The relaxation of a single knotted ring polymer is studied by Brownian dynamics simulations. The relaxation rate lambda_q for the wave number q is estimated by the least square fit of the equilibrium time-displaced correlation function to a double exponential decay at long times. The relaxation rate distribution of a single ring polymer with the trefoil knot appears to behave as lambda_q=A(1/N^)x for q=1 and lambda_q=A'(q/N)^x' for q=2 and 3, where x=2.61, x'=2.02 and A>A'. The wave number q of the slowest relaxation rate for each N is given by q=2 for small values of N, while it is given by q=1 for large values of N. This crossover corresponds to the change of the structure of the ring polymer caused by the localization of the knotted part to a part of the ring polymer.
We present computer simulations to examine probability distributions of gyration radius for the no-thickness closed polymers of N straight segments of equal length. We are particularly interested in the conditional distributions when the topology of the loop is quenched to be a certain knot, K. The dependence of probability distribution on length, N, as well as topological state K are the primary parameters of interest. Our results confirm that the mean square average gyration radius for trivial knots scales with N in the same way as for self-avoiding walks, where the cross-over length to this…
We present a computer simulation study of the compact self-avoiding loops as regards their length and topological state. We use a Hamiltonian closed path on the cubic-shaped segment of a 3D cubic lattice as a model of a compact polymer. The importance of ergodic sampling of all loops is emphasized. We first look at the effect of global topological constraint on the local fractal geometry of a typical loop. We find that even short pieces of a compact trivial knot, or some other under-knotted loop, are somewhat crumpled compared to topology-blind average over all loops. We further attempt to exa…
We compare Monte Carlo simulations of knotted and unknotted polymers whose ends are connected to two parallel walls. The force $f$ exerted on the polymer is measured as a function of the separation $R$ between the walls. For unknotted polymers of several monomer numbers $N$, the product $fN^ν$ is a simple function of $R/N^ν$, where $ν\simeq 0.59$. By contrast, knotted polymers exhibit strong finite size effects which can be interpreted in terms of a new length scale related to the size of the knot. Based on this interpretation, we conclude that the number of monomers forming the knot scales as…
We investigate the Rouse dynamics of a flexible ring polymer with a prime knot. Within a Monte Carlo approach, we locate the knot, follow its diffusion, and observe the fluctuations of its length. We characterise a topological time scale, and show that it is related to a self-reptation of the knotted region. The associated dynamical exponent, $z_T=2.32\pm.1$, can be related to that of the equilibrium knot length distribution and determines the behaviour of several dynamical quantities.
We investigate the effect of knot type on the properties of a ring polymer confined to a slit. For relatively wide slits, the more complex the knot, the more the force exerted by the polymer on the walls is decreased compared to an unknotted polymer of the same length. For more narrow slits the opposite is true. The crossover between these two regimes is, to first order, at smaller slit width for more complex knots. However, knot topology can affect these trends in subtle ways. Besides the force exerted by the polymers, we also study other quantities such as the monomer-density distribution ac…
We discuss the scattering function of a Gaussian random polygon with N nodes under a given topological constraint through simulation. We obtain the Kratky plot of a Gaussian polygon of N=200 having a fixed knot for some different knots such as the trivial, trefoil and figure-eight knots. We find that some characteristic properties of the different Kratky plots are consistent with the distinct values of the mean square radius of gyration for Gaussian polygons with the different knots.
We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, $R_G$, and that of the region containing the knot, $R_{G,k}$, are small compared to the distance b between the obstacles, the knot is weakly localised and $R_G$ scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where $R_G > b > R_{G,k}$, the geometry of the polymer becomes branched. When $R_{G,k}…
Common experience tells us that a knot significantly weakens the polymer strand in which it is tied, which in turn leads to more facile chain rupture under tensile loading. Using first-principles molecular dynamics calculations we describe the dynamical evolution of the radicals that form after chain rupture of a single knotted alkane molecule in their very early stages of life. They are able to recombine, to form cyclic alkanes and to undergo disproportionation phenomena with nearby chain segments. The breaking of a single knotted polymer chain under mechanical loading is thus predicted to re…
We investigate the sedimentation of knotted polymers by means of stochastic rotation dynamics, a molecular dynamics algorithm that takes hydrodynamics fully into account. We show that the sedimentation coefficient s, related to the terminal velocity of the knotted polymers, increases linearly with the average crossing number n_c of the corresponding ideal knot. To the best of our knowledge, this provides the first direct computational confirmation of this relation, postulated on the basis of experiments in "The effect of ionic conditions on the conformations of supercoiled DNA. I. sedimentatio…
Two types of average structures of a single knotted ring polymer are studied by Brownian dynamics simulations. For a ring polymer with N segments, its structure is represented by a 3N -dimensional conformation vector consisting of the Cartesian coordinates of the segment positions relative to the center of mass of the ring polymer. The average structure is given by the average conformation vector, which is self-consistently defined as the average of the conformation vectors obtained from a simulation each of which is rotated to minimize its distance from the average conformation vector. From e…
The statistical mechanics of a long knotted collapsed polymer is determined by a free-energy with a knot-dependent subleading term, which is linked to the length of the shortest polymer that can hold such knot. The only other parameter depending on the knot kind is an amplitude such that relative probabilities of knots do not vary with the temperature $T$, in the limit of long chains. We arrive at this conclusion by simulating interacting self-avoiding rings at low $T$ on the cubic lattice, both with unrestricted topology and with setups where the globule is divided by a slip link in two loops…
Inspired by how certain proteins "sense" knots and entanglements in DNA molecules, here we ask if there exist local geometric features that may be used as a read-out of the underlying topology of generic polymers. We perform molecular simulations of knotted and linked semiflexbile polymers and study four geometric measures to predict topological entanglements: local curvature, local density, local 1D writhe and non-local 3D writhe. We discover that local curvature is a poor predictor of entanglements. In contrast, segments with maximum local density or writhe correlate as much as 90% of the ti…
The topological effects on the thermal properties of several knot configurations are investigated using Monte Carlo simulations. In order to check if the topology of the knots is preserved during the thermal fluctuations we propose a method that allows very fast calculations and can be easily applied to arbitrarily complex knots. As an application, the specific energy and heat capacity of the trefoil, the figure-eight and the $8_1$ knots are calculated at different temperatures and for different lengths. Short-range repulsive interactions between the monomers are assumed. The knots configurati…
By performing Monte Carlo sampling of $N$-steps self-avoiding polygons embedded on different Bravais lattices we explore the robustness of universality in the entropic, metric and geometrical properties of knotted polymer rings. In particular, by simulating polygons with $N$ up to $10^5$ we furnish a sharp estimate of the asymptotic values of the knot probability ratios and show their independence on the lattice type. This universal feature was previously suggested although with different estimates of the asymptotic values. In addition we show that the scaling behavior of the mean squared radi…
Many experiments have been done to determine the relative strength of different knots, and these show that the break in a knotted rope almost invariably occurs at a point just outside the `entrance' to the knot. The influence of knots on the properties of polymers has become of great interest, in part because of their effect on mechanical properties. Knot theory applied to the topology of macromolecules indicates that the simple trefoil or `overhand' knot is likely to be present with high probability in any long polymer strand. Fragments of DNA have been observed to contain such knots in exper…
Monte Carlo simulations are used to study the conformational behavior of a semiflexible polymer confined to cylindrical and conical channels. The channels are sufficiently narrow that the conditions for the Odijk regime are marginally satisfied. For cylindrical confinement, we examine polymers with a single knot of topology $3_1$, $4_1$, or $5_1$, as well as unknotted polymers that are capable of forming S-loops. We measure the variation of the free energy $F$ with the end-to-end polymer extension length $X$ and examine the effect of varying the polymer topology, persistence length $P$ and cyl…
The effects of entanglement in solutions and melts of unknotted ring polymers have been addressed by several theoretical and numerical studies. The system properties have been typically profiled as a function of ring contour length at fixed solution density. Here, we use a different approach to investigate numerically the equilibrium and kinetic properties of solutions of model ring polymers. Specifically, the ring contour length is maintained fixed, while the interplay of inter- and intra-chain entanglement is modulated by varying both solution density (from infinite dilution up to \approx 40…
Several nontrivial properties are shown for the mean square radius of gyration $R_K^2$ of ring polymers with a fixed knot type K. Through computer simulation, we discuss both finite-size and asymptotic behaviors of the gyration radius under the topological constraint for self-avoiding polygons consisting of N cylindrical segments with radius r. We find that the average size of ring polymers with a knot K can be much larger than that of no topological constraint. The effective expansion due to the topological constraint depends strongly on the parameter r which is related to the excluded volume…
Numerical studies of the average size of trivially knotted polymer loops with no excluded volume are undertaken. Topology is identified by Alexander and Vassiliev degree 2 invariants. Probability of a trivial knot, average gyration radius, and probability density distributions as functions of gyration radius are generated for loops of up to N=3000 segments. Gyration radii of trivially knotted loops are found to follow a power law similar to that of self avoiding walks consistent with earlier theoretical predictions.
We study by Monte Carlo simulations a model of knotted polymer ring adsorbing onto an impenetrable, attractive wall. The polymer is described by a self-avoiding polygon (SAP) on the cubic lattice. We find that the adsorption transition temperature, the crossover exponent $ϕ$ and the metric exponent $ν$, are the same as in the model where the topology of the ring is unrestricted. By measuring the average length of the knotted portion of the ring we are able to show that adsorbed knots are localized. This knot localization transition is triggered by the adsorption transition but is accompanied b…
The thermal properties of coarse grained knotted polymers containing two kinds of monomers $A$ and $B$ fluctuating in a solution are investigated on a simple cubic lattice using the Wang-Landau MC algorithm. These knots have a more complex phase diagram than knots formed by homopolymers, including the possible presence of metastable states. Two different setups are considered: i) charged block copolymers in a ion solution and ii) neutral copolymers with the $A$ monomers above and the $B$ monomers below the theta point. A precise interpretation of the peaks observed in the plots of the specific…
Hydrodynamic interactions as modeled by Multi-Particle Collision Dynamics can dramatically influence the dynamics of fully flexible, ring-shaped polymers in ways not known for any other polymer architecture or topology. We show that steady shear leads to an inflation scenario exclusive to ring polymers, which depends not only on Weissenberg number but also on contour length of the ring. By analyzing velocity fields of the solvent around the polymer, we show the existence of a hydrodynamic pocket which allows the polymer to self-stabilize at a certain alignment angle to the flow axis. This self…
We study stress relaxation in bi-disperse entangled polymer solutions. Shorter chains embedded in a majority of longer ones are known to be oriented by coupling to them. We analyze the mechanism for this both by computer simulation and theoretically. We show that the results can be understood in terms of stress fluctuations in a polymer melt and chain screening. Stress fluctuations are frozen on the relaxation time of the longer chains, and these will induce strong orientational couplings in the shorter chains.
The non-linear stress-strain relation for crosslinked polymer networks is studied using molecular dynamics simulations. Previously we demonstrated the importance of trapped entanglements in determining the elastic and relaxational properties of networks. Here we present new results for the stress versus strain for both dry and swollen networks. Models which limit the fluctuations of the network strands like the tube model are shown to describe the stress for both elongation and compression. For swollen networks, the total modulus is found to decrease like (V_0/V)^{2/3} and goes to the phantom …
The relaxation of polymers around and below their glass transition temperature is governed by a range of correlated unit processes with a wide range of timescales. The fast deformation rates of shock loading can negate a significant fraction of these processes resulting in the dynamical glass transition in rubbers. In this letter we report the inverse, a transient melting of glassy polymer under shock loading. The large deviatoric stresses near the shock front induce fast transitions in backbone dihedral angles and a stress relaxation characteristic of polymer melts. This is followed by the sl…
Fracture initiation in glassy polymers with no notch is studied together with the evolution at zero stress in the glassy state. Confocal microscopy observations and auto-correlation methods are used to characterize specimens of polymethyl methacrylate (PMMA) loaded at room temperature and subsequently unloaded. The evolution of the morphology and the location of the cracking submitted to elongation rate up to 8% are reported and analyzed during the zero-stress relaxation. The crackling (longitudinal crack, transverse crack and crazes) takes place mainly within a 10micron thickness layer from t…
Mixtures of polymers of varying topologies and stiffnesses display complex emergent rheological properties that often cannot be predicted from their single-component counterparts. For example, entangled blends of ring and linear polymers have been shown to exhibit enhanced shear thinning and viscosity, as well as prolonged relaxation timescales, compared to pure solutions of rings or linear chains. These emergent properties arise in part from the synergistic threading of rings by linear polymers. Topology has also been shown to play an important role in composites of flexible (e.g., DNA) and s…
We propose a theoretical framework for dealing with a transient polymer network undergoing small deformations, based on the rate of breaking and re-forming of network crosslinks and the evolving elastic reference state. In this framework, the characteristics of the deformed transient network at microscopic and macroscopic scales are naturally unified. Microscopically, the breakage rate of the crosslinks is affected by the local force acting on the chain. Macroscopically, we use the classical continuum model for rubber elasticity to describe the structure of the deformation energy, whose refere…
Modeling the Dielectric Relaxation in Semicrystalline Polymers -- Understanding the Role of the Interphase between the Amorphous and Crystalline Domains
We study the effect of physical aging on the mechanical properties of a model polymer glass using molecular dynamics simulations. The creep compliance is determined simultaneously with the structural relaxation under a constant uniaxial load below yield at constant temperature. The model successfully captures universal features found experimentally in polymer glasses, including signatures of mechanical rejuvenation. We analyze microscopic relaxation timescales and show that they exhibit the same aging characteristics as the macroscopic creep compliance. In addition, our model indicates that th…
Dielectric relaxation spectra of block polymers containing sequential type-A dipoles are considered. Spectra of a specific set of block copolymers can be combined to isolate the dynamic cross-correlation between the motions of two distinct parts of the same polymer chain. Unlike past treatments of this problem, no model is assumed for the underlying polymer dynamics.
Using different segmental dynamics and relaxation, characteristics of the interface growth is examined in an electrophoretic deposition of polymer chains on a three (2+1) dimensional discrete lattice with a Monte Carlo simulation. Incorporation of faster modes such as crankshaft and reptation movements along with the relatively slow kink-jump dynamics seems crucial in relaxing the interface width. As the continuously released polymer chains are driven (via segmental movements) and deposited, the interface width $W$ grows with the number of time steps $t$, $W \propto t^β,$ ($β\sim 0.4$--$0.8)$,…
The calculation of discrete or continuous relaxation time spectra from rheometric measurables of polydisperse polymers is an ill-posed problem. In this paper, a curve fitting method for solving this problem is presented and compared to selected models from the literature. It is shown that the new method is capable of correctly predicting the molecular mass distributions of linear polydisperse polymer melts as well as their relaxation time spectra.
The viscoelastic effect on the hydrodynamic relaxation in semidilute polymer solutions is investigated. From the linearized two-fluid model equations, we predict that the dynamical asymmetry coupling between the velocity fluctuations and the viscoelastic stress influences on the hydrodynamic relaxation process, resulting in a wave-number-dependent shear viscosity.
Cooperative polymer dynamics at the crossover between the unentangled and the entangled regimes: theoretical predictions of scattering and linear shear relaxation for polyethylene melts
The dynamic-mechanical response of flexible polymer networks is studied in the framework of tube model, in the limit of small affine deformations, using the approach based on Rayleighian dissipation function. The dynamic complex modulus G* is calculated from the analysis of a network strand relaxation to the new equilibrium conformation around the distorted primitive path. Chain equilibration is achieved via a sliding motion of polymer segments along the tube, eliminating the inhomogeneity of the polymer density caused by the deformation. The characteristic relaxation time of this motion <tau>…
We show that the density and temperature dependences of the $α$-relaxation time of several glassforming polymers can be described through a single scaling variable $X=e(ρ)/T$, where $e(ρ)$ is well fitted by a power law $ρ^x$, $x$ being a species-specific parameter. This implies that ``fragility'' is an intrinsic, density-independent property of a glassformer characterizing its super-Arrhenius slowing down of relaxations, and it leads us to propose a modification of the celebrated Angell plot.
The dynamics of structural relaxation in a model polymer glass subject to spatially-homogeneous, time-periodic shear deformation is investigated using molecular dynamics simulations. We study a coarse-grained bead-spring model of short polymer chains below the glass transition temperature. It is found that at small strain amplitudes, the segmental dynamics is nearly reversible over about $10^4$ cycles, while at strain amplitudes above a few percent, polymer chains become fully relaxed after a hundred cycles. At the critical strain amplitude, the transition from slow to fast relaxation dynamics…
Glassy polymers show strain hardening: at constant extensional load, their flow first accelerates, then arrests. Recent experiments have found this to be accompanied by a striking and unexplained dip in the segmental relaxation time. Here we explain such behavior by combining a minimal model of flow-induced liquefaction of a glass, with a description of the stress carried by strained polymers, creating a non-factorable interplay between aging and strain-induced rejuvenation. Under constant load, liquefaction of segmental motion permits strong flow that creates polymer-borne stress. This slows …
We present a unified theory for the longitudinal dynamic response of a stiff polymer in solution to various external perturbations (mechanical excitations, hydrodynamic flows, electrical fields, temperature quenches ...) that can be represented as sudden changes of ambient/boundary conditions. The theory relies on a comprehensive analysis of the non--equilibrium propagation and relaxation of backbone stresses in a wormlike chain. We recover and substantially extend previous results based on heuristic arguments. Intriguing new experimental implications are pointed out.
Using molecular simulations and theory, we develop an explicit mapping of the contribution of molecular relaxation modes in glassy thermosets to the shear modulus, where the relaxations were tuned by altering the polarity of side groups. Specifically, motions at the domain, segmental, monomer, and atomic levels are taken from molecular dynamics snapshots and directly linked with the viscoelasticity through a framework based in the lattice dynamics of amorphous solids. This unique approach provides direct insight into the roles of chemical groups in the stress response, including the timescale …
Glassy polymers show strain hardening: at constant extensional load, their flow first accelerates, then arrests. Recent experiments under such loading have found this to be accompanied by a striking dip in the segmental relaxation time. This can be explained by a minimal nonfactorable model combining flow-induced melting of a glass with the buildup of stress carried by strained polymers. Within this model, liquefaction of segmental motion permits strong flow that creates polymer-borne stress, slowing the deformation enough for the segmental (or solvent) modes to then re-vitrify. Here we presen…
