Session B49: Focus Session: Long-time, Entangled Dynamics in Polymers - Rods, Rings, Fibers, Particle Tracking

An improved dissipative particle dynamics model for simulation of entangled polymers

We develop an improved polymer model to capture entanglements within the DPD framework by using simplified bond-bond repulsive interactions to prevent bond crossings. We show that structural and thermodynamic properties can be improved by applying a segmental repulsive potential (SRP) that is a function of the distance between the midpoints of the segments, rather than the minimum distance between segments. The alternative approach, termed the modified segmental repulsive potential (mSRP), is shown to produce chain structures and thermodynamic properties that are similar to the softly-repulsive, flexible chains of standard DPD. Parameters for the mSRP are determined from topological, structural and thermodynamic considerations. The effectiveness of the mSRP in capturing entanglements is demonstrated by calculating the diffusion and mechanical properties of an entangled polymer melt. This improved DPD method was used in simulations for entangled polymer networks to explore impact of branched architectures on the mechanical response to the tensile and compressive deformation.   

Theory of the nonlinear rheology of topologically entangled rod fluids

Our first-principles microscopic theory of the tube confinement field and dynamics of topologically entangled rod fluids is extended to describe nonlinear rheology. Stress generically weakens tube constraints, resulting in a competition between reptation and transverse activated entropic barrier hopping. For a step-strain deformation, four distinct nonlinear relaxation regimes are predicted with increasing strain amplitude: quiescent-like reptation, strongly accelerated reptation due to stress-induced tube dilation, relaxation dominated by lateral barrier hopping, and, beyond a critical strain of order unity, an initially complete destruction of the tube constraint (microscopic yielding) followed by a re-entanglement process with complex kinetics and multi-step stress relaxation. A theory for continuous start up shear has also been formulated. In the nonlinear regime, deformation-rate-dependent stress overshoots are predicted. In the nonequilibrium steady state, strong shear thinning occurs determined largely by the rate-dependent dilated tube diameter. At very high Weissenberg numbers, a stress plateau in the flow curve is predicted and relaxation is controlled by a convective-constraint-release-like process that self-consistently emerges within the theory.   

Theoretical and computational studies of entangled rod-coil block copolymer diffusion

Despite continued interest in the thermodynamics of rod-coil block copolymers for functional nanostructured materials in organic electronics and biomaterials, relatively few studies have investigated the dynamics of these systems which are important for understanding diffusion, mechanics, and self-assembly kinetics. Here, the diffusion of coil-rod-coil block copolymers through entangled melts is simulated using the Kremer-Grest molecular dynamics model, demonstrating that the mismatch between the curvature of the rod and coil blocks results in dramatically slower reptation through the entanglement tube. For rod lengths near the tube diameter, this hindered diffusion is explained by a local curvature-dependent free energy penalty produced by the curvature mismatch, resulting in a rough energy surface as the rod moves along the tube contour. Compared to coil homopolymers which reptate freely along the tube, rod-coil block copolymers undergo an activated diffusion process which is considerably slower as the rod length increases. For large rods, diffusion of the rod through the tube only occurs when the coil blocks occupy straight entanglement tubes, which requires ``arm retraction'' as the dominant relaxation mechanism.   

Self-Consistent Field Theory of Gaussian Ring Polymers

Ring polymers, being free from chain ends, have fundamental importance in understanding the polymer statics and dynamics which are strongly influenced by the chain end effects. At a glance, their theoretical treatment may not seem particularly difficult, but the absence of chain ends and the topological constraints make the problem non-trivial, which results in limited success in the analytical or semi-analytical formulation of ring polymer theory. Here, I present a self-consistent field theory (SCFT) formalism of Gaussian (topologically unconstrained) ring polymers for the first time. The resulting static property of homogeneous and inhomogeneous ring polymers are compared with the random phase approximation (RPA) results. The critical point for ring homopolymer system is exactly the same as the linear polymer case, $\chi N$ = 2, since a critical point does not depend on local structures of polymers. The critical point for ring diblock copolymer melts is $\chi N \approx$ 17.795, which is approximately 1.7 times of that of linear diblock copolymer melts, $\chi N \approx$ 10.495. The difference is due to the ring structure constraint.   

Self-consistent field theory for polymer dynamics

We develop a self-consistent field theory (SCFT) for polymer dynamics. We reformulate a Rouse model for interacting monomers as a dynamical functional integral over field variables, using standard techniques. Novel aspects include our use of the functional Fokker-Planck equation to describe single-chain dynamics, and our extremization of the functional integral, resulting in a set of self-consistent equations for the time-dependent monomer density and the mean force field on a monomer. Our theory is distinct from published dynamical SCFTs that combine elements of equilibrium SCFT with phenomenological dynamical evolution schemes. The time scale in our theory is known exactly; no phenomenological kinetic coefficient needs to be introduced. Dynamical quantities in our theory have analogs in equilibrium SCFT, allowing sophisticated numerical techniques developed for equilibrium SCFT to be applied directly to study the dynamics. Our approach is flexible and can be used, for example, to study polymer melt dynamics. To test the self-consistent nature of the theory in space-time, we examine the simple case of the dynamics of trapped, interacting particles, and binary mixtures, in one dimension.   

Structure Formation in Semi-Dilute Polymer Solution during Electrospinning

In our recent work it was shown that longitudinal stretching of electrospun highly entangled semi-dilute polymer solution caused by jet hydrodynamic forces, transforms the topological network to an almost fully-stretched state within less than 1 mm from the jet start (PRE, 2011). Further evolution of the polymer network is related to a disentanglement of polymer chains and transformation of the topological network structure. As was sown by Malkin et al., (Rheol. Acta, 2011) high deformation rate of a topological polymer network, results in reptations of macromolecules caused by uncompensated local forces, whereas Brownian motion effect is negligible. Based on this conclusion, we examine the disentanglement process, using a mechanical pulley-block system assembled from multiple pulleys suspended by elastic springs, and taut string connecting two blocks. Each pulley corresponds to a topological knot; the taut string corresponds to a reptated chain; the springs correspond to surrounded polymer chains; and the blocks correspond to local deformation force. It turned out that the system is sensitive to system parameters. The pulleys can approach each other and the string stops to move. Such a behavior corresponds to formation of bundle of knots of entangled chains. In other conditions, the string continuously moves while the pulleys did not approach each other which corresponds to disentanglement of polymer chains. These experiments clarify the disentanglement kinetics in rapid-deformed polymer system.   

Edge electrospinning from a fluid-filled bowl for high throughput production of quality nanofibers

We present a stationary, edge-cylinder geometry for high throughput electrospinning that utilizes a reservoir filled with polymer solution and a concentric cylindrical collector [\textit{Nanotechnology} \textbf{22}, 345303 (2011)]. In this ``bowl'' electrospinning configuration, under high voltage initiation, multiple jets spontaneously form on the fluid surface, rearrange until they are approximately equidistant along the reservoir-edge and spin towards the collector, producing high quality fibers after the voltage is reduced to a working value. The technique produced poly(ethylene oxide) nanofibers with average diameter of 225 nm and a demonstrated throughput $\sim$40 times higher than traditional single-needle electrospinning. The electric field patterns generated by traditional, bowl, and our previously reported edge-plate [\textit{Polymer} \textbf{51}, 4928 (2010)] geometries show significant similarity in field magnitude and gradient along a path towards the collector, which may underlie the ability to form similar quality fibers. We discuss how the interaction between fluid properties and the applied electric field determines the effective flow rate, jet stability versus time, throughput, and fiber quality.   

Size-dependent behavior of as-spun nanofiber vs. polymer molecular weight

The size-dependent behavior of nano-objects is a well-known and widely-accepted phenomenon, but up to now it has no satisfactory explanation. From the physical point of view, such a behavior is to be related to an internal scale parameter which is comparable with the scale of the system. Recently Ji et al. showed that the elastic moduli of polystyrene nano-fibers of different molecular weights can be described by one universal curve as a function of fiber radius, scaled by radius of gyration $R_g $ (EPL, 84, 56002, 2008). However, the crossover to the size-dependent behavior in the above dependence occurs at $R \mathord{\left/ {\vphantom {R {R_g }}} \right. \kern-\nulldelimiterspace} {R_g }\sim 25-30$, therefore the radius of gyration, $R_g $, is too small in order to play a role of the required scale parameter. This discrepancy requires an explanation. Utilizing a number of well-known scaling dependences, on the base of the model of confinement mechanism of polymer nanofiber reinforcement, proposed by us earlier, it is demonstrated that elastic modulus of polymer nanofibers is described by the function $F\left[ {\left( {R \mathord{\left/ {\vphantom {R {R_g }}} \right. \kern-\nulldelimiterspace} {R_g }} \right)^\alpha } \right]$ ($\alpha \sim 1.5)$. This function, conforming to experimental observation, increases at small argument values, whereas for large argument values tends to value of the bulk elastic modulus, in doing so the crossover scale to the size-dependent behavior also agrees with experimental data.   

Mechanical properties and morphology of polymer gels

Understanding morphology and mechanical response of polymeric gels is of particular importance to design materials with required energy dissipation characteristics. We will present our latest results for polymer gels based on 1) self-assembled block copolymers and 2) chemically cross-linked polymers. The dissipative particle dynamics (DPD) was used to predict morphology in good agreement with atomic force microscopy. We have performed DPD non-equilibrium oscillatory shear calculations predicting elastic modulus of unentangled gels that correlates well with experimental rheology data. However, this methodology fails to predict mechanics of entangled polymer networks due to unphysical chain crossing brought by the soft potentials used in DPD simulations. Recently, we have introduced an improved segmental repulsion potential that removes the bond crossing allowing for reptation dynamics. The improved DPD method was used in simulations for entangled gels to explore impact of branched architecture of solvent on the mechanical response to the tensile deformation. Novel architectures of solvent resulting in a dramatic increase of the elastic modulus were identified. The topological analysis was applied to understand contributions of chemical cross-links and entanglements to the stress.   

Rapid, accurate single particle tracking based on radial symmetry center determination

Accurately tracking particles in images is a crucial task for applications as diverse as super-resolution microscopy, membrane biophysics, and microrheology. Tracking errors can easily propagate into flawed conclusions about mechanisms underlying particle dynamics. The commonly used method of locating the center of a particle by direct fitting a Gaussian function to a measured intensity profile is very accurate, but is computationally intensive and not generalizable to non-point-like particles. Its slowness is a necessary consequence of numerically searching a large parameter space. I introduce a new approach to sub-pixel particle tracking based on exploiting radial symmetry, valid for any radially symmetric particle intensity profile. I provide an algorithm that analytically, non-iteratively calculates the best-fit symmetry center to determine the particle location. Over a wide range of signal-to-noise ratios, this approach yields accuracies nearly identical to those of Gaussian fitting with execution times over two orders of magnitude faster and with greater robustness in the presence of nearby particles. This algorithm is tested on simulated data as well as real images from several experimental systems, including colloidal assemblies and single fluorescent proteins.   

Molecular Simulations of Passive Particle Rheology

In this work, we demonstrate that nanoscale viscoelastic properties of polymer melts can be obtained from molecular dynamics (MD) simulations by using an approach analogous to the experimental passive microrheology. We carry out MD simulations of a system consisting of a probe particle that is embedded in a polymer melt represented using the bead-spring model. The mean squared displacement of the probe particle determined from these MD simulations is analyzed to calculate the storage and the loss moduli of the medium. Our results indicate that calculation of the viscoelastic properties from the straightforward Generalized Stokes Einstein Relationship leads to unphysical values for these quantities in the high frequency regime. We show that this problem can be alleviated by accounting for the inertial effects in the system. Our particle rheology simulation results are quantitatively compared with the literature results that were obtained from other simulation approaches. Results will also be presented for the effects of the particle size, rigidity and the chain length on the viscoelastic properties.   

NMR-based Molecular Rheology of Entangled Polymers in Bulk and in Nanoscopic Confinement

We demonstrate the use of simple proton low-field NMR to probe the validity of the tube model of polymer dynamics. The method yields a time-domain measure of the segmental orientation autocorrelation function $C(t)$, which in turn is directly related to the stress relaxation modulus $G(t)$, thus providing a true molecular measure of rheologically relevant quantities. The fixed-tube model does not describe actual data well, and current work focuses on deuteron labeling schemes to investigate the relevance of contour-length fluctuation (CLF) or constraint release (CR) effects. As first results, we found that unexpectedly, CR processes are responsible for modified chain modes faster than actual reptation [1], and also that the dynamics is inhomogeneous along a given chain, stressing also the significance of CLF. We also present recent results for melt dynamics in nanoscopic confinement of long cylindrical channels of 20-400 nm diameter [2]. We consistently observe a fraction of chains whose dynamics is less isotropic on long time scales, i.e., in the Doi-Edwards regimes III (reptation) and IV (disentangled dynamics)\\[0pt] [1] F. Vaca Ch{\'a}vez, K. Saalw{\"a}chter, {\em Phys. Rev. Lett.} {\bf 104}, 198305 (2010), [2] S. Ok et al., {\em Macromolecules} {\bf 43}, 4429 (2010)