An iterative method for hydrodynamic interactions in Brownian dynamics simulations of polymer dynamics

Effect of Hydrodynamic Interactions and Flow on Charge Transport in Redox-Active Polymer Solutions

Redox-active polymers (RAPs) are a subclass of polyelectrolytes that can store charge and undergo redox self-exchange reactions. RAPs are of great interest in the field of redox flow batteries (RFBs) due to their ability to quickly charge and discharge, their chemical modularity, and their molecular size. However, designing RAPs for efficient charge transport at the molecular level requires a fundamental understanding of the charge transport mechanisms that occur in RFBs. Previous work from our group has explored these mechanisms, and in this paper, we seek to improve upon the previous model by incorporating both hydrodynamic interactions (HIs) and out-of-equilibrium dynamics, which are both highly pertinent to flow battery systems. We use a hybrid Brownian dynamics and Monte Carlo simulation to model redox-active polymer chains in both dilute and semidilute solutions. This model is used to show that HI is an important feature when charge hopping is not the major mechanism for charge displacement and leads to more rapid segmental and translational motion of polymer chains that expedites charge transport at low polymer concentrations. We demonstrate that strong extensional flows may result in either enhanced or decreased transport depending on the fraction of charges present on the RAP chain. We show that flow not only can promote charge transport by extending polymer conformations but can also suppress nonadjacent charge hopping processes that are important for transport at high charge fractions. Shear flows can similarly enhance charge transport through chain extension, but tumbling dynamics lead to oscillatory displacements that become dominant features with high charge fractions and strong flows.

Brownian dynamics simulations of bottlebrush polymers in dilute solution under simple shear and uniaxial extensional flows

We study the dynamics of bottlebrush polymer molecules in dilute solutions subjected to shear and uniaxial extensional flows using Brownian dynamics simulations with hydrodynamic interaction (HI). Bottlebrush polymers are modeled using a coarse-grained representation, consisting of a set of beads interacting pairwise via a purely repulsive potential and connected by finitely extensible nonlinear springs. We present the results for molecular stretching, stress, and solution viscosity during the startup of flow as well as under steady state as a function of side chain length while keeping the backbone length fixed. In extensional flow, the backbone fractional extension and the first normal stress difference decrease with an increase in side chain length at a fixed Weissenberg number (Wi). Using simulation results both in the presence of and in the absence of HI, we show that this is primarily a consequence of steric interaction resulting from the dense grafting of side chains. In shear flow, we observe a shear-thinning behavior in all cases, although it becomes less pronounced with increasing side chain length. Furthermore, nonmonotonicity in the backbone fractional extension is observed under shear, particularly at high Wi. We contextualize our simulation results for bottlebrush polymers with respect to existing studies in the literature for linear polymers and show that the unique dynamical features characterizing bottlebrush polymers arise on account of their additional molecular thickness due to the presence of densely grafted side chains.

Orientationally Averaged Version of the Rotne-Prager-Yamakawa Tensor Provides a Fast but Still Accurate Treatment of Hydrodynamic Interactions in Brownian Dynamics Simulations of Biological Macromolecules

The Brownian dynamics (BD) simulation technique is widely used to model the diffusive and conformational dynamics of complex systems comprising biological macromolecules. For the diffusive properties of macromolecules to be described correctly by BD simulations, it is necessary to include hydrodynamic interactions (HIs). When modeled at the Rotne-Prager-Yamakawa (RPY) level of theory, for example, the translational and rotational diffusion coefficients of isolated macromolecules can be accurately reproduced; when HIs are neglected, however, diffusion coefficients can be underestimated by an order of magnitude or more. The principal drawback to the inclusion of HIs in BD simulations is their computational expense, and several previous studies have sought to accelerate their modeling by developing fast approximations for the calculation of the correlated random displacements. Here, we explore the use of an alternative way to accelerate the calculation of HIs, i.e., by replacing the full RPY tensor with an orientationally averaged (OA) version which retains the distance dependence of the HIs but averages out their orientational dependence. We seek here to determine whether such an approximation can be justified in application to the modeling of typical proteins and RNAs. We show that the use of an OA-RPY tensor allows translational diffusion of macromolecules to be modeled with very high accuracy at the cost of rotational diffusion being underestimated by ∼25%. We show that this finding is independent of the type of macromolecule simulated and the level of structural resolution employed in the models. We also show, however, that these results are critically dependent on the inclusion of a non-zero term that describes the divergence of the diffusion tensor: when this term is omitted from simulations that use the OA-RPY model, unfolded macromolecules undergo rapid collapse. Our results indicate that the orientationally averaged RPY tensor is likely to be a useful, fast, approximate way of including HIs in BD simulations of intermediate-scale systems.

An Orientationally Averaged Version of the Rotne-Prager-Yamakawa Tensor Provides A Fast But Still Accurate Treatment Of Hydrodynamic Interactions In Brownian Dynamics Simulations Of Biological Macromolecules

The Brownian dynamics (BD) simulation technique is widely used to model the diffusive and conformational dynamics of complex systems comprising biological macromolecules. For the diffusive properties of macromolecules to be described correctly by BD simulations, it is necessary to include hydrodynamic interactions (HI). When modeled at the Rotne-Prager-Yamakawa (RPY) level of theory, for example, the translational and rotational diffusion coefficients of isolated macromolecules can be accurately reproduced; when HIs are neglected, however, diffusion coefficients can be underestimated by an order of magnitude or more. The principal drawback to the inclusion of HIs in BD simulations is their computational expense, and several previous studies have sought to accelerate their modeling by developing fast approximations for the calculation of the correlated random displacements. Here we explore the use of an alternative way to accelerate calculation of HIs, i.e., by replacing the full RPY tensor with an orientationally averaged (OA) version which retains the distance dependence of the HIs but averages out their orientational dependence. We seek here to determine whether such an approximation can be justified in application to the modeling of typical proteins and RNAs. We show that the use of an OA RPY tensor allows translational diffusion of macromolecules to be modeled with very high accuracy at the cost of rotational diffusion being underestimated by ∼25%. We show that this finding is independent of the type of macromolecule simulated and the level of structural resolution employed in the models. We also show, however, that these results are critically dependent on the inclusion of a non-zero term that describes the divergence of the diffusion tensor: when this term is omitted from simulations that use the OA RPY model, unfolded macromolecules undergo rapid collapse. Our results indicate that the orientationally averaged RPY tensor is likely to be a useful, fast approximate way of including HIs in BD simulations of intermediate-scale systems.

Hydrodynamic interactions in topologically linked ring polymers

Despite decades of interdisciplinary research on topologically linked ring polymers, their dynamics remain largely unstudied. These systems represent a major scientific challenge as they are often subject to both topological and hydrodynamic interactions (HI), which render dynamical solutions either mathematically intractable or computationally prohibitive. Here we circumvent these limitations by preaveraging the HI of linked rings. We show that the symmetry of ring polymers leads to a hydrodynamic decoupling of ring dynamics. This decoupling is valid even for nonideal polymers and nonequilibrium conditions. Physically, our findings suggest that the effects of topology and HI are nearly independent and do not act cooperatively to influence polymer dynamics. We use this result to develop highly efficient Brownian dynamics algorithms that offer enormous performance improvements over conventional methods and apply these algorithms to simulate catenated ring polymers at equilibrium, confirming the independence of topological effects and HI. The methods developed here can be used to study and simulate large systems of linked rings with HI, including kinetoplast DNA, Olympic gels, and poly[n]catenanes.

Simulation of semidilute polymer solutions in planar extensional flow via conformationally averaged Brownian noise

The dynamics and rheology of semidilute polymer solutions in strong flows are of great practical relevance. Processing applications can in principle be designed utilizing the relationship between nonequilibrium polymer conformations and the material properties of the solution. However, the interplay between concentration, flow, hydrodynamic interactions (HIs), and topological interactions which govern semidilute polymer dynamics is challenging to characterize. Brownian dynamics (BD) simulations are particularly valuable as a way to directly visualize how molecular interactions arise in these systems and are quantitatively comparable to single-molecule experiments. However, such simulations are often computationally intractable and are limited by the need to calculate the correlated Brownian noise via decomposition of the diffusion tensor. Previously, we have introduced an iterative conformational averaging (CA) method for BD simulations which bypasses these limitations by preaveraging the HI and Brownian noise in an iterative procedure. In this work, we generalize the CA method to flowing semidilute solutions by introducing a conformation dependent diffusion tensor and a strain dependent approximation to the conformationally averaged Brownian noise. We find that this approach nearly quantitatively reproduces both transient and steady state polymer dynamics and rheology while achieving an order of magnitude computational acceleration. We then utilize the CA method to investigate the concentration and flow rate dependence of polymer dynamics in planar extensional flows. Our results are consistent with previous experimental and simulation studies and provide a detailed view of broad conformational distributions in the semidilute regime. We observe interconversion between stretched and coiled states at steady state, which we conjecture occur due to the effect of concentration on the conformation dependent polymer drag. Additionally, we observe transient flow-induced intermolecular hooks in the startup of flow which lead to diverse and unique stretching pathways.

Ring polymer dynamics and tumbling-stretch transitions in planar mixed flows

The properties of dilute polymer solutions are governed by the conformational dynamics of individual polymers which can be perturbed in the presence of an applied flow. Much of our understanding of dilute solutions comes from studying how flows manipulate the molecular features of polymer chains out of equilibrium, primarily focusing on linear polymer chains. Recently there has been an emerging interest in the dynamics of nonlinear architectures, particularly ring polymers, which exhibit surprising out-of-equilibrium dynamics in dilute solutions. In particular, it has been observed that hydrodynamics can couple to topology in planar elongational and shear flows, driving molecular expansion in the nonflow direction that is not observed for linear chains. In this paper, we extend our understanding of dilute ring polymer dynamics to mixed flows, which represent flow profiles intermediate between simple shear or planar elongation. We map the conformational behaviors at a number of flow geometries and strengths, demonstrating transitions between coiled, tumbling, and stretched regimes. Indeed, these observations are consistent with how linear chains respond to mixed flows. For both linear and ring polymers, we observe a marked first-order-like transition between tumbling and stretched polymers that we attribute to a dynamic energy barrier between the two states. This manifests as bimodal extension distributions in a narrow range of flow strengths and geometries, with the primary difference between rings and linear chains being the presence of molecular expansion in the vorticity direction.

Nanofluid Dynamics of Flexible Polymeric Nanoparticles Under Wall Confinement

Describing the hydrodynamics of nanoparticles in fluid media poses interesting challenges due to the coupling between the Brownian and hydrodynamic forces at the nanoscale. We focus on multiscale formulations of Brownian motion and hydrodynamic interactions (HI) of a single flexible polymeric nanoparticle in confining flows using the Brownian Dynamics method. The nanoparticle is modeled as a self-avoiding freely jointed polymer chain that is subject to Brownian forces, hydrodynamics forces, and repulsive interactions with the confining wall. To accommodate the effect of the wall, the hydrodynamic lift due to the wall is included in the mobility of a bead of the polymer chain which depends on its proximity to the wall. Using the example of a flexible polymeric nanoparticle, we illustrate temporal dynamics pertaining to the colloidal scale as well as the nanoscale.

Optimization of flow-focusing devices for homogeneous extensional flow

We present a methodology for the shape optimization of flow-focusing devices with the purpose of creating a wide region of homogeneous extensional flow, characterized by a uniform strain-rate along the centerline of the devices. The numerical routines employed include an optimizer, a finite-volume solver, and a mesh generator operating on geometries with the walls parameterized by Bézier curves. The optimizations are carried out for devices with different geometric characteristics (channel aspect ratio and length). The performance of the optimized devices is assessed for varying Reynolds numbers, velocity ratio between streams, and fluid rheology. Brownian dynamics simulations are also performed to evaluate the stretching and relaxation of λ-DNA molecules in the devices. Overall, the optimized flow-focusing devices generate a homogeneous extensional flow over a range of conditions typically found in microfluidics. At high Weissenberg numbers, the extension of λ-DNA molecules in the optimized flow-focusing devices is close to that obtained in an ideal planar extensional flow with an equivalent Hencky strain. The devices presented in this study can be useful in microfluidic applications taking advantage of homogeneous extensional flows and easy control of the Hencky strain and strain-rate.

Multi-resolution polymer Brownian dynamics with hydrodynamic interactions

A polymer model given in terms of beads, interacting through Hookean springs and hydrodynamic forces, is studied. A Brownian dynamics description of this bead-spring polymer model is extended to multiple resolutions. Using this multiscale approach, a modeller can efficiently look at different regions of the polymer in different spatial and temporal resolutions with scalings given for the number of beads, statistical segment length, and bead radius in order to maintain macro-scale properties of the polymer filament. The Boltzmann distribution of a Gaussian chain for differing statistical segment lengths gives a diffusive displacement equation for the multi-resolution model with a mobility tensor for different bead sizes. Using the pre-averaging approximation, the translational diffusion coefficient is obtained as a function of the inverse of a matrix and then in closed form in the long-chain limit. This is then confirmed with numerical experiments.