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 % volume occupancy) and ring topology (by considering unknotted and trefoil-knotted chains). The equilibrium metric properties of rings with either topology are found to be only weakly affected by the increase of solution density. Even at the highest density, the average ring size, shape anisotropy and length of the knotted region differ at most by 40% from those of isolated rings. Conversely, kinetics are strongly affected by the degree of inter-chain entanglement: for both unknots and trefoils the characteristic times of ring size relaxation, reorientation and diffusion change by one order of magnitude across the considered range of concentrations. Yet, significant topology-dependent differences in kinetics are observed only for very dilute solutions (much below the ring overlap threshold). For knotted rings, the slowest kinetic process is found to correspond to the diffusion of the knotted region along the ring backbone.
Understanding how entanglements affect the behaviour of polymeric complex fluids is an open challenge in many fields. To elucidate the nature and consequence of entanglements in dense polymer solutions, we propose a novel method: a "dynamical entanglement analysis" (DEA) to extract spatio-temporal entanglement structures from the pair-wise displacement correlation of entangled chains. By applying this method to large-scale Molecular Dynamics simulations of linear and unknotted, nonconcatenated ring polymers, we find a strong and unexpected cooperative dynamics: the footprint of mutual entrainm…
We argue that the mean crossing number of a random polymer configuration is simply a measure of opacity, without being closely related to entanglement as claimed by several authors. We present an easy way of estimating its asymptotic behaviour numerically. These estimates agree for random walks (theta polymers), self-avoiding walks, and for compact globules with analytic estimates giving $\log N, a-b/N^{2ν-1},$ and $N^{1/3}$, respectively, for the average number of crossings per monomer in the limit $N\to \infty$. While the result for compact globules agrees with a rigorous previous estimate, …
We use Brownian dynamics simulations and advanced topological profiling methods to characterize the out-of-equilibrium evolution of self-entanglement in linear polymers confined into nano-channels and under periodic compression. We distinguish two main forms of entanglement, geometrical and topological. The latter is measured by the number of (essential) crossings of the physical knot detected after a suitable bridging of the chain termini. The former is instead measured as the average number of times a linear chain appears to cross itself when viewed under all projections, and is irrespective…
We propose a theoretical approach to quantitatively account for the role of entanglement in the nucleation of polymer melts, which is the unique feature of polymer differentiated from small molecules. By performing molecular dynamics simulations, we obtain the nucleation barriers of polymer systems with different entanglement densities, which exhibits an opposite trend compared to the prediction of the classic nucleation theory (CNT). To amend the deficiency of the CNT in polymer crystallization, we introduce the entanglement free energy to reflect the role of entanglement in polymer nucleatio…
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…
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}…
Polymer fibers consist of macromolecules oriented along the fiber axis. Better alignment of chains leads to an increased strength of the fiber. It is believed that the key factor preventing formation of a perfectly oriented fiber is entanglement of polymers. We performed large-scale computer simulations of uniaxial stretching of semicrystalline ultrahigh molecular weight polyethylene. We discovered that there is an optimal number of entanglements per macromolecule necessary to maximize chain orientation in a fiber. Polymers that were entangled too strongly formed less oriented fibers. On the o…
We develop topological methods for characterizing the relationship between polymer chain entanglement and bulk viscoelastic responses. We introduce generalized Linking Number and Writhe characteristics that are applicable to open linear chains. We investigate the rheology of polymeric chains entangled into weaves with varying topologies and levels of chain density. To investigate viscoelastic responses, we perform non-equilibrium molecular simulations over a range of frequencies using sheared Lees-Edwards boundary conditions. We show how our topological characteristics can be used to capture k…
Large-scale simulations of thermal welding of polymers are performed to investigate the rise of mechanical strength at the polymer-polymer interface with the welding time. The welding process is in the core of integrating polymeric elements into devices as well as in thermal induced healing of polymers; processes that require development of interfacial strength equal to that of the bulk. Our simulations show that the interfacial strength saturates at the bulk shear strength much before polymers diffuse by their radius of gyration. Along with the strength increase, the dominant failure mode cha…
Polymer melts undergoing large deformation by uniaxial elongation are studied by molecular dynamics simulations of bead-spring chains in melts. Applying a primitive path analysis to strongly deformed polymer melts, the role of topological constrains in highly entangled polymer melts is investigated and quantified. We show that the over-all, large scale conformations of the primitive paths (PPs) of stretched chains follow affine deformation while the number and the distribution of entanglement points along the PPs do not. Right after deformation, PPs of chains retract in both directions paralle…
Dried droplets of polymer solutions of different molecular weights and concentrations leave various types of 'coffee-ring' patterns. These patterns are consequence of contact line motion. We have observed for very low molecular weight the droplet periphery part contains spheroidal structures whereas central part contains continuous layer. For higher molecular weight contact line exhibits 'stick-slip' motion. Whereas for very high molecular weight contact line delays enough to start motion and finally moves uninterruptedly leaving a continuous layer of polymer. We have explained this phenomena …
Entanglement of polymer chains is ubiquitous in elastomers, gels, and biological tissues. While the effects of chain entanglement on elasticity and viscoelasticity of polymer networks have been intensively studied, it remains elusive how chain entanglement affects fracture and fatigue of polymer networks. In this paper, using polyacrylamide hydrogels as a model material, we systematically compare fracture toughness and fatigue threshold of polymer networks with various levels of chain entanglement. We find that the fracture toughness and fatigue threshold of an unentangled polymer network are …
The stretchability of polymeric materials is critical to many applications such as stretchable electronics and soft robotics, yet the stretchability of conventional cross-linked linear polymers is limited by the entanglements between polymer chains. We show using molecular dynamics simulations that cross-linked ring polymers are significantly more stretchable than cross-linked linear polymers. Compared to linear polymers, the entanglements between ring polymers do not act as effective cross-links. As a result, the stretchability of cross-linked ring polymers is determined by the maximum extens…
Reptation theory has been highly successful in explaining the unusual material properties of entangled polymer solutions. It reduces the complex many-body dynamics to a single-polymer description where each polymer is envisaged to be confined to a tube through which it moves in a snake-like fashion. For flexible polymers, reptation theory has been amply confirmed by both experiments and simulations. In contrast, for semiflexible polymers experimental and numerical tests are either limited to the onset of reptation, or were performed for tracer polymers in a fixed, static matrix. Here we report…
Self-healing of polymer films often takes place as the molecules diffuse across a damaged region, above their melting temperature. Using molecular dynamics simulations we probe the healing of polymer films and compare the results with those for thermal welding of homopolymer slabs. The two processes differ in their interfacial structure since damage leads to increased polydispersity and more short chains. A polymer sample was cut into two separate films that were then held together in the melt state. The recovery of the damaged film was followed as time elapsed and polymer molecules diffused a…
The interplay of geometrical and topological entanglement in semiflexible knotted polymer rings confined inside a spherical cavity is investigated using advanced numerical methods. By using stringent and robust algorithms for locating knots, we characterize how the knot length lk depends on the ring contour length, Lc and the radius of the confining sphere, Rc . In the no- and strong- confinement cases we observe weak knot localization and complete knot delocalization, respectively. We show that the complex interplay of lk, Lc and Rc that seamlessly bridges these two limits can be encompassed …
Molecular dynamics simulations were performed for a polymer melt. In quiescent states, the inter-chain interaction energy supported by each particle takes relatively large values persistently for long times if the particle is close to an entanglement. Thus, if the interaction is averaged over appropriate time intervals, we can detect active spots on the chains produced by entanglement constrains. If rapid shearing is applied, these active regions subsequently become bended in eventual zig-zag shapes of the chains. We also demonstrate that stress overshoot occurs with onset of disentanglement.
We have investigated the viscosity and the plateau modulus of actin solutions with a magnetically driven rotating disc rheometer. For entangled solutions we observed a scaling of the plateau modulus versus concentration with a power of 7/5. The measured terminal relaxation time increases with a power 3/2 as a function of polymer length. We interpret the entanglement transition and the scaling of the plateau modulus in terms of the tube model for semiflexible polymers.
We present results of molecular dynamics simulations of very long model polymer chains analyzed by various experimentally relevant techniques. The segment motion of the chains is found to be in very good agreement with the repatation model. We also calculated the plateau-modulus G_N. The predicitions of the entanglement length N_e from G_N and from the mean square displacements of the chains segments disagree by a factor of about 2.2(2), indicating an error in the prefactor in the standard formula for G_N. We show that recent neutron spin echo measurements were carried out for chain lengths wh…
Circuit topology refers to the arrangement of interactions between objects belonging to a linearly ordered object set. Linearly ordered set of objects are common in nature and occur in a wide range of applications in economics, computer science, social science and chemical synthesis. Examples include linear bio-polymers, linear signaling pathways in cells as well as topological sorts appearing in project management. Using a statistical mechanical treatment, we study circuit topology landscapes of linear polymer chains with intra-chain contacts as a prototype of linearly sorted objects with int…
We report simulation results on melts of entangled linear polymers confined in a free-standing thin film. We study how the geometric constraints imposed by the confinement alter the entanglement state of the system compared to the equivalent bulk system using various observables. We find that the confinement compresses the chain conformation uniaxially, decreasing the volume pervaded by the chain, which in turn reduces the number of the accessible inter-chain contact that could lead to entanglements. This local and non-uniform effect depends on the position of the chain within the film. We als…
In this article we study from a non-perturbative point of view the entanglement of two directed polymers subjected to repulsive interactions given by a Dirac $δ-$function potential. An exact formula of the so-called second moment of the winding angle is derived. This result is used to provide a thorough analysis of entanglement phenomena in the classical system of two polymers subjected to repulsive interactions and related problems. No approximation is made in treating the constraint on the winding angle and the repulsive forces. In particular, we investigate how repulsive forces influence th…
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…
In this work a new strategy is proposed in order to build analytic and microscopic models of fluctuating polymer rings subjected to topological constraints. The topological invariants used to fix these constraints belong to a wide class of the so-called numerical topological invariants. For each invariant it is possible to derive a field theory that describes the statistical behavior of knotted and linked polymer rings following a straightforward algorithm. The treatment is not limited to the partition function of the system, but it allows also to express the expectation values of general obse…
The interplay between the topological and geometrical properties of a polymer ring can be clarified by establishing the entanglement trapped in any portion (arc) of the ring. The task requires to close the open arcs into a ring, and the resulting topological state may depend on the specific closure scheme that is followed. To understand the impact of this ambiguity in contexts of practical interest, such as knot localization in a ring with non trivial topology, we apply various closure schemes to model ring polymers. The rings have the same length and topological state (a trefoil knot) but hav…
We have performed accurate dynamic light-scattering measurements near critical demixing points of solutions of polystyrene in cyclohexane with polymer molecular weight ranging from 200,000 to 11.4 million. Two dynamic modes have been observed, ''slow'' and ''fast'', that result from a coupling between diffusive relaxation of critical fluctuations of the concentration and visco-elastic relaxation associated with the entanglement network of the polymer chains. The coupling with the visco-elastic mode causes an additional slowdown of the critical mode on top of the uncoupled diffusion mode. By im…
Large-scale molecular simulations are performed to investigate tensile failure of polymer interfaces as a function of welding time $t$. Changes in the tensile stress, mode of failure and interfacial fracture energy $G_I$ are correlated to changes in the interfacial entanglements as determined from Primitive Path Analysis. Bulk polymers fail through craze formation, followed by craze breakdown through chain scission. At small $t$ welded interfaces are not strong enough to support craze formation and fail at small strains through chain pullout at the interface. Once chains have formed an average…
We derive a thermodynamically consistent framework for incorporating entanglement dynamics into constitutive equations for flowing polymer melts. We use this to combine the convected constraint release (CCR) dynamics of Ianniruberto-Marriccui into a finitely-extensible version of the Rolie-Poly model, and also include an anisotropic mobility as in the Giesekus model. The reversible dynamics are obtained from a free energy that describes both a finitely-extensible conformation tensor and an ideal gas of entanglements along the chain. The dissipative dynamics give rise to coupled kinetic equatio…
Monte Carlo computer simulations are used to study the segregation behaviour of two polymers under cylindrical confinement. Using a multiple-histogram method, the conformational free energy, F, of the polymers was measured as a function of the centre-of-mass separation distance, λ. We examined the scaling of the free energy functions with the polymer length, the length and diameter of the confining cylinder, the polymer topology (i.e. linear vs ring polymers), and the packing fraction and size of mobile crowding agents. In the absence of crowders, the observed scaling of F(λ) is similar to tha…
The coupled dynamics of entangled polymers which span a broad time and length scales govern the unique viscoelastic properties of polymers. To follow chain mobility by numerical simulations from the intermediate Rouse and reptation regimes to the late time diffusive regime, highly coarse grained models with purely repulsive interactions between monomers are widely used since they are computationally the most efficient. Here using large scale molecular dynamics simulations the effect of including the attractive interaction between monomers on the dynamics of entangled polymers melts is explored…
The classical rheological theories of entangled polymeric liquids are built upon two pillars: Gaussian statistics of entanglement strands and the assumption that the stress arises exclusively from the change of intramolecular configuration entropy. We show that these two hypotheses are not supported by molecular dynamics simulations of polymer melts. Specifically, the segment distribution functions at the entanglement length scale and below deviate considerably from the theoretical predictions, in both the equilibrium and deformed states. Further conformational analysis reveals that the intrac…
Polymeric films with greater impact and ballistic resistance are highly desired for numerous applications, but molecular configurations that best address this need remain subject to debate. We study the resistance to ballistic impact of thin polymer films using coarse-grained molecular dynamics simulations, investigating melts of linear polymer chains and star polymers with varying number 2<=f<=16 and degree of polymerization 10<=M<=50 of the arms. We show that increasing the number of arms f or the length of the arms M both result in greater specific penetration energy within the parameter ra…
Extensive molecular simulations are applied to characterize the equilibrium dynamics, entanglement topology, and nonlinear extensional rheology of symmetric ring-linear polymer blends with systematically varied ring fraction $ϕ_R$. Chains with degree of entanglement $Z\approx14$ mixed to produce 10 well-entangled systems with $ϕ_R$ varying from neat linear to neat ring melts. Primitive path analysis are used to visualize and quantify the structure of the composite ring-linear entanglement network. We directly measure the quantity of ring-linear threading and linear-linear entanglement as a fun…
Polymer composites are ubiquitous in biology and industry alike, owing to their emergent desirable mechanical properties not attainable in single-species systems. At the same time, polymer topology has been shown to play a key role in tuning the rheology of polymeric fluids. However, how topology impacts the rheology of composites remains poorly understood. Here, we create composites of rigid rods (microtubules) polymerized within entangled solutions of flexible linear and ring polymers (DNA). We couple linear and nonlinear optical tweezers microrheology with confocal microscopy and scaled par…
The effects of entanglement and chain orientation on strain hardening in glassy polymers are separated by examining mixtures of chains with different lengths. Simulations show that the orientation of a molecule of a given chain length is the same in monodisperse systems and bidisperse mixtures, even when entangled and unentangled chains are mixed. In addition, the stress in mixtures is equal to the weighted average of the stresses in monodisperse systems. These results indicate that chains contribute independently to strain hardening, that chain orientation is determined by local interactions …
We visualize entanglements in polymer melts using molecular dynamics simulation. A bead at an entanglement interacts persistently for long times with the non-bonded beads (those excluding the adjacent ones in the same chain). The interaction energy of each bead with the non-bonded beads is averaged over a time interval $τ$ much longer than microscopic times but shorter than the onset time of tube constraints $τ_{\rm e}$. Entanglements can then be detected as hot spots consisting of several beads with relatively large values of the time-averaged interaction energy. We next apply a shear flow wi…
In systems characterized by a rough potential energy landscape, local energetic minima and saddles define a network of metastable states whose topology strongly influences the dynamics. Changes in temperature, causing the merging and splitting of metastable states, have non trivial effects on such networks and must be taken into account. We do this by means of a recently proposed renormalization procedure. This method is applied to analyze the topology of the network of metastable states for different polypeptidic sequences in a minimalistic polymer model. A smaller spectral dimension emerges …
In polymer physics it is typically assumed that excluded volume interactions are effectively screened in polymer melts. Hence, chains could be described by an effective random walk without excluded volume interactions. In this letter, we show that this mapping is problematic by analyzing the occurrence of knots, their spectrum and sizes in polymer melts, corresponding random walks and chains in dilute solution. The effective random walk severely overrates the occurrence of knots and their complexity, particularly when compared to melts of flexible chains, indicating that non-trivial effects du…
Primitive path analyses of entanglements are performed over a wide range of chain lengths for both bead spring and atomistic polyethylene polymer melts. Estimators for the entanglement length N_e which operate on results for a single chain length N are shown to produce systematic O(1/N) errors. The mathematical roots of these errors are identified as (a) treating chain ends as entanglements and (b) neglecting non-Gaussian corrections to chain and primitive path dimensions. The prefactors for the O(1/N) errors may be large; in general their magnitude depends both on the polymer model and the me…
We study the motion of dispersed nanoprobes in entangled active-passive polymer mixtures. By comparing the two architectures of linear vs. unconcatenated and unknotted circular polymers, we demonstrate that novel, rich physics emerge. For both polymer architectures, nanoprobes of size smaller than the entanglement threshold of the solution move faster as activity is increased and more energy is pumped in the system. For larger nanoprobes, a surprising phenomenon occurs: while in linear solutions they move qualitatively as before, in active-passive ring solutions nanoprobes {\it decelerate} wit…
We investigate the aggregation and phase separation of thin, living T. tubifex worms that behave as active polymers. Randomly dispersed active worms spontaneously aggregate to form compact, highly entangled blobs, a process similar to polymer phase separation, and for which we observe power-law growth kinetics. We find that the phase separation of active polymerlike worms does not occur through Ostwald ripening, but through active motion and coalescence of the phase domains. Interestingly, the growth mechanism differs from conventional growth by droplet coalescence: the diffusion constant char…
The work addresses the analogy between trivial knotting and excluded volume in looped polymer chains of moderate length, $N<N_0$, where the effects of knotting are small. A simple expression for the swelling seen in trivially knotted loops is described and shown to agree with simulation data. Contrast between this expression and the well known expression for excluded volume polymers leads to a graphical mapping of excluded volume to trivial knots, which may be useful for understanding where the analogy between the two physical forms is valid. The work also includes description of a new method …
We use computer simulations to compare the dynamical behaviour of torus and even-twist knots in polymers under tension. The knots diffuse through a mechanism similar to reptation. Their friction coefficients grow linearly with average knot length for both knot types. For similar complexity, however, the torus knots diffuse faster than the even twist knots. The knot-length auto-correlation function exhibits a slow relaxation time that can be linked to a breathing mode. Its timescale depends on knot type, being typically longer for torus than for even-twist knots. These differences in dynamical …
The statistical mechanics of single polymer knots is studied using Monte Carlo simulations. The polymers are considered on a cubic lattice and their conformations are randomly changed with the help of pivot transformations. After each transformation, it is checked if the topology of the knot is preserved by means of a method called pivot algorithm and excluded area (in short PAEA) and described in a previous publication of the authors. As an application of this method the specific energy, the radius of gyration and heat capacity of a few types of knots are computed. The case of attractive shor…
By combining molecular dynamics simulations and topological analyses with scaling arguments, we obtain analytic expressions that quantitatively predict the entanglement length $N_e$, the plateau modulus $G$, and the tube diameter $a$ in melts that span the entire range of chain stiffnesses for which systems remain isotropic. Our expressions resolve conflicts between previous scaling predictions for the loosely entangled [Lin-Noolandi: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^3$], semiflexible [Edwards/de Gennes: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^2$], and tightly-entangled [Morse: $G\ell…
The Kremer-Grest (KG) model is a standard for studying generic polymer properties. Here we have equilibrated KG melts up to and beyond $200$ entanglements per chain for varying chain stiffness. We present methods for estimating the Kuhn length corrected for incompressibility effects, for estimating the entanglement length corrected for chain stiffness, for estimating bead frictions and Kuhn times taking into account entanglement effects. These are the key parameters for enabling quantitative, accurate, and parameter free comparisons between theory, experiment and simulations of KG polymer mode…
We use molecular dynamics simulations to investigate the microscopic and macroscopic response of model polymer networks to uniaxial elongations. By studying networks with strands lengths ranging from $N_s=20$ to 200 we cover the full crossover from cross-link to entanglement dominated behavior. Our results support a recent version of the tube model which accounts for the different strain dependence of chain localization due to chemical cross-links and entanglements.
