Bead-rod Polymer Research Articles

A Penalty Method to Model Particle Interactions in DNA-Laden Flows

We present a hybrid fluid-particle algorithm to simulate flow and transport of DNA-laden fluids in microdevices. Relevant length scales in microfluidic systems range from characteristic channel sizes of millimeters to micron scale geometric variation (e.g., post arrays) to 10 nanometers for the length of a single rod in a bead-rod polymer representation of a biological material such as DNA. The method is based on a previous fluid-particle algorithm in which long molecules are represented as a chain of connected rods, but in which the physically unrealistic behavior of rod crossing occurred. We have extended this algorithm to include screened Coulombic forces between particles by implementing a Debye-Hückel potential acting between rods. In the method an unsteady incompressible Newtonian fluid is discretized with a second-order finite difference method in the interior of the Cartesian grid domain; an embedded boundary volume-of-fluid formulation is used near boundaries. The bead-rod polymer model is fully coupled to the solvent through body forces representing hydrodynamic drag and stochastic thermal fluctuations. While intra-polymer interactions are modeled by a soft potential, polymer-structure interactions are treated as perfectly elastic collisions. We demonstrate this method on flow and transport of a polymer through a post array microchannel in 2D where the polymer incorporates more realistic physical parameters of DNA, and compare to previous simulations where rods are allowed to cross. We also show that the method is capable of simulating 3D flow in a packed bed micro-column.

A Hard Constraint Algorithm to Model Particle Interactions in DNA-Laden Flows

We present a new numerical method for particle interactions in polymer-fluid models of DNA-laden flows. The DNA is represented by a bead-rod polymer model and is fully-coupled to the fluid. The main objective in this work is to properly model polymer-polymer and polymer-surface interactions by enforcing the physical rod-rod and rod-surface non-crossing constraints. Our new method is based on a rigid constraint algorithm whereby rods elastically bounce off one another to prevent crossing, similar to our previous algorithm used to model polymer-surface interactions.

Stress, birefringence, and conformational relaxation of an initially straight stiff bead-rod polymer

The stress and optical relaxation of an initially straight stiff polymer chain are studied through Brownian dynamics simulations (based on a semiflexible bead-rod model) covering a broad range of time scales and polymer lengths. The strong stress component sigma11 (where "1" is the direction of the original alignment) is shown to be associated with the chain's longitudinal relaxation while the weak stress component sigma22 = sigma33 is shown to depend on the chain's transverse relaxation. The two independent stress components follow a different relaxation; this anisotropy is shown to result from the participation of the different relaxation modes in the transverse direction. The chain's optical relaxation is shown to be affected by the longitudinal dynamics only. The early relaxation of the strong stress component sigma11 and that of the chain's optical properties constitute a universal behavior--i.e., valid for any stiffness of the bead-rod chain, since at the early times the bending forces do not affect the longitudinal dynamics. Based on the knowledge of the physical mechanism and the chain's conformational behavior, we predict and explain the polymer stress and optical relaxation. A nonlinear stress-optic law (valid for any time and chain stiffness) is derived based on the identified relation of the chain configuration with the optical properties and the polymer stress. A coarse-grain model describing extended semiflexible bead road chains is also derived.

Brownian dynamics simulation of linear polymers under elongational flow: Bead–rod model with hydrodynamic interactions

Brownian dynamics (BD) simulations of a linear freely jointed bead–rod polymer chain with excluded volume (EV) interaction have been performed under elongational flow with and without the use of fluctuating hydrodynamic interactions (HI). The dependence of the chain size, shape and intrinsic elongational viscosity on the elongational rate ε̇ are reported. A sharp coil–stretch transition is observed when ε̇ exceeds a critical value, ε̇c. The inclusion of the HI leads to a shift in the coil–stretch transition to higher flow values. Chain deformation due to elongational flow is observed to first consist of the alignment of the chain with the direction of flow without significant chain extension followed by additional alignment of the bond vectors with the flow direction and chain extension as flow rate is increased further. The distribution function for the chain’s radius of gyration becomes significantly broader within the transition region which implies an increase in fluctuations in the chain size in this region. The structure factors parallel and perpendicular to the flow direction illustrate different elongational rate dependencies. At high rates, the structure factor in the direction of the flow exhibits an oscillating dependence which corresponds to the theoretically predicted shape for a rigid-rod model. The mean squared orientation of each bond within the chain with respect to the flow direction as function of bond number is nearly parabolic in shape with the highest degree of orientation found within the chain’s interior. The dependence of the critical elongational rate, ε̇c, on the chain length, N, is observed to be ε̇c∼N−1.96 when hydrodynamic interactions are not employed and ε̇c∼N−1.55 when they are invoked. These scaling exponents agree well with those obtained in previous BD simulations of bead-FENE (i.e., finitely extensible nonlinear elastic) spring chains as well as with the theoretical predictions of ε̇c∼N−2 and ε̇c∼N−1.5 without and with hydrodynamic interactions based on the Rouse and Zimm models, respectively.

The conformation of semi-rigid polymers during flow through a fixed fiber bed

Linear conservative birefringence measurements of dilute and semi-dilute solutions of the semi-rigid polymer xanthan gum in a glycerine and water solvent are conducted during flow through a disordered fiber bed. Even though the plug flow through the bed contains no average velocity gradient our results show that the xanthan gum molecules stretch, on average, in the direction of flow. The polymer stretch or birefringence increases with the number of fiber interactions within the bed until a steady-state conformation is attained. The degree of stretch increases monotonically with increasing De pore (the ratio of a characteristic polymer relaxation time and the characteristic flow time in a bed pore). The results show a number of qualitative differences when compared to those for the flow of flexible polyisobutylene (PIB) (Evans et al., J. Fluid Mech., 28 (1994) 319) through the same fiber bed. In particular, xanthan gum attains a steady-state conformation after convecting 12 pore-lengths in the bed (compared to approximately 25 for the PIB solution) and this value is independent of De pore whereas it increased with increasing De pore for the flexible polymer. Additionally the steady-state birefringence of the xanthan gum solutions displays no critical De pore at which the birefringence increases dramatically (as does the flexible polymer system) and instead exhibits a slow growth with increasing De pore. These unique results are interpreted and understood using novel Brownian dynamics simulations of a semi-rigid bead-rod polymer model flowing through a simulated stochastic porous bed flow field created via a spectral expansion. Our results show that the simulations can predict most of the experimental results, at least qualitatively, and in some instances quantitatively, and therefore provide an important tool for understanding these complicated phenomena.

Computer simulations of polymer chain relaxation via Brownian motion

Numerical simulations are employed to study the Brownian motion of a bead-rod polymer chain dissolved in a solvent. An investigation is conducted of the relaxation of the stress for an initially straight chain as it begins to coil.For a numerical time step δt in the simulations, conventional formulae for the stress involve averaging large ±O(1/(δt)1/2) contributions over many realizations, in order to yield an O(1) average. An alternative formula for the stress is derived which only contains O(1) contributions, thereby improving the quality of the statistics.For a chain consisting of n rods in a solvent at temperature T, the component of the bulk stress along the initial chain direction arising from tensions in the rods at the initial instant is $k\hat{T}\times n(\frac{1}{3}n^2 + n +\frac{2}{3})$. Thus the bead-rod model yields results very different from other polymer models, such as the entropic spring of Flory (1969), which would assign an infinite stress to a fully aligned chain. For rods of length l and beads of friction factor $\hat{\zeta}$ the stress decays at first on $O(\hat{\zeta}\hat{l}^2/k\hat{T}\times 1/n^2)$ time scales. On longer time scales, this behaviour gives way to a more gradual stress decay, characterized by an $O(k\hat{T}\times n)$ stress following a simple exponential decay with an $O(k\hat{T}/\hat{\zeta}\hat{l}^2\times 1/n^2)$ rate. Matching these two limiting regimes, a power law decay in time t is found with stress $O(k\hat{T}\times n^2\times (k\hat{T}\hat{t}/\hat{\zeta}\hat{l}^2)^{-1/2})$. The dominant physical processes occurring in these separate short, long and intermediate time regimes are identified.

Anisotropic friction and excluded volume effects in freely jointed bead–rod polymer chain models

We have studied the kinematics and rheological properties of undiluted polymers by modeling the fluids as a collection of interacting Kramers freely jointed bead–rod chains with anisotropic friction using Brownian dynamics simulations. The anisotropic friction depends on the instantaneous configuration of the system. The resulting stochastic differential equations have multiplicative noise. The correlation of the ‘‘Langevin’’ Brownian forces appearing in these equations turns out to be anisotropic as well. The excluded volume effect, often ignored in kinetic theory, plays an important role in the behavior of short polymer chains. The chain kinematics may be distorted in its absence. In this work, we also discuss the effects of two types of excluded volume (EV) forces (hard-sphere and soft-repulsive forces) on rheological properties of concentrated polymer solutions.