Damage Mechanisms of Plasticized Cellulose Acetate under Tensile

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Damage Mechanisms of Plasticized Cellulose Acetate under Tensile Deformation Studied by Ultrasmall-Angle X‑ray Scattering Agathe Charvet,* Caroll Vergelati, Paul Sotta, and Didier R. Long*

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Laboratoire des polymères et Matériaux Avancés, UMR 5268 Solvay/CNRS, Solvay in Axel’One, 87 avenue des Frères Perret, 69192 Saint-Fons, France ABSTRACT: We consider the microscopic mechanisms of damaging in plasticized cellulose acetate under tensile stress. We show how they appear and develop during the course of deformation until failure. By using scanning transmission electron microscopy, we observe the presence of cavities and the coexistence of homogeneous and fibrillar crazes. Ultrasmall-angle X-ray scattering experiments allow for describing the onset of damaging and the growth of crazes and for measuring the volume fractions of damages as well as their shapes and sizes at different stages during deformation. We propose that damages are initiated by the nucleation of cavities in the vicinity of pre-existing impurities. We then show that their initial growth after nucleation is blocked at a size of order 100 nm by strain hardening in their immediate vicinity where deformation and stress are amplified. Increasing the stress further leads to a new growth regime for a small fraction of crazes. Ultimate failure is due to the propagation of this small number of crazes and not to the accumulation of crazes and a coalescence process. We propose that this growth process is the consequence of homogeneous nucleation of new cavities just in front of existing ones. The volume fractions of damage remain very low, of order of 10−4 at failure. The strain hardening behavior appears to be the key for preventing an early failure of the material and conferring high ductility to the material.

1. INTRODUCTION In view of their promising properties such as biodegradability and renewability,1 great effort has been dedicated to investigating cellulose acetate polymers in order to expand their fields of potential applications. For cellulose acetate with a 2.45 substitution degree (DS), the existence of a narrow temperature window between melting and degradation makes melt processing only possible with a sufficient amount of an external plasticizer.2 Corresponding polymer/plasticizer blends are amorphous.2 Recent studies have been carried out for characterizing the miscibility,3−6 plasticization,7 and mechanical properties of commercial cellulose acetate with a DS of 2.45.8−11 It was noticed that the tensile behavior of plasticized cellulose acetate below T g strongly depends on the experimental temperature, the nature of the plasticizer, and the polymer chain orientation induced by the injection process.8 Depending on these parameters, the polymer can be brittle or ductile, with strain hardening occurring from 8% of true strain during the tensile experiment.8 Strain hardening has also been observed in highly entangled amorphous polymers such as polycarbonate (PC), poly(methyl)methacrylate (PMMA), or polyvinyl chloride (PVC).12−14 It can be defined as stress increase at high deformation13,15−21 with a characteristic slope (strain hardening modulus ESH) of order of magnitude 107−108 Pa well below the glass transition. The origin of this phenomenon still remains under debate.16−19,21−30 It has long been attributed, for example, by Haward,16 to the entropic elasticity of an © XXXX American Chemical Society

entangled or cross-linked rubbery network. This theoretical approach was widely adopted and supported by experiments showing an increase of hardening modulus with increasing entanglement or cross-linking densities.18,19,31 The entropic model qualitatively reproduces the functional form of stress− strain curves but deviates from some experimental results. These flaws in the entropic interpretation of ESH were not seriously considered and challenged until they were illustrated in experiments performed by van Melick et al.13 These authors have shown that ESH measured over a wide range of temperature decreases linearly with temperature and becomes very small near the glass transition temperature,32 which goes against an entropic model. In addition, entropic models cannot explain the orders of magnitude of the experimentally measured ESH, which is found to be around 10−100 times larger than what would be expected from any realistic entanglement density.24 A more fundamental flaw of entropic models is also that, unlike rubber, glass is not ergodic. In the glassy state, the conformational entropy of polymer chains is much less than its equilibrium value. Thermal activation is not sufficient to allow chains to sample conformations freely, and rearrangements occur mainly under active deformation at a frequency that scales with the strain rate.33 These open questions about glassy strain hardening were summarized by Received: April 26, 2019 Revised: July 17, 2019

A

DOI: 10.1021/acs.macromol.9b00858 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Kramer17 and, after him, multiple authors performed simulations and developed new molecular-based theories.13,20,24,25,28,30,34,35 The key new insights are that strain hardening is nonentropic but fundamentally viscoelastoplastic24 and seems to be controlled by many of the same mechanisms that control plastic flow.35,36 A series of recent articles based on simulations and experiments are in agreement with the idea that the strain hardening is correlated to molecular interactions which control flow stress.26 The same physics regarding yield stress and strain hardening are involved on the microscopic scale26,27 and corresponds to molecular interactions rather than entanglements. It is now generally accepted that polymer chains orient during plastic deformation, which results in anisotropic materials with enhanced properties in the orientation direction.37 Both the stress flow and strain hardening modulus are higher when deformation enhances the orientation produced by prestrain.18,19,29,38,39 The behavior of polymers at large strain values plays a key role in determining their failure mode and mechanical performances. Experimentally, it was observed that polymers which exhibit larger strain hardening, such as PC, are tougher and tend to undergo ductile rather than brittle deformation. This improvement in ductility was explained by Meijer and Govaert20 by the idea that strain hardening suppresses strain localization or shear banding. However, the question of the microscopic mechanisms of damage related to failure remains open. Do these defects appear from the beginning of strain hardening or just before breaking? Do they lead to failure by the propagation of a single defect or by accumulation of these defects? A burst of detailed experimental studies on micromechanisms of deformation in amorphous polymers, particularly on the morphology of crazes that are generated when these materials are submitted to tensile stresses, have been published in the past decades.40−48 Some correlations between these micromechanisms of deformation and macroscopic mechanical properties have been established from experimental and theoretical studies, but, to the best of our knowledge, few relations between the presence of strain hardening and damage mechanisms have been proposed.20,49 It is suggested that the morphology of crazes depends on the toughness of polymers. Polymers considered to be “brittle” such as polystyrene (PS) will deform preferentially by crazing.40 The more brittle these materials are, the coarser is the internal structure of the crazes.40,44 Conversely, polymers considered to be “ductile” such as PC deform preferentially by shear banding or by homogeneous deformation bands (craze-like zones filled with the homogeneous stretched polymer).41,42,44 The different craze structures have also been discussed using an “entanglement model”44 which describes the transition from fibrillated crazes to homogeneous deformation bands when entanglement densities increase. Crazing has been an intense subject of research over the past 50 years.46,48,50−56 However, the initiation step remains poorly understood. The most-used concept of craze nucleation is the “meniscus-instability model”46,48,50 proposed by Argon and Salama57 and originally developed by Taylor.58 Monnerie et al.55 suggest that crazes initiate from a defect which induces stress concentration in its vicinity. Local stress concentration generates plastic deformation in a thin polymer layer surrounded by the bulk glassy polymer which has not yet reached its yield point. The behavior of this thin polymer layer under deformation can be compared to the mechanism of

meniscus instability which induces the growth of the craze by the coalescence of voids.46,48,55 More recently, Michler44 proposed a different scenario of craze nucleation. They supposed that it is preceded by a localized plastically deformed zone where cavitation takes place. This pre-existing structure made of softer material domains is related to the entanglement network.44 The craze growth mechanism is often described as a two-step process: craze tip propagation (according to the mechanism of meniscus instability as described above) and craze opening (related to chain entanglements48,50,55). The failure mechanism in polymers has often been assimilated to crack opening. Two theoretical models based on linear elasticity mechanics proposed by Griffith and Irwin describe the effect of loading near a defect.55 Griffith59 considers the elastic energy stored near the defect, while Irwin60 uses the stress distribution around the defect.61,62 Brown63 have then used these criteria to describe the plastic failure which occurs ahead of the tip when all fibrils inside the craze have broken. It is shown that the toughness varies like the squared breaking force of chain molecules and the squared density of entangled strands at the interface. The Brown−Kramer’s description of craze−crack transition63−65 is based on the fracture mechanics approach. They have calculated either critical stress intensity or a strain energy release rate of the mechanically anisotropic fibrillar network within the craze. The strain energy release rate for crack opening is described as being proportional to the entanglement density. Most experimental studies on damage in amorphous polymers are carried out by transmission electron microscopy (TEM) or by small- or ultrasmall-angle X-ray scattering (SAXS, USAXS)66−69 on samples after tensile failure. Only few authors, such as Stoclet et al.,70 follow the evolution of damage during deformation by in situ USAXS analysis. With this technique, the appearance of defects can be observed on a mesoscopic scale. In their study, Stoclet et al. propose a qualitative mechanism of damage in poly(lactic acid) under tensile deformation. This type of study was also performed by Mourglia-Seignobos et al.71 in neat polyamide 6.6 during fatigue tests. These authors propose a quantitative fit of USAXS intensities obtained at different steps in fatigue experiments, which allows them to obtain the sizes and volume fractions of the observed damage. The authors then propose a quantitative mechanism of damage in polyamide 6.6, from the initiation step up to failure. They propose that cavities nucleate in the amorphous phase and grow until they are blocked by semicrystalline disorder. The corresponding typical sizes are in between 10 and 100 nm. Breaking then occurs by accumulation of these defects. In this work, we shall focus on the nature of damage mechanisms in amorphous plasticized cellulose acetate. Strain hardening is thought to delocalize deformation and prevent defect propagation.20 We aim at giving a more specific content to this picture. Our purpose is to describe as quantitatively as possible the initiation and growth of damage as well as the mechanisms leading to final failure. In order to characterize microscopic damage mechanisms, we have monitored the damage in plasticized cellulose acetate all along tensile tests, and particularly during the strain hardening regime. Scanning TEM (STEM) and USAXS were combined in order to observe a wide spectrum of damage sizes, ranging from tens of nanometers to tens of micrometers. We were thus able to study and describe damage occurrence right from the origin. We B

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molecular weight of pure cellulose acetate is measured at Mw = 73 260 g/mol. A small increase of the molecular weight is observed on plasticized cellulose acetate due to melt processing. Cellulose acetates are semiflexible polymers. Their persistence length have been measured by various techniques in the literature including neutron scattering, SAXS, light scattering in solutions, or calculated by molecular dynamics simulations. Experimental results depend on the degree of substitution of the considered cellulose acetate and on the considered solvent. The persistence length of cellulose acetate with DS 2.5 is found to be 13 nm in N,Ndimethylacetamide and 8 nm in acetone,74 whereas the value of 13 nm was obtained by numerical simulations.75 2.2. Extrusion and Injection Molding. DEP-plasticized (DEPpCDA) or TA-plasticized cellulose diacetate (TA-pCDA) samples were prepared by extrusion. The desired amounts of the plasticizer and CDA powder (dried in vacuum at 70 °C during 24 h) are fed into a twin-screw Clextral extruder (L/D ratio 48). The temperature of the screw profile was kept between 170 and 200 °C,76 the screw speed was 400 rpm, the throughput was kept constant at 20 kg/h, and the residence time was approximately 5 min. Because of thermosensitivity of cellulose acetate, the screw profile was adapted to limit self-heating. After extrusion, plates with dimensions 350 × 100 × 4 mm3 were injected with a Billion 750T BI-MAT injection press. The mold temperature was kept at 80 °C and included a v-shaped runner with a ramp. Based on observations on glass fiber-reinforced semicrystalline polymers, we assume that injection molding into plates leads to a homogeneous orientation of polymer chains.77 Tensile bars with dimensions 92 × 14 × 4 mm3 and a radius of curvature of 71 were directly cut from the plates at different angles θ ranging from 0° to 90° with respect to the main flow direction. The specimen geometry (hourglass shape) was specifically designed in order to study the strain hardening behavior. 2.3. Characterization Methods. 2.3.1. Tensile Tests. The Young’s modulus E, yield stress σy, and strain hardening modulus ESH of studied samples were obtained by tensile tests carried out on a Zwick/Roell Z050 universal testing machine equipped with a 50 kN load cell and a video-controlled material testing system (VidéoTraction, Apollor, Vandoeuvre)78 and a thermally controlled chamber. This technique is based on the measurement and active control of local deformation in a representative volume element (defined by dot markers on the tensile specimen). Samples were strained at a constant strain rate ε̇ = 10−3 s−1 (strain rate defined as ε̇ = L̇ /L0) and at different temperatures 40, 60, and 80 °C (which are below the Tg of the samples). Each test was performed on five different specimens. 2.3.2. Microscopic Observations. Damage morphologies were observed by STEM. Tensile specimens were cut in the center region below the failure surface, as shown in Figure 1. The surface was then

have followed an approach similar to that presented by Stoclet et al.70 and Mourglia-Seignobos et al.,71 which consists of performing USAXS analyzes at different strain values during tensile deformation. Quantitative measurements of scattered intensities allow us to determine the evolution of parameters such as the number density, shape, and size of damages during deformation, thereby enabling us to propose a quantitative mechanism of initiation, growth, and failure in plasticized cellulose acetate systems. In this paper, the strain hardening regime observed under tensile measurements in plasticized cellulose acetate systems as well as the influence of various parameters such as temperature and the nature and content of the plasticizer will first be described. Then, micrographs of damage morphologies obtained by STEM after failure will be presented. Next, the scattered intensities obtained by USAXS at different strain values will be modeled quantitatively. The effect of various parameters on the evolution of damage will be discussed. Finally, damage mechanisms observed in plasticized cellulose acetate will be discussed, highlighting the different stages, namely, initiation, growth, and rupture.

2. MATERIALS AND METHODS 2.1. Materials. Four samples were used in this study, with two plasticizing agentstriacetin (TA) and diethyl phthalate (DEP)at two plasticizer contents (15 and 20 wt %). With this level of the plasticizer content, the plasticized polymer can be considered as homogeneous. The miscibility limit between cellulose acetate and DEP or TA has been studied by Bao et al.3,72 on samples prepared by solvent casting and by using differential scanning calorimetry (DSC) measurements, dynamic thermal mechanical analysis, and small-angle neutron scattering. In the case of DEP, this limit was found around 25 wt % and with TA around 20 wt %.72 As a consequence, our samples are within the miscibility regime. In this manuscript, we adopt the following nomenclature: DEP15CDA for cellulose acetate plasticized with 15 wt% of DEP, TA15CDA for cellulose acetate plasticized with 15 wt% of TA, and the same formalism for cellulose acetate plasticized with 20 wt% of plasticizer (DEP20-CDA and TA20-CDA). More generally, for plasticized cellulose acetate samples, we use the nomenclature DEPpCDA and TA-pCDA. TA is a biosourced plasticizer, and DEP is a common plasticizer of cellulose acetate which constitutes a reference for this work as it is usually the case in the literature. DEP and TA were obtained from Sigma-Aldrich (Saint Quentin Fallavier, France). Cellulose acetate with an average DS of 2.45 (CDA) was supplied by Acetow GmbH (Freiburg, Germany). CDA with a DS of 2.45 was chosen because, according to Kamide and Saito,73 cellulose acetate has a processing temperature (Tm = 505 K) lower than its degradation temperature (Td = 509 K) only in the limited range of DS ≈ 2.5. The corresponding polymers can be melt processed and injected with external plasticizers. The corresponding glass transition temperatures and molecular weight of these plasticized polymers are given in Table 1. Molecular masses were measured by gel permeation chromatography (GPC) using a protocol established by GBU Acetow, Solvay, in order to optimize the measurement of cellulose acetate masses. The

Figure 1. Schematic representation of the area observed by scanning electron microscopy. trimmed at 23 °C using a microtome (Leica, Germany) with a diamond knife in order to obtain a mirror surface. Ultrathin sections of 80 nm were cut by ultramicrotomy at room temperature with a diamond knife. Sections were then picked up on a copper grid (200 mesh) and imaged using a Zeiss Ultra 55 STEM in the following conditions: accelerating tension 30 kV, diaphragm aperture size 20 μm, and work distance ≈ 4 mm in dark field. 2.3.3. USAXS Measurements. USAXS was used to characterize the polymer damage from the nanometer up to the near micrometer scale. USAXS experiments were carried out on the high brilliance beamline (ID02) at the European Synchrotron Radiation Facility (ESRF,

Table 1. TA-pCDA and DEP-pCDA Glass Transition Temperatures (Tg) Measured by Modulated DSC and Molecular Weight (Mw) Measured by GPC DEP-pCDA Tg (°C) (obtained by MDSC) Mw (g/mol) (obtained by GPC)

TA-pCDA

15 wt %

20 wt %

15 wt %

20 wt %

136 96 270

124 94 840

138 91 890

118 89 440 C

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Macromolecules Grenoble, France) with a wavelength λ = 0.995 Å. X-ray beam illuminated sample area was about 100 × 100 μm2. 2D detector Rayonix MX-170HS (detector area 170 × 170 mm2) was used. Three sample-to-detector distances d from 1 to 21 m were used. For d = 21 m, the range of scattering vector q = 4π sin θ/λ (where 2θ is the scattering angle) is 10−3 nm−1 < q < 10−2 nm−1; for d = 8 m, it is 10−2 nm−1 < q < 10−1 nm−1; for d = 1 m, 10−1 nm−1 < q < 10 nm−1. “Empty frames” were recorded and subtracted. The acquisition time was 5 s. The partial scattering curves corresponding to each distance are normalized by the thickness and transmission of each sample and by the incident beam intensity, and then regrouped without additional adjustment to give the scattered intensity I(q), that is, the scattering cross-section per unit volume of the sample, expressed in mm−1. The scattering by homogeneous, dilute, and spherical objects (particles or cavities) of radius R randomly dispersed in an otherwise homogeneous polymer matrix exhibits two characteristic regimes. In the so-called Porod regime,79 for qR > π, I(q) ≅ Bq−4, where B = 2πβ2ρe2NS, in which β2 = 7.8 × 10−24 mm2 is the electron crosssection, ρe2 is the squared electron density difference between the polymer matrix and the objects, S = 4πR2 is the surface area, and N is the number density of the objects. In the so-called Guinier approximation, for qR < π, I(q) ≅ G exp(−q2R2/5), where G = β2ρe2NV2, in which V = 4πR3/3 is the volume of one object. Beaucage and et al.79 have proposed a so-called global unified scattering function which approximates the scattering of a spherical particle and implements these two regimes erf 10 2 2 ji − q R zyz zz + B I(q) = G expjjjj z q4 k 5 {

3. RESULTS AND DISCUSSION 3.1. Tensile Behavior. The true stress−strain curves obtained in plasticized CDA samples under uniaxial tension at a true axial strain rate ε̇ = 1 × 10−3 s−1 are shown in Figure 2. For each sample, specimens were cut at three different angles θ with respect to the main injection direction (see Section 2.2). The temperatures of 40 and 80 °C (which are below the glass transition temperature Tg) were chosen to study the ductile behavior. The behavior of all polymers is typical of glassy polymers. Stars in Figure 2c indicate the strain values within the strain hardening regime, where USAXS measurements were performed. The initial Young’s moduli E (determined as the initial slopes of stress−strain curves) characterize the stiffness of the polymers. Yield (yield stress σy) occurs at the end of the viscoelastic regime and is subsequently followed by a small drop of the stress called strain softening and by a long plastic regime.82 As in other ductile amorphous polymers (such as PC, PMMA, or PVC), the last stage shows increasing strain hardening.83 This stage continues until the specimens reach their critical stress value at failure. Experimental values are reported in Table 3. Figure 2 shows that the macroscopic orientation of polymer chains, defined by angle θ, which significantly influences the yield stress as well as the strain hardening observed at large deformations.8,21,39,84 Increasing angle θ systematically results in a decrease of yield stress and strain hardening modulus ESH, measured by the slope over the last 5% of strain. From Figure 2a, the strain hardening modulus measured on DEP15-CDA decreases from 95 MPa at θ = 0° down to 70 MPa at θ = 90°. Polymers stretched in the same direction as the macroscopic chain orientation (θ = 0°) exhibit a higher stiffness. This decrease in properties when increasing the θ angle is observed regardless of the plasticizer, the composition, or the experimental temperature. By comparing these results with those presented in the work of Charvet et al.,8 it is possible to consider the influence of the plasticizer content on strain hardening. For the same distance to Tg, and T − Tg, both the strain hardening modulus ESH and the elongation at break εr in systems with 15 and 20 wt % of the plasticizer are of the same order of magnitude. For example, TA15-CDA (Tg = 138 °C) with θ = 0° stretched at 60 °C has ESH = 132 MPa, while TA20-CDA (Tg = 118 °C) with θ = 0° stretched at 40 °C has ESH = 120 MPa. Systems plasticized with TA are more sensitive to the increase in temperature than those plasticized with DEP. At 60 °C, strain hardening moduli are of the same order of magnitude whatever the plasticizing agent is.8 Conversely, systems with different plasticizers show different values for the strain hardening moduli measured at 80 °C. Systems plasticized with TA have lower ESH moduli than those plasticized with DEP. At θ = 0°, the TA15-CDA system has ESH = 77 MPa, while the DEP15-CDA system has ESH = 95 MPa. To understand the ductility observed under tensile experiments on plasticized cellulose acetate samples, it is essential to characterize the structure of the damage at micro- and mesoscales. 3.2. Microscopic Observations of Damage. Microscopic investigations have been performed by STEM. Because of the high sensitivity of cellulose acetate to electron irradiation, microscopic investigations are very difficult to perform. The focus must be done quickly in order to prevent

12

( ( )) qR

(1)

where erf(x) is the error function,80 which approaches zero when x < 1 and 1 when x > 1. When a material contains several sets of uncorrelated, polydisperse cavities or structures, each set should be modeled by a global unified scattering function integrated over the size distribution, and the contributions of all sets must then be added together to recover the overall sample scattering.81 The fitting of the scattered intensities was performed with Igor Pro. The electron density (number of electron per volume) of plasticized CDA is given by

ρTA ‐ CDA = ωCDA ρCDA + ωTA ρTA

(2)

ρDEP ‐ CDA = ωCDA ρCDA + ωDEPρDEP

(3)

where ωCDA, ωTA, and ωDEP are the volume fractions of CDA, TA, and DEP, respectively. ρCDA, ρTA, and ρDEP are the electron densities of CDA, TA, and DEP, respectively, and can be expressed as ρi = nidiNA/ Mi, where ni is the molar number of electrons (nCDA = 130, nTA = 116 and nDEP = 118), Mi is the molar mass (MCDA = 246 g/mol, MTA = 218 g/mol, and MDEP = 222 g/mol) per monomer or molecule of species i, di is the density of component i, (dCDA = 1.28 g/mm3, dTA = 1.16 g/mm3, and dDEP = 1.19 g/mm3) and NA = 6.022 × 1023 mol−1 is Avogadro’s number. The obtained electron densities are summarized in Table 2.

Table 2. Electron Densities of Plasticized Cellulose Acetate Samples ρ (mm−3) plasticizer content (wt %) 0 15 20 100

DEP-pCDA

TA-pCDA

× × × ×

4.02 × 1020 4.00 × 1020 3.72 × 1020

4.07 4.03 4.02 3.81

1020 1020 1020 1020

D

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Figure 2. Tensile stress−strain curves obtained at 0.001 s−1 and at (a,b) 80 and (b,c) 40 °C. (a−c) DEP-plasticized CDA and (b−d) TAplasticized CDA. θ is the angle of the sample axis (and stretching direction) with respect to the main injection direction (see Section 2.2). Black curves: θ = 0°, red curves: θ = 45°, blue curves: θ = 90°. Stars indicate the points at which USAXS measurements were carried out.

Table 3. Mechanical Properties of Plasticized Cellulose Acetate Polymersa materials 40 °C

DEP20-CDA

TA20-CDA

80 °C

DEP15-CDA

TA15-CDA

θ (deg) 0 45 90 0 45 90 0 45 90 0 45 90

σy (MPa)

E (MPa) 2900 2200 2200 2700 2000 1700 2100 1800 1600 2100 1800 1400

± ± ± ± ± ± ± ± ± ± ± ±

400 160 260 400 200 175 330 220 160 500 150 85

41 37.7 35 40 36.9 32.7 28 26 24 29 26 22

± ± ± ± ± ± ± ± ± ± ± ±

1 0.5 1 1 0.5 0.5 2 1 1 4 1 1

ESH (MPa) 124 126 136 120 136 131 95 79 70 77 57 37

± ± ± ± ± ± ± ± ± ± ± ±

5 19 34 22 19 8 15 6 21 7 9 7

εr (%) 22 34 38 26 34 43 29 36 38 14 32 38

± ± ± ± ± ± ± ± ± ± ± ±

0 3 5 5 4 1 6 2 3 9 2 3

Data obtained from Figure 2. θ is the angle of the sample axis (and stretching direction) with respect to the main injection direction (see Section 2.2). The strain hardening modulus ESH is measured by the slope over the last 5% of strain in each curve. a

structure of crazes can be either a fibrillar network where fibrils are separated by microvoids (as seen in Figure 3f) or homogeneous (as seen in Figure 3b). In the latter case, no apparent structure is revealed and the interior of the craze is fully composed of elongated polymers.41 It is observed that the morphologies and sizes of the crazes depend on the plasticizer and on the orientation of the macroscopic polymer chains, measured by angle θ. In the case of TA-plasticized polymers, increasing angle θ leads to a transition from homogeneous to fibrillar crazes. Their proportion and size also increase as θ increases, as shown in Figure 3d−f. At θ = 0°, TA-pCDA exhibits several homogeneous crazes whose sizes can reach around 1 μm in length and 0.1 μm in thickness and a small

polymer degradation. Same sensitivity is described in the literature with PMMA.44 This issue may explain why no microscopic studies on cellulose acetate polymers have yet been published. Large cavities with diameters in between 100 nm and 2 μm have been observed by X-ray tomography and optical microscopy in all materials used in this study, even in reference, unstretched samples. These cavities are probably generated during the injection process. Micrographs reported in Figure 3 show different types of damage observed after failure at 80 °C for cellulose acetate plasticized with 15 wt % of plasticizer. Crazing is observed in all polymers (Figure 3). Crazes propagate perpendicular to the tensile direction.44 The interior E

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Figure 3. STEM micrographs: damage in plasticized cellulose acetate observed after tensile failure at 80 °C: (a−c) plasticized with DEP at θ = 0°− 45°−90°, respectively; (d−f) plasticized with TA at θ = 0°−45°−90°, respectively. Cavities are observed in all micrographs. Homogeneous and coarsely fibrillated crazes coexist when θ ≥ 45°. Larger damage defects are observed in TA-plasticized CDA. The double arrow in each micrograph indicates the tensile direction.

number of fibrillated crazes whose largest sizes are around 4 μm long and 0.3 μm thick. At θ = 90°, an increase of the volume fraction of fibrillar crazes is observed, and their sizes are typically 2 μm in length and 0.3 μm in thickness. The resolution of the microscope allows for determining these sizes within a typical error bar of ±0.05 μm. Regarding the influence of the plasticizer, it is observed that TA-pCDA exhibits larger number of crazes than DEP-pDA. In the latter case, crazes are more homogeneous. A very small number of fibrillar crazes is only observed at θ = 45°. The sizes of fibrillar crazes are found to be around 1.5 μm long and 0.1 μm thick. By contrast with TA-plasticized polymers, increasing angle θ in DEP-pCDA leads to a decrease of the craze thickness, as shown in Figure 3c. The craze thickness does not exceed 0.05 μm at θ = 90° and 0.2 μm at θ = 0°. Another feature is observed at θ = 45° for TA-pCDA. Crazes seem to be organized in shear bands at 45° with respect to the direction of applied stress which corresponds to the injection direction (i.e., to the alignment of the polymer chains). Plasticized cellulose acetate samples after tensile failure at 60 °C were also analyzed by STEM, but no damage was observed on the micrographs. It is supposed that the size of damage is not compatible with the microscope resolution or that the contrast between structural damage (homogeneous crazes) and the polymer matrix is too small. From these observations, the different types of damages observed in plasticized cellulose acetate are schematized in Figure 4. The X axis indicates the tensile direction.

Figure 4. Schematic representation of the different structures of damage observed in plasticized cellulose acetate. The tensile direction is along X. Zoom of a fibrillated craze.

This schematic representation will be used for analyzing the scattered intensities measured by USAXS. The first category of defects consists of elongated cavities (microvoids) observed within fibrillar crazes. Category 2 includes both the homogeneous and fibrillar crazes. Finally, category 3 corresponds to large cavities formed during the injection process. Microscopic analyses give access to an average size and structure of crazes but only in a quite limited representative area. Moreover, objects smaller than 100 nm can hardly be observed. Therefore, a quantitative study of the damages must be complemented by USAXS measurements. 3.3. Characterization of Damage by USAXS. All polymers were analyzed by USAXS measurements after tensile F

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Macromolecules experiments at 40, 60, and 80 °C at different levels of strain. As shown in Figure 5, it is observed that the initial nearly isotropic

It is observed that the scattered intensities parallel to the tensile direction increase with the deformation as a consequence of increasing damage. Two different evolutions of intensities are observed depending on DEP or TA plasticizing agents. At small q values, the scattered intensity parallel to the tensile direction of DEP-pCDA does not increase with the strain, whereas in TA-pCDA, the intensity at small q values goes on increasing with the deformation. At 32% of true strain, intensity in TA-pCDA is two decades higher in the small q value region than DEP-pCDA. The increase of the intensity parallel to the tensile direction indicates the appearance of damage elongated in the direction normal to the tensile direction, that is, crazes (defects 2 in Figure 4). The scattered intensities perpendicular to the tensile direction remain constant until failure in both cases, at the exception of TA-pCDA from 32% of deformation for which a bump is observed at q values comprised between 0.06 and 0.2 nm−1. It confirms the appearance of elongated cavities oriented in the tensile direction, that is, the formation of fibrillar crazes (defect 1 in the Figure 4). From these graphs, we can observe that the plasticizer has an influence on the damage mechanisms. The TA plasticizer seems to favor the development and the growth of damages in the polymers. The number and the sizes of crazes seem to be larger in TA-pCDA than in DEP-pCDA polymers. Figure 7 reports the normalized scattered intensities parallel and perpendicular to the tensile direction at different angles θ (0°−45° and 90°) obtained after failure at 60 °C on sample TA15-CDA. By comparing the scattered intensities of the TA15-CDA sample broken at 60 °C at the three different

Figure 5. 2D USAXS patterns for TA15-CDA θ = 90° before the tensile experiment (a) and after failure at 80 °C (b). Tensile direction is horizontal.

scattering becomes anisotropic after failure. The scattering from crazes has already been described by Paredes and Fischer85 and refined by Brown and Kramer.86 It results in high anisotropic patterns having the form of two elongated streaks approximately perpendicular to each other, as observed in Figure 5. Figure 6 reports the scattered intensities measured in the direction normal and parallel to the tensile direction by USAXS experiments for samples TA20-CDA and DEP20CDA. The measurement has been carried out during the tensile experiment at 40 °C until failure. The contributions of the different damage morphologies identified by microscopic analysis are plotted on the graph using notations I1, I2, and I3.

Figure 6. Evolution of the USAXS intensities in directions parallel (a−c) and perpendicular (b−d) to the tensile direction on (a,b) DEP20-CDA and (c,d) TA20-CDA samples with an angle θ = 90° at different strain values during tensile experiments at T = 40 °C. Black arrows indicate different fitting contributions. G

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Macromolecules

Figure 7. USAXS-scattered intensities in directions parallel (in blue) and perpendicular (in green) to the tensile direction on TA15-CDA samples after failure at T = 60 °C for different angles θ (a) θ = 0°, (b) θ = 45°, and (c) θ = 90°. The arrows indicate the small bump observed in the direction normal to the applied stress.

Figure 8. USAXS intensities in directions parallel (in blue) and perpendicular (in green) to the tensile direction in DEP15-CDA samples with θ = 90° after failure at 60 (a) and 80 °C (b).

angles θ, different evolutions of the scattered intensities parallel to the tensile direction are observed for θ = 0° and θ ≥ 45°. The slight bump observed on scattered intensities perpendicular to the tensile direction for samples at θ ≥ 45° indicates the occurrence of crazes with fibrils. It means that angle θ between the tensile direction and the injection direction has an influence on the number and sizes of damage. When θ = 0°, scattered intensities are smaller which indicates that smaller and fewer damage are developed in the polymer. The increase of angle θ appears to be accompanied by an increase of the size and number of crazes. Figure 8 shows the normalized scattered intensities for the DEP15-CDA system at θ = 90° broken at 60 and 80 °C. It is observed in the q range values 7 × 10−2 < q < 3 × 10−1 nm−1 that the evolution of the scattered intensity perpendicular to the tensile direction measured on DEP15-CDA after failure at 80 °C exhibit a bump which is attributed to the apparition of elongated cavities inside crazes corresponding to fibrils, whereas it is not the case when scattered intensities are measured on DEP15-CDA after failure at 60 °C. A transition from homogeneous crazes to fibrillated crazes is observed when increasing the temperature. It is also found that the intensity measured on the sample broken at 80 °C is one decade higher than those in the case of the broken sample at 60 °C. This increase in intensity with temperature corresponds to the increase of the volume fraction and sizes of crazes. These results confirm the influence of the plasticizer, temperature, and angle θ on the damage mechanisms and

morphologies as it was observed by microscopic analysis. In order to propose a quantitative description of these damage mechanisms, the scattered intensities obtained by USAXS measurements are then fitted by theoretical equations. 3.4. Interpretation of the Scattered Intensities. From the 2D patterns, the scattered intensities are integrated along the tensile direction (para) [in an interval of the azimuth angle (−10, +10°)], and perpendicular to it (perp), [interval (80, 100°)]. Figure 9 gives a schematic representation of the general form of global scattered intensities in both parallel (direction x, blue curve) and perpendicular (direction z, green curve) directions. The contributions from all four types of defects as identified from STEM observations and schematized in Figure 4 are included. When q > 2π/c (region 1 in Figure 9), scattering comes from the small elongated cavities which are in between fibrils inside fibrillated crazes. These voids are modeled on average by uniaxial ellipsoids elongated along the tensile direction with long radius d and small radius c, as schematized in Figure 4. The corresponding scattered intensity is anisotropic. It gives access to long radius d in the tensile direction (para, X) and to short radius c in the direction perpendicular to the tensile direction (perp, Z). The scattered intensities in both directions are expressed as: H

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Macromolecules and β=

Lmax Dmin − Dmax Lmin Lmax − Lmin

(8)

The size distribution P(L) [where P(L) dL is the number of cavities per unit volume with a size comprised between L and L + dL] is described by a power law P2(L) = P2L−α

(9)

The distribution P(L) is normalized in such a way that the number density of crazes of type 2 is given by N2 = Figure 9. Schematic diagram of complete scattered intensity of a polymer with different damage morphologies as represented in Figure 4. The contributions from all three types of defects are included: (1) small elongated cavities (interior structure of crazes), (2) homogeneous and fibrillated crazes, and (3) large cavities induced by the injection process. The red dash curve is a schematic representation of the isotropic intensity in the initial state (before tensile deformation).

ij qd jj i −q2d 2 y erf 10 jj zz 2 2 2 j j z + 4.5 Ipara1(q) = β ρpol V1 N1jjexpjj jj j 5 zz (qd)4 { jj k k

12

( )

2

Iperp (q) = β ρpol

2

1

ij qc jj i −q2c 2 y erf 10 jj jj zz z + 4.5 V1 N1jjexpjj jj j 5 zz (qc)4 jj k { k 2

12

( )

ϕ2 =

(10)

∫L

Lmax

V2(L)P2(L) dL

(11)

min

where V2 = 4πLD /3 is the ellipsoid volume. Thus, in region 2, the scattered intensity is expressed by following equations ij jj i −q2L2 y j j zz zz Ipara 2(q) = β 2ρpol 2 φ 2P2 V2 2L−αjjjexpjjj j j Lmin jj k 5 z{ j k qL 12 y zz erf 10 zz zzzdL + 4.5 4 zz (qL) zz {

∫

(4)

Lmax

( )

jij j ji −q2D2 zy zz Iperp2(q) = β ρpol φ P2 V2 L jjexpjjj jj j 5 zz Lmin jj k { k qD 12 y zz erf 10 zz zzzdL + 4.5 4 zz (qD) zz {

where N1 is the total number density, and V1 = 4πdc2/3 is the volume of small elongated cavities. The corresponding volume fraction is ϕ1 = N1V1. Following this, we shall make simplifying assumption that c = γd, where γ is a number smaller than one. When q is in the range 2π/D < q < 2π/c (region 2 in Figure 9), small isotropic and elongated voids described previously are unresolved, and the scattering can only reveal the global craze structure. Crazes are filled with polymer with a volume fraction of air φ, so that the contrast factor can be written as ρe2 = φ2ρpol2, where ρpol is the average electron density of the polymer matrix. We assume the value φ = 0.25 which is coherent with the polymer volume fraction used by MourgliaSeignobos et al.71 in polyamide and with the volume fraction of air within a fibrillar craze found by Michler40 in PS. Crazes are modeled by uniaxial oblate ellipsoids with radius (half thickness) L along the tensile direction (X) and larger radius D in the perpendicular direction, as represented in Figure 4. Microscopic observations (Figures 3 and 4) have shown large distributions of craze sizes. To describe the resulting scattered intensity, some hypotheses must be done. We assume that larger radius D (respectively L) varies between extremum values Dmin and Dmax (respectively, Lmin and Lmax) and that D is a linear function of L

2

2

2

∫

Lmax

(12)

2 −α j j

( )

(13)

With D related to L by eq 6. In the region of q < 2π/D (region 3 in Figure 9), the scattered intensity is nearly isotropic. It comes from the response of large spherical cavities formed during the injection process. The radius of these cavities is comprised between Rmin and Rmax. The size distribution is also expressed by a power law P3(R ) = P3R−α1

2

2

∫R

I3(q) = β ρpol P3

erf

(6)

+ 4.5

With D − Dmin ϵ = max Lmax − Lmin

P2(L) dL

min

2

(5)

D(L) = ϵL + β

Lmax

And the corresponding volume fraction of crazes ϕ2 is given by

yz zz zz zz zz zz {

yz zz zz zz zz zz {

∫L

R max

qR 12 10 4

( )

(qR )

2 −α1j j

V3 R

min

jij j ij −q2R2 yz zz jjexpjjj jj j 5 zz jj k { k

(14)

yz zz zz zzdR zz zz {

(15)

V3 is the volume of spherical cavities with radius R. The number density of large cavities is given by

(7) I

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Macromolecules N3 =

∫R

R max

P4(R ) dR

occurs below 10% of true strain. The volume fraction and sizes of these nucleated crazes remain very small, in between 4 × 10−6 and 10−5 with a size which does not exceed 100 nm in length for both polymers. The quantitative USAXS analysis gives access to the number density N2 of these crazes. It is found that at 10% true strain, about 1010 small crazes per mm3 have already nucleated. This number is of the same order of magnitude as for the pre-existing defects (N3) as shown in Table 4. Once the crazes have nucleated, the number density of crazes does not evolve anymore up to failure. Their growth seems to be blocked by an internal process within the material. For instance, in DEP20-CDA at θ = 90° submitted to a tensile experiment at 40 °C (Figure 6a,b), the maximum size of crazes increases from 100 to 260 nm in length and from 60 to 70 nm in thickness, and their volume fraction only increases from 1.4 × 10−5 to 6 × 10−5 between 10 and 34% of deformation (see Table 4). We propose to explain this moderate growth by the presence of strain hardening, which blocks or slows down craze growth. This mechanism will be discussed in the Results and Discussion section. When the macroscopic applied stress becomes sufficiently high, the larger crazes present in the sample can finally grow, which constitutes a second growth regime. Different growth kinetics are observed in this second growth regime depending on the plasticizing agent. In the case of samples plasticized with DEP, the growth of large craze becomes quickly unstable and catastrophic for the polymer properties. The second growth regime is too fast, in the time scale of the experiment, to be observed in USAXS or it involves too small a number of crazes. The analysis of these polymers after failure indicates that the maximum craze sizes are of the order of 200 nm long and 70 nm thick with volume fractions of the order of 6 × 10−5. When cellulose acetate is plasticized with TA, a second growth regime is observed. A second population of crazes appears which scatters at small q values and corresponds to crazes with the largest sizes, as reported in Table 5. This population is experimentally observed by the second bump on the scattered intensity parallel to the tensile direction, as shown in Figure 6c. It corresponds to the large crazes with dimensions DD and LL in Table 5. This reveals that a small proportion of the (largest) crazes formed by nucleation of cavities around the pre-existing defects during the first growth regime will start to grow faster. The USAXS analyses indicate that between 106 and 107 crazes per mm3 belong to this second family (N′2), as reported in Table 5. This constitutes a small fraction of the total number of crazes in the sample, which is in the order 1010. Craze sizes are multiplied by 6 compared to the first family, as reported in Table 5. These crazes then continue to grow until failure occurs at about 40% of true strain. The analysis of these samples after failure indicates that these crazes reach maximum sizes of 3 μm long and 500 nm thick with a volume fraction of 9 × 10−4. The corresponding growth rates can be estimated at 5 nm/s. In the Results and Discussion section, we will propose a mechanism of growth which describes this acceleration. 3.4.3. Influence of the Polymer Chain Orientation. Figure 7 reports the normalized scattered intensities parallel and perpendicular to the tensile direction obtained after failure at 60 °C. All fitting parameters are given in Table 6. Different evolutions of the scattered intensities parallel to the tensile direction are observed for θ = 0° and θ ≥ 45°. In

(16)

min

And the corresponding volume fraction ϕ3 is given by ϕ3 =

∫R

R max

V3(R )P3(R ) dR

(17)

min

Altogether, the global scattered intensities are the sums of all contributions.81 Ipara(q) = I0(q) + Ipara1(q) + Ipara 2(q) + I3(q)

(18)

Iperp(q) = I0(q) + Iperp (q) + Iperp (q) + I3(q)

(19)

1

2

The abovementioned discussion illustrates the importance of combining microscopic observations and scattering experiments. While direct observations enable identifying the various types and typical size and structure of damage defects, the analysis of scattering curves enable a quantitative determination of the evolution of associated parameters. 3.4.1. Reference Unstretched Samples. The scattering intensities observed in all polymers in the reference, unstretched state, that is, before the tensile experiment, have been fitted with eq 15. This scattering is due to cavities resulting from the injection process. This population of cavities is modeled with a size distribution exponent α1 = 3.8 and radii ranging from Rmin = 1 nm to Rmax = 1.5 μm. For specimens cut in the 0° and 90° directions, a small amount of cavities elongated in the direction perpendicular to the direction of injection is observed. Their contributions are added to the fitted intensities in the form of ellipsoidal cavities. The volume fraction of initial damages does not exceed 10−5, and the number density of initial defects N3 is of order 1010 to 1011 mm−3. 3.4.2. Evolution of Damage under Tensile Deformation. In this study, we are interested in the mechanisms of initiation and propagation of damages under tensile deformation. Figure 6 shows the parallel and perpendicular scattered intensities at different stages of the deformation at 40 °C for DEP20-CDA and TA20-CDA systems at θ = 90°. Experimental data have been fitted by global scattered intensity, eqs 18 and 19. The analysis of experimental scattered intensities provides the size and volume fraction of each type of damage. Adjustable parameters are given in Tables 4 and 5. Regarding the micromechanisms of damage in plasticized cellulose acetate systems, two regimes can be observed during the tensile deformation. The first regime corresponds to the nucleation of small crazes. The nucleation of these crazes Table 4. Sizes and Volume Fractions of Damage Corresponding to Fitting Parameters Used at Different Strains for DEP20-CDA with θ = 90° in Figure 6, α = 3.8 1st craze family strain (%)

Dmin (nm)

Dmax (nm)

Lmin (nm)

Lmax (nm)

10 15 20 24 28 32 34

5 5 5 5 5 5 5

50 70 75 75 90 110 130

1 1 1 1 2 2 2

30 30 30 30 30 32 35

Φ2 1.4 3.3 5.4 7.2 5.6 5.4 6.0

× × × × × × ×

10−5 10−5 10−5 10−5 10−5 10−5 10−5

N2 (mm−3) 1 5 7 9 2 1 1

× × × × × × ×

1010 1010 1010 1010 1010 1010 1010 J

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Macromolecules

Table 5. Sizes and Volume Fractions of Damage Corresponding to Fitting Parameters Used at Different Strains for TA20- with θ = 90° in Figure 6, α = 3.8 1st craze family strain (%)

Dmin (nm)

Dmax (nm)

Lmin (nm)

Lmax (nm)

15 24 28 32 36 40

5

80 90 90 100 100 100

2 2 3 3 4 4

40 35 35 35 35 35

2nd craze family Φ2 1.3 3.5 5.7 6.2 6.1 2.6

× × × × × ×

10−5 10−5 10−5 10−5 10−5 10−4

N2 (mm−3)

DDmin (nm)

DDmax (nm)

LLmin (nm)

LLmax (nm)

9 × 109 1010 6 × 109 7 × 109 5 × 109 2 × 1010

N/A 100 100 100 100 120

600 800 1300 1500 1700

50 50 45 35 35

250 250 250 250 230

Φ2′ 2.2 4.1 5.0 1.3 6.3

× × × × ×

10−5 10−5 10−5 10−4 10−4

N′2 (mm−3) 106 2 × 106 106 4 × 106 107

Table 6. Size and Volume Fraction of Different Damages Corresponding to Fitting Parameters Used in Figure 7 for TA15CDA Samples after Failure at 60 °C, α = 3.8 1st craze family

2nd craze family

θ (deg)

Dmin (nm)

Dmax (nm)

Lmin (nm)

Lmax (nm)

Φ2

N2 (mm−3)

DDmin (nm)

0 45 90

10 8 5

300 80 70

5 5 3

25 25 20

1.9 × 10−6 5.5 × 10−5 3.6 × 10−5

2 × 107 3 × 109 5 × 109

N/A 80 70

DDmax (nm)

LLmin (nm)

LLmax (nm)

Φ2

N′2 (mm−3)

500 550

30 28

160 180

8.9 × 10−5 8.3 × 10−5

1 × 107 2 × 107

Table 7. Sizes and Volume Fractions of Different Damages Corresponding to Fitting Parameters Used in Figure 8a 1st craze family

2nd craze family

T (°C)

Dmin (nm)

Dmax (nm)

Lmin (nm)

Lmax (nm)

Φ2

N2 (mm−3)

DDmin (nm)

DDmax (nm)

LLmin (nm)

LLmax (nm)

Φ2′

N′2 (mm−3)

60 80

10 5

115 80

3 4

35 30

7.1 × 10−5 3.5 × 10−4

6 × 109 3 × 1010

N/A 80

1200

30

150

4.9 × 10−4

2 × 107

The two α and φ2 values correspond to two distinct size distributions.

a

10−4. However, it was observed earlier that in TA20-CDA, the volume fraction of crazes at failure is found to be of the order of 9 × 10−4 and crazes sizes can reach 3 μm in length and 500 nm in thickness. Increasing the plasticizer content leads to an increase of craze sizes and volume fractions but does not affect the craze morphologies and the existence of only one craze population with moderate size in DEP-pCDA and two populations of crazes in TA-pCDA. Fibrillar crazes are observed in TA-pCDA, and smaller homogeneous crazes are observed in DEP-pCDA. An increase of the volume fractions and sizes of crazes is observed as θ changes from 45° to 90° in TA20-CDA, while it is not the case in TA15-CDA. Therefore, the micromechanism of deformation seems to be more sensitive to the orientation of polymer chains as the plasticizer content increases. 3.4.5. Influence of Temperature. It has been observed that temperature affects the ductility of plasticized cellulose acetate by decreasing the strain hardening modulus. STEM observations have also revealed a difference in the morphologies of damage as temperature increases. In samples broken at 80 °C, crazes were visible on the microscopic images, whereas no damage was observed for the same samples broken at 60 °C. USAXS analyses performed after tensile failure has confirmed these differences in the structure of the damage. Figure 8 shows the normalized scattered intensities for the DEP15-CDA system at θ = 90° broken at 60 and 80 °C. Adjustable parameters are given in Table 7. In the case of the samples broken at 60 °C (Figure 8a), a single size distribution is used to fit the contribution of crazes. Moreover, no bump is observed on the scattered intensity perpendicular to the tensile direction (green curve in Figure

the latter case, two size distributions of crazes describe the scattered intensities, whereas one is sufficient with θ = 0°. There is also an increase of the volume fraction of craze. At θ = 0°, the volume fraction of craze φcrazes is around 2 × 10−6 and at θ ≥ 45°, the volume fraction of crazes increases up to 1 × 10−4. Few differences are observed between the cases θ = 45° or θ = 90°. The slight bump observed on scattered intensities perpendicular to the tensile direction for samples at θ > 45° indicates the occurrence of fibrillated crazes with a volume fraction 2 × 10−6. Increasing θ leads to an increase of the volume fraction and of the size of the crazes, as observed by microscopic analysis. It is observed that when the injection direction is parallel to the tensile direction (θ = 0°), and whatever the plasticizing agent is, the materials are more rigid (Young’s modulus and strain hardening modulus are higher and elongation at break is smaller) and break by propagation of single craze as described above for polymers plasticized with DEP. 3.4.4. Influence of the Plasticizer Content. Figure 6c,d shows the scattered intensities of TA20-CDA (Tg = 120 °C8) during tensile experiments at 40 °C, and Figure 7c shows the scattered intensities of TA15-CDA (Tg = 140 °C8) after tensile failure at 60 °C, both at θ = 90°. The sizes and volume fractions of crazes formed during the tensile experiment in TA15-CDA systems are lower than those measured in TA20-CDA, as reported in Tables 5 and 6. At a sufficiently high stress value (approximately 50 MPa), a second family of crazes with sizes about 4 times larger also appears in TA15-pCDA. The postmortem analysis of these samples indicates that the maximum craze sizes are of the order of 1 μm long and 300 nm thick with volume fractions of the order of K

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Macromolecules 8a), which means that no fibril is observed. In the case of the sample broken at 80 °C (Figure 8b), two size distributions are used to describe the contribution of crazes. In addition, in the q range, 7 × 10−2 < q < 3 × 10−1 nm−1; the scattered intensity perpendicular to the tensile direction exhibits a marked bump which is attributed to the appearance of elongated cavities inside crazes. The volume fraction ϕ1 indicates the presence of 7 × 10−5 fibrillar crazes at failure at 80 °C. A transition from homogeneous crazes to fibrillated crazes is observed as the experiment temperature increases. It is also found that the intensity levels measured on the sample broken at 80 °C are one decade higher than those of the sample broken at 60 °C. This increase in intensity with temperature indicates that the volume fraction of crazes increases. Indeed, the volume fraction of crazes in the sample after failure at 80 °C is found to be about 8 × 10−4, while it is about 7 × 10−5 after failure at 60 °C. The increase of experimental temperature leads to a decrease of the second growth regime kinetics, and a second population of larger crazes is observed even in DEPpCDA samples.

testing, as reported in Tables 4 and 5. It is observed that the number of the nucleated crazes no longer varies with the deformation until failure, see Table 4. The small increase of the volume fraction Φ2 and of the sizes of these crazes with the deformation indicates that this initial craze nucleation is followed by a slow craze growth without new nucleation, as reported in Table 4. At 10% of true strain, the system DEP20-CDA at θ = 90° exhibits a volume fraction of crazes Φ2 = 1 × 10−5 with a maximum craze length of 100 nm. At 34% of true strain (failure), the volume fraction is increased up to Φ2 = 6 × 10−5 with a maximum craze length of 250 nm. In semicrystalline polymers, Mourglia-Seignobos et al.71 have shown that the craze growth is blocked by the crystalline phase. Because no crystalline phases exist in our amorphous polymers, we could think that the craze growth would lead to rapid brittle failure soon after the appearance of a single cavity after 10% of deformation. This is not what was observed. We suggest that ductile behavior observed in our polymers is due to strain hardening which blocks the propagation of cracks. When the applied stress is sufficiently high, we observe that sample breaks due to brutal crack propagation. Experimentally, this second growth regime is observed when the macroscopic stress reaches 50 MPa approximately, see Table 5 and Figure 2. USAXS measurements have shown, in TA-pCDA at θ = 90° samples (see Figure 6), that a second family of crazes is observed after 24% of true strain approximately which corresponds to a macroscopic stress of about 45 MPa. This second family is deduced from the appearance of a second bump at small q values in the scattered intensity parallel to the tensile direction. The number of these crazes per unit volume is found to be of order N2′ ≅ 106 mm−3, as reported in Table 5. The small volume fraction of crazes observed, even after failure, indicates that this second craze family cannot result from a mechanism of craze coalescence. The distance between two crazes is too large. We assume that this second family of larger crazes results from a very small proportion of crazes initially nucleated (first regime) which grow faster than the rest of the initial craze population. These crazes go on growing until the growth rate of one of them accelerates sufficiently to break the sample within the time frame of the experiment. This second growth regime is so rapid in DEP-pCDA that it leads to a “brittle” failure of the sample as soon as one craze starts to grow faster. We are not able to observe the formation of a second family of larger crazes. It is possible that their number is too small, for example, smaller than 104 mm−3 for being observable in USAXS experiments. Note that this regime is observed in DEP-pCDA samples for experiments performed at 80 °C (see Figure 8). Conversely, for TA-plasticized samples, this second growth regime appears, and we are able to observe the evolution of this second family of larger crazes. The presence of this second regime explains why the volume fraction and the sizes of the crazes are larger in TA-pCDA samples (reaching 0.1% at failure for the sample TA20-CDA at θ = 90° with a maximal craze size of 3 μm in length and 0.5 μm in thickness) as compared to ones in DEP-pCDA samples for which the second growth regime is not observed for deformations performed at 60 °C. We have shown that different parameters can influence this damage kinetics. The increase of the experiment temperature leads to an increase of the sizes and the volume fractions of crazes. Even for polymers plasticized with DEP, the size of the crazes can reach 3 μm in length, and the volume fraction is

4. INTERPRETATION AND DISCUSSION 4.1. Summary of the Experimental Results. Tensile experiments on plasticized cellulose acetate samples have revealed the presence of an important strain hardening regime above 8% of true strain for samples with 15 and 20 wt % of the plasticizer.8 It is shown that the strain hardening moduli are higher for DEP-pCDA samples than TA-pCDA samples and decrease with the increase of angle θ and with the temperature of the experiment, as typically observed in the literature.39,87 Concerning the damage morphologies observed in these samples, STEM observations have been carried out after failure at 80 °C and indicate the presence of homogeneous crazes41 and a small amount of fibrillated crazes41 growing preferentially in the direction normal to the applied stress in DEP-pCDA and the coexistence of homogeneous crazes and fibrillated crazes in TA-pCDA,44 as observed in Figure 3. It also appears that the increase of angle θ leads to an increase of the volume fraction and the size of crazes. In TA-pCDA samples, the increase of this angle θ reveals a transition from homogeneous crazes to fibrillated crazes with a progressive increase of the volume fraction of fibrillated crazes, as shown in Figure 3. Similar transitions of the damage morphology have been observed in the literature.44 The analysis of the damage microstructure by USAXS measurements confirms the results obtained by scanning electron microscopy. The best resolution of X-ray techniques allows us to observe objects at much smaller scales than those observed in microscopy, that is, of the order of 10 nm.88 The influence of temperature and the evolution of the microstructure during a tensile experiment can be investigated. The analysis of the normalized scattered intensity curves makes it possible to highlight the damage initiation and propagation mechanisms in plasticized cellulose acetate samples.70,71 It is observed that the damage mechanism can be described by a two-step mechanism. During the first step, USAXS analysis have shown that small crazes with sizes comprised between 10 and 100 nm nucleate simultaneously within the time resolution of our experiment before 10% of true strain. The number of these nucleated crazes per unit volume is found to be of order N2 ≅ 1010 mm−3, which is in the same order of magnitude as that of the number of impurities initially present in the sample with N3 ≅ 1010 mm−3 measured on samples before tensile L

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Macromolecules found to be in order of 8 × 10−4. The second regime of craze growth described above is observed in DEP-pCDA samples when the temperature of the experiment is 80 °C, as shown in Figure 8. The macroscopic orientation of the polymer chains (θ) also influences this kinetics. When the chains are oriented in the tensile direction (θ = 0°), the failure is catastrophic for all the studied systems including those plasticized with TA. The second growth regime and, consequently, the second craze family are not observed, as shown in Figure 7. Regarding the influence of the plasticizer content, it is observed that its increase leads to an increase of the craze sizes and of their volume fraction. However, the typology of the crazes remains similar for a given plasticizer. With DEP, the crazes remain homogeneous and for TA, the coexistence between homogeneous and fibrillated crazes is still observed, as discussed in Section 3.4.4. We consider now more detailed interpretations of the physical mechanisms leading to damaging and ultimate failure. Several key issues need to be explained:

γp3 π 16 * (p) = ζhom 3 Π2

Nucleation is thermally activated. The nucleation time of a craze is related to the experimental nucleation barrier ζ by ij ζ yz zz τ = τ0 expjjj j kBT zz k {

where τ0 is a microscopic time of order 10 s, kB is the Boltzmann constant 1.38 × 10−23 m2·kg·s−2·K−1, and T is the experiment temperature. Nucleation is a process which takes place parallelly in a relevant volume V. The nucleation rate in a system of volume V is an extensive quantity. The nucleation frequency ν in a sample is thus given by ν ≈ τ0−1

ij −ζ yz V zz expjjj 3 j kBT zz ac k {

97

(23)

Volume V is not necessarily the whole volume of the sample but may be the volume where stress is intensified as a consequence of the shape of the sample or in the vicinity of solid particles (impurities). Volume a3c is the volume, where an elementary cavitation event takes place. For relevant experimental time scales, a3c is typically of order 10−27 m3. The relevant time scale of nucleation under tensile stress is around τexp = 102 s which is the duration of the experiment. According to eq 23, this corresponds to a typical nucleation free energy barrier given by ij τexp V yz + ln 3 zzz ζ = kBT jjjln j τ ac z{ 0 k

(24)

The first term in parenthesis on the right hand side of eq 25 is of order 30. The second depends on the relevant volume V. Note that the dependence of ζ on the considered volume is logarithmic and is thus small. If we assume that V is of order 10−22 m3 (which is typically the volume where stress is intensified in the vicinity of an existing cavity in the sample as we shall see later), the second term is of order 10. The relevant energy barrier for nucleation in our problem is then of order ζ = 40kBT = 1.6 × 10−19 J. Note that if volume V is macroscopic, for example, of order 10−9 m3, which may be the relevant volume in an experiment of cavitation in pure water, for instance,90−92 the second term in eq 24 is of order 40. Thus, for being metastable for a duration of about 100 s, a sample of volume 10−9 m3 requires free energy barriers for cavitation of order 70kBT (2.8 × 10−19 J), whereas systems of sizes of volume 10−22 m3 requires free energy barriers of only 40kBT. The nucleation barrier in our samples can be estimated from eq 21 using experimental values of the parameters. The surface tension of plasticized cellulose acetate is found to be around 4.0 × 10−2 J/m2 at 20 °C.98 Crazes in our experiments nucleate at stress values of order of 30 MPa. The local negative pressure Π may be smaller but similar to the applied stress (in the vicinity of a solid particle for instance) which would give a theoretical nucleation barrier ζ*hom of order 10−18 J. We find thus that there is a discrepancy of about a factor of 5−10 regarding the theoretical homogeneous nucleation barrier ζ*hom of a craze as compared to the experimental value ζ deduced from the observed nucleation times. On the basis of our experiments, it follows that homogeneous nucleation of cavities cannot be the relevant mechanism for damage nucleation. We propose that cavity

In the following sections, we discuss the different steps of the damaging mechanism, based on experiments and physical interpretations. The elementary nucleation process is discussed in Section 4.2. The ensuing different regimes, which correspond to the slow growth of crazes followed by the acceleration of the growth kinetics up to failure, are described in Section 4.3. In Section 4.4, we discuss other interpretations for similar issues proposed in the literature and how they differ from the one proposed in our manuscript. 4.2. Nucleation of Crazes. According to the classical theory of homogeneous nucleation, cavitation occurs when the isotropic component of the stress in the bulk material, that is the negative pressure, denoted Π in what follows, exceed a given value. The free energy of homogeneous nucleation contains a term corresponding to the release of elastic energy, which favors nucleation of a craze, and a surface energy term, which tends to prevent nucleation. For a given Π value, the free energy of nucleation has a maximum for a critical cavity diameter ac beyond which a cavity becomes unstable and grows, while below ac, it will collapse.89−92 The critical diameter ac is given by

4γp Π

(22) −12

1 What is the mechanism responsible for the first appearance of crazes? 2 What is the mechanism which stabilizes damaging between its initiation at about 30 MPa applied stress (i.e., 10% deformation) and 60 MPa (breaking)? 3 How can we explain the stabilization of the number of crazes at about ∼1010/mm3 after their appearance at 10% deformation? 4 How can we explain that the majority of crazes do not grow (about 99.99% of them) upon increasing the stress and that only a tiny minority grow and are responsible for ultimate failure? 5 What is the growth mechanism of crazes that is responsible for ultimate failure?

ac =

(21) 89−96

(20)

where γp is the surface tension of the material. Typically, for accessible time scales, ac is of order 1 nm.90−92 The corresponding theoretical free energy barrier for homogeneous nucleation is given by M

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angle is large, and in any case larger than 90°. According to Young−Dupré relation, this corresponds to γI − γIP < 0. This condition may be realized between strongly polar polymer-like plasticized cellulose acetate (surface tension γP of about 4.0 × 10−2 J/m2 at 20 °C98 with roughly equivalent polar and dispersive components) and a nonpolar impurity. These impurities which need yet to be identified may be residues due to partial combustion resulting from the high temperature processing conditions. When the contact angle θ in a flat interface is larger than π/2 (non-wetting interface), it can be further increased by the roughness of the impurity, as schematized in Figure 12.103,104

nucleation in our experiments is heterogeneous, that is, it initiates at interfaces between the polymer matrix and preexisting impurities. This point of view is consistent with USAXS experiments which show that the number of nucleated crazes is very close to the number of initial defects (cavities and impurities induced by injection) observed before tensile analysis. Our point of view is that crazes during tensile tests nucleate on the same impurities which lead to the formation of bubbles during injection. Heterogeneous nucleation is very often observed, for example, in water for which the pressure at cavitation is very frequently much smaller than the theoretical prediction deduced from the standard nucleation theory,92,99 corresponding to a reduction by a factor of 10 of the observed free energy barrier as compared to the theoretical one. At an interface between the polymer and a solid particle, the free-energy barrier of nucleation is reduced by a factor f(θ) which depends on contact angle θ between the polymer and the impurity (see Figure 10)100,101 * = ζhom * f (θ ) ζimp

(25)

where the function f(θ) is given by f (θ ) =

(2 − cos θ )(1 + cos θ )2 4

Figure 12. Measure of the contact angle θ between the rough surface of the impurity and the polymer.

(26)

In the case of a nonwetting interface, the polymer does not penetrate the surface roughness, which increases the contact angle θ between the polymer and the impurity and consequently decreases the interfacial adhesion. According to the Cassie−Baxter model,105 the interfacial tension can be expressed by the following equation γIP

rough

= γP(1 − ε) + εγIP and γI

rough

= εγI

(27)

ε is the roughness of the surface impurity, as reported in Figure 12. By replacing γIP by γIPrough we find

Figure 10. Measure of the contact angle θ between the interface polymer−impurity.

cos(θrough) = ε cos(θ ) − (1 − ε)

(28)

As ε and/or cos (θ) are small, cos(θrough) ≅ ε − 1, and θrough ≅ π − 2ε . A moderate roughness ε = 0.5 is sufficient to reduce the free energy barrier by about a factor 5−6 which is the order of what is required so that

The evolution of f(θ) as a function of θ (in rad) is reported in Figure 11. The contact angle is given by the Young−Dupré relation102 cos θ = (γI − γIP)/γP where γI and γP are the surface tensions of the impurity and of the polymer, respectively, and γIP is the interfacial tension between the polymer and the impurity. Figure 11 shows that the factor f(θ) may reduce the nucleation barrier by several orders of magnitude if the contact

* = ζhom * f (φ ) = 1.6 × 10−19 J = ζ ζimp rough

(29)

4.3. Controlled Growth by the Strain Hardening. The growth of these cavities is the second stage of the damage process and can be divided in two steps. 4.3.1. First Regime. Once a cavity has nucleated, the local elastic stress around it relaxes. The growth of the cavity is driven by the relaxation of the polymer density towards its equilibrium value at the considered temperature. The cavity may be assumed to be spherical and grows rapidly.91,92 In liquids such as water, the cavity would grow without limit reaching observable macroscopic size on a short time scale. However, the polymer which we consider is different from liquids in an essential way. It exhibits strain hardening at large deformation amplitudes. Therefore, during the growth of the cavity, the tangential stress increases as a consequence of the tangential strain undergone by the polymer during this bubble inflation process and of the strain hardening behavior of the considered material. Once the tangential stress reaches a value of the order

Figure 11. Evolution of f(θ) as a function of θ. N

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Macromolecules of a few Π, where Π is the applied stress at large distance from the bubble, the stress in the vicinity of the cavity becomes able to equilibrate the applied stress on a larger scale. The growth of the cavity is blocked. Note that in the case of a polymer which does not exhibit strain hardening, the growth would go on unimpeded: the sample would break after the nucleation of a single cavity, the growth of which nothing would stop. Figure 13 gives a schematic representation of the first regime of craze

50% also. The scattered intensity, which is proportional to R6 is expected to increase by about a factor of 10 which is what is observed in Figure 6a. Thus, the increase of the scattering intensity in this figure upon increasing deformation can be interpreted by a linear increase of the size of the cavities with the applied stress. However, upon increasing stress further, we observe in Figure 6c that cavities grow again in a way that cannot be accounted for by eq 31. This second growth process leads to a large increase of the cavities, from typically 60 nm in diameter to 1 μm and more, up to macroscopic failure. We assume that this second step growth process is due to the nucleation of new cavities in the vicinity of already existing cavities. 4.3.2. Subsequent Growth of Crazes. We consider here this second-step growth process. After a nucleated cavity is blocked at a size of order 60 nm, we assume that this cavity may grow by nucleation of new cavities in its vicinity and more specifically in its equatorial neighborhood, where stress is intensified. This growth process may take place once the macroscopic stress has reached a sufficiently high value. Volume V influenced by the effect of stress intensification in the vicinity of an existing cavity is proportional to the size of this craze itself (V ∝ D3). The larger the craze, the larger the volume submitted to an intensified stress and the larger the nucleation rate in the neighborhood of a cavity. This mechanism is described by schematic in Figure 14. Craze

Figure 13. Schematic representation of the first step of cavity growth influenced by the strain hardening.

growth, where Π is the macroscopic imposed stress, σtan is the local stress in the equator position of the cavity, a(0) is the initial cavity diameter and R is the final diameter that the cavity can reach. Radius R of the cavity goes on growing until the tangential stress value become equivalent to the macroscopic stress applied on the sample with an intensification effect,106 σtan ≅ κΠ, where κ is a number larger than but of order 1. We obtain the following equation σtan =

R − a(0) ESH ≈ κΠ a(0)

Figure 14. Schematics of the growth crazes by nucleation of cavities in the equator positions and followed by a mechanism of coalescence.

(30)

ESH is the strain hardening modulus. One may assume that a(0) ≈ 20 nm which is the typical distance for an interface to recover bulk glassy properties.107 At smaller distances from the interface, the polymer is in a mobile state and not glassy, thereby unable to display strain hardening and to bear high stress. Thus, we can deduce diameter R of the cavities R=κ

Π a(0) + a(0) ESH

growth is proportional to the nucleation rate of cavities in the volume of polymer submitted to an intensified stress. As a consequence, larger crazes grow more rapidly than smaller ones. The cavitation frequency in the vicinity of a given craze is given by eq 24, where the volume V is of order 10−22 m3. For a nucleation rate of order 0.01/s, the nucleation barrier must be of order ζ = 40kBT = 1.6 × 10−19 J as calculated above. This nucleation free energy barrier corresponds to a local stress Π of order 100 MPa according to eq 22. This is the stress level which is expected in the vicinity of a cavity, where the macroscopic stress is intensified by a factor of order 1.5 or 2 depending on the shape of the cavity, whereas the macroscopic stress in this regime is about 50 MPa. As a consequence, homogeneous nucleation just ahead of already existing cavities allows us for explaining the growth of cavities upon increasing the stress from 30 MPa up to 60 MPa, for which, typically, our samples break. Let us consider in more detail the growth kinetics due to new cavity nucleation ahead of the growing one. The growth of a craze can be expressed as a function of nucleation time τ, defined by eq 23, of new cavities in the vicinity of already existing cavities, where stress is intensified. The corresponding region has volume V ∝ D3, where D is the diameter of the considered cavity. The nucleation rate per unit volume is defined by ∼1/ac3τ. The nucleation rate of new cavities in the vicinity of a current craze of size D is given by ν ≈ D3/ac3τ.

(31)

As an example, the measurement of R in DEP20-CDA at θ = 90° during a tensile experiment at 40 °C has been carried out. We observed that the maximum size of R is of order 60 nm which is compatible with the estimate that can be deduced from eq 31. We propose that cavities nucleate on impurities and that they grow rapidly until they reach a size of order 60 nm at which their growth is blocked by the strain hardening behavior of the polymer in the vicinity of the growing bubble. Note that once the cavities are blocked, their diameters grow linearly with the applied stress according to eq 31. This growth takes place without new damage and is an elastoplastic effect associated to the strain hardening in the vicinity of the cavity. This result is consistent with what is observed in USAXS experiments as can be observed in Figure 6a. Between 15% deformation and 35% of deformation, the stress increases by 50%. According to eq 31, the diameter increases thus by about O

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At a local level, there is certainly a distribution of stress in the vicinity of the cavities. All cavities are not submitted exactly to the same stress, in particular, as a consequence of a disordered distribution of cavities. Those for which the stress is slightly larger grow much more rapidly. It allows for explaining why we observe a small fraction of large craze in the case of TA-pCDA before failure. It allows also for understanding why we do not observe intermediate steps between crazes of moderated sizes and final failure in some samples (i.e., DEPpCDA) because the fraction of craze leading to ultimate failure may be too small to be observed in USAXS. It would be the case, for instance, if their number is smaller than 104 mm−3. 4.4. Discussion. Crazing has been the subject of intense research over the past 50 years. It is often described in the literature that damage mechanisms are strongly related to polymer entanglements. Regarding the craze initiation, Michler44 have suggested recently that the development of a craze is preceded by the formation of a localized plastic deformation zone. As this zone develops, the hydrostatic stress increases, and, when exceeding a critical stress level, cavitation will take place. They assumed that these “precrazes” are characterized by a domain-like structure, where weak and localized mobile domains exist between the entanglement networks. These structures are only visible after “straininginduced contrast-enhanced” pretreatment using solution or vapors of chemicals such as osmium tetroxide.44 The most common approach used to describe the craze initiation is the stress concentration effect in the vicinity of the pre-existing defect in the polymer sample.48 It had been proposed long ago by Argon108 that craze growth takes place by nucleation ahead of the craze tip, in an analogous way as the model we propose here. Then, a few years later, Argon argued that this mechanism is not relevant for several reasons.109,110 The first reason is that this mechanism may be incompatible with the formation of fibrils. The second reason is that nucleation would not be possible at the observed stress levels. Our point of view is that nucleation of new cavities ahead of the craze tip as we discuss in this manuscript is perfectly consistent with the apparition of fibrils. Indeed, between neighboring cavities, thin polymer films are formed. Once they reach a thickness of order 10 nm, they become unstable due to disjoining pressure effects and they are expected to rupture.101 This mechanism should lead to a fibrillike structure. Regarding the second reason, we find in our experiments that the growth rate of crazes is perfectly compatible with homogeneous nucleation of cavities ahead of already existing cavities. The latter is supposed to have nucleated initially on impurities at an applied stress of about 30 MPa. A strong indication for this is that their number is constant throughout the experiment and that this number is equal to the number of cavities created during the injection process. We assume that the same impurities which allow for the creation of cavities during injection lead also to the cavitation of crazes under applied stress. Their number has been found to be of order 1010 mm−3 in our samples. The alternative mechanism to nucleation ahead of the craze is the so-called meniscus instability.57,111 This mechanism has been proposed by Taylor58 for describing hydrodynamic instabilities between two plates when they are pulled away one from the other at a small angle. This mechanism may be relevant for a polymer in the molten state. However, reconciling this mechanism with the strain hardening behavior of polymers such as those studied here is not obvious. Instead

Once a new cavity has nucleated in the vicinity of a craze, it grows until it reaches a size of order ξ ≈ 100 nm, where it is blocked by the strain hardening mechanism. The evolution of the craze size D(t) obeys then to the following equation dD D 3ξ ≈ 3 dt ac τ

(32)

which leads to ÄÅ ÉÑ ac 3 ÅÅÅ 1 1 ÑÑÑ t ÅÅ 2 ÑÑ ≈ − 2 Å Ñ 2 ÅÅÇ ξD (0) ξD (t ) ÑÑÖ τ

(33)

where D(0) ≈ 100 nm is the size of the initial craze at 15% of deformation. When D(t) → +∞, the breaking time t∞ starting from a craze of size of D(0) is then obtained t∞ ≈

ac 3 ξD2(0)

τ (34)

One sees thus that if the nucleation time τ at the considered stress level is of order τr = 108 s, the sample breaks in a time t∞ ≈ 100 s (assuming that ac ≈ 1 nm; ξ ≈ 100 nm and D(0) ≈ 100 nm). In order to consider how sharp the transition is between no growth at all and a fast growth process, consider the following relations regarding the nucleation times at different stress 1 levels: ln(τ ) ∝ ln(τ0) + α 2 , with α being a constant, and Π is Π

1

the local stress and ln(τr) ∝ ln(τ0) + α Π 2 with τr = 108 s, and r

the local stress at breaking Πr = 100 MPa. Thus, we deduce the following equation ÉÑ ÄÅ 2 ÉÑ ÄÅ 2 ÑÑ ÑÑ ÅÅ Π jij τr zyzÅÅÅÅ Π r jij τ zyz r Ñ Å Ñ Å lnjj zz ≈ lnjj zzÅÅ 2 − 1ÑÑ ≈ 46ÅÅ 2 − 1ÑÑÑÑ jτ z j τ0 zÅÅ Π ÑÑÖ ÑÑÖ ÅÅÇ Π (35) k r{ k {Ç which expresses the evolution of the nucleation time as a function of the applied stress. Increasing the stress leads to an abrupt decrease of the nucleation time τ and consequently, the craze−crack transition becomes immediate when a critical stress value is reached. The rupture time t∞ given by eq 34 is highly sensitive to the stress, as shown in Figure 15. When the local stress is 100 MPa, the nucleation time is equal to τr = 108 s, and the macroscopic breaking time is t∞ ≈ 100 s. By decreasing the local stress down to 85 MPa, we obtain t∞ ≈ 1010 s which is infinite for all practical purposes.

Figure 15. Evolution of ln(τ/τr) as a function of the macroscopic stress-applied σ. P

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The critical stress level of failure observed experimentally (of order a few 107 Pa) are significantly higher than the contribution that could have entanglements (∼105 Pa). This same argument had allowed Meijer and Govaert20 to reject their involvement in the strain hardening mechanism. It appears that strain hardening plays a key role in stabilizing the defects once they nucleate. As discussed, it is difficult to reconcile the strain hardening mechanism and the meniscus instability mechanism for craze growth. However, we do not rule out the relevance of this mechanism in polymers which do not exhibit strain hardening, which form the majority of the polymers studied in the literature regarding crazing.46,112 For polymers which exhibit strong strain hardening, we expect this effect to block the growth of crazes. Upon increasing the stress further, the crazes can grow by homogeneous nucleation ahead of them. This mechanism could be studied in the future by similar experiments such as those performed by Vanel et al.93 In their experiments, the authors monitor cavitation events by microphones, with the resolution of single events. Such experiments could be performed on bulk polymer materials in order to better describe the individual events of damaging and craze growth.

of being liquid under the applied deformation, the considered polymer undergoes strain hardening which may prevent the meniscus instability to take place. This point of view is supported by the fact that most of the crazes do not grow once they have reached a size of order 50−100 nm. We find indeed that at least 99.99% of cavities do not grow at all except for a linear increase of their diameter with the applied stress. Only a tiny fraction of crazes grow in the second step as described above, involving new damages in the material, which may be accounted for by nucleation of cavities ahead of the already existing ones. The strongly nonlinear growth kinetics as a function of the local stress and of the size of the cavities explains why only a small fraction undergo this second step growth process. For describing this growth kinetics, strongly nonlinear constitutive relations are used in the literature such d that dt γ ∝ σ m, where exponent m may be as large as 20 or more.46,112 These constitutive relations derive from an Eyring picture.113 As such, they involve an “activation volume v” which has no clear physical interpretation (for a discussion at this regard, see refs114−116 and references therein) but is used for fitting the data, such as plastic flow for instance. Our point of view is that the corresponding growth kinetics can be explained by a nucleation mechanism which does not require the introduction of an ill-defined “activation volume”. This nucleation model involves only the polymer surface energy. The growth of crazes involves the stress hardening behavior of the considered polymer and the nucleation model. The nucleation process that we have discussed in this manuscript leads naturally to very strong nonlinear dependences of the growth rate as a function of the applied stress. Note that an important role is often attributed to fibrils in the literature, in particular, for bearing stress. This point of view seems difficult to reconcile with experiments performed over the past 20 years regarding dynamics in thin suspended films. Thin polymer films of thicknesses smaller than 20 nm have a very low glass transition temperature.117−119 Fibrils that are one dimensional objects of about 10 nm in diameter are expected to have an even lower glass transition temperature than ultrathin films. Based on these recent experimental results, one would expect fibrils to be in a mobile state, above their glass transition. As such they should not be able to undergo strain hardening and to contribute significantly to the mechanical response. In particular, one may expect that their contribution is insufficient to prevent stress intensification ahead of the craze as we consider in this manuscript which is the only assumption that we need for our discussion. Fibrils are expected to play a role for craze widening by pulling the polymer from the matrix as the craze widens.46,112 The assumed molten state of the fibrils in this manuscript is not in contradiction with this picture because the free surface of the matrix in a craze is also in the molten state as experiments in thin films show. Note that no experiment yet has been able to probe directly the mechanical or dynamical state of fibrils. It is often argued that strain hardening is due to entanglements. Indeed, fibrils are made of entangled polymer under large elongation. However, recent theoretical, experimental, and numerical simulation studies indicate that strain hardening is related to long-lived molecular interactions such as those observed in the glassy state.25,34 As a consequence, we do not expect fibrils to support a large fraction of the stress. Our point of view therefore has been that fibrils do not bear enough stress in order to prevent stress intensification ahead of the crazes.

5. CONCLUSIONS In this paper, we show that under the effect of tensile stress, plasticized cellulose acetate damage mechanisms take place in two main stages. The first stage is associated with heterogeneous cavity nucleation in the vicinity of pre-existing impurities (i.e., related to the injection process). We find indeed that homogeneous nucleation is impossible in cellulose acetate because the free energy barrier of nucleation is too high at the stress values (∼30 MPa) for which cavitation is observed experimentally. On the other hand, in the vicinity of an impurity, the nucleation energy is lowered by factor f(θ) depending on the contact angle between the polymer and the impurity. In addition, the surface of these impurities may be rough, which further lowers the free energy barriers of nucleation. The free energy barrier calculated for homogeneous nucleation is compatible for heterogeneous nucleation to take place at the considered stress values. Experimentally, we have observed that the craze growth can be described as a two regime process. The first slow growth regime is observed. We propose to explain this moderate growth by the presence of strain hardening. Once the cavity is nucleated, the tensile stress around it relaxes. Under the effect of the macroscopic stress, the cavity starts to grow. During the growth of the cavity, the tangential stress increases as a consequence of the strain hardening behavior of our material. Thus, the cavity can grow until it reaches a size κΠ R ≈ E a(0) + a(0) ≈ 60 nm . Once the diameter of a cavity SH

reaches this value, the tangential stress reaches a value of order κΠ, where κ is a number of order 2. This tangential stress in the vicinity of the cavity equilibrates the stress applied on a larger scale. Upon increasing the stress further, the diameter of the cavities increases proportionally to the stress without further damage in the first step. In the second step, the size of cavities starts to increase faster than in the linear growth regime. We have observed indeed that crazes grow much faster when the applied macroscopic tensile stress reaches a sufficiently high value. We propose that this growth is governed by a mechanism of successive nucleations of new cavities in the vicinity of the existing Q

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(13) van Melick, H. G. H.; Govaert, L. E.; Meijer, H. E. H. On the origin of strain hardening in glassy polymers. Polymer 2003, 44, 2493−2502. (14) Richeton, J.; Ahzi, S.; Daridon, L.; Rémond, Y. A formulation of the cooperative model for the yield stress of amorphous polymers for a wide range of strain rates and temperatures. Polymer 2005, 46, 6035−6043. (15) Cross, A.; Haward, R. N.; Mills, N. J. Post yield phenomena in tensile tests on poly(vinyl chloride). Polymer 1979, 20, 288−294. (16) Haward, R. N. Strain hardening of thermoplastics. Macromolecules 1993, 26, 5860−5869. (17) Kramer, E. J. Open questions in the physics of deformation of polymer glasses. J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 3369− 3371. (18) Arruda, E. M.; Boyce, M. C. Evolution of plastic anisotropy in amorphous polymers during finite straining. Int. J. Plast. 1993, 9, 697−720. (19) Arruda, E. M.; Boyce, M. C.; Quintus-Bosz, H. Effects of initial anisotropy on the finite strain deformation behavior of glassy polymers. Int. J. Plast. 1993, 9, 783−811. (20) Meijer, H. E. H.; Govaert, L. E. Mechanical performance of polymer systems: the relation between structure and properties. Prog. Polym. Sci. 2005, 30, 915−938. (21) Senden, D. J. A.; van Dommelen, J. A. W.; Govaert, L. E. Strain Hardening and Its Relation To Bauschinger Effects in Oriented Polymers. J. Polym. Sci., Part B: Polym. Phys. 2010, 48, 1483−1494. (22) Jatin; Sudarkodi, V.; Basu, S. Investigations into the origins of plastic flow and strain hardening in amorphous glassy polymers. Int. J. Plast. 2014, 56, 139−155. (23) Hoy, R. S.; Robbins, M. O. Strain hardening of polymer glasses: Effect of entanglement density, temperature, and rate. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 3487−3500. (24) Hoy, R. S.; Robbins, M. O. Strain Hardening of Polymer Glasses: Entanglements, Energetics, and Plasticity. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2008, 77, 031801. (25) Hoy, R. S.; Robbins, M. O. Strain Hardening in Polymer Glasses: Limitations of Network Models. Phys. Rev. Lett. 2007, 99, 117801. (26) Robbins, M. O.; Hoy, R. S. Scaling of the Strain Hardening Modulus of Glassy Polymers with the Flow Stress. Polym. Phys. 2009, 47, 1406−1411. (27) Govaert, L. E.; Engels, T. A. P.; Wendlandt, M.; Tervoort, T. A.; Suter, U. W. Does the Strain Hardening Modulus of Glassy Polymers Scale with the Flow Stress? J. Polym. Sci., Part B: Polym. Phys. 2008, 46, 2475−2481. (28) Wendlandt, M.; Tervoort, T. A.; Suter, U. W. Strain-hardening modulus of cross-linked glassy poly(methyl methacrylate). J. Polym. Sci., Part B: Polym. Phys. 2010, 48, 1464−1472. (29) Senden, D. J. A.; Krop, S.; van Dommelen, J. A. W.; Govaert, L. E. Rate- and Temperature-Dependent Strain Hardening of Polycarbonate. J. Polym. Sci., Part B: Polym. Phys. 2012, 50, 1680−1693. (30) Senden, D. J. A. Strain Hardening and Anisotropy in Solid Polymers; Technische Universiteit Eindhoven: Eindhoven, 2013. (31) Chui, C.; Boyce, M. C. Monte Carlo Modeling of Amorphous Polymer Deformation: Evolution of Stress with Strain. Macromolecules 1999, 32, 3795−3808. (32) Hasan, O. A.; Boyce, M. C. Energy storage during inelastic deformation of glassy polymers. Polymer 1993, 34, 5085−5092. (33) Hoy, R. Modeling strain hardening in polymer glasses using molecular simulations. In Polymer Glasses; Roth, C. B., Ed.; Taylor & Francis Group: Boca Raton, 2016; pp 425−450. (34) Hoy, R. S.; Robbins, M. O. Strain hardening of polymer glasses: Effect of entanglement density, temperature, and rate. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 3487−3500. (35) Hoy, R. S. Why is Understanding Glassy Polymer Mechanics So Difficult? J. Polym. Sci., Part B: Polym. Phys. 2011, 49, 979−984. (36) Hoy, R. S.; O’Hern, C. S. Viscoplasticity and large-scale chain relaxation in glassy-polymeric strain hardening. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2010, 82, 041803.

cavities. The larger the craze, the larger the volume of the polymer ahead at craze tips undergoing intensified stresses, and therefore the higher the nucleation rate of new small cavities in its vicinity, which leads to an acceleration of the craze growth. As soon as a craze reaches a sufficiently large size, or that the local stress level is sufficiently high, this craze becomes unstable (at tensile test time scales) so that it propagates as a crack resulting in the rupture of the sample. This mechanism happens so quickly for DEP-plasticized cellulose acetate samples that it leads to rapid breakage of the material, or with a too small number of crazes per unit volume, preventing us from observing the phenomenon. In the case of TA-plasticized polymers, however, we have observed this second growth regime on a small population of crazes using USAXS. The scattered intensities revealed the presence of a second family of crazes with sizes well above those of the first family which grow only linearly with the applied stress.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.C.). *E-mail: [email protected] (D.R.L.). ORCID

Paul Sotta: 0000-0002-4378-0858 Didier R. Long: 0000-0002-3013-6852 Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Klemm, D.; Heublein, B.; Fink, H.-P.; Bohn, A. Cellulose: Fascinating Biopolymer and Sustainable Raw Material. Angew. Chem., Int. Ed. 2005, 44, 3358−3393. (2) Zugenmaier, P. 4. Characteristics of cellulose acetates- 4.1 Characterization and physical properties of cellulose acetates. Macromol. Symp. 2004, 208, 81−166. (3) Bao, C. Y.; Long, D. R.; Vergelati, C. Miscibility and Dynamical Properties of Cellulose Acetate/Plasticizer Systems. Carbohydr. Polym. 2015, 116, 95−102. (4) Scandola, M.; Ceccorulli, G. Viscoelastic Properties of Cellulose Derivatives: 1 Cellulose Acetate. Polymer 1985, 26, 1953−1957. (5) Scandola, M.; Ceccorulli, G. Viscoelastic Properties of Cellulose Derivatives 2. Effect of Diethylphthalate on the Dynamic Mechanical Relaxations of Cellulose Acetate. Polymer 1985, 26, 1958−1962. (6) Sousa, M.; Brás, A. R.; Veiga, H. I. M.; Ferreira, F. C.; de Pinho, M. N.; Correia, N. T.; Dionísio, M. Dynamical Characterization of a Cellulose Acetate Polysaccharide. J. Phys. Chem. B 2010, 114, 10939− 10953. (7) Fordyce, C. R.; Meyer, I. W. A. Plasticizers for Cellulose Acetate and Cellulose Acetate Butyrate. Ind. Eng. Chem. 1940, 32, 1053− 1060. (8) Charvet, A.; Vergelati, C.; Long, D. R. Mechanical and ultimate properties of injection molded cellulose acetate/plasticizer materials. Carbohydr. Polym. 2019, 204, 182−189. (9) Mohanty, A. K.; Wibowo, A.; Misra, M.; Drzal, L. T. Development of renewable resource-based cellulose acetate bioplastic: Effect of process engineering on the performance of cellulosic plastics. Polym. Eng. Sci. 2003, 43, 1151−1161. (10) Carollo, P.; Grospietro, B. 5.5 Plastic materials. Macromol. Symp. 2004, 208, 335−352. (11) Zugenmaier, P. 4. Characteristics of cellulose acetates- 4.1 Characterization and physical properties of cellulose acetates. Macromol. Symp. 2004, 208, 81−166. (12) Kierkels, J. T. A. Tailoring the Mechanical Properties of Amorphous Polymers; Eindhoven University of Technology, 2006. R

DOI: 10.1021/acs.macromol.9b00858 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules (37) Ge, T.; Robbins, M. O. Anisotropic plasticity and chain orientation in polymer glasses. J. Polym. Sci., Part B: Polym. Phys. 2010, 48, 1473−1482. (38) Botto, P. A.; Duckett, R. A.; Ward, I. M. The yield and thermoelastic properties of oriented poly(methyl methacrylate). Polymer 1987, 28, 257−262. (39) Rawson, F. F.; Rider, J. G. A Correlation Of Young’s Modulus With Yield Stress In Oriented Poly(Vinyl Chloride). Polymer 1974, 15, 107−110. (40) Michler, G. H. Electron microscopic investigations of the structure of crazes in polystyrene. Colloid Polym. Sci. 1985, 263, 462− 474. (41) Michler, G. H. Crazes in amorphous polymers I. Variety of the structure of crazes and classification of different types of crazes. Colloid Polym. Sci. 1989, 267, 377−388. (42) Michler, G. H. Correlation between craze formation and mechanical behaviour of amorphous polymers. J. Mater. Sci. 1990, 25, 2321−2334. (43) García Gutiérrez, M. C.; Michler, G. H.; Henning, S.; Schade, C. Micromechanical Behavior of Branched Polystyrene as Revealed by in Situ Transmission Electron Microscopy and Microhardness. J. Macromol. Sci., Part B: Phys. 2001, 40, 797−812. (44) Michler, G. H. Amorphous Polymers. Electron Microscopy of Polymers, Springer Laboratory; Springer Berlin Heidelberg: Berlin, Heidelberg, 2008. (45) Donald, A. M.; Kramer, E. J.; Bubeck, R. A. The Entanglement Network and Craze Micromechanics in Glassy Polymers. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 1129−1141. (46) Kramer, E. J. Microscopic and Molecular Fundamentals of Crazing. Crazing in Polymers; Advances in Polymer Science; Springer, 1983; Vols. 52−53. (47) Henkee, C. S.; Kramer, E. J. Crazing and Shear Deformation in Crosslinked Polystyrene. J. Polym. Sci., Polym. Phys. Ed. 1984, 22, 721−737. (48) Kramer, E. J.; Berger, L. L. Fundamental Processes of Craze Growth and Fracture. Crazing in Polymers; Advances in Polymer Science; Springer, 1990; Vols. 91−92. (49) Deblieck, R. A. C.; Van Beek, D. J. M.; Remerie, K.; Ward, I. M. Failure mechanisms in polyolefines: The role of crazing, shear yielding and the entanglement network. Polymer 2011, 52, 2979−2990. (50) Kambour, R. P. A review of crazing and fracture in thermoplastics. Macromol. Rev. 1973, 7, 1−154. (51) Friedrich, K. Crazes and Shear Bands in Semi-Crystalline Thermoplastics. Crazing in Polymers; Advances in Polymer Science; Springer, 1983; Vols. 52−53, pp 226−274. (52) Dettenmaier, M. Intrinsic Crazes in Polycarbonate: Phenomenology and Molecular Interpretation of a new Phenomenon. Crazing in Polymers; Advances in Polymer Science; Springer, 1983; Vols. 52− 53. (53) Bucknall, C. B. New criterion for craze initiation. Polymer 2007, 48, 1030−1041. (54) Plummer, C. J. G.; Donald, A. M. Crazing mechanisms and craze healing in glassy polymers. J. Mater. Sci. 1989, 24, 1399−1405. (55) Halary, J. L.; Lauprêtre, F.; Monnerie, L. 10. Damage and Fracture of Solid Polymers. Polymer Materials: Macroscopic Properties and Molecular Interpretations; John Wiley & Sons, Inc.: Hoboken, New Jersey, 2011; pp 185−210. (56) Doll, W.; Konczol, L. Micromechanics of Fracture under Static and Fatigue Loading: Optical Interferometry of Crack Tip Craze Zones. Crazing in Polymers; Advances in Polymer Science; Springer, 1990; Vols. 91−92, pp 137−217. (57) Argon, A. S.; Salama, M. M. Growth of crazes in glassy polymers. Philos. Mag. 1977, 36, 1217−1234. (58) Taylor, G. I. The Instability of Liquid Surfaces when Accelerated in a Direction Perpendicular to their Planes. I. Proc. R. Soc. London, Ser. A 1950, 201, 192−196. (59) Griffith, A. A. The Phenomena of Rupture and Flow in Solids. Philos. Trans. R. Soc. London, Ser. A 1921, 221, 163−198.

(60) Irwin, G. R. Analysis of Stresses and Strains near the End of a Crack Traversing a Plate. J. Appl. Mech. 1957, E24, 351−369. (61) Döll, W.; Könczöl, L.; Schinker, M. G. Size and mechanical properties of craze zones at propagating crack tips in poly(methyl methacrylate) during fatigue loading. Polymer 1983, 24, 1213−1219. (62) Döll, W. Kinetics of crack tip craze zone before and during fracture. Polym. Eng. Sci. 1984, 24, 798−808. (63) Brown, H. R. A molecular interpretation of the toughness of glassy polymers. Macromolecules 1991, 24, 2752−2756. (64) Xu, D.-B.; Hui, C.-Y.; Kramer, E. J.; Creton, C. A micromechanical model of crack growth along polymer interfaces. Mech. Mater. 1991, 11, 257−268. (65) Sha, Y.; Hui, C. Y.; Ruina, A.; Kramer, E. J. Detailed simulation of craze fibril failure at a crack tip in a glassy polymer. Acta Mater. 1997, 45, 3555−3563. (66) Mills, P. J.; Kramer, E. J.; Brown, H. R. Real time small-angle Xray scattering from polystyrene crazes during fatigue. J. Mater. Sci. 1985, 20, 4413−4420. (67) Salomons, G. J.; Singh, M. A.; Bardouille, T.; Foran, W. A.; Capel, M. S. Small-angle X-ray scattering analysis of craze-fibril structures. J. Appl. Crystallogr. 1999, 32, 71−81. (68) Lode, U.; Pomper, T.; Karl, A.; von Krosigk, G.; Cunis, S.; Wilke, W.; Gehrke, R. Development of crazes in polycarbonate, investigated by ultra small angle X-ray scattering of synchrotron radiation. Macromol. Rapid Commun. 1998, 19, 35−39. (69) Brown, H. R.; Sindoni, Y.; Kramer, E. J.; Mills, P. J. Diffraction studies of craze structure. Polym. Eng. Sci. 1984, 24, 825−832. (70) Stoclet, G.; Lefebvre, J. M.; Séguéla, R.; Vanmansart, C. In-situ SAXS study of the plastic deformation behavior of polylactide upon cold-drawing. Polymer 2014, 55, 1817−1828. (71) Mourglia-Seignobos, E.; Long, D. R.; Odoni, L.; Vanel, L.; Sotta, P.; Rochas, C. Physical Mechanisms of Fatigue in Neat Polyamide 6,6. Macromolecules 2014, 47, 3880−3894. (72) Bao, C. Y. Cellulose Acetate/Plasticizer Systems: Structure, Morphology and Dynamics. PhD Thesis, University of Lyon, 2015. (73) Kamide, K.; Saito, M. Effect Of Total Degree Of Substitution On Molecular Parameters Of Cellulose Acetate. Eur. Polym. J. 1984, 20, 903−914. (74) Kamide, K.; Saito, M.; Suzuki, H. Persistence length of cellulose and cellulose derivatives in solution. Makromol. Chem. Rapid Commun. 1983, 4, 33−39. (75) Kroon-Batenburg, L. M. J.; Kruiskamp, P. H.; Vliegenthart, J. F. G.; Kroon, J. Estimation of the Persistence Length of Polymers by MD Simulations on Small Fragments in Solution. Application to Cellulose. J. Phys. Chem. B 1997, 101, 8454−8459. (76) Warth, H.; Mülhaupt, R.; Schätzle, J. Thermoplastic Cellulose Acetate and Cellulose Acetate Compounds Prepared by Reactive Processing. J. Appl. Polym. Sci. 1997, 64, 231−242. (77) Rolland, H.; Saintier, N.; Wilson, P.; Merzeau, J.; Robert, G. In situ X-ray tomography investigation on damage mechanisms in short glass fibre reinforced thermoplastics: Effects of fibre orientation and relative humidity. Composites, Part B 2017, 109, 170−186. (78) G’Sell, C.; Jonas, J. J. Determination of the plastic behaviour of solid polymers at constant true strain rate. J. Mater. Sci. 1979, 14, 583−591. (79) Beaucage, G.; Kammler, H. K.; Pratsinis, S. E. Particle size distributions from small-angle scattering using global scattering functions. J. Appl. Crystallogr. 2004, 37, 523−535. (80) Abramowitz, M.; Stegun, I. A.; Romer, R. H. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover Publications, Inc: New York, 1970; p 297. (81) Sorensen, C. M.; Oh, C.; Schmidt, P. W.; Rieker, T. P. Scaling description of the structure factor of fractal soot composites. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 58, 4666−4672. (82) van Breemen, L. C. A.; Engels, T. A. P.; Klompen, E. T. J.; Senden, D. J. A.; Govaert, L. E. Rate- and temperature-dependent strain softening in solid polymers. J. Polym. Sci., Part B: Polym. Phys. 2012, 50, 1757−1771. S

DOI: 10.1021/acs.macromol.9b00858 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules (83) Kierkels, J. T. A. Tailoring the Mechanical Properties of Amorphous Polymers. PhD Thesis, Eindhoven University of Technology, 2006. (84) Govaert, L. E.; van Melick, H. G. H.; Meijer, H. E. H. Temporary toughening of polystyrene through mechanical preconditioning. Polymer 2001, 42, 1271−1274. (85) Paredes, E.; Fischer, E. W. On the Mechanism of Fibril Growth in Polycarbonate and Poly(methyl Methacrylate) Crazes. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 929−930. (86) Brown, H. R.; Kramer, E. J. Craze microstructure from smallangle x-ray scattering (SAXS). J. Macromol. Sci., Part B: Phys. 1981, 19, 487−522. (87) Bridle, C.; Buckley, A.; Scanlan, J. Mechanical Anisotropy of Oriented Polymers. J. Mater. Sci. 1968, 3, 622−628. (88) Zafeiropoulos, N. E.; Davies, R. J.; Schneider, K.; Burghammer, M.; Riekel, C.; Stamm, M. The Relationship between Craze Structure and Molecular Weight in Polystyrene as Revealed by μSAXS Experiments. Macromol. Rapid Commun. 2006, 27, 1689−1694. (89) Landau, L. D.; Lifshitz, E. M. Chapter XVSurfaces. Statistical Physics, 3rd ed.; Butterworth-Heinemann: Oxford, 1980; pp 517−537. (90) Fisher, J. C. The Fracture of Liquids. J. Appl. Phys. 1948, 19, 1062−1067. (91) Herbert, E.; Caupin, F. The limit of metastability of water under tension: theories and experiments. J. Phys.: Condens. Matter 2005, 17, S3597−S3602. (92) Herbert, E.; Balibar, S.; Caupin, F. Cavitation pressure in water. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2006, 74, 041603. (93) Vanel, L.; Ciliberto, S.; Cortet, P.-P.; Santucci, S. Timedependent rupture and slow crack growth: elastic and viscoplastic dynamics. J. Phys. D: Appl. Phys. 2009, 42, 214007. (94) Blander, M.; Katz, J. L. Bubble nucleation in liquids. AIChE J. 1975, 21, 833−848. (95) Pomeau, Y. Fundamental problems in brittle fracture: unstable cracks and delayed breaking. C. R. Mec. 2002, 330, 249−257. (96) Pauchard, L.; Meunier, H. Instantaneous and time-lag breaking of a two-dimensional solid rod under a bending stress. Phys. Rev. Lett. 1993, 70, 3565−3568. (97) Kramers, H. A. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 1940, 7, 284−304. (98) Rustemeyer, P. History of CA and evolution of the markets. Macromol. Symp. 2004, 208, 1−6. (99) Ma, M.; Vijayan, K.; Hiltner, A.; Baer, E.; Im, J. Shear yielding modes of polycarbonate. J. Mater. Sci. 1989, 24, 2687−2696. (100) Colton, J. S.; Suh, N. P. The nucleation of microcellular thermoplastic foam with additives: Part I: Theoretical considerations. Polym. Eng. Sci. 1987, 27, 485−492. (101) de Gennes, P. G. Wetting: statics and dynamics. Rev. Mod. Phys. 1985, 57, 827−863. (102) Bico, J.; Thiele, U.; Quéré, D. Wetting of textured surfaces. Colloids Surf., A 2002, 206, 41−46. (103) de Gennes, P.-G.; Brochard-Wyart, F.; Quéré, D. Special Interfaces. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; Springer New York: New York, NY, 2004; pp 215−259. (104) Oliver, J. F.; Huh, C.; Mason, S. G. Resistance to spreading of liquids by sharp edges. J. Colloid Interface Sci. 1977, 59, 568−581. (105) Cassie, A. B. D.; Baxter, S. Wettability of porous surfaces. Trans. Faraday Soc. 1944, 40, 546−551. (106) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon Press, 1986. (107) Dequidt, A.; Long, D. R.; Merabia, S.; Sotta, P. Mechanical Properties of Polymers and Nano-Composites Close to the Glass Transition. In Polymer Glasses; Connie, B. R., Ed.; CRC Press Boa Raton, 2016; pp 301−354. (108) Argon, A. S. A theory for the low-temperature plastic deformation of glassy polymers. Philos. Mag. 1973, 28, 839−865. (109) Argon, A. S. Craze initiation in glassy polymers - Revisited. Polymer 2011, 52, 2319−2327. (110) Argon, A. S.; Hannoosh, J. G. Initiation of crazes in polystyrene. Philos. Mag. 1977, 36, 1195−1216.

(111) Fields, R. J.; Ashby, M. F. Finger-like crack growth in solids and liquids. Philos. Mag. 1976, 33, 33−48. (112) Donald, A. M. Crazing. In The Physics of Glassy Polymers; Haward, R. N., Young, R. J., Eds.; Springer Netherlands: Dordrecht, 1997; pp 295−341. (113) Argon, A. S.; Bessonov, M. I. Plastic flow in glassy polymers. Polym. Eng. Sci. 1977, 17, 174−182. (114) Dequidt, A.; Conca, L.; Delannoy, J.-Y.; Sotta, P.; Lequeux, F.; Long, D. R. Heterogeneous Dynamics and Polymer Plasticity. Macromolecules 2016, 49, 9148−9162. (115) Conca, L.; Dequidt, A.; Sotta, P.; Long, D. R. Acceleration and Homogenization of the Dynamics during Plastic Deformation. Macromolecules 2017, 50, 9456−9472. (116) Long, D. R.; Conca, L.; Sotta, P. Dynamics in glassy polymers: The Eyring model revisited. Phys. Rev. Mater. 2018, 2, 105601. (117) Roth, C. B.; Pye, J. E.; Baglay, R. R. Correlating glass transition and physical aging in thin polymer films. In Polymer Glasses; Roth, C. B., Ed.; CRC Press Boca Raton, 2016; pp 181−203. (118) Baglay, R. R.; Roth, C. B. Communication: Experimentally determined profile of local glass transition temperature across a glassy-rubbery polymer interface with a Tg difference of 80 K. J. Chem. Phys. 2015, 143, 111101. (119) Baglay, R. R.; Roth, C. B. Local glass transition temperature Tg(z) of polystyrene next to different polymers: Hard vs. soft confinement. J. Chem. Phys. 2017, 146, 203307.

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DOI: 10.1021/acs.macromol.9b00858 Macromolecules XXXX, XXX, XXX−XXX