Entangled Polymer Melts Research Articles

Diffusion of Sticky Nanoparticles in a Polymer Melt: Crossover from Suppressed to Enhanced Transport

The self-diffusion of a single large particle in a fluid is usually described by the classic Stokes–Einstein (SE) hydrodynamic relation. However, there are many fluids where the SE prediction for nanoparticles diffusion fails. These systems include diffusion of nanoparticles in porous media, in entangled and unentangled polymer melts and solutions, and protein diffusion in biological environments. A fundamental understanding of the microscopic parameters that govern nanoparticle diffusion is relevant to a wide range of applications. In this work, we present experimental measurements of the tracer diffusion coefficient of small and large nanoparticles that experience strong attractions with unentangled and entangled polymer melt matrices. For the small nanoparticle system, a crossover from suppressed to enhanced diffusion is observed with increasing polymer molecular weight. We interpret these observations based on our theoretical and simulation insights of the preceding article (paper 1) as a result of a ...

Theory and Simulation of Attractive Nanoparticle Transport in Polymer Melts

We theoretically study the diffusion of a single attractive nanoparticle (NP) in unentangled and entangled polymer melts based on combining microscopic “core–shell” and “vehicle” mechanisms in a dynamic bond percolation theory framework. A physical picture is constructed which addresses the role of chain length (N), degree of entanglement, nanoparticle size, and NP–polymer attraction strength. The nanoparticle diffusion constant is predicted to initially decrease with N due to the dominance of the core–shell mechanism, then to cross over to the vehicle diffusion regime with a weaker N dependence, and eventually plateau at large enough N. This behavior corresponds to decoupling of NP diffusivity from the macroscopic melt viscosity, which is reminiscent of repulsive NPs in entangled melts, but here it occurs for a distinct physical reason. Specifically, it reflects a crossover to a transport mechanism whereby nanoparticles adsorb on polymer chains and diffuse using them as “vehicles” over a characteristic d...

Strain localization during squeeze of an entangled polymer melt under constant force

We explore the nonlinear rheological behavior of an entangled melt under one-dimensional squeeze at constant squeezing force. In the absence of any meniscus failure, we find according to the particle-tracking velocimetric observations that the nonlinear response during creep takes place through strain localization. Specifically, the creep is observed to speed up due to the reduced viscosity that first takes place at the polymer/wall interface. Apart from the massive apparent wall slip, the particle-tracking velocimetry reveals strain localization in the bulk, presumably due to localized yielding of the entanglement network. Thus, the combination of rheometric information and knowledge about the strain field has illustrated a common feature concerning the nonlinear response of well-entangled polymers to large deformation, i.e., shear banding during yielding.

Identification of thermal shear bands in a low molecular weight polymer melt under oscillatory strain field

We present real-time micro-thermal measurements of the response of viscous fluids (low molecular weight unentangled and entangled polymer melts) submitted to an oscillatory mechanical shear strain (in conditions of conventional viscoelastic measurements). We show that thermal changes occur at the early steps of the applied deformation. A succession of thermodynamic states is identified showing the formation of non-uniform temperature shear bands along the strain direction. These thermal shear bands indicate the coexistence of cold and warm zones appearing in phase with the deformation. The synchronism of the temperature variation with the mechanical strain reveals a reversible process of elastic type indicating that viscous liquids might exhibit thermoelastic behaviors.

A singular perturbation study of the Rolie-Poly model

A singular perturbation analysis for the Rolie-Poly model is presented for the limit of large relaxation time. The model describes entangled linear polymer melts and time scales for a Rouse time and reptation time. First, the case of the Rouse time approaching zero is named the nonstretching model, and the addition of a solvent model gives the small parameter to be the ratio of the retardation time to the slow reptation time. This analysis is instructive in formulating the perturbation analysis for the full Rolie-Poly model without the solvent, where the small parameter is the ratio of Rouse time to reptation time. The resulting formulae elucidate the characteristic features of thixotropic yield stress fluids, including yield stress hysteresis, delayed yielding and long term persistence of a decreased viscosity after cessation of flow. Two fundamental initial value problems are solved: startup of shear flow and cessation of shear flow. The scalings lead naturally to simplified dynamical systems that describe regimes of fast, slow and yielded dynamics, and show how the combination of these regimes can be used to describe the entire flow evolution.

Nanorheology of Entangled Polymer Melts.

We use molecular simulations to probe the local viscoelasticity of an entangled polymer melt by tracking the motion of embedded nonsticky nanoparticles (NPs). As in conventional microrheology, the generalized Stokes-Einstein relation is employed to extract an effective stress relaxation function G_{GSE}(t) from the mean square displacement of NPs. G_{GSE}(t) for different NP diameters d are compared with the stress relaxation function G(t) of a pure polymer melt. The deviation of G_{GSE}(t) from G(t) reflects the incomplete coupling between NPs and the dynamic modes of the melt. For linear polymers, a plateau in G_{GSE}(t) emerges as d exceeds the entanglement mesh size a and approaches the entanglement plateau in G(t) for a pure melt with increasing d. For ring polymers, as d increases towards the spanning size R of ring polymers, G_{GSE}(t) approaches G(t) of the ring melt with no entanglement plateau.

Molecular Dynamics Investigation of the Relaxation Mechanism of Entangled Polymers after a Large Step Deformation

The chain retraction hypothesis of the tube model for nonlinear polymer rheology has been challenged by the recent small-angle neutron scattering (SANS) experiment (Wang, Z.; Lam, C. N.; Chen, W.-R.; Wang, W.; Liu, J.; Liu, Y.; Porcar, L.; Stanley, C. B.; Zhao, Z.; Hong, K.; Wang, Y., Fingerprinting Molecular Relaxation in Deformed Polymers. Phys. Rev. X 2017, 7, 031003). In this work, we further examine the microscopic relaxation mechanism of entangled polymer melts after a large step uniaxial extension by using large-scale molecular dynamics simulation. We show that the unique structural features associated with the chain retraction mechanism of the tube model are absent in our simulations, in agreement with the previous experimental results. In contrast to SANS experiments, molecular dynamics simulations allow us to accurately and unambiguously determine the evolution of the radius of gyration tensor of a long polymer chain after a large step deformation. Contrary to the prediction of the tube model, our simulations reveal that the radius of gyration in the perpendicular direction to stretching increases monotonically toward its equilibrium value throughout the stress relaxation. These results provide a critical step in improving our understanding of nonlinear rheology of entangled polymers.

Primitive Path Analysis and Stress Distribution in Highly Strained Macromolecules.

Polymer material properties are strongly affected by entanglement effects. For long polymer chains and composite materials, they are expected to be at the origin of many technically important phenomena, such as shear thinning or the Mullins effect, which microscopically can be related to topological constraints between chains. Starting from fully equilibrated highly entangled polymer melts, we investigate the effect of isochoric elongation on the entanglement structure and force distribution of such systems. Theoretically, the related viscoelastic response usually is discussed in terms of the tube model. We relate stress relaxation in the linear and nonlinear viscoelastic regimes to a primitive path analysis (PPA) and show that tension forces both along the original paths and along primitive paths, that is, the backbone of the tube, in the stretching direction correspond to each other. Unlike homogeneous relaxation along the chain contour, the PPA reveals a so far not observed long-lived clustering of topological constraints along the chains in the deformed state.

Dynamics of Nanoparticles in Entangled Polymer Solutions.

The mean square displacement ⟨r2⟩ of nanoparticle probes dispersed in simple isotropic liquids and in polymer solutions is interrogated using fluorescence correlation spectroscopy and single-particle tracking (SPT) experiments. Probe dynamics in different regimes of particle diameter (d), relative to characteristic polymer length scales, including the correlation length (ξ), the entanglement mesh size (a), and the radius of gyration (Rg), are investigated. In simple fluids and for polymer solutions in which d ≫ Rg, long-time particle dynamics obey random-walk statistics ⟨r2⟩:t, with the bulk zero-shear viscosity of the polymer solution determining the frictional resistance to particle motion. In contrast, in polymer solutions with d < Rg, polymer molecules in solution exert noncontinuum resistances to particle motion and nanoparticle probes appear to interact hydrodynamically only with a local fluid medium with effective drag comparable to that of a solution of polymer chain segments with sizes similar to those of the nanoparticle probes. Under these conditions, the nanoparticles exhibit orders of magnitude faster dynamics than those expected from continuum predictions based on the Stokes-Einstein relation. SPT measurements further show that when d > a, nanoparticle dynamics transition from diffusive to subdiffusive on long timescales, reminiscent of particle transport in a field with obstructions. This last finding is in stark contrast to the nanoparticle dynamics observed in entangled polymer melts, where X-ray photon correlation spectroscopy measurements reveal faster but hyperdiffusive dynamics. We analyze these results with the help of the hopping model for particle dynamics in polymers proposed by Cai et al. and, on that basis, discuss the physical origins of the local drag experienced by the nanoparticles in entangled polymer solutions.

Extensional Strain Hardening Induced by π-π Interactions in Barely Entangled Polymer Chains: The Curious Case of Poly(4-vinylbiphenyl).

Aromatic π-π interactions between phenyl groups of adjacent chains in poly(4-vinylbiphenyl) (PVBP) have profound effects on the dynamics of this polymer. We report two unexpected nonlinear viscoelastic responses of PVBP when subjected to uniaxial flow. One is the unprecedented observation of extensional strain hardening (SH) in a barely entangled polymer melt. An even more intriguing finding is that SH of lightly (or even barely) entangled melts occurs at strain rates one order of magnitude below the coil-stretch transition predicted by Rouse theory (ϵ[over ˙]_{H}=0.5/τ_{R}).We postulate that this behavior is due to a molecular rearrangement mechanism (supported by x-ray diffraction measurements) that involves flow-induced π-π stacking of the phenyl groups, which results in an enhancement of the friction coefficient between polymer chains.

Rheology and structural changes of plasticized zeins in the molten state

Zeins, storage proteins from maize, are suitable for making biobased thermoplastic materials. The rheological behavior of a commercial zein plasticized with 20 w% glycerol was studied in the molten state by steady-state flow experiments in extrusion conditions and oscillatory rheometry. For low residence times, a shear-thinning viscoelastic behavior was observed, with G″ exceeding G′. After 300 s at 130 °C, the complex viscosity |η ∗| = 7 × 103 ω −0.46 was found to be similar to that of thermoplastic polymer melts used in fused deposition modeling. However, the ratio between the exponents of the power laws describing G′(ω) and G″(ω) did not meet the typical value of 2 for entangled polymer melts. Moreover, for longer residence times, the viscosity increased and a gelation phenomenon was observed with a crossing over of G′(ω) and G″(ω). Gel times ranged from 6000 s at 120 °C to 1700 s at 150 °C. The evolution of the macromolecular structure assessed by sodium dodecyl sulfate-polyacrylamide gel electrophoresis and high-performance size exclusion chromatography suggested that this gelation phenomenon involves various types of covalent and non-covalent cross-links. Disulfide bonds played a significant role in gelation kinetics despite a very low cysteine residue content in the protein primary structure (about 1 mol%). These results suggested that plasticized zeins initially behave like a low-viscosity non-entangled polymer melt, before cross-linking progressively led to a continuous network.

Diffusion dynamics of nanoparticle and its coupling with polymers in polymer nanocomposites

Diffusion dynamics of nanoparticles in polymer nanocomposites is investigated by coarse-grained molecular dynamics simulations. We report a strong polymer chain length dependent non-Gaussian diffusion of nanoparticles (NPs) in entangled polymer melt. By analyzing the NP motion and the polymer relaxation at different length and time scales, we find that the non-Gaussian diffusion of NPs originates from a dynamic coupling at a specific length scale with the melt polymer chain segments that possess similar size with NP. Our results provide a microscopic understanding of the complex particle environment interactions and a detailed mechanism for nanoparticle diffusion in entangled polymer melt.

Stability analysis of bilayer polymer fiber spinning process

The linear stability of a bilayer fiber spinning process is analyzed for axisymmetric disturbances under isothermal conditions. The co-extruded fiber consists of two concentric layers of rheologically stratified polymeric fluids. While the core fluid is an entangled polymer melt described using the eXtended Pom-Pom (XPP) model, the sheath layer is a low molecular weight unentangled polymer modeled as an Upper Convected Maxwell (UCM) fluid. The analysis indicates that, for fast extensional flows, such an arrangement enhances the critical draw ratio of the process over that found from the spinning of XPP fluid alone, thus delaying the onset of draw resonance and stabilizing the system. Further, the range of flow Deborah number for stable spinning is found to be much broader than that for the UCM fluid alone. For low to moderately elastic flows, the stability behavior is governed mainly by the rheology of the more-elastic core layer (XPP fluid), whereas for the highly elastic flows, the stability behavior is dominated by the less-elastic sheath layer (UCM fluid). The sensitivity of the stability diagram with respect to various spinning flow parameters, like the relative fraction of core and sheath layers, the rheological stratification of two fluids, and the spine-line dimensions, has also been examined in order to identify the region of enhanced stability such that the draw resonance is suppressed.

Detailed analysis of Rouse mode and dynamic scattering function of highly entangled polymer melts in equilibrium

We present large-scale molecular dynamics simulations for a coarse-grained model of polymer melts in equilibrium. From detailed Rouse mode analysis we show that the time-dependent relaxation of the autocorrelation function (ACF) of modes p can be well described by the effective stretched exponential function due to the crossover from Rouse to reptation regime. The ACF is independent of chain sizes N for N/p < Ne (Ne is the entanglement length), and there exists a minimum of the stretching exponent as N/p → Ne. As N/p increases, we verify the crossover scaling behavior of the effective relaxation time τeff,p from the Rouse regime to the reptation regime. We have also provided evidence that the incoherent dynamic scattering function follows the same crossover scaling behavior of the mean square displacement of monomers at the corresponding characteristic time scales. The decay of the coherent dynamic scattering function is slowed down and a plateau develops as chain sizes increase at the intermediate time and wave length scales. The tube diameter extracted from the coherent dynamic scattering function is equivalent to the previous estimate from the mean square displacement of monomers.

Equation of State Based Slip Spring Model for Entangled Polymer Dynamics

A mesoscopic, mixed particle- and field-based Brownian dynamics methodology for the simulation of entangled polymer melts has been developed. Polymeric beads consist of several Kuhn segments, and their motion is dictated by the Helmholtz energy of the sample, which is a sum of the entropic elasticity of chain strands between beads, slip springs, and nonbonded interactions. The entanglement effect is introduced by the slip springs, which are springs connecting either nonsuccessive beads on the same chain or beads on different polymer chains. The terminal positions of slip springs are altered during the simulation through a kinetic Monte Carlo hopping scheme, with rate-controlled creation/destruction processes for the slip springs at chain ends. The rate constants are consistent with the free energy function employed and satisfy microscopic reversibility at equilibrium. The free energy of nonbonded interactions is derived from an appropriate equation of state, and it is computed as a functional of the local density by passing an orthogonal grid through the simulation box; accounting for it is necessary for reproducing the correct compressibility of the polymeric material. Parameters invoked by the mesoscopic model are derived from experimental volumetric and viscosity data or from atomistic molecular dynamics simulations, establishing a "bottom-up" predictive framework for conducting slip spring simulations of polymeric systems of specific chemistry. The mesoscopic simulation methodology is implemented for the case of cis-1,4-polyisoprene, whose structure, dynamics, thermodynamics, and linear rheology in the melt state are quantitatively predicted and validated without a posteriori fitting the results to experimental measurements.

Complex dynamics of capillary imbibition of poly(ethylene oxide) melts in nanoporous alumina

Capillary penetration of a series of entangled poly(ethylene oxide) melts within nanopores of self-ordered alumina follows an approximate t1/2 behavior according to the Lucas-Washburn equation; t is the time. However, the dependence on the capillary diameter deviates from the predicted proportionality to d1/2; d is the pore diameter. We observed a reversal in the dynamics of capillary rise with polymer molecular weight. Chains with 50 entanglements (Mw ≤ 100 kg/mol) or less show a slower capillary rise than theoretically predicted as opposed to chains with more entanglements (Mw ≥ 500 kg/mol) that display a faster capillary rise. Although a faster capillary rise has been predicted by theory and observed experimentally, it is the first time to our knowledge that a slower capillary rise is observed for an entangled polymer melt under conditions of strong confinement (with 2Rg/d = 1). These results are discussed in the light of theoretical predictions for the existence of a critical length scale that depends on the molecular weight and separates the microscopic (d < d*) from the macroscopic (d > d*) regime.

Retraction of "Origin of Stress Overshoot during Startup Shear of Entangled Polymer Melts".

Using Brownian dynamics simulation, we determine the chain orientation and stretching and their connection to stress overshoot in an entangled polymer melt undergoing startup shear at rates lower than the reciprocal of the Rouse time yet higher than the reciprocal of the reptation time. In this rate regime, the prevailing tube theory attributes the stress overshoot to alignment of the primitive chain. In contrast, our results reveal that there is substantial chain stretching that persists well beyond the Rouse time, and it contributes significantly to the initial stress growth. In particular, stress overshoot is found to be primarily due to chain retraction after considerable stretching rather than chain orientation.

Retraction of “Coupled Effect of Orientation, Stretching and Retraction on the Dimension of Entangled Polymer Chains during Startup Shear”

The theoretical description of how entangled polymers undergo large deformation has been at the heart of polymer dynamics. Over the years, the polymer physics community has adopted the tube model as the standard paradigm for describing polymer deformation and flow in both the linear and nonlinear regimes. The tube model envisions an ensemble of chains that undergo reptation and fluctuation inside smooth confining tubes on time scales of the Rouse relaxation time. It provides a reasonable and appealing description of the equilibrium and near equilibrium dynamics of entangled polymer melts and concentrated solutions. Some of the predictions based on this paradigm are also in apparent agreement with macroscopic rheological measurements for large deformations. However, some discrepancies with the tube model have been noted in the literature. For example, the time for the onset of time-strain superposibility in experiments has been shown to be much longer than the Rouse time anticipated based on the tube model. More recently, S.-Q. Wang’s group using particle-tracking velocimetry reported a number of nonlinear rheological phenomena that questioned the validity of the tube-model assumptions. The theoretical implications of the reported phenomena have been controversial.

Molecular dynamics simulations of polymer crystallization under confinement: Entanglement effect

We carried out molecular dynamics simulations to study the crystallization of polymer melts subjected to confinement between two parallel walls. Two types of walls, bare and grafted walls, are studied. The bare walls are slippery for chain motions, and the interactions between polymer chains and the walls are moreover chosen either attractive or repulsive. The crystallization in the case of bare walls generally consists of surface-induced processes close to the walls followed by homogeneous nucleation in both the boundary and middle regions. In the case of grafted walls, parts of polymer chains residing close to the walls are adhesive to the surfaces and become permanent graft points. We find that the surface-induced crystallization is increasingly suppressed with increasing grafting density. At high grafting densities, only crystallization in the middle regions is observed. We calculated the spatial distribution of entanglement lengths and related it to the crystallization behavior. The entanglement length close to the walls is found to decrease with increasing grafting density, as the adhesion points act as effective entanglement knots. In light of our recent results that less entangled polymer melts lead to faster crystallization and higher crystallization order, we now show that this conclusion stands also for the case of confined polymer melts. Our results suggest entanglements to be an universal factor towards the understanding of polymer crystallization under different situations, in particular at supercooling.

Orientational Cross-Correlation in Entangled Binary Blends in Primitive Chain Network Simulations

It has been reported from the molecular dynamics simulation that the orientational cross-correlation (OCC) has a significant contribution to the relaxation modulus of polymers. Remarkably, the OCC contribution has been found to be universal with respect to the molecular weight and even to its distribution for unentangled polymers [Cao; Likhtman Phys. Rev. Lett. 2010, 104, 207801]. In this study, the OCC in entangled bidisperse polymer melts was evaluated by multichain slip-link simulations. The obtained OCC was similar to that in monodisperse systems in its intensity. On the other hand, the time development reflected the motion of short and long chains, and consequently the universality was not observed when the molecular weights of two components are well-separated and the long chain fraction is small. Comparison to the results obtained by the other slip-link model suggests that the cross-correlation is induced by the force balance and the fluctuation at the entanglement.