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.     

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.     

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.     

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

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

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.     

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     

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.     

The creation and spatial structure of end-linked bimodal polymer networks: a Monte Carlo study

The creation and spatial structure of end-linked bimodal polymer networks was investigated by means of the three-dimensional bond-fluctuation model. The portion of long and short chains was varied. The curing process was found to be reaction-controlled within our simulation parameters. The spatial distribution of the cross-linkers in the created networks revealed no large-scale inhomogeneities. Investigations of the pair-correlation functions of the center of mass of the short chains were consistent with the random distribution of the cross-linkers.     

Toughness of polymeric networks: The limited importance of crosslink density

Four series of polymeric model networks were prepared with bimodal chain length distribution between crosslink points and two types of dangling chains as network defects. In the last series the crosslink density was changed without a large change in the chemical composition. The fracture toughness of those networks were compared with that of the defect–free networks. The fracture toughness of the various networks is surprisingly little influenced by the introduction of defects. Neither bimodality, nor dangling chains, nor a high sol fraction alters the toughness of the network. A good correlation between K_Ic and the weight fraction of polyether is observed. A much smaller dependence of K_Ic on the strand density can be deduced. The yield stress is high and approximately invariant for all systems studied. It is concluded that the toughness of a polymeric network does not seem to be influenced by its perfection and only to a small extent by its degree of crosslinking.     

Fracture of model polyurethane elastomeric networks

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

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.     

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.     

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.     

Coexistence of two kinds of fluorinated hydrogenated micelles as building blocks for the design of bimodal mesoporous silica with two ordered mesopore networks

A simple and effective route has been developed for the synthesis of bimodal (3.6 and 9.4 nm) mesoporous silica materials that have two ordered interconnected pore networks. Mesostructures have been prepared through the self-assembly mechanism by using a mixture of polyoxyethylene fluoroalkyl ether and triblock copolymer as building blocks. The investigation of the R(F)(8)(EO)(9)/P123/water phase diagram shows that in the considered surfactant range of concentrations the system is micellar (L(1)). DLS measurements indicate that this micellar phase is composed of two types of micelles; the size of the first one at around 7.6 nm corresponds unambiguously to the pure fluorinated micelles. The second type of micelles at higher diameter consists of fluorinated micelles that have accommodated a weak fraction of P123 molecules. Thus, in this study the bimodal mesoporous silica is really templated by two kinds of micelles.     

1H multiple-quantum nuclear magnetic resonance investigations of molecular order in polymer networks. II. Intensity decay and restricted slow dynamics

We present an approach towards the analysis of the intensity decay in proton multiple-quantum experiments on polymeric networks in terms of slow fluctuations of the residual dipole-dipole coupling tensor. Solutions for individual spin pairs as well as the three-spin system of methyl groups are derived, and the influence of the cycle time of the multiple-quantum pulse sequence is evaluated. The multiple-quantum strategy discussed herein features the advantage that the magnitude of the fluctuating part of the residual dipole-dipole coupling constant and the correlation time of the slow process can be determined independently of the integral residual coupling constant as well as its distribution. The theory is applied to experiments on end-linked poly(dimethylsiloxane) model networks with mono- and bimodal chain length distributions, where it is found that, for all samples, correlation times of the slow processes average to about 1 ms, and that the magnitude of the fluctuating part of the dipole-dipole coupling is significantly smaller than the average dipole-dipole coupling constant. This observation is interpreted in terms of considerably restricted reorientations of topological constraints.     

Fabrication and characterization of multidimensional tungsten oxide networks

Approaches for preparation of porous oxides with different length scales are currently available[1?3]. Materials with hierarchical pore systems at two or three different length scales have also been prepared[4?7]. Such materials are important both for the…     

Detection of heterogeneities in dry and swollen polymer networks by proton low-field NMR spectroscopy

We report on the implementation of modern proton multiple-quantum NMR methods for the characterization of molecular order and dynamics in polymer networks and melts on cost-efficient low-resolution low-field NMR instrumentation. The method permits the extraction of chain order parameter distributions, and is therefore sensitive to a heterogeneity of cross-links and other topological constraints. Data from samples with bimodal network chain length distributions acquired at 20 MHz (0.47 T) are in quantitative agreement with results obtained at high field (500 MHz). It is shown for the first time that the chain order distribution is broadened upon swelling, providing evidence for the nontrivial nature of the swelling process.     

Segmental and crosslink point motion in networks

¹³C- and ³¹P-NMR spin lattice relaxation in the rotating frame have been measured on a series of networks prepared from monodisperse and deliberately bimodal poly(propylene glycols) (PPG) crosslinked with tris(4-isocyanatophenyl) thiophosphate. The T_1pC minima correspond to loss maxima in the DMTA (Dynamic Mechanical Testing) measured at 10Hz. The T_1p^P minima fall at higher temperatures than those of T_1p^C for the same network indicating that these crosslinks lag the segments in frequency of motion at a given temperature. The carbon relaxation is biphasic below T_g of the segments indicating two relaxation domains which we assign to bulklike PPG segments and PPG segments proximal to he crosslink. Lineshape analysis by a diffusional model indicates crosslink reorientation is not isotropic until well above T_g. Relaxation and lineshapes for the bimodal networks indicate that junctions are not uniformly plasticized by the segments.     

基因网络前馈环路中的涨落共振

应用化学主方程和线性涨落近似方法,重点研究了前馈环路（FFL）对外界输入弱信号的响应,特别考察了它的涨落共振现象．研究发现Z基因的FR行为很大程度上依赖于FFL的协同性：协同FFL中Z的FR曲线呈明显的单峰,而非协同FFL中该曲线出现明显双峰．由于振荡信号常常在实际应用中用来探测网络的调控结构,因此可以利用涨落共振曲线的定性差别来区分FFL网络的性能．     

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.