Modelling the elastic behaviour of real chains in polymer networks

Monte-Carlo simulations of the elastic behaviour of PDMS networks, using a realistic (R-I-S) network-chain model, are able to reproduce experimentally observed deviations from Gaussian network behaviour in uniaxial extension. The finite extensibility of the network chains is shown to cause non-affine deformation of the mean-square network-chain end-to-end distance, even at moderate sample deformations (λ≈1.5). An increase in the proportion of fully-extended chains with increasing macroscopic strain gives rise to a steady decrease in the rate of network free-energy change, causing a reduction in the network modulus at moderate macroscopic strains. There is no need to invoke a transition from affine to phantom chain behaviour as deformation increases.     

Molecular aspects of polymer network deformation - small angle neutron scattering and NMR studies

Poly(1,4-butadiene) networks obtained by a 4-functional random cross-linking reaction over a broad range of polymer concentration were studied by small angle neutron scattering(SANS), ²H NMR and Monte Carlo(MC) simulation in the isotropic and uniaxially deformed state. The defect structure of the networks has been characterized by MC simulation of the cross-linking reaction. The anisotropy of the radius of gyration in deformed networks determined from SANS has been analyzed by the theory of Ullman. It was found that the number of active cross-links per chain is in agreement with MC and that the chain deformation follows phantom behaviour. The local orientation as measured by ²H NMR is related to the global anisotropy of the network by a MC calculation of oriented chains. The ²H NMR line shape of the deformed network is analyzed in terms of two relaxation processes arising from interior parts of the chains and from segments at chain ends. The mobility of both decrease with strain. It was found that the orientation connected to the first process shows the classical strain dependence of rubber elasticity, whereas the second exhibits a weaker dependence on strain.     

The theory of non-linear molecular orientation and stress in polymer fluids

Non-linear stress, and orientation characteristics for polymer fluids (melts, solutions) composed of chain macromolecules of finite length have been derived. Freelyjointed chains with inverse Langevin statistics have been assumed, and their behaviour in potential hydrodynamic fields analyzed. Numerical calculations have been performed for uniaxial extensional flow in a wide range of flow rates (and stresses). In the range of small stresses, orientation is a linear function of stress. At higher stresses, orientation factor levels off, asymptotically approaching unity.Flow orientation characteristics significantly differ from those derived from affine deformation of polymer networks. This difference is a natural consequence of constraints imposed by network junctions on chain deformation.This work is dedicated to Professor Hanns-Georg Kilian on his 60th birthday in appreciation of his contribution to Polymer Physics.     

Segmental orientation in entangled chains

Slip-link model of an entangled chain is used to calculate average orientation of chain segments. The results in the asymptotic regime of very long chains prove linear dependence of optical anisotropy on stress despite complex stress-strain relation. The linear stress-optical law is predicted both for a single chain and a model network subjected to uniaxial deformation. The calculated stress exhibits non-linearity in Mooney-Rivlin plot. Effects due to entanglements are proportional to assumed number of slip-links per chain.     

Chain segment ordering in uniaxially strained networks: A deuterium NMR investigation

The deuterium NMR (²H-NMR) is used for probing the chain segment orientation in polymer networks under uniaxial stress. The method is based on the observation of an incomplete time averaging of quadrupolar interactions affixed to deuterated segments. The samples are end-linked polydimethylsiloxane networks. The ²H-NMR experiments are performed either on labelled network chains or an labelled probe polymer chains dissolved in the network. The basic results are the following: — The induced uniaxial order is related to a uniaxial dynamics of chain segments around the direction of the applied constraint. — A permanent orientation is observed on free polymer chains dissolved in the deformed networks. — The mean degrees of orientational order induced along short and long chains in bimodal networks are the same. These experimental facts appear as evidences for cooperative orientational couplings between chain segments in the deformed networks.     

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     

Segmental orientations and deformation mechanism of a poly(ether‐block‐amide) film

Simultaneous measurements of microscopic infrared dichroism, mesoscale deformation, and macroscopic stress have been made for a microphase‐separated film of poly(ether‐block‐amide) 4033 during uniaxial stretching at temperatures between 30 and 91 °C, well below the melting point of the hard polyamide‐12 (PA) domains. Before the onset of dramatic microstructural alterations, the true stress–strain relationship on the mesoscale can be described with an interpenetrating network model, and poly(tetramethylene oxide) (PTMO) soft segments undergo affine deformation. Beyond a threshold strain at which stress from the soft network becomes larger than that from the hard network, plastic deformation occurs in the hard PA domains, and this is accompanied by the downward derivations of the true stress and molecular orientation of PTMO blocks from the model predictions. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1161–1167, 2005     

Liquid crystalline triblock copolymers. Mechanical behaviour and orientation under uniaxial strain

The stress‐strain and orientation behaviour of side‐chain liquid crystalline(SLCP) ABA triblock copolymers with a backbone of polystyrene‐block‐1,2‐polybutadiene‐block‐polystyrene and a cyanobiphenyl mesogen in the side chain was investigated in dependence of molecular weight. The polymer shows the behaviour of a thermoplastic liquid crystalline elastomer(TPLCE) in the nematic phase in a region between the glass transitions of the polystyrene block and the SLCP. The ultimate properties and E‐modulus is lower than for conventional thermoplastic elastomers. Under uniaxial strain liquid crystalline order perpendicular to the direction of strain is induced.     

Structural developments in synthetic rubbers during uniaxial deformation by in situ synchrotron X‐ray diffraction

The molecular orientation and strain‐induced crystallization of synthetic rubbers—polyisoprene rubber, polybutadiene rubber, and butyl rubber [poly(isobutylene isoprene)]—during uniaxial deformation were studied with in situ synchrotron wide‐angle X‐ray diffraction. The high intensity of the synchrotron X‐rays and the new data analysis method made it possible to estimate the mass fractions of the strain‐induced crystals and amorphous chain segments in both the oriented and unoriented states. Contrary to the conventional concept, the majority of the molecules (50–75%) remained in an unoriented amorphous state at high strains. Each synthetic rubber showed a different behavior of strain‐induced crystallization and molecular orientation during extension and retraction. Our results confirmed the occurence of strain‐induced networks in the synthetic rubbers due to the inhomogeneity of the crosslink distribution. The strain‐induced networks containing microfibrillar crystals and oriented amorphous tie chains were responsible for the ultimate mechanical properties. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 956–964, 2004     

Rejuvenation of PLLA: Effect of plastic deformation and orientation on physical ageing in poly(l‐lactic acid) films

Poly(l ‐lactic acid) (PLLA) is a bio‐degradable polyester which exhibits brittle behaviour due to relatively fast physical ageing of the amorphous phase. This work describes the effects of thermal rejuvenation and molecular orientation of the amorphous phase on this physical ageing process. Uniaxial compression testing showed that physical ageing of the amorphous phase increases the yield stress and the associated strain softening response, both contributing to the observed embrittlement of PLLA in tension. Molecular orientation at constant crystallinity was applied by uniaxial and biaxial plastic deformation just above the glass transition temperature, up to plastic strains of 200% to avoid strain‐induced crystallisation. Using stress‐relaxation experiments combined with tensile testing, both as a function of ageing time, it is shown that both uniaxial and biaxial plastic deformation in excess of 150% plastic strain, decelerates and possibly prohibits the physical ageing process. The oriented monofilaments and films have improved mechanical properties such as stiffness, strength and strain‐to‐break, which were not affected by physical ageing during the whole testing period (40 days). In addition, plastic deformation to higher draw ratios and/or higher temperatures strongly enhanced crystallinity and resulted in PLLA monofilaments and films that also exhibited tough behaviour, not affected by physical ageing. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 2233–2244     

Orientational birefringence of uniaxially compressed PMMA networks in the unswollen state

Swollen PMMA networks show significant deviations from the theory of rubber elasticity. The unusual mechanical, optical and swelling properties have been explained by the existence of associations, well known for PMMA solutions.Unswollen PMMA networks show stress optical properties which are very similar to the networks swollen in carbon tetrachloride which is known as a strong complexing solvent. This indicates that associations exist in the bulk material. Swelling in carbon tetrachloride does not change the content of associations, only the glass transition temperature is reduced.Increasing the amount of crosslinking agent proves that the stress optical coefficient depends on the crosslinking density. This is contrary to the theoretical predictions. An explanation given by the effect of short network chains is not satisfactory. However, the consideration of a crosslinking chain instead of a crosslinking point gives a sufficient explanation. An oriented crosslinking chain gives a positive contribution to the total optical anisotropy while a PMMA chain shows a change from negative values (hindered rotation) to positive values (free rotation).Part I in Colloid & Polym Sci 262:182 (1984)Deceased Sept. 4, 1984     

Stress-optical properties of bimodal polymer networks

The optical properties of bimodal poly(dimethylsiloxane) (PDMS) networks were studied with special emphasis on the non-linear stress optical properties exhibited by these materials. In particular the effect of chain length, and junction functionality on the strain induced birefringence was investigated. It is shown that for the non-linear properties to clearly manifest themselves a critical concentration of short chains is essential and that the junctions are tetra-functional. However, all bimodal compositions studied were found to exhibit a non-linear variation of birefringence with strain irrespective of the junction functionality. The optical properties of the unimodal networks were found to vary linearly with stress and strain as expected. The transition from non-linear to linear optical behavior on increasing the molecular weight of the short chains is also established.     

Polydomains in liquid-crystal elastomers

In liquid-crystal elastomers, the simultaneous presence of rubber elasticity due to the crosslinked backbone chains and of optical birefringence due to the mesogens in the side chains lead to exceptional physical properties. An elastic deformation of the network influences the order of the mesogens and, therefore, the optical properties. A theory based on a Landau-de Gennes expansion of the free energy is proposed. In the opaque polydomain phase, the local orientation is given by a compromise between the external mechanical field and a local anchoring interaction. As the field is increased, it becomes energetically favorable for the mesogens to align parallel to the mechanical field, and a transition to a transparent monodomain structure occurs. Results for the average orientation, the stress and the chain conformation are given.     

Molecular orientation behavior of poly(vinyl chloride) as influenced by the nanoparticles and plasticizer during uniaxial film stretching in the rubbery stage

The structural evolution during uniaxial stretching of poly(vinyl chloride) films was studied using our real time spectral birefringence stretching machine. The effect of clay loading and the amount of plasticizer as well as the rate effects on the birefringence development and true mechanical response are presented with a final model summarizing the molecular phenomena during stretching. Mechano‐optical studies revealed that birefringence correlated with mechanical response (stress, strain, work) nonlinearly. This was primarily attributed to the preexisting strong network of largely amorphous chains connected via small crystallites that act as physical crosslinking points. These crystallites are not easily destroyed during the high‐speed stretching process as evidenced from the birefringence–true strain curves along with the X‐ray crystallinity measurements. At high speeds, the amorphous chains do not have enough time to relax and hence attain higher orientation levels. The crystallites, however, orient more efficiently when stretched at slow speeds. Apparently, some relaxation of the surrounding amorphous chains helps rotate the crystallites in the stretching direction. Overall birefringence is higher at high stretching speeds for a given true strain value. When the nanoparticles are incorporated, the orientation levels are increased significantly for both the crystalline and amorphous phases. Nanoplatelets increase the continuity of the network because they have strong interaction with the amorphous chains and/or crystallites. This in turn helps transfer the local stresses to the attached chains and increase the orientation levels of the chains. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 724–742, 2005     

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.     

Relationship of deformation behavior to thermal transitions of ethylene/styrene and ethylene/octene copolymers

The stress‐strain response of low‐crystallinity ethylene‐octene (EO) and ethylene‐styrene (ES) copolymers with 7–20 mol % comonomer was compared over a temperature range that spanned the glass‐transition and crystal melting regions. Above the onset temperature of the glass transition, the copolymers exhibited elastomeric behavior with low initial modulus, uniform deformation to high strains, and high recovery after the stress was released. In the glass‐transition range, an initial low‐stress elastomeric response was followed by a distinct “bump” in the stress‐strain curve. On the basis of the temperature and rate dependence of the stress‐strain curve, local strain‐rate measurements, local temperature changes, and recovery characteristics, the “bump” was identified as high strain yielding. Hence, the stress‐strain curve sequentially exhibited the features of elastomeric and plastic deformation. Following high strain yielding, strain hardening dramatically increased the fracture strength. This behavior was defined as elastomeric‐plastic. Elastomeric‐plastic behavior in the broad glass‐transition range constituted a gradual transition from elastomeric behavior at higher temperatures to low‐temperature plastic behavior with high modulus and macroscopic necking. Because of the lower glass‐transition temperature of EO, ?40 °C as compared with ?10 °C for ES, the onset of elastomeric‐plastic behavior occurred at a significantly lower temperature. The concept of a network of flexible chains with fringed micellar crystals serving as the multifunctional junctions that provides the structural basis for elastomeric behavior of low‐crystallinity ethylene copolymers was extended to elastomeric‐plastic behavior by considering a network with a fraction of rigid, glassy chains. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 40: 142–152, 2002     

Nonequilibrium Monte Carlo simulation of lattice block copolymer chains subject to oscillatory shear flow

This paper has extended nonequilibrium Monte Carlo (MC) approach to simulate oscillatory shear flow in a lattice block copolymer system. Phase transition and associated rheological behaviors of multiple self-avoiding chains have been investigated. Stress tensor has been obtained based upon sampled configuration distribution functions. At low temperatures, micellar structures have been observed and the underlying frequency-dependent rheological properties exhibit different initial slopes. The simulation outputs are consistent with the experimental observations in literature. Chain deformation during oscillatory shear flow has also been revealed. Although MC simulation cannot account for hydrodynamic interaction, the highlight of our simulation approach is that it can, at small computing cost, investigate polymer chains simultaneously at different spatial scales, i.e., macroscopic rheological behaviors, mesoscopic self-assembled structures, and microscopic chain configurations.     

Relation between stress and segmental orientation during mechanical cycling of a natural rubber‐based compound

X‐ray diffraction technique is used to evaluate the evolution of the segmental orientation in a natural rubber sample during mechanical uniaxial stretching and in the Gaussian regime condition. The method proves to be sensitive enough for testing the validity of the stress‐optical law. Measurements are performed at different temperatures and show that the dependence of the orientation parameter upon elongation is very close to the prediction of classical phantom theories. On the contrary, a significant Mooney–Rivlin correction is needed to account for the stress–strain relation. Consequently, systematic deviations from the stress‐optical law are observed, in particular for elongations below 2. These deviations are adverse from predictions of both the constrained and the diffuse junction theories. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 943–950