Experimental studies and molecular modelling of the stress-optical and stress-strain behaviour of poly(ethylene terephthalate). Part II: Molecular modelling of birefringence of poly(ethylene terephthalate) networks

The Monte-Carlo (MC) approach of Paper I is developed to predict the birefringence of PET. An extension of the modelling of polarisability is used that accounts for chain flexibility and for structural units containing several types of bonds. The rotational-isomeric-state (RIS) model for PET chains in melts is employed to calculate the polarisability of the terephthaloyl segment and of each of the 27 possible conformations of the five skeletal bonds of the glycol segment. These polarisabilities then enable the birefringent properties of drawn PET melts to be predicted. A method of pre-averaging the individual glycol polarisabilities that greatly reduces the length of the calculation is shown to be valid.It is found that shorter PET chains produce higher values of birefringence (Δñ) for a given deformation. This trend is due to greater proportions of the chains reaching higher conformational extensions and therefore becoming more oriented. In disagreement with Kuhn and Grün theory, Δñ is not linearly related to λ² − λ⁻¹. This non-linear behaviour is related to the non-linear behaviour of the orientation functions of the terephthaloyl and glycol segments, 〈P₂(cos ζ_ter)〉 and 〈P₂(cos ζ_gly)〉, with λ² − λ⁻¹. The non-linear behaviour of 〈P₂(cos ζ_ter)〉 with λ² − λ⁻¹ was confirmed experimentally in Paper I, where the measured values of 〈P₂(cos ζ_ter)〉 were found to be closely predicted by the present MC modelling.In the present paper, the predicted values of Δñ, calculated according to various published values of bond polarisabilities are presented and discussed. They will be used in Paper III to model the measured birefringence of drawn PET.     

Experimental characterisation and further molecular modelling of the conformational and elastic behaviour of poly(ethylene terephthalate) network chains

The Monte-Carlo (MC) method developed to model the elastomeric stress-strain behaviour of polyethylene (PE), poly(dimethyl siloxane) (PDMS) and poly(ethylene terephthalate) (PET) networks and the stress-optical behaviour of PE networks is now developed to investigate further the stress-strain behaviour of PET networks. Accurate infrared (IR) spectrometry measurements have been used to determine the populations of gauche conformers in the glycol residues of PET chains in melts. The proportion of gauche states was found to be 76%, consistent with the rotational energy difference of −4.16 kJ mol⁻¹ between trans and gauche states used previously.The greater conformational flexibility of the PET chain compared with the PE chain leads to lower network moduli and smaller deviations from Gaussian and affine network behaviour. Previous results are briefly reviewed and new comparisons of the elastic behaviour of PET and PE chains are made using normalised plots. Subsequent publications will apply the present results to interpreting the measured stress-strain and the stress-optical properties of entangled PET melts.     

Molecular modelling of the elastic behaviour of poly(ethylene terephthalate) network chains

The Monte-Carlo (MC) method developed to model the elastomeric stress-strain behaviour of polyethylene (PE) and poly(dimethyl siloxane) (PDMS) networks and the stress-optical behaviour of PE networks is now applied to the stress-strain behaviour of poly(ethylene terephthalate) (PET) networks. In keeping with the previous results for PE and PDMS networks, increases in the proportions of fully extended chains with macroscopic deformation are found to give rise to steady decreases in the rates of Helmholtz energy changes, causing reductions in moduli at moderate macroscopic deformations. There is no need to invoke a transition from affine to phantom chain behaviour as deformation increases.By using rotational-isomeric-state (RIS) models of the network chains and the MC method, stress-strain behaviour can be related to chemical structure. In this respect, the greater conformational flexibility of the PET chain leads to lower network moduli and smaller deviations from Gaussian network behaviour than for PE networks. In addition, the stiff, aromatic section of the PET repeat unit structure is seen to endow particular characteristics on the end-to-end distribution functions of PET chains. These characteristics are taken fully into account in evaluating the elastomeric properties of the PET networks. Subsequent publications will apply the present results to interpreting the measured stress-strain and the stress-optical properties of entangled PET melts.     

Experimental studies and molecular modelling of the stress-optical and stress-strain behaviour of poly(ethylene terephthalate). Part I: Infra-red spectroscopic investigation and modelling of chain conformation and orientation changes on drawing

Quantitative infra-red data on oriented poly(ethylene terephthalate) (PET) films have been used to determine the changes in the proportions of the trans and gauche conformers of the glycol residues and the development of molecular orientation of the terephthaloyl groups as functions of draw ratio. The results are compared with predictions for the stretching of a rubber-like network based on rotational-isomeric-state (RIS) Monte Carlo (MC) modelling. It is shown that both sets of data are consistent with stretching an entangled molecular network of about 10 PET monomer units per network chain. The onset of crystallisation at a draw ratio of 2.5 affects the glycol trans-gauche conformer contents but has no detectable effect on terephthaloyl orientation.     

Molecular modelling of the stress-strain behaviour of elastomers in pure shear

It is demonstrated that experimental stress-strain data for several types of polymer network under pure shear display an approximately universal behaviour. The Monte-Carlo network-modelling method of Stepto, Taylor and Cail is extended to treat stress-strain behaviour in pure shear and its predictions are in good agreement with the experimental data. The predictions of Gaussian network theory are shown to be seriously in error.     

Molecular Modelling of the Conformational Flexibility and Elastic Behaviour of Short Poly(ethylene terephthalate) Chains

The Monte-Carlo (MC) rotational-isomeric-state (RIS) method developed previously to model the stress–strain behaviour of poly(ethylene terephthalate) (PET) is now applied to short PET chains of between 42 and 84 skeletal bonds. The effects on the radial and probability density distribution functions of the long, flexible virtual bond used to represent the terephthaloyl unit are investigated. The distribution functions generated, based on finite samples of chains, are found to be discontinuous with subsidiary maxima in their tails. The discontinuities lead to uncertainties in the simulated network elasticity properties and, in order to reduce the uncertainty, it is necessary to truncate the distributions at the values of r where they first become discontinuous. The stress–strain properties calculated from the truncated distributions are shown to be consistent with those presented previously for longer PET chains, for which discontinuities in the sampled functions are not apparent.     

Error analysis and modelling of non-linear stress-strain behaviour in measuring dynamic mechanical properties of polymers with the Rheovibron

The Rheovibron is widely used for measuring the dynamic mechanical properties of polymers. Poor reproducibility and data variation which have been considered inherent to the Rheovibron operation necessitate both instrument modifications and modelling of the instrument operation for quantitative interpretation of the experimental data. The instrument modifications that have been made in this study include electronic modifications, a new clamp design that eliminates sample mounting and slippage problems, and the design of an environmental chamber, which, along with a nitrogen purge, eliminates moisture condensation on the sample at low temperatures. A previously-developed mathematical model for the instrument has also been extended to account for the effect of the static stress and the stress rate imposed on the sample during measurement. Based on a simple constitutive relation for non-linear stress-strain behaviour of polymers, this model has been applied to a number of polymeric materials including polyethylene, polyimide, poly (ethylene terephthalate), and polypropylene, among others. The instrument modifications and modelling of the influence of static stress and the stress rate imposed on the sample during dynamic mechanical testing allows for reproducible (within 5%) and accurate use of the Rheovibron.     

Mechanical and optical properties of block polymers I. Polyester-urethanes

The extraordinary physical properties exhibited by block polymers have been ascribed to the presence of a multiphase microstructure in which the higher modulus phase acts as a quasi-crosslink or filler particle. Recently, the optical examination of these materials combined with mechanical testing has given investigatiors new insight into this complex morphology. In this investigation, simultaneous stress, strain, and birefringence data have been collected on a block polyester urethane elastomer. Stress-softening is observed in cyclic stress-strain experiments, giving rise to significant hysteresis in the stressstrain and birefringence-stress curves. In these tests, prestrain causes a large increase in the stress-optical coefficient, but has little effect on the strain-optical coeffcient. As the temperature is increased, the strain-optical coefficent decreases while the stress-optical coefficient increases with the latter exhibiting a larger temperature dependence than predicted by the kinetic theory of rubber elasticity. The good agreement between the mechanical-optical response of polyester urethanes and that of other block polymer systems provides further evidence of their morphological similarity.     

Mechanical response of amorphous and semicrystalline poly(ethylene terephthalate) and modelling in frame of quasi point defect theory

Abstract

The thermomechanical activation of deformation in amorphous and (38%crystalline) semicrystalline poly(ethylene terephthalate) (PET) has been investigated using dynamic mechanical tests and large strain experiments. Activation parameters for both materials are determined in the neighbourhood of the glass transition temperature using stress relaxation experiments. The complete mechanical behaviour of amorphous PET is then analysed in the frame of a molecular model or ‘quasi point defect’ theory. With this aim, a new method is proposed, based on only three isochronal measurements, and leading to the determination of the set parameters. This method makes it possible to reproduce the stress–strain curve over a range of temperatures, as well as the relevant activation parameters. Finally some qualitative explanations are given for the mechanical behaviour of semicrystalline PET.     

A rheo-optical analysis of converging wedge flow for estimation of stress-optical coefficient

Results are presented from a rheo-optical study of a poly(dimethyl siloxane) (PDMS) fluid in a converging wedge flow cell at room temperature for estimation of the linear stress-optical coefficient. The linearity of the stress and polarizability or refractive index tensors were checked by flow birefringence measurements in pure extensional flow along the centerline of the converging flow cell that extended well into the non-Newtonian region. The stresses were evaluated using (a) shear and extension rates from laser Doppler anemometry (LDA) velocity measurements and (b) information obtained from orientation angle as a test of constitutive equations. All measurements were optical (i.e., non-intrusive), and a linear stress-optical coefficient of 1.475 × 10⁻¹⁰ Pa⁻¹ was obtained using a two-term Goddard-Miller model for room temperature flow of a PDMS fluid in combined shear and extension in a converging wedge flow cell.     

Experimental and modelling of supercritical oil extraction from rapeseeds and sunflower seeds

The supercritical oil extraction from oleaginous seeds (sunflower and rapeseeds) is presented here through experimental and modelling results. The experimental setup allows an accurate following of the mass of the oil extracted and to derive the experimental influences of pressure, temperature and supercritical CO₂ flowrate on the extraction curves. These parameters are very sensitive and highlight the necessity of precise optimisation of experimental conditions. In order to complete the behaviour of supercritical fluids extraction, an improved modelling is proposed. The modelling basic equations are based on others modelling published previously. In this work, the determination of several parameters comes from correlations and the other constants are fitted with all the experimental results. Thus the modelling is more representative and predictive as other ones. The modelling results present a good agreement with the experimental results, and hence it can be used for the dimensioning of some extraction autoclaves.     

Predicting the stress-strain behaviour of zeolite-cemented sand based on the unconfined compression test using GMDH type neural network

Stabilizing sand with cement is considered to be one of the most cost-effective and useful methods of in-situ soil improvement, and the effectiveness is often assessed using unconfined compressive tests. In certain cases, zeolite and cement blends have been used; however, even though this is a fundamental issue that affects the settlement response of a soil, very few attempts have been made to assess the stress-strain behaviour of the improved soil. Also, the majority of previous studies that predicted the unconfined compressive strength (UCS) of zeolite cemented sand did not examine the effect of the soil improvement variables and strain concurrently. Therefore, in this paper, an initiative is taken to predict the relationships for the stress-strain behaviour of cemented and zeolite-cemented sand. The analysis is based on using the unconfined compression test results and Group Method of Data Handling (GMDH) type Neural Network (NN). To achieve this end, 216 stress-strain diagrams resulting from unconfined compression tests for different cement and zeolite contents, relative densities, and curing times are collected and modelled via GMDH type NN. In order to increase the accuracy of the predictions, the parameters associated with successive stress and strain increments are considered. The results show that the suggested two and three hidden layer models appropriately characterise the stress-strain variations to produce accurate results. Moreover, the UCS values derived from this method are much more accurate than those provided in previous approaches. Moreover, the UCS values derived from this method are much more accurate than those provided in previous approaches which simply proposed the UCS values based on the content of the chemical binders, compaction, and/or curing time, not considering the relationship between stress and strain. Finally, GMDH models can be considered to be a powerful method to determine the mechanical properties of a soil including the stress-strain relationships. The other novelty of the work is that the accuracy of the prediction of the strain-stress behaviour of zeolite-cement-sand samples using the GMDH models is much higher than that of the other models.     

Hierarchical modelling of polymeric materials

Within the last 20 years, computer simulations of materials have evolved from an academic curiosity to a predictive tool for addressing structure-property-processing-performance relations that are critical to the design of new products and processes. Chemical engineers, with their problem-oriented thinking and their systems approach, have played a significant role in this development.The computational prediction of physical properties is particularly challenging for polymeric materials, because of the extremely broad spectra of length and time scales governing structure and molecular motion in these materials. This challenge can only be met through the development of hierarchical analysis and simulation strategies encompassing many interconnected levels, each level addressing phenomena over a specific window of time and length scales.In this paper we will briefly discuss the fundamental underpinnings and example applications of new methods and algorithms for the hierarchical modelling of polymers. Questions to be addressed include: How can one equilibrate atomistic models of long-chain polymer melts at all length scales and thereby predict thermodynamic and conformational properties reliably? How can one quantify the structure of entanglement networks present in these melts through topological analysis and relate it to rheological properties? Are there ways to predict the microphase-separated morphology and stress-strain behaviour of multicomponent block copolymer-based materials, such as pressure sensitive adhesives? Is it possible to anticipate changes in the barrier properties of glassy amorphous polymers used in packaging applications as a consequence of modifications in the chemical constitution of chains?     

Anharmonicity of Quasi-Lattice Vibrations in Glasses of the As(Sb)–O–I(Br,Cl) System in the Framework of the Fluctuation Hole Model

Parameters of the free volume theory (such as the fraction of the fluctuation free volume, the hole volume, the energy of hole formation, and the maximum internal pressure) are calculated from the experimental data on the density, microhardness, glass transition temperature, and longitudinal and transverse ultrasonic velocities. The anharmonicity of particle vibrations and the nonlinearity of interatomic interaction forces in vitreous oxides and oxyhalides are considered in the framework of the free volume concept. The relation of the critical strains of the interatomic bonds and the volume strain of the glass network to the lattice Grüneisen parameter is analyzed. The quantities calculated from the thermodynamic Grüneisen equation with the formulas relating the Grüneisen parameter to the Poisson ratio and the fraction of the fluctuation free volume are compared with those determined from the dependences of the elastic modulus on microhardness. A correlation between the parameters of the free volume model and the models based on modern structural concepts of inorganic glasses is discussed.     

Viscoelasticity of some engineering plastics analyzed with the modified stress-optical rule

The complex Young's modulus, E*(ω), and the complex strain-optical coefficient, O*(ω), of poly(ether sulfone) (PES), polysulfone (PSF), and polyethermide (PEI), were measured over the frequency range 1 to 130 Hz. The data were analyzed with a modified stress-optical rule: The Young's modulus was decomposed into two complex functions, E(ω) and E(ω); the modified stress-optical coefficient, C_R and C_G, associated with the rubber (R) and glass (G) components, respectively, were determined. The results for six polymers, including polystyrene, poly(α-methyl styrene), and bisphenol A polycarbonate were compared with each other. One of the coefficients, C_R, equivalent to the stress-optical coefficient in melts, mainly depended on the way in which phenyl groups were connected to the chain. The other, C_G, was in the range of 20 to 40 Brewsters, and did not strongly depend on the details of polymer structure. The component function, E(ω), which was located in the glassy region and originated from the high glassy modulus, was almost the same in shape when plotted against ω with double logarithmic scales. The R component, E(ω), located at the long time end of the glass-to-rubber transition zone, was slightly sensitive to the molecular structure of polymers.     

On the interrelation of rubber elastic stress-strain curves under different types of strain

With specially shaped samples it is possible to study the stress-strain relation σ(γ) in simple shear up to large shear deformation γ. Measurements of σ(γ) on synthetic polyisoprene, IR 305, differently crosslinked with dicumylperoxide, are reported and compared with those for uniaxial extension on the same materials. Using the simple deformation geometry including shearbands proposed in the meander model (1), both stress-strain relations can be converted into each other. These results favor the applied superstructural model and moreover show that the deformation behavior of rubbers is governed by one material law, σ(γ).     

Multi-scale modelling of progressive damage and failure behaviour of 2D woven SiC/SiC composites

In this paper, a multi-scale modelling approach has been developed to predict the progressive damage and failure behaviour of 2D woven SiC/SiC composites. At the tow scale, non-linear tow properties have been determined by a micromechanics-based damage model, in which two scalar damage variables were introduced to characterize the fibre-dominated and matrix-dominated damage, respectively. Based on periodic boundary conditions, a meso-scale unit cell model has been established to simulate the macroscopic stress-strain responses and progressive damage processes of the composite under uniaxial tensile, compressive and in-plane shear loadings, respectively. In the numerical method, the non-linear properties of constituent materials have been implemented by the user defined subroutine, USDFLD of the finite element package, Abaqus. The numerical results and their comparisons with experimental stress-strain curves have been presented. The failure mechanisms of the composite under each loading have been also discussed. The high efficiency and prediction accuracy of the model make it possible to analyse large scale woven composites.     

Unusual stress-strain properties of natural rubber vulcanizates with high primary molecular weight

Summary Natural rubber vulcanizates with high primary molecular weight were prepared by mixing raw rubber and dicumyl peroxide in benzene followed by freeze-drying. The stress-strain properties of these vulcanizates were quite different from those prepared by conventional mastication method. The important characteristics of these vulcanizates is high tensile modulus, high tensile strength, large hysteresis loss, and higher degree of strain-induced crystallization. The difference in the stress-strain behavior between these rubbers and conventional vulcanizates are discussed from the standpoint of the network structure.     

Dispersive forces of particle–surface interactions: direct AFM measurements and modelling

Fundamentals of particle–particle interaction are of great interest in agglomeration processes. Particle adhesion depends on dispersive forces (van der Waals force), local chemical bindings, Coulomb force and capillary attractions. Additionally, surface properties like roughness, adsorption layers and surface chemistry strongly affect adhesion forces. van der Waals interactions are poorly understood because popular ab initio force calculations for molecules like density functional theory (DFT) often do not lead to proper results. van der Waals forces are difficult to measure directly. We present direct measurements of particle–particle and particle–surface interactions in the gas phase carried out with an atomic force microscope (AFM). Special emphasis is given to a proper statistical treatment of the data. For modelling of particle adhesion, we use computer-assisted empirical potential methods. Parameters like adsorbed water and surface roughness are considered. We extract parameters for weak interactions from the Lifshitz theory and gas adsorption data. Adsorbing molecules can be used as probes to measure dispersive forces. Studying surface and particle properties combined with computer-assisted modelling is a basic requisite to reach the aim of predicting particle–particle interactions in industrial processes.     

Effect of network structure on the stress-strain behaviour of endlinked PDMS elastomers

Endlinked poly(dimethylsiloxane) (PDMS) networks synthesized from telechelic precursor chains of different molar mass were prepared with varying volume fractions of non-reactive chains acting as solvent. Uniaxial extension and compression measurements were performed on these networks to investigate their stress-strain behaviour. The effect of the network structure and of solvent on the stress-strain behaviour is examined by controlling the extent of crosslinks and entanglements during the network synthesis. The master curve of the Rubinstein and Panyukov non-affine slip-tube (NAST) model provide an adequate fit to most of the extension and compression data. Furthermore, the crosslink and entanglement parameters of the NAST model (G_c and G_e) are found to be in general in reasonable agreement with the 2C₁ and 2C₂ parameters of the Mooney-Rivlin continuum model applied to the extension data. For high molar mass precursor chains, the entanglement contribution to the modulus surpasses the crosslink contribution.