Biaxial extension measurements on bimodal elastomeric networks

Bimodal elastomeric networks were prepared by tetrafunctionally endlinking mixtures of short and long hydroxyl-terminated poly (dimethylsiloxane) chains having number-average molecular weights of 500 and 18,000 g/mol^?1, respectively, over a composition range of 0–98 mol % of the short chains. Biaxial extension (compression) measurements were carried out by inflation of circular sheets of these materials at 23°C. The unimodal network (0 mol % short chains) showed the same behavior reported by other workers for noncrystal-lizable networks: as the compression increased, the reduced stress or modulus [f*] went through a rather slight maximum followed by a gradual leveling off to the rupture point. The bimodal networks, however, showed much more pronounced maxima with another, significant increase in [f*] at very high compressions. This final increase is presumably due to non-Gaussian effects from the very limited extensibility of the short chains, and thus parallels the upturns in [f*] frequently reported for bimodal networks at very high elongations.     

Model polyurethane elastomers prepared from noncrystallizable poly(propylene oxide) chains

Elastomers of controlled molecular structure were prepared from hydroxyl-terminated atactic poly(propylene oxide) (PPO) chains having number-average molecular weights M_n in the range 800–4360 g mole^?1. The chains were end-linked into noncrystallizable trifunctional networks using a specially prepared aromatic triisocyanate. The networks thus obtained were studied with regard to their stress–strain isotherms in the unswollen state, in elongation at 25°C, and with regard to their equilibrium swelling in benzene at 61°C. Values of the modulus in the limit at high deformation were in good agreement with corresponding results previously obtained on networks of poly(dimethylsiloxane) (PDMS). This is of considerable importance since use of the widely used “plateau modulus” as a measure of interchain entangling would suggest that the networks of PPO would have a much higher density of such entanglements than would the corresponding networks of PDMS. The close similarity between the moduli of the two types of networks therefore argues against the idea that such entanglements make large contributions to the equilibrium elastomeric properties of a polymer network. These values of the high deformation modulus are also in good agreement with recent molecular theories as applied to the nonaffine deformation of a “phantom” network. The values of the low deformation modulus were considerably smaller than the values predicted for an affine deformation, however, suggesting that the junction points were not firmly embedded in the network structure. This is presumably due to the relatively low degree of chain-junction entangling in the case of relatively short network chains. The swelling equilibrium results were in very good agreement with the new theory of network swelling developed by Flory.     

Elastomeric properties of end-linked networks of high cross-link functionality. Accounting for possible changes in effective functionality with extent of reaction and chain-length distribution

This study reanalyzes some elastomeric properties in elongation reported for poly(dimethylsiloxane) (PDMS) networks of high cross-link functionality which had been prepared by using multifunctional siloxane oligomers to end link vinyl-terminated PDMS chains. The extent of reaction of the vinyl end groupsP _vi spanned the range of 0.40 to 0.95. These networks had elongation moduli that significantly exceeded the values predicted by the Flory-Erman theory, except at very low values ofP _vi. Trends in their stress-strain isotherms, as characterized by the Mooney-Rivlin constants 2C ₂ and the ratio 2C ₂/C₁, also appeared to be different from those predicted by theory. Neglected in such standard analyses, however, was the fact that the segments between cross-links along the junction precursor molecules can themselves act as short network chains, contributing to the modulus and giving a strongly bimodal distribution of both network chain lengths and cross-link functionalities. Of particular interest is the apparent change in functionality with extent of reaction and chain length distribution. The results thus obtained do suggest strong dependence of the observed values of the phantom modulus on the network chain-length distribution, particularly at very small values of the ratio of the length of the short chains to the long ones. Calculations based on recognition of these complications can be used to characterize more realistically the deformation of such networks. The results give much better agreement with experiment. Such behavior could be an important characteristic of elastomeric networks in general.Also, a preliminary attempt was made to bridge theory with experiment based on Kloczkowski, Mark, and Erman's recent theory of fluctuations of junctions in regular bimodal networks. The agreement between theory and experiment thus obtained is rather satisfactory and lends further support to assumptions that take into account the possibly bimodal nature of these high-functionality networks.     

Elastomers having high functionality cross links viewed as bimodal networks. An alternative interpretation in terms of bimodal chain-length distributions

Several groups have now prepared poly(dimethylsiloxane) networks of high cross-link functionality by end-linking vinyl-terminated chains by means of Si? H groups in siloxane oligomers (CH₃)₃SiO[SiHCH₃O]_xSi(CH₃)₃. The elongation moduli of these networks were generally found to be considerably larger than the values predicted from the functionality and number density of the cross links (based on the stoichiometry of the end-linking reaction). Not all the Si? H groups in an oligomer are used in the end-linking reaction, however, and the segments between cross-links can themselves act as short network chains. The connectivity of these short chains to the long ones, in what is essentially a bimodal distribution, has been neglected in analyses to date. They are taken into account in the present analysis, giving much better agreement between experiment and theory. The stress-strain behavior for such very short chains can be characterized by the use of Monte Carlo methods and the Fixman-Alben non-Gaussian distribution. This alternative analysis seems useful in reproducing the experimental observations, but further experimental and theoretical will be required to remove some remaining ambiguities. © 1995 John Wiley & Sons, Inc.     

Rouse's dynamics of networks with pendant chains

A model to describe the dynamics of networks with linear pendant chains has been formulated based on the properties of ensembles of micronetworks, using the Rouse model. This development indicates that the terminal relaxation time of pendant chains with relatively large molecular weight scales with the square of the molecular weight of those chains. On the other hand, when the molecular weight of pendant and elastically active chains are comparable, a nearly exponential growth of the terminal relaxation time with the molecular weight is predicted. The main predictions of the model are compared with experimental results of model poly(dimethyl siloxane) (PDMS) networks, with controlled amounts of linear pendant chains of known molecular weight. The terminal relaxation time of these networks was estimated from the values of the loss modulus G″(ω) measured experimentally. An exponential dependence on the molecular weight of pendant chains was derived for the terminal relaxation time. This behavior is in good agreement with the predictions of our model for micronetworks, provided that the friction coefficient scales linearly with the number of entanglements. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1121–1130, 1999     

Model polyurethane networks prepared from poly(ethylene oxide) and poly(tetramethylene oxide)

Polyurethane elastomers of known degrees of cross-linking were prepared from hydroxylterminated poly(ethylene oxide) (PEO) and poly(tetramethylene oxide) (PTMO) chains having numberaverage molecular weights in the range 880–6820 g mol^?1. The chains were end-linked into “model” trifunctional networks using a specially prepared aromatic triisocyanate. The networks thus obtained were studied with regard to their stress-strain isotherms in both the unswollen and swollen states, in elongation at 25°, and with regard to their equilibrium swelling in benzene at 57.9°. Values of the modulus in the limit at high deformation were in good agreement with corresponding results previously obtained on trifunctional networks of poly(dimethylsiloxane) (PDMS). Since PEO has a much higher value of the plateau modulus in the uncross-linked state, this agreement indicates that inter-chain entanglements do not contribute significantly to the equilibrium modulus of an elastomeric network. These values of the high deformation modulus are also in good agreement with recent molecular theories as applied to the non-affine deformation of a “phantom” network. The swelling equilibrium results were in very good agreement with the new theory of network swelling developed by Flory.     

Moduli of model elastomeric networks prepared from polymer chains having a high plateau modulus,and their interpretation in terms of the constrained-chain model

High-molecular weight polybutadiene chains having approximately 47% cis-1,4 units and 45% trans-1,4 units were crosslinked through their carbon-carbon double bonds using p-bis(dimethylsilyl) benzene as crosslinking agent and chloroplatinic acid as catalyst. This particular polymer was chosen because the high plateau modulus it exhibits in the un-crosslinked state is taken to indicate large numbers of chain entanglements, and stress–strain measurements on such networks have frequently been interpreted with the assumption that the trapping of such entanglements during crosslinking should contribute significantly to their modull. It is shown in the present investigation that such results are equally well interpreted in terms of the new constrained-chain theory of rubbery elasticity. © 1993 John Wiley & Sons, Inc.     

Computational annealing of simulated unimodal and bimodal networks

A conjugate gradient Monte Carlo algorithm was used to simulate the annealing of two and three dimensional end-linked unimodal and bimodal polydimethylsiloxane networks. Equilibrium is satisfied at every crosslink during network energy minimization resulting in distinct differences in network characteristics from classical assumptions. Annealed unimodal networks were found to retain the uniformly dispersed arrangement of crosslinks generated during the crosslinking algorithm. Radial distribution functions of chain vector lengths for various unimodal systems show a shift in the mean chain length from the rms length prior to annealing to shorter lengths upon annealing. Short chains in bimodal networks cluster during the annealing process in agreement with experimental investigations of short chain agglomeration in the literature. This work provides the first predictions of bimodal chain network clustering via simulated network formation and demonstrates the critical role of network annealing in determining the initial configurations of deformable elastomeric networks. This information is extremely useful in the development of accurate constitutive models of bimodal networks.     

Elastic behavior of bimodal poly(dimethylsiloxane) networks

The rubberlike elastic behavior of bimodal poly(dimethylsiloxane) (PDMS) networks was investigated by the Monte Carlo simulation method and enumeration calculation method on the basis of the rotational‐isomeric‐state (RIS) model. These bimodal PDMS networks consist of short chains (chain length from 10 to 20) as well as long chains (chain length equal to 150). For long PDMS chains, through generating many PDMS conformations in the equilibrium state using the Monte Carlo simulation method we can obtain the average Helmholtz free energy and the average energy. For short PDMS chains with chain lengths from 10 to 20, as the total number of conformations is only from 6.56 × 10³ to 3.87 × 10⁸, we adopt the enumeration calculation method. The deformation is partitioned nonaffinely between the long and short chains, and this partitioning can be determined by requiring the free energy of the deformed network to be minimized. Chain dimensions and thermodynamic statistical properties of bimodal PDMS networks at various elongation ratios are discussed. We find that elastic force f increases with elongation ratio λ; the energy contribution f_u to elastic force is significant, and the ratio of ranges from 0.15 to 0.36 at T = 343 K. In the meantime, elastic force f increases with the average energy 〈U〉. The energy change in the process of tensile elongation is taken over, which has been ignored in previous theories. Our calculations may provide some insights into the phenomena of rubberlike elasticity of bimodal networks. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 40: 105–114, 2002     

Fracture of model polyurethane elastomeric networks

Fracture properties of model elastomeric networks of polyurethane have been investigated with a double‐edge notch geometry. The networks were synthesized from monodisperse end‐functionalized polypropylene glycol precursors and a trifunctional isocyanate. All reagents were carefully purified and nearly defect‐free ideal networks were prepared at a stoichiometry very close to the theoretical one. Three networks were prepared: an unentangled network of short chains (M_n = 4 kg mol^?1), an entangled network of longer chains (M_n = 8 kg mol^?1) and a bimodal network with 8 kg mol^?1 and 1 kg mol^?1 chains. The presence of entanglements was found to increase significantly the toughness of the rubber, in particular at room temperature, relative to the bimodal networks and to the short chains network. Fracture experiments were carried out at different strain rates and temperatures and showed for all three networks a marked decrease in fracture toughness with increasing temperature and decreasing strain rate which mirrored reasonably well the rate and temperature dependence of tan δ, the dissipative factor. However the proportionality factor between tan δ, and G_IC was very material dependent and the shift factors obtained for the master curves of the viscoelastic properties could not be used to build fracture energy master curves. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010     

Experimental study of molecular entanglement in polymer networks

Elastomeric molecular networks have been prepared by endlinking polydimethyl siloxane molecules having functional chain ends, both in the presence of an unreactive polymeric diluent and in the undiluted state. Values of tensile (Young) modulus were found to be in good agreement with the simple molecular theory of rubberlike elasticity for networks prepared in a highly diluted state. For concentrated systems the modulus was anomalously high, however. The discrepancy can be attributed to chain entanglements. A second interpenetrating network was introduced into networks formed in the diluted state by replacing the diluent polymer by reactive polymer, which was then gelled in situ. The modulus of these combined networks was much higher than the sum of the moduli of the constituent networks, implying a large contribution from molecular entanglements. © 1994 John Wiley & Sons, Inc.     

Dependence of viscoelastic spectrum width on the structure of model imperfect networks prepared by endlinking

Using the theory of branching processes, structural parameters such as the molecular weights of elastically active network chains (EANCs), including dangling chains, backbone EANCs and dangling chains of networks built up by the alternating polyaddition of a bi- and trifunctional monomer, are characterized. The theory is compared with viscoelastic data on polyurethane networks prepared from poly(oxypropylene)triols and diisocyanate at various initial ratios of functional groups; in the calculation, the distribution of functionalities of the triols used and the possible incompleteness of the reaction is taken into account. The comparison reveals that both the length of the backbone EANC and the length of dangling chains contribute to the total width of the retardation spectrum.     

Thermomechanics of bimodal end-linked polydimethylsiloxane networks with one of the chains much short

Summary A novel equation of state for bimodal networks with extremely short chains as solid intrusions is developed proceeding from the van der Waals theory of molecular networks. Stress-strain and thermoelastic measurements on bimodal end-linked polydimethylsiloxane model networks can satisfactorily be described. The physical reasons behind that representation will be discussed.Dedicated to Professor Dr. K. Ueberreiter on his 70th birthday.     

Monte Carlo simulations of the spatial structure of end-linked bimodal polymer networks: part II

The article presents the results of Monte Carlo simulations of bimodal networks performed with the Bond-Fluctuation-Algorithm. First the sol-fractions of networks with different ratios of short chains were studied and found to be always less than 2%. Concerning clustering behaviour, we saw that while random networks always form a main cluster containing more than 95% of all chains, simulated networks with less than 80% short chains do not form a main cluster. The density profiles during the swelling process show that clustering is reflected in a lower swelling degree and a sharper transition zone between the inner part and the boundary regions of the network. Finally, comparing the density distributions of crosslinkers of unimodal and bimodal networks, we found that all unimodal networks have a more ordered structure in their interior than in the melt. On the other hand, bimodal networks, where the ratio between long and short chains leads to equal masses of the fractions, show a superposition of two separate density distribution peaks, leading to a broader distribution than the Gaussian distribution found for a melt.     

Model networks of end-linked polydimethylsiloxane chains. XI. Use of very short network chains to improve ultimate properties

Elastomeric networks were prepared by tetrafunctionally end-linking mixtures of various proportions of relatively long and very short polydimethylsiloxane (PDMS) chains. The former had a number-average molecular weight of 18,500 and the latter either 660 or 220 g mole^?1. The series of (unfilled) bimodal networks thus prepared were studied in elongation to the rupture point at 25°C, and in swelling equilibrium in benzene at room temperature. Elasticity constants characterizing the Gaussian regions of the stress–strain isotherms, and values of the degree of equilibrium swelling were used to evaluate the most recent molecular theories of rubberlike elasticity. The isotherms also gave values of the elongation at which the modulus begins to increase anomalously because of limited chain extensibility, and values of the elongation and nominal stress at the point of rupture. These results were interpreted in terms of the known configurational characteristics of the constituent PDMS chains. Values of the energy or work required for rupture were used as an overall measure of the “toughness” of the networks. The very short chains were found to give a marked increase in toughness, through an increase in ultimate strength without the usual corresponding decrease in maximum extensibility. A variety of additional experiments will be required in order to elucidate the molecular origins of this important effect.     

On the relevance of the 8‐chain model and the full‐network model for the deformation and failure of networks formed through photopolymerization of multifunctional monomers

Crosslinked networks were synthesized by copolymerization of mono‐functional tert‐butyl acrylate (tBA) with diethyleneglycol dimethacrylate (DEGDMA) or polyethylene glycol dimethacrylates (PEGDMA). By varying the chain length and concentration of the difunctional PEGDMA, we obtained tBA‐PEGDMA copolymer networks while by varying the concentration of difunctional DEGDMA, we obtained tBA‐DEGDMA crosslinked networks. The various materials were submitted to large deformations through uniaxial tension tests. For moderate weight percent of crosslinking agent, up to 20%, the networks showed standard S‐shape stress–strain curves, characteristic of rubber‐like elasticity. Two macromolecular models, the 8‐chain model and the full‐network model, were applied to fit the uniaxial tensile response of the materials. Both models provide good representations of the overall uniaxial stress–strain response of each material. After fitting to stress–strain data, the network models were employed to predict the shear modulus and the elongation at break. Neither the 8‐chain nor the full network model were capable of predicting the failure strain or shear modulus, indicating these models are best used to describe stress–strain relations rather than predict mechanical properties for the network polymers considered here. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1226–1234, 2008     

Size-based separation of synthetic polyelectrolytes in entangled polymer solution capillary electrophoresis: the effect of binary mixtures of separating polymers differing in molecular mass

The influence on the electrophoretic behavior of polystyrenesulfonates of the percentage of high-molecular-mass chains in an entangled poly(ethylene oxide) solution having a bimodal molecular mass distribution has been investigated and compared with the results obtained for similar solutions of unimodal molecular mass distribution. The comparisons between the different separating polymer solutions were made at a constant total mass concentration, so as to keep constant the mesh size and to highlight the sole effect of the network dynamics. The use of binary polymer mixtures of two different molecular masses but of same nature can be a convenient alternative to modulate the dynamics of the network and the viscosity of the separating medium. A 20-30% content of high-molecular-mass chains in an entangled poly(ethylene oxide) solution having a binary molecular mass distribution appears to be a good compromise for a moderate viscosity and a good separation selectivity in comparison with a solution containing only chains of high molecular mass at the same concentration.     

Elastomeric Properties of Polysiloxane Networks. Bimodal Elastomers that are Spatially Inhomogeneous and Others that are Very Broadly Multimodal

End‐linking poly(dimethylsiloxane) was used to prepare bimodal elastomers networks so as to have inhomogeneous nanostructures, and also to prepare others having very broadly multimodal chain‐length distributions. Macroscopic phase separation, probably high crosslink density clusters, was observed to occur in some of the bimodal networks. The mechanical properties in simple extension and in equilibrium swelling were measured. The bimodal elastomers that were not obviously inhomogeneous showed very good mechanical properties, but the macroscopically phase‐separated networks, and the broadly multimodal network were weak. Analysis of the Mooney‐Rivlin profiles suggests that the reinforcing mechanism could have a structural component in addition to that from the limited extensibilities of the short chains. The mechanical properties and the extents of swelling support the cluster conjecture, in accord with previous morphological studies on spatially‐inhomogeneous polysiloxane elastomers.     

The effect of network chain-length distribution,specifically bimodality,on strain-induced crystallization

Polyurethane elastomers were prepared from a series of poly(ethylene oxide) samples by end-linking the chains into “model” trifunctional networks. The molecular weight M_c between crosslinks in such networks is simply the number-average molecular weight M_n of the precursor polymer. End-linking samples separately gave networks with unimodal distributions of network chain lengths, whereas end-linking mixtures of two samples having very different values of M_n gave bimodal distributions with average values of M_c equal to the average value of M_n for the two samples. Stress-strain isotherms in elongation were obtained for these networks, both unswollen and swollen to various extents. Strain-induced crystallization was manifested in elastic properties that changed significantly with changes in temperature. Swelling has more complicated effects, since it causes deformation of the network chains as well as melting of some of the crystallites. Comparisons among stress-strain isotherms at constant M_c indicate that bimodality facilitates strain-induced crystallization.     

Model networks of end-linked polydimethylsiloxane chains

Summary Elastomeric networks of high extensibility were prepared by end-linking mixtures of vinyl-terminated polydimethylsiloxane chains having molecular weights of approximately 600 and 11,000 g mol^–1, with silanes chosen to give junction functionalities ranging from 3 to 8. The resulting bimodal networks were studied in elongation, at 25 °C, to their rupture points, and in swelling equilibrium in benzene at room temperature. The elongation moduli [f ^*] were found to be in satisfactory agreement with previous results obtained by end-linking hydroxyl-terminated polydimethylsiloxane chains. Values of [f ^*] at low and moderate deformations gave relatively low values of the ratio of elasticity constants 2C ₂/2C ₁, which is a measure of the extent to which the elongation changes from approximately affine to nonaffine as the elongation increases. The low values obtained for this ratio are presumably due to diminished interpenetration of configurational domains in the case of very short chains. In spite of its small magnitude, 2C ₂/2C ₁ does show some decrease with increase in , as predicted by the recent molecular theory of rubberlike elasticity developed by Flory. The swelling equilibrium results were also found to be in satisfactory agreement with theory. The elongation moduli increased significantly at high elongations, and the values of the elongation at which the upturn was first discernible were very nearly independent of , This is consistent with the interpretation of this anomalous behaviour in terms of limited chain extensibility. The maximum extensibility generally decreased somewhat with increase in and this caused a decrease in both the ultimate strength and the toughness of the elastomer, as measured by the energy required for rupture.