Reptation Model Research Articles

Dilatational-plasticity opens a new mechanistic pathway for macromolecular transport across glassy interfaces

Interdiffusion-based macromolecular transport across glassy interfaces is reportedly achieved at high temperatures in accordance with the classical model of reptation. Here, for the first time, we report a new mechanistic pathway for achieving solid-state glassy joining by triggering rapid macromolecular acceleration through mechanical deformation. Large-scale molecular simulations reveal that active plastic deformation in glassy polymers, at temperatures well below the bulk (and surface) glass transition temperatures Tgb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {T}_g^b$$\\end{document} (and Tgs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {T}_g^s$$\\end{document}), causes segmental translations of macromolecules leading to interfacial interpenetrations, and the formation of new entanglements. The mechanistic basis for this new type of bonding is identified as molecular-scale dilatations and densifications during deformation-induced mobility. The reported insights open promising avenues for achieving quick, strong, and energetically less-intensive joining of polymeric glasses across various sectors.

Fast Field-Cycling Nuclear Magnetic Resonance Relaxometry of Perfluorosulfonic Acid Ionomers and Their Perfluorosulfonyl Fluoride Precursors Membranes.

The spin-lattice relaxation rates (R1) of fluorine nuclei in perfluorosulfonic acid (PFSA) ionomer membranes and their precursor solid perfluorosulfonyl fluoride (PFSF) were measured by fast field-cycling (FFC) NMR relaxometry. The XRD profiles of PFSA and PFSF are similar and show a characteristic peak, indicating the alignment of main chains. While the SAXS profiles of the PFSA membranes show two peaks, those of the solid PFSF lack the ionomer peak which is characteristic of hydrophilic side chains in the PFSA ionomer membranes. The Larmor frequency dependence of R1 obeys power law and the indices are dependent on the sample and temperature. The indices of the PFSA membranes change from -1/2 to -1 along with the Larmor frequency and temperature dependence decrease, which is consistent with the generalized defect diffusion model. Estimated activation energies are in good agreement with those obtained from dynamical mechanical analysis and dielectric spectroscopy, indicating the segmental motion of the backbones as the common origin of these observations. On the other hand, the index changes to -3/4 in the case of the PFSFs, which has been predicted by the reptation model.

Field-driven polyelectrolyte-polymer collisions in nanochannels.

Even though dilute (unentangled) polymer solutions cannot act as gel-like sieving media, it has been shown that they can be used to separate DNA molecules in capillary electrophoresis. The separation then comes from sporadic, independent DNA-polymer collisions. We study polymer-polymer collisions in nanochannels (i.e., channels that are smaller than the normal size of the polymers), a situation where a polyelectrolyte is forced to migrate "through" isolated uncharged molecules during electrophoresis. We use Langevin dynamics simulations to explore the nature of these collisions and their effect on the net motion of the two polymer chains. We identify several types of collisions, including some that are unique to nanochannels. When the uncharged polymer is much larger than the polyelectrolyte, the system is reminiscent of gel electrophoresis, and we propose a modified empirical reptation model to explain the data, with an orientation factor that depends on the tube diameter. We also observe that the duration of a collision is a non-monotonic function of the polymer size ratio when the two chains are of comparable size, a surprising resonance-like phenomenon, which, combined with the asymmetric nature of molecular conformations during collision, suggests possible ratchet-like mechanisms that could be used to sort polyelectrolytes in nanodevices.

The Challenges Facing the Current Paradigm Describing Viscoelastic Interactions in Polymer Melts.

Staudinger taught us that macromolecules were made up of covalently bonded monomer repeat units chaining up as polymer chains. This paradigm is not challenged in this paper. The main question raised in polymer physics remains: how do these long chains interact and move as a group when submitted to shear deformation at high temperature when they are viscous liquids? The current consensus is that we need to distinguish two cases: the deformation of "un-entangled chains" for macromolecules with molecular weight, M, smaller than Me, "the entanglement molecular weight", and the deformation of "entangled" chains for M > Me. The current paradigm stipulates that the properties of polymers derive from the statistical characteristics of the macromolecule itself, the designated statistical system that defines the thermodynamic state of the polymer. The current paradigm claims that the viscoelasticity of un-entangled melts is well described by the Rouse model and that the entanglement issues raised when M > Me, are well understood by the reptation model introduced by de Gennes and colleagues. Both models can be classified in the category of "chain dynamics statistics". In this paper, we examine in detail the failures and the current challenges facing the current paradigm of polymer rheology: the Rouse model for un-entangled melts, the reptation model for entangled melts, the time-temperature superposition principle, the strain-induced time dependence of viscosity, shear-refinement and sustained-orientation. The basic failure of the current paradigm and its inherent inability to fully describe the experimental reality is documented in this paper. In the discussion and conclusion sections of the paper, we suggest that a different solution to explain the viscoelasticity of polymer chains and of their "entanglement" is needed. This requires a change in paradigm to describe the dynamics of the interactions within the chains and across the chains. A brief description of our currently proposed open dissipative statistical approach, "the Grain-Field Statistics", is presented.

The transformation between different modes of motion of particles in steady aeolian transport

Aeolian sediment transport is an important morphodynamic process that drives geomorphological evolution, affects the climate system, and induces various natural disasters. The creep load involves both particles of a sufficiently large size that cannot be easily lifted to the upper air, as well as smaller particles that can transition between different modes of motion, although the contribution of the latter has rarely been investigated. In this study, we investigate the features of the saltation cycle, namely the time period from the moment a particle is entrained into the air to when the particle is again trapped in the bed, and the transformation mechanism between creep, reptation, and saltation by directly simulating the steady-state sand transport of particles within the saltation regime. The results indicate that a saltation cycle generally contains one to tens of individual hop processes, with the mean and standard deviation of the hop times decreasing with the increasing friction velocity, owing to the presence of more energetic particles at the bed surface. Over half of the transported particles move in creep or reptation model because a moving particle generally experiences frequent transition between creep, reptation, and saltation. Furthermore, the fundamental reason for the change in the mode of motion is the strong impact velocity dependence of the total coefficient of restitution during the rebound process. Current rebound models predict a reasonable coefficient of restitution only if the particle impacts the bed at a moderate velocity; however, their predictions deviate from the actual situation at relatively low or high impact velocities. These findings enhance our understanding of the motion dynamics of particles within wind-blown sand movements, and further reduce the uncertainty in predicting their mass flux in different modes of motion.

Rheology and self-healing of amine functionalized polyolefins

The rheological and self-healing behavior of a class of catalytically synthesized amine-functionalized polyolefins is investigated. We demonstrate that these materials possess tunable rheological properties according to the molecular weight and display autonomous self-healing. The linear viscoelastic properties are modeled using a tube-based model developed by Hawke et al. [J. Rheol., 60, 297–310, (2016)] to calculate several model parameters that describe the individual chain dynamics. The self-healing response is described by findings from the reptation model as well as recent theory on associating polymer networks with reversible bonds. The cooperation between experiments, modeling, and theory provide insight into designing new materials with programmable rheological properties and superior self-healing ability.

Determination of the molecular weight distribution of ultrahigh molecular weight polyethylene from solution rheology

We investigate the linear rheology of ultrahigh molecular weight polyethylene (UHMWPE) solutions with the aim of determining the molecular weight distribution of the polymer. The UHMWPE is dissolved in oligo-ethylene in order to avoid issues related to unfavorable interactions with the solvent. To prepare the solutions, UHMWPE, solvent, and a fixed amount of antioxidants are mixed by means of a corotating twin-screw microcompounder. All prepared solutions are within the concentrated regime, as confirmed by the scaling laws of the main rheological parameters (plateau modulus, relaxation time, and zero-shear viscosity) with concentration. Based on the viscoelastic response of the solutions, we adopt a heuristic approach to extrapolate the linear viscoelastic behavior of the melt, according to a time-concentration superposition principle. Such a technique allows us to span many decades of angular frequency, eventually attaining the terminal relaxation regime. The latter is difficult to achieve by direct measurements in the molten state because of experimental issues such as extremely long experimental times and thermal limits. The viscoelastic spectrum of the melt is used to obtain the molecular weight distribution (MWD) according to the time-dependent diffusion/double reptation model. The MWD of UHMWPE evaluated by using this approach agrees well with data obtained from gel permeation chromatography.

DNA topology dictates emergent bulk elasticity and hindered macromolecular diffusion in DNA-dextran composites

Polymer architecture plays critical roles in both bulk rheological properties and microscale macromolecular dynamics in entangled polymer solutions and composites. Ring polymers, in particular, have been the topic of much debate due to the inability of the celebrated reptation model to capture their observed dynamics. Macrorheology and differential dynamic microscopy (DDM) are powerful methods to determine entangled polymer dynamics across scales; yet, they typically require different samples under different conditions, preventing direct coupling of bulk rheological properties to the underlying macromolecular dynamics. Here, we perform macrorheology on composites of highly overlapping DNA and dextran polymers, focusing on the role of DNA topology (rings versus linear chains) as well as the relative volume fractions of DNA and dextran. On the same samples under the same conditions, we perform DDM and single-molecule tracking on embedded fluorescent-labeled DNA molecules immediately before and after bulk measurements. We show DNA-dextran composites exhibit unexpected nonmonotonic dependences of bulk viscoelasticity and molecular-level transport properties on the fraction of DNA comprising the composites, with characteristics that are strongly dependent on the DNA topology. We rationalize our results as arising from stretching and bundling of linear DNA versus compaction, swelling, and threading of rings driven by dextran-mediated depletion interactions.

ALTERNATIVE USE OF THE SENTMANAT EXTENSIONAL RHEOMETER TO INVESTIGATE THE RHEOLOGICAL BEHAVIOR OF INDUSTRIAL RUBBERS AT VERY LARGE DEFORMATIONS

ABSTRACT Extensional deformations represent an effective stimulus to explore the rich rheological response of branched polymers and elastomers, enabling the design of polymers with specific molecular structure. However, probing the polymer behavior at large deformations is often limited by the experimental devices. We here present an alternative use of the Sentmanat Extensional Rheometer (SER) that allows Hencky strain units much larger than the maximum value achievable, ∼3.6. The proposed procedure consists of an oblique positioning of the sample in the measuring area. If a small inclination of the sample is used, the departure from the ideal uniaxial flow is negligible at Hencky strains <1, and nearly zero for larger values. Experimental results in the linear viscoelastic regime are compared with the double reptation model in order to discern polydispersity and branching effects, whereas the extensional rheology data are contrasted with the molecular stress function theory (MSF), revealing important information about the polymer structure, especially on the long-chain branching (LCB). Finally, the analysis of sample failure upon elongation allowed us to correlate the polymer structure to the rheological behavior during mixing processes.

A multiscale analysis framework for formation and failure of the thermoplastic interface

Formation and failure of the interface determine the quality of thermoplastic composite structures. However, accurate prediction of interface strength evolution during advanced manufacture is still challenging due to the multiscale feature of thermoplastics, which hinders the design of composites processing and analysis of structure reliability. Here, we develop a multiscale model for formation and failure of thermoplastic interface. The reptation model of polymer chains is used to describe the chain density evolution at the interface during manufacture. Then, by releasing the rigid-bond assumption in the traditional freely-jointed-chain model, the Lennard-Jones potential function is incorporated to model polymer chain scission. Finally, chain-length dispersity, individual chain scission, and chain density reduction are incorporated to model the interface failure. The model predictions are in good agreement with the experimental results of composites welding, indicating that the developed model can be used to analyze the interface strength evolution of thermoplastics in the advanced manufacturing processes.

Surrogate Modeling with Gaussian Processes for an Inverse Problem in Polymer Dynamics

When rheological models of polymer blends are used for inverse modeling, they can characterize polymer mixtures from rheological observations. This requires repeated evaluation of potentially expensive rheological models. We explored surrogate models based on Gaussian processes (GP-SM) as a cheaper alternative for describing the rheology of polydisperse binary blends. We used the time-dependent diffusion double reptation (TDD-DR) model as the true model; it takes a 5-dimensional input vector specifying the binary blend as input and yields a function called the relaxation spectrum as output. We used the TDD-DR model to generate training data of different sizes [Formula: see text], via Latin hypercube sampling. The optimal values of the GP-SM hyper-parameters, assuming a separable covariance kernel, were obtained by maximum likelihood estimation. The GP-SM interpolates the training data by design and offers reasonable predictions of relaxation spectra with uncertainty estimates. In general, the accuracy of GP-SMs improves as the size of the training data [Formula: see text] increases, as does the cost for training and prediction. The optimal hyper-parameters were found to be relatively insensitive to [Formula: see text]. Finally, we considered the inverse problem of inferring the structure of the polymer blend from a synthetic dataset generated using the true model. Surprisingly, the solution to the inverse problem obtained using GP-SMs and TDD-DR was qualitatively similar. GP-SMs can be several orders of magnitude cheaper than expensive rheological models, which provides a proof-of-concept validation for using GP-SMs for inverse problems in polymer rheology.

Two-stage model of prepreg bonding interface formation in automated fiber placement process

While Automated Fiber Placement (AFP) of thermoset matrix composites are widely used in the aviation industry, there is little conclusive research on the relationship between the physical model of bonding interface formation process and the actual bonding strength between prepreg layers formed in AFP process. Although massive amounts of experimental data on prepreg tack have been achieved from existing research, engineers are unable to use these data as a decisive criterion in choosing process parameters. In this research, a prepreg layup physical model based on reptation model and viscoelastic mechanical model is built, in which the bonding interface formation process is divided into two stages, namely, diffusion and viscous stage. Layup-peeling experiments are conducted via a special designed high-speed layup experimental platform so that practical AFP process parameters can be imitated, and a logarithmic curve of layup velocity-peeling energy under different layup pressure is achieved. The slope of the logarithmic curve and the surface morphology of the sample after peeling prove the correctness of the established model. Simultaneously, the experimental data proves that when prepreg is peeled off, the transition from the cohesive failure mode to the interface failure mode occurs at the laying speed between 100 mm/s and 200 mm/s. These results can be used as a reference for choosing AFP process parameters to realize the balance between good bonding quality and harmless separation of adjacent prepreg layers.

Study of Allowable Interlaminar Normal Stress Based on the Time-Temperature Equivalence Principle in Automated Fiber Placement Process.

Automatic fiber placement (AFP) is a type of labor-saving automatic technology for forming composite materials that are widely used in aviation and other fields. In this process, concave surface delamination is a common defect, as existing research on the conditions for this defect to occur is insufficient. To predict the occurrence of this defect, the concept of allowable interlaminar normal stress is proposed to define its occurrence conditions, and based on this concept, probe tests are carried out using the principle of time–temperature equivalence. Through the laying speed/allowable normal stress curve measured in the probe experiment, the physical meaning of allowable normal stress is discussed. At the same time, the measured curve is quantitatively analyzed, combined with viscoelastic theory and the molecular diffusion reptation model, and the dominating effect in the formation of a metal/prepreg layer and prepreg/prepreg layer is determined. Finally, the experimental data are used to guide the parameter selection in an automatic placement engineering case and prove its correctness.

Synergistic ionic interactions in EMAA ionomer blends: A rheological and mechanical property investigation

The rheological and mechanical (tensile) properties of ethylene methacrylic acid (EMAA) ionomers neutralized by Na, Li, and Zn are investigated in binary combinations of two pure ionomers with different counterions. Molecular weight, methacrylic acid content, degree of neutralization, and ion type are tested as experimental parameters to investigate the composition that leads to the largest enhancement in rheological and mechanical properties. Binary cation blend components with low molecular weight and high neutralization reveal enhancement in rheological and mechanical properties beyond a linear mixing combination of material properties such as zero-shear viscosity, relaxation time, and Young’s modulus. The sticky reptation model is applied to EMAA ionomer blends of binary counterions to calculate the timescales of chain dynamics delayed by ionic associations. The ionic clusters of binary cations have longer-lasting ionic associations forming a reinforced reversible crosslink network. The experimental analysis consists of linear and nonlinear viscoelastic experiments, uniaxial extension, and dynamic mechanical analysis, which reveal the conditions and extent of synergism seen in binary EMAA ionomer blends.

Linear and nonlinear viscoelasticity of self-associative hydrogen-bonded polymers

We conducted a nonequilibrium dissipative particle dynamics simulation on the linear and nonlinear viscoelastic behaviors of polymer melt associated with hydrogen bonds. The effects of the number of hydrogen bonding groups and the strength of hydrogen bonding on viscoelasticity were examined. The simulation results show that the polymers with strongly associating hydrogen bonds exhibit supramolecular networks, and their motion displays some similarities to that of polymer melts following the Reptation model. The strain hardening is due to the strain-induced stretching of polymer chains in the supramolecular networks. The number and lifetime of hydrogen bonding have a combined impact on the dynamic and linear viscoelastic behaviors of self-associative polymers. The storage moduli increase linearly with increasing the numbers of hydrogen bonding groups as the lifetime of hydrogen bonds is comparable to or larger than the relaxation time of polymer chains. In contrast, hydrogen bonding has a less pronounced influence on the dynamics and viscoelasticity as the lifetime of hydrogen bonds is minimal. The results are closely aligned with the experimental findings and provide a deep insight into the viscoelasticity of self-associative polymers via hydrogen bonds.

A Dual-Phase Approach to Reveal the Presence and the Impact of the TLL Transition in Polymers Melts. Part I: Predicting the Existence of Boyer’s TLL Transition from the Vogel–Fulcher Equation

R.F. Boyer recognized the manifestations of a T > Tg transition-relaxation as early as 1963 and named it TLL, the liquid-liquid transition. He suggested that it was due to the melting of “local order”, a controversial issue conflicting with the dominant theories at the time, led by P. Flory, which asserted a structure-less liquid state for melts. At the same time as this controversy unrolled, de Gennes published his reptation model of polymer physics which, after some modifications and ramifications, quickly became the new paradigm to describe the dynamic properties of polymer flow. The new model of reptation has no theoretical arguments to account for a T > Tg transition occurring in the melt; hence, the current consensus about the existence of TLL is still what it was already in 1979, that it is probably an artifact only existing in the imagination of Boyer. In part I of this paper on the TLL transition we mathematically derive the existence and the characteristics of TLL from a dual-phase description of the free volume using a modification of the Vogel-Fulcher equation (VF), a well-known formulation of the temperature dependence of the viscosity of polymer melts. This new expression of the VF formula, that we call the TVF equation, permits to determine that TLL is in an iso-free volume and iso-enthalpic state when M, the molecular weight, varies. The data analyzed by the TVF equation are the dynamic rheological results for a series of monodispersed, un-entangled polystyrene samples taken from the work of Majesté. The new analysis also permits to put in evidence the existence of a new transition, which we call Mmc, approximately located at Mmc ≈ Mc/10, where Mc is the molecular weight for entanglement. A Dual-Phase interpretation of Mmc is proposed.

Diffusion of Thin Nanorods in Polymer Melts.

The diffusion of monomerically thin nanorods in polymer melts is studied by molecular dynamics simulations. We focus on the systems where chains are long enough to screen the hydrodynamic interactions such that the diffusion coefficient D ∥ for the direction parallel to the rod decreases linearly with increasing rod length l. In unentangled polymers, the diffusion coefficient for the direction normal to the rod exhibits a crossover from D ⊥ ~ l -2 to D ⊥ ~ l -1 with increasing l, corresponding to a progressive coupling of nanorod motion to the polymers. Accordingly, the rotational diffusion coefficient D R ≈ D ⊥ l -2 ~ l -4 and then D R ~ l -3 as l increases. In entangled polymers, D ⊥ and D R are suppressed for l larger than the entanglement mesh size a. D ⊥ ~ l -3 and D R ~ l -5 for l sufficiently above a in agreement with de Gennes' rod reptation model.

Cooperative Chain Dynamics of Tracer Chains in Highly Entangled Polyethylene Melts.

By neutron spin echo spectroscopy, we have studied the center of mass motion of short tracer chains on the molecular length scale within a highly entangled polymer matrix. The center of mass mean square displacements of the tracers independent of their molecular weight is subdiffusive at short times until it has reached the size of the tube d; then, a crossover to Fickian diffusion takes place. This observation cannot be understood within the tube model of reptation, but is rationalized as a result of important interchain couplings that lead to cooperative chain motion within the entanglement volume ∼d^{3}. Thus, the cooperative tracer chain motions are limited by the tube size d. If the center of mass displacement exceeds this size, uncorrelated Fickian diffusion takes over. Compared to the prediction of the Rouse model we observe a significantly reduced contribution of the tracer's internal modes to the spectra corroborating the finding of cooperative rather than Rouse dynamics within d^{3}.

Strategy for reducing molecular ensemble size for efficient rheological modeling of commercial polymers

We develop a method for efficient prediction of linear and nonlinear rheology of polydisperse polymers by judicious selection of a small number of representative polymer molecules from the large ensemble of chains comprising the molecular weight and branching distribution. Specifically, we use a numerical inversion of the double reptation model to select five or six representative molecular species to optimally fit the linear rheology of commercial polymers and then use the regressed parameters to compute the nonlinear rheology via the Rolie-double-poly (RDP) model. The method is tested for several model systems, namely, two polydisperse linear polystyrene (PS) samples described in Shivokhin et al. [Polym. Eng. Sci. 56, 1012–1020 (2016)] and Munstedt [J. Rheol. 24, 847–867 (1980)], a commercial linear low density polyethylene (LLDPE), and a low density polyethylene (LDPE) whose rheological properties are measured in the current study. For the linear polymers including the PS samples and the LLDPE, the predictions of the “representative” RDP model of the start-up extensional rheology are comparable to those of the “full” RDP model based on all the species drawn from the gas permeation chromatography characterization. The method then successfully predicts the linear rheology of blends of LLDPE and LDPE using the same representative molecules found by fitting each of the pure polymers. Further fitting the extensional rheology of the LDPE requires using the “priorities” q i and stretch relaxation times τ s , i of the representative molecules as adjustable parameters, whose values are then held fixed when predicting the extensional rheology of blends of the LDPE with the LLDPE roughly as successfully as does branch-on-branch model. The reduction in the number of representative polymer species offers new opportunities for faster simulations of flowing polymers, as well as for the prediction of segmental orientation to be used in the modeling of flow-induced crystallization.

Connecting the Stimuli-Responsive Rheology of Biopolymer Hydrogels to Underlying Hydrogen-Bonding Interactions.

Many biopolymer hydrogels are environmentally responsive because they are held together by physical associations that depend on pH and temperature. Here, we investigate how the pH and temperature responses of the rheology of hyaluronan hydrogels are connected to the underlying molecular interactions. Hyaluronan is an essential structural biopolymer in the human body with many applications in biomedicine. Using two-dimensional infrared spectroscopy, we show that hyaluronan chains become connected by hydrogen bonds when the pH is changed from 7.0 to 2.5 and that the bond density at pH 2.5 is independent of temperature. Temperature-dependent rheology measurements show that because of this hydrogen bonding the stress relaxation at pH 2.5 is strongly slowed down in comparison to pH 7.0, consistent with the sticky reptation model of associative polymers. From the flow activation energy, we conclude that each polymer is cross-linked by multiple (5–15) hydrogen bonds to others, causing slow macroscopic stress relaxation, despite the short time scale of breaking and reformation of each individual hydrogen bond. Our findings can aid the design of stimuli-responsive hydrogels with tailored viscoelastic properties for biomedical applications.