The slip-link model: A constitutive equation for general biaxial extension of polymer melts

Starting from a slip-link model, we derive a general strain measure for polymer melts. Predictions are compared to experimental data of uniaxial, planar, ellipsoidal, and equibiaxial extensions of a polyisobutylene (PIB) melt. The data do not support Doi's assumption that the tube diameter remains unchanged by deformation. The relative tube diameter and its inverse, the molecular stress function, can be extracted directly from the data. At small strains, the relative cross section of the tube is inversely proportional to the average stretch of the tube.     

Tube relaxation: A necessary concept in the dynamics of strained polymers

A theory for the relaxation of large strains in polymer melts is outlined. It is based on the Doi–Edwards slip-link model and the new concept of tube relaxation. Self-consistency makes this concept necessary when polymer melts are concerned. The discrepancy with previous non-self-consistent theories is not negligible and should be easy to observe experimentally. Special attention is given to the overall size and to the mean orientation of a labeled N-chain in a step-strained melt of P-chains. Using semiquantitative arguments, we predict different regimes depending on the respective values of N and P, and propose approximate evolution equations for each case. These equations may be used to design and interpret a variety of fluorescence, infrared, or NMR polarization measurements, as well as small-angle neutron scattering experiments.     

The Rheology of Linear and Long-chain Branched Polymer Melts

By generalising the Doi-Edwards tube model to the Molecular Stress Function theory, the non-linear rheology of polymer melts can be described quantitatively. The strain-hardening of linear polymer melts in extensional flows can be accounted for by a strain energy function, which reflects the increase of strain energy due to tube squeeze. In comparison to linear polymer melts, long-chain branched polymer melts show enhanced strain-hardening. This is due to the fact that while the backbone of the branched macromolecule is stretched by deformation, side chains are compressed. It is demonstrated that the experimentally observed slope of the elongational viscosity after inception of strain-hardening depends on the ratio β of total molar mass to backbone molar mass as predicted by the model. The steady-state (plateau) value of the elongational viscosity depends on the maximum relative stretch, , which can be supported by chain segments and which represents the maximum elastic energy storable in the polymeric system.     

Flow-alignment and rheology of polymer melts: Computation of the single-link orientational distribution function

The single-link orientational distribution function and the space-averaged stresses in the fluid are computed for the case of steady shear flow of polymer melts. The computation is achieved with Galerkin's method with spherical harmonics and Euler polynomials as trial functions. The stress components become power functions of shear rate when the latter is large. The single-link orientational distribution function f solves the Fokker-Planck equation subject to a boundary condition for f at the chain ends. A solution is obtained for every shear rate and ratio of the orientational and one-dimensional diffusion coefficient. It is demonstrated that the Fokker-Planck equation with appropriate boundary condition is useful in order to predict the flow-alignment and stresses in good agreement with experimental data as well as with recent results of a nonequilibrium molecular dynamics computer simulation on polymer melts.     

The theory of non-linear molecular orientation and stress in polymer fluids

Non-linear stress, and orientation characteristics for polymer fluids (melts, solutions) composed of chain macromolecules of finite length have been derived. Freelyjointed chains with inverse Langevin statistics have been assumed, and their behaviour in potential hydrodynamic fields analyzed. Numerical calculations have been performed for uniaxial extensional flow in a wide range of flow rates (and stresses). In the range of small stresses, orientation is a linear function of stress. At higher stresses, orientation factor levels off, asymptotically approaching unity.Flow orientation characteristics significantly differ from those derived from affine deformation of polymer networks. This difference is a natural consequence of constraints imposed by network junctions on chain deformation.This work is dedicated to Professor Hanns-Georg Kilian on his 60th birthday in appreciation of his contribution to Polymer Physics.     

Transition of polymers from rubbery elastic state to fluid state

On increasing the temperature of a polymer, the transition of the polymer from a rubbery elastic state to a fluid state could occur. The transition temperature is termed the fluid temperature of the polymer, T _f, which has a direct relationship with the polymer molecular weight. As one of polymer parameters, T _f is as important as the glass transition temperature of a polymer, T _g. Moreover, special attention to T _f should be paid for polymer processing. In research on the transition of a polymer from a rubbery elastic state to a fluid state, the concept of T _f would be more reasonable and more effective than the concept of T _l,l because it is neglected in the concept of T _l,l in that the molecular weight of a polymer may affect the transition of the polymer. In this paper the discussion on the fluid temperature involves the characters of polymers, such as the deformation—temperature curve, the temperature range of the rubbery state and the shear viscosity of polymer melt. From the viewpoint of the cohesional state of polymers, the transition of a polymer from a rubbery elastic state to a fluid state responds to destruction and construction of the cohesional entanglement network in the polymer. The relaxing network of polymer melt would be worthy to be considered as an object of study. __________ Translated from Huaxue Tongbao (Chemistry), 2008,71(3) (in Chinese)     

Dynamics of supramolecular associative polymer networks at the interplay of chain entanglement,transient chain association,and chain‐sticker clustering

The dynamic mechanical properties of supramolecular associative polymer networks depend on the average number of entanglements along the network‐forming chains, N_e, and on their content of associative groups, f . In addition, there may be further influence by aggregation of the associative groups into clusters, which, in turn, is influenced by the chemical structure of these groups, and again by N_e and f of the polymer. Therefore, the effects of these parameters are interdependent. To conceptually understand this interdependency, we study model networks in which (a) N_e, (b) f , and (c) the chemical structure of the associative groups are varied systematically. Each network is probed by rheology. The clustering of the associative groups is assessed by analyzing the rheological data at the end range of frequency covered and by comparison of the number of supramolecular network junctions with the maximum possible number of binary transient bonds. We find that if the total number of the network junctions, which can be formed either by interchain entanglement or by interchain transient associations, is greater than a threshold of 13, then the likelihood of cluster formation is high and the dynamics of supramolecular associative polymer networks is mainly controlled by this phenomenon. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1209–1223     

Application of the ISM EoS for polymer melts

A method for predicting an analytical equation of state for polymer melts from surface tension and liquid state density at the freezing temperature (γ_f,ρ_f), as scaling constants, is presented. B₂(T) follows a promising corresponding-states principle. Calculation of α(T) and b(T), the two other temperature-dependent constants of the equation of state, are made possible by scaling. As a result, γ_f and ρ_f are sufficient for the calculation of thermophysical properties of polymer melts. We applied the procedure to predict specific volumes of polyethylene glycol (PEG), polypropylene glycol (PPG), polypropylene (PP) and polyvinylchloride (PVC) at compressed state with temperature range from 298.15 to 423.15 K and pressures up to 200 MPa. The experimental specific volumes were correlated satisfactorily with our procedure and the results are within 3%.     

Orientation of cellulose crystallites in regenerated cellulose fibres under tensile and bending loads

The degree of orientation in regenerated cellulose fibres with a diameter of 36μm was determined using position-resolved synchrotron X-ray microbeam diffraction. The fibres were characterized in unstrained condition, under tensile strain, and in bending. A homogeneous distribution of the degree of crystalline orientation (Herman’s orientation factor f _c = 0.85) across the fibre thickness was found in the unstrained fibre. The degree of orientation of cellulose crystallites increased in a linear manner with increasing tensile strain applied to the fibre. Also in bending, a linear relationship between applied strain and the degree of crystalline orientation was found, where f _c increased in tension and decreased in compression. This linear relationship was found to be valid for both the tensile and the compressive zone of the bent fibre.     

Surface tension and density measurements of the molten YCl3-KCl binary system

The surface tension and the density of the YCl₃-KCl system were determined as a function of temperature (1073–1248 K) by the maximum bubble pressure technique. These results are represented by linear empirical equations as functions of temperature. At a constant temperature, the surface tensions of YCl₃-KCl melts show a negative deviation from Guggenheim's equation for ideal solutions.     

Influence of initial polymer concentration in solution and weight-average molecular weight on the drawing behavior of polyethylenes

The influence of initial polymer concentration in solution (c), weight-average molecular weight (M_ω), and drawing temperature on the solid-state drawing behavior of linear polyethylenes was investigated. Optimum conditions, with respect to maximum attainable draw ratio, are observed in isothermal drawing experiments. Moreover, it is shown that high maximum attainable draw ratios can also be obtained upon multistage drawing of UHMW-PE (ultrahigh-molecular-weight polyethylene, M_ω > 10⁶ g/mol) gel films cast from concentrated solutions. The high maximum attainable draw ratio in combination with the high molecular weight (M_ω > 10⁶ g/mol) and polymer concentration (c = 10% w/v) is of particular interest because it results in tapes or fibers with a high Young's modulus (100 GPa) and tensile strength (2.5–3.5 GPa). It is also shown that the maximum attainable draw ratio of polyethylenes scales with the Bueche parameter (c · M_ω) to the ?0.5 power. This experimental observation indicates that intermolecular interactions not only dominate the rheological properties of polyethylene melts and concentrated solutions, but also strongly influence the solid-state drawing behavior of linear polyethylenes.     

Effect of the functionality of junctions on the elasticity of polymacromonomer networks: Computer simulation

The elastic properties of polymer networks formed via the radical polymerization of macromonomers with two polymerizable end groups are studied via computer simulation. It is shown that variation in the average functionality of network junctions, f _avg, in a wide range (∼5–55) leads to a significant change in the shear modulus of the network. According to experiments with real networks (gels of poly(ethylene oxide) macromonomers), the shear modulus increases as f _avg increases. This effect is not due only to a decrease in the fluctuations of positions of network junctions. The main cause of the increase in the modulus is that the modulus component due to interaction between polymer chains (entanglements) increases as the functionality of junctions in the investigated networks increases. The conclusion is made that these networks gain entanglements during the formation of network junctions with high functionality rather than inherit them from the solution of macromonomer chains.     

Crazing and shear deformation in crosslinked polystyrene

Thin films of polystyrene (PS) are bonded to copper grids and crosslinked with electron irradiation. When the films are strained in tension regions of local plastic deformation, either crazed or plane stress deformation zones (DZs), nucleate and grow from dust particles. the nature of the local deformation, as well as the local extension ration λ, is determined by transmission electron microscopy. The behavior of the PS glass is consistent with its being a network of molecular strands of total density v = v_E + v_X, where v_E is the entangled strand density inferred from melt elasticity measurements of uncrosslinked PS and v_X is the density of crosslinked strands determined from the ratio of the applied electron dose to the electron dose for gelation. when v is less than 4 × 10²⁵ m^?3 (<1.3v_E), only crazes are observed whose microstructure is similar to those in uncrosslinked PS. As v increases from 4 × 10²⁵ to 8 × 10²⁵ m^?3 (from 1.3v_E to 2.5v_E) shear deformation begins to compete with crazing. As v increases above 8 × 10²⁵ m^?3, only shear DZs are observed, the strain in which becomes progressively more diffuse as v increases. The λ in the crazes and DZs correlate well with λ_max, the maximum extension ratio of a strand in a network of density v computed using the Porod×Kratky model. For crazes ln(λ) ? 0.9 ln(λ_max) and for DZs ln(λ) ? 0.55 ln(λ_max). The strain at which crack nucleation is first observed increases as v increases from <5% in uncrosslinked PS with v = 3.3 × 10²⁵ m^?3 to >20% in PS with v = 33 × 10²⁵ m^?3 (v = 10v_E); crosslinking to still higher crosslink densities, e.g., v = 14v_E, results in cracks which propagate in a catastrophic manner at low applied strains. An optimum v thus exists, one not too high to suppress local shear ductility but high enough to suppress crazes which can act as crack nucleation sites. these results are compared with previous results on a variety of linear homopolymers, copolymers, and polymer blends that are characterized by a wide range of v (v = v_E). The transitions from crazing to crazing plus shear and from crazing plus shear to shear only take place at almost identical values of v. In addition the correlation between λ in the crazes and DZs and λ_max for a single network strand is the same for both classes of polymers. This agreement implies that chain scission is the major mechanism by which strands in the entanglement network are removed in forming fibril surfaces. Craze suppression, by either increasing v in the crosslinked polymer or v_E in the uncrosslinked ones, is due to the extra energy required to break more main-chain bonds to form these surfaces.     

Thermoelasticity of networks in swelling equilibrium with single component diluents: Specific polymer-diluent interactions

Using thermoelastic measurements, specific polymer-diluent interactions have been demonstrated for atactic poly(vinyl alcohol) networks swollen in water. This was done by following a thermodynamic parameter as a function of network swelling, which was controlled by varying the extent of network crosslinking. At low crosslinking (high swelling) the ratio of the energy component f_e of the force to the total force f was found to be perhaps a little less than ?0.38 for poly(vinyl alcohol) swollen in water to less than 0.36 volume fraction of polymer. The method of evaluation is new and should be applicable to other polymer single component diluent systems.     

Structural developments in synthetic rubbers during uniaxial deformation by in situ synchrotron X‐ray diffraction

The molecular orientation and strain‐induced crystallization of synthetic rubbers—polyisoprene rubber, polybutadiene rubber, and butyl rubber [poly(isobutylene isoprene)]—during uniaxial deformation were studied with in situ synchrotron wide‐angle X‐ray diffraction. The high intensity of the synchrotron X‐rays and the new data analysis method made it possible to estimate the mass fractions of the strain‐induced crystals and amorphous chain segments in both the oriented and unoriented states. Contrary to the conventional concept, the majority of the molecules (50–75%) remained in an unoriented amorphous state at high strains. Each synthetic rubber showed a different behavior of strain‐induced crystallization and molecular orientation during extension and retraction. Our results confirmed the occurence of strain‐induced networks in the synthetic rubbers due to the inhomogeneity of the crosslink distribution. The strain‐induced networks containing microfibrillar crystals and oriented amorphous tie chains were responsible for the ultimate mechanical properties. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 956–964, 2004     

Plastic deformation of glassy polymethylene: Computer-aided molecular-dynamic simulation

Molecular-dynamic simulation of low-temperature plastic deformation (T _def = 50 K, T _def/T _g ≤ 0.3) is studied for glassy polymethylene under the regime of active uniaxial compression and tension for a cell composed of 64 chains containing 100 -CH₂ groups in each (as united atoms) and with periodic boundary conditions. Thirty-two such cells are created, and, in each cell, polymethylene chains in the statistical coil conformation are independently constructed. The cells are subjected to isothermal uniaxial compression at T _def = 50 K by ɛ = 30% and by ɛ = 70% under uniaxial tension. In the course of loading, a σ-ɛ diagram is recorded, while the mechanical work spent on deformation, the changes in the overall potential energy of the system, and the contributions from various potential interactions (noncovalent van der Waals bonds, chemical links, valence and torsional angles) are estimated. The results are averaged over all 32 cells. The relaxation of stored potential energy and residual strain after complete unloading of the deformed sample is studied. The relaxation of stored energy and residual strain is shown to be incomplete. Most of this energy and strain is stored in the sample at the deformation temperature for long period. The conformational composition of chains and the average density of polymer glass during loading are analyzed. Simulation results show that inelastic deformations commence not with the conformational unfolding of coils but with the nucleation of strain-bearing defects of a nonconformational nature. The main contribution to the energy of these defects is provided by van der Waals interactions. Strain-bearing defects are nucleated in a polymer glass during tension and compression primarily as short-scale positive volume fluctuations in the sample. During tension, the average density of the glass decreases; during compression, this parameter slightly increases to ɛ ≈ 8% and then decreases. An initial increase in the density indicates that, during compression and at ɛ < 8%, coils undergo compactization via an increase in chain packing. During compression, the concentration of trans conformers remains unchanged below ɛ ≈ 8% and then decreases. During compression, it means that in a glass, coils do not increase their sizes at strains below ɛ ≈ 8%. During tensile drawing, coils remain unfolded below ɛ ≈ 35%; at higher strains, coils become enriched with trans conformers or unfold. At this stage, the concentration of trans conformers linearly increases. The development of a strain-induced excess volume (strain-bearing defects) entails an increase in the potential energy of the sample. Under the given conditions of deformation, nucleation of strain-bearing defects and an increase in their concentration are found to be the only processes occurring at the initial stage of loading of glassy polymethylene. The results of computer-aided simulation are compared with the experimental data reported in the literature.     

Properties of simple models of polymer networks. A Monte Carlo simulation

A model polymer network was constructed from branched chains. Each chain was built on a simple cubic lattice forming a star-branched polymer consisting of f = 3 arms of equal lengths. The fragment of network under consideration consisted of 1, 2 and 3 star polymers with different topology of connections. The only potential used was excluded volume (athermal chains). The properties of the network were determined by the means of computer simulations using the classical Metropolis sampling algorithm (local micromodifications of chain conformation). The behaviour of linear chains of the same molecular weight was also studied as a state of reference. The influence of attaching the next star-branched chain to the network on its static and dynamic properties was studied. The short-time dynamic behaviour of chain fragments was determined and discussed.     

Molecular composites made of ionic poly(p‐phenylene terephthalamide) and poly(4‐vinylpyridine): Deformation modes

Deformation modes were examined on strained thin films of a series of molecular composites containing ionically modified rodlike molecules of poly(p‐phenylene terephthalamide) (PPTA) dispersed in a polar polymer matrix. The rigid molecules were a modified form of PPTA in which the H atom of the amide group was replaced, on 30 mol % of the monomer units, by an ionic propane sulfonate group. The polar polymer matrix of these composites was the flexible‐coil polymer, poly(4‐vinylpyridine). Ionic interactions between the two components increased the effective entanglement strand density and produced changes in the deformation modes. The observed changes were dependent on the relative concentration of the two components and on the nature of the counterion. With K⁺ as the counterion, the induced deformation mode changed from pure crazing, as in the matrix polymer, to combined crazing and shear deformation at 5 wt % of the ionic polymer and to essentially pure shear deformation as the concentration increased to 15 wt %. However, when Ca²⁺ was the counterion, pure shear deformation developed at a concentration of only 5 wt %. This effect was attributed to a greater ionic interaction and to a higher effective strand density of the composites, when monovalent K⁺ was replaced by divalent Ca²⁺. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 429–436, 2003     

Molecular orientation in hot-drawn poly(ethylene terephthalate) films as determined from the intrinsic polarized fluorescence

The groundstate-stable dimers in the non-crystalline regions of uniaxially drawn poly(ethylene terephthalate) (PET) films were used as chain-intrinsic fluorescent labels for studying the orientation distribution in the non-crystalline regions. As far as indicated by the spectra and the fluorescence decay law, the fluorescent group remains unchanged when the sample is uniaxially oriented by drawing above the glass transition temperature. In contrary to the behaviour of physically incorporated probe molecules, the orientation coefficient f ₂ ^F of the dimers is proportional to the amorphous orientation coefficientf ₂ ^A ; concerning the fluorescence signal, amorphous includes all the material outside the perfect crystal.During deformation, the orientation coefficientf ₂ ^F follows approximately a superposition curve of crystallite-like orientation, separable in the initial range of stretching ratio <2.5, and of true-amorphous orientation of the affine network type that becomes noticably at >3.At temperatures closely aboveT _g, and within the selected range of stretching parameters, the fluorescence intensity of PET remains nearly constant with increasing stretching ratio; at >2.5, where the maximum crystallite orientation is achieved and the increase of amorphous orientation becomes noticably, a slight augmentation of the dimer concentration is observed.     

Rouse–Bueche theory and the calculation of the monomeric friction coefficient in a filled system

Direct experimental access to the monomeric friction coefficient (ζ₀) relies on the availability of a suitable polymer dynamics model. Thus far, no method has been suggested that is applicable to filled systems, such as filled rubbers or microphase‐segregated A–B–A thermoplastic elastomers (TPEs) at T_g,B < T < T_g,A. Building upon the procedure proposed by Ferry for entangled and unfilled polymer melts, the Rouse–Bueche theory is applied to an undiluted triblock copolymer to extract ζ₀ from the linear behavior in the rubber‐glass transition region, and to estimate the size of Gaussian submolecules. When compared at constant T – T_g, the matrix monomeric friction factor is consistent with the corresponding value for the homopolymer melt. In addition, the characteristic Rouse dimensions are in good agreement with independent estimates based on the Kratky–Porod worm‐like chain model. These results seem to validate the proposed approach for estimating ζ₀ in filled systems. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 1437–1442