A Common Multiscale Feature of the Deformation Mechanisms of a

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A Common Multiscale Feature of the Deformation Mechanisms of a Semicrystalline Polymer L. Farge,*,† S. André,*,† F. Meneau,‡ J. Dillet,† and C. Cunat† †

Laboratoire d’Energétique et de Mécanique Théorique et Appliquée CNRS UMR 7563, University of Lorraine, 2, avenue de la Forêt de Haye TSA 60604-54518 Vandoeuvre-lès-Nancy, Cedex, France ‡ Soleil synchrotron, Swing Beamline L’Orme des Merisiers, 91190 Saint-Aubin, France ABSTRACT: The deformation mechanisms of a semicrystalline polymer (SCP) subjected to tensile loading were experimentally studied from nano to micro and macro length scales. The paper’s focus is on the development of anisotropic features at all scales during deformation. At the nanoscale, original results are presented based on in-situ SAXS (small-angle X-ray scattering) measurements. At the microscale, previously obtained results using in-situ ISLT (incoherent steady light transport) and postmortem SRXTM (synchrotron radiation X-ray tomographic microscopy) are briefly summarized. For all these techniques, original data treatments were carried out for quantifying the evolution of anisotropy in the material’s microstructure. At the macroscopic scale, a 3D digital image correlation technique was used for determining all the components of the Hencky strain tensor in the center of the necking region. New results are presented in this field for the studied SCP in terms of the quantity D = εL/|εT|: ratio of the longitudinal strain (along the drawing axis) to the transverse strain (perpendicular to the drawing axis). The main result of this study lies in a plot of variable D as a function of εL. This plot evidences the same three clearly distinct regimes as those obtained using SAXS, ISLT, and SRXTM experiments to measure anisotropy quantitatively at the microstructural level. This result proves that everything occurring at the microscale gives a signature at the macroscale and hence opens up new routes for the modeling of the mechanical behavior of such materials.

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INTRODUCTION In view of the very complex association of amorphous and crystalline domains of so-called semicrystalline polymers (SCPs), a full description of deformation mechanisms escapes, up to now, to a unified explicative scheme. Their multiscale character explains this difficulty and first because all available experimental techniques must be used by researchers, which means different probing scales, different nature of information (from purely qualitative to quantitative), different physical processes (and therefore modeling) to get it, different instruments and technologies to master, etc. A second obstacle lies in the huge heterogeneity of the microstructure of SCP studied specimens. This is a common feature of materials with glass transition, which is due to an extremely high sensitivity to the elaboration process: initial liquid state, cooling rates and temperature gradients in the bulk, nature of the macromolecules, presence of additional chemical species. Regarding the first point, the bibliography is very rich in observations and measurements made at different levels, especially thanks to recent improvements in the achievable spatial resolution. Starting from the lower scales, WAXS (wideangle X-ray scattering) and SAXS (small-angle X-ray scattering) techniques take advantage of the high energy available from synchrotron sources to produce X-rays for diffraction interaction. They cover the 1−100 nm scale and allow for insitu studies using tensile stages.1−4 Techniques allowing for © XXXX American Chemical Society

visualization of microstructure and digital image treatment as well are presently promoting a great step in our knowledge. Microscopy (scanning electron microscopy, transmission electron microscopy) observation is one of them,5 but the role of SRXTM (synchrotron radiation X-ray tomographic microscopy) will play a key role in the next decades as illustrated by recent results.6,7 Spatial resolution of the order of 1 μm or less is now achievable, and in-situ testing conditions will be the next advance. Quantitative treatment of 3D volumes brings valuable information to analyze the cavitation phenomenon6,8,9 or alternative mechanisms in the case of not cavitating materials.7 For nonopaque (turbid) materials, light scattering techniques and especially IPSLT (incoherent polarized steady light transport) allow for probing the mesoscopic scale with interesting features like dynamic, nonintrusive, and in-situ testing on conventional testing machines with low-cost equipment.10 At the macroscopic scale, thermal imaging was also used on SCPs in order to catch thermal effects as the thermodynamic manifestation of microstructural reorganization.11 Finally, video extensometry was used to study macroscopic scale effects through volume strain measurements, especially in the perspective of analyzing cavitation.12 Received: September 24, 2013 Revised: November 26, 2013

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dx.doi.org/10.1021/ma4019747 | Macromolecules XXXX, XXX, XXX−XXX

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Regarding the second difficulty mentioned above to study SCPs, the strategy we used was to focus our studies during more than 10 years on the same SCP, a commercial grade of high-density polyethylene (HDPE) with continued supply. This HDPE material has been studied with most of the techniques cited before, taking advantage of in-situ conditions whenever possible. All results presented were obtained in tensile test conditions. The Material section (section 1) will give the current knowledge we have on this material. In section 2, results will be presented, which are related to the microstructure evolution in HDPE deformed under uniaxial tension, especially by monitoring quantities describing its anisotropy. These results come from investigations based on SAXS (nanoscale), SRXTM (microscale), and ISLT (incoherent steady light transport: microscale). For all these techniques the raw information corresponds to intensity patterns, the shapes of which depend on the microstructural anisotropy. The specific image treatments that were applied to the SAXS, ISLT, and SRXTM data for quantifying the microstructure anisotropy degree at the nano and micro scales as a function of the longitudinal strain (along the drawing axis) will be given in section 2. The DIC (digital image correlation) technique allows nowadays investigating more precisely the material’s macroscopic deformation state. We present in section 3 original results regarding volume strain measurements obtained through 3D-DIC measurements performed on two perpendicular faces of a specimen during a continuous tensile test. Contrarily to previous studies,12,13 the results presented here using 3D-DIC clearly show the experimental errors that can be made using in-plane video extensometry. Similar use of 3D-DIC to the one presented in this paper has been made by Grytten et al.14 It was applied to a talc/elastomer modified propylene compound and produced data on true stress/true strain curves, Poisson’s ratio, and volumetric strains. Such measurements will be presented here for HDPE. From all these results, we present in section 4 an original result which proves that a consistent signature can be drawn from all observations from the nanometric scale (X-ray investigation) to the millimetric scale (macroscopic strain measurements). In section 5, in view of all the results previously presented and of the main results drawn from the literature, we discuss the microstructural transformations that occur in the material during the drawing process.

determined from the correlation function K(x), calculated by the cosine transformation of the intensity profiles.15,16 The structural characteristics of our material at room temperature were found to be long period of the stack of 26.8 nm, crystalline phase thickness 18.5 nm, and amorphous layer thickness 8.2 nm. The crystalline volume fraction deduced from these measurements was found to be 69% (confirming DSC results). The morphological configuration of the undeformed HDPE under cross-polarized microscope does not reveal any spherulitic structure at the mesoscale. Quantitative characterizations reported below have been performed at various scales in the center of the necking region when subjecting the material to tensile deformation (see Figure 1).

1. MATERIAL AND SPECIMEN GEOMETRY The SCP studied in this work is a HDPE produced by Röchling Engineering Plastics KG (grade ‘‘500 Natural’’). The specimens were manufactured by extrusion process and supplied in 6 mm thickness sheets. Information from the supplier indicates molecular weight and density of 500 000 g/mol and 0.95 g/ cm3, respectively. Differential scanning calorimetry yields a crystallinity index of about 68 wt % for the specimen. The yield stress is 33 MPa, corresponding to a 0.1 value for the yield strain. Synchrotron SAXS experiments reported in the next section have permitted the morphological characterization of the undeformed material. With an appropriate oven, a fusion experiment has been realized up to 140 °C. Integrated SAXS intensity profiles were calculated from the pattern recorded during the heating and cooling stages, from which the average background scattering in the melting state was retrieved. The average morphological parameters of the lamellar stacks were

2. CHARACTERIZATIONS AT THE MICROSCALE: QUANTIFICATION OF THE MICROSTRUCTURE ANISOTROPY Results obtained using three techniques will be reviewed here with two driving goals: (i) to concentrate on results relating to the key aims of this study but also (ii) to give additional information on the material studied in this work in order to prepare the elements of the discussion. The first one concerns in-situ SAXS experiments performed with a synchrotron light source and hence characterization of the HDPE material at the nanoscale. These results have not been published yet and deserve some developments here accounting for the quantitative treatment, which has been made within the scope of the paper to determine the orientation distribution functions of the scatterers. The two other ones concern in-situ light scattering experiments and postmortem X-tomography, both of them

Figure 1. Experimental arrangement. On the right: enlargement of the central region of the dog-bone-shaped specimen with axis labeling.

In almost all experiments, the initial cross section in the center of the dog-bone-shaped specimens was 6 × 6 mm2 in order to favor pure bulk mechanisms. In order to promote uniaxial stress state in the center of the specimen, a 6 mm length flat part was machined on the specimen lateral faces (see Figure 1). In the whole paper, axis 1 corresponds to the tensile axis. Axes 2 and 3 respectively correspond to the width and thickness directions (see right part of Figure 1). The specimen deformation evolution is characterized through the true longitudinal strain ε11 = ln(l/l0).

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Figure 2. In-situ SAXS patterns at different strain levels.

dealing with a micrometric characterization and having been already fully discussed in other places.7,9,10 Therefore, in these two last cases, only a brief description will be made to explain how the quantitative measurement of a medium anisotropy index is obtained as a function of the deformation level of the SCP. SAXS (Small-Angle X-ray Scattering) Experiments. Xray scattering experiments have been performed at the French synchrotron source “Soleil” on the “Swing” beamline. The energy of the beam was set to 6 keV, and the detector/sample distance was adjusted to 0.17 or 6 m respectively for WAXS or SAXS measurements. Deformation mechanisms were studied in situ under tension, using a mini tensile stage (Kammrath & Weiss, 5 kN) allowing for deformation of samples having the same cross section as used in classical testing machines (for displacement rates up to 20 μs−1). According to the specific objective of this article, the results presented here will focus only on the anisotropic characterization of scatterers from SAXS data in the [0.03−0.8 nm−1] q-range. Figure 2 presents several patterns recorded at different deformation states. As it has been observed by many authors,1,17−23 the SAXS patterns reveal a very strong increase in intensity when reaching the yield point compared to the initial undeformed state, an anisotropy of the image in transverse direction (see image for ε11 = 0.09), a return back to global isotropic behavior (ε11 = 0.45), followed by a more and more pronounced anisotropic pattern in the tensile direction. The elliptical shape of the patterns results from both the development of anisotropic scatterers during tension and evolving orientational distribution. It is not straightforward to extract information from such SAXS patterns where the scatterers are not known (evolving shape, unknown structure, evolving orientational distribution, evolving size, polydispersity, etc.), and there is no unique pathway to make some anisotropy analysis. The choice that has been made to analyze those patterns (100 images acquired in the range 0 ≤ ε11 ≤ 1.9) is the following:22 (i) Averaged intensity profiles over a small angle (less than ±10°) are calculated in the tensile direction 1 and in the perpendicular transverse direction 2. (ii) The log−log plots of these profiles as a function of the modulus of the scattering vector q (see Figure 3 as an example corresponding to image ε11 = 0.7 in Figure 2) are typical for systems of (polydispersed) oblate ellipsoids or platelets24 and have slopes between −2 and −4. Starting from high q value, the intensity decreases with a particular scattering power law which is supposed to reflect the signature of primary particles. Going down to low values makes the intensity more sensitive to interparticle correlations or structure factor (generally identified with a clear peak). A characteristic point at q1 ≈ 0.1 nm−1 can be put in evidence as resulting from a break in slope (Figure 3). This point is determined precisely as a maximum in a Kratky-

Figure 3. Intensity versus q profiles for transverse and tensile directions (ε11 = 0.7).

like plot qA ← max[qnI(q)], where n was set to 3.25 here, a value that corresponds to equal slopes of the large q regime q > qA, for both vertical (longitudinal) and horizontal (transversal) profiles. In such systems, the origin of this crossover is very difficult to establish. It can be the signature of a change in particle sizes (form factor effect) or of a change in interparticle correlations (structure factor effect). Although we believe here that it is due to the presence of primary particles (because it appears roughly independent of the strain and hence of the scatterer’s concentration), it is possible to focus on this particular point to study the sole effect of the orientational distribution function. (iii) For all peak positions qA(ε) at all deformation stages, we can plot the angular distribution of intensity over 360° (Figure 4). This plot exhibits a wavy shapeaccording to the elliptical shape of the patternsand is directly proportional to the orientational distribution function (odf) expressed as a function of the plane angle in the detector frame (see Méheust et al.25 and the Appendix in Van der Beek et al.26 for full explanations). The mathematical form we use for the odf was found in Bihannic et al.27 where it was used to weight the form factor of disk-shaped objects randomly oriented in dilute 3D systems in order to simulate 2D intensity patterns. (iv) The so-called ellipsoid probability density function (or odf) has been used directly to fit the I(Ψ) curve at a given qA (Figure 4). This function is ⎡⎛ b ⎞ 2 ⎤−3/2 2 2 PDFellipsoid(Ψ) = ⎢⎜ ⎟ cos (Ψ) + sin (Ψ)⎥ ⎣⎝ a ⎠ ⎦

(1)

and depends on the angle Ψ between horizontal and vertical directions of the frame detector and on the axis length ratio C


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Figure 5. Tomographic images and 2D Fourier transforms those images (obtained by averaging the results for 500 equiv tomographic views). (a) Amicro SRXTM < 0: microstructure elongated perpendicularly to the tensile axis. (b) Amicro SRXTM > 0: microstructure elongated along the tensile axis.

Figure 4. Rescaled angular intensity at ε11 = 0.7 with curve fitting.

b/a. It can be easily derived from the formula (5) given in Bihannic et al.27 in the case of azimuthal symmetry (symmetry around the drawing axis) of the specimen (ϕ = 0) which can be easily considered here. The curve-fitting procedure (Figure 4) allows the identification of the minor semiaxis to major semiaxis ratio b/a, which is considered to give a quantitative anisotropy characterization of the medium. (v) This can be made at all strain states (for all acquired patterns) to produce the anisotropy index Anano SAXS(ε11) obtained by SAXS (nanoscale), defined as follows ⎧ a /b − 1 < 0 when b/a > 1 nano ASAXS (ε11) = ⎨ ⎩1 − b/a > 0 when b/a < 1

tensile direction. It is possible to observe intensity peaks in the patterns. It shows that the size distribution of the microstructural objects present in the real image is relatively narrow. The analysis of Fourier transform patterns makes it possible to clearly show characteristics that may be subtly disseminated in the whole real space image. For instance, the microstructural anisotropy is very difficult to distinguish in the image corresponding to Figure 5a, top (low strain level). In reciprocal space, by contrast, the local microstructural anisotropy is clearly highlighted by the shape of the Fourier transform image (Figure 5a, bottom). For very deformed states (see Figure 5b, top), the microstructural anisotropy clearly appears even in the real space images. It corresponds to a well-established fibrillar state. However, in real space, it is not easy to figure out any straightforward and simple procedure that could allow for quantifying the degree of anisotropy. In reciprocal space, by contrast, an anisotropy index can easily be defined: I − IV micro (ε11) = H ASRXTM IH + IV (3)

(2)

which is negative when anisotropy is in the transverse 2 direction and positive in the tensile 1 direction. The plot Anano SAXS(ε11) is given in Figure 8 and will be discussed further. SRXTM (Synchrotron Radiation X-ray Tomographic Microscopy) Experiments. The experiments were performed on the TOMCAT beamline at the Swiss Light Source (Paul Scherrer Institute). The 3D volume reconstructions were obtained by mapping a series of X-ray projections acquired over 180° of sample rotation. At the maximum allowable optical magnification, a voxel size of 0.38 μm in each dimension was achieved. The material’s internal structure can then be reconstructed with a spatial resolution of about 1 μm.7 The microstructure anisotropy analysis was performed in reciprocal space. Noise was reduced by calculating average Fourier transforms of the images over 500 equiv tomographic views corresponding to parallel planes along the drawing axis. These planes are taken in the material core region. Examples of such tomographic views are shown in Figure 5 for low (Figure 5a, top) and high (Figure 5b, top) strain levels. For small strain levels, the 2D Fourier transform patterns are clearly anisotropic and elongated along the drawing axis (Figure 5a, bottom). It shows that at the micrometric level the microstructure is oriented perpendicularly to the drawing axis. For very deformed states, the 2D Fourier transform patterns become oriented perpendicularly to the drawing axis (Figure 5b, bottom). The microstructure is then elongated along the

IV and IH are respectively the intensity values of the Fourier transform images taken on the vertical and on the horizontal axis at the first peak location. If Amicro SRXTM > 0, the microstructure is elongated along the drawing direction. If Amicro SRXTM < 0, it is transversely oriented. It should be noted that unlike ISLT and SAXS measurements, only postmortem experiments can be performed with the SRXTM technique. ISLT (Incoherent Steady Light Transport) Experiments. Starting from the yield point, the HDPE studied in this work whitens during tensile deformation. The deformationinduced whitening phenomenon is due to the appearance in the medium of micrometric microstructural discontinuities. Because their size is comparable to the visible wavelengths, all wavelengths of visible light are scattered approximately identically (Mie scattering).28 It results in the whitening phenomenon. The principle of an in-situ incoherent steady light transport (ISLT) experiment is relatively simple. A laser beam (wavelength λ = 635 nm, spot diameter 50 μm) is D


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focused in the center of the necking region of the HDPE material while it is being deformed. The backscattered intensity patterns are recorded with a digital camera. Figure 6 shows

In the case of the considered material, it was clearly shown that the light scatterers have a micrometric size.10 The backscattered intensity figures are clearly not isotropic (Figure 6). This results from the anisotropy in the scattering medium at the micrometric scale. In order to quantify the anisotropy developed within the material, an anisotropy index Amicro ISLT similar to the one defined for SRXTM study can be introduced. It is based on the image intensities taken in two perpendicular directions at ρ = 2lTR: micro AISLT (ε11) =

IV − IH IH + IV

(4)

IH is taken for θ = 0 ± 5° (or 180 ± 5°). IV is taken for θ = 90 ± 5° (or 270 ± 5°). The interpretation of this anisotropy index is the same as for nano micro Amicro SRXTM and ASAXS; AISLT > 0: orientation along the drawing micro direction, AISLT < 0: transverse orientation. This anisotropy index definition comes directly from the theory of light transport in scattering media (see section “Image Anisotropy Analysis” in the paper of Blaise et al.7). The orientation of the ISLT backscattered intensity patterns is the same as that of the microstructure. In the case of SAXS and SRXTM techniques, the analysis was performed in reciprocal space in which the intensity patterns are oriented perpendicularly with respect to the microstructure orientation. This is the reason why the numerator in eq 3 is IH − IV while it is the opposite quantity in eq 4. micro micro The plots Anano SAXS(ε11), AISLT (ε11), and ASRXTM(ε11) are given in Figure 10 and will be discussed further.

Figure 6. Example of backscattered intensity figures for a semicrystalline polymer: (a) ε11 = 0.17, Amicro ISLT < 0 microstructure elongated micro > 0 perpendicularly to the tensile axis. (b) ε11 = 1.6, AISLT microstructure elongated along the tensile axis.

backscattered intensity patterns corresponding to small (Figure 6a) and high (Figure 6b) strain levels. On the upper part of the figure, we show the backscattered intensity patterns as they appear to the naked eye. On the lower part, the backscattered intensity patterns are represented using pseudocolors. The transport length (lTR) corresponds to a kind of mean free path of photons inside the medium, which is defined in the diffusion approximation (multiple scattering events make equiprobable directions for photons path). The spatial extension of the backscattered intensity pattern is thus directly related to this parameter. An appropriate image treatment was developed in our laboratory for extracting the transport length from the backscattered intensity pattern.9−11,29 This quantity decreases when the amount of scattering in the material increases. So the lTR measurement allows for an indirect quantification of the deformation-induced whitening phenomenon. Using polarized light, it is also possible to evaluate the size of the microstructural discontinuities that scatter the visible light.30

3. CHARACTERIZATION OF STRAIN AT THE MACROSCALE USING 3D DIGITAL IMAGE CORRELATION Information on the Measurement Technique. We now proceed to the presentation of new results regarding volume strain measurements obtained using 3D-digital image correlation (3D-DIC) applied to a random patterning sprayed (speckle) on the specimen subjected to tensile testing. Figure 1 shows the initial geometry of the specimen. The necking phenomenon appears in the central area of the specimen. In this region, in the undeformed state, the width and thickness have the same 6 mm value. The image correlation technique needs post-treatment and cannot be used in real time for monitoring the tensile test strain rate. For that purpose, we have used a video extensometer (videotraction) based on

Figure 7. (a) Undeformed specimen as observed by camera 1, (b) ε11 map on the lateral face, (c) ε11 map on the main face, (d) ε33 map on the lateral face, and (e) ε22 map on the main face. E


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Different values for ε22 and ε33 can also result from the specimen shape. For the considered dog-bone specimen the dimensions are constant along the 3 direction but decrease along the 2 direction. However, in order to reduce this effect, a flat part was machined in the center of the lateral faces (see Figure 1). The ratio of longitudinal extension to average transversal (lateral) contraction is defined by D = ε11/|εt| where the average transversal strain is εt = (ε22 + ε33)/2. In the present work, D must be seen as a geometrical indicator of how the material is being deformed at the macroscopic scale during the test. In response to a given Δε11 increment of ε11 (ε11 → ε11 + Δε11), the smaller the D value, the larger the contraction of the specimen lateral dimensions (thickness and width). Naturally, in the linear elastic regime, D is constant and D−1 is equivalent to the Poisson’s ratio. Also, D is directly derived from the volume strain measurement, normalized with respect to the applied strain: εv/ε11 = 1 − 2D−1. With the tensile force F, it is possible to determine the true stress, ratio of the force F to the actual cross section during the test: σ = F/S = (F/S0) exp[−(ε22 + ε33)] = (F/S0) exp[−2εt]. This equation for true stress is valid only if stress concentrations near the notch tips are neglected. It should also be noted that after the yield point the neck shoulders can also induce stress concentrations. Sometimes, only ε11 is measured. In that case, for calculating the true stress, εt , can be evaluated by using the constant volume assumption and writing 2εt = ε11. Experimental Results for the Volume Strain. Figure 8 shows the evolution of the volume strain as measured during

marker tracking. This device allows the measurement of ε11 in a point situated in the center of the necking region. This measurement is made in real time. It is then possible to monitor the tensile machine for obtaining a constant longitudinal strain rate in the point where ε11 is measured: ε̇11 = 1.25 × 10−3 s−1. 3D-DIC (Aramis-GOM Instruments) measurements are based on the stereocorrelation principle. The two optical centers of the camera lenses are placed along a direction parallel to the drawing axis (Figure 1). Each of the two cameras records images in the central specimen region where two adjacent faces of the specimen are observable (Figure 7). It is therefore possible to perform strain measurements for the specimen face corresponding to the (1,2) plane (lower face in Figure 7a, main face of the specimen) and to the (1,3) plane (upper face in Figure 7a, lateral face of the specimen). The speckle (see Figure 7a) was applied to the specimen using an air brush. We have used 40 × 40 pixels facets and a 8 pixels step size (distance between two adjacent facets). The strain reference length is then 16 pixels.31 One pixel approximately corresponds to 13 μm on the specimen. On the (1,2) plane, it is possible to obtain ε11 and ε22 maps (see Figures 7c and 7e). On the (1,3) plane, the measurement of ε11 and ε33 can be made (see Figures 7b and 7d). Let us note that a similar experimental procedure was already used by Grytten et al. for studying ductile thermoplastics.14 Naturally, the markers used by the video extensometer are placed on one of the two faces of the specimen on which the strain field was not measured by 3D-DIC. The measurement was obtained up to a true strain value of approximately ε11 = 1.8 in the center of the necking region. The stretch ratio is then roughly 600%. During the tensile test, 600 images were recorded by each camera. In Figure 7b−e, it is possible to observe that the strain is highly localized in the necking region. There, because of the neck shoulders, the specimen deformation state is very complex. However, it is possible to show experimentally that in the cross section corresponding to the center of the necking region the deformation state is homogeneous. Moreover, in this region, the specimen curvature is zero. The three axes that are naturally associated with the specimen (axes 1, 2, 3 shown in Figure 1) are then the constant deformation principal axes. So the measurement of the three longitudinal strains ε11, ε22, and ε33 is sufficient for fully determining the Hencky strain tensor (εH). In order to reduce the measurement noise, ε11, ε22, and ε33 were evaluated over the cross section corresponding to the center of necking region (indicated by black rectangles in Figure 7b−e where these quantities are uniform. It corresponds to an averaging over 42 measurements along the specimen thickness as well as along the specimen width. Starting from ε11, ε22, and ε33, several quantities can be obtained for specifically characterizing some aspects of the deformation state. The logarithmic volume strain is given by εV = ln(V/V0) = ε11 + ε22 + ε33 = tr(εH). Volume strains are often evaluated using 2D optical devices, in particular Videotraction that was originally developed by G’sell et al.13 and is now widely used. With such optical devices, measurements are only possible on one single face of the specimen: ε11 and only ε22 (or ε33) can be evaluated. For calculating the volume strain, it is then necessary to make the assumption of a transversally isotropic deformation leading to ε22 = ε33. The validity of this assumption will be addressed in this work. It should be noted that this assumption is not simply that the material is transversally isotropic.

Figure 8. Volume strain measurements. LF: lateral face; MF: main face; 2F: values calculated with measurements taken on the two faces.

the tensile test. In order to allow the evaluation of the measurement reproducibility, three different tests are presented. In Figure 8 these tests are respectively denoted test A, test B, and test C. Controlled by video extensometry, the strain rate in the center of the necking region (where the measurement is made) is the same for these three tests. During the whole tensile test, 600 experimental points were obtained. However, in order to distinguish clearly between the different curves, we have only plotted one experimental point out of ten. The volume strain was calculated in three different manners to highlight the errors that can be made when performing these measurements: (i) LF (lateral face), measurement only on the lateral face (1,3) and then assuming that ε22 = ε33 proved here as a biased evaluation; (ii) MF (main face), measurement only on the main face (1,2) and then assuming that ε33 = ε22, again proved here as a biased evaluation; (iii) 2F (2 faces), measurement using the two separate measures of ε22 and ε33 taken on the two studied faces leading to a correct evaluation. F


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Figure 9. (a) Volume strain and true stress curves. (b) Volume strain for low strain levels.

10−5 × √3 ≈ 1.3 × 10−4. The error bars in Figure 5b correspond to ±3Sd. For the high strain levels, because of the speckle degradation, the measurement uncertainty is higher. Strain range E can surely be identified as the elastic stage where the slope of the curve εv versus ε11 is 0.123. It corresponds to a 0.44 value for Poisson’s ratio. In strain range I, from ε11 ≈ 0.03 to ε11 ≈ 0.35, the increase of εv is much smaller. In this strain range (see Figure 9b), it is not possible to observe any abrupt change of the volume strain dependency on ε11: εv increases very slightly in an approximately linear fashion. The slope is 0.0238, which corresponds to a 0.49 value for Poisson’s ratio. Next in strain range II, the volume strain rises at an increasing rate up to ε11 ≈ 1.05. Finally, in strain range III, the slope of the curve decreases. At the end of the test, the volume strain is almost constant. In strain range III, strong strain hardening effect can be observed on the true stress/true strain curve. At the microstructural scale, it can be assumed that deformation induced matter reorganization phenomena could lead to volume strain changes. In the following section, we try to build a bridge between the volume strain changes and the evolution of microstructural anisotropy characterized at both the micrometric and nanometric scales.

Toward the end of the test, the measurement noise increases. It is a consequence of the speckle degradation due to high strain levels. In Figure 8, it can be seen that the volume strain values obtained only with measurements taken on the main face differ greatly to those corresponding to the lateral face. It clearly shows that for the studied specimen the correct value for the volume strain can only be evaluated by using two perpendicular faces of the specimen. Recalling the equation for volume strain, εv = ln(V/V0) = ε11 + ε22 + ε33, it is clear that the difference between the volume strain values that were separately calculated on the two specimen faces can have two origins: (1) different results for the two ε11 values that were respectively measured on the main face and on the lateral face; (2) different values for ε22 (measured on the main face) and for ε33 (measured on the lateral face). It is possible to check that the difference between the two ε11 values remains smaller than 5 × 10−3 during the test (curves not shown there). So the latter assumption is correct: the volume strain difference results from different transverse strain values: ε22 ≠ ε33. In Figure 9a, we show on the same plot the true strain/true stress curve and the evolution of the volume strain. Naturally, the true stress was evaluated with the two different values ε22 and ε33 measured on the two perpendicular faces. At the end of the test, σ is 88.5 MPa. With the single measurement of ε22 on the main specimen face and the assumption ε33 = ε22, the biased value for σ would be 100.7 MPa (relative error 12%). For the alternative case (measurement of ε33 and assumption ε22 = ε33), it would be 77.9 MPa. Using the constant volume assumption, it would be 103.0 MPa. In Figure 9, the volume strain is the average value corresponding to the three tests shown in Figure 8. Four regimes can be distinguished on the volume strain curve. The first one (strain range E, up to ε11 ≈ 0.015) is associated with a rapid increase of the volume strain. It is easier to identify region E in Figure 9b, which corresponds to an enlargement of the curve for the low strain levels. For evaluating the measurement noise, we have carried out 3D DIC measurements from images successively acquired by the two cameras for a specimen clamped between the grips of the tensile machine but without applying any force. By analyzing the obtained strain fields, it was found that the standard deviation for the measurement noise is 5 × 10−4. Over the cross section, an averaging over 42 measurements is made. With the assumption of a Gaussian distribution of the error on strain measurement, the standard deviation for the averaged strain is 5 × 10−4/√42 ≈ 7.7 × 10−5. Three strain measurements are needed for evaluating the volume strain. The standard deviation for the volume strain is then Sd = 7.7 ×

4. RELATIONSHIP BETWEEN MICROSTRUCTURAL ANISOTROPY AND MACROSCOPIC STRAIN micro In Figure 10, we show the dependency of Anano SAXS, AISLT , and Amicro on the strain value ε . All these measurements were SRXTM 11 obtained in the center of the necking region where the strain

Figure 10. Relation between the microstructural anisotropy index evolutions (left axis) and the macroscopic deformation mechanisms characterized by D (right axis). G


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faces of the specimen would also be significantly different. This was not observed (see section 3 part “Experimental Results for the Volume Strain”). It shows that the highlighted macroscopic anisotropy results from material properties and not from specimen geometry. In some works,12,32 negative volume strains were observed. These works are based on 2D optical strain measurements made on the main specimen face. For calculating the volume strain, the assumption was made that the two transverse strains were the same: ε33 = ε22. This could explain why negative volume strains were observed when 3D-DIC never confirms that. This metrological remark is of importance as for the material considered in the present study, measurements made only on the main specimen face (see MF curves in Figure 8) would have led to the erroneous conclusion that the material undergoes compaction in the strain range [ε11 ≈ 0.3−ε11 ≈ 0.5]. Microstructure Evolution and Macroscopic Strain Measurements. Considering the previously presented experimental results and the main results found in the literature, the material microstructural transformations that occur during the tensile test are analyzed here: On the one hand, the microstructural state of SCPs is very well-known in the undeformed state: the material is made of stacks of lamellae having a favored direction of growth. The lamellae are separated by amorphous regions (8.2 nm thick for the considered material). At the specimen scale, the orientation of the lamellae stacks is uniformly distributed. In a given direction the material rigidity mainly results from the lamellae oriented along that direction. On the other hand, the fibrillar microstructure of very deformed SCPs is also very well documented: the lamellae are fragmented into small blocks coupled by tie molecules. The chains are oriented along the drawing axis. Deformation mainly results from chain stretching in the amorphous phase leading to a strong strain hardening as can be observed at the end of the true stress/true strain curve (Figure 9a). However, the deformation mechanisms corresponding to the transformation from the lamellar to the fibrillar microstructure are still not completely known. Strobl and collaborators have studied deformation mechanisms for SCPs by performing true stress/true strain curves at constant strain rate coupled with the study of the microstructure evolution by WAXS. By analyzing the changes of both the differential compliance and recovery properties, it was found that after the yield point, three different strain ranges can be highlighted for describing the microstructure evolution.33−36 The transitions between these three regimes occur at critical strain values. In the papers of the Strobl’s research group: the corresponding points are denoted: B (ε11 ≈ 0.1, approximately the yield point), C (ε11 ≈ 0.4−0.6, slightly smaller37 than 0.4 for HDPE), and D (ε11 ≈ 1.1). It was found that these points are independent of temperature or strain rate.38 These three strain ranges are each associated with distinct deformation mechanisms at microstructure scale: (i) From B to C: collective slips of crystalline blocks inside the lamellae. Each block still belongs to a certain lamella; the lamellar structure is then globally preserved. (ii) From C to D: lamellae fragmentation and beginning of fibrillation. (iii) After D: chain disentanglement, which results in a finite truly irreversible deformation even if the material is annealed at a temperature close to the melting point. At this final state the material has reached the fibrillar morphology, the medium is then strongly anisotropic: the microstructure is elongated along the drawing direction.

rate was constant and where the volume strain was also micro micro evaluated. In short, Anano SAXS, AISLT , and ASRXTM clearly exhibit the same behaviors, which means a same signature of the physical processes at all submicrometric scales. Three different regimes can be observed. First, the anisotropy indexes are negative: the microstructure appears anisotropic, elongated along the transverse direction, at both the nanometric and micrometric scales. Next the index values increase rapidly until a strain value roughly between ε = 1 and ε = 1.5. It corresponds to a reorientation of the microstructure toward the drawing direction. Finally, at the end of the test, the anisotropy index values remain approximately constant or increase slightly. Not considered further is the elastic regime well-evidenced on the volume strain curve (strain range E of Figure 9b), which corresponds to a very small strain range and cannot be associated with any change of the microstructure. The three highlighted regimes correspond then to the evolution of the microstructure anisotropy, and it is interesting to wonder about potential consequences on the macroscopic deformation mechanisms. As previously mentioned, the influence of the microstructure anisotropy on the geometrical transformation of the specimen (in the center of the necking region) can be analyzed through the macroscopic quantity D = ε11/|εt|. In Figure 10, D is represented on the same plot as the three indexes that are used for characterizing the microstructure anisotropy evolution. It is striking that the D quantity exhibits an evolution very comparable to those of the indexes corresponding to the microstructure anisotropy changes. The three regimes that can be seen in Figure 9a for the volume strain are even easier to distinguish on the D plot (Figure 10): (i) strain range I: up to ε11 ≈ 0.35, D decreases; (ii) strain range II: from ε11 ≈ 0.35 to ε11 ≈ 1.05, the slope of the curve first increases and next is approximately constant; (iii) strain III: starting from ε11 ≈ 1.1, a marked decrease of the curve slope can be observed. However, although all curves behave in a similar way, an order seems visible through the slight shift of the curves with respect to the strain magnitude. The microstructure orientation changes occur first (earlier) for smaller strain levels at the nanometric scale (SAXS) and in second position for the micrometric case (ISLT) last (later) for the millimeter scale (3D-DIC Volume Strain measurements). Such observation is more difficult to assert for the SRXTM curve (micrometric scale) since the material has undergone relaxation (postmortem measurements), but anyway it still falls between nanometric and millimetric observations.

5. DISCUSSION Macroscopic Anisotropy. For the considered HDPE material, although the specimen initial width and thickness are the same, it is clearly not possible to assume that the two transverse strains remain equal during a tensile test in order to evaluate either the true stress or the volume strain. This anisotropic behavior can be considered as resulting from the geometrical constraints imposed on the microstructure during the extrusion process and the liquid to solid state transformation. In principle, different deformation states on the two specimen faces could also result from the shape of the dogbone specimen (see section 3, part “Information on the Measurement Technique”). However, specimen geometry effects would not only lead to different values for ε22 and ε33. The two ε11 values that were obtained on the two perpendicular H


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transition from lamellar to fibrillar morphology is associated with a strong volume strain increase (see Figure 9a). During that process, crystallized chains are transferred into the amorphous phase, which has a smaller density than the crystalline phase. It results in macroscopic material dilatation. However, intensification of the cavitation phenomenon may also be responsible for the volume strain increase. In particular, the reshaping of the existing nanovoids in the tensile direction may be associated with an increase of their volume. Peterlin’s model42 is often cited for describing the lamellar/fibrillar transition, but it inherently implies cavitation and thereby material macroscopic dilatation.39 It should however be noted that SRXTM measurements have shown that at the resolution of the measurement (1 pixel = 0.38 μm) no micrometric voids appear to be present in the material.7 Strain Range III [ε11 > 1.1]. In strain range III, no strong change of the microstructure anisotropy can be observed: the fibrillar structure is well-established. Deformation mainly results from disentanglement of tie molecules between crystalline blocks resulting in a strong strain hardening at the end of the true stress/true strain curve. These deformation mechanisms are associated with a moderate increase of the volume strain followed by a plateau (Figure 9a). Whitening Phenomenon. In the case of SCPs, the whitening phenomenon is generally attributed to the development of micrometric cavities in the material that scatter visible light in the Mie regime. Smaller cavities (as those that can be detected by SAXS) would not scatter all the wavelengths of the visible spectrum in an equivalent way.28 For the considered HDPE, previous studies7,9,10 have shown that the whitening phenomenon begins for a strain value of about 0.08 and mainly occurs in the range [ε ≈ 0.08−ε ≈ 0.25]. In the same strain range [ε ≈ 0.03−ε ≈ 0.25] (see Figure 9b), it is not possible to observe any abrupt change of the volume strain dependency on ε11 that could indicate any sudden development of micrometric cavities in the material. By contrast, εv remains small and varies in an approximately linear fashion: the slope is 0.0238, which corresponds to a 0.49 value for the Poisson’s ratio. These results reinforce the conclusions of our previous studies that have shown that no simple and straightforward links between scattering by individual cavities and the whitening phenomenon for SCPs could be established.10 In particular, SRXTM measurements have shown that no micrometric cavities are present in the material.7

However, the above analysis does not incorporate the cavitation phenomenon. Cavitation is generally highlighted by means of SAXS measurements that allow for the detection of voids with sizes in the range 1−100 nm. It was shown that in the case of SCPs the nanovoiding phenomenon does not occur in a homogeneous way within the material; it depends on the local orientation of the lamellae.18,39 Galeski et al.40 have outlined the mechanisms describing the nanovoiding phenomenon under imposed uniaxial tension at the lamellae scale for small strain levels: The nanovoiding phenomenon occurs first between lamellae that are transversally oriented with respect to the tensile axis for a strain level close to the yield point (≈ point B in the works of Strobl’s group). Their shape is also transversally oriented with respect to the drawing axis. It results from a confinement effect. The average distance between the centers of the nanocavities is of the order of a few tens of nanometers.23,41 For higher strain levels, the nanocavities reorient toward the drawing direction.2,17−23 By analyzing simultaneously recorded SAXS and WAXS patterns,2,17,20 it was found that the reorientation of nanovoids occurs at the same time as the onset of the destruction of the lamellar morphology (≈ point C in the works of Strobl’s group). The reshaping of the nanovoids is forced by fragmentation and reorientation of crystals around which voids were initially embedded. Strain Range I [ε11 ≈ 0.1−ε11 ≈ 0.35]. For our material, as expected, the SAXS detected nanovoids are transversally oriented when they first appear in the material. In the framework of the aforementioned interpretation given by Galeski et al. of the nanovoids formation mechanisms, the specific regions where the nanovoids first appear correspond to the areas embedded between lamellae perpendicularly oriented with respect to the drawing axis.40 In these regions, the distance between the centers of the nanocavities is smaller than the spatial resolution of the experimental tools that were used for analyzing the material at the micrometric level. As a consequence, regions where nanovoiding occurs appear like areas having an average density lower than that of the rest of the material. These regions are perpendicularly oriented with respect to the drawing direction and could then be associated with the micrometric objects observed through ILST and SRXTM. At this stage, the nanovoids have still not changed their orientation, which proves that the lamellar structure is globally preserved. It agrees with the conclusions of Strobl’s research group, which shows that the lamellar structure is preserved for strain range I. The very small increase of the volume strain proves that the lamellar structure strongly limits the cavitation phenomenon: only nanovoids embedded between lamellae lying normal to the tensile stress direction can appear in the material. At the macroscopic level, the decrease of the quantity D = ε11/|εt| shows that the microstructural deformation mechanisms promote the specimen lateral contraction. This implies a decrease of the transverse size of the lamellae that are perpendicularly oriented with respect to the drawing axis, probably by kinking.20,41 Strain Range II [ε11 ≈ 0.35−ε11 ≈ 1.1]. In this strain range, at both nanometric and micrometric scales, the microstructure strongly reorients toward the drawing direction through lamellar fragmentation. In particular, the shape of nanovoids evolves: they become elongated along the drawing direction. Strain range II can then be associated with the transition from lamellar to fibrillar structure. Our measurements show that the

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CONCLUSION For a high density polyethylene material deformed in tension, the evolution of the macroscopic deformation state was characterized by measuring the specimen transverse strains functions of the applied longitudinal strain. The measurement was obtained in the necking region. It was found that after the yield point the deformation process takes place in three distinct stages. In addition, the microstructure evolution of the same specimen was experimentally studied at nano and micro scales, focusing especially on the anisotropy evolution, as developed by the microstructure. A common feature at all scales has been clearly evidenced and can be readily observed with precise 3D strain measurements. This is probably an important fact to consider in view of behavioral model construction. Trying to reproduce the anisotropic behavior is a challenge that could be a relevant approach to integrate the hierarchy of microstructural transformations. I


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(27) Bihannic, I.; Baravian, C.; Duval, J. F. L.; Paineau, E.; Meneau, F.; Levitz, P.; de Silva, J. P.; Davidson, P.; Michot, L. J. J. Phys. Chem. B 2010, 114, 16347−16355. (28) Van de Hulst, H. C. Light Scattering by Small Particles; Dover Publications: Mineola, NY, 1981. (29) Baravian, C.; André, S.; Renault, N.; Moumini, N.; Cunat, C. Rheol. Acta 2008, 47, 555−564. (30) Dillet, J.; Baravian, C.; Caton, F.; Parker, A. Appl. Opt. 2006, 45, 4669−4678. (31) Eriksen, R.; Berggreen, C.; Boyd, S. W.; Dulieu-Barton, J. M. EPJ Web of Conferences 2010, 6, 31013. (32) Brusselle-Dupend, N.; Cangemi, L. Mech. Mater. 2008, 40, 743− 760. (33) Hiss, R.; Hobeika, S.; Lynn, C.; Strobl, G. Macromolecules 1999, 32, 4390−4403. (34) Men, Y.; Strobl, G. J. Macromol. Sci., Phys. 2001, 40, 775−796. (35) Men, Y.; Strobl, G. Macromolecules 2003, 36, 1889−1898. (36) Hong, K.; Rastogi, A.; Strobl, G. Macromolecules 2004, 37, 10165−10173. (37) Fu, Q.; Men, Y.; Strobl, G. Polymer 2003, 44, 1927−1933. (38) Hobeika, S.; Men, Y.; Strobl, G. Macromolecules 2000, 33, 1827−1833. (39) Galeski, A. Prog. Polym. Sci. 2003, 28, 1643−1699. (40) Galeski, A.; Argon, A. S.; Cohen, R. E. Macromolecules 1988, 21, 2761−2770. (41) Krumova, M.; Henning, S.; Michler, G. H. Philos. Mag. 2006, 86, 1689−1712. (42) Peterlin, A. Plastic deformation of crystalline polymers. In Polymeric Materials; Baer, E., Ed.; American Society for Metals: Metals Park, OH, 1975; pp 175−195.

AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (L.F.). *E-mail [email protected] (S.A.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the help of I. Bihannic for careful critical reading of the SAXS data treatment part. Providing beam time and financial support of the experiments by the synchrotron source facility at Soleil (SWING Beamline) is gratefully acknowledged. We also thank Alain Gerard, a technician at LEMTA lab, for tedious preparation of the tensile HDPE specimen for all light scattering, microtomography, WAXS/SAXS, and macroscopic DIC experiments over the past years.

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A Common Multiscale Feature of the Deformation Mechanisms of a

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