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.     

Polyamide 6,6 fibers: Tensile deformation behavior and chain scission

The strain dependence of the elastic, anelastic and plastic components of deformation energy was determined by means of cyclic stress-strain experiments for a set of polyamide 6,6 fibers obtained by different processing techniques. Small angle X-ray scattering experiments revealed that the deformation of the supermolecular lattices of the fibers, consisting of crystalline lamellae and amorphous regions, was identical to the macroscopic deformation of the sample.ESR experiments showed that deformation gives rise to chain rupture events obviously occurring in the amorphous regions in all fibers above a critical strain level. The strain dependence of the free radical concentration was found to agree closely with the corresponding behavior of the plastic deformation energy. This indicates that chain rupture events influence stress-strain properties, particularly at large strains. The absolute values of the experimentally determined plastic deformation energy and of the theoretical value, however, calculated from the number and energy balance of ruptured chains, disagree strongly. Possible explanations are free radical recombinations and secondary dissipative processes resulting from chain rupture.     

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.     

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.     

基于弹性统计理论的硅橡胶纳米复合材料本构关系的建立

基于考虑了悬垂链的橡胶弹性统计模型,通过引入应变放大因子,建立了硅橡胶纳米复合材料的基于微观机制的本构关系,其中利用硅橡胶分子信息(分子量M、乙烯基含量wt_(Vi)%)、乙烯基反应程度(q)估算获得本构方程中的交联点间链段分子量(Mc),网络链(network strands)体积分数(Φ)等参数,通过拟合确定了与纳米粒子相关的部分参数(初始应变放大因子X_0,极限应变放大因子X_∞,衰减因子z),对掺杂白炭黑的单组分及长短链配合硅橡胶拉伸应力-应变数据进行拟合,在采用相同X_∞,z值情形下,拟合曲线仍能与实测值符合较好(拟合的Adj.R-Square值分别为0.99576、0.99596)。基于微观物理机制的本构关系能够成为联系微观分子结构参数与宏观应力的桥梁,本文工作有望为更有针对性地改进和优化硅橡胶的性能提供依据。     

Conformational elasticity theory of chain molecules

This paper develops a conformational elasticity theory of chain molecules, which is based on three key points: (i) the molecular model is the rotational isomeric state (RIS) model; (ii) the conformational distribution function of a chain molecule is described by a function of two variables, the end-to-end distance of a chain conformation and the energy of the conformation; (iii) the rule of changes in the chain conformational states during deformation is that a number of chain conformations would vanish. The ideal deformation behavior calculated by the theory shows that the change in chain conformations is physically able to make the upward curvature of the stress-strain curve at the large-scale deformation of natural rubber. With the theory, different deformation behaviors between polymers with different chemical structures can be described, the energy term of the stress in the deformations can be predicted, and for natural rubber the fraction of the energy term is around 13%, coinciding with the experi     

Monte Carlo calculations on polypeptide chains. VIII. Distribution functions for the end-to-end distance and radius of gyration for hard-sphere models of randomly coiling poly(glycine) and poly(L-alanine).

A Monte Carlo study of the distribution functions for the end-to-end distance and radius of gyration for hard-sphere models of poly(glycine) and poly(L-alanine) random coils has been conducted in the chain-length range n = 3 to 100 monomer units for both unperturbed chains and chains perturbed by long-range interactions (excluded volume effects). The distribution functions for the radius of gyration in all cases have been very precisely calculated, those for the perturbed end-to-end distance less precisely, and those for the unperturbed end-to-end distance least precisely. Empirical distribution functions of the form W(p) = ap-b exp(-cp-d) for the reduced end-to-end distance p = r/"r-2"-one-half and a similar form for the reduced radius of gyration could be least-squares fit to the Monte Carlo data. The expansion factors alpha-r and alpha-s were calculated vs. chain length and were used to test various versions of the two-parameter theory of the excluded volume effect. To be consistent with the chain-length dependence of alpha-r and alpha-s as determined by the Monte Carlo calculations, each of these theories required two different binary cluster integrals, a beta-r based on alpha-r and a beta-s based on alpha-s, both of which were strongly chain-length dependent. Both of these results suggest that the two-parameter theory is not applicable to the models used in this study. It was also found that, except for very short chain lengths, plots of ln alphs-r vs. ln n were linear, and thus that alpha-r could be estimated for long chain lengths. Comparison of these estimates with the experimental data on four polypeptide chains in one-earth solvents that the hard-sphere models used in this study yield expansion factors that do not seriously overestimate the magnitude of the excluded volume effect.     

Nonlinear elasticity of rodlike macromolecules in condensed state

The finite deformation elasticity of gels of stiff rodlike polymer chains is discussed theoretically. Unlike a rubber composed of flexible polymer chains, the elasticity of this system is energetic, and arises from the elastic deformation of the stiff chains which are forced to bend under macroscopic strain. The stress-strain curve of this system is shown to be nonlinear even if the bending of the rods is small, and has a characteristic S-shaped form which is unlike that of a rubber. The effect of orientational order is also discussed.     

Relation between Energy Contribution to Elastic Force and Strain Dependence of Modulus for Polybutadiene Vulcanizates

It is well known that the modulus G = r/(λ - λ^?2) varies with deformation, thus deviating from the predictions of statistical theories of rubber elasticity which require it to be constant. It has also been found that there is a nonnegligible energy contribution to the elastic force. It is postulated that these two phenomena are related because both arise from energetic interaction between chains.

Based on the lateral order of chains indicated by x-ray fiber diagrams of elongated noncrystallizable elastomers, it is suggested that energetic interaction of chains is induced by strain orientation. Proportionality between these two is assumed. The orientation distribution functions of end-to-end vectors and of statistical chain segments are considered. The proportionality constants are determined from the energy contribution to the strain dependence of the coefficient of thermal expansion. With the aid of these constants the modulus, corrected for energy contribution, is calculated. The observed and calculated elongation dependence of G agree reasonably well.

It is concluded that an energy interaction between aligned chains can account for the deviation of the observed stress elongation relation from the predictions of entropy elasticity theories.     

Conformational Entropy as a Means to Control the Behavior of Poly(diketoenamine) Vitrimers In and Out of Equilibrium

Control of equilibrium and non‐equilibrium thermomechanical behavior of poly(diketoenamine) vitrimers is shown by incorporating linear polymer segments varying in molecular weight (MW) and conformational degrees of freedom into the dynamic covalent network. While increasing MW of linear segments yields a lower storage modulus at the rubbery plateau after softening above the glass transition (T_g), both T_g and the characteristic time of stress relaxation are independently governed by the conformational entropy of the embodied linear segments. Activation energies for bond exchange in the solid state are lower for networks incorporating flexible chains; the network topology freezing temperature decreases with increasing MW of flexible linear segments but increases with increasing MW of stiff segments. Vitrimer reconfigurability is therefore influenced not only by the energetics of bond exchange for a given network density, but also the entropy of polymer chains within the network.     

Kinetics of interior loop formation in semiflexible chains

Loop formation between monomers in the interior of semiflexible chains describes elementary events in biomolecular folding and DNA bending. We calculate analytically the interior distance distribution function for semiflexible chains using a mean field approach. Using the potential of mean force derived from the distance distribution function we present a simple expression for the kinetics of interior looping by adopting Kramers theory. For the parameters, that are appropriate for DNA, the theoretical predictions in comparison with the case are in excellent agreement with explicit Brownian dynamics simulations of wormlike chain (WLC) model. The interior looping times (tauIC) can be greatly altered in the cases when the stiffness of the loop differs from that of the dangling ends. If the dangling end is stiffer than the loop then tauIC increases for the case of the WLC with uniform persistence length. In contrast, attachment of flexible dangling ends enhances rate of interior loop formation. The theory also shows that if the monomers are charged and interact via screened Coulomb potential then both the cyclization (tauc) and interior looping (tauIC) times greatly increase at low ionic concentration. Because both tauc and tauIC are determined essentially by the effective persistence length [lp(R)] we computed lp(R) by varying the range of the repulsive interaction between the monomers. For short range interactions lp(R) nearly coincides with the bare persistence length which is determined largely by the backbone chain connectivity. This finding rationalizes the efficacy of describing a number of experimental observations (response of biopolymers to force and cyclization kinetics) in biomolecules using WLC model with an effective persistence length.     

聚乙烯型尾形链构象统计的Monte Carlo研究

以聚乙烯型尾形链为例 ,根据链的实际键长、键角和Θ条件下每一链段处于不同旋转异构态的条件概率 ,用MonteCarlo模拟方法生成样本链分子 ,计算了尾形链的均方末端矩及其分量、构象数 ,并与完全计算法及理论推导值作了比较 ,讨论了尾形链构象数和沿边界方向及法线方向均方末端矩分量随链段数n及键角Φ的变化 .     

Computer simulation study of the global phase behavior of linear rigid Lennard-Jones chain molecules: comparison with flexible models

The global phase behavior (i.e., vapor-liquid and fluid-solid equilibria) of rigid linear Lennard-Jones (LJ) chain molecules is studied. The phase diagrams for three-center and five-center rigid model molecules are obtained by computer simulation. The segment-segment bond lengths are L = sigma, so that models of tangent monomers are considered in this study. The vapor-liquid equilibrium conditions are obtained using the Gibbs ensemble Monte Carlo method and by performing isobaric-isothermal NPT calculations at zero pressure. The phase envelopes and critical conditions are compared with those of flexible LJ molecules of tangent segments. An increase in the critical temperature of linear rigid chains with respect to their flexible counterparts is observed. In the limit of infinitely long chains the critical temperature of linear rigid LJ chains of tangent segments seems to be higher than that of flexible LJ chains. The solid-fluid equilibrium is obtained by Gibbs-Duhem integration, and by performing NPT simulations at zero pressure. A stabilization of the solid phase, an increase in the triple-point temperature, and a widening of the transition region are observed for linear rigid chains when compared to flexible chains with the same number of segments. The triple-point temperature of linear rigid LJ chains increases dramatically with chain length. The results of this work suggest that the fluid-vapor transition could be metastable with respect to the fluid-solid transition for chains with more than six LJ monomer units.     

Novel amphiphilic network polymers consisting of nonpolar,short primary polymer chains and polar,long crosslink units: Influence of characteristic dangling chains on swelling behavior of resulting amphiphilic gels

Novel amphiphilic network polymers consisting of nonpolar, short primary polymer chains and polar, long crosslink units were prepared, and the swelling behavior of resulting amphiphilic gels is discussed by focusing on the influence of characteristic dangling chains; that is, benzyl methacrylate (BzMA) was copolymerized with tricosaethylene glycol dimethacrylate [CH₂?C(CH₃)CO(OCH₂CH₂)₂₃OCOC(CH₃)?CH₂, PEGDMA‐23] in the presence of lauryl mercaptan as a chain‐transfer agent because BzMA forms nonpolar, short primary polymer chains and PEGDMA‐23 as a crosslinker contains a polar, long poly(oxyethylene) unit. The enhanced incorporation of dangling chains into the network polymer was brought by shortening the primary polymer chain length, and copolymerization with methoxytricosaethylene glycol methacrylate, a mono‐ene counterpart of PEGDMA‐23, enforced the incorporation of flexible dangling poly(oxyethylene) chains into the network polymer, although the former dangling chains as terminal parts of primary poly(BzMA) chains were rather rigid. Then, the influence of characteristic dangling chains on the swelling behavior of amphiphilic gels was examined in mixed solvents consisting of nonpolar t‐butylbenzene and polar methanol. The profiles of the solvent‐component dependencies of the swelling ratios were characteristic of amphiphilic gels. The introduction of dangling poly(oxyethylene) chains led not only to an increased swelling ratio but also to sharpened swelling behavior of amphiphilic gels. The swelling response of amphiphilic gels was checked by changing the external solvent polarity. The dangling chains with freely mobile end segments influenced the swelling response of gels. The amphiphilic gels with less entangled, collapsed crosslink units exhibited faster swelling response than the ones with more entangled, collapsed primary polymer chains. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2192–2201, 2004     

Spatial self-organization of comb macromolecules

The properties and the self-assembly of single comb macromolecules in solution were studied. The elastic properties of a polymer chain with a high density of side chains forming a cylindrical brush were discussed, in particular, its persistence length was calculated. The cases of brushes with flexible and rigid side chains, as well as brushes with two types of incompatible side segments, were considered. It was shown that brushes with rodlike dangling chains have a higher rigidity. In addition, a comb macromolecule with the hydrophobic main chain and hydrophilic side chains was considered. Such a macromolecules in a selective solvent forms a globule with the hydrophobic core and a soluble shell. The specific feature of the globule is its ability to acquire nonspherical spatial forms. Problems related to the stability and transformation of globule shape are discussed in detail.     

Vector length distributions and polarizabilities of effective polymer chains: a statistical segment approach

The freely orienting model of a polymer chain is generalized by considering the distribution of vector lengths and polarizabilities of the statistical segments in the chain with a constant number of skeletal bonds in each of the segments The bonds in the segments are assumed to exist in their RIS (Rotational Isomeric States) conformations. The segment is characterized, i.e., its end-to-end length and polarizability distributions are computed. Bond polarizabilities, as determined by Denbigh, have been used for polyethylene and poly(cis-isoprene), and are assumed to be independent of the environment. Two methods are used to compute the chain length distribution from the length distribution of statistical segments: (i) an exact method, using a modified version of Chandrasekhar’s approach, originally formulated for chains of segments having constant length; and (ii) an alternative approach, which considers the series expansion of the Helmholtz Free Energy of an isolated chain, making the analysis computationally more viable without significant loss in accuracy. The averages of the chain end-to-end length distributions have been computed at 373 K for poly(cis-isoprene) and at 403 and 413 K for polyethylene. Also, chain polarizability is determined from the distribution of statistical segment lengths and polarizabilities. The results are in a form that can be used to obtain stress-deformation and optical anisotropy-deformation relationships of assemblies of chains, such as crosslinked networks.     

平面壁限制的高分子链末端距及其概率分布

The problem of polymer chains near an impenetrable plane is investigated by means of the probability method. It is shown that the 2kth moment of the reduced normal component of the end-to-end distance A2k only depends on the reduced distance to the plane of the first segment AZ0, here, A=l－ 1· , n is the chain length, l is the bond length and fixed to be unity, which can be expressed as A2k=f(AZ0). When AZ0≈ 0, A2k is the maximum(A2k=k!), then it decreases rapidly and soon reaches the minimum with the increase of AZ0, afterwards A2k goes up gradually and reaches the limit value [(2k－ 1)× (2k－ 3)× … × 1]/2k when AZ0 is large enough. Suggesting that the polymer chain can be significantly elongated for small Z0 and contracted for an intermediate range of Z0 due to the barrier. The distribution of the end-to-end distance also depends on the distance Z0 to the plane of the first segment.     

石墨烯/聚乙烯复合材料及其拉伸性能的分子动力学模拟

利用分子动力学方法,模拟石墨烯/聚乙烯复合材料的微观结构和性能,并采用单轴拉伸模拟方法研究石墨烯/聚乙烯复合材料的拉伸性能.结果表明,在石墨烯/聚乙烯复合材料平衡构型中,聚乙烯基体分子在石墨烯表面处形成多层吸附层,吸附层处于动态稳定状态,层内分子可以发生扩散迁移.吸附层内聚乙烯分子发生"吸附固化"现象,分子弯曲程度减弱,发生有序排列,且在垂直于石墨烯方向的运动性能受到抑制.拉伸模拟结果表明,石墨烯能够提高聚乙烯材料的拉伸性能.在弹性区和屈服区,石墨烯阻碍了复合材料在垂直于拉伸方向的压缩变形,聚乙烯分子"吸附固化"结构保持稳定,引起体系整体应力的迅速升高.在软化区,由于石墨烯发生剧烈弯曲,"吸附固化"结构发生破坏,最终引起体系应力迅速减小.在弹性区和屈服区,体系应变主要引起了非键相互作用的改变.在软化区之后,应变主要导致了体系内分子成键相互作用的改变.应变速率能够提高复合材料的屈服应力,而不改变复合材料应力应变的整体趋势.     

Interactions between polymer brush-coated spherical nanoparticles: the good solvent case

The interaction between two spherical polymer brushes is studied by molecular dynamics simulation varying both the radius of the spherical particles and their distance, as well as the grafting density and the chain length of the end-grafted flexible polymer chains. A coarse-grained bead-spring model is used to describe the macromolecules, and purely repulsive monomer-monomer interactions are taken throughout, restricting the study to the good solvent limit. Both the potential of mean force between the particles as a function of their distance is computed, for various choices of the parameters mentioned above, and the structural characteristics are discussed (density profiles, average end-to-end distance of the grafted chains, etc.). When the nanoparticles approach very closely, some chains need to be squeezed out into the tangent plane in between the particles, causing a very steep rise of the repulsive interaction energy between the particles. We consider as a complementary method the density functional theory approach. We find that the quantitative accuracy of the density functional theory is limited to large nanoparticle separation and short chain length. A brief comparison to Flory theory and related work on other models also is presented.     

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.