Reptation Model Research Articles

Cooperative dynamics in polymer melts from the unentangled to the entangled regime

The dynamics of polymer liquids present a rich variety of phenomena with characteristic scaling exponents that change across the transition from unentangled to entangled dynamics as the degree of polymerization of the polymer increases beyond a specific value. Entanglements are the result of topological interactions that have their origin in hard-core repulsive interactions between chains, which prevent the chains from crossing themselves and other chains. Entanglements are present in liquids of polymers with any degree of polymerization, but they only affect the dynamics of long chains. Conventional theoretical approaches to describing entangled and unentangled dynamics (i.e. the Rouse theory and the reptation model) are formally incompatible and do not encompass the dynamics across the transition. In this paper, we extend our approach for the cooperative dynamics of a group of polymers across the transition from unentangled to entangled dynamics. We test the robustness of the approach by comparing theoretical predictions against simulation trajectories of unentangled and entangled polyethylene melts. We find a good agreement between theory and simulations in both regimes.

Polymer dynamics from synthetic polymers to proteins

Starting from the standard model of polymer motion — the Rouse model — we briefly present some key experimental results on the mesoscopic dynamics of polymer systems. We touch the role of topological confinement as expressed in the reptation model and discuss in some more detail processes limiting the confinement. In the second part we relate to some new developments concerning the measurement of large-scale internal dynamics of proteins by neutron spin echo.

Review of the role of the interphase in the control of composite performance on micro- and nano-length scales

In fiber reinforced composites (FRCs), exhibiting heterogeneous structure at multiple length scales, the interphase phenomena at various length scales were shown to be of pivotal importance for the control of the performance and reliability of such structures. Various models based on continuum mechanics were used to describe effects of the macro- and meso-scale interphase on the mechanical response of laminates and large FRC parts, satisfactorilly. At the micro-scale, the interphase is considered a 3D continuum with ascribed average properties. Number of continuum mechanics models was derived over the last 50 years to describe the stress transfer between matrix and individual fiber with realtively good success. In these models, the interphase was characterized by some average shear strength, τa, and elastic modulus, Ea. On the other hand, models for tranforming the properties of the micro-scale interphase around individual fiber into the mechanical response of macroscopic multifiber composite have not been generally successfull. The anisotropy of these composite structures are the main reasons causing the failure of these models. The strong thickness dependence of the elastic modulus of the micro-scale interphase suggested the presence of its underlying sub-structure. On the nano-scale, the discrete molecular structure of the polymer has to be considered. The term interphase, originally introduced for continuum matter, has to be re-defined to include the discrete nature of the matter at this length scale. The segmental immobilization resulting in retarded reptation of chains caused by interactions with solid surface seems to be the primary phenomenon which can be used to re-define term interphase on the nano-scale. Thus, the Rubinstein reptation model and a simple percolation model were used to describe immobilization of chains near solid nano-particles and to explain the peculiarities in the viscoleastic response of nano-scale “interphase.” It has also been shown that below 5 nm, Bernoulli–Euler continuum elasticity becomes not valid and higher-order elasticity along with the proposed reptation dynamics approach can provide suitable means for bridging the gap in modeling the transition between the mechanics of continuum matter at the micro-scale and mechanics of discrete matter at the nano-scale.

Estimativa do tamanho da trinca em corpo de prova com linha de solda: simulação x resultado experimental

Este trabalho tem por objetivo apresentar os resultados obtidos por simulação numérica realizada com o aplicativo desenvolvido neste laboratório, o qual permite prever o tamanho da trinca em corpo de prova com linha de solda. Os resultados da simulação serão comparados com os resultados experimentais obtidos a partir de corpos de prova moldados pelo processo de injeção em diferentes temperaturas e caracterizados a partir de ensaio de tração. O aumento do tamanho da trinca em produtos moldados pelo processo de injeção pode provocar a diminuição da tensão de ruptura efetiva na região de sua existência devido à diminuição da área de contato ou de interpenetração. Este problema pode ser de grande importância em peças estruturais, pois pode reduzir a tensão efetiva de trabalho. A partir de equações que descrevem: o resfriamento de polímeros, a reologia de polímeros, a difusão e a interpenetração molecular e tomando-se como base o modelo de reptação descrito por De Gennes, determinou-se o perfil da difusão molecular na interface de junção dos fluxos e a interpenetração ao longo da espessura, bem como se determinou a interpenetração crítica utilizando-se o tamanho da trinca. Por fim, com a diminuição da temperatura de injeção, o corpo de prova solidifica mais rapidamente, diminuindo a difusão e a interpenetração molecular na interface da linha de junção. Com o aumento da interpenetração molecular, aumenta-se a tensão máxima de ruptura do polímero termoplástico Polystyrol 158 K da Basf. Determinou-se uma dependência entre a interpenetração máxima no centro do corpo de provas e a interpenetração crítica correspondente à trinca obtida por ensaios de tração em função da variação da temperatura. Assim, foi desenvolvido um aplicativo que descreve com muita coerência a dimensão da trinca.

Thermodynamic, Transport, and Viscoelastic Properties of PBX-9501 Binder: A Molecular Dynamics Simulations Study

Atomistic molecular dynamics simulations were performed on a low-molecular-weight nitroplasticized Estane® mixture representative of the binder used in PBX-9501. Pressure-volume-temperature (PVT) behavior over a wide range of pressures and temperatures above the order-disorder temperature (ODT) of Estane was determined and represented with the empirical Tait and Sun equations of state. The effect of temperature, pressure, and plasticization on transport properties of the mixture was also examined. A combination of molecular dynamics simulations and theoretical reptation models was used to predict the shear stress relaxation modulus G(t) of PBX-9501 binder at 473 K and 1 atm pressure. Data obtained from simulations of the model PBX-9501 binder presented here can be utilized to predict the temperature and pressure dependence of the shear stress relaxation modulus for temperatures above the ODT.

Entanglement Relaxation Time in Polyethylene: Simulation versus Experimental Data

A combined computer simulation and experimental study describing the viscoelastic properties of linear polyethylene is presented. For the simulation, a set of C1000 polyethylene models were equilibrated using advanced Monte Carlo moves. Then, MD trajectories were calculated. From these simulations the entanglement molecular weight, Me, and the entanglement relaxation time, τe, were directly obtained. By introducing the experimental value for the plateau modulus and the simulated values for Me and τe into the reptation model, one finds that the derived curves of the relaxation shear modulus nicely coincide with the experimental ones.

Electrophoretic migration of proteins in semidilute polymer solutions

We present a systematic study of the electrophoretic migration of 10-200 kDa protein fragments in dilute-polymer solutions using microfluidic chips. The electrophoretic mobility and dispersion of protein samples were measured in a series of monodisperse polydimethylacrylamide (PDMA) polymers of different molecular masses (243, 443, and 764 kDa, polydispersivity index <2) of varying concentration. The polymer solutions were characterized using rheometry. Prior to loading onto the microchip, the polymer solution was mixed with known concentrations of SDS (SDS) surfactant and a staining dye. SDS-denatured protein samples were electrokinetically injected, separated, and detected in the microchip using electric fields ranging from 100 to 300 V/cm. Our results show that the electrophoretic mobility of protein fragments decreases exponentially with the concentration c of the polymer solution. The mobility was found to decrease logarithmically with the molecular weight of the protein fragment. In addition, the mobility was found to be independent of the electric field in the separation channel. The dispersion is relatively independent of polymer concentration and it first increases with protein size and then decreases with a maximum at about 45 kDa. The resolution power of the device decreases with concentration of the PDMA solution but it is always better than 10% of the protein size. The protein migration does not seem to correspond to the Ogston or the reptation models. A semiempirical expression for mobility given by van Winkle fits the data very well.

Dynamical properties of semidilute solutions of hydrogen-bonded supramolecular polymers

The dynamical properties of semidilute solutions of supramolecular polymers formed from molecular recognition directed association between monomers bearing complementary hydrogen bonding groups were investigated by rheological and dynamic light scattering experiments. The steady-state flow curves showed a shear banding type instability, namely the occurrence of a stress plateau sigma(p) above a critical shear rate .gamma(c) . The values of sigma(p)and .gamma(c) were found to be of the same order of magnitude as those of the elastic plateau modulus and the inverse stress relaxation time, respectively. The above features are in agreement with the theoretical predictions based on the reptation model. Dynamic light scattering experiments showed the presence in the autocorrelation function of the concentration fluctuations of a slow viscoelastic relaxation process that is likely to be of Rouse type.

A molecular based model for polymer viscoelasticity: Intra- and inter-molecular variability

We develop dynamic equations for rubber viscoelasticity based on a stick-slip continuum molecular-based model. The model developed is a continuum tube reptation model in which a chemically cross-linked (CC) system of molecules act as constraint box per unit volume for a physically constrained (PC) system of molecules. The CC-system carries along the PC-system during instantaneous step deformations. The subsequent relaxation of the PC-system is determined by the configuration of the CC-system, its own configuration and confirmation, and external force fields. Conversely, the deformation of the PC-system acts as an internal variable affecting the deformations of the constraining CC-system. We model the relationship between these processes to derive a model of viscoelasticity in rubber deformation. In developing a relaxation process for the PC-system, we start from the fact that the PC-system is composed of long molecular chains. The dynamics of these molecular chains are developed by modelling them as chains of beads connected by springs, which represent inter-molecular potentials. Various segments of the molecular chains relax at different rates. In addition, variability in relaxation times across molecular chains is permitted.

Direct Measurement of the Intermolecular Forces Confining a Single Molecule in an Entangled Polymer Solution

We use optical tweezers to directly measure the intermolecular forces acting on a single polymer imposed by surrounding entangled polymers (115 kbp DNA, 1 mg/ml). A tubelike confining field was measured in accord with the key assumption of reptation models. A time-dependent harmonic potential opposed transverse displacement, in accord with recent simulation findings. A tube radius of 0.8 microm was determined, close to the predicted value (0.5 microm). Three relaxation modes (approximately 0.4, 5, and 34 s) were measured following transverse displacement, consistent with predicted relaxation mechanisms.

Diffusion in the two-dimensional necklace model for reptation

An extension of a recently introduced one-dimensional model, the necklace model, is used to study the reptation of a chain of N particles in a two-dimensional square lattice. The mobilities of end and middle particles of a chain are governed by three free parameters. This new model mimics the behavior of a long linear and flexible polymer in a gel. Noninteracting and self-avoiding chains are considered. For both cases, analytical approximations for the diffusion coefficient of the center of mass of the chain, for all values of N , are proposed. The validity of these approximations for different values of the free parameters is verified by means of Monte Carlo simulations. Extensions to higher dimensions are also discussed.

The effect of molar mass distribution on extrudate swell of linear polymers

Extrudate swell is an important viscoelastic feature of polymer melt processing. Experimental work with linear polymers has identified extrudate swell to depend strongly on average molar mass and molar mass distribution. Unfortunately there was no method identified to distinguish the relative effects of average mass and polydispersity. This paper revisits previously published experimental data on high-density polyethylenes and proposes a model to understand the coupled effect of these two molecular variables. The model connects findings from double reptation models via the recoverable compliance to extrudate swell. The reported results provide support to the model and lead to a better understanding of the important role of high-mass fractions in polydisperse polymers.

Dynamics of free chains in polymer nanocomposites

The dynamics of entangled polymeric chains in a polymer filled with nanoparticles is studied by means of molecular dynamics simulations of a model system. The primary objective is to study to what extent the reptation of polymers not in direct contact with fillers is modified with respect to the neat material. To this end, two systems are considered: A regular filled material in which the filler-polymer affinity is controlled, and a system in which the beads in contact with the filler at the beginning of the production phase of the simulation are tethered to the filler surface. This second system represents the limit case of long polymer-filler attachment time. In this case attention is focused on the free chains of the melt. The dynamics in the two models is different. In the filled system uniform slowing down for all Rouse modes is observed. The effect varies monotonically with the filler-polymer affinity. Up to saturation, this behavior may be captured by usual models with an effective, affinity-dependent, friction coefficient. In the system with grafted chains, the free chain Rouse dynamics is identical to that in the neat material, except for the longest modes which are significantly slowed down. More interestingly, the dynamics of the free chains depends in a nonmonotonic way on the polymer-filler affinity, although the free chains do not come in direct contact with the filler. This effect is due to small changes in the structure of the polydisperse brush upon modification of the affinity. Specifically, the density of the brush and the amount of interpenetration of free and grafted chains depend on the filler-polymer affinity. The use of a reptation model with modified tube diameter to capture this behavior is discussed.

Self-Diffusion of Entangled Linear and Circular DNA Molecules: Dependence on Length and Concentration

Self-diffusion coefficients (D) of DNA molecules of varying length and concentration were measured by tracking the Brownian motion of individual fluorescently labeled tracer molecules. Four possible cases were examined: linear tracer molecules surrounded by linear molecules (L-L), circular tracers surrounded by linears (C-L), linear tracers surrounded by circles ( L-C), and circles surrounded by circles (C-C). With 6 and 11 kilobasepair (kbp) DNA D was largely insensitive to topology and varied consistent with Rouse scaling (D L -1 C -0.5 ). In contrast, with 25 and 45 kbp DNA topology had a strong influence. At 1 mg/mL we found DC-C > DL-C. DL-L. DC-L.I n theL-L, L-C, and C-C cases a crossover from scaling consistent with the Rouse model to scaling consistent with the reptation model (D L -2 C -1.75 ) was observed at 6 times the molecular overlap concentration. In contrast, DC-L decreased much more steeply with concentration, indicating that a process much slower than reptation governs that case.

Cooperative polymer dynamics under nanoscopic pore confinements probed by field-cycling NMR relaxometry

Reptational dynamics of bulk polymer chains on a time scale between the Rouse mode relaxation time and the so-called disengagement time is not compatible with the basic thermodynamic law of fluctuations of the number of segments in a given volume. On the other hand, experimental field-cycling NMR relaxometry data of perfluoropolyether melts confined in Vycor, a porous silica glass of nominal pore dimension of 4 nm, closely display the predicted signatures for the molecular weight and frequency dependences of the spin-lattice relaxation time in this particular limit, namely T1 proportional M-1/2nu1/2. It is shown that this contradiction is an apparent one. In this paper a formalism is developed suggesting cooperative chain dynamics under nanoscopic pore confinements. The result is a cooperative reptational displacement phenomenon reducing the root-mean-squared displacement rate correspondingly but showing the same characteristic dependences as the ordinary reptation model. The tube diameter effective for cooperative reptation is estimated on this basis for the sample system under consideration and is found to be of the same order of magnitude as the nominal pore diameter of Vycor.

Application of reptation models to modeling of rheological behavior of polymer nanocomposites

Characteristic rheological behaviors of polymer nanocomposites were studied in shear flow and uniaxial elongational flow. Solid-like plateau storage modulus, strong shear thinning at low frequency regions, and strain hardening at elongational flow were observed. Especially, strain hardening was clearly observed for polymers without long chain branches if nanoparticles were homogeneously dispersed in the polymer matrix and interactions between nanoparticles and surrounding polymer molecules were sufficiently strong. Reptation models were used to model the nanoscale dynamics of nanoparticles and macromolecular chains, and the characteristic rheological behavior of nanocomposites could be explained. Brownian dynamics simulation of Doi-Edwards reptation model was applied and two particle constraint coefficients were introduced to express the influence of nanoparticles on molecular orientation and reptational diffusion of polymer chains. In the simulation, stress tensor including link tension coefficient which characterizes anisotropic friction coefficient of the molecular chain was used to obtain material functions by assuming that the anisotropy of friction was altered by the presence of nanoparticles. Additional frictional force between polymer chains and nanoparticles was considered and the suitable relaxation process and chain stretch were incorporated by considering the full chain geometry of polymer molecular chains. All the reptation models considering the effect of nanoparticles were verified by comparing the theoretical results with experimental data for polymer nanocomposites in shear and elongational flows.

Polymer dynamics: from synthetic polymers to proteins

In polymeric materials, the structures, the macroscopic mechanical and rheological properties, and the phase changes are determined to a high degree by thermal motion of the atoms and molecules. Most of the relevant dynamics take place on mesoscopic lengths and timescales in between the picosecond atomic scale and the macroscopic frame. Offering the proper space–time observation window, neutron spin echo spectroscopy uniquely addresses these motions. We briefly present some key experimental results on the mesoscopic dynamics of polymer systems. We address briefly the standard model of polymer motion, the Rouse model, the role of topological confinement as expressed in the reptation model and, finally, processes limiting the confinement – we discuss contour length fluctuations and constraint release of entangled chains. Very recently it also became possible to directly identify large-scale internal dynamics of proteins by neutron spin echo. We report the results of these pioneering studies, which are most likely to initiate further experiments on the large-scale motions of proteins and their relation to protein function.

Hydrophobically modified associating polyacrylamide (HAPAM) synthesized by micellar copolymerization at high monomer concentration

A series of hydrophobically modified polyacrylamides and their hydrolyzed derivatives containing both long hydrophobic groups C18 (<1.2 mol%), and carboxylic groups (20 mol%) have been prepared by micellar polymerization at high monomer concentration (19 wt%) and post-hydrolysis. 1H and 13C NMR elucidation of polymer microstructures displayed a good agreement with feed ratio. Post-hydrolysis process seems to be a more effective route to control the hydrolysis degree. Physico-chemical parameters determination by automatic continuous mixing (ACM) techniques in dilute solution shows the effect of the blocky distribution on the reinforcement of intramolecular hydrophobic association. Rheological measurements show potential thickening properties according to sticky reptation model.

Coarse-Grained Brownian Dynamics Simulations of Electrophoresis of DNA Molecules from Generalized Reptation Models

We derive a course-grained Brownian dynamics simulation algorithm for the motion of DNA chains under an external electric field. We generalize a model for bulk rheological response of entangled polymers by Schieber et al. [J. Rheol. 2003, 47, 213], allowing both rheology and electrophoresis to be modeled under the same theoretical framework. The effect of the external electric field is included by modifying the strand free energy, which then influences both the monomer sliding dynamics between entanglements and the renewal of entanglements at the chain ends. We also modify the free energy to include the finite extensibility of the chain segments and account for changes in local chain mobility with chain configuration. The resulting model gives a detailed account of the chain dynamics on length scales above the tube diameter and is suitable to model a wide range of field conditions. Simulation results for a constant-field mobility are presented, which fully characterize the model under these conditions. We ensure that the model captures known scaling regimes from analytic theory and compare the predictions to a set of experimental data for constant-field mobility of double-stranded DNA in agarose gels at a range of gel concentrations. This data comparison provides a method to characterize DNA−matrix systems, providing possibilities for modeling of more general field conditions.

Neutron Spin Echo for the Exploration of Large Scale Macromolecular Dynamics

In soft materials the structures, the macroscopic mechanical and rheological properties and the phase changes are determined to high degree but thermal motion of the atoms and molecules. Most of the relevant dynamics takes place on mesoscopic lengths and time scales in between the picosecond atomic scale and the macroscopic frame. Offering the proper space time observation window, neutron spin echo (NSE) spectroscopy uniquely addresses these motions. In this short review we briefly present some key experimental results on the mesoscopic dynamics of polymer systems. We address the standard model of polymer motion, the Rouse model, the role of topological confinement as expressed in the reptation model and finally, processes limiting the confinement–we discuss contour length fluctuations of entangled chains. Very recently it became also possible to directly identify large scale internal dynamics of proteins by neutron spin echo. We report the results of these pioneering studies, which most likely will initi...