Applications of a two-network model for crosslinks and trapped entanglements

The dependence of stress and birefringence on strain in uniaxial extension of crosslinked rubbers can accurately be described by a model in which the properties of a network of crosslinks and a network of trapped entanglements are additive. The crosslink network is neo-Hookean and the entanglement network can conveniently be described by the Mooney-Rivlin equation. When the crosslinks are introduced in a state of strain near the glass transition temperature, the two networks have different reference undeformed states; they can be distinguished by appropriate measurements in the state of ease where their associated stresses are equal and opposite and in the state of deformation where the cross-links were introduced and make no contribution to the stress. When partial relaxation is permitted before crosslinking, trapping probabilities can be calculated for both relaxed and unrelaxed entanglements and compared with the Langley theory. The results are consistent with the terminal mechanism of relaxation in the tube theory of Doi and Edwards.     

Equilibrium swelling and solvation studies on crosslinked polyacrylamides

Polyacrylamide (PA) crosslinked with four different crosslinking agents, triethyleneglycol dimethacrylate (TEGDMA), N,N′‐methylene bisacrylamide (NNMBA), hexanediol dimethacrylate (HDDMA) and divinylbenzene (DVB), with mole percents ranging from 5 to 20, was prepared by solution polymerization and subjected to swelling and solvation studies. Solubility parameters and cohesive energy density were determined from swelling studies. Molecular weight between crosslinks for these systems were determined by Flory–Rehner analysis. There is a discontinuous volume change for 10% NNMBA and HDDMA crosslinked PA, 15% TEGDMA crosslinked PA and 10 and 15% DVB crosslinked PA in solvent mixtures of acetic acid and water due to phase change occurring at this stage. The hydrogels exhibit inhomogeneous crosslink distribution due to multiple crosslinking, cyclization and network irregularity owing from arising from entanglements. As the percentage of crosslinking increases, crosslinks become more homogeneous due to a decrease in entanglements. Copyright © 2004 Society of Chemical Industry     

Solute diffusion in swollen membranes. IV. Theories for moderately swollen networks

A theoretical model was developed to describe solute transport through moderately swollen networks. The model is an extension of the previous analysis of Peppas and Reinhart (1983), and it describes the normalized effective diffusion coefficient of the solute through the network as a function of the equilibrium degree of swelling, Q, the hydrodynamic radius of the solute, r_s, the number average molecular weight between crosslinks, M¯_c, and a function f(ξ) of the mesh size ξ, which takes into consideration barriers due to crosslinks, entanglements, etc. For the development of this model, the Cohen–Turnbull (1958) free volume theory was modified to incorporate topological and mobility characteristics from de Gennes' analysis (1979).     

Preparation and moduli of model polymer networks

Polystyrene containing a small number of randomly spaced p-substituted secondary amine groups has been prepared. This polymer has been crosslinked in solution at concentrations in the range 5–20%, by hexamethylene diisocyanate, to form gels. The moduli of these gels were measured and the results compared to theoretical predictions from rubber elasticity theory. The dependence of the moduli on the concentration at which crosslinking takes place has been explained in terms of chemical crosslinks, closed loops and entanglements. The results support a value for the ideal modulus of nkT, where n is the number of crosslinks per cm³.     

Parameters governing strain induced crystallization in filled natural rubber

A series of in situ synchrotron X-ray diffraction experiments are performed during the stretching of weakly and highly vulcanized carbon black (CB), silica and grafted silica filled natural rubber sample (NR). Conversely to literature, Mullins effect observed after one stretching cycle modifies the strain induced crystallization (SIC) behaviour of the sample. The onset of crystallization is ruled by the strain amplification induced by the filler presence. Moreover, fillers (CB and silica) behave as additional crosslinks into NR network, through filler-rubber interactions that either accelerate or slow down the crystallization rate depending on NR matrix chemical crosslink density. This is consistent with the assumption that effective network density, which is due to chemical crosslinks, entanglements, and filler-rubber interactions, controls the crystallization rate.     

Deformation behavior of poly(dimethyl siloxane) networks. I. Applicability of various theories to modulus prediction

The shear moduli of end-linked poly(dimethyl siloxane) networks were measured as a function of mol wt between chemical crosslinks (M_c) and the dilution-at-cure (C_x). The results were interpreted in terms of theories that take into account contributions from trapped entanglements. It was found that the network deformation follows the predictions of the theory of “phantom” networks, with an added contribution from trapped entanglements. At high mol wts of the precursor polymer, network imperfections play a role, as seen in the frequency dependence of the shear modulus. This results in deviation from predicted behavior. © 1993 John Wiley & Sons, Inc.     

The interpretation of orientation — strain relationships in rubbers and thermoplastics

The orientation—strain behaviour typical of non-crystalline polymers is examined with reference to new techniques for determining molecular orientation parameters in a variety of crosslinked and thermoplastic materials, as well as the long-established methods based on the optical anisotropy of natural rubber. It is shown that the affinely-deforming ‘random chain’ assumed by conventional theory is not successful in predicting the relationship between chain orientation and strain unless the network is assumed to change significantly with strain. The implications of this observation are discussed in terms of the behaviour of crosslinks and entanglements: the conventional view of anisotropy needs to be supplemented by an approach in which orientation and strain are seen as distinct aspects of a polymer's response to the stress imposed upon it.     

Crosslinking of ultra-high molecular weight polyethylene in the melt by means of 2,5-dimethyl-2,5-bis(tert-butyldioxy)-3-hexyne: 2. Crystallization behaviour and mechanical properties

Ultra-high molecular weight polyethylene, was crosslinked in the melt at 180°C by means of 2,5-dimethyl-2,5-bis(tert-butyldioxy)-3-hexyne. It is shown that crosslinks and entanglements, whether trapped or not, contribute to the same degree to the decrease in crystallinity of the polyethylene upon crosslinking. Crosslinking introduced a very stable type of radical into the polyethylene. It was concluded that an almost completely crosslinked material with high crystallinity and therefore good mechanical properties, could be obtained by using as little as 0.2–0.3 wt% of peroxide.     

Theory of the variation of relaxational properties of polymers during network formation

Recent work on the change of mechanical and dielectric relaxational properties accompanying the formation of a polymer network from reactive oligomers is reviewed. The theoretical considerations cover the essential features of the behaviour observed in different polymer systems. The theory offers new possibilities for obtaining information about the chemical kinetics, physical structure, and structure formation during crosslinking, by means of relaxation studies. Analysis of the theory of the kinetics of structure formation suggests the following picture of the process of crosslinking. Shortlv before the gel point, there appears a state of ‘kinetic entanglement’, which is completed at gel; chemical crosslinks then start to form. Beyond the gel point the contribution of entanglements to the mechanical behaviour is insignificant. However, fluctuation restrictions on crosslinks, and the topological restrictions on network chains play an important role in the reduction of the front-factor and in the increase of the Mooney-Rivlin C₂-term during network formation.     

Preparation of high-strength poly(vinyl alcohol) fibers by crosslinking wet spinning

High-strength poly(vinyl alcohol) (PVA) fiber was obtained by the crosslinking wet-spinning technique, which is an improved technique of the conventional non-crosslinked type wet-spinning of PVA. High tensile strength as well as high Young's modulus was achieved by introduction of the borate ion-aided crosslinks during the coagulation process. The drawability of the as-spun fiber greatly depends on the fiber thickness. The thinner the fiber, the higher the drawability. Since thinner fiber is subject to a very high shear rate on extrusion, the crosslinks introduced are believed to maintain topological memory of the oriented chains, which have a low density of entanglements. This allows drawing the fiber to a higher draw ratio. The strength and Young's modulus of the resultant highly drawn PVA fiber were achieved to be 22 g/d (2.3 GPa) and 430 g/d (50 GPa), respectively. The mechanism of the spinning was discussed and the spinning condition was carefully examined in order to optimize the final mechanical properties of the PVA fibers.     

Influence of side‐chain structures on the viscoelasticity and elongation viscosity of polyethylene melts

Metallocene‐catalyzed, low‐density and linear low‐density polyethylenes with similar melt indexes were used to investigate how side‐chain structures influence the elongation viscosity and viscoelastic properties. The viscoelastic properties were determined with a rotation rheometer, while the elongation viscosities were acquired by using isothermal fiber spinning. The Phan‐Thien‐Tanner (PTT) model was also used to understand how the side‐chain structure affects the elongation behavior. Experimental results demonstrate that the log G′ vs. log G″ plot can qualitatively describe the effects of the side chain branch on the rheological properties of polyethylene melts. According to the results determined by the PTT model, low‐density polyethylene (LDPE) has low elongation viscosities at high strain rates. This low elongation viscosity can be attributed to the fact that LDPE has high shear thinning behavior. The long‐chain branching tends to increase entanglements, thereby enhancing the storage modulus, elongation viscosity and shear‐thinning behaviors. Uniform side‐chain distribution lowers the entanglements, which results in a low storage modulus, elongation viscosity and shear‐thinning behavior.     

Preparation and properties of physically crosslinked sodium carboxymethylcellulose/poly(vinyl alcohol) complex hydrogels

A series of physically crosslinked complex hydrogels of poly(vinyl alcohol) (PVA) and sodium carboxymethylcellulose (CMC) were prepared via physical mixing and a freeze/thaw technique. The morphology of the CMC/PVA complex gels was analyzed with differential scanning calorimetry and wide‐angle X‐ray diffraction. It was found that the crystallinity and melting temperature of the complex gels decreased, whereas the glass‐transition temperature increased, with an increase in the content of CMC. The reswelling of the complex gels was pH‐responsive and relied on the content of CMC and the freeze/thaw cycles. A network structure model of the complex gel was presented. PVA crystalline regions served as physical crosslinks; the interaction between CMC and PVA resulted in intramolecular entanglements. It was also found that the model drug hemoglobin was released completely from the complex hydrogels in 4 h, and its release rate increased with an increase in the content of CMC. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2008     

Role of chain entanglements on fiber formation during electrospinning of polymer solutions: good solvent, non-specific polymer-polymer interaction limit

Chain entanglements are one of many parameters that can significantly influence fiber formation during polymer electrospinning. While the importance of chain entanglements has been acknowledged, there is no clear understanding of how many entanglements are required to affect/stabilize fiber formation. In this paper, polymer solution rheology arguments have been extrapolated to formulate a semi-empirical analysis to explain the transition from electrospraying to electrospinning in the good solvent, non-specific polymer-polymer interaction limit. Utilizing entanglement and weight average molecular weights (M_e, M_w), the requisite polymer concentration for fiber formation may be determined a priori, eliminating the laborious trial-and-error methodology typically employed to produce electrospun fibers. Incipient, incomplete fiber formation is correctly predicted for a variety of polymer/solvent systems at one entanglement per chain. Complete, stable fiber formation occurs at ≥2.5 entanglements per chain.     

Interpenetrating polymer networks with controllable intermolecular hydrogen bonding: Swelling behavior

Summary The swelling behavior in different solvents of IPNs based on poly(butyl acrylate) and modified polystyrene containing strong proton-donor hydroxyl groups is studied. The experimental data of the volume fractions of the networks in the swollen state V_expt at equilibrium are compared with those expected from the Thiele-Cohen equation (V_TC), which assumes that the two networks are independent and no additional crosslinks form during the formation of the IPNs. For the IPNs without inter-component hydrogen bonding, V_expt in toluene are in good agreement with V_TC. As the hydroxyl content increases, the deviation of V_expt from V_TC increases indicating more crosslinks form in the process of IPN formation. This deviation in the solvents of proton-acceptor type becomes smaller but still apparent. This result implies that hydrogen bonding between the components is destroyed by the solvents but the physical entanglements formed during the formation of the IPNs remain.     

Model for a scission-crosslinking process with both H and Y crosslinks

We present a mathematical model that may predict average molecular weights and weight fraction of solubles for polymer chains subjected to irradiation, where both scission and crosslinking are present. Two types of crosslinks are allowed: four-armed or H crosslinks, and three-armed or Y crosslinks. This is a departure from the more traditional models for irradiation, where only H crosslinks are allowed. The model is valid both in the pregel and postgel regions, and may be applied to chains with any known distribution of molecular weights. We compare predictions from the model with experimental data on polydimethylsiloxane treated with electron beams at different doses, and show that the introduction of Y crosslinks leads to a marked improvement in the quality of the predictions as compared with a more traditional model where only H type crosslinks are allowed.     

The effect of temperature and deformation rate on the hot-drawing behavior of porous high-molecular-weight polyethylene fibers

The nature of the deformation process involved in hot drawing of porous high-molecular-weight polyethylene was examined by apparent elongational viscosity measurements at drawing temperatures between 100°C and 150°C and deformation rates in the range of 10^?6–10^?3 m/s. The temperature dependence of the apparent elongational viscosity revealed three distinguishable intervals with different activation energies. In the range of 100–133°C, the activation energy amounted to 50 kJ/mol, indicating that hot-drawing in this region proceeds by a sliding motion of separate fibrillar units. The interval between 133°C and 143°C was characterized by an activation energy of about 150 kJ/mol. Moreover, the porous character of the polyethylene fibers was found to decrease in the drawing process above 133°C. These observations were ascribed to an aggregation of the elementary fibrils upon hot-drawing due to partial melting at the surface of the fibrils. At temperatures above 143°C the activation energy was strongly affected by the initial morphology and the draw ratio of the fibers and amounted to values in the range of 200–600 kJ/mol. Molecular orientation in this region is accomplished by a slippage of individual chains, with entanglements acting as semipermanent crosslinks. Decreasing of the rate of elongation in the drawing process resulted in premature fiber breakage, indicating that the crosslinking action of the entanglements is limited by the time scale of the process.     

Effect of network structure on the stress-strain behaviour of endlinked PDMS elastomers

Endlinked poly(dimethylsiloxane) (PDMS) networks synthesized from telechelic precursor chains of different molar mass were prepared with varying volume fractions of non-reactive chains acting as solvent. Uniaxial extension and compression measurements were performed on these networks to investigate their stress-strain behaviour. The effect of the network structure and of solvent on the stress-strain behaviour is examined by controlling the extent of crosslinks and entanglements during the network synthesis. The master curve of the Rubinstein and Panyukov non-affine slip-tube (NAST) model provide an adequate fit to most of the extension and compression data. Furthermore, the crosslink and entanglement parameters of the NAST model (G_c and G_e) are found to be in general in reasonable agreement with the 2C₁ and 2C₂ parameters of the Mooney-Rivlin continuum model applied to the extension data. For high molar mass precursor chains, the entanglement contribution to the modulus surpasses the crosslink contribution.     

Elastic properties of swollen real polymer networks

Summary In this paper a new set of stress-strain relations (uniaxial, equi-biaxial, unequibiaxial extensions and pure shear) for swollen networks were presented. Which are derived from the molecular theory of rubber elasticity with constraints of junctions and trapped entanglements and with crosslinks, trapped entanglements and carbon black-polymer interactions. They successed in relating the elastic equation of state to the volume fraction of polymer in swollen networks by three molecular parameters C₁, C₂ and C₃. The relation of stress-strain for uniaxial extension was verified by experiments. It is shown that this relation can successfully predict the dependence of V2 on the C₁, C₂ and C₃, and the contribution of modulus for swollen networks from the trapped entanglement which it shows that role of entanglements can only approach to a limited value, never to zero.     

Macromolecular entanglement. IV. Modulus of swelling acrylic fibers as a means of studying macromolecular entanglements

In a swelling state at a temperature of incipient dissolution, an acrylic fiber is highly elastic just like a rubber. A system of entangled macromolecular chains is considered analogous to a cross-linked rubber network and from the theory of rubber elasticity, the measured extension modulus of an acrylic fiber at the terminal swelling state may be used to calculate the number of entanglements. By means of a capillary rheometer, a series of model acrylic fibers with varied degrees of entanglements were prepared. The entanglements present in these model fibers were then measured by the other two thermal methods (SDSC and SSS/SS methods) already established in our laboratory. The die swell ratios were also measured microscopically at the exit of the capillary rheometer. The experimental results of entanglements measured by the various methods were well correlated, at least qualitatively. © 1994 John Wiley & Sons, Inc.     

Molecular modeling of cellulose in amorphous state part II: effects of rigid and flexible crosslinks on cellulose

It is essential to understand the molecular level response of crosslinked cellulose chain segments upon deformation, in order to develop new agents which convey high durable-press (DP) rating to cellulose fibers with minimal strength loss. In this work models of amorphous cellulose crosslinked with both rigid and flexible crosslinks were constructed computationally for this purpose. Rigid crosslinks bound cellulose molecular segments together and blocked the chain slippage, providing cellulose models with a higher initial modulus and better elastic response. However, the loss of the chain slippage led to stress being distributed unevenly among cellulose chains. Chains in some regions were subjected higher stress and these regions were opened up much more than the rest of the cellulose, which presumably caused models to fail. When conformationally flexible crosslinks were used, breaking strain of cellulose was not significantly reduced but deformation recovery was not improved either, in comparison with the models of untreated cellulose. Conformational transitions were observed in the flexible crosslinks during extension. These results help to explain how and why rigid crosslinks work to provide wrinkle resistant properties and why they also lower tensile strength, and that just using a conformationally flexible crosslinking will not provide any recovery.