Stress-strain Isotherms Research Articles

The use of model polyisobutylene networks to study strain-induced crystallization

Model elastomeric networks were prepared by trifunctionally end linking hydroxyl-terminated chains of polyisobutylene having number-average molecular weights in the range 10−3Mn = 2.4 – 10.7 g mol−1. Their stress-strain isotherms in elongation at 25°C in the unswollen state showed significant increases or upturns in modulus at high elongations, due to strain-induced crystallization. Increase in degree of cross-linking (decrease in Mn) was found to decrease the elongation required to initiate crystallization and the maximum extensibility, but to increase the magnitude of the upturn in modulus.

Characterization of elastomeric network structures using the effects of swelling on stress-strain isotherms and the extents of swelling at thermodynamic equilibrium

Swelling has long been used in the study of elastomeric networks. In the case of stress-strain measurements on swollen networks, the most important observation was the decrease of the Mooney constant 2C2, which is widely used to represent the magnitude of the departure of an observed stress-strain isotherm from that predicted by the simplest molecular theories. Swelling equilibrium data were generally analysed by the Flory-Rehner equation for affine networks simply to obtain a measure of the degree of cross-linking. The advent of model networks whose structures are well known and of a new theory due to Flory constitute great improvements for the study of rubberlike elasticity. In this, new context, swelling can be used to obtain additional important molecular information through both uniaxial extension of swollen samples and swelling equilibrium. In particular, the relation between the main parameter κ characterizing the local constraints on junction fluctuations in the Flory theory, the extent of swelling, and the molecular weight between cross-links can be probed.

Polyisobutylene model elastomers prepared using demonstrably complete end-linking reactions

Linear polyisobutylene (PIB) molecules with either hydroxyl or isopropenyl groups at the chain ends were prepared using cationic polymerizations with bifunctional initiator-chain transfer agents. Extensive spectroscopic analyses confirmed the essentially perfect difunctionality of the two types of polymers. The former polymers were end-linked using an aromatic triisocyanate, and the latter by means of a tetrafunctional silane. The resulting trifunctional and tetrafunctional model PIB networks were found to have absolutely negligible sol fractions, which demonstrates that the end-linking reactions used to prepare them were essentially complete. The networks were studied with regard to their equilibrium stress-strain isotherms in uniaxial extension at 25 °C. The results thus obtained are in satisfactory agreement with theory and yield no evidence whatever for significant elastic contributions from inter-chain entanglements.

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

AbstractElastomeric 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.

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

AbstractElastomers of controlled molecular structure were prepared from hydroxyl‐terminated atactic poly(propylene oxide) (PPO) chains having number‐average molecular weights Mn 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.

Model Elastomeric Networks Prepared from Poly(caprolactone)

Model networks of the polyurethane-type were prepared by end-linking hydroxyl-terminated poly(e-caprolactone) chains with an aromatic triisocyanate. The chains had number-average molecular weights Mn in the range 570—1950g mol−1, and the reactions were carried out in the undiluted state. The trifunctional networks thus obtained were studied with regard to their stress-strain isotherms in both the unswollen and swollen states, in elongation at 25°C. Values of the high deformation modulus fell between the limits predicted for the non-affine deformation of a “phantom” network and the affine deformation of a network in which the cross-links are firmly embedded in their surroundings. As expected, the values of the low deformation modulus were much closer to the affine limit.

Model networks of end-linked polydimethylsiloxane chains. IX. Gaussian, non-Gaussian, and ultimate properties of the trifunctional networks

Model elastomeric networks were prepared by end-linking hydroxyl-terminated polydimethylsiloxane chains with a trialkoxy silane. Mixtures of various proportions of relatively long and very short chains were employed, since the resulting ’’bimodal’’ networks are of unique importance in characterizing non-Gaussian effects related to limited chain extensibility. Stress–strain isotherms of these trifunctional networks gave values of the elongation at which the modulus begins to increase anomalously, and values of the elongation and modulus at the point of rupture. These results were interpreted in terms of calculated values of the maximum extensibilities of the chains, and were compared with previously reported results on the corresponding tetrafunctional networks in order to characterize the effects of crosslink functionality on these properties. In addition, the elasticity constants characterizing the Gaussian regions of the isotherms, and values of the degree of equilibrium swelling were used to evaluate the most recent molecular theories of rubberlike elasticity, particularly with regard to how the elastic effectiveness of the network chains depends on chain length and extent of deformation.

Model networks of end-linked polydimethylsiloxane chains. VII. Networks designed to demonstrate non-Gaussian effects related to limited chain extensibility

End-linking polymer chains by means of a multifunctional cross-linking agent provides an ideal way for obtaining elastomeric networks of any desired distribution of chain lengths. In the present investigation, this technique was employed to give polydimethylsiloxane networks consisting of various proportions of relatively long and very short network chains. The stress–strain isotherms of these networks generally showed an anomalous increase in the modulus at high elongation, and the increase persisted at high temperatures and high degrees of swelling. This non-Gaussian effect was quantitatively correlated with the limited extensibility of the network chains; specifcally, the increase in modulus was found to begin at approximately 60%–70% of the maximum extensibility of the network chains, and network rupture at 80%–90%. The elongation at which the increase becomes evident increases with decrease in the proportion of the very short chains, thus verifying the nonaffine nature of the deformation at high elongation. In addition, the elasticity constants characterizing the Gaussian regions of the isotherms, and the values of the degree of equilibrium swelling were used to evaluate the most recent molecular theories of rubberlike elasticity, particularly with regard to the structure factor relating the modulus of an elastomer to the average length or molecular weight of the network chains.

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.

Recent studies of rubberlike elasticity

New synthetic techniques are now available for the preparation of “model” polymer networks of known and carefully controlled molecular structure. The preparation of such networks and their utilization in elucidating molecular aspects of rubberlike elasticity are illustrated using highly elastomeric, unfilled networks of polydimethylsiloxane. In one such preparative technique, hydroxyl-terminated chains are end-linked using either a tetrafunctional or trifunctional silicate, in either the undiluted state or in solution. Because of the selectivity and completeness of the reaction, the molecular weight and molecular weight distribution of the network chains are to good approximation those of the chains prior to their end-linking into the network structure. The experimentally determined stress-strain isotherms and degrees of equilibrium swelling of the resulting networks are then used to evaluate the molecular theories of rubberlike elasticity, particularly with regard to the dependence of the elastic modulus and ultimate properties on the degree of cross linking and the cross-link functionality. Another preparative technique employs random and block copolymers containing reactive vinyl groups in known amount and known distribution along the chains. Selectively cross-linking through these sites then yields model networks ranging from random to highly non-random in chain length distribution. These networks are particularly useful for characterizing the effects of very short chains on the average degree of cross-linking or average network chain length as obtained from stress-strain and swelling equilibrium measurements.

Model networks of end-linked polydimethylsiloxane chains. IV. Elastomeric properties of the tetrafunctional networks prepared at different degrees of dilution

Highly elastomeric ’’model’’ networks were prepared by end-linking polydimethylsiloxane chains of known molecular weight, using a tetrafunctional orthosilicate. The reactions were carried out in both the undiluted state and in solution, with n-dodecane, cyclic dimethylsiloxane oligomer, and linear dimethylsiloxane oligomer as diluents. The networks thus prepared 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 room temperature. The results obtained did not depend significantly on the amount of diluent present during the preparation of the networks, which indicates that the end-linking reaction employed must be very nearly complete even in the undiluted state. Values of the modulus in the reaction employed must be very nearly complete even in the undiluted state. Values of the modulus in the limits of large and small elongation were obtained from the stress–strain isotherms and were found to be in satisfactory agreement with theoretical predictions for nonaffine and affine deformations, respectively. The results from the swelling equilibrium measurements were found to fall between these two extremes. They are in excellent agreement, however, with the results of a very recent theory of network swelling in which the degree of nonaffiness depends on both the network structure and the equilibrium degree of swelling. The relatively good agreement between experiment and theory suggests that interchain entanglements do not make a large contribution to the equilibrium elastomeric properties of a polymer network.

Model networks of end-linked polydimethylsiloxane chains. III. Effect of the functionality of the cross-links

Fractions of hydroxyl-terminated polydimethylsiloxane (PDMS) of relatively narrow molecular weight distribution were cross-linked by reacting the hydroxyl chain ends, in the undiluted state, with a trifunctional orthosilicate. Stress–strain isotherms were obtained on the resulting networks in elongation at 25° C, and were represented in terms of the equation [f*]=2C1+2C2α−1, where [f*] is the reduced stress or modulus and α is the elongation of the network. The present results on the trifunctional PDMS networks were compared with previously reported results on the corresponding tetrafunctional networks, thereby characterizing the effect of changing the network functionality φ from four to three. The observed changes were found to be in good agreement with predictions based on the most recent molecular theories of rubberlike elasticity. Specifically, experiment and theory were found to be in good agreement with regard to the factor which relates the modulus 2C1 to the structure of the network, and with regard to the ratio 2C2/2C1 which is a measure of the extent to which the elastic deformation changes from affine to nonaffine with increasing stress.

The Effect of Swelling on the Elasticity of Polyisobutylene Networks at High Elongation

Sulfur-cured, unfilled networks of butyl rubber (essentially polyisobutylene, PIB) were swollen to varying degrees with a nonvolatile diluent. A study in elongation of their rupture points was carried out at sufficiently low temperature to permit strain-induced crystallization in the unswollen state. The stress—strain isotherms thus obtained differed somewhat from those similarly obtained for crystallizable networks of cis-1, 4-polybutadiene, in that for example the strain-induced crystallization never caused an abrupt decrease in the modulus. The marked increases in modulus which did occur at high elongations in the case of the unswollen PIB networks were found to diminish, and eventually to be suppressed entirely, with increased swelling. Both the ultimate strength and maximum extensibility decreased substantially with an increase in swelling. The fractional decreases in both quantities were much larger than for typical noncrystallizable networks of poly(dimethylsiloxane), but the ultimate strength was found to be much more sensitive to the reinforcing effect of the strain-induced crystallization.

Effect of Strain-Induced Crystallization on the Elastomeric Properties of Cis-1,4-Polybutadiene Networks

Abstract Polymer networks, when studied at very high elongations, frequency show anomalous stress-strain isotherms in that they exhibit values of the modulus or “reduced force” [ƒ*] which increase markedly with increasing elongation. Such isotherms depart appreciably from the form predicted by the molecular theories of rubberlike elasticity and from the Mooney—Rivlin representation adopted from phenomenological arguments as well. For this reason, the interpretation of the increase in [ƒ*] at high elongations has been of great interest for a considerable period of time. For several decades now, this behavior has generally been attributed to the limited extensibility of the network chains. Critical examination of various published results pertinent to this question, in conjunction with more definitive experimental studies reported recently, however, support the alternative suggestion that such atypical isotherms are due to strain-induced crystallization. One experiment particularly relevant to this issue is the study of the stress-strain isotherms of a polymer network as a function of temperature. Such experiments have been carried out on natural rubber, but the relatively poor thermal stability of this polymer and the extent to which its melting point is increased by elongation make it essentially impossible to study the high elongation stress-strain relationships for this polymer at a temperature sufficiently high to ensure complete suppression of strain-induced crystallinity. Similar experiments carried out on networks of a more suitable polymer, polyisobutylene, gave stress-strain isotherms showing the upturn in the reduced force at low temperature, but not at higher temperatures, thus strongly implicating strain-induced crystallization as the origin of this anomalous behavior. These experiments, however, suffered somewhat from the one shortcoming that the increase in temperature required to suppress the upturn in [ƒ*] also decreased the maximum extensibility of the network to below the elongation at which the upturn occurred at the lower temperatures.

The effect of swelling on the stress-strain isotherms and ultimate properties of polydimethylsiloxane networks in elongation

Gamma radiation was used to prepare three cross-linked, unfilled samples of highly elastomeric polydimethylsiloxane. Portions of each sample were studied in elongation to their rupture points, at 30 °C, in both the unswollen state and swollen with low molecular weight dimethylsiloxane fluid. Values of the volume fraction ν2 of polymer in the networks ranged from 1.00 to 0.40. None of the stress-strain isotherms obtained showed any upturn in the reduced force or modulus [f*] at high elongation, an observation in agreement with the conclusion that such increases in [f*], when observed, are due to strain-induced crystallization. All of the isotherms were well represented by the semi-empirical equation [f*]=2C1+2C2α−1, whereα is the elongation, and 2C1 and 2C2 are the Mooney-Rivlin constants. Values of 2C2, which serves as a measure of the extent to which [f*] varies with elongation, showed a decrease with decrease in ν2, and generally also with decrease in degree of cross-linking. The ultimate properties reported were the valuesλr and [f*], of the total elongation and the reduced force, respectively, at the rupture point. Decrease in degree of cross-linking causes a significant increase inλr and a significant decrease in [f*]r; decrease in ν2, however, has only a relatively small effect onλr and [f*]r.

Model networks of end-linked polydimethylsiloxane chains. I. Comparisons between experimental and theoretical values of the elastic modulus and the equilibrium degree of swelling

Two samples of hydroxyl-terminated polydimethylsiloxane having molecular weights the order of 2×104 and 1×103 g mol−1 were separated into a total of six fractions of relatively narrow molecular weight distribution. Portions of one of the unfractionated polymers and each of the fractions were cross-linked by reacting the hydroxyl chain ends, in the undiluted state, with a tetrafunctional orthosilicate. The resulting networks of end-linked chains were studied with regard to their stress–strain isotherms in elongation at 25°C and their equilibrium swelling in benzene at room temperature. Values of the elastic modulus obtained from the isotherms support theoretical arguments that fluctuations in the network chain vectors reduce the value of the modulus to approximately one-half of the value predicted for affine deformations of chain vectors constrained in their fluctuations by cross-links firmly embedded in the network medium. Values of the equilibrium degree of swelling of the networks calculated on this basis are also in good agreement with experiment. The networks formed by end-linking relatively short chains have small values of the semiempirical constant 2C2 used as a measure of the departure of an observed stress–strain isotherm from the form predicted by theory. Although this observation is consistent with the suggestion that such end-linked networks have a much smaller number of interchain entanglements than do randomly cross-linked networks, other evidence and arguments unfortunately do not support this assumption.

The Effect of Temperature on the Stress-Strain Isotherms of Polyisobutylene Networks at High Elongation

Abstract Three chemically cured, unfilled polyisobutylene networks were studied in elongation to their rupture points at several temperatures in the range 15–80°C. At the lower temperatures, the stress-strain isotherms obtained were similar to those reported for unfilled networks of natural rubber and several other polymers in that they showed an anomalous, marked increase in the modulus or reduced force at high elongations. At intermediate temperatures, this upturn in the force is significantly diminished, however, and the networks rupture at lower elongations. At the higher temperatures, there is no evidence whatever for an upturn in the force, and the elongation at rupture is further reduced. These results on polyisobutylene networks implicate strain-induced crystallization rather than limited chain extensibility as the origin of the pronounced increase in the force at high elongations. The reinforcement provided by the crystallites formed in this manner would account for the increased extensibility of the networks at the lower temperatures. Since natural rubber is very similar to polyisobutylene in its crystallizability, the anomalous stress-strain isotherms widely reported for networks of this polymer are thus probably also due largely, if not entirely, to network crystallization.

The Constants 2C1 and 2C2 in Phenomeno-Logical Elasticity Theory and Their Dependence on Experimental Variables

Abstract There has been a very large number of experimental studies of the elastic properties of amorphous polymer networks in uniaxial deformation, particularly elongation, over the past few decades. In spite of these efforts, and related theoretical investigations, major questions remain unanswered with regard to stress-strain isotherms over the range of moderate deformations. The major question in this area is the molecular basis or interpretation of the phenomenological constants 2C1 and 2C2 used to represent the experimentally observed isotherms. It is hoped that the present survey of the results of the most important of these experimental studies, particularly in regard to the effect of experimental variables on 2C1 and 2C2 will encourage additional work, leading to an unambiguous solution to this important unsolved problem in the area of rubberlike elasticity.

Stress - strain isotherms and thermoelastic properties of crosslinked amorphous polyoxyethylene in compression

Stress-strain isotherms at a number of temperatures in the range 45–85°C have been determined for a compressed, amorphous polyoxyethylene network swollen with a constant amount of non-volatile diluent. The observed isotherms showed much smaller departures from the form indicated by the molecular theories of rubberlike elasticity than is generally the case for polymer networks in elongation. The thermoelastic behaviour of the network was characterized by the dependence of the stress on temperature at constant deformation. The value of the temperature coefficient of the unperturbed dimensions of polyoxyethylene calculated from these data is in good agreement with values obtained both from thermoelastic measurements on polyoxyethylene networks in elongation, and from viscosity-temperature studies on isolated polyoxyethylene chains in a thermodynamically good solvent.

Elasticity theory. IV. The stress–strain relation

AbstractThe stress–strain relation is analyzed in terms of the gyration tensor where the sum is over all N atomic groups comprising the elastomeric material. All groups are taken for convenience to have the same mass. This tensor is linear in the strain, and furthermore it may be evaluated from a configuration integral. The definition of the strain by use of Sαβ gives new insight into the connection between statistical and continuum mechanics, and further clarifies the significance of “affine deformation.” Stress components are obtained as derivatives of the (Helmholtz) free energy A with respect to the invariants of Sαβ. The first invariant is just the radius of gyration S2 of the network, and the dependence of A on S2 alone provides a basis for analysis of stress–strain isotherms.