On Material Equations in Second Gradient Electroelasticity

The concern of this work is the derivation of material conservation and balance laws for second gradient electroelasticity. The conservation laws of material momentum, material angular momentum and scalar moment of momentum on the material manifold are derived using Noether's theorem and the exact conditions under which they hold are rigorously studied. The corresponding balance laws are also presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Statistical mechanics of temporary polymer networks

The statistical theory of temporary polymer networks developed in parts I and II of this work is closed by a discussion on dynamical effects, including comparison with experiments. After a short review of the theory developed in parts I and II we discuss the situation in which a prescribed velocity gradient is imposed on the physical system formed by the temporary network and the solvent. The dynamical equation for the 2nd moments (derived from a generalized bead-spring model) contains the configuration-dependent transition probability, which depends on the second moments in a complex way. An approximate solution of these equations is obtained from a computer program. It shows the experimentally observed behavior, in particular the stress overshoot maximum after a sudden start of the flow.The derivation of the above equations contains simplifying assumptions which, however, leave the essential physics intact. The most important assumptions concern the relaxation time approach and the decay and formation processes. These were supposed to be dilute enough to break up the many-junction processes into one-junction processes. Both assumptions are analogous to frequently used assumptions in Boltzmann's kinetic theory.     

Single-chain dynamics in frozen polymer networks

In this article, we consider the dynamics of short chains in frozen polymer networks, at different degrees of swelling. The crosslinked networks are bimodal, at monomer densities ranging from 0.51 in the dry state to 0.098 at the maximum degree of swelling. The 3D bond-fluctuation method is used to simulate the polymers, and we use a Monte Carlo method for monomer moves which has been presented in an earlier publication (Nedelcu and Sommer, J Chem Phys 130:204902, 2009). Crossover scaling functions for the terminal relaxation times τ _d and diffusion coefficients D of the short chains are observed to approach de Gennes’ reptation predictions, similar to what has already been observed for long chains diffusing in polymer melts. However, the mean square displacement of a middle monomer of a chain in high-density networks shows only two regimes, first subdiffusive with a scaling exponent close to 1/4, followed by a smooth transition to free diffusion at longer times.     

Theory of polymer networks with closed cycles

Summary The probabilistic theory of crosslinked polymer systems developed by the author is generalized to allow for application to systems with cycles. Based on the two parameters — the crosslinking densityl (or the fraction of reacted functionalities) and the fraction of junctions closing cycles — topological characteristics of such a system are found. It is shown that the presence of cycles increases the critical value of the crosslinking density, whereas the critical value of the reduced crosslinking density ought to remain equal to 2. Then from the introduced principle of macroscopical uniformity, is found to be a function of crosslinking densityl.

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Zusammenfassung Die vom Verfasser entwickelte probabilistische Theorie der vernetzten Polymersysteme wird so verallgemeinert, daß Systeme mit Zyklen eingeschlossen sind. Man findet topologische Kenngrößen solcher Systeme, die von den beiden Parametern Vernetzungsdichtel (bzw. Häufigkeit der Netzstellen) und Häufigkeit der Zyklen bildenden Bindungen abhängen. Es wird gezeigt, daß durch das Vorhandensein von Zyklen der kritische Wert der Vernetzungsdichte erhöht wird, wohingegen die reduzierte Vernetzungsdichte den kritischen Wert 2 behalten sollte. Weiter wird mittels des eingeführten Prinzips der makroskopischen Einheitlichkeit abgeleitet, daß eine Funktion der Vernetzungsdichte sein muß.

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Necessary conditions for shear thickening in associating polymer networks

We study shear-thickening phenomena observed in thermoreversible gels of telechelic associating polymers on the basis of the transient network theory. We introduce nonlinear models for the elongational properties of the middle chain; the tension along middle chains contains a nonlinear term with an amplitude A that diverges if the chain is fully stretched. Models for the dissociation rate of end chains from network junctions are also introduced; the dissociation rate is coupled to the middle-chain tension by a coupling constant g. The balance between the nonlinearity strength and coupling intensity determines the necessary condition for thickening. A strong nonlinearity in the tension caused by a large A leads to thickening if the dissociation rate of the end chains is weakly coupled to the middle-chain tension. In contrast, if it is strongly coupled, the end chains easily dissociate upon a small nonlinear elongation of the middle chains, thus leading to thinning. We present theoretical phase diagrams showing the boundary of the thickening/thinning transition in the A–g plane, and we show the existence of a critical point for thickening in these diagrams.     

Constitutive equations for non-affine polymer networks with slippage of chains

A model is derived for isothermal three-dimensional deformation of polymers with finite strains. A polymer fluid is treated as a permanent network of chains bridged by junctions (entanglements). Macro-deformation of the medium induces two motions at the micro-level: (i) sliding of junctions with respect to their reference positions that reflects non-affine deformation of the network, and (ii) slippage of chains with respect to entanglements that is associated with unfolding of back-loops. Constitutive equations are developed by using the laws of thermodynamics. Three important features characterize the model: (i) the symmetry of relations between the elongation of strands and an appropriate configurational tensor, (ii) the strong nonlinearity of the governing equations, and (iii) the account for the volumetric deformation of the network induced by stretching of chains. The governing equations are applied to the numerical analysis of extensional and shear flows. It is demonstrated that the model adequately describes the time-dependent response of polymer melts observed in conventional rheological tests.     

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.     

Stress relaxation due to chain fractures in transient polymer networks

Aspects of a network model for concentrated dispersions are applied to polymer networks. It is shown how network deformation caused by network fracture affects the macroscopic stress.     

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.     

Statistical mechanics of temporary polymer networks I. The equilibrium theory

In part I of this work (the present article) the equilibrium state of temporary polymer networks is treated in the framework of thermodynamics and statistical mechanics. The network is described as an open system. Thereby we use a modified spring-bead model in which the beads represent junctions that decay and reform thus adding a viscous component to the assumed elastic behaviour of the permanent network. The relevant statistical equation — analogous to Liouville's equation — is solved. The grand-canonical probability density function and two of three equations of state are derived. Explicit formulae are given for several relevant probabilities. For instance the probabilityw (z)dz that a network chain connecting two junctions has a contour length betweenz andz +dz is given by the Wien type formulaw(z) =A z ³ exp {–B z} whereA andB do not depend onz.     

Predicting thermal shape memory of crosslinked polymer networks from linear viscoelasticity

The viscoelastic behavior of an amorphous shape memory polymer network and its dependence on time and temperature were measured by dynamic mechanical analysis. The resulting thermo-mechanical behavior was modeled and implemented in a commercial finite element code. The ability of the resulting thermomechanical model to simulate and, eventually, predict the shape storage and shape recovery of the material was evaluated against experimental shape memory thermomechanical torsion data in a large deformation regimen. The simulations showed excellent agreement with experimental shape memory thermomechanical cycle data. This demonstrates the dependence of the shape recovery on time and temperature. The results suggest that accurate predictions of the shape recovery of any amorphous polymer networks under any thermomechanical conditions combination solely depends on considering the material viscoelasticity and its time–temperature dependence.     

Effects of fluctuations of cross-linking points on viscoelastic properties of associating polymer networks

We study the effects of fluctuations of cross-linking points, or junctions, on the dynamic mechanical and viscoelastic properties of polymer networks formed by multisticker associating polymers on the basis of a single-chain approach. Fluctuations of junctions are implemented by introducing virtual springs. We consider two possible cases for the treatment of virtual springs: the direct contribution from virtual springs is either neglected or included in the stress. We show that, if neglected, the fluctuation of junctions decreases (or softens) the dynamic modulus over a wide range of frequencies. This result agrees qualitatively with the result of several multichain models that predicts the decrease of the static or plateau modulus. We also show that the fluctuation accelerates the associative Rouse mode at low frequencies originating from the association/dissociation process of stickers. These results are apparently reasonable, but it is expected that there are some errors arising from thermodynamical inconsistency due to the neglect of virtual-spring contribution from the total stress against the virtual work principle and the second law of thermodynamics. On the other hand, if the direct contribution from the virtual springs is included in the stress, thermodynamics is satisfied but the plateau modulus does not change, contrary to the multichain prediction. The softening occurs only at the low-frequency regime. Thus, each method has both merits and demerits, and hence the treatment of junction fluctuations in the framework of single-chain approaches requires care and further investigation.     

Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts

Summary An entangled network such as a polymer melt or a concentrated solution is here described by a set of two simultaneous equations. One of them is a balance of entanglements, the other gives the stress in the classical form of aMaxwell equation.The balance of entanglements contains both an entanglement generation term which depends on the distance from equilibrium and an entanglement destruction term which depends on the stress level. The parameters appearing in theMaxwell and the balance equations are made to depend in a specified way on the existing number of entanglements.The model is here tested by comparison with existing data of steady-state elongational and shear viscosity of polymer melts.With 6 figures