Small strain behavior of the semi-crystalline polymers has been widely studied during the last forty years because of their large range of application under visco-elastic solicitations. Moreover, a large part of these studies has been dedicated to polyethylene (PE) mainly because of its massive production and its relative simplicity. Its microstructure is known to be organised at different scales. At the mesoscale, it is formed of spherulites with a characteristic size around 1–10 μm.1–4 These spherulites are composed of radially stacked lamellae and interlamellar amorphous regions.
The analysis of the spherulite deformation is very complex because of their complicated morphology. Under tensile solicitation, the lamella stacks set in the equator regions, perpendicular to the tensile direction, are submitted to local stretching whereas the lamellae parallel to the tensile direction, are moved close due to Poisson effect.5
The spherulite deformation is undoubtedly influenced at the nanoscale by the lamella parameters such as the crystallinity, the crystallite thickness, and the intrinsic properties of the amorphous and the crystalline phases.
It is also clear that the local stress can be different from the macroscopic applied stress. Therefore, the overall deformation of the spherulite depends both on the local properties and on the local stress. It is very difficult to obtain quantitative information on the local stress, but it is likely correlated to lamella organization, entanglements and tie molecule which are known to be involved in stress transmission between the crystalline and the amorphous layers.6 They are also probably involved in the stress transmission at the scale of the spherulite. Their concentration is function of the crystallization conditions and the molecular parameters.7–10
The fact that all the PE do not all exhibit the same spherulite deformation was reported in the work of Hay and Keller.11 The spherulite deformation has been observed by optic microscopy. They have suggested that the macrodeformation of the spherulite during yielding could be classified into two different groups: homogeneous and heterogeneous. In the first case, all parts of a given spherulite extend simultaneously in constant proportion. In the heterogeneous case, the spherulites yield in one part only, the rest of the microstructure remaining unaltered. The same results were obtained on Nylon 6 by Galeski and Argon.12 Thanks to transmission electron microscopy, the spherulite deformation was observed and found to be heterogeneous, with a significant occurrence of cavitation in the equatorial regions during yielding. Lin and Argon13 have reported that in all cases, the microdeformations in the spherulite were nonuniform because of the radial orientation of plastically anisotropic crystallites and the presence of the amorphous region.
The aim of this work is to evaluate the influence of the local mechanical coupling (or stress transmission) on the deformation of the equatorial regions, but also on the entire spherulite deformation and thus on the macroscopic behavior.
Experiments are performed on several polyethylenes with controlled molecular parameters and subjected to different thermal treatments. Indeed, molecular topology differences coupled with various crystallization conditions allow obtaining wide range of microstructural characteristics such as crystallinity, crystallite thickness and also stress transmitter (ST) density. The local deformation in the spherulite will be analyzed following up the deformation of the lamella stacks from SAXS in situ tensile tests performed at the European synchrotron radiation facility (ESRF).
Four polyethylenes, obtained using the Philips method with a chromium oxide leading to a medium molecular weight Mw (between 180 and 230 kDa), are studied. They differ from their molecular topology and so can be classified into two different groups: PE A and PE B [respectively with a hexene co-unit (C6) content of 1.8 mol % and 0.8 mol %] (see Table 1) are considered as “branched” due to their significant C6 co-unit concentration.14 The PE C and PE D (0.1 and 0.2 mol %, respectively) are qualified as “quasi-linear” due to their lower C6 concentration. In each category, the polyethylenes differ from their crystallinity: the one of the PE A (49%) is lower than the one of the PE B (54%), and the same gap exists between the crystallinity of the PE C and PE D (65 and 69%, respectively). Beside these four polyethylenes, an ultra high molecular weight polyethylene (UHMWPE) is also studied. It is totally linear but because of its ultra high molecular weight (Mw is ca. 9 × 106 g/mol), its crystallinity in the quenched state is relatively low (49%).
Table 1. Initial Characteristics of the Different Polyethylenes
Five hundred micrometer-thick sheets are obtained from pellets molding between aluminum foils in a press at 170 °C. Then, the polymer sheets are quenched in water at a rate of ∼30 °C/s. Concerning the UHMWPE, 1 mm-thick sheets are obtained from industrial ones provided by Ticona. The sheets are then remelted at 170 °C before being quenched. To modify the microstructure, isothermal crystallizations are performed with two different processes. Samples designated hereafter as “annealed” are heated from their quenched-state to a temperature close to the crystallization temperature (see the Table 2) and are held in these conditions in a thermostatic oil bath for about 15 hours.
Table 2. Characteristics and Natural Draw Ratios of the Different Polyethylenes
The samples called “isotherm” are remelted at 170 °C in an oven before being cooled in a thermostatic oil bath at a temperature close to the crystallization temperature (see the Table 2) and held in these conditions during fifteen hours. Samples are tightly wrapped to avoid oil contamination from the thermostatic bath. Moreover, the infra-red technique does not show peak at 1720 cm−1 relating to the carbonyl and consequently probes that the oxidation is insignificant.
To characterize the mechanical properties of the samples, tensile tests have been carried out. Dumbbell-shaped samples of 22 mm gauge length, about 5 mm width, and about 0.5 mm of thickness (1 mm for the UHMWPE) were cut from the sheets. A series of tests was performed with a MTS machine at 23 °C. They were carried out with a crosshead-speed constant, fixed to have an initial strain rate of 5 × 10−3 s−1 and to obtain nominal stress/strain curves. These tests allow measuring in particular the natural draw ratio (λn) with the method of tangents.6
Differential Scanning Calorimetry (DSC)
The thermal analysis of the samples was conducted using an indium-calibrated Perkin–Elmer DSC7 apparatus. Five to eight milligram samples were cut from the polymer sheets, and placed into aluminum pans. The melting thermograms were recorded at a heating rate of 5 °C/min, under nitrogen flow. The crystallinity (Xc) was calculated at ±1% using eq 1:
where ΔHf is the specific heat of fusion of the specimen and is the heat of fusion of a perfect crystal and equal to 290 J/g.15
Small-angle X-Ray scattering experiments were carried out on the CRG D2AM beam-line of the ESRF (Grenoble France).
The measurements were performed using a wave length of 1.54 Å allowing an observation range in q from 0.05 to 0.95 nm−1. The SAXS patterns were recorded thanks to a 2D detector and then analyzed computing the intensity by azimuthal integration over sectors of ±7.5°. The data corrections (dark current, flat field response and tapper distortion) were carried out using the software available on the beamline. Silver behenate was used for the q-range calibration.
The long period (Lp) was calculated from the maximum of the diffuse intensity corrected by the Lorentz factor [Iq2 = f(q)] with the relation:
qmax corresponding to the peak maximum.
The thicknesses of the amorphous layer (La) and the crystallite thickness (Lc) were deduced, with a precision of ±10%, from the long period Lp and the crystallinity (Xc) (measured by DSC) thanks to the following relations:
With ρc the crystalline density equals to 1.003 g cm−3 and ρ the density of the sample.16
Thanks to a homemade miniature stretching device, in situ tensile tests were performed on dumbbell-shaped samples of gauge length 6.5 mm, width about 4 mm, and thickness of about 0.5 mm (1 mm for the UHMWPE). The samples were stretched at an initial strain rate of 6.4 × 10−4 s−1. The symmetric displacement of the two clamps allows probing the same zone during the tensile test. The pattern acquisition time was chosen to have a good compromise between the pattern quality and the smallest deformation as possible during the acquisition. Hence the acquisition time was 5s, which corresponded to a deformation about 0.3%.
Estimation of ST Density
Elements as Tie Molecules (TMs), entanglements and possibly the partial percolation of the crystalline phase participate to the network which is likely essential to transmit the applied stress through the lamella stacks (Fig. 1). Their mechanical contribution can hardly be dissociated by experimental methods, that is why we consider that they all belong to the stress transmission.
Contrary to the crystallinity Xc or the long period Lp, the density of ST is difficult to quantify. To our knowledge, direct measurement of the ST concentration is impossible. However, several indirect approaches have been proposed to qualify their density. Strain hardening -measured in compression- on true stress/strain curve can be used as an indicator of the macromolecular network as it was found to be attributed to both chain entanglement density and TM concentration.17, 18 Beside, the natural draw ratio is also considered as a ST concentration indicator since it is correlated to the density of tie molecules and entanglements.9, 19 The neck width measured on the nominal stress/strain curve is also representative of the density of the STs.6 Beside these three indirect measurement methods, Brown and coworkers have proposed a statistical model which predicts the probability for a chain to be tied. It was based on the hypothesis that the gyration diameter had to be superior to 2Lc + La20 or 2Lp21 to make the molecule tying.
In this study, we have chosen to use the natural draw ratio λn, but the neck width leads to equivalent results. Then the values of natural draw ratio are compared with those calculated with the Brown's model choosing 2Lp as the criterion of the calculation. The criterion 2Lc+ La has also been studied and it gives similar results.
The values of natural draw ratio λn and those of the Brown's model are reported in the Table 2.
λn is plotted versus the ST density calculated with Brown's model on Figure 2. An excellent correlation is highlighted. Hence, both ST density evaluations give the same characterization of the materials and it can be concluded that the natural draw ratio is well representative of the ST density.
To analyse the influence of the crystallization conditions and the co-unit contents on the density of ST, λn is plotted against the co-unit content on Figure 3; as λn evolution is opposed to ST density evolution, the scale has been inversed. Looking at the influence of thermal treatment (see arrows on Fig. 3), the results are consistent with the ones of Seguela8 or Tarin and Thomas.22 Indeed, it is found that the isotherm treatment prevents the formation of STs thanks to a better folding of the chains during the slow crystallization. It is worth noticing that annealing treatment leads to the same trend but with a much lower efficiency. As expected, quenching allows obtaining high ST density favoring the formation of a disordered structure.
Beside, the influence of the co-unit content can also be studied (see the dotted lines on Fig. 3). As expected, a high co-unit content allows obtaining a higher ST density than a low co-unit content. Indeed, a branch can be considered as a defect that can not be incorporated in the crystal.15, 23, 24 Hence, during chains folding, co-units are rejected in the amorphous phase, favoring the formation of tie molecules.
The evolution of the ST density can also be analysed against the crystallinity (see Fig. 4). It is found that he lower the crystallinity is, the higher the ST density is.
Follow-Up of the Deformation
As described by Hubert et al.,5 the two-dimensional SAXS initial pattern shows an isotropic scattering ring that is characteristic of periodic lamella stacks randomly oriented (see Fig. 5). During stretching at small strains, the lamella stacks perpendicularly oriented to the tensile direction (equatorial lamellae) are submitted to local stretching whereas the lamellae parallel to the tensile direction (polar lamellae), are moved close due to Poisson effect.5 Hence, the initial scattering ring turns into an oval-shaped, indicating that the perpendicular long spacing Lp increases whereas parallel long spacing Lp decreases (see Fig. 5). Concerning UHMWPE, even if it does not exhibit spherulitic structure, the same analysis can be proposed since the lamella stacks are isotropically distributed and thus, the two extreme lamella orientations also exist.
In this study, our analysis is limited exclusively to the deformation of the lamellae initially perpendicular to the tensile direction. First, because it is more directly interpretable and also because it corresponds to the local deformation measured in the tensile direction. It has to be mentioned that only the intensity diffused by the lamellae with a normal perpendicular to the incident beam can be detected. However, they are considered to be representatives of all the lamellae perpendicular to the tensile direction.
The SAXS patterns have been analysed to quantify the local deformation. The intensity has been computed by azimuthal integration over a sector of ±7.5°, in the horizontal zones of the pattern, that is, in the zones perpendicular to the tensile axis, before being subjected to the Lorentz correction. The curves Iq2 versus q can thus be plotted (see Fig. 6).
The scattered intensity has not been normalized by the sample thickness as the sole position of the peak is sufficient to perform our analysis. Consequently, It can be seen that the scattered intensity decreases along the tensile test. As expected, it is observed that the peak moves to the small q, which indicates that the long period increases when submitted to tensile solicitation.
Comparison Between Local Strains and Macroscopic Strains
As the long period can be measured for all the patterns along the tensile test, the deformation ε = ΔLp/Lp can be calculated and followed-up.
The local strain εp = ΔLp/Lp can thus be analyzed versus the macroscopic nominal strain. The two cases illustrated on Figure 7(a,b) are two extreme cases of deformation behavior in the set of samples.
The local strain of the PE D isotherm and linear is plotted versus the macroscopic nominal strain on Figure 7(a). For very small strains, before yielding, a linear relation between the local strain and the macroscopic deformation is highlighted. Then a change of slope appears before the yield stress which has been correlated to the initiation of cavitation [see Fig. 7(a)]. Just after, the acquisition had to be stopped because the scattered intensity was too strong. Indeed, voids scatter much more than the lamella stacks and thus the intensity scattered by the lamellae can not be deconvoluted precisely.
The behavior of the PE A (branched) quenched is then plotted on Figure 7(b). All along its drawing, this sample does not show any phenomenon of cavitation. Until 30%, a linear relation between local and macroscopic strains can be observed. After yielding, the long period Lp is no more measurable, which indicates the initiation of the fibrillar transformation. It can be noticed that in this case, the linear relation goes on for larger strains than in the case of the PE D (linear) isotherm.
The attention will be focused exclusively on the visco-elastic behavior, where the relation between εlocal and εmacro is linear for all the PE. This relation remains linear while the initiation of the fibrillar transformation or the cavitation have not occurred yet.
For each sample, the ratio εlocal /εmacro is measured [see Figure 7(a,b)].
Deformation of the Equatorial Region Versus the Deformation of the Polar Region
To analyze the influence of the crystallinity on small strain behavior, the ratios εlocal/εmacro of all the samples are plotted against the crystallinity on Figure 8.
The ratio εlocal/εmacro is found to be approximately constant, or slightly decreasing with the crystallinity, and the medium ratio is equal to 0.5. This value is quite surprising as it means that the deformation of the equatorial regions in the tensile direction is twice less deformed than the whole sample. It implies also that the spherulite deformation is not homogeneous. In a simplistic model, a serie and parallel coupling could be associated respectively to the equatorial and the polar regions, thus the equatorial regions should be the softer. The consequences are important. Such a modelling implies that the local modulus continuously increases from the equatorial to the polar regions. As a first hypothesis, we can consider that the stress is homogeneous. The equatorial regions should be then the most deformed. The experimental results are clearly in contradiction with these assertions.
They reveal that the equatorial regions are twice less deformed than the global material and suggest that the polar regions should be much more deformed to accommodate the macroscopic deformation. This conclusion is not intuitive but is not in contradiction with the previous results found in the literature which show that the equatorial regions are strongly deformed but only after yielding.11, 12
To propose a plausible explanation for the experimental results, the simple model described previously has to be reconsidered. Indeed, if the polar regions were correctly modelled by a parallel model, the local modulus of the polar regions Epolar should be very high, much higher than the medium modulus of the PE. Taking the modulus of the crystalline phase Ec equal to 7 GPa,25 the amorphous modulus Ea equal to 5MPa and a crystallinity of 50%, the modulus Epolar reaches 3500 MPa.
Therefore, the polar regions would rather be closer to a Takayanagi's model indicated on the Figure 9. This model takes into account the fact that the crystallites can be radially prefractured and separated by a small amount of amorphous phase (A2). The interspherolitic zones are also taken into account in this small amount of amorphous phase. The amorphous phase A1 indicated on the Figure 9(a) represents the amorphous phase present in the lamella stacks.
Only 2% content of A2 leads to decrease the modulus of the polar lamella stacks enough to be inferior to the medium modulus: with a global crystallinity of 50%, when increasing the introduced amount of amorphous phase from 0 to 2%, the local modulus decreases from 3500 MPa to 230 MPa.
The equatorial lamella stacks have now to be considered. First, they are organized identically as the polar lamella stacks. However, the orientation of the tensile direction leads to propose a serie coupling model [see Fig. 9(b)].
The local modulus of the equatorial regions can thus be calculated. Taking the same values of Ec, Ea and Xc, the local modulus Eeq equals 10 MPa. This value is very low, much lower than the medium modulus of the PE and the calculated modulus of the polar regions. Therefore, this modulus is not high enough to explain the experimental results.
The local deformation εlocal can be expressed with the local stress σlocal and the local modulus Elocal as followed:
Therefore, to explain the results two cases can be considered: the local modulus of the equatorial regions Eeq is higher than the one of the polar regions Epolar (which is not the current prediction of the Takayanagi's model), or the local stress applied on the equatorial regions σeq is much lower than the one on the polar regions σpolar.
Considering the first hypothesis, it exists reasonable physical reasons which could advocate for the idea that the equatorial region is stiffer than the polar one. Indeed, during the process of crystallization, it is suspected that the simultaneous formation of two neighbouring crystals can generate the formation of tie molecules that are finally nearly tight in the solid state. These partially tight tie molecules can strongly affect the amorphous modulus Ea of the equatorial lamella stacks as the modulus of a tight chain is around 185GPa.26
For a 50% crystallinity PE, only a content of 0.15% of tight tie molecules in parallel in the amorphous phase is necessary to increase the modulus of the amorphous phase from 5 MPa to 250 MPa. It leads then to a modulus of the equatorial stack close to the one of the average material (around 450 MPa).
To be consistent with the experimental data a very strong amorphous modulus (250 MPa) has to be chosen. It is worth noticing that these tight tie molecules affect Ea as long as they are along the tensile direction. Therefore, the amorphous modulus of the polar lamellae should be significantly less affected. The amorphous phase would be then strongly anisotropic and its average value would be much lower than 250 MPa.
It is also possible that the second hypothesis plays a significant role so that the amorphous modulus could be a little lower. Indeed, the distribution of the stresses in the spherulite can be heterogeneous and then explain that the polar regions are more deformed than the equatorial regions. It is possible that the specific arrangement of the lamellae in the spherulite induces a local stress in the polar regions σpolar higher than in the equatorial regions σeq. The following discussion will propose few arguments to support this hypothesis.
Discussion: influence of the crystallinity and the ST density on the deformation of the equatorial regions of the different PE.
On the Figure 8, the second surprising observation is that the ratio εlocal/εmacro is approximately constant whatever the chosen PE. However, their molecular and microstructural parameters such as the crystallinity and the ST density are very different, and they were supposed to affect the deformation of the spherulite.
Considering that the equatorial stacks behave approximately as a serie coupling model, an increase in the hard phase content has much less influence on the global modulus than an increase in the soft phase modulus. Consequently, it can be considered that the modulus of the equatorial regions is essentially and strongly affected by the tight tie molecules. Therefore, for a constant local stress, the higher the ST density is, and so the lower the crystallinity is, the less deformed the equatorial lamella stacks should be.
However, the ratio εlocal/εmacro is found to be constant for all the PE, which leads to conclude that an other contrary phenomenon affects this local equatorial deformation. An explanation can be proposed using the following relation:
The constant is found experimentally equal to 0.5.
When the crystallinity increases, Emacro increases while Eeq decreases because of the decrease of the ST density. Hence increases. To keep constant the ratio εlocal/εmacro, the ratio has to decrease. Hence, the local stress respectively to the macroscopic stress should not be constant as it was assumed before.
Therefore, the distribution of the stresses in the spherulite could be function of the crystallinity. The higher the crystallinity is, the less stressed the equatorial lamellae are. Hence, the increase of the crystallinity has the effect to unload the equatorial regions probably thanks to a partial percolation of the crystallites.
Finally, the ST density and the crystallinity have opposite effects on the local deformation of the equatorial lamella stacks, which explain the constancy of the ratio εlocal/εmacro for all the PE (see Fig. 10).
It can be also noticed that the ratio εlocal/εmacro slightly decreases with the crystallinity. Therefore, it seems that the crystallinity (and thus the possible partial percolation) would have a stronger influence on the ratio εlocal/εmacro than the density of tie molecules.
The lamella stack deformation depends on their orientation with respect to the loading axis. The lamella stacks perpendicularly oriented to the tensile direction (equatorial regions) are submitted to local stretching whereas the lamellae parallel to tensile direction (polar regions), are moved close. Obviously, the local deformations in all the regions of the spherulite are controlled by the microstructural parameters such as the crystalline and the amorphous lamella thicknesses, and the molecular parameters. The influence of the ST density and the crystallinity on the axial spherulite deformation is investigated in this work.
Various polyethylenes with different molecular parameters are subjected to different thermal treatments and a wide range of microstructural parameters and ST density is obtained. The density of STs is well quantified by the natural draw ratio and calculated by the Brown's model.
From SAXS in situ tensile tests, the local deformation that takes place in the equatorial regions is measured. Comparing the local deformation to the macroscopic deformation allows detecting that the deformation of the spherulite is not homogeneous. Indeed, the ratio εlocal/εmacro is found to be nearly constant and equal to 0.5. This value indicates that the equatorial regions are less deformed than the polar regions in the tensile direction. Two main causes can be considered to explain this result. First, the local modulus of the equatorial lamella stacks can be strongly increased due to the presence of a small amount of tight tie molecules. Secondly, due to the specific arrangement of the lamellae in the spherulite, the equatorial regions are possibly less stressed than the polar regions.
The constancy of the ratio εlocal/εmacro has been found to result from two opposite phenomena: when increasing the crystallinity, the ST density decreases which causes the decrease of the equatorial local modulus and then the increase of the deformation. In addition when increasing the crystallinity, a possible partial percolation of the crystalline phase would lead to unload the equatorial lamella stacks and then decreases their deformation.
The authors thank Jenny Faucheu and Emilie Planes for their help in the experiments at the ESRF.