Micromechanical modeling of the elastic properties of semicrystalline polymers: A three‐phase approach

The mechanical performance of semicrystalline polymers is strongly dependent on their underlying microstructure, consisting of crystallographic lamellae and amorphous layers. In line with that, semicrystalline polymers have previously been modeled as two and three‐phase composites, consisting of a crystalline and an amorphous phase and, in case of the three‐phase composite, a rigid‐amorphous phase between the other two, having a somewhat ordered structure and a constant thickness. In this work, the ability of two‐phase and three‐phase composite models to predict the elastic modulus of semicrystalline polymers is investigated. The three‐phase model incorporates an internal length scale through crystalline lamellar and interphase thicknesses, whereas no length scales are included in the two‐phase model. Using linear elastic behavior for the constituent phases, a closed form solution for the average stiffness of the inclusion is obtained. A hybrid inclusion interaction model has been used to compute the effective elastic properties of polyethylene. The model results are compared with experimental data to assess the capabilities of the two‐ or three‐phase composite inclusion model. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010     

Crystalline microstructure and mechanical properties of crosslinked EPDM aged under gamma irradiation

The morphology and the mechanical properties at room temperature of crosslinked EPDM irradiated or not have been studied. It has been shown that these materials are composed of two phases: semicrystalline zones with a crystallinity ratio of 20% and mainly amorphous zones. The semicrystalline zones make a continuous path through the film and therefore control the mechanical properties of the material below the melting temperature. As irradiation (in the tested range of irradiation dose) and crosslinking degree have no significant influence on the arrangement and proportion of the crystalline lamellae, all samples have nearly the same mechanical behavior at small strains. At large strains, the interactions between amorphous and crystalline parts in semicrystalline zones play the main role in the mechanical response; irradiation, by degradation of these interactions, leads to a smaller hardening phenomenon and a decrease in elongation at break. From an application point of view, in spite of the low crystallinity fraction of these materials, the presence of an important number of crystallites, as evidenced by SAXS measurements, strongly limits the consequences of irradiation on the mechanical properties. However, the mechanical reinforcement strongly depending on the presence of these crystallites, it is therefore highly sensitive to temperature: this can be an important issue for the applications of these materials since their use temperature is close to the crystallite melting temperature. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 97–105, 2010     

Mechanical relaxations in polypropylene: A new experimental and theoretical approach

Accurate dynamic mechanical measurements have been performed on semicrystalline isotactic polypropylene over wide ranges of temperature and frequency. A mechanical model has been used to analyze experimental results in order to separate the behavior of amorphous and crystalline phases. The two main α and β relaxation processes have been analyzed. The β relaxation, related to the glass-rubber transition of the amorphous fraction, has been studied with the help of a physical model. The behavior is similar to that of a wholly amorphous polymer, with two characteristics: a high rubbery plateau, indicating a crosslinking effect by the crystalline phase, and a strong effect of interfaces in shear strain. Experimental data suggest the α relaxation originates within the crystalline phase and that it can be attributed to diffusion of defects. The amorphous phase plays an important role in this process, because it has to adapt itself by cooperative movements to respect the compatibility of deformations of the two phases. The formalism developed here rationalizes experimental results obtained with samples having different thermal histories.     

Effects of thermal history on the rigid amorphous phase in poly(phenylene sulfide)

The rigid amorphous phase of semicrystalline poly(phenylene sulfide) (PPS) has been studied as a function of thermal history using scanning calorimetry, dielectric relaxation, density, and small-angle x-ray scattering (SAXS). Based on the new heat of fusion of perfect crystalline PPS, which is 26.7±0.8 cal/gram, the weight fraction of rigid amorphous phase is shown to be nearly twice as large as previously reported [1]. The mass fraction of the rigid amorphous phase ranges from 0.24 to 0.42 and is dependent upon thermal treatment. We have taken the approach of assuming a three-phase model for the morphology of semicrystalline PPS consisting of crystalline lamellae, mobile amorphous, and rigid amorphous components. Using this three-phase model, we determine that the average density of the rigid amorphous fraction is 1.325 g/cc, which is slightly larger than the density of the mobile amorphous phase fraction and was insensitive to thermal history. From the SAXS long period, the layer thicknesses of the mobile amorphous phase, rigid amorphous phase, and crystal lamellae were estimated. Only the lamellar thickness shows a systematic variation with thermal history, increasing with melt or cold crystallization temperature, or with decreasing cooling rate.     

A study of melting of low density polyethylene by in situ WAXD

Analysis of low density polyethylene using in situ wide-angle X-ray diffraction shows that the melting process takes place over a broad temperature range with continuous destruction of lamellar laterals. This process exhibits decreasing rate as the temperature increases, indicating that in general each lamellae has less order in outer regions than in central ones. The shifts in angular position of the amorphous halo as a function of the temperature show that the amorphous phase in a semicrystalline sample is less disordered compared to that in an overall melted sample. As the temperature increases the unit cell parameter a increases, generating a crystalline phase density reduction.     

The effect of supramolecular structure on deformation and relaxation transitions in ethylene-1-hexene copolymers crystallized from solution

Structure and mechanical properties of three samples based on ethylene-1-hexene copolymers prepared by crystallization from decalin solutions were studied. The rate of decrease in the true yield stress was shown to depend on temperature. This behavior was assumed to be due to the involvement in deformation of the “intermediate phase” that is located at the lamellar surface between the crystalline and amorphous phases.     

On the predictability of the high-frequency elastic modulus of semicrystalline polymers as function of crystallinity

The relation of the high-frequency elastic moduli of semicrystalline polymers to volume fraction crystallinity is correctly described by the Hashin-Shtrikman theory, without any disposable constants, as a function of the ratio of the modulus of the amorphous to that of the crystalline phase. Hence the (high-frequency) reduced modulus of semicrystalline polymers is largely a function of the temperature T/T_g. The importance of T/T_m for the modulus of the crystalline phase precludes the existence of a single universal reduced modulus versus temperature curve.     

The “Relaxed” state in a semicrystalline polymer: Experimental characterization and modeling

This work addresses the general issue of the mechanical behavior of the confined amorphous phase in rubbery semicrystalline polymers. Even far above the glassy transition temperature, the amorphous phase in semicrystalline polymers is known to remain constrained by crystals and is less mobile than a purely amorphous polymer close to its equilibrium rubbery state. The aim of this paper, based on Polyamide 11, is to investigate the existence and significance of a relaxed state in the amorphous phase of a semicrystalline polymer far above T _g. A strain-rate independent tensile curve (called the “asymptotic curve”) is evidenced below a critical strainrate, consistently with a fully relaxed state of the rubbery amorphous phase. Nevertheless, a contradictory mechanical phenomenology was observed at the same time (hysteretic unloading, relaxation, and creep involving the same strain-rates as the “asymptotic” loading regime), suggesting joint amorphous and crystalline processes. Modeling of this paradoxical behavior is attempted, based on the experimental results. The first one-dimensional simulations are presented. Published in Russian in Vysokomolekulyarnye Soedineniya, Ser. A, 2008, Vol. 50, No. 5, pp. 797–808. This article was submitted by the authors in English.     

Morphology and modulus–temperature behavior of semicrystalline poly(ethylene terephthalate) (PET)

The influence of the thermal history on the morphology and mechanical behavior of PET was studied. The degree of crystallinity (density measurements) and the morphological structure (electron microscopy and small-angle x-ray diffraction) depend on the crystallization temperature. The viscoelastic parameters obtained from the modulus–temperature curves are mainly determined by the morphology of the samples. The glass-transition temperature, T_i, is a function of the crystallinity and the crystallization temperature. It is maximum for a crystallinity between 0.34 and 0.39 for a sample crystallized isothermally between 120 and 150°C. This dependence on crystallization conditions is ascribed to the conformation of the amorphous chain segments between the crystalline lamellae as well as the concentration and the molecular weight of the polymer material rejected during isothermal crystallization. Both factors are supposed to be temperature-dependent. The value of the rubbery modulus is a function of both the volume concentration of the crystalline lamellae and the structure of the interlamellar amorphous regions (chain folds, tie molecules, chain ends, and segregated low molecular weight material). Annealing above the crystallization temperature of isothermally crystallized samples has a marked influence on their morphology and mechanical behavior. The morphological structure and the viscoelastic properties of annealed PET samples are completely different from those obtained with samples isothermally crystallized at the same temperature.     

The mechanical moduli of lamellar semicrystalline polymers

Bounds on the elastic constants are derived for semicrystalline polymers whose local morphology is lamellar. Local response matrices (stiffness and compliance) are formulated in three dimensions that simultaneously incorporate uniform in-plane strain and additive forces from layer to layer of crystalline and amorphous phases and uniform stress and additive displacements normal to the lamellar surfaces. Spatial averaging of the stiffness and compliance matrices under the assumption of axially symmetric orientation gives the upper and lower bounds on the longitudinal and transverse tensile moduli and the axial and transverse shear moduli as functions of the separate phase elastic constants, the volume percent crystallinity, and the moments of the orientation 〈cos²θ〉 and 〈cos⁴θ〉. The bounds are much tighter than the Voight upper and Reuss lower bounds that do not recognize phase geometry. Using the known crystal elastic constants of polyethylene, sample calculations on isotropic unoriented materials show that the divergence of bounds at high crystallinity necessitated by the extreme crystal anisotropy shows up only at very high crystallinity. At low temperature the bounds are tight enough to specify G₁, the amorphous modulus, from the measured G and the known crystal elastic constants. At higher temperatures and lower G, the bounds are not tight enough for this purpose but the shear modulus versus crystallinity and temperature data are well fitted by the lamellar lower bound using a temperature-dependent, crystallinity-independent G₁.     

Micro-Macro Scale Relationship in Creep Deformation of Ultra High Molecular Weight Polyethylene

Macroscopic and microscale creep deformations of UHMWPE were investigated by using in situ SAXS.A methodology for the measurement of the local creep deformation of inter-lamellar amorphous phase has been proposed.The local strain of inter-lamellar amorphous phase (εa) and macroscopic strain (εmacro) were evaluated and they were compared to study the relationship between macroscopic and microscale creep deformation of UHMWPE.Both of them exhibit two deformation regions against creep time.The entanglements show a strong impact on both the macroscopic and local inter-lamellar amorphous phase creep behavior and they can be well correlated to the molecular weight between two entanglements estimated from strain-hardening modulus.Compared to the macroscopic creep deformation,local inter-lamellar amorphous layers have a smaller creep deformation.From the local creep measurement,the apparent modulus of inter-lamellar amorphous phase can also be estimated (200 ＜ Ma ＜ 500 MPa).These values are much higher than the Young's modulus of bulk amorphous PE,which can be well explained by the confinement of the lamellar stacks and the enhancement of the amorphous phase with the relatively high concentration of entanglements.This study provides a useful means and quantitative data for achieving the scale transition between the micro and the macro structural levels for the study of viscos-elastic deformation.     

Molecular dynamics in PVDF/PVA blends as revealed by dielectric loss spectroscopy

The relaxation behavior of a series of compatible poly(vinylidene fluoride) (PVDF) and poly(vinyl acetate) (PVA) blends has been investigated by dielectric spectroscopy in a broad frequency and temperature range. Blends with PVDF content higher than 60% in weight are semicrystalline. Semicrystalline blends show a relaxation (α_c) occurring in the crystalline phase of PVDF. Both semicrystalline and amorphous blends exhibit two processes, α and β associated to the overall segmental dynamics and to localized motions in the amorphous phase, respectively. For high PVDF content samples, the β relaxation exhibits an anomalous behavior characterized by a crossover from segmental to local dynamics, upon decreasing temperature, attributed to confinement effects taking place in PVDF segregated regions. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1653–1661, 2007     

Blends of propylene-ran-ethylene and propylene-ran-(1-butene) copolymers: Crystal superstructure and mechanical properties

Blends of isotactic propylene-ran-ethylene (EP) and propylene-ran-(1-butene) (BP) copolymers with various comonomer content (2-3.1 wt.% ethylene, 9.9 wt.% 1-butene), were prepared in Brabender internal mixer at various compositions (25/75, 50/50, 75/25). Static, impact and dynamic mechanical behavior of copolymers and their blends was investigated. The crystalline structure was studied by DSC and SAXS analysis. For all copolymers the lamellar thickness, crystallinity degree and glass transition temperature are lower than those of iPP homopolymer, depending on the comonomer content. It was found that the copolymers exhibit improved impact strength as compared to plain iPP, due to lower crystallinity and higher mobility of chains within amorphous component. Moreover, the elastic modulus as well as the yield behavior of the examined samples resulted to depend primarily on the amount of the crystalline phase and the thickness of the lamellar crystals, respectively. A linear dependence of yield stress on the logarithm of reciprocal lamellar thickness was observed for blends and copolymers, supporting the concept of thermal nucleation of dislocations which control the crystallographic slip processes initiated at the yield point. The blends of BPS with either EPS or EP2 display complete miscibility in the entire range of composition and their mechanical properties are intermediate between those of plain components, changing gradually with the composition.     

Micromechanisms involved in the atypical tensile behavior observed in polyamide 11 at high temperature

Even far above the glass transition temperature, the amorphous phase in semicrystalline polymers is known to be constrained by crystals and less mobile than a pure amorphous polymer close to its equilibrium rubbery state. The aim of this paper devoted to Polyamide 11 was to investigate the existence and significance of a relaxed state in the amorphous phase of a semicrystalline polymer far above T_g. It focuses on the high temperatures, low strain‐rates, and small deformation ranges. A nonstrain‐rate dependent tensile curve (called “asymptotic curve”) was evidenced below a critical strain‐rate, consistently with reaching a fully relaxed state of the rubbery amorphous phase. Nevertheless, paradoxical mechanical features were observed at the same time (nonstrain‐rate dependent but hysteretic unloading, relaxation, and creep involving same strain‐rates as the asymptotic loading regime). Micromechanisms (orientation of primary crystals, creation of local hexagonal arrangements, orientation, and relaxation of the amorphous phase) were analyzed from DSC and X‐ray experiments. It suggested distinct amorphous and crystalline contributions depending on the loading path and therefore highlighted paradox of the mechanical behavior. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3046–3059, 2007     

The dielectric constant of lamellar semicrystalline polymers

The problem of representing the dielectric constant of semicrystalline polymers in terms of the dielectric constants and volume fractions of the constitutent crystalline and amorphous phases is considered. For locally lamellar morphology, bounds based on uniform electric and displacement fields are derived. The equations also include the degree of crystal orientation as a parameter. For unoriented polymers the bounds are considerably tighter than the Hashin–Shtrikman bounds, the latter being the best possible without knowledge of phase geometries. The bounds presented here are sufficiently tight to represent the dielectric constant with practical accuracy for a number of examples of semicrystalline polymers. A treatment is also given of the dielectric constant where the lamellar morphology is further specified as being organized into spherulite-like structures. These bounds are somewhat tighter than the lamellar bounds.     

Strain hardening of polycarbonate in the glassy state: Influence of temperature and molecular weight

This study is concerned with the temperature and molecular weight dependence of the strain-hardening behavior of polycarbonate. It is shown that the strain-hardening modulus reduces with increasing temperature and decreasing molecular weight. This result is interpreted in terms of temperature accelerated relaxation of the entanglement network. Moreover, it is shown that frozen-in orientations, induced by homogeneous deformations above the glass transition temperature, lead to anisotropic yield behavior that can be fully rationalized (and modelled) in terms of a superimposed stress contribution of the prestrained network. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2041–2049, 2004     

Chloroform sorption in nanoporous crystalline and amorphous phases of syndiotactic polystyrene

The transport of chloroform in films of atactic polystyrene and of semicrystalline syndiotactic polystyrene in its nanoporous form (δ‐form) has been investigated by gravimetric analysis. Experimental tests have been performed at 35, 49, and 56 °C and at several vapor pressures ranging from 0.5 to 100 Torr. Nonequilibrium lattice fluid prediction of the amorphous sorption behavior was used to enucleate the sorption contribution of the crystalline nanoporous phase from semicrystalline samples. Sorption behavior of the crystalline phase has been interpreted on the basis of Langmuir equation. Moreover, the chloroform sorption at low activities in the crystalline nanoporous phase has been predicted by using Grand Canonical Monte Carlo molecular simulations. Isosteric heats of sorption were also experimentally evaluated for the crystalline phase, and compared with the corresponding prediction of molecular simulation. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 8–15, 2008     

Crystallization and chain conformation of semicrystalline and amorphous polymer blends studied by wide‐angle and small‐angle scattering

We examine the crystallization and chain conformation behavior of semicrystalline poly(ethylene oxide) (PEO) and amorphous poly(vinyl acetate) (PVAc) mixtures with wide‐angle X‐ray diffraction (WAXD), small‐angle X‐ray scattering (SAXS), and small‐angle neutron scattering (SANS) experiments. For blends with PEO weight fractions (wt_PEO) greater than or equal to 0.3, below the melting point of PEO, the WAXD patterns reveal that crystalline PEO belongs to the monoclinic system. The unit‐cell parameters are independent of wt_PEO. However, the bulk crystallinity determined from WAXD decreases as wt_PEO decreases. The scattered intensities from SAXS experiments show that the systems form an ordered crystalline/amorphous lamellar structure. In a combination of WAXD and SAXS analysis, the related morphological parameters are assigned correctly. With the addition of amorphous PVAc, both the average amorphous layer thickness and long spacing increase, whereas the average crystalline layer thickness decreases. We find that a two‐phase analysis of the correlation function from SAXS, in which the scattering invariant is linearly proportional to the volume fraction of lamellar stacks, describes quantitatively the crystallization behavior of PEO in the presence of PVAc. When wt_PEO is close to 1, the samples are fully spaced‐filled with lamellar stacks. As wt_PEO decreases from 1.0 to 0.3, more PVAc chains are excluded from the interlamellar region into the interfibrillar region. The fraction outside the lamellar stacks, which is completely occupied with PVAc chains, increases from 0 to 58%. Because the radius of gyration of PVAc with a random‐coil configuration determined from SANS is smaller than the average amorphous layer thickness from SAXS, we believe that the amorphous PVAc chains still persist with a random‐coil configuration even when the blends form an ordered structure. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2705–2715, 2001     

Deformation in lamellar and crystalline structures: in situ simultaneous small‐angle X‐ray scattering and wide‐angle X‐ray diffraction measurements on polyethylene terephthalate fibers

Changes in the lamellar and crystalline structures were followed as a function of applied stress to understand the influence of the interactions between the crystalline and amorphous domains on the fiber properties. We observed a reversible transformation from a structure giving a four‐point small‐angle pattern to a structure giving a two‐point pattern; these structures corresponded to the lamellae with oblique and normal lamellar surfaces, respectively. The characteristics of these two structures such as the stack diameter, stack height, and tilt angle were different and were determined by the processing conditions and did not change when the fiber was elastically deformed. The structure giving a two‐point pattern was probably the load‐carrying lamellar entity in these fibers. The diameter of the lamellar stacks, tilt angle of the lamellae, and the strain in the lamellar spacing appeared to have the most influence on properties such as tenacity and dimensional stability. The long‐spacing strain, which is about the same as the fiber strain, determined the modulus at low elongation as well as ultimate elongation. These indicate that the lamellar stacks have at least as much influence as the interfibrillar chains on fiber properties. Structural features that determine the performance in semicrystalline polymers were investigated by analyzing four generations of polyethylene terephthalate fibers. Some of the fiber properties correlate with changes in the crystalline domains such as the crystalline orientation, size, and unit cell dimensions. Fibers in which the crystalline strain was large because of their strong linkages to the amorphous chains, and better load transfer, had the highest modulus and lowest ultimate elongation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1538–1553, 2003     

Shear-induced topology changes in liquid crystals of the soybean lecithin/DDAB/water system

The viscoelastic behavior of the two different liquid crystalline lamellar phases and the liquid crystalline cubic phase of the mixed soybean lecithin/DDAB system in water was studied through rheology, with mechanical parameters studied as a function of composition. The swollen or diluted lamellar region is formed by vesicles, and its characteristic flow curve presents two-power law regions separated by a region where viscosity passes through a maximum. Yield stress and shear-dependent flow behavior were also observed. The microstructure suffers transformation under shear stress, and rheological response shifts from thixotropic to antithixotropic loops. Similar rheological behavior has been observed for samples in the collapsed or concentrated lamellar region, at the water-rich corner of the phase diagram. Vesicle formation may therefore occur by shearing the initial stacked and open bilayers. However, concentrated lamellar samples in the water-poor part of the phase diagram are less sensitive to shear effects and show plastic behavior and thixotropy. All lamellar samples manifest high elasticity. The dynamic responses of both lamellar topologies, i.e., vesicles and open bilayers, are comparable and exhibit an infinite relation time. The bicontinuous cubic, liquid crystalline phase is highly viscous. Its dynamic response cannot be modeled by a Maxwell model.