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.     

Swelling behavior and mechanical properties of endlinked poly (dimethylsiloxane) networks and randomly crosslinked polyisoprene networks

Measurements of the equilibrium degree of swelling and of the equilibrium modulus were performed on poly(dimethylsiloxane) networks (PDMS) and on polyisoprene vulcanizates. The results support the concept that topological interactions between network chains, e.g. entanglements or the like, have a large influence on the rubber elastic behavior, at least within a certain range of network densities.PDMS networks having network chains of different lengths and varying functionlities of the crosslinks were prepared in bulk by endlinking fractionated ,-divinyl PDMS via multifunctional hydrogen-siloxanes (f=3 to 22). Natural rubber (NR) and synthetic liquid polyisoprene (IR) were cured in bulk with various amounts of dicumyl peroxide to give randomly crosslinked samples.The experimentally determined moduli and degrees of swelling were compared with theoretical predictions based on the phantom network theory and affine network theory, taking into account only chemical crosslinks. The observed discrepancies can be traced back to a contribution of topological interactions (trapped entanglements) to the total effective network density. The modulus and swelling data are consistent, thus ruling out non-equilibrium effects.     

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 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.     

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.     

Interpretation of the moduli of trifunctional model elastomers of polydimethylsiloxane

A survey is made of published experimental values of the elongation moduli of trifunctional model networks of polydimethylsiloxane [? Si(CH₃)₂O? ]. The results thus obtained are interpreted in light of recent molecular theories of rubberlike elasticity, particularly with regard to predicted values of the modulus, the degree of nonaffineness of the deformation, the relevance of the network sol fraction, and the efficiency and possible importance of the trapping of interchain entanglements. The most important result of this study is the conclusion that it is not necessary to invoke equilibrium contributions from interchain entanglements in order to interpret the experimental results presently available in the literature.     

Unimodal and bimodal networks of poly(dimethyl-siloxane) in pure shear

Model networks of poly(dimethylsiloxane) (PDMS) were prepared by tetrafunctionally endlinking hydroxyl-terminated chains of various molecular weights. Some networks were prepared from mixtures of chains so as to yield a bimodal distribution of network chain lengths and, in some cases, these networks were prepared in solution. The stress–strain behavior of these unimodal and bimodal networks was studied in pure shear, which was imposed by stretching a sheet of the network having a large ratio of width to length in the direction perpendicular to the width. The pure-shear moduli of both types of networks generally were found to depend markedly on strain. Stress–strain isotherms for unimodal networks prepared from chains of one or the other of two molecular weights were well interpreted using the constrained-junction model of Flory and Erman. The bimodal networks showed large increases in the pure-shear modulus at high strains which were similar to those reported for uniaxial extension and compression. Endlinking in solution decreases the modulus in general and its upturn in particular, presumably because of diminished chain-junction entangling.     

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.     

The use of model networks with reptating chains to study extraction efficiencies

It is possible to prepare “model” elastomeric networks having known values of the molecular weight M_c between crosslinks by endlinking functionally terminated polymer chains having number-average molecular weights M_n equal to the desired values of M_c. If chains having chemically inert groups at both ends are intentionally included during the preparation of such a system, they will remain unattached, merely reptating through the subsequently formed network structure. This technique was used to prepare a series of tetrafunctional polydimethylsiloxane (PDMS) networks having essentially the same degree of crosslinking (10^?3M_c = 11.3 g mol^?1) and constant amount of diluent in the form of unattached PDMS chains having molecular weights of 10^?3M_d = 26.4, 18.6, 15.8, 9.8, 6.7, 1.2, and 0.70 g mol^?1. Because of the very high mobility of PDMS, it was also possible to introduce essentially the same amount of the same diluents into already formed PDMS networks having the same M_c. Extractions carried out using tetrahydrofuran at room temperature showed that the diluent (“sol fraction”) introduced by swelling the network is more easily removed than that present during the endlinking, possibly because of less convoluted arrangements within the network structure. Chains with the largest values of M_d which were present during the endlinking were found to be very difficult to remove entirely. It is therefore extremely important to carry out exhaustive extractions to obtain reliable values of network sol fractions, particularly when such data are to be used to estimate extents of reaction in the preparation of end-linked elastomers.     

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.     

Elasticity and structure of crosslinked polymers

Summary The subject of this work is to determine the dynamic and static elastic properties of polymer networks and to compare the experimental results with theoretical predictions.It is found that poly(dimethylsiloxane) (PDMS) networks on the one hand and polyisoprene (IR) and natural rubber (NR) networks on the other hand show different behaviour in frequency dependence of the storage modulusG is observed in the range 10^–3 – 1 Hz, contrary to IR or NR. The reason for this is the absence of entanglements in PDMS.Comparing the measured static moduli with those calculated by rubber elasticity theory, we found that the front factorA· <r ²>/<r ²>₀ is near to 0.5 in case of PDMS, but near to I for IR/NR networks. Both parts of the front factor may possibly cause this difference. As the PDMS chains possess a relatively high mobility we can assume that fluctuations of the junction points are less restricted than in IR/NR. This causes a structure factor A smaller than 1 in agreement with Flory's recent theory.According to James and Guth, the network chains tend to contract during the crosslinking process. This will be more likely in the networks of entanglement-free, highly mobile PDMS chains than in IR or NR rubbers. Hence the memory term <r ²>/<r ²>₀ is smaller for PDMS than for IR/NR.Both alternative explanations are based on the different mobility of the chains considered. It may be assumed that the front factor is influenced by both effects and not only by the fluctuations of crosslinks.Dedicated to Professor Dr. K. Ueberreiter on his 70th birthday     

A New Molecular Theory of Non‐Linear Viscoelasticity for Vitrifiable (or Crystallizable) Thermoplastic Polyurethane Elastomers

The molecular theory of non‐linear viscoelasticity for vitrifiable thermoplastic polyurethane elastomers (VTPUE) is a refinement and extension of viscoelastic theory of thermoplastic elastomers and polyurethanes to glassy transition, a structural model and a mechanism of vitrification for glassy polymers were proposed. Five kinds of constituent chains with Nagai chain constraint consisting of soft‐domains, hard‐domains, and entanglements are used as the elementary structural and statistical ensemble units for the correlation of molecular and phase‐domain structures to the static and dynamic mechanical behaviors. So the influences of non‐Gaussian in character, the phase separation of domain, the network topology of structure, the affined deformation of constituent chains, and the thermal history are all taken into account in the constituent chains of the theory. Free energies of deformation for the VTPUE segment copolymer were calculated by the statistical mechanics with the probability distribution functions of the sizes for the five kinds of constituent chains. Then the static constitutive equations and modulus of four types of deformation and the dynamic shear viscosity, modulus and loss tangent of VTPUE are derived from the proposed theory. The theory is successful in relating the molecular chain parameters C₁₀₀, C₀₂₀, and C₂₀₀ to the constitutive equations and modulus under large deformations and the micro‐domain structure to the complex shear viscosity and modulus and the loss tangent. The dynamic shear modulus and loss tangent of VTPUE are related to the domain structures through the fraction of hard segments (W_h), the molecular weight of soft segment (M_ns), and the growth dimensional parameters of hard and soft domains (β). Two series of linear VTPUE copolymers (ES and ET) with different fractions(W_h) of hard segments and molecular weight (M_ns) of soft segments were prepared. Their static and dynamic mechanical properties were studied by uni‐axial extension and dynamic analysis tests. Then the constitutive equation at uni‐axial extension and the expressions of shear modulus and loss tangent are verified by these experimental data, and excellent agreement between the theory and experiments is achieved. It is shown, that the proposed theory can predict the viscoelastic behavior of vitrifiable thermoplastic polyurethanes.     

Swelling equilibrium and sorption kinetics of urethane-modified bismaleimides elastomer

The swelling equilibrium and diffusion kinetics in various solvents of the maleimide-terminated polyurethanes (UBMIs) and of the triol and tetraol-crosslinked polyurethanes (PU) were studied. The polymer volume fraction of the UBMIs at swelling equilibrium is much higher than that of the tetraol-crosslinked PU networks for the same type of polyol used in the PU. It was explained by the high functionality of the UBMIs produced in the network structure. Furthermore, the molecular weight between crosslinks (M_c) has been calculated from the swelling model and the results exhibit good agreement with the proposed network structure. The early time sorption kinetic data were obtained to investigate the diffusion mechanism of the solvent in the networks. The solubility, diffusion coefficients, and permeability of the solvent in UBMI networks were found to be lower than in the multiol-crosslinked PU networks. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 1747–1755, 1997     

Model networks of end-linked polydimethylsiloxane chains. XIV. Stress-strain,thermoelastic, and birefringence measurements on the bimodal networks at very low temperatures

Bimodal networks consisting of very short and relatively long poiydimethylsiloxane (PDMS) chains were studied from 30 to ?52°C in an attempt to elucidate the anomalous increases in modulus [f^*] exhibited by such elastomeric materials at high elongations. Temperature was found to have very little effect on (i) the elongation at which the upturn in [f^*] becomes discernible, (ii) the elongation at which rupture occurs, and (iii) the total increase in [f^*] up to the rupture point. The standard force-temperature (“thermoelastic”) plots were linear, but gave values of the energetic contribution to the total force which were significantly smaller than those universally obtained on unimodal, long-chain PDMS networks. Birefringence-temperature relations were also found to be linear, and yielded values of the optical-configuration parameter and its temperature coefficient which were in satisfactory agreement with the corresponding values reported for unimodal PDMS networks. These results indicate that even at very low temperatures the observed increases in modulus (and consequent improvements in ultimate strength) are due to non-Gaussian effects arising from limited chain extensibility, rather than from intermolecular reinforcing effects such as strain-induced crystallization.     

Swelling and Tensile Properties of Tetra‐Polyethylene glycol via Coarse‐Grained Molecular Models

Coarse‐grained (CG) implicit‐solvent potentials are developed for tetra‐polyethylene glycol (PEG) at different water concentrations using the iterative Boltzmann inversion (IBI) technique. The resulting potentials are used to study the swelling and tensile properties of tetra‐PEG gels at various swelling degrees φ_m. Two types of network topologies are considered, one “ideal” with a defect‐free diamond connectivity and the other “realistic” as simulated from an experimentally based cross‐linking process. Equilibrium swelling results for the realistic Tetra‐PEG networks are consistent with available experimental data, while those for the ideal tetra‐PEG networks exhibit much larger swelling. The realistic networks have higher Young's modulus E _m at the same φ_m than ideal networks due to the presence of trapped entanglements. Uniaxial deformation results of realistic networks show that E _m increases with degree of swelling, in accord with experimental results. The Young's moduli of gels at different φ_m confirm that the CG potentials developed by IBI are most suited to predict swelling states commensurate with the φ_m values at which the potentials were calibrated. A more generic, coarser potential, based on matching the persistence length of atomistic PEG chains in water, is able to produce a similar swelling behavior of an ideal diamond network.

     

Network formation in polyurethanes due to allophanate and biuret formation: Gel fraction and equilibrium modulus

Gel fraction and equilibrium elastic modulus of networks formed from α, ω-dihydroxypoly(oxypropylene) and bis(4-isocyanatophenyl)methane (MDI) as a result of side reactions were compared with theoretical predictions based on the theory of branching processes (Macromolecules, 23, 1774 (1990)). If isocyanate is in excess, trifunctional allophanate and biuret groups are formed. The experimental sol fraction can be correlated with theoretical predictions. The resulting networks are rather weak and the measured equilibrium modulus can be correlated with the theoretical values calculated from the concentration of elastically active network chains if the value of the front factor is close to that of a phantom network without a trapped entanglements contribution.     

Extraction and sorption studies using linear polymer chains and model networks

Model networks have been prepared by tetrafunctionally endlinking linear polydimethylsiloxane (PDMS) chains having molecular weights M_n in the range 2000–15,000 g mole^?1. The first series of networks were prepared from mixtures containing known amounts of unreactive linear PDMS chains with molecular weights M_d between 1000 and 16,000 g mole^?1. Rates of extraction were used to estimate diffusion coefficients; as expected, they were found to increase with increase in molecular weight M_c = M_n between crosslinks, but to decrease with increase in M_d. The ease with which all of such a diluent could be removed showed the same dependence on M_c and M_d. A second series of networks was prepared from the same reactive PDMS chains without diluents. Sorption and extraction studies using the same diluents were then carried out. The diffusion coefficients for sorption were found to be in the range (1.7–15.0) × 10^?12 m² s^?1 and depended on both M_c and M_d. The amount of diluent absorbed at equilibrium was between 10 and 70%, which is in good agreement with predictions from the Flory equation for dilation in networks, with account of constraints on crosslink fluctuations.     

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     

The modulus of randomly-crosslinked poly(dimethylsiloxane) networks

In several published studies, randomly crosslinked networks were prepared from poly(dimethylziloxane) by the selective crosslinking of vinyl side chains with a silicon-hydride crosslinking agent. Stress-strain measurements on these elastomers gave values of the elongation modulus in the limits of small and large deformations which exceeded those predicted by the Flory-Erman theory. Although these unexpectedly large values at the small-strain limit have frequently been attributed to contributions from trapped entanglements, the present analysis interprets them as simply arising from contributions from short chains inadvertently introduced from the silicon-hydride crosslinking agent. In this interpretation there is a bimodal distribution of network chain lengths and, possibly, of crosslink functionalities as well. The present analysis gives results in good agreement with experiment.     

EQUILIBRIUM SWELLING PROPERTIES OF LIGHTLY CROSSLINKED ISOTROPIC NETWORKS: NATURAL RUBBER CROSSLINKED IN DRY STATE

Lightly crosslinked natural rubber (NR) networks have been characterized by equilibrium swelling in toluene. A good agreement between the equilibrium swelling crosslink-junction densities (μ_E) and the values expected from the stoichiometry of dicumyl peroxide decomposition (μ_c) has been obtained using Flory's early equation of state. The applicable value of χ = 0.37 was found to compare equally well and is in tandem with the previously reported literature values of χ = 0.391 and χ = 0.35 from swelling and heat of mixing measurements, respectively.

At low crosslink-junction densities i.e., μ_E ≤ 4.2 mM.l^?1 corresponding on average to 4 crosslink-junctions per cis 1,4-poly-isoprene chain and below, network formation is found to be incomplete. The approach used also provides a simple way of differentiating lightly crosslinked networks from the state of gelation.