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.     

“Belt-loop” model of chain entanglement

A simple model of an entangled chain is proposed. Statistical properties of the model are examined based on the partition function derived to include geometrical constraints imposed by entanglements. The model chain statistics results, for long chains, in a modified Gaussian function. The new statistics applied in the affine network theory yield stress-strain dependence, which qualitatively agrees with experimental data obtained for uniaxial extension and compression. Non-linear Mooney-Rivlin plots with a maximum appearing in the compression region are predicted for unswollen networks. With increasing swelling, non-linearity decreases. The proposed explanation of these phenomena is based on the restraints imposed on entangled chains, rather than on network junctions, unlike in the Flory-Erman theory. No arbitrary parameters are involved in the model.     

Orientation of the α- and γ-modification of elastic polypropylene at uniaxial stretching

New metallocene catalysts applied to propylene polymerization expand the range of properties of polypropylene (PP), resulting in semi-crystalline materials having crystallinities below 60% up to X-ray amorphous highly elastic ones. To date the origin of the unique elastic mechanical behavior of such low crystalline PP is not completely understood. Therefore, the microscopic orientation of those PPs due to uniaxial stretching was investigated using wide-(WAXS) and small-angle X-ray scattering (SAXS). The aim of this study was to correlate these orientations or changes in the developed fiber textures with the macroscopic stress-strain behavior. This includes efforts to come closer to the main question of the nature of the physical cross-links in these not chemical cross-linked homopolymers, which is the reason for the high elastic behavior. Therefore, high molecular weight metallocene PPs showing different crystallinities (0-36%) were stretched to several elongations and the structural changes during the deformation were recorded by X-ray scattering. Stress-strain measurements show the great potential of these PPs as a thermoplastic rubber material. For quantitative analysis and discussion of the polymer chain orientations, the orientation functions were calculated. Correlations between the orientation functions and the stress-strain curves allow an interpretation of the macroscopic behavior on a microscopic scale. A higher cross-linking density in elongated samples indicates that the network, which is responsible for the elasticity, mainly built up by strain-induced morphology changes and chain orientations.     

Polyethylene networks crosslinked in solution: Preparation,elastic behavior,and oriented crystallization. I. Crosslinking in solution

A study has been made of the crosslinking of linear polyethylene in solution. Networks containing a low number of trapped entanglements and elastically ineffective chain ends were prepared by crosslinking high molecular weight linear polyethylene in 1,2,4-trichlorobenzene solutions with dicumyl peroxide at 120°C. No syneresis was observed during crosslinking except at high peroxide concentrations. The networks were characterized by swelling measurements, infrared spectroscopy, and differential scanning calorimetry. The crosslinking efficiency, calculated from swelling, was found to be proportional to the square of the polymer volume fraction. The proportionality constant was 0.8, indicating close to unit efficiency for undiluted polymer. Chemical modification of the polyethylene chains by attachment of peroxide and solvent fragments was of the order of one foreign unit per elastically active network chain, depending on peroxide and polymer concentration. Sol–gel analysis indicated that no chain scission occurred. These results are shown to be consistent with a “cage” mechanism for crosslinking. The possible topological consequence of this mechanism, preferential crosslinking of entanglements, is discussed. The concentration of trapped entanglements was also found to be proportional to the square of the polymer volume fraction. The proportionality constant corresponds to a molecular weight between entanglements of 4000 for the undiluted polymer, which is close to the value of 4200 found for networks prepared from the undiluted polymer. Since the results obtained are based mainly on the use of the swelling equation, different aspects of the applicability of this equation for the evaluation of the crosslinking process are discussed. As regards the reference dimensions, which should be known for a quantitative application of the elastic theory, the results strongly support the use of the dimensions of the network chains after completion of crosslinking.     

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

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

The dynamics of single chains within a model polymer melt

Discontinuous molecular dynamics simulations are performed on a system containing 32 hard chains of length 192 at a volume fraction of phi = 0.45 to explore the idea that localized entanglements have a significant effect on the dynamics of the individual chains within an entangled polymer melt. Anomalous behavior can still be observed when studying the dynamics of the individual chains, although increased time averaging causes the anomalous relaxation-memory-release behavior that was observed previously in the system to smooth out. First, the individual chain mean squared displacements and apparent diffusion coefficients are calculated, and a wide distribution of diffusive behavior is found. Although the apparent diffusion coefficient curve averaged over all chains displays the predicted long-time diffusive behavior, the curves for the individual chains differ both qualitatively and quantitatively. They display superdiffusive, diffusive, and subdiffusive behavior, with the largest percentage of chains exhibiting superdiffusive behavior and the smallest percentage exhibiting the predicted diffusive behavior. Next, the individual chain end-to-end vector autocorrelation functions and relaxation times are determined, and a wide distribution of stress relaxation behavior is found. The times when the end-to-end vector autocorrelation functions relax completely span almost an order of magnitude in reduced time. For some chains, the end-to-end vector autocorrelation function relaxes smoothly toward zero similar to the system average; however, for other chains the relaxation is slowed greatly, indicating the presence of additional entanglements. Almost half of the chains exhibit the anomalous behavior in the end-to-end vector autocorrelation function. Finally, the dynamic properties are displayed for a single chain exhibiting anomalous relaxation-memory-release behavior, supporting the idea that the relaxation-memory-release behavior is a single-chain property.     

Stochastic entangled chain dynamics of dense polymer solutions

We propose an adjustable-parameter-free, entangled chain dynamics model of dense polymer solutions. The model includes the self-consistent dynamics of molecular chains and solvent by describing the former via coarse-grained polymer dynamics that incorporate hydrodynamic interaction effects, and the latter via the forced Stokes equation. Real chain elasticity is modeled via the inclusion of a Pincus regime in the polymer's force-extension curve. Excluded volume effects are taken into account via the combined action of coarse-grained intermolecular potentials and explicit geometric tracking of chain entanglements. We demonstrate that entanglements are responsible for a new (compared to phantom chain dynamics), slow relaxation mode whose characteristic time scale agrees very well with experiment. Similarly good agreement between theory and experiment is also obtained for the equilibrium chain size. We develop methods for the solution of the model in periodic flow domains and apply them to the computation of entangled polymer solutions in equilibrium. We show that the number of entanglements Π agrees well with the number of entanglements expected on the basis of tube theory, satisfactorily reproducing the latter's scaling of Π with the polymer volume fraction φ. Our model predicts diminishing chain size with concentration, thus vindicating Flory's suggestion of excluded volume effects screening in dense solutions. The predicted scaling of chain size with φ is consistent with the heuristic, Flory theory based value.     

Microscopic theory of rubber elasticity

A microscopic integral equation theory of elasticity in polymer liquids and networks is developed which addresses the nonclassical problem of the consequences of interchain repulsive interactions and packing correlations on mechanical response. The theory predicts strain induced softening, and a nonclassical intermolecular contribution to the linear modulus. The latter is of the same magnitude as the classical single chain entropy contribution at low polymer concentrations, but becomes much more important in the melt state, and dominant as the isotropic-nematic liquid crystal phase transition is approached. Comparison of the calculated stress-strain curve and induced nematic order parameter with computer simulations show good agreement. A nearly quadratic dependence of the linear elastic modulus on segmental concentration is found, as well as a novel fractional power law dependence on degree of polymerization. Quantitative comparison of the theory with experiments on polydimethylsiloxane networks are presented and good agreement is found. However, a nonzero modulus in the long chain limit is not predicted since quenched chemical crosslinks and trapped entanglements are not explicitly taken into account. The theory is generalizable to treat the structure, thermodynamics and mechanical response of nematic elastomers.     

Entanglements and elasticity in polybutadiene networks

Polybutadiene networks were prepared by peroxide crosslinking of monodisperse 1,4-polybutadienes both in solution and in bulk. The effect of the entangled sol fraction on the elastic modulus of high-molecular-weight polybutadiene was observed in stress relaxation measurements. Sol fraction was shown to make a large contribution to the Mooney–Rivlin 2C₂ term. This effect was also observed on the molecular level in NMR spin-spin relaxation measurements. For networks crosslinked in bulk the stress relaxation measurements suggest the presence of trapped entanglements. The 2C₂ term is insensitive to sol extraction in these networks. NMR spin-lattice relaxation measurements in the rotating frame at 4.68 kHz verify the presence of additional effective crosslinks in these networks.     

Network architecture and modulus of miscible heteropolymer blends

The architecture and entanglement density of a multicomponent polymer network composed of miscible, flexible chains is related to the corresponding properties and concentrations of the pure components. It is assumed that the coupling frequency between different chain species is proportional to their fractional participation in the blend and that the entanglement ability of polymers can be affected by the presence of heteropolymer neighbors. The theory, equally applicable to temporary and permanent networks, estimates the molecular weight between entanglements and their total number along a chain in the blend. This information is used in establishing a mixing law for the rubbery plateau modulus of a fluid polymer blend and in deriving a relationship for the equilibrium modulus of an interpenetrating polymer network containing trapped entanglements and dangling segments. The theoretical predictions are compared with experimental results from the literature on several miscible polymer blends.     

Transient Changes in Topology and Energy on Extension of Polybutadiene Networks

The theory of elasticity of polymer networks has been developed along two lines. The phenomenological approach leads to the Mooney-Rivlin relation between stress and extension ratio for uniaxial extension. The statistical theory of elasticity, based on a model for polymer molecules, predicts a similar relation with one of the constants zero. Actual elastic properties of rubbers do not agree fully with either theory.

Experimental results are reported obtained with quantitatively cured polybutadiene and polyisoprene vulcanizates. These data are near-equilibrium results through the use of a cyclic stress sequence which largely eliminates the influence of long-time creep. The dependence of the initial modulus and the parameters of the Mooney-Rivlin relation on the chemical nature and the degree of branching of the polymer, the type of cross-links, and temperature has been investigated. A possible relation between the energy component of the elastic force and one of the parameters is discussed.

These results as well as those in the literature refer to irreversible processes. It is proposed that this irreversibility results from friction accompanying slippage of chain entanglements. This mechanism is compatible with the observed dependences. It is concluded that the variation of elastic properties with elongation is due to changes in network topography.

Some observations are made on the topological changes of vulcanizate networks at very high elongations. Similarities are pointed out between reinforcement by stress crystallization and by addition of carbon black. The effect of blacks is attributed mainly to preferential adsorption on the carbon particles of short network chains which become overstressed at high deformation. On adsorption the kinetic energy of these particular chains will be dissipated in the form of heat of adsorption.

Examples are given of the applicability of F. Bueche's relation between extension of the sample and that of the elastomer matrix in a filled vulcanizate. This equation differs from that based on Einstein's relation for the viscosity of suspensions, which has been shown to be applicable in other filled rubbers. The difference between the two relations may be associated with the absence or presence of chemical bonding of elastomer to filler.

Network topography has an influence on the ultimate properties of vulcanizates. Polybutadiene samples in which the cross-links are single bonds break at lower elongation than those with equal concentrations of cross-links consisting of 18-atom chains.     

Microscopic theory of orientational order, structure and thermodynamics in strained polymer liquids and networks

A microscopic integral equation theory of the segmental orientational order parameter, structural correlations and thermodynamics of strained polymer solutions, melts and networks has been developed. The nonclassical problem of the consequences of intermolecular excluded volume repulsions and chain connectivity is addressed. The theory makes several novel predictions, including effective power law dependences of the orientational order parameter on monomer concentration and chain degree of polymerization, and strain hardening of the bulk modulus. The predictions of a nearly classical strain dependence, and supralinear scaling with segment concentration, of the strain-induced nematic order parameter is in agreement with nuclear magnetic resonance experiments. The absolute magnitudes of the a priori calculated orientational order parameter agree with simulations and experiments to within a factor of 2. The possible complicating influence of "trapped entanglements" in crosslinked networks is discussed. Extensions of the theory are possible to treat the mechanical response of flexible polymer liquids and rubbers, and the structure, thermodynamics, and mechanical properties of strained liquid crystal forming polymers.     

Shear and extensional rheology of entangled polymer melts: Similarities and differences

This work extends our previous understanding concerning the nonlinear responses of entangled polymer solutions and melts to large external deformation in both simple shear and uniaxial extension. Many similarities have recently been identified for both step strain and startup continuous deformation, including elastic yielding, i.e., chain disentanglement after cessation of shear or extension, and emergence of a yield point during startup deformation that involves a deformation rate in excess of the dominant molecular relaxation rate. At a sufficiently high constant Hencky rate, uniaxial extension of an entangled melt is known to produce window-glass-like rupture. The present study provides evidence against the speculation that chain entanglements tie up into "dead knots" in constant-rate extension because of the exponentially growing chain stretching with time. In particular, it is shown that even Instron-style tensile stretching, i.e., extending a specimen by applying a constant velocity on both ends, results in rupture. Yet, in the same rate range, the same entangled melt only yields in simple shear, and the resulting shear banding is clearly not a characteristic of rupture. Thus, we conclude that chain entanglements respond to simple shear in the manner of yielding whereas uniaxial extension is rather effective in causing some entanglements to lock up, making it impossible for the entanglement network to yield at high rates.     

Synthesis and aggregation behavior of grafted maleic acid copolymers

Grafted SMA containing poly(styrene-co-maleic anhydride)-g-(poly(ethylene glycol) monomethyl ether) (SMA-PEG) and its hydrophobically modified products poly(styrene-co-maleic anhydride)-g-(poly(ethylene glycol) monomethyl ether & dodecyl) (SMA-PEG+C(12)) and poly(styrene-co-maleic anhydride)-g-(dodecyl) (SMA-C(12)) were prepared using a single batch method. Their adsorption and rheology behavior was investigated using equilibrium surface tension and rheological techniques. The adsorption parameters, saturation surface excess concentration (Γ(max)), and the minimum area (A(min)) of these copolymers were evaluated. The results show that Γ(max) increased and A(min) correspondingly decreased with increasing hydrophobicity. Aggregation standard free energy of SMA-PEG+C(12) and SMA-C(12) suggested that increased hydrophobicity enhanced the tendency for aggregation to occur. The distinctive differences in the macroscopic appearance were shown by aqueous samples of the copolymers. The samples of SMA-M behaved as Newtonian fluids at all concentrations (from 1.0 wt% to 20.0 wt%), indicating that there were no macromolecular chain entanglements or interactions between aggregates in solution. For SMA-PEG+C(12), at concentrations above 10.0 wt%, the presence of cross-links between aggregates is presumed to be the reason for the viscoelastic behavior. Solid-like elastic behavior could occur at low concentration (5.0 wt%) of SMA-C(12), suggesting the formation of networks by inter-chain aggregation of the hydrophobic dodecyl chains.     

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

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

阴离子疏水缔合共聚物/阳离子蠕虫状胶束协同增粘自组装网络

将阴离子疏水缔合丙烯酰胺共聚物P(NaAMC14S-b-AM)与阳离子蠕虫状胶束十六烷基三甲基溴化铵/水杨酸钠(CTAB/NaSal)在水溶液中自组装制备了新型的缔合增粘体. 由稳态剪切和动态流变实验结果得出: 自组装体系在80 ℃下仍具有显著的协同增粘效应, 其流变行为符合Maxwell模型. 同蠕虫状胶束相比, 自组装体系的稳态模量G0、力学松弛时间τR和缠结点密度ν都有增加, 由此分析缔合体系中两组分间形成了相互缠结的网络结构, 在链缠结处共聚物主链上的疏水侧链嵌入到了蠕虫状胶束的内核.     

高分子熔体和浓溶液流变性能的研究

基于多重缠结网络结构模型和高分子链上缠结点在流动中可进行动态解缠和再缠结的多重蠕动机理,用统计力学和动力学相结合的方法,分别计算出了缠结链组的末端距分布函数;处于缠结状态下高分子链构象统计分布函数;受力下聚合物熔体粘弹性形变自由能和解除外力下高分子挤出体可回复性粘弹性形变自由能,提出了高分子挤出体可回复形变的粘弹性分子理论。推导出的高分子熔体的回忆函数、简单剪切流下的本构方程和物料函数,并采用一种新的方法测定出物料的四种参数： η0、 GN0、 n′和 a。对于高分子挤出体,可回复性粘弹性形变由快速弹性形变和慢速粘弹性形变两者组成,当把两种形变量的复合结构参数－分子链的反式构象分数引入两种形变自由能表达式后,就从理论上得到了可回复形变量同挤出胀大比间的定量表达式,从而建立起一个具有分子链结构参数的新的挤出胀大比方程,可回复形变量同挤出条件（温度、挤出速率和量以及口模长径比不同的挤出机）和树脂结构特征（分子量及分布）的关系式以及在特殊情形下的简化表达式,并用几种高分子熔融体系的挤出胀大比和可回复性形变量的实验数据对理论进行验证,理论方程同实验数据较好的符合。     

Dual cross-linked networks hydrogels with unique swelling behavior and high mechanical strength: based on silica nanoparticle and hydrophobic association

A series of physically cross-linked hydrogels composed poly(acrylic acid) and octylphenol polyoxyethylene acrylate with high mechanical strength are reported here with dual cross-linked networks that formed by silica nanoparticles (SNs) and hydrophobic association micro-domains (HAMDs). Acrylic acid (AA) and octylphenol polyoxyethylene acrylate with 10 ethoxyl units (OP-10-AC) as basic monomers in situ graft from the SNs surface to build poly(acrylic acid) hydrophilic backbone chains with randomly distributed OP-10-AC hydrophobic side chains. The entanglements among grafted backbone polymer chains and hydrophobic branch architecture lead to the SNs and HAMDs play the role of physical cross-links for the hydrogels network structure. The rheological behavior and polymer concentration for gelation process are measured to examine the critical gelation conditions. The correlation of the polymer dual cross-linked networks with hydrogels swelling behavior, gel-to-sol phase transition, and mechanical strength are addressed, and the results imply that the unique dual cross-linking networks contribute the hydrogels distinctive swelling behavior and excellent tensile strength. The effects of SNs content, molecular weight of polymer backbone, and temperature on hydrogels properties are studied, and the results indicate that the physical hydrogel network integrity is depended on the SNs and HAMDs concentration.     

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.     

Effects of polymer chain length and stiffness on phase separation dynamics of semidilute polymer solution

Three-dimensional dissipative particle dynamics (DPD) simulations were performed to investigate the phase separation dynamics of semidilute polymer solutions with different polymer chain length and stiffness. For the polymer solution composed of shorter and more flexible chains, a crossover of the domain growth exponent from 1/3 to 2/3 was observed during the course of phase separation, indicating that the growth mechanism altered from diffusion to interfacial-tension driven flow. When the chain flexibility was kept the same but the chain was lengthened to allow for the chain entanglement to occur, the growth exponent changed to 1/4 in the diffusion-dominating coarsening regime while the growth exponent remained 2/3 in the flow-dominating regime. When the chain length was kept short but the stiffness was increased, the growth exponent became 1/6 in the diffusion-dominating regime and little effect was observed in the flow-dominating coarsening regime. The slow down of the phase separation dynamics in the diffusion-dominating coarsening could be explained by that the polymer chains could only perform wormlike movement when chain entanglements occurred or when the chain motion was limited by chain stiffness during phase separation. Moreover, when both the effects of chain length and stiffness were enhanced, polymer networks composed of longer and stiffer chains appeared and imposed an energy barrier for phase separation to occur. As a result, the polymer solution with stiffer and longer chains required a larger quench depth to initiate the phase separation and caused the delay in crossover of the coarsening mechanism from diffusion to flow.