A molecular theory of stress–strain relationship of spherulitic materials

A primitive molecular theory for stress–strain relationship of spherulitic polymers is presented based on a consideration of changes in conformational free energy in the tie chains and floating chains located between crystalline lamellae within an ideal spherulite which is assumed to undergo an affine deformation. Numerical stress–strain curves are calculated as a function of temperature, crystallinity, and tie chain fraction.     

Crystal orientation in a semicrystalline polymer in relation to deformation of spherulites

A mathematical development interrelating the orientation distribution functions of three kinds of orientation units for a polymer spherulite (i.e., a crystal lamella, a crystallite, and a given reciprocal lattice vector of the crystallite) is formulated on the basis of series expansions of the distribution functions in generalized spherical harmonies. Two types of uniaxial deformation models of a polyethylene spherulite, taking account of micronecking and untwisting of crystal lamellae, and of chain tilting and untwisting of crystal lamellae, respectively, both in addition to affine deformation of the lamellae are discussed. The models are tested by comparison of the theoretical orientation distribution functions of some reciprocal lattice vectors of the crystallite with the results of x-ray diffraction experiments.     

Crystal orientation in a semicrystalline polymer in relation to deformation of polymer spherulites. II. Orientation distribution function of crystallites within crystal lamella as a function of lamellar orientation

A model relating crystal orientation in a semicrystalline polymer to the deformation of polymer spherulites is proposed. The distribution function for orientation of crystallites within crystal lamellae is assumed to be a function of lamellar orientation. In addition to the orientation of crystal lamellae in affine fashion, several parameters are introduced to characterize the untwisting of the crystal lamellae and the four different types of orientations of the crystallites within the crystal lamellae in the undeformed and deformed states of the spherulite. The model was tested by experiments in uniaxial stretching of a low-density polyethylene. The theoretical distributions of orientation of given reciprocal lattice vectors of the crystallites, such as the reciprocal lattice vectors of the (110) and (200) crystal planes, are compared with the results of x-ray diffraction experiments. It was found that the most important factors in fitting the model to experimental results are: (a) the fraction of crystallites having random orientation within lamella and, in turn, representing the degree of imperfection of the lamella in the undeformed state; (b) the ease of transition of crystal orientation within lamella from b-axis orientation parallel to the lamellar axis to two types of c-axis orientations (type C_a and type C_r) parallel to the stretching direction; and (c) the fraction of crystallites having orientation in type C_r (unfolding mechanism) rather than type C_a (rotation mechanism).     

Crystal and amorphous orientation behavior of poly(chlorotrifluoroethylene) films in relation to crystalline superstructure

The relationship between the crystalline superstructure of polymer films and molecular orientation was studied in cold-drawn poly(chlorotrifluoroethylene) films by wide-angle x-ray diffraction, birefringence, and depolarized light scattering. By changing crystallization conditions, specimens with almost identical crystallinity but different crystalline superstructures were obtained; i.e., (1) a structure having a random array of crystallites, (2) a superstructure having a rod-like orientation correlation of the chains (a prespherulitic and sheaf-like superstructure), and (3) spherulitic superstructure. Upon stretching of specimens, crystallites initially randomly arranged orient with their chain axes along the stretching direction in accord with simple affine deformation. The amorphous chains also orient along the stretching direction. The orientation behavior of the specimens having the rod-like superstructure is similar to that of the specimens with a random array of crystallites, indicating that the interaction between the crystallites in the superstructure is relatively weak. The molecular orientation behavior of the spherulitic specimens, however, strongly deviates from simple affine deformation owing to strong interaction of the crystallites in the spherulites. The deviation can be interpreted in terms of spherulite deformation and of internal reorientation of chains within deformed spherulites.     

Relationship between piezoelectricity and superstructure in highly oriented poly(vinylidene fluoride) film prepared by zone drawing

The oriented superstructure of poly(vinylidene fluoride) is controlled by using a forced-quenching type of zone drawing apparatus. Systematic variation of the weight fraction χ(I) of form-I crystals and the orientation function f_a of amorphous chains shows that the piezoelectricity increases with increasing χ(I) and f_a. A change in the state of molecular aggregation during poling is also effective in increasing the piezoelectricity and the orientation of the crystal b axis along the poling direction. Equations relating piezoelectricity to the form-I crystallinity, the orientation of amorphous chains, and the orientation of the crystal b axis along the poling direction are derived. These are based on a mechanical model having regions of taut tie molecules in parallel with composite regions consisting of crystalline and amorphous blocks in series.     

Crystal stability and mechanical properties of drawn polymers: A statistical-mechanical approach

The partition function is formulated by the generating function method for a stacked lamellar model of alternating crystalline and amorphous layers. The random-walk problem of enumerating statistical weights for conformations of amorphous chains confined by two parallel walls is solved for the body-centered cubic lattice as a generalization of the one-wall model treated by Roe. The mean lengths of loop, tie, and cilia chains and the free energy of the system are calculated for the random-reentry and-bridge model as a function of the distance h of separation between crystal layers and as a function of the number N of loop chains in a crystal block as the basic structural element of the system. The mean length of amorphous chains decreases at a given thickness l of the crystal layers with decreasing h or with increasing N. The free energy of the system exhibits no minimum with respect to N, showing that the folded-chain crystal is thermodynamically stable, especially for relatively small l. Additionally, it is shown that another requirement for stabilizing relatively small crystals (small l) is the formation of an aggregate structure of crystals, whereas a large single-crystal (large l) is relatively stable, irrespective of h and N. Furthermore, a theoretical model is developed to calculate the force and elastic modulus of a highly deformed stacked system, assuming that the only change in the configurations of amorphous chains within the interlamellar regions is due to deformation, except for scission of tie chains having fewer segments than are needed to span the interlamellar distance of the deformed system. It becomes evident that taut tie chains are effective in increasing the modulus of the stacked system.     

Rate of thermooxidation and polymer morphology

A model of the thermooxidation of semicrystalline polymers involving two processes simultaneously is described. The first process is initiated by oxygen dissolved in the interspherulitic amorphous pahse. The second is related to a slower rate of polymer chain scission m the interlamellar amorphous pahse, which is controlled by oxygen diffusion into the spherulite. The presented model describes an exampble of isotactic polypropylene and predicts the effect of polymer morphology on the process of thermooxidation. It leads to the conclusion that the initial rate of polymer chain scission depends on the amount of interspherulitic amorphous phase fraction but the slower rate, considered in steady state conditions of oxygen diffussion into the interlamellar amorphous phase, depends depends on spherulite size according to the derived equation.     

NMR study of the tie-chain length distribution in oriented polymers

Wide-line NMR has been used in an investigation of noncrystalline (amorphous) regions in oriented semicrystalline polymers. Nylon 6 was chosen as a model material. The tie-chain length distribution function, the fraction of tie chains in the total number of chains in the crystallite cross section, and the relative number of taut tie chains have been determined. The data on the tie-chain length distribution are used in discussing specific features of vitrification of the amorphous regions in oriented polymers and in prediction macroscopic mechanical properties.     

Plasticity of Semicrystalline Polymers

Semi-crystalline polymers can be deformed up to a very high strain. The deformation process involves frequently a complete molecular rearrangement of the chain-folded lamellar morphology into a more or less chain-unfolded fibrillar microstructure. This transformation is likely to occur through an intermediate state of high molecular disorder at a local scale. It led to the formulation of a concept of strain-induced melting-recrystallization process as a main mechanism of the structure transformation. In contrast, several structural features occurring at moderate plastic strains are relevant to strictly crystallographic processes. The plastic deformation process of semicrystalline polymers and the micromechanisms involved are discussed. A critical discussion of experimental findings is made to point out the strength or the deficiency of the various argumentations. It is demonstrated that the crystallographic slip mechanisms, including slips: transverse and along the chains are the basic deformation mechanisms in the deformation sequence, active at all strain levels. Direct microscopic evidence of chain slip activity even at well advanced stages of the deformation process is presented. In contrary, the melting-recrystallization seems to be restricted to the high-strain stage accompanied by chain unfolding and perhaps limited to only a small fraction of the crystalline phase. In addition the experimental results demonstrates clearly that the cavitation, necessary in the Peterlin's model, is really unessential in producing high deformation and appearance of the final highly oriented structure. This can be effectively accomplished with only crystallographic mechanisms employed. A very important role in the deformation sequence is played by the partially reversible shear deformation of amorphous interlamellar layers, producing not only high orientation of amorphous component but also influencing deeply the deformation of crystalline phase, since both phases are strongly connected and must deform simultaneously and consistently.     

An analytical technique for measuring relative tie-molecule concentration in polyethylene

While consideration of the crystalline domains have long dominated research in understanding the properties of semicrystalline polymers, a satisfactory understanding of crack growth in these materials can only be realized by developing corresponding analytical tools to characterize the amorphous region. Since slow stable cracks in these materials preferentially form between crystalline lamellae, the role of tie molecules—the amorphous chains that bridge crystalline lamellae—are particularly important in this regard. Unfortunately, there is no method readily available for quantitative assessment of tie molecules. Through deformation and subsequent chlorination of polyethylene films, it is demonstrated that infrared dichroism can be used to determine relative tie-molecule concentration. Using this technique, one can a priori predict which resin in a series having comparable densities but widely varying molecular weights or comonomer distributions exhibits better crack resistance.     

The fatigue process in highly oriented nylon 6 fibers

The micromechanism of the fatigue process in highly oriented nylon 6 fibers is discussed on the basis of changes in mechanical and structural properties during fatiguing. Experimental results show that the fatigue process can be divided into two stages. The characteristic features in the initial period are increases in breaking strength, long period, and molecular orientation, and a reduction in dye penetration. In the second period, after about 500 cycles, breaking strength and orientation decrease slightly, and the long period, permanent strain, and dye penetration increase with duration of fatiguing. It is demonstrated that the structural changes mainly occur in the amorphous regions of the fiber structure. The structural and mechanical changes in the initial period lead to the conclusion that the initial cyclic strain causes strain hardening caused by extended tie chains which do not rupture. A combination of load bearing by tie chains and sliding motion of the fibrillar elements can explain the progressive degradation of the fiber during the second stage of fatiguing.     

Tensile deformation of polyamide (PA) 6 probed by rheo-optical near-infrared (NIR) spectroscopy

A rheo-optical near-infrared (NIR) spectroscopy, based on the combination of NIR spectroscopy and mechanical analysis, was applied to polyamide (PA) 6 samples consisting of bundled amorphous chains. Sets of strain-dependent NIR spectra as well as tensile stress of dried and wet treated PA 6 samples were collected during the mechanical elongation of the samples. The spectra were then subjected to two-dimensional (2D) correlation analysis to elucidate fine features of the spectral changes. An asynchronous correlation peak develops between the bands at 2355 and 2300 nm due to the combination modes of CH₂ groups arising from the rubbery amorphous chain and rigid crystalline lamella of the dried PA 6, respectively. It therefore indicates that during the tensile deformation, the orientation of the amorphous chain is induced first to cause the elastic deformation. Further elongation results in the rotation of the crystalline lamella connected with the amorphous chain. This correlation intensity apparently increases by the wet treatment, suggesting that water molecule in the PA 6 disrupts the H-bonding interaction between the adjacent polymer chains and thus makes the polymer more flexible. Accordingly, it is likely the H-bonding between the polymer chains works in a manner somewhat similar to cross-linked polymers, which substantially effects on the mechanical property of the PA 6.     

Orientation of the remaining amorphous chains during crystallization: Isotactic polystyrene

The orientation of statistical chain segments in the amorphous regions of a semicrystalline polymer can be characterized quantitatively by x-ray diffraction. A calculated background (which is orientation independent) must be subtracted, which requires that the intensity profile of the amorphous halo be measured on an absolute scale. Application is illustrated to samples of isotactic polystyrene which were isothermally crystallized at fixed elongation. These samples differ from those of the previous study due to the unintentional introduction of a small amount of crystallinity (0.25% to 1%) which serves as tie points in the drawing operation, giving a stronger amorphous orientation. Preferred orientation of the amorphous chain segments is rapidly depleted during the first few percent of crystallization. The crystallite orientation distribution is quite sharp, and broadens only slowly with crystallization. These results imply that, for the crystallization conditions employed, the crystallite orientation must be determined by nuclei which are formed at the earliest stages of the crystallization process.     

THE RHEO-OPTICAL FTIR STUDY OF POLYESTERPOLYETHER SEGMENTED COPOLYMER

The polyester-polyetber segmented copolymer has been investigated by rheo-optical FTIR during stretehing for the informastion on strain induced crystallizatinn of soft segment chains, hard and soft chain orientation.At room temperature 15℃, the soft segment chains of polyester-polyether being to crystallize at 220% strain and the degree of crystallixation increa(?)e with draw ratio, but there will not be any soft segment chains crystallization above 21℃even at bigber strain. The average orientation of hard segment chains are higher than that of the soft chain at high strain level, both are positive oriented into the stretching direction at allover strain level. This indicates that the b(?)d (?)ent ch(?)ins are dispersed into the elastomeric phase and without forming spherulite.     

Phase Structures of Nascent Polyethylene Powder Studied by Wideline Proton NMR

The wideline proton NMR spectra of polyethylene powder samples were analyzed in terms of contributions from three components： （1） a rigid part with immobile chains, （2） a soft region with liquid-like character which produces a Lorentzian contribution to the spectrum, and （3） an intermediate region in which the rotation of methylene groups about C-C bonds is partially hindered. The relative mass fractions as well as chain mobilities varied greatly among samples produced by different polymerization techniques. The NMR crystallinity agreed well with that estimated by WAXD and was much higher than DSC crystallinity, indicating an inclusion of the contribution from a crystalline-amorphous interphase. The crystalline defects in the rigid part could be significantly affected by processing parameters when employing the same type of polymerization technique. The intermediate region in the NMR spectra was analyzed according to the comparison between bimodal high density polyethylene and corresponding linear unimodal one. It was found that the mass fraction of the NMR interphase could be an indication of the percentage of tie molecules between crystalline lamellae and thus may significantly affect the mechanical properties of polymeric material.     

Crystal orientation in a semicrystalline polymer in relation to deformation of polymer spherulites. III. Small-angle light scattering

Small-angle polarized light scattering from a deformed three-dimensional spherulite is formulated on the basis of the deformation model proposed in Part II of this series. The intensity distribution of scattered light is discussed chiefly for the cross-polarization condition, the so-called H_v polarization, as a function of elongation of the spherulite. In the undeformed state, the scattered intensity distribution forms the typical fourleaf clover pattern, and the intensity decreases with increasing fraction of crystals oriented randomly (type R crystals) within the crystal lamellae of the spherulites. In a system composed of type R crystals and folded-chain crystals (type B crystals) within the lamellae, the four-leaf pattern moves to the horizontal zone near the equator with increasing elongation of the spherulite, and, simultaneously, extends to some extent to the vertical zone near the meridional direction as a parameter measuring the ease of lamellar untwisting increases. In a system composed, in addition to type R and type B crystals, of crystals transformed from type B to type C_a and type C_r due to tilting and unfolding of polymer chains, respectively, within the crystal lamellae an eight-leaf pattern appears, even at small elongation up to about 30%. Each lobe of the eight-leaf pattern undergoes a characteristic change with increasing elongation. In both systems, the scattered intensity increases with sharpening of orientation distribution of crystals within the crystal lamellae.     

Plastic deformation of polypropylene. IV. Wide-angle x-ray scattering in the neck region

Wide-angle x-ray scattering (WAXS) patterns of two polypropylene samples, a quenched sample drawn at 21°C and an annealed sample drawn at 100°C, were investigated in a range of values of draw ratio λ very closely spaced through the neck region. In both cases, a range of small λ where deformation occurred by spherulite deformation was followed by one of higher λ where microfibrils were formed. The contribution to the WAXS pattern of microfibrils could be clearly distinguished from that of deformed spherulites because of the better orientation parallel to the draw direction of the former as compared to the latter. Additionally, for a drawing temperature of 21°C, microfibrils crystallize in the “smectic” phase as compared to the monoclinic phase for the initial sample and deformed spherulites. At this temperature, plastic deformation proceeds through the spherulite deformation mechanism up to λ = 1.4 accompanied by an increase in chain orientation with increasing λ. For λ > 1.4 plastic deformation appears to occur exclusively through microfibril formation. For drawing at 100°C, spherulite deformation is accompanied by very little change in chain orientation up to λ = 2, where microfibril formation begins. For λ > 2 (T_d = 100°C) plastic deformation is accompanied by both microfibril formation and some spherulite deformation as reflected by changes in both orientation and crystallite size. At this temperature the lateral crystallite size in the microfibrils is related to the long period according to the “equilibrium crystallite shape” previously found for annealed polypropylene.     

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.     

Radical formation during tensile deformation of highly drawn crystalline poly[p-(2-hydroxyethoxy)benzoic acid] fibers

The micromechanism of tensile deformation of poly[p-(2-hydroxyethoxy)benzoic acid] fibers is discussed on the basis of a detailed esr study of radical formation. The concentration of primary phenoxy radicals, which were detected during deformation at room temperature as a direct indicator of main-chain rupture, was determined by extrapolating the radical decay curves at various strains to zero time. The relation between the initial radical concentration and the strain is well expressed by the cumulative normal distribution curve. By use of this relation and a model of fiber structure, the distribution of the contour length of tie chains was determined. No radicals were detected during a second stretching cycle until the maximum strain in the first run was exceeded. The deformation model which includes alternating crystalline and amorphous regions connected by tie chains, a distribution of contour lengths of tie chains, and a phase transformation of molecular chains in the crystalline region accounted fairly well for the observed stress–strain behavior of monofilaments in first and second stretching cycles. The comparison between the observed and the calculated radical concentration suggests that statistical factors and other deformation mechanisms have to be taken into account.     

The competition between crystallization and demixing in polymer blends. IV. Detection of composition inhomogeneities around growing spherulites

In polymer blends of an amorphous and a semicrystalline component, the crystallization kinetics and the resulting morphology are heavily determined by the diffusion ability of the whole chains and by the dwelling site of the amorphous polymer. Depending on the relative rates of spherulite growth and chain diffusion, radial composition profiles around the growing spherulites and a gradual increase of the melt bulk composition can develop. The resulting change in composition, particularly at the crystallization front, causes a corresponding temporal variation of the spherulite growth rate. In the present article, two experimental techniques are introduced to prove the existence and to determine the course of these concentration profiles. They are based on the composition dependences of the spherulite growth rate and the number density of primary nuclei. Their efficiency is demonstrated by measurements on PVDF/PEA blends. The blend composition at the crystal growth front was found to change by absolute 25%, and the width of the profile can amount to up to 70 μm. © 1996 John Wiley & Sons, Inc.