Entanglement Tube Research Articles

Cooperative Dynamics of Highly Entangled Linear Polymers within the Entanglement Tube.

We present a quantitative comparison of the dynamic structure factors from unentangled and strongly entangled poly(butylene oxide) (PBO) melts. As expected, the low molecular weight PBO displays Rouse dynamics, however, with very significant subdiffusive center-of-mass diffusion. The spectra from high molecular weight entangled PBO can be very well described by the dynamic structure factor based on the concept of local reptation, including the Rouse dynamics within the tube and allowing for non-Gaussian corrections. Comparing quantitatively the spectra from both polymers leads to the surprising result that their spectra differ only by the contribution of classical Rouse diffusion for the low molecular weight melt. The subdiffusive component is common for both the low and high molecular weight PBO melts, indicating that in both melts the same interchain potential is active, thereby supporting the validity of the Generalized Langevin Equation approach.

Elasticity of Highly Entangled Polymer Networks and Gels: Review of Models and Theory of Nonaffine Deformations

The main models of phantom and topologically entangled polymer networks are surveyed. A theory of anisotropic and nonaffine deformation of both swollen and deswollen (with partial solvent removal) strongly entangled polymer networks in athermal and θ-solvents has been developed. It is shown that under weak anisotropic deformations of the deswollen network, the entanglement tube consists of fractal loopy globules. In a θ-solvent, slight deformations of the network lead to a decrease in the overlap of loopy globules without changing their sizes. Deformations of swollen networks, as well as strong deformations of deswollen networks, are described in terms of the slip-tube model. An effective Hamiltonian has been derived that determines the entropy of fractal loopy globules. Based on the Hamiltonian, it is shown that topological constraints can be described using the polymer–quantum diffusion analogy. The connection between topological and quantum entanglements is demonstrated.

Screening confinement of entanglements: Role of a self-propelling end inducing ballistic chain reptation.

Synthetic and natural nanomaterials with self-propelling mechanisms continue to be explored to boost chain mobility beyond normal reptation in the crowded environments of entangled chains. Here we employ scaling theory and numerical simulations to demonstrate that activating one chain end of a singular or isolated chain boosts entanglement-constrained chain reptation from the one-dimensional diffusive mobility as described by the de Gennes-Edwards-Doi model to ballistic motion along the entanglement tube contour. The active chain is effectively screened from the constraint of entanglements on length scales exceeding the tube size.

Investigation of temperature-dependent electrical parameters in a Schottky barrier diode with multi-walled carbon nanotube (MWCNT) interface

We research the electrical parameters of the MWCNT/n-6H-SiC Schottky barrier diode (SBD) as a function of temperature. Voltage-dependent current and the capacitance measurements of the diode have been made between the temperatures of 300–480 K. The surface of the diode coated with CNT by drop drying method is examined by Scanning Electron Microscopy (SEM) and Raman Spectroscopy. It has been observed that MWCNTs coat on the semiconductor randomly and as entangled tubes, and the intensity in the D line is higher than in the G line. The ideality factor and barrier heights obtained from Thermionic Emission (TE) theory are in the range of 1.64–1.07 and 0.76–1.10 eV, respectively. The ideality factor and barrier heights of the produced diode are strongly related to temperature. By using Cheung-Cheung and Norde methods, series resistance, ideality factor, and barrier height parameters are calculated depending on temperature. It is effective in changing the series resistance of MWCNTs used as interface material. Capacitance-voltage (C–V) measurements of the MWCNT/n-6H-SiC Schottky barrier diode are made at different frequencies at 300 K and a frequency of 1 MHz depending on different temperature. The increase of the produced diode capacitance at low frequencies is associated with the interface states.

Validation and Refinement of Unified Analytic Model for Flexible and Semiflexible Polymer Melt Entanglement.

We combine molecular dynamics simulations and topological analyses (TA) to validate and refine a recently proposed unified analytic model [Hoy, R. S.; Kröger, M. Phys. Rev. Lett. 2020, 124, 147801] for the reduced entanglement length, tube diameter, and plateau modulus of polymer melts. While the functional forms of the previously published expressions are insensitive to the choice of the TA method and N e -estimator, obtaining better statistics and eliminating all known sources of systematic error in the N e -estimation alters their numerical coefficients. Our revised expressions quantitatively match bead-spring simulation data over the entire range of chain stiffnesses for which systems remain isotropic, semiquantitatively match all available experimental data for flexible, semiflexible, and stiff polymer melts (including new data for conjugated polymers that lie in a previously unpopulated stiffness regime), and outperform previously developed unified scaling theories.

Using Simulation to Evaluate a Tube Perception Algorithm for Bin Picking

Bin picking is a challenging problem that involves using a robotic manipulator to remove, one-by-one, a set of objects randomly stacked in a container. In order to provide ground truth data for evaluating heuristic or machine learning perception systems, this paper proposes using simulation to create bin picking environments in which a procedural generation method builds entangled tubes that can have curvatures throughout their length. The output of the simulation is an annotated point cloud, generated by a virtual 3D depth camera, in which the tubes are assigned with unique colors. A general metric based on micro-recall is proposed to compare the accuracy of point cloud annotations with the ground truth. The synthetic data is representative of a high quality 3D scanner, given that the performance of a tube modeling system when given 640 simulated point clouds was similar to the results achieved with real sensor data. Therefore, simulation is a promising technique for the automated evaluation of solutions for bin picking tasks.

Concentration dependent debundling and single tube dispersions of pristine multiwalled carbon nanotubes functionalized with double tail phospholipids

Multiwalled carbon nanotubes (MWNTs) exist as aggregates of highly entangled tubes due to large aspect ratios and strong Van der Waals interactions among them in their native states. In order to render them suitable for any application, MWNTs need to be separated and dispersed uniformly in a solvent preferably as individual tubes. In the present work, it is demonstrated that a double tail lipid such as 1, 2-dipalmitoyl-sn-glycero-3-phosphoethanolamine (DPPE) is capable of dispersing MWNTs in ethanol. Ultra-stable suspensions were obtained by optimizing two key parameters: DPPE to MWNT weight ratio (ε) and MWNT concentration (c). Stability of the suspensions increased with the increasing ε value up to an optimum point (ε = 1.8) and then decreased drastically beyond that. CNT dispersions with ε = 1.8 were extremely stable (with a Zeta potential of 108.26 ± 2.15 mV) and could be retained in suspended form up to 3 months. Effect of MWNT concentration on disaggregation was very significant and stable suspensions could be formed for MWNT concentrations only below 0.14 mg ml−1. Above this concentration, no stable dispersions could be obtained even with ε = 1.8. Compression isotherms of Langmuir monolayers of the DPPE functionalized MWNTs spread at the air water interface were highly repeatable, suggesting that the MWNTs in dispersion were present as separate tubes coated with phospholipids. SEM micrographs of the Langmuir–Blodgett (LB) films, deposited at high surface pressures on silicon wafers, show that MWNTs remain as single nanotubes with no signs of reaggregation. TEM micrographs of MWNT suspensions indicated random adsorption of DPPE on MWNTs. Our work makes it possible to explore potential applications of LB films of MWNTs (stabilized by DPPE) in the development of conducting thin films for sensor applications or as supports to immobilize catalysts for heterogenous reactions.

Unified Analytic Expressions for the Entanglement Length, Tube Diameter, and Plateau Modulus of Polymer Melts.

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, Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{3}], semiflexible [Edwards-de Gennes: Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{2}], and tightly entangled [Morse, Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{1+ϵ}] regimes, where ℓ_{K} and p are, respectively, the Kuhn and packing lengths. We also find that maximal entanglement (minimal N_{e}) coincides with the onset of local nematic order.

High-performance fiberglass/epoxy reinforced by functionalized CNTs for vehicle applications with less fuel consumption and greenhouse gas emissions

Carbon Nanotubes (CNTs) is among the most promising nanofiller materials that could be used for enhancing the properties of fiberglass/epoxy laminates for vehicle industries with less CO2 emission (the key player in the climate change). However, usually the commercialized CNTs are supplied in the shape of heavily entangled tubes what leads to random dispersion of CNTs in the polymer matrix and decrease in their performance, especially at industrial scale. Within this frame, the chemical functionalization process was used in the present research to avoid this problem and to modify the surface properties of CNTs at the same time, thus improving compatibility and solubility of CNTs in epoxy solutions. Afterwards, probe sonicator (pre-dispersion), ultrasonic path (main dispersion), mechanical mixer (mixed CNTs/Epoxy solutions with hardener), and vacuum infiltration (to remove air bubbles) were used to disperse functionalized CNTs with different concentrations (in the range 0.05–0.4 wt%) in the epoxy-hardener solutions. Then, vacuum-assisted resin transfer technique followed by curing process were used to prepare 4 layers-fiberglass/CNTs/epoxy panels. The mechanical and impact properties of the prepared panels were tested according to ASTM D7025 and ISO 6603-2 standards, respectively. Also, thermal behavior of the panels was investigated using thermogravimetric (TG-DTG). Finally, the environmental performance in terms of greenhouse gas emission (GHGE) was evaluated according to ISO-14040 standard, taking the resulting strength and changes in density into account. The results showed that 0.35 wt% of FCNTs were enough to improve the strength of panels by ~60%, compared to pure sample. Which means that weight structure of vehicles can decrease by 23%. Also, fuel consumption and GHGE can decrease significantly by 16% and ~26%, respectively. In addition, thermal stability and energy impact absorption at the same concentration of CNTs were improved by 5% and 31%, respectively.

Effect of Chain Length Dispersity on the Mobility of Entangled Polymers.

While nearly all theoretical and computational studies of entangled polymer melts have focused on uniform samples, polymer synthesis routes always result in some dispersity, albeit narrow, of distribution of molecular weights (Đ_{M}=M_{w}/M_{n}∼1.02-1.04). Here, the effects of dispersity on chain mobility are studied for entangled, disperse melts using a coarse-grained model for polyethylene. Polymer melts with chain lengths set to follow a Schulz-Zimm distribution for the same average M_{w}=36 kg/mol with Đ_{M}=1.0 to 1.16, were studied for times of 600-800 μs using molecular dynamics simulations. This time frame is longer than the time required to reach the diffusive regime. We find that dispersity in this range does not affect the entanglement time or tube diameter. However, while there is negligible difference in the average mobility of chains for the uniform distribution Đ_{M}=1.0 and Đ_{M}=1.02, the shortest chains move significantly faster than the longest ones offering a constraint release pathway for the melts for larger Đ_{M}.

An experimental study of the composite CNT/copper coating

This paper presents experimental results on the preparation and investigation of the carbon nanotubes–copper composite material. Carbon nanotubes (CNTs) were synthesized on silicon substrates by the chemical vapor deposition (CVD) method and then filled with copper by evaporation from a melting pot in a vacuum. Copper evenly covered both the surface of the entangled tubes and the free substrate surface between the tubes. To improve the adhesion of tubes and matrix material, a carbon substructure was grown on the surface of tubes by adding working gas plasma to the CNT synthesis area. It is proposed to use a copper coating as a diffusion barrier upon subsequent filling of the reinforcing CNT frame by a carbide-forming materials matrix with predetermined physico-mechanical and tribological properties.

Superhydrophobic and Flexible Microporous Polymer Paper

AbstractA flexible microporous polymer paper (MPP) is prepared with the polymer tubes obtained from the Sonogashira coupling reaction of 1,4‐diiodotetrafluorobenzene and 1,3,5‐triethynylbenzene. MPP has an interconnected network structure of polymer tubes and shows a hierarchical pore structure. Relatively large pores are formed among entangled polymer tubes, while micro‐ and mesopores are generated in the tube walls. MPP is foldable, mechanically stable, and stretchable. When the folding stress is removed, a wrinkle forms along the folded line, as observed in common papers composed of cellulose pulps. The mechanical stability of MPP results from electrostatic interactions between electron deficient tetrafluorobenzene units and electron rich triethynylbenzene units in the polymer. The tensile stress–tensile strain curve shows that MPP can elongate up to 15% strain. MPP is superhydrophobic, showing a water contact angle of 162° and can be used for the filtration of small molecules such as a dye and iodine.

Relation between Cross-Link Fluctuations and Elasticity in Entangled Polymer Networks

Computer simulation data on junction fluctuations in end-linked model networks are analyzed using the framework of the constraint junction model and the slip tube model for entangled networks. It is argued that cross-linking freezes in conformational fluctuations of chains and cross-links, which creates a small preload of entangled strands. This can be modeled using a “confinement degree of polymerization” Nc(N) with a weak dependence on the degree of polymerization N. The confined motion of network junctions with f active connections to the network is well described by using the exact solution of the constrained junction model with a constraining term that is given by the superposition of f/2 entanglement tubes with confinement Nc(N). It is shown that the modulus of end-linked model networks is the sum of a phantom and an entanglement contribution, whereby the entanglement contribution shows the same correction Nc(N) as both the cross-link fluctuations and the time average bond orientations. The results ...

A force-level theory of the rheology of entangled rod and chain polymer liquids. I. Tube deformation, microscopic yielding, and the nonlinear elastic limit.

We employ a first-principles-based, force-level approach to construct the anharmonic tube confinement field for entangled fluids of rigid needles, and also for chains described at the primitive-path (PP) level in two limiting situations where chain stretch is assumed to either be completely equilibrated or unrelaxed. The influence of shear and extensional deformation and polymer orientation is determined in a nonlinear elastic limit where dissipative relaxation processes are intentionally neglected. For needles and PP-level chains, a self-consistent analysis of transverse polymer harmonic dynamical fluctuations predicts that deformation-induced orientation leads to tube weakening or widening. In contrast, for deformed polymers in which chain stretch does not relax, we find tube strengthening or compression. For all three systems, a finite maximum transverse entanglement force localizing the polymers in effective tubes is predicted. The conditions when this entanglement force can be overcome by an externally applied force associated with macroscopic deformation can be crisply defined in the nonlinear elastic limit, and the possibility of a "microscopic absolute yielding" event destroying the tube confinement can be analyzed. For needles and contour-relaxed PP chains, this force imbalance occurs at a stress of order the equilibrium shear modulus and a strain of order unity, corresponding to a mechanically fragile entanglement tube field. However, for unrelaxed stretched chains, tube compression stabilizes transverse polymer confinement, and there appears to be no force imbalance. These results collectively suggest that the crossover from elastic to irreversible viscous response requires chain retraction to initiate disentanglement. We qualitatively discuss comparisons with existing phenomenological models for nonlinear startup shear, step strain, and creep rheology experiments.

Entanglement Density Tunes Microscale Nonlinear Response of Entangled Actin

We optically drive a microsphere at constant speed through entangled actin networks of 0.2 - 1.4 mg/ml at rates faster than the critical rate controlling the onset of a nonlinear response. By measuring the resistive force exerted on the microsphere during and following strain we reveal a critical concentration $c^{*}\simeq0.4$ mg/ml for nonlinear features to emerge. For $c>c^{*}$, entangled actin stiffens at short times with the degree of stiffening $S$ and corresponding timescale $t_{stiff}$ scaling with the entanglement tube density, i.e. $S\sim t_{stiff}\sim d_t^{-1}\sim c^{3/5}$. The network subsequently yields to a viscous regime with the yield distance $d_y$ scaling linearly with yield force $f_y$ and inversely with the entanglement length ($f_y\sim d_y\sim l_{e}^{-1} \sim c^{2/5}$). Stiffening and yielding dynamics are consistent with recent theoretical predictions for nonlinear cohesive breakdown of entanglements. We further show that above $c^{*}$ force relaxation proceeds via slow filament disengagement from dilated tubes coupled with $\sim$10x faster lateral hopping, with the corresponding concentration dependences in agreement with recent theoretical predictions for entangled rigid rods.

Network dynamics in nanofilled polymers.

It is well accepted that adding nanoparticles (NPs) to polymer melts can result in significant property improvements. Here we focus on the causes of mechanical reinforcement and present rheological measurements on favourably interacting mixtures of spherical silica NPs and poly(2-vinylpyridine), complemented by several dynamic and structural probes. While the system dynamics are polymer-like with increased friction for low silica loadings, they turn network-like when the mean face-to-face separation between NPs becomes smaller than the entanglement tube diameter. Gel-like dynamics with a Williams–Landel–Ferry temperature dependence then result. This dependence turns particle dominated, that is, Arrhenius-like, when the silica loading increases to ∼31 vol%, namely, when the average nearest distance between NP faces becomes comparable to the polymer's Kuhn length. Our results demonstrate that the flow properties of nanocomposites are complex and can be tuned via changes in filler loading, that is, the character of polymer bridges which ‘tie' NPs together into a network.

Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers.

A scaling model of self-similar conformations and dynamics of nonconcatenated entangled ring polymers is developed. Topological constraints force these ring polymers into compact conformations with fractal dimension df = 3 that we call fractal loopy globules (FLGs). This result is based on the conjecture that the overlap parameter of subsections of rings on all length scales is the same and equal to the Kavassalis-Noolandi number OKN ≈ 10-20. The dynamics of entangled rings is self-similar and proceeds as loops of increasing sizes are rearranged progressively at their respective diffusion times. The topological constraints associated with smaller rearranged loops affect the dynamics of larger loops through increasing the effective friction coefficient but have no influence on the entanglement tubes confining larger loops. As a result, the tube diameter defined as the average spacing between relevant topological constraints increases with time t, leading to "tube dilation". Analysis of the primitive paths in molecular dynamics simulations suggests a complete tube dilation with the tube diameter on the order of the time-dependent characteristic loop size. A characteristic loop at time t is defined as a ring section that has diffused a distance equal to its size during time t. We derive dynamic scaling exponents in terms of fractal dimensions of an entangled ring and the underlying primitive path and a parameter characterizing the extent of tube dilation. The results reproduce the predictions of different dynamic models of a single nonconcatenated entangled ring. We demonstrate that traditional generalization of single-ring models to multi-ring dynamics is not self-consistent and develop a FLG model with self-consistent multi-ring dynamics and complete tube dilation. This selfconsistent FLG model predicts that the longest relaxation time of nonconcatenated entangled ring polymers scales with their degree of polymerization N as τrelax ~ N7/3, while the diffusion coefficient of these rings scales as D3d ~ N-5/3. For the entangled solutions and melts of rings, we predict power law stress relaxation function G(t) ~ t-3/7 at t < τrelax without a rubbery plateau and the corresponding viscosity scaling with the degree of polymerization N as η ~ N4/3. These theoretical predictions are in good agreement with recent computer simulations and are consistent with experiments of melts of nonconcatenated entangled rings.

Crossover between activated reptation and arm retraction mechanisms in entangled rod-coil block copolymers.

Using a coarse-grained slip-spring model, the dynamics of rod-coil block copolymers is explored over a wide parameter space to fully capture the crossover between the short rod (activated reptation) and long rod (arm retraction) limits. An analytical, closed-form expression for curvilinear diffusion by activated reptation was derived by separating the drag into individual components for the rod and coil block. Curvilinear diffusion in the intermediate rod regime, where both mechanisms are important, was then found to be faster than predicted when both mechanisms are independently combined. The discrepancy in the crossover regime arises because the rod-coil copolymer's exploration of space is not accurately described by either a coil homopolymer (assumed by activated reptation) or a rod homopolymer (assumed by arm retraction). This effect is explored by tracking the rod orientation as the polymer reptates, confirming that the polymer reptates along a path that becomes more rodlike as the rod fraction is increased. Thus, activated reptation under-predicts diffusion because the rod can choose reptation paths that are more extended than the coil homopolymer by renewal of the entanglement tube from the ends. Arm retraction under-predicts diffusion because minor rotations of the rod allow some motion before full retractions of the coil block. Finally, more familiar 3-dimensional center-of-mass diffusion measurements are related to the curvilinear diffusion analysis because the ratio of these two quantities varies smoothly between the coil and rod homopolymer limits as the reptation path becomes more extended.

Self-Diffusion and Constraint Release in Isotropic Entangled Rod–Coil Block Copolymers

Understanding dynamic relaxation mechanisms in self-diffusion and constraint release processes of rod–coil block copolymers is important for many technological applications that employ neat melts or concentrated solutions. Using a model system composed of poly(alkoxyphenylenevinylene) rods and polyisoprene coils, reptation theories of entangled rod–coil block copolymers are investigated in the isotropic melt state. Self-diffusion was measured by forced Rayleigh scattering using a red laser line and a new blue photoswitchable dye that allow operation above the bandgap of most semiconducting polymers. In contrast to previous tracer studies where the diffusion of rod–coils through a coil homopolymer matrix is slowed relative to coil homopolymers because of a mismatch in the curvature of the rod and coil entanglement tubes, slowed diffusion is only present in self-diffusion measurements above a critical molecular weight. An activated reptation mechanism with constraint release is proposed as a modification to...

Tube Curvature Slows the Motion of Rod-Coil Block Copolymers through Activated Reptation.

Understanding the dynamics of molecules with complex shapes is important as researchers develop advanced materials using hybrid molecules. This study applies a slip-spring model to visualize and quantify the entangled dynamics of rod-coil block copolymers. The parameters of the model are determined by matching with molecular dynamics simulation results. By monitoring the positions of polymers along the entanglement tube, rod-coil copolymers are shown to disfavor configurations where the rod occupies curved portions of the tube of randomly varying curvature created by the coil ends. This confirms that reptation of copolymers occurs by an activated mechanism and is the first demonstration of the activation barriers that have been previously inferred through diffusion measurements by simulation and experiment. The barriers to diffusion are further quantified by considering the curvilinear motion of ring polymers, and their effect on diffusion is quantitatively captured by considering one-dimensional motion along an entanglement tube with a rough free energy potential.