Molecular models of polymer networks and constitutive equations of rubber elasticity

Equilibrium elastic behaviour of crosslinked polymers (networks) is analysed both from the macroscopic (phenomenological) and microscopic (molecular) points of view. Molecular mechanisms possibly responsible for non-linear rubber elasticity are discussed. They include: conformational entropy of network chains at small and large extensions, hindered rotation of such chains and intra-chain point interactions, dipole-dipole interactions between asymmetric chain segments, and topological constraints leading to chain entanglement.     

Network topology and the theory of rubber elasticity

The earliest investigations on rubber elasticity, commencing in the 19th century, were necessarily limited to phenomenological interpretations. The realisation that polymers consist of very long molecular chains. commencing c. 1930, gave impetus to the molecular theory of rubber elasticity (1932-). according to which the high deformability of an elastomer, and the elastic force generated by deformation, stem from the configurations accessible to long molecular chains. Theories of rubber elasticity put forward from 1934-1946 relied on the assumption that the junctions of the rubber network undergo displacements that are affine in macroscopic strain. The theory of James and Guth (1947) dispensed with this premise, and demonstrated instead that the mean positions of the junctions of a ‘phantom’ network consisting of Gaussian chains devoid of material properties are affine in the strain. The vital significance of the distinction between the actual distribution of chain vectors in a network and their distribution if the junctions would be fixed at their mean positions went unnoticed for nearly 30 years. Experimental investigations, commencing with the incisive work of Gee in 1946. revealed large departures from the relationship of stress to strain predicted by the theories cited. This discrepancy prompted extensive studies, theoretical and experimental, during succeeding years. Inquiry into the fundamentals of polymer networks, formed for example by interlinking very long polymer molecules, exposed the need to take account of network imperfections, typically consisting of chains attached at only one end to a network junction. Various means were advocated to make corrections for these imperfections. The cycle rank ζ of the network has been shown (1976) to be the fundamental measure of its connectivity, regardless of the junction functionality and pattern of imperfections. Often overlooked is the copious interpenetration of the chains comprising typical elastomeric networks. Theories that attempt to represent such networks on a lattice are incompatible with this universal feature. Moreover, the dense interpenetration of chains may limit the ability of junctions in real networks to accommodate the fluctuations envisaged in the theory of phantom networks. It was suggested in 1975 that departures from the form predicted for the elastic equation of state are due to constraints on the fluctuations of junctions whose effect diminishes with deformation and with dilation. Formulation of a self-consistent theory based on this suggestion required recognition of the non-affine connection between the chain vector distribution function and the macroscopic strain in a real network, which may partake of characteristics of a phantom network in some degree. Implementation of the idea was achieved through postulation of domains of constraint affecting the equilibrium distribution of fluctuations of network junctions from their mean positions. This led in due course to a theory that accounts for the relationship of stress to strain virtually throughout the ranges of strain accessible to measurement. The theory establishes connections between structure and elastic properties. This is achieved with utmost frugality in arbitrary parameters.     

单轴受压下水泥土全应力-应变关系曲线研究

为研究水泥土的力学性能和方便工程应用,采用内蒙古黄河灌区周边典型的粉质土配制水泥土,并进行了不同水泥掺入比5％,10％,15％和20％,不同养护龄期7d、14d、21 d、28 d、60d和90d下的无侧限抗压强度室内试验,得到了大量应力-应变关系曲线.通过采用Matlab可视化曲线拟合工具界面cftool进行分析,建立了一种全新的水泥土应力-应变关系的非线性模型,该模型采用logistic函数拟合曲线的上升段,采用复合双曲线函数拟合曲线的下降段.研究结果表明:该非线性模型简单实用,能够很好地与试验数据进行吻合,并较传统的非线性模型拟合效果更好.     

高强珊瑚混凝土(HSCC)单轴受压性能试验研究

骨料强度低、缺陷明显导致珊瑚混凝土难以应用于高强度或特殊防护结构。采用小粒径级配的珊瑚砂完全取代粗骨料制备高强珊瑚混凝土(HSCC),系统测试了HSCC的基本力学性能、单轴受压性能和孔隙率等。结果表明:优化骨料粒径和配合比制备的HSCC强度等级为C105,轴心抗压强度与立方体抗压强度的比值在0.86~0.94;未掺加纤维的HSCC失效模式为爆炸性破坏,掺加体积率为0.43%的聚丙烯纤维(PPF)的HSCC表现出明显的延性破坏特征,破坏后在20%~30%极限强度时进入残余强度转折点;HSCC孔隙率约在4.7%~6.9%,干表观密度为2 230~2 310 kg/m³,表面宏观孔洞较少,通过扫描电镜发现骨料边缘填充情况较好。     

The structure and elasticity of polyurethane networks: 1. Model networks of poly(oxypropylene) triols and diisocyanate

The equilibrium mechanical and optical behaviour of networks prepared from poly(oxypropylene) triols (PPT) and 4,4′-diphenylmethane diisocyanate (MDI) at various initial molar ratios of reactive groups, $\text{r}{}_{\mathrm{<mtext>H</mtext>}}\text{=}\text{[}\text{OH}\text{]}\text{[}\text{NCO}\text{]}$, in the range 0.6 < r_H < 1.75 have been investigated. The reaction proceeded at 80°C and in the presence of an organotin catalyst to a high conversion of minotirty groups (0.96–1.0). A comparison between experimental and theoretical (based on the theory of branching processes) dependences of the equilibrium modulus, G_e, on r_H or on the sol content, w_s, led to the following conclusions: (1) In the range r_H ? 1 the theory adequately describes experimental dependences, while in the range r_H < 1 excess crosslinking takes place, obviously due to the formation of allophanate groups. (2) In PPT networks with a higher molecular weight (M_n = 2630), G_e is about twice the theoretical value for the front factor A = 1, which is interpreted as a contribution due to permanent interchain interactions. (3) In networks of lower PPT (M_n = 708), experimental G_e values lie between the theoretical ones calculated for A = 1 (affine deformation crosslinks) and $\text{A =}\text{1}\text{3}$ (phantom network). (4) The difference between the eperimental and theoretical moduli for $\text{A =}\text{1}\text{3}$ may be adequately described by using Langley's concept of trapped entanglement contribution (for networks of the longer triol quite satisfactorily, for those of shorter PPT with some systematic deviation) with the same proportionality constant. (5) For correlations between the experiment and theory a generalized plot of the reduced modulus of the weight fraction of the gel proved to be useful.     

Modeling single chain elasticity of single-stranded DNA: A comparison of three models

The dominant models of polymers, i.e., the freely jointed chain (FJC) model, the wormlike chain (WLC) model and the freely rotating chain (FRC) model are modified by integrating the inherent single-molecule elasticity obtained from quantum mechanics (QM) ab-initio calculations. The QM modified models have been utilized to generate fitting curves for single-stranded DNA obtained in organic solvent. The analyses on the deviation between the fitting curve and the experimental force curve demonstrate that the QM-FRC and QM-FJC model are suitable for ssDNA, but not the QM-WLC model. We also find that one repeating unit of ssDNA is corresponding to a Kuhn segment in QM-FJC model or two rotating units in QM-FRC. Having close correlation to the inherent elasticity and real molecular structure of the polymer, QM-FJC and QM-FRC are emerging as structure relevant models.     

单轴压缩固化泥炭土的弹塑性损伤模型试验研究

为了研究固化泥炭土的力学性能,将机制砂作为填充材料,水泥和磷石膏作为胶结材料,针对昆明泥炭土开展了一系列固化试验。根据固化土体的无侧限抗压强度试验和单轴循环加-卸载试验,讨论了泥炭土在不同水泥掺量和磷石膏掺量下的固化效果。基于损伤理论和应变等效性假设,建立了单轴压缩状态下固化泥炭土的弹塑性损伤模型。研究结果表明:固化泥炭土抗压强度随水泥掺量的增加而增大,同时随着磷石膏掺量的增加,强度增长速率表现为先增大后减小的趋势。固化泥炭土的应力-应变滞回曲线呈下部不闭合的新月形,且在塑性开始和接近破坏阶段不闭合区域面积较大,说明这两个阶段产生较大的能量损耗。最后,考虑水泥掺量和磷石膏掺量的影响,根据试验结果采用曲线拟合的方法,得到了固化泥炭土弹塑性损伤本构模型的各项参数。     

Elasticity of real networks: Comparison of molecular theory with experiments

Predictions of the theory of elasticity of real networks are compared with results of experiments. The shape of the stress-strain curve for networks in the dry and swollen states and over wide ranges of strain, both in tension and compression, agrees with results of calculations based on the theory. The theory is also in close agreement with results of multiaxial stress-strain experiments and with the predictions of the degree of crosslinking obtained from measurements of the modulus. The theory may additionally be applied to the analysis of birefringence. The assumption of the linear additivity of the elastic and mixing free energies in a swollen network leads to results which are in agreement with experimental findings on different crosslinked systems.     

Deformation studies of polymers by time resolved Fourier transform infra-red spectroscopy: 1. Development of the technique

We have constructed a miniature closed-loop servo-controlled hydraulic tensile tester interfaced to our fast scanning Fourier transform infra-red spectrometer. Macroscopic information in the form of stress-strain curves can be collected along with the microstructural changes in the form of infra-red data. This technique in conjunction with our newly developed time resolved Fourier transform spectroscopy package has been used to follow structural changes in macromolecules.     

Polypropylene–rubber blends: 1. The effect of the matrix properties on the impact behaviour

The effect of matrix properties, i.e. crystallinity and molecular weight, on the impact behaviour of polypropylene–EPDM blends was studied. The blends were made on a twin-screw extruder. The impact strength was determined as a function of temperature, using a notched Izod impact test. The matrix crystallinity was varied by varying the matrix isotacticity, and ranged from 33 to 50 wt%.

With increasing temperature the polymers show a sharp brittle–ductile transition. This brittle–ductile transition temperature (T_bd) shifts to higher temperatures with increasing crystallinity of the polypropylene. However, the balance of properties and the modulus–T_bd relationship were better with blends made with higher crystalline PP.

The matrix molecular weight was decreased by treating a high molecular weight PP–EPDM (85/15 vol%) master blend with peroxide. In this way blends were obtained with a high MFI and a small rubber particle size. The matrix MFI of the blends thus obtained ranged from 2 to 30 dg min⁻¹. With decreasing matrix molecular weight the T_bd increased. The peroxide treated blends exhibited a considerably lower T_bd than comparable blends made in the standard way with a similarly small particle size. Peroxide treatment of a master blend is an effective method of preparing blends with a high MFI, small particle size and good ductility.     

The role of branching architecture in shape memory semi-IPNs: Shape memory effect and tube model analysis

The impact of branching architecture of one continuous uncrosslinked phase on properties of classic shape memory semi-interpenetrating polymer networks (semi-IPNs) was explored. Crosslinked poly (methyl methacrylate) (PMMA)/star-shaped polyethylene glycol (PEG) (PMMA/SPEG) semi-IPNs and PMMA/linear PEG (PMMA/LPEG) semi-IPNs were synthesized with the same PEG content. Mechanical properties, phase structure, thermal properties, dynamic mechanical properties, and shape memory properties of these two semi-IPNs systems were compared. Due to the better compatibility of SPEG in the PMMA network, which was derived from little crystallization compared with PMMA/LPEG semi-IPNs, PMMA/SPEG semi-IPNs exhibited a combination of large tensile strength and high elongation at break. PMMA/SPEG semi-IPNs, which had little crystallization exhibited superior shape recovery versus PMMA/LPEG semi-IPNs, which had more crystallization. Moreover, the higher the crystallinity in PMMA/PEG semi-IPNs was the worse long-term temporary shape retention. Based on tube model theory, the high shape recovery capacity of PMMA/SPEG semi-IPNs is mainly ascribed to the retraction of free PEG arms, which is entropically favorable and thermally activated due to the fluctuations of the path length. This result is supported by stress relaxation analysis and the influence of long shape fixity time on shape fixity ratio for these two systems.     

玄武岩复合材料筋增强ECC受拉性能及裂缝控制机理

工程水泥基复合材料(ECC)因其高韧性和多缝开裂特性成为研究热点,纤维复合材料(FRP)因具有抗拉强度高、密度小、耐腐蚀性好等优点而受到广泛关注。为研究玄武岩复合材料(BFRP)筋增强ECC(BFRP-ECC)的受拉性能以及筋材对基体的裂缝控制机理,考虑了基体类别和配筋率等因素,对ECC狗骨试件、BFRP-ECC和BFRP-砂浆薄板试件进行了单轴拉伸试验,同时借助数字图像相关法(DIC)技术获得了试件受拉过程中的全场应变和开裂状态,基于Richard的弹塑性应力应变公式提出了BFRP-ECC单轴受拉应力应变本构模型。结果表明：BFRP-ECC的极限拉应力随配筋率的增加而增大;ECC基体对复合材料的受拉性能增强效果优于砂浆基体,同时以ECC为基体的复合材料在裂缝间距和宽度控制上都明显优于以砂浆为基体的复合材料;BFRP筋能增加BFRP-ECC裂缝处的桥连应力,减小裂缝间距和宽度,增加裂缝数量。本文建立的BFRP-ECC单轴受拉应力应变本构模型与试验数据吻合良好,较好地反映了BFRP-ECC受拉应力应变关系。     

ABA triblock copolymer-based model networks in the bulk: Effect of the number of arms on microphase behavior

The microphase separation behavior in the strong segregation limit of model networks based on cross-linked ABA triblock copolymers in the bulk was investigated by calculating analytically the Gibbs free energies of the various possible morphologies: The spherical, the cylindrical, the lamellar, the reverse cylindrical, the reverse spherical and the disordered. In addition to the elastic and interfacial components, a free energy expression describing the interaction between the network junctions was also introduced. Free energy minimization allowed the construction of a phase diagram with axes the junction functionality and the composition. The phase diagram shows that an increase in the junction functionality leads to an expansion of the region of spheres at the expense of cylinders, and to an eventual elimination of reverse spheres in favor of reverse cylinders. The domain sizes for each morphology were also calculated analytically.     

A new organofunctional methoxysilane bilayer system for promoting adhesion of epoxidized rubber to zinc: Part 1: Optimization of practical adhesion

Practical adhesion of rubber to zinc cords is measured for various zinc silanization treatments. Aminopropyltrimethoxysilane (APS) alone or both APS and mercaptopropyltrimethoxysilane (γ-MPS) were used as coupling agents for zinc and epoxidized natural rubber (ENR). It is shown that special chemical and physical conditions are required to provide strong practical adhesion. With APS alone, best results are obtained when the zinc is silanized with 1% APS in isopropanol–water, cured at 110 °C for 1 h in air and vulcanized with ENR (20% epoxy groups) at 170 °C for 15 min. The adherence is further improved by double silanization with γ-MPS (0.5%) and then APS (1%). In both cases, the concentration of silanes, the cure temperature of the silane layers, the ageing and acidity of the silane solutions are the main parameters which must be thoroughly optimized. In the case of double silanization, the highest practical adhesion seems to be correlated with a true γ-MPS/APS bilayer structure which is achieved for a specific application of the γ-MPS. It is suggested that this layer is bound to zinc through its thiol function, and the APS layer (on the γ-MPS layer) is cross-linked to γ-MPS and ENR through silanol and epoxy groups, respectively. When the silanized zinc surface was vulcanized and the rubber then peeled off the zinc surface, XPS analysis of the bare zinc areas indicates a rupture characteristic of an adhesive failure in the case of silanization by APS alone, and a cohesive failure in the case of the double silanization.     

Analysis of variation of molecular parameters of natural rubber during vulcanization in conformational tube model. II. Influence of sulfur/accelerator ratio

A natural rubber (NR) with a conventional sulfur cure system and a ratio of sulfur/accelerator (Ω) equal to 3 was investigated. The network structure of the NR during vulcanization was analyzed using a model of rubber elasticity based on the tube concept, which was applied to the treatment of the stress–strain measurements. The influence of cure time and temperature on the chemical crosslinks density was analyzed. The values were compared with those obtained by means of an equilibrium volume swelling measurement. The differences between samples of NR cured with Ω = 3 and 1.5 were analyzed. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 2747–2755, 1999     

Experiments on the small-strain behaviour of crosslinked natural rubber: 2. Extension and compression

Experiments were carried out to characterize the small-strain tension and compression behaviour of dicumyl peroxide crosslinked natural rubber. Strains that were smaller by an order of magnitude than any reported previously on natural rubber were achieved. Our results support the contention that the compression and extension moduli of natural rubber are different. A new finding is reported: that is, the moduli in tension and compression do not become constant but rather they increase significantly as zero deformation is approached.     

Polyurethane-polysiloxane interpenetrating polymer networks: 1. A polyether urethane-poly(dimethylsiloxane) system

The synthesis and properties of a polyether urethane network based on Adiprene L-100, of a poly(dimethylsiloxane) network and of nine interpenetrating polymer networks based on these polymers were investigated. To form the latter materials, the prepolymers were mixed and crosslinked simultaneously, but by separate mechanisms. Comparison of the network solubility parameters suggested marked incompatibility. Optical microscopy, dynamic mechanical analysis and the tensile testing indicated gross phase separation. From 90 to 50% of the polyether urethane component, this network was continuous and the poly(dimethylsiloxane) was present as dispersed phases. From 40 to 10% of the polyether urethane, the situation was reversed. Some degree of interchain mixing at phase boundaries was detected by ¹³C nuclear magnetic resonance spectroscopy.     

On Mooney and Mooney stress relaxation. II. A nonlinear viscoelastic description: Experimental and numerical results

It has been investigated whether the stress build‐up and the stress relaxation involved in a Mooney test, with subsequent Mooney stress relaxation, can be described by nonlinear viscoelastic theory, more particularly the Wagner constitutive model. For this purpose, the viscoelastic behavior of three nonvulcanized EPDM materials, with similar Mooney viscosity but varying elasticity, has been studied. Relaxation time spectra were obtained from dynamic mechanical experiments, from which the step‐strain stress‐relaxation modulus was calculated. Stress build‐up experiments were performed with a cone and plate system in order to obtain the so‐called damping function (a measure for the deformation sensitivity) of the materials. Using these material functions, the Mooney test was successfully described with the Wagner constitutive model. Experimental and theoretical Mooney stress‐relaxation rates are in close agreement. The predicted Mooney viscosity is up to 25% lower than the measured value. This may be due to nonideal conditions during the Mooney test, such as inhomogeneous heating and secondary flows, and to inaccuracy of the damping function. The model calculations confirm the strong experimental dependence of Mooney measurements on small variations in instrumental conditions such as geometry, rotation speed, and so forth. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 1220–1233, 1999     

Rubber as additives to lower thermal expansion coefficient of plastics: 1. Morphology and properties

A substantial approach was proposed to design polymer alloys with very low coefficient of linear thermal expansion (CLTE). The large reduction in CLTE is not based on an addition of low-thermal-expansion filler to suppress the bulk expansion, but on the fine control of the polymer alloy's micro-morphology so that the expansion is preferentially toward the thickness direction. In this study, rubber was used as an additive to tune the thermal expansion behavior of various plastics. Although the rubber has a high thermal expansion coefficient, it was found that, when the rubber domains are deformed into microlayers and co-continuous with the plastic matrix, the CLTE of the polymer alloy parallel to the microlayer directions could be reduced to a very low level. Various influencing factors including rubber concentration, viscosity ratio, interfacial adhesion as well as the domain size were investigated.