational savings over the standard Cholesky decomposition. The above algorithm was used to obtain scaling exponents for various static and zero shear rate dynamical properties, which are found to be consistent with theoretical and/or experimental predictions.

Effect of confinement on DNA dynamics in microfluidic devices

The dynamics of dissolved long-chain macromolecules are different in highly confined environments than in bulk solution. A computational method is presented here for detailed prediction of these dynamics, and applied to the behavior of ∼1–100 μm DNA in micron-scale channels. The method is comprised of a self-consistent coarse-grained Langevin description of the polymer dynamics and a numerical solution of the flow generated by the motion of polymer segments. Diffusivity and longest relaxation time show a broad crossover from free-solution to confined behavior centered about the point H≈10Sb, where H is the channel width and Sb is the free-solution chain radius of gyration. In large channels, the diffusivity is similar to that of a sphere diffusing along the centerline of a pore. For highly confined chains (H/Sb≪1), Rouse-type molecular weight scaling is observed for both translational diffusivity and longest relaxation time. In the highly confined region, the scaling of equilibrium length and relaxation t...

Theory of shear-induced migration in dilute polymer solutions near solid boundaries

In this work, a continuum theory is developed for the behavior of flowing dilute polymer solutions near solid surfaces, using a bead-spring dumbbell model of the dissolved polymer chains. Hydrodynamic interactions between the chains and the wall lead to migration away from the wall in shear flow. At steady state, this hydrodynamic effect is balanced by molecular diffusion; an analytical expression for the resulting concentration profile is derived. It is shown that the depletion layer thickness is determined by the normal stresses that develop in flow and can be much larger than the size of the polymer molecule. The transient development of this depletion layer is also studied, as well as the spatial development downstream from an entrance. Numerical and similarity solutions in these cases show that the developing concentration profile generally displays a maximum at an intermediate distance from the wall.

Effects of boundaries on pattern formation : catalytic oxidation of CO on platinum

The effect of boundaries on pattern formation was studied for the catalytic oxidation of carbon monoxide on platinum surfaces. Photolithography was used to create microscopic reacting domains on polycrystalline foils and single-crystal platinum (110) surfaces with inert titanium overlayers. Certain domain geometries give rise to patterns that have not been observed on the untreated catalyst and bring to light surface mechanisms that have no analog in homogeneous reaction-diffusion systems.

Coarse Brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation

We demonstrate how time integration of stochastic differential equations (i.e., Brownian dynamics simulations) can be combined with continuum numerical analysis techniques to analyze the dynamics of liquid crystalline polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the approach analyzes the (unavailable in closed form) “coarse” macroscopic equations, estimating the necessary quantities through appropriately initialized, short “bursts” of Brownian dynamics simulation. Through this approach, both stable and unstable branches of the equilibrium bifurcation diagram are obtained for the Doi model of LCPs and their “coarse stability” is estimated. Additional macroscopic computational tasks enabled through this approach, such as coarse projective integration and coarse stabilizing controller design, are also demonstrated.

Symmetric diblock copolymer thin films confined between homogeneous and patterned surfaces: Simulations and theory

We have investigated the ability of a simple phenomenological theory to describe the behavior of symmetric diblock copolymer thin films confined between two hard surfaces. Prior knowledge of the morphology in the confined films is crucial for applying this theory to predict the phase diagram of such systems. Taking advantage of our observations in Monte Carlo simulations, we use the theory to construct phase diagrams for thin films confined between patterned-homogeneous surfaces, and obtain good agreement with our results of simulations. Two conditions are essential for obtaining long-range ordered perpendicular lamellae: a lower stripe-patterned surface with the surface pattern period Ls comparable to the bulk lamellar period L0, and an upper neutral or weakly preferential surface. We have also examined the undulation of perpendicular lamellae between two hard surfaces. For the cases of two homogeneous (preferential) surfaces and patterned-preferential surfaces, our calculations using the phenomenologica...

Wall slip and the nonlinear dynamics of large amplitude oscillatory shear flows

Large amplitude oscillatory shear flows of polymer melts between parallel plates may exhibit complicated nonperiodic responses characteristic of quasiperiodicity or chaos. This complex time dependence is related to the wall slip exhibited by these materials. We use simple models for the fluid elasticity and slip to theoretically and computationally study the nonlinear dynamics of melts in oscillatory shear. The results indicate that both fluid elasticity and a dynamic (e.g. memory‐slip) model for the wall slip are necessary for nonperiodic dynamics to occur. Furthermore, when elasticity and a dynamic slip model are coupled, many qualitative features of the dynamics observed in the experiments can be reproduced. In particular, asymmetric periodic responses exhibiting even harmonics are found, as well as quasiperiodic and chaotic motions. Particularly interesting is the prediction of multiple stable periodic motions for a given set of parameters, depending on initial conditions. As a special case, the model...