Dynamics of semiflexible polymers in a flow field

Cross-streamline migration and near-wall depletion of elastic fibers in micro-channel flows

The complex dynamics of elastic fibers in viscous fluids are central to many biological and industrial systems. Fluid-structure interactions underlying these dynamics govern the shape and transport of flexible fibers, and understanding these interactions can help tune flow properties in applications such as microfluidic separation, printing and clogging. In this work, we use slender-body theory to study micromechanical dynamics that arise from the coupling between the elastic backbone of a fiber and the local straining flow that contributes to filament flipping and cross-streamline migration. The resulting transverse drift is unbiased in either direction in simple shear flow. However, a non-uniform shear rate results in bias towards regions of high shear, which we connect to the shape transitions during flips. We discover a depletion layer that forms near the boundaries of pressure-driven channel flow due to the competition between such a cross-streamline drift and steric exclusion from the walls. Finally, we develop scaling laws for the curvature of filaments during flip events, demonstrating the origin of the drift bias in non-uniform flows, and confirm this behavior from our simulations. Put together, these results shed light on the role of a local and dominant coupling between elasticity and viscous resistance in dictating long-term dynamics and transport of elastic fibers in confined flows.

Dynamics of flexible filaments in oscillatory shear flows

The fluid-structure interactions between flexible fibers and viscous flows play an essential role in various biological phenomena, medical problems, and industrial processes. Of particular interest is the case of particles freely transported in time-dependent flows. This work elucidates the dynamics and morphologies of actin filaments under oscillatory shear flows by combining microfluidic experiments, numerical simulations, and theoretical modeling. Our work reveals that, in contrast to steady shear flows, in which small orientational fluctuations from a flow-aligned state initiate tumbling and deformations, the periodic flow reversal allows the filament to explore many different configurations at the beginning of each cycle. Investigation of filament motion during half time periods of oscillation highlights the critical role of the initial filament orientation on the emergent dynamics. This strong coupling between orientation and deformation results in new deformation regimes and novel higher-order buckling modes absent in steady shear flows. The primary outcome of our analysis is the possibility of suppression of buckling instabilities for certain combinations of the oscillation frequency and initial filament orientation, even in very strong flows. We explain this unusual behavior through a weakly nonlinear Landau theory of buckling, in which we treat the filaments as inextensible Brownian Euler-Bernoulli rods whose hydrodynamics are described by local slender-body theory.

Self-Attractive Semiflexible Polymers under an External Force Field

The dynamical response of a tethered semiflexible polymer with self-attractive interactions and subjected to an external force field is numerically investigated by varying stiffness and self-interaction strength. The chain is confined in two spatial dimensions and placed in contact with a heat bath described by the Brownian multi-particle collision method. For strong self-attraction the equilibrium conformations range from compact structures to double-stranded chains, and to rods when increasing the stiffness. Under the external field at small rigidities, the initial close-packed chain is continuously unwound by the force before being completely elongated. For double-stranded conformations the transition from the folded state to the open one is sharp being steeper for larger stiffnesses. The discontinuity in the transition appears in the force-extension relation, as well as in the probability distribution function of the gyration radius. The relative deformation with respect to the equilibrium case along the direction normal to the force is found to decay as the inverse of the applied force.

Periodic deformation of semiflexible colloidal chains in eccentric time-varying magnetic fields

Elastic filaments driven out of equilibrium display complex phenomena that involve periodic changes in their shape. Here, the periodic deformation dynamics of semiflexible colloidal chains in an eccentric magnetic field are presented. This field changes both its magnitude and direction with time, leading to novel nonequilibrium chain structures. Deformation into S-, Z-, and 4-mode shapes arises via the propagation and growth of bending waves. Transitions between these morphologies are governed by an interplay among magnetic, viscous, and elastic forces. Furthermore, the periodic behavior leading to these structures is described by four distinct stages of motion that include rotation, arrest, bending, and stretching of the chain. These stages correspond to specific intervals of the eccentric field's period. A scaling analysis that considers the relative ratio of viscous to magnetic torques via a critical frequency illustrates how to maximize the bending energy. These results provide new insights into controlling colloidal assemblies by applying complex magnetic fields.

Wall-anchored semiflexible polymer under large amplitude oscillatory shear flow

The properties of semiflexible polymers tethered by one end to an impenetrable wall and exposed to oscillatory shear flow are investigated by mesoscale simulations. A polymer, confined in two dimensions, is described by a linear bead-spring chain, and fluid interactions are incorporated by the Brownian multiparticle collision dynamics approach. At small strain, the polymers follow the applied flow field. However, at high strain, we find a strongly nonlinear response with major conformational changes. Polymers are stretched along the flow direction and exhibit U-shaped conformations while following the flow. As a consequence of confinement in the half-space, frequency doubling in the time-dependent polymer properties appears along the direction normal to the wall.

Buckling, crumpling, and tumbling of semiflexible sheets in simple shear flow

As 2D materials such as graphene, transition metal dichalcogenides, and 2D polymers become more prevalent, solution processing and colloidal-state properties are being exploited to create advanced and functional materials. However, our understanding of the fundamental behavior of 2D sheets and membranes in fluid flow is still lacking. In this work, we perform numerical simulations of athermal semiflexible sheets with hydrodynamic interactions in shear flow. For sheets initially oriented near the flow-vorticity plane, we find buckling instabilities of different mode numbers that vary with bending stiffness and can be understood with a quasi-static model of elasticity. For different initial orientations, chaotic tumbling trajectories are observed. Notably, we find that sheets fold or crumple before tumbling but do not stretch again upon applying greater shear.

Dynamics and fragmentation of small inextensible fibres in turbulence

The fragmentation of small, brittle, flexible, inextensible fibres is investigated in a fully developed, homogeneous, isotropic turbulent flow. Such small fibres spend most of their time fully stretched and their dynamics follows that of stiff rods. They can then break through tensile failure, i.e. when the tension is higher than a given threshold. Fibres bend when experiencing a strong compression. During these rare and intermittent buckling events, they can break under flexural failure, i.e. when the curvature exceeds a threshold. Fine-scale massive simulations of both the fluid flow and the fibre dynamics are performed to provide statistics on these two fragmentation processes. This gives ingredients for the development of accurate macroscopic models, namely the fragmentation rate and daughter-size distributions, which can be used to predict the time evolution of the fibre size distribution. Evidence is provided for the generic nature of turbulent fragmentation and of the resulting population dynamics. It is indeed shown that the statistics of break-up is fully determined by the probability distribution of Lagrangian fluid velocity gradients. This approach singles out that the only relevant dimensionless parameter is a local flexibility which balances flow stretching to the fibre elastic forces. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Understanding conformational and dynamical evolution of semiflexible polymers in shear flow

A clear description of the conformational and dynamical evolution of polymer chains in shear flow is the fundamental basis of microfluidic separations and macroscopic rheological behaviors. We employ graph theory analysis to analyze the local deformation and dynamics of linear polymer chains with different rigidities in shear flow based on the simulation trajectories that record the instantaneous conformations and dynamics. Our results show that all semiflexible chains experience quasi-periodic tumbling motions when the shear strain overwhelms the U-shape (or S-shape) deformation energy barrier. More interestingly, the contact map provides solid evidence for the asymmetric deformation in the whole tumbling motion. In the stretching process: at small and intermediate shear strength, flexible polymers show a quasi-affine deformation while semiflexible ones are initially unfolded from the center of the chains, then both of them follow the extension with half dumbbell- or dumbbell-like ends; at high shear strength, all polymer chains present only a dumbbell-like extension. In the collapse process, all chains prefer to initiate the folding from chain ends. This finding can facilitate our understanding on how semiflexible polymer chains relax and dissipate the stress in shear flow.

Morphological transitions of elastic filaments in shear flow

The morphological dynamics, instabilities, and transitions of elastic filaments in viscous flows underlie a wealth of biophysical processes from flagellar propulsion to intracellular streaming and are also key to deciphering the rheological behavior of many complex fluids and soft materials. Here, we combine experiments and computational modeling to elucidate the dynamical regimes and morphological transitions of elastic Brownian filaments in a simple shear flow. Actin filaments are used as an experimental model system and their conformations are investigated through fluorescence microscopy in microfluidic channels. Simulations matching the experimental conditions are also performed using inextensible Euler-Bernoulli beam theory and nonlocal slender-body hydrodynamics in the presence of thermal fluctuations and agree quantitatively with observations. We demonstrate that filament dynamics in this system are primarily governed by a dimensionless elasto-viscous number comparing viscous drag forces to elastic bending forces, with thermal fluctuations playing only a secondary role. While short and rigid filaments perform quasi-periodic tumbling motions, a buckling instability arises above a critical flow strength. A second transition to strongly deformed shapes occurs at a yet larger value of the elasto-viscous number and is characterized by the appearance of localized high-curvature bends that propagate along the filaments in apparent "snaking" motions. A theoretical model for the as yet unexplored onset of snaking accurately predicts the transition and explains the observed dynamics. We present a complete characterization of filament morphologies and transitions as a function of elasto-viscous number and scaled persistence length and demonstrate excellent agreement between theory, experiments, and simulations.

Efficient simulation of semiflexible polymers with stiff bonds

We investigate the simulation of stiff (extensible) and rigid (inextensible) semiflexible polymers in solution. In particular, we focus on polymers represented as chains of beads, interconnected by bonds with a low to zero extensibility, and significant persistence in the bond orientations along the chain, whose dynamical behavior is described by the Langevin equation. We review the derivation of the pseudopotential needed for rigid bonds. The efficiency of a number of routines for such simulations is determined. We propose a routine for handling rigid bonds which is, for longer chains, substantially more efficient than the existing ones. We also show that for extensible polymers, the Rouse modes can be exploited to achieve highly efficient simulations. At realistic values for the extensibility, e.g., that of double-stranded DNA, the simulations are orders of magnitude faster than those for rigid bonds. With increasing stiffness, however, the allowable time step and hence the efficiency decreases, until a crossover point is reached below which the routines with rigid bonds are more efficient; we present a numerical estimate of this crossover point.

A dissipative particle dynamics study of a flexible filament in confined shear flow

In this paper, we investigate the dynamics of a tethered flexible filament due to fluid flow inside a microchannel. We use the finite sized dissipative particle dynamics (FDPD) approach to model this problem. The flexible filament is modeled as a bead-spring system with both extensional and flexural rigidity. The influence of flow rate and bending stiffness on the filament dynamics is studied in terms of the different conformational modes obtained. The competing effects of the hydrodynamic force and elastic force in the presence of Brownian thermal effects of comparable order influence the mode shapes of the filament. The dynamics of the filament motions are further analyzed using proper orthogonal decomposition. An important consequence of the dynamics of the filament is that it causes cross-flow in the micro-channel, which could potentially be exploited in micro-mixing and pumping applications. The cross stream fluid transport is observed to be more pronounced for higher bending stiffness.

Conformational Properties of Active Semiflexible Polymers

The conformational properties of flexible and semiflexible polymers exposed to active noise are studied theoretically. The noise may originate from the interaction of the polymer with surrounding active (Brownian) particles or from the inherent motion of the polymer itself, which may be composed of active Brownian particles. In the latter case, the respective monomers are independently propelled in directions changing diffusively. For the description of the polymer, we adopt the continuous Gaussian semiflexible polymer model. Specifically, the finite polymer extensibility is taken into account, which turns out to be essential for the polymer conformations. Our analytical calculations predict a strong dependence of the relaxation times on the activity. In particular, semiflexible polymers exhibit a crossover from a bending elasticity-dominated dynamics to the flexible polymer dynamics with increasing activity. This leads to a significant activity-induced polymer shrinkage over a large range of self-propulsion velocities. For large activities, the polymers swell and their extension becomes comparable to the contour length. The scaling properties of the mean square end-to-end distance with respect to the polymer length and monomer activity are discussed.

Buckling transition of a semiflexible filament in extensional flow

An analytical expression for the fluctuation-rounded stretch-coil transition of semiflexible polymers in extensional flows is derived. The competition between elasticity and tension is known to cause a buckling instability in filaments placed near hyperbolic stagnation points and the effect of thermal fluctuations on this transition has yet to receive full quantitative treatment. Motivated by the findings of recent experiments as well as our simulations, we solve for the amplitude of the first buckled mode near the onset of the instability. This reveals a stochastic supercritical bifurcation, which is in excellent agreement with full numerical simulations.

Effects of thermal noise on the transitional dynamics of an inextensible elastic filament in stagnation flow

We investigate the dynamics of a single inextensible elastic filament subject to anisotropic friction in a viscous stagnation-point flow, by employing both a continuum model represented by Langevin type stochastic partial differential equations (SPDEs) and a dissipative particle dynamics (DPD) method. Unlike previous works, the filament is free to rotate and the tension along the filament is determined by the local inextensible constraint. The kinematics of the filament is recorded and studied with normal modes analysis. The results show that the filament displays an instability induced by negative tension, which is analogous to Euler buckling of a beam. Symmetry breaking of normal modes dynamics and stretch-coil transitions are observed above the threshold of the buckling instability point. Furthermore, both temporal and spatial noise are amplified resulting from the interaction of thermal fluctuations and nonlinear filament dynamics. Specifically, the spatial noise is amplified with even normal modes being excited due to symmetry breaking, while the temporal noise is amplified with increasing time correlation length and variance.

Dynamics of a semiflexible polymer or polymer ring in shear flow

Polymers exposed to shear flow exhibit a remarkably rich tumbling dynamics. While rigid rods rotate on Jeffery orbits, a flexible polymer stretches and coils up during tumbling. Theoretical results show that in both of these asymptotic regimes the corresponding tumbling frequency f(c) in a linear shear flow of strength γ scales as a power law Wi(2/3) in the Weissenberg number Wi = γτ, where τ is a characteristic time of the polymer's relaxational dynamics. For a flexible polymer these theoretical results are well confirmed by a large body of experimental single molecule studies. However, for the intermediate semiflexible regime, especially relevant for cytoskeletal filaments like F-actin and microtubules, the situation is less clear. While recent experiments on single F-actin filaments are still interpreted within the classical Wi(2/3) scaling law, theoretical results indicated deviations from it. Here we perform extensive Brownian dynamics simulations to explore the tumbling dynamics of semiflexible polymers over a broad range of shear strength and the polymer's persistence length l(p). We find that the Weissenberg number alone does not suffice to fully characterize the tumbling dynamics, and the classical scaling law breaks down. Instead, both the polymer's stiffness and the shear rate are relevant control parameters. Based on our Brownian dynamics simulations we postulate that in the parameter range most relevant for cytoskeletal filaments there is a distinct scaling behavior with f(c) τ* = Wi(3/4)f(c)(x) with Wi = γτ* and the scaling variable x = (l(p)/L)(Wi)(-1/3); here τ* is the time the polymer's center of mass requires to diffuse its own contour length L. Comparing these results with experimental data on F-actin we find that the Wi(3/4) scaling law agrees quantitatively significantly better with the data than the classical Wi(2/3) law. Finally, we extend our results to single ring polymers in shear flow, and find similar results as for linear polymers with slightly different power laws.

Dynamics of a polyelectrolyte in simple shear flow

The configurational dynamics of a polyelectrolyte (PE), subjected to a simple shear flow, is studied using Brownian dynamics (BD) and Dissipative Particle Dynamics (DPD) simulations of a bead-spring model with explicit counterions. We explore the effect of counterion condensation on the tumbling and extension of PEs by varying the shear rates for a range of values of the electrostatic coupling parameter A (which is defined as the ratio of the Bjerrum length to the size of the monomer). In all cases, the power spectrum of Rs(t) (which characterizes the projected length of the PE in the flow direction as a function of time) exhibits a power law decay at high frequencies, similar to that for a dumbbell in shear flow. For lower values of A (A ~ 2), the tumbling of the PE is periodic and is always associated with folding and stretching, which is in contrast to the oscillatory transition between the extended and globular states seen at higher values of A (A ~ 15). We observe that for A ~ 2 the tumbling frequency decreases and the average tumbling time increases with hydrodynamic interaction (HI). For A > 15, we observe a critical shear rate γ[combining dot]c below which there is considerable counterion condensation and the PE remains in the globular state with a structure akin to that of a neutral polymer in poor solvent. The γ[combining dot]c and the behavior of the PE above the critical shear rate are dependent on the HI. For a given shear rate, when there is considerable condensed counterion fluctuation, the PE extends as a whole and then collapses by the formation of folds with no observable periodicity in tumbling. When the condensed counterion fluctuations are suppressed, the polymer exhibits periodic tumbling. Simulation artifacts resulting from the implicit nature of the solvent and that due to boundary conditions are discussed by comparing the BD results with that obtained from the DPD simulations incorporating Ewald summation for electrostatics.

Direct observation of the dynamics of semiflexible polymers in shear flow

The flow behavior of polymeric liquids can be traced back to the complex conformational dynamics of polymer molecules in shear flow, which poses a major challenge to theory and experiment alike due to the inherently large number of degrees of freedom. Here we directly determine the configurational dynamics of individual actin filaments with varying lengths in a well defined shear geometry by combining microscopy, microfluidics, and a semiautomated moving stage. This allows the identification of the microscopic mechanisms and the derivation of an analytical model for the dynamics of individual filaments based on the balance of drag, bending, and stochastic forces.

Dynamics of the primary cilium in shear flow

In this work, the equilibrium shape and dynamics of a primary cilium under flow are investigated by using both theoretical modeling and experiment. The cilium is modeled as an elastic beam that may undergo large deflection due to the hydrodynamic load. Equilibrium results show that the anchoring effects of the basal body on the cilium axoneme behave as a nonlinear rotational spring. Details of the rotational spring are elucidated by coupling the elastic beam with an elastic shell. We further study the dynamics of cilium under shear flow with the cilium base angle determined from the nonlinear rotational spring, and obtain good agreement in cilium bending and relaxing dynamics when comparing between modeling and experimental results. These results potentially shed light on the physics underlying the mechanosensitive ion channel transport through the ciliary membrane.

Fluctuations, dynamics, and the stretch-coil transition of single actin filaments in extensional flows

Semiflexible polymers subject to hydrodynamic forcing play an important role in cytoskeletal motions in the cell, particularly when filaments guide molecular motors whose motions create flows. Near hyperbolic stagnation points, filaments experience a competition between bending elasticity and tension and are predicted to display suppressed thermal fluctuations in the extensional regime and a buckling instability under compression. Using a microfluidic cross-flow geometry, we verify these predictions in detail, including a fluctuation-rounded stretch-coil transition of actin filaments.

Tension dynamics and viscoelasticity of extensible wormlike chains

The dynamic response of prestressed semiflexible biopolymers is characterized by the propagation and relaxation of tension, which arises due to the near inextensibility of a stiff backbone. It is coupled to the dynamics of contour length stored in thermal undulations but also to the local relaxation of elongational strain. We present a systematic theory of tension dynamics for stiff yet extensible wormlike chains. Our results show that even moderate prestress gives rise to distinct Rouse-like extensibility signatures in the high-frequency viscoelastic response.

Hydrodynamic interactions between two semiflexible inextensible filaments in Stokes flow

Hydrodynamic interactions between two semiflexible inextensible filaments are shown to have a significant impact on filament buckling and their subsequent motion in Stokesian fluids. In linear shear flow, hydrodynamic interactions lead to filament shear dispersion that depends on the filament aspect ratio and the initial filament separation. In linear extensional flow, hydrodynamic interactions lead to complex filament dynamics around the stagnation point. These results suggest that hydrodynamic interactions need to be taken into account to determine the self-diffusion of non-Brownian semiflexible filaments in a cellular flow [Y.-N. Young and M. J. Shelley, Phys. Rev. Lett. 99, 058303 (2007)].

Interactions of semiflexible filaments and molecular motors

This paper summarizes the results of numerical simulations of the interaction of a pair of biofilaments mediated by a molecular motor. The filaments are modeled as flexible rods, and the results are applicable to microtubules, which are relatively stiff, as well as to much softer filaments, such as actin. The results provide insight into the effects of flexibility on cytoskeleton formation and the rheology of semiflexible filament networks. The simulations are based on a nonlinear elasticity equation. The results show that flexibility enhances the tendency of filaments to align. The enhancement in turn favors the formation of large-scale structures in multifilament systems. Simulations for soft filaments show that the action of the motor can result in the formation of multiple loops of the filaments as a result of buckling, which can affect the structure of a cross-linked network and thereby its rheology. The estimate for the minimal buckling length as a function of the motor speed, the viscosity of the solvent, and the bending stiffness of the filament is derived analytically.

Stretch-coil transition and transport of fibers in cellular flows

It is shown that a slender elastic fiber moving in a Stokesian fluid can be susceptible to a buckling instability--termed the "stretch-coil" instability--when moving in the neighborhood of a hyperbolic stagnation point of the flow. When the stagnation point is part of an extended cellular flow, it is found that immersed fibers can move as random walkers across time-independent closed-streamline flow. It is also found that the flow is segregated into transport regions around hyperbolic stagnation points and their manifolds, and closed entrapment regions around elliptic points.

Tension dynamics in semiflexible polymers. II. Scaling solutions and applications

In part I [O. Hallatschek, preceding paper, Phys. Rev. E 75, 031905 (2007)] of this contribution, a systematic coarse-grained description of the dynamics of a weakly bending semiflexible polymer was developed. Here, we discuss analytical solutions of the established deterministic partial integro-differential equation for the spatiotemporal relaxation of the backbone tension. For prototypal experimental situations, such as the sudden application or release of a strong external pulling force, it is demonstrated that the tensile dynamics reflects the self-affine conformational fluctuation spectrum in a variety of intermediate asymptotic power laws. Detailed and explicit analytical predictions for the tension propagation and relaxation and corresponding results for common observables, such as the end-to-end distance, are obtained.