The chemo-mechanical coupling behavior of hydrogels incorporating entanglements of polymer chains

To describe precisely the chemo-mechanical coupling behavior of hydrogels, a general form of free energy density function is presented by considering chain entanglements and functionality of junctions. We use the chemical potential of the solvent and the deformation gradient of the network as the independent variables of the developed free energy function, and implement this material model in the finite element package, ABAQUS, to analyze several examples of chemo-mechanical equilibrium deformation behaviors of hydrogels. The influence of chain entanglements and junction functionality on the chemo-mechanical behavior of hydrogels is addressed based on our simulation. With the coded subroutine UHYPER, this work may provide a numerical tool to study complex phenomena in hydrogels.     

Some thermal and rheological properties of homogeneous interpenetrating polymer networks

Summary Homogeneous interpenetrating polymer networks, (IPNs), consisting of methacrylic and epoxy networks, were obtained at various compositions from a simultaneous polymerization of DGEBAMA and DGEBA. For each composition, the glass transition occurs at a well-defined temperature which is lower than the weighted average of the glass transition temperatures of each component. Tensile experiments showed a change of mechanical behaviour above some critical strain value. This phenomenon was corroborated by stress relaxation tests which allowed the determination of a complete relaxation below a critical strain. This strain is increasing with the temperature and decreasing with the crosslink density. Such a property disappeared after the addition of grafting molecules which prevented both networks from any relative sliding. In this way this behaviour appears to be a specific property of interpenetrating networks.With 10 figures     

A theory for large deformation and damage of interpenetrating polymer networks

Elastomers and gels can be formed by interpenetrating two polymer networks on a molecular scale. This paper develops a theory to characterize the large deformation and damage of interpenetrating polymer networks. The theory integrates an interpenetrating network model with the network alteration theory. The interpenetration of one network stretches polymer chains in the other network and reduces its chain density, significantly affecting the initial modulus, stiffening and damage properties of the resultant elastomers and gels. Double-network hydrogels, a special type of interpenetrating polymer network, have demonstrated intriguing mechanical properties including high fracture toughness, Mullins effects, and necking instability. These properties have been qualitatively attributed to the damage of polymer networks. Using the theory, we quantitatively illustrate how the interplay between polymer-chain stiffening and damage-induced softening can cause the Mullins effect and necking instability. The theory is further implemented into a finite-element model to simulate the initiation and propagation of necking instability in double-network hydrogels. The theoretical and numerical results are compared with experimental data from multiple cyclic compressive and tensile tests.     

A constitutive model of polyacrylate interpenetrating polymer networks for dielectric elastomers

A physically based method is proposed to represent interpenetrating polymer networks and their electromechanical behavior. The mechanical behavior of the material is nonlinear elastic and the electromechanical coupling arises from electrostatic effects often called the Maxwell stress effect. Ha et al. have synthesized interpenetrating polymer networks (IPNs) that invalidate the need for an external pre-stretch mechanism in dielectric elastomers. IPNs of acrylic elastomer and 1, 6-hexanediol diacrylate were successfully synthesized to create free-standing films with preserved prestretch. This results in a dual polymer network, with one polymer network in tension and the other in compression. The prestretch is preserved chemically in the dominant network. The internal prestretch is accompanied by an overall stiffening of the dual polymer network leading to compromised actuation strains. A mechanistically simple representation of the networks is proposed by means of a model of two springs in parallel, replaced by an equivalent single spring. A material parameter is introduced to account for the effect of the weight percent of the secondary network. The effect of the additive on the preserved prestretch in the primary network and hence the overall stress strain response is determined. Specifically, a modified Ogden strain energy function is proposed that describes the mechanical behavior of the new interpenetrating polymer network. The electromechanical response of the material is described using a previously presented constitutive formulation that works well for single network polymers. The model results indicate that ideally an interpenetrating polymer network DE should not stiffen when the secondary network is formed to avoid reduced actuation strains.     

Ultrasonic investigation of interpenetrating networks of poly(methyl methacrylate) and polyurethane

Interpenetrating networks are the most recent development in polymeric blend materials. Due to the crosslinking of both the continuous and dispersed phases, a high degree of molecular mixing is achieved in these materials. Notwith-standing that poly(methyl methacrylate)-polyurethane (PMMA-PUR) interpene-trating and semi-interpenetrating networks have been extensively investigated by Meyer et al., ultrasonic relaxation technique has been applied here for the first time. These materials were found to be highly ultrasound absorbing.It is observed that ultrasonic absorption has a peak at a particular composition of PMMA-PUR interpenetrating network. The absorption coefficient increases with frequencyf. The absorption is of relaxational nature and is not due to the scattering of ultrasonic waves by the domains of the dispersed phase. At every composition of the interpenetrating network, the/f ² vs.f curve indicates the presence of a relaxation frequency below 2 MHz and that the absorption increases with the temperature at some compositions which indicates the presence of thermal relaxation. An attempt is made to relate the absorption with the relaxation of pendent groups of polyurethane in the continuous phase.     

Convection and diffusion of polymer network strands

A review of several important constitutive equations is herein conducted with an eye towards determining those most suitable for use in modelling polymer melt processing. General principles are invoked for a priori screening of the equations without needing detailed comparison of the model predictions with experimental data. These principles, which are derived from continuum mechanics, thermodynamics and molecular kinetic theory, and dela with convection and diffusion of entangled polymer strands during flow, are: (1) During sudden deformations, the stress is a unique function of the total strain. (2) The second law of thermodynamics holds for all deformations. (3) The constitutive equation can be derived from a plausible molecular model which describes the convection and diffusion. (4) The model parameters can be determined by a reasonable number of rheometric experiments. Based on these principles, it is concluded that separable free energy models are the most promising. These are either BKZ integral models with a kernel factorable into a time-dependent and a strain-dependent part. or sets of Maxwell-type differential equations that employ a generalized convected derivative, and that are linear in stress in the absence of flow.     

Capillary flow behavior of bicomponent polymer blends

The flow behavior of bicomponent polymer blends of four types of polymers (polypropylene, polystyrene, high-density polyethylene and polymethyl-methacrylate) was examined using a capillary extrusion rheometer. The viscosity of the blend was generally less than the value calculated by the theoretical or empirical additivity rules proposed in previous reports, whereas the entrance pressure loss, which is considered to be an effect of elasticity, was larger than the estimated value. Thus the variation of the viscosity with blending ratio was inversely proportional to the variation in the elastic property. The cross-section of the material extruded in a roughly dispersed state showed an annularly stratified flow pattern in which the lower viscosity component polymer appeared to form the outer skin layer. However, the observation that the viscosity of the properly blended material at certain blending ratios was sometimes lower than that of either homopolymer could not be explained.     

On the viscosity-temperature behavior of polymer melts

Based on the free volume concept and the equation by Doolittle, an empirical equation is offered for the flow activation energy, E ^*, for polymer melts for the range of over 150°C above glass transition temperature, T _g. This E ^* represents the temperature coefficient of viscosity for the Newtonian region which is also equal to the value measured at constant shear stress for non-Newtonian flow. Data show that the E ^* of linear polymers approaches a constant value for a temperature range above T _g+150°C. Data on 17 polymers are correlated. The proposed equation for this region predicts the E ^* of polymer melts from the volume expansion coefficient, _l, above T _g and also from the T _g.Correlations have also been developed between E ^* and _l and between E ^* and T _g by simplifying the equation by use of the Simha-Boyer expression. A polymer having a lower _l or higher T _g generally has a higher E ^*. However, more satisfactory results are obtained by calculating E ^* from both _l and T _g. The E ^* calculated is found to agree with measurements within the experimental precision of about ±1 Kcal/mole.The effects of polymer composition, molecular weight, branching and microstructure on E ^* are also discussed. These factors influence E ^* in the way in which they effect _l and T _g.     

Impact fracture characterization of polymer with ductile behavior

The characterization of ductile polymers with ductile fracture behavior is still a controversial subject. Presently, two approaches based on fracture mechanics have been proposed for the evaluation of impact failure of plastics, i.e., the method using the concept of essential fracture work, and the crack initiation and propagation approach. This paper discusses the validity of the fracture parameters used in these approaches. Additional experimental results and evidence are presented; they indicate that the concept of essential work of fracture cannot be used to characterize the fracture property of plastics. At high loading rates, the fracture energy of plastics can decrease with crack extension, resulting in a variable essential work of fracture. It is also found that the work dissipated in the outer plastic zone as formulated in the concept does not only reflect the non-essential work but is also related to the fracture process.Instead, the approach based on two material parameters should be used to evaluate the impact fracture behavior. Crack initiation and propagation energies should be considered to account for the variation of material's fracture behavior during crack propagation so as to include time rate and size scale effects.     

Microstructure and melt flow behavior of a starch-based polymer

Results on some physical properties and on melt processing of a starch-based polymer under steady-state shearing are presented. A peculiar microstructure involving a strong pseudoplastic behavior at high shear rates as well as yield stress at lower ones is discussed. A model is proposed to explain the characteristic viscoelastic behavior of this material based on hydrophylic and hydrophobic interactions between starch and vinyl-alcohol copolymers.In spite of the highly structured and composite nature of this class of materials, the full body of results reveals that they can be easily processed by means of common manufacturing techniques involving melt pumping and die forming. A comparison with a low density polyethylene (LDPE) grade for film blowing is also shown.     

Rheological behavior of a semi-crystalline polymer during isothermal crystallization

The theological behavior of a molten semi-crystalline polymer, namely, a high density polyethylene (HDPE), was investigated during isothermal crystallization from the melt, using dynamic oscillatory experiments at 1 tad/s in a parallel plates rheometer. The theological results were compared with those obtained from differential scanning calorimetry in the same conditions. During the crystallization, the molten and crystallizing polymer provides a useful model for filled polymers, the crystalline phase being the filler and the liquid phase being the matrix. In most cases, the filler can be considered to be spherical shaped (spherulites). Owing to the amorphous phase linking liquid and crystallites, the adhesion between matrix and filler in this system is perfect. The filler content increases continuously during the crystallization. This model might be used to test laws relating the theological parameters to the volume fraction of filler. Problems related to the rheometry for such systems are discussed and the key parameters insuring reproducibility and accuracy in the measurements are pointed out. The relative sensitivity of the various theological parameters (storage and loss moduli, loss angle) to structural changes of the liquid has been out forward. Some preliminary equations relating the variation of these parameters to the volume fraction of filler, through the use of simple fractal exponents have been derived and discussed in comparison with laws provided by various authors.     

Stress relaxation behavior of PMMA/PS polymer blends

In this work, the stress relaxation behavior of PMMA/PS blends, with or without random copolymer addition, submitted to step shear strain experiments in the linear and nonlinear regime was studied. The effect of blend composition (ranging from 10 to 30 wt.% of dispersed phase), viscosity ratio (ranging from 0.1 to 7.5), and random copolymer addition (for concentrations up to 8 wt.% with respect to the dispersed phase) was evaluated and correlated to the evolution of the morphology of the blends. All blends presented three relaxation stages: a first fast relaxation which was attributed to the relaxation of the pure phases, a second one which was characterized by the presence of a plateau, and a third fast one. The relaxation was shown to be faster for less extended and smaller droplets and to be influenced by coalescence for blends with a dispersed phase concentration larger than 20 wt.%. The relaxation of the blend was strongly influenced by the matrix viscosity. The addition of random copolymer resulted in a slower relaxation of the droplets.     

Strain hardening behavior of polymer blends with fibril morphology

We investigate the rheological behavior of the polymer blends with fibril morphology, with special focus on the effect of fibril morphology on the extensional properties under uniaxial extension. We add a small amount of the dispersed phase to the matrix, and control the blend morphology by changing the viscosity ratio. When the fibril morphology is maintained, the blend shows not only a significant increase of the extensional viscosity but the strain hardening behavior. The extensional viscosity increases depending on the aspect ratio of the fibers, while the strain hardening behavior originates from the restricted stretching of deformable fibers, which has been confirmed theoretically by introducing the concept of rigidity of the fiber. It suggests a way to induce the strain hardening behavior by introducing deformable fibrils into the matrix, that is, by the design of polymer blends with a small amount of dispersed phase such that the fibril structure is maintained.     

Evaluation by means of stress relaxation (after a step strain) experiments of the viscoelastic behavior of polymer melts in uniaxial extension

 As is widely acknowledged, morphology in most materials is far more sensitive to extensional than to shear deformations but, unfortunately, due to the experimental difficulties involved, there are no non-destructive, morphology probing techniques in such flows, i.e., the equivalent of stress relaxation and oscillatory experiments in shear flows. This paper tries to overcome some of those drawbacks by proposing an experimental technique that allows stress relaxation experiments after a step strain in uniaxial extension to be performed. The benefits of this technique are twofold: (a) while the deformation is small enough for the response to be in the linear viscoelastic regime it constitutes a probe of the microstructure of the material and (b) it allows the departure to the non-linear regime to be studied, useful, for example, for the definition of the damping function in uniaxial extensional flow or for the study of the response of materials to fast transient flows with a strong extensional component, such as contraction flows. In this work the proposed technique, which requires a correction to the apparent (theoretical) strain rate in order to allow the calculation of the true Hencky strains attained during the strain step, is tested and validated for two polyisobutylene melts. Received: 9 April 2001 Accepted: 26 July 2001     

Molecular network model for the rheology of gelling polymer solutions

A molecular network model is proposed to describe the rheology of macromolecular solutions undergoing chemical or physical gelation. The model is based on the Bird—Carreau network model [1] with the addition of chemical reaction kinetics to predict the formation of chemical crosslinks among the polymer molecules in solution. The goal is to provide a framework for describing the rheology of gels, that are currently used as fracturing fluids in oil well simulation, formed from polymer solutions that are crosslinked by the addition of transition metal ions. The model has the ability to predict an increase in storage modulus with time, shear thinning viscosity, stress overshoot upon the inception of shear flow, and viscosity changes during the simulation of flow histories that are representative of those encountered in fracturing operations.     

Influence of microstructural parameters on mechanical behavior of polymer gels

The swelling deformation behavior of polymer gels is often described in terms of the Flory–Rehner framework, in which the Flory–Rehner free energy function is based on the simplest affine network model, does not take entanglements into account. However, the real polymer networks have many chain entanglements. In this paper, a new hybrid free energy function composed of the Edwards–Vilgis slip-link model and the Flory–Huggins solution theory is presented for the prediction of the influence of chain entanglements on mechanical behavior of gels. The simulation results of mechanical behavior in free swelling, uniaxial extension, biaxial constraint and simple shear are presented. It is shown that in the nonentangled state, this new hybrid free energy function reduces to the Flory–Rehner free energy function; in the entangled state, the influence of entanglements on the mechanical behavior of gels is significant, the more entangled networks exhibit higher stress.     

Effective viscoelastic behavior of particulate polymer composites at finite concentration

Polymeric materials usually present some viscoelastic behavior. To improve the mechanical behavior of these materials, ceramics materials are often filled into the polymeric materials in form of fiber or particle. A micromechanical model was proposed to estimate the overall viscoelastic behavior for particulate polymer composites, especially for high volume concentration of filled particles. The method is based on Laplace transform technique and an elastic model including two-particle interaction. The effective creep compliance and the stress and strain relation at a constant loading rate are analyzed. The results show that the proposed method predicts a significant stiffer response than those based on Mori-Tanaka's method at high volume concentration of particles.     

Long-term behavior of mineral-filled polymer composites modeled by discrete spectra

A discrete spectra transformation technique is used for the processing and analysis of long-term stress relaxation and creep compliance data of mineral-filled polymer composites. A non-linear regression simultaneously adjusts the parameters of N discrete relaxation or retardation spectra. For small N the solution is insensitive to the choice of regression starting value sets. From the relaxation time spectrum a corresponding discrete retardation spectrum and creep compliance can be calculated using the Laplace transform and vice versa. The analysis of long-term (more than 1200 days) both relaxation and retardation experimental data demonstrates the applicability of the transformation technique. Comparisons of the experimental and calculated spectra are given. The influence of the filler amount is demonstrated.     

Peculiarities of rheological behavior of filled polymer melts in uniaxial stretching

The peculiarities of theological behavior of filled polymer melts in uniaxial extension in a wide range of strain rates (from 2× 10^–5 to 1 × 10^–1 s^–1) have been studied. Linear polyethylene and 1,4-polybutadiene containing up to 21.5 vol.% of carbon black, silica, calcium carbonate or glass fibers were used. It has been found that the transition from uniform to nonunion stretching due to the neck formation is typical of all specimen compositions, when they approach steady-state straining. Depending on the structure and rheological characteristics of the compositions general conditions for this transition have been established. The general regularities for varying the rheological characteristics of filled polymers in the course of their uniform stretching have been recognized. These regularities depend on the molecular characteristics of the polymer matrix and the presence in the compositions of the structural framework of high disperse filler or the network formed by the entangled fibers. Using polyethylene compositions it has been shown that the introduction of small amounts of disperse or fibrous fillers can give rise to acceleration of the relaxation process in filled polymers.     

A network theory of constrained elastic recovery in concentrated polymer solutions

Summary The stress-strain-history relations derived byGreen andTobolsky for a relaxingGaussian molecular network with temporary junctions, generalized to allow for a distribution of junction mean lifetimes, have previously been used to calculate the elastic recovery which occurs in a polymer solution in a state of steady shear flow when all the stress components are instantaneously made zero. The equations predict that the instantaneous part of the recovery involves, in addition to the expected shear recovery, an expansion in directions normal to the previous lines of flow.The present paper contains the corresponding calculations for the case in which this expansion is not allowed to take place, the liquid being constrained so that planesx ₂=const. move rigidly in directions parallel to thex ₁-axis during recovery as well as during the steady shear flow;x ₁,x ₂,x ₃ denote coordinates relative to a rectangularCartesian coordinate system fixed in space. It is shown that the relation between shear stress and the time integral of shear strain is of the linear type much studied in the literature; the calculation of the behaviour ofp ₁₁–p ₂₂, the difference of two normal stress components, is new. It is shown that ifp ₂₁, the shear stress component, be instantaneously made zero (following steady shear flow),p ₁₁–p ₂₂ also decreases instantaneously but in general takes time to reach the value zero; this is true even for the case in which all junctions have the same mean lifetime when the instantaneous shear recovery is not followed by any delayed recovery. The magnitudes of both instantaneous and ultimate shear recovery are calculated; it is found that the latter differs by a factor of two from that calculated on the hypothesis that the stress tensor in steady shear flow is an isotropic function of the strain tensor describing the ultimate shear recovery. The transient behaviour associated with the start of steady shear flow is also considered. Inertial forces are neglected throughout.

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Zusammenfassung Die Beziehungen zwischen der Spannung und dem Zeitintegral der Verformung, die vonGreen undTobolsky für ein sich entspannendes Gaußsches Netzwerk mit zeitweiligen Vernetzungen abgeleitet wurden, werden mit Berücksichtigung der Verteilung der mittleren Lebensdauer der Vernetzungen verallgemeinert. Diese Beziehungen sind schon früher benutzt worden, um die elastische Erholung zu berechnen, die erfolgt, wenn in einer Polymerlösung, die sich in einem Zustand stationärer Scherströmung befindet, alle Kräfte momentan gleich Null gesetzt werden. Die Gleichungen sagen aus, daß der Momentanteil der Erholung nicht nur eine voraussichtliche Schererholung, sondern auch eine Ausbreitung in den Richtungen normal zu den Strömungslinien einschließt.Die vorliegende Abhandlung enthält die diesbezügliche Rechnung für den Fall, wo diese Ausbreitung nicht stattfinden kann, da die Flüssigkeit so umschlossen ist, daß die Ebenen normal zurx ₂-Achse sich unnachgiebigerweise in den Richtungen parallel mit derx ₁-Achse während der Erholung als auch während der stationären Scherströmung bewegen.x ₁,x ₂, undx ₂ bezeichnen die Koordinaten bezüglich eines räumlich festen rechtwinkeligen, cartesischen Achsenkreuzes. Man erhält eine lineare Beziehung zwischen der Schubspannung und dem Zeitintegral der Schiebung, die im Schrifttum schon viel behandelt worden ist. Die Berechnung des Verhaltens der Differenz der zwei Normalspannungskomponentenp ₁₁ undp ₂₂ ist neu. Es wird gezeigt, daß wenn, bei stationärer Scherströmung, die Schubspannungskomponentep ₂₁ momentan Null gesetzt wird, dannp ₁₁ bisp ₂₂ auch momentan abnimmt, aber im allgemeinen ist eine gewisse Zeit erforderlich, den Nullwert zu erreichen. Das trifft sogar dann zu, wenn alle Vernetzungen dieselbe mittlere Lebensdauer haben und wo der momentanen Schuberholung nicht eine verzögerte Erholung folgt. Die Beträge der momentanen und endgültigen Schererholung wurden berechnet. Die so ermittelte endgültige Schererholung beträgt die Hälfte derer, die man, von der Annahme ausgehend, daß der Spannungstensor der stationären Scherströmung eine isotrope Funktion des Verzerrungstensors, der die endgültige Schererholung beschreibt, sei, berechnet. Das vorübergehende Verhalten am Anfang der stationären Scherströmung ist auch in Betracht gezogen. Dagegen sind Trägheitskräfte durchweg vernachlässigt.