Strain Hardening and Strain Softening of Reversibly Cross-linked Supramolecular Polymer Networks

The large amplitude oscillatory shear behavior of metallo-supramolecular polymer networks formed by adding bis-Pd(II) cross-linkers to poly(4-vinylpyridine) (PVP) in dimethyl sulfoxide (DMSO) solution is reported. The influence of scanning frequency, dissociation rate of cross-linkers, concentration of cross-linkers, and concentration of PVP solution on the large amplitude oscillatory shear behavior is explored. In semidilute unentangled PVP solutions, above a critical scanning frequency, strain hardening of both storage moduli and loss moduli is observed. In the semidilute entangled regime of PVP solution, however, strain softening is observed for samples with faster cross-linkers (k(d) ～ 1450 s(-1)), whereas strain hardening is observed for samples with slower cross-linkers (k(d) ～ 17 s(-1)). The mechanism of strain hardening is attributed primarily to a strain-induced increase in the number of elastically active chains, with possible contributions from non-Gaussian stretching of polymer chains at strains approaching network fracture. The divergent strain softening of samples with faster cross-linkers in semidilute entangled PVP solutions, relative to the strain hardening of samples with slower cross-linkers, is consistent with observed shear thinning/shear thickening behavior reported previously and is attributed to the fact that the average time that a cross-linker remains detached is too short to permit the local relaxation of polymer chain segments that is necessary for a net conversion of elastically inactive to elastically active cross-linkers. These and other observations paint a picture in which strain softening and shear thinning arise from the same set of molecular mechanisms, conceptually uniting the two nonlinear responses for this system.     

A molecular dynamics simulation study on polymer networks of end-linked flexible or rigid chains

The differences in formation and structural properties of polymer networks consisting of end-linked flexible or rigid chains were studied by molecular dynamics simulation. Networks were formed from monodisperse, linear, short, flexible or rigid chains with functional end groups and a stoichiometric ratio of trifunctional cross-linkers. The rigid chains had a rodlike shape defined by an angle potential, while the flexible chains had no angle potential. In order to understand the influence of chain rigidity, all parameters of precursor chains (length, reactivity, bond potential, nonbonding potential) were the same, with the exception of the angle potential. The system density rho, corresponding to the concentration of monomer in solvent, was varied from 0.01 to 0.11. Different network structures resulting from the different processes of network formation were observed. Simulations showed that the flexible chains created an inhomogeneous network on a large scale via microgel cluster formation, in agreement with experimental observations, whereas the rigid chains rapidly created a homogeneous network in the entire system volume without first generating microgel clusters, with the additional difference that they gave rise to mutually interpenetrating networks at the local scale.     

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.     

Dependence of the moduli of random bimodal networks on chain-length distribution

There have now been a number of experimental studies on the preparation and elastomeric properties of random bimodal networks of polydimethylsiloxane. The mole per cent of the short chains and their molecular masses covered a wide range, thus resulting in various polydisperse chain-length distributions. The networks were studied with regard to their stress-strain isotherms in elongation, and values of their moduli in the large-deformation (phantom) limit were found to depend on the chain-length distribution. This Important result is in disagreement with phantom network theory, which assumes the elastic modulus is dependent only on the mean value of chain lengths such that the cycle rank of the network is preserved. The effective functionality of the long chains was found to depend on the number of short chains present. Better agreement with experiment was obtained when account was taken of the connectivity of the very short chains, in what is essentially a bimodal distribution of both network chain lengths and cross-link functionalities. Relevant here is the fact that as the degree of chemical cross-linking Increases, the shear modulus G moves away from the affine limit, toward the phantom limit. This increase toward phantom behavior is presumably due to the fact that the mutual interspersion of chains is less when the chains are shorter, even in the small-strain region.     

Characterization of polystyrene networks by small-angle neutron scattering

Polystyrene networks prepared by anionic polymerization have been characterized by small-angle neutron scattering. Two kinds of systems have been examined: (A) networks with labelled branch points allowing characterization of the spatial distribution of crosslinking points; (B) networks containing a low proportion of chains labelled with perdeuterated polystyrene in order to characterize the conformation of individual elastic chains of the polymeric network. The dependence of the results on swelling and uniaxial extension is discussed.     

Polymer networks by molecular dynamics simulation: Formation, thermal, structural and mechanical properties

A molecular dynamics simulation method is presented and used in the study of the formation of polymer networks. We study the formation of networks representing the methylene repeating units as united atoms. The network formation is accomplished by cross-linking polymer chains with dedicated functional end groups. The simulations reveal that during the cross-linking process, initially branched molecules are formed before the gel point; approaching the gel point, larger branched entities are formed through integration of smaller branched molecules, and at the gel point a network spanning the simulation box is obtained; beyond the gel point the network continues to grow through the addition of the remaining molecules of the sol phase onto the gel (the network); the final completion of the reaction occurs by intra-network connection of dangling ends onto unsaturated cross-linkers. The conformational properties of the strands in the undeformed network are found to be very similar with the conformational properties of the chains before cross-linking. The uniaxial deformation of the formed networks is investigated and the modulus determined from the stress-strain curves shows reciprocal scaling with the precursor chain length for networks formed from sufficiently large precursor chains (N ≥ 20).     

Divergent Shear Thinning and Shear Thickening Behavior of Supramolecular Polymer Networks in Semidilute Entangled Polymer Solutions

The steady shear behavior of metallo-supramolecular polymer networks formed by bis-Pd(II) cross-linkers and semidilute entangled solutions of poly(4-vinylpyridine) (PVP) in dimethyl sulfoxide (DMSO) or N,N-dimethyl formamide (DMF) is reported. The steady shear behavior of the networks depends on the dissociation rate and association rate of the cross-linkers, the concentration of cross-linkers, and the concentration of the polymer solution. The divergent steady shear behavior-shear thinning versus shear thickening-of samples with identical structure but different cross-linker dynamics (J. Phys. Chem. Lett. 2010, 1, 1683-1686) is further explored in this paper. The divergent steady shear behavior for networks with different cross-linkers is connected to a competition between different time scales: the average time that a cross-linker remains open (τ(1)) and the local relaxation time of a segment of polymer chain (τ(segment)). When τ(1) is larger than τ(segment), shear thickening is observed. When τ(1) is smaller than τ(segment), only shear thinning is observed.     

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.     

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.     

Investigation of coagulation process of wet-spun sodium alginate polymannuronate fibers with varied functionality using organic coagulants and cross-linkers

Alginate fibers composed of mannuronate blocks were manufactured via a wet spinning process by varying organic coagulants and cross-linkers. As cross-linkers, both ionic (Ca²⁺, Ba²⁺, and Al³⁺) and covalent (citric acid) were used, and as coagulants, dimethyl sulfoxide (DMSO), dimethylformamide (DMF), and tetrahydrofuran (THF) were selected. Fiber properties depended on the ionic size, valence, and chemical structure of the cross-linkers. Among the used cross-linkers, Al³⁺ being the smallest ion, could diffuse inside the polymer solution easily and form metal-carboxylate coordination with three mannuronate chains. However, Ca²⁺ and Ba²⁺ both were very large compared to it, and polymer chains were positioned at about 48° for Ca²⁺ and in a parallel manner for Ba²⁺ after forming inter and intrachain metal-carboxylate bonds. Citric acid underwent an esterification reaction with the hydroxyl groups of mannuronate chains. Coagulants stabilized the cross-linking process and formed hydrogen bonds with the polymer chains. Depending on the cross-linkers and coagulants, the fiber diameter ranged from 217 μm to 830 μm and tensile modulus ranged from 88 MPa to 2 MPa. Ca²⁺, and then Ba²⁺ ions were more effective as cross-linkers since they produced fibers with superior mechanical properties followed by citric acid and Al³⁺. For all cross-linkers except citric acid, when DMSO was used as a coagulant, the tensile modulus was the highest. This indicates that DMSO better stabilized the cross-linking process during coagulation. In phosphate buffer saline (PBS), only Ba²⁺ cross-linked fibers could retain their original structure for 24 h, and fiber formed with coagulant DMSO swelled the least because of its compact structure. It also lost the least percentage of weight during 6 weeks. Thermal properties of the samples were also different as Ba²⁺ and Ca²⁺ cross-linked samples were more resistive to high temperature than other samples. 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay illustrated the non-cytotoxicity of all the manufactured fibers.     

Correlated fluctuations in polymer networks

Polymer micronetworks with Cayley tree and random topologies are compared. The fluctuations of junctions in random networks exhibit significant departures from the mean value corresponding to those in Cayley tree topology. Correlations among the fluctuations are examined for the two types of networks. Correlations are stronger, and cover longer distances along the chain contours in random networks, compared to those observed in Cayley trees. This is a consequence of the non-uniform spatial distribution of connected vertices in a random network, as opposed to Cayley trees in which chains extend between junctions located in successive tiers only. The relaxation spectra of the two types of networks are compared. The cumulative distribution of frequencies obeys a power law dependence on frequency, with exponents of 1.4 and 1.1 for trifunctional Cayley tree and random networks, respectively. These exponents are significantly smaller than the Debye value of 3 for regular three dimensional crystals.     

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.     

Synthesis of cell-permeable stapled BH3 peptide-based Mcl-1 inhibitors containing simple aryl and vinylaryl cross-linkers

We report the synthesis of a series of distance-matching aryl and vinylaryl cross-linkers for constructing stapled peptides containing cysteines at i,i+7 positions. Langevin dynamics simulation studies helped to classify these cross-linkers into two categories: the rigid cross-linkers with narrower S–S distance distribution and the flexible cross-linkers with wider S–S distance distribution. The stapled Noxa BH3 peptides with the flexible distance-matching cross-linkers gave the highest degree of helicity as well as the most potent inhibitory activity against Mcl-1. However, the stapled peptides with the highest hydrophobicity showed the most efficient cellular uptake. Together, this work illustrates the divergent nature of binding affinity and cellular uptake, and the vital importance of choosing appropriate cross-linkers in constructing stapled peptides with the drug-like properties.     

THE SPATIAL DISTRIBUTION AND MOMENTS OF CIRCULAR FREELY JOINTED CHAIN

The spatial distribution function and second moments of circular freely jointed chain are derived based on an analytical method. The circular Gauss chain, which is simple for long chains, is compared with the circular freely jointed chain, which is exact for short chains. It is shown that the Gauss chain model predicts a more compact configurational distribution than the exact freely jointed chain. The two chain models, however, become closer to each other when the chain length increases. It is found that the difference of the mean square radius of gyration calculated with these two chain models is a constant, independent of the chain length.     

Novel cross-linked homogeneous polyacrylamide gels with improved separation properties: investigation of the cross-linker functionality

Polyacrylamide (PAAm) gels were synthesized using cross-linkers with their potential functionality (twice the number of double bonds of a cross-linker) varying from six to sixteen. Improved electrophoretic separation and highly desirable porosity and sieving properties were observed for most of the PAAm gels containing novel cross-linkers. An increase in the potential functionality of cross-linkers used in PAAm gels was an important factor, influencing the pore size and pore size distribution of the network.     

Dependence of the elastomeric properties of bimodal networks on the lengths and amounts of the short chains

Bimodal elastomeric networks were synthesized by tetrafunctionally end-linking very short and relatively long hydroxyl-terminated chains of poly(dimethylsiloxane). Decrease in the molecular weight of the short chains (from 880 to 660 to 460 g mol^?1) generally results in significant increases in ultimate strength and energy for rupture. Decrease in the number of short chains, however, causes these two quantities to go through a maximum. Too many short chains gives essentially a brittle thermoset, and this precludes any upturn in the modulus from the limited extensibility of the short chains. Having too few short chains makes their limited extensibility irrelevant since the entire macroscopic deformation can then be taken up by the long chains present in the network.     

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.     

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.     

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.     

Characterisation of UV-cured acrylate networks by means of hydrolysis followed by aqueous size-exclusion combined with reversed-phase chromatography

UV-cured networks prepared from mixtures of di-functional (polyethylene-glycol di-acrylate) and mono-functional (2-ethylhexyl acrylate) acrylates were analysed after hydrolysis, by aqueous size-exclusion chromatography coupled to on-line reversed-phase liquid-chromatography. The mean network density and the fraction of dangling chain ends of these networks were varied by changing the concentration of mono-functional acrylate. The amount and the molar-mass distribution of the polyethylene-glycol chains between cross-links (M(XL)) and polyacrylic acid (PAA) backbone chains (the so-called kinetic chain length (kcl)) in the different acrylate networks were determined quantitatively. The molar-mass distribution of kcl revealed an almost linear dependence on the concentration of mono-functional acrylate. Analysis of the starting materials showed a significant concentration of mono-functional polyethylene-glycol acrylate. In combination with the analysis of the extractables of the UV-cured networks (polymers not attached to the network, impurities that originate from the photo-initiator and unreacted monomers), more insight in the total network structure was obtained. It was shown that the UV-cured networks contain only small fractions of residual compounds. With these results, the chemical network structure for the different UV-cured acrylate polymers was expressed in network parameters such as the number of PAA units which are cross-linked, the degree of cross-linking, and the network density, which is the molar concentration of effective network chains between cross-links per volume of the polymers. The mean molar mass of chains between chemical network junctions (M(C)) was calculated and compared with results obtained from solid-state NMR and DMA. The mean molar mass of chains between network junctions as determined by these methods was similar.