How Melt-Stretching Affects Mechanical Behavior of Polymer Glasses

Article pubs.acs.org/Macromolecules

How Melt-Stretching Affects Mechanical Behavior of Polymer Glasses Gregory D. Zartman, Shiwang Cheng, Xin Li, Fei Lin, Matthew L. Becker, and Shi-Qing Wang* Morton Institute of Polymer Science and Engineering, University of Akron, Akron, Ohio 44325-3909, United States ABSTRACT: This work takes a simple phenomenological approach to the questions of when, how, and why a brittle polymer glass turns ductile and vice versa. Perceiving a polymer glass as a hybrid, we recognize that both the primary structure formed by van der Waals forces (network 1) and chain network (i.e., the vitrified entanglement network) (network 2) must be accounted for in any discussion of the mechanical responses. To show the benefit of this viewpoint, we first carried out well-defined melt-stretching experiments on four common polymer glasses (PS, PMMA, SAN, and PC) in a systematic way either at a fixed Hencky strain rate to a given degree of stretching at several temperatures or at a given temperature to different levels of stretching using the same Hencky rate. Then we attempted to preserve the effect of meltstretching on the chain network structure by rapid thermal quenching. Subsequent room-temperature tensile extension of these melt-stretched amorphous polymers reveals something universal: (a) along the direction of the melt-stretching, the brittle glasses (PS, PMMA, and SAN) all become completely ductile; (b) perpendicular to the melt-stretching direction, the ductile glass (PC) becomes brittle at room temperature. We suggest that the transformations (from brittle to ductile or ductile to brittle) arise from either geometric condensation or dilation of load bearing strands in the chain network due to the melt-stretching. Regarding a polymer glass as a structural hybrid, we also explored two other cases where the ductile PC becomes brittle at room temperature: (1) upon aging near the glass transition temperature; (2) when blended with PC of sufficiently low molecular weight. These results indicate that (i) the strengthening of the primary structure by aging can raise the failure stress σ* to a level too high for the chain network to sustain and (ii) the PC blend becomes brittle upon weakening the chain network by dilution with short chains.

I. INTRODUCTION Polymer glasses fail under sufficient tensile deformation in either a brittle manner or ductile fashion, as shown by extensive studies of the past several decades.1−3 After many pioneering studies on deformation, yielding, and failure processes of glassy polymers, several groups4−20 are still actively looking for the origin of ductility in polymer glasses and strategies to avoid brittle fracture. In general, there have been three leading treatments of, or approaches to, the subject of failure behavior of polymer glasses. The earliest studies resort to the Ludwik21− Davidenov-Wittman22−Orowan23 (LDWO) hypothesis as illustrated in Figure 1a concerning the brittle−ductile transition (BDT), which assumes that brittle fracture and plastic flow are independent processes. Here σy is the stress, at which yielding (i.e., plastic deformation) of the primary structure (formed by intersegmental van der Waals interactions) takes place, and the brittle strength σb is associated with the effective strength of backbone bonds. In this picture, the strength and areal density of chemical bonds determine the level of the brittle strength σb. Moreover, the yield strength σy corresponds to a critical tensile stress at which the networking due to van der Waals interactions breaks down.24 At the brittle-to-ductile transition temperature Tbd, the yield strength equals the brittle strength: σb(Tbd) = σy(Tbd) = σB. © 2012 American Chemical Society

On the basis of the LDWO hypothesis, Vincent performed early studies on the BDT. According to a diagram similar to Figure 1a, Vincent discussed25 the relationship between the yield strength σy and brittle strength σb by examining the effects of strain rate and material variables such as molecular weight Mw, crystallinity, additives, cross-linking, side groups, and molecular orientation on the BDT. For a dozen common polymer glasses, Vincent24 found an approximate linear relationship between σB and the backbone bond areal density φ. His analysis implied that the brittle fracture at BDT was due to chain scission although the experimentally observed σB ∼ 100 MPa is lower by a factor of 100 than26 an estimate of φf b. On the other hand, Vincent showed25 that biaxially meltstretched poly(methyl methacrylate) (PMMA) turns ductile at room temperature in tensile extension. Since the melt deformation cannot appreciably increase the value of φ, the observed correlation could not explain why some brittle polymers turn ductile upon melt-stretching. Similarly, Bersted27 claimed that whether a polymer glass is brittle or ductile “depends chiefly on the intrinsic strength of the material in Received: May 11, 2012 Revised: August 2, 2012 Published: August 10, 2012 6719

dx.doi.org/10.1021/ma300955h | Macromolecules 2012, 45, 6719−6732

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tension, which is postulated to be dependent on the entanglement network density.” However, this line of research to focus on the relationship between the yield strength σy and brittle strength σb was often discounted: There was a widely accepted view28 that one must directly study crack propagation and the characteristics associated with such microstructural features as crazing29−34 and shear banding phenomena.35 Nevertheless, we note that Ward praised the merit of the LDWO approach in his book.36 Moreover, within the LDWO analysis, some other studies emphasized that the yielding of the primary structure due to van der Waals forces arose from sufficient molecular mobility under the particular test conditions.37 In other words, segmental relaxation characteristics of glassy polymers were thought to dictate their failure behavior.38 Researchers have gone so far as to correlate ductile failure with the existence of a secondary (beta or gamma) transition below Tg,38−42 although Vincent showed25 many examples to the contrary. Aharoni43 argued that ductile failure occurs when the physical volume of an entanglement strand is lower than some activation volume and suggested that the ultimate mechanical properties of polymers below Tg are not necessarily dictated by relaxational phenomena and the size of the submolecular entities undergoing these relaxations. The second treatment focused on the characteristics of craze formation and crack propagation. Pioneering studies of Kramer 31−33 and others 27,43 revealed a great deal of experimental information correlating the entanglement density to the characteristics of crazes. In particular, they observed that polymers with high entanglement network density tend to be ductile. The entanglement density is the number of entanglement strands per unit volume given by νE = ρNa /Me = 1/plent 2

(1)

where ρ is mass density, Me the entanglement molecular weight, lent the entanglement spacing, and Na the Avogadro constant. Here the second equality defines the packing length p. Denoting the theoretical maximum chain extension λmax between entanglements as λmax = Le /lent = lent /lK ≡

ne

(2)

where Le is the chain contour length of an entanglement strand, lK is the Kuhn length, and ne is the number of Kuhn segments in an entanglement strand, Kramer and co-workers observed that brittle polymer glasses tend to have a larger λmax. Their extensive investigations31−33,44−56 on the characteristics of craze indicated that the extension in the craze zone, λcraze, is lower for a polymer with lower λmax. They further noted44 that the tensile stress σfib exerted on fibrils in the craze is enhanced by a factor of λcraze relative to the applied tensile stress S, i.e., σfib = λcrazeS. Thus, “a useful way to increase the 'brittle’ fracture stress and decrease the ductile-to-brittle transition temperature of a glassy polymer is to decrease its entanglement contour length Le” according to eq 2.44 However, it was noted45 that PMMA does not follow the trend of λcraze ∼ λmax. A third approach is being actively pursued currently by a number of groups.4−20 It is based on an observation that in compression ductile polymers tend to show a quantitatively different stress−strain curve from that of brittle polymers. Specifically, these studies focused the characteristics of strain hardening, which refers to the fact that in the post-yielding stage the stress grows with increasing strain. The surging interest9−14,17,20 to understand the origin of the so-called strain

Figure 1. (a) A phase diagram based on the LDWO hypothesis.21−23,36 A ductile glass becomes brittle at the brittle−ductile transition (BDT) temperature Tbd, defining a unique level of brittle strength σB. (b) Our view of polymer glasses as a “double network”, i.e., a hybrid made of a primary structure due to nonbonded van der Waals interactions and an underlying load-bearing network due to chain uncrossability. Here the open circles represent polymer segments that are covalently bonded to one another and also interact among themselves through van der Waals (vdW) forces denoted by the (blue) dots. It is through vdW interactions (blue dots) that the primary structure can be regarded as a different load-bearing network from the chain network during linear response. (c) In contrast to (a), at temperature T < Tbd, the primary vdW structure is dominant in the sense that its breakdown may result in severe tensile strain localization, i.e., crazing where the chain network is greatly extended. Above Tbd, the primary structure is weak. It is the chain network that bears the load and determines when macroscopic yielding takes place, often in the form of shear strain localization. (d) In tensile extension, the extensional stress σ can be projected onto two orthogonal directions as shown. In the inclined plane of area A′, there is shear stress σ12 = (σ/ 2) sin 2θ, where the true stress σ is related to the engineering stress σengr as σengr(L/L0). It is assumed that the interpenetrating chains could free themselves from one another more readily along an inclined plane than in a tensile manner. 6720


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hardening was prompted by Kramer’s reminder57 that the origin of the strain hardening is not entropic rubber elasticity, as often regarded in the past.58,59 The recent studies were further motivated by the speculation that strain hardening might prevent brittle failure. For example, it was concluded7 that “stable or unstable neck growth (is) dependent on the ratio between yield stress and hardening modulus.” However, such studies do not directly address the question of how a polymer glass transitions from being ductile to being brittle. To determine the effectiveness of each previous approach, it is instructive to review several intriguing phenomena. (a) Ductile polymers including PC can turn brittle upon effective aging60 through adequate annealing. (b) Application of sufficient hydrostatic pressure causes a brittle polymer such as PS to become ductile.28,61−66 (c) Most polymer glasses have a BDT. In other words, lowering the temperature sufficiently can cause a polymer glass to switch from ductile failure to brittle fracture.30,31 Conversely, at sufficiently high temperatures brittle polymer glasses often turn ductile. (d) Brittle glasses such as PS can become ductile if they are first subjected to sufficient predeformation at temperatures well above Tg.67−78 In particular, PS can be as ductile as PC when tested in tensile deformation along the direction of melt-stretching, as we will demonstrate systematically in this paper. (e) Thick specimens of PC can also undergo brittle failure. Since aging, pressurization, variation of the sample dimensions, or temperature change (below Tg) does not alter the chain packing appreciably, the alleged correlation between ductility and entanglement density alone is insufficient to explain phenomena (a), (b), and (c) listed above. If one has to apply the idea of the LDWO hypothesis depicted in Figure 1a, one is then forced to say that the ductility due to pressurization arose from lowering of σy since the pressure cannot alter the carbon−carbon bond strength. But the experiment involving pressurization had only revealed increased yield strength of PS due to pressure.28 Thus, the LDWO hypothesis depicted by Figure 1a clearly has trouble in accounting for the pressure effect. There is a lack of consensus about the origin of phenomenon (d), i.e., about why brittle polymer glasses can turn ductile upon predeformation. Van Melick et al. suggested79 that “the observed ductility of polystyrene originates from the reduction of strain softening and not from the molecular orientation” whereas others emphasized the importance of molecular orientation to enhance strain hardening13,14 and reduce the severity of crazing.78 Several studies on premelt deformation of brittle glasses also conclude that molecular orientation is responsible for the change in ductility.80,81 One study even concluded long ago70 that this effect is specific to the microstructure of PS because the molecular packing of the predeformed ductile PS is different from that found for the brittle isotropic PS. Other brittle polymer glasses such as PMMA sometimes show brittle fracture after predeformation,82−85 although Vincent had also reported25 elongation up to 110% before breaking of biaxially stretched PMMA instead of the typical few percent for the isotropic PMMA glass. Lastly, regarding the phenomenon (e), one can avoid the geometric effects in uniaxial extension of polymer glasses by fixing the sample dimensions when other physical parameters such as temperature and molecular composition are varied. Thus, in the present study, we will fix the specimen geometry for all the samples under investigation.

Our representation of polymer glasses differs from the phase diagram of Figure 1a. We think of an amorphous polymer glass as a hybrid depicted in Figure 1b. The double network arises as follows.86 In the melt state, there is the entanglement network that renews on the time scale of reptation87 in quiescence. Upon cooling down below the glass transition temperature Tg, this chain network (arising from the uncrossability) freezes, becoming embedded in the vitrification of all segments, where another network emerges due entirely to intermolecular interactions, i.e., the van der Waals (vdW) forces. In what follows, we shall call this nonbonded network the “primary structure” in polymer glasses. With decreasing temperature, it becomes more difficult for the polymer segments to hop out of their confinement potentials so that the primary structure turns stronger. Thus, our phase diagram in Figure 1c is different from Figure 1a. Just below Tg, the primary structure, made of the vdW “bonds” and denoted by the dots, is weak and can readily be destroyed during tensile extension of ductile polymer glasses. After the primary structure yields at a few percentage of tensile deformation, the mechanical response is dictated by the chain network. Frequently, in this ductile regime, i.e., at T > Tbd, upon the breakdown of the weak primary structure, a polymer glass suffers shear yielding88 when the tensile stress σ projects a sufficiently high shear stress σ12 to cause reconstruction of the chain network. Figure 1d describes the decomposition of the tensile stress along an inclined plane. This shear yielding initiates global necking, involving one or two propagating front(s) where the yielded region89 keeps expanding into the yet-to-yield region(s). At T < Tbd, the primary structure is dominant in the sense that its collapse involves sufficiently high stresses to initialize extreme strain localization in the chain network known as crazing. In a craze, the chain network undergoes considerable tensile extension, yet away from the crazes the glass is hardly deformed. Brittle fracture immediately ensues because the fibrils made of the stretched chain network are weak and subject to a higher stress as noted by Kramer and co-workers.44 We apply the new “phase diagram” of Figure 1c instead of Figure 1a as a guide to study yielding and failure behaviors of all polymer glasses of high molecular weight. In particular, we explore the possibility to rationalize the effects of meltstretching and aging on the deformation, yielding, and failure behavior of polymer glasses. This paper is separated into an experimental part and a theoretical interpretation/analysis part. Our experiments are of two stages: First, well-defined melt-stretching tests of the brittle PS, PMMA, and SAN are carried out to various degrees. Second, after thermal quenching of these melt-stretched polymers, we carry out room-temperature tensile extension tests to show that these brittle glasses can all turn ductile to resemble PC. Few previous studies90 have systematically controlled the melt strength condition and examined the effects of melt-stretching on the failure behavior of polymer glasses. The controlled melt-stretching allows us to reveal a correlation between the mechanical properties in the glassy state and the state of the chain network that has been affected by the melt-stretching. Moreover, perhaps for the first time, we demonstrate that by melt-stretching PC, the same ductile PC can undergo completely brittle fracture as the isotropic PS does when extended at room temperature in the direction perpendicular to the melt-stretching direction. In the second interpretative part, we describe the effect of melt-stretching in 6721


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were found using a TA Instruments Q2000 DSC ramping at 10 °C/ min to well above Tg followed by cooling at the same rate. Melt-stretching of the brittle polymer samples were carried out at least 20 deg above their glass temperatures using a first generation SER fixture mounted on an Advanced Rheometric Expansion System (ARES) rotational rheometer equipped with a Rheometric Scientific Oven accurate to 0.1 °C. In order to prepare uniform specimens to use on the SER, pellets were compression molded using a Carver press and a custom mold that allows sheets with dimensions of 120 mm × 120 mm × 0.5 mm to be made. Dimensions of 8 mm × 25 mm × 0.5 mm were then cut from the large sheets, and the ends of the specimens were adhered to the SER using commercially available Krazy Glue. After the Krazy Glue dried, the temperature of the oven was increased to the desired temperature, and the sample was allowed to equilibrate for 5 min. Subsequently, the samples were extended under prescribed conditions. At the end of melt-stretching, the samples were quickly quenched toward room temperature with a mist of cool water from a spray bottle. After melt-stretching, samples were approximately 0.2 mm × 3 mm in their cross section, with a length between clamps (L0) of 15 mm. The samples where then placed in an Instron at room temperature and extended at a rate of 1 or 5 mm/min, respectively. Except for the data presented in Figure 5a that were collected with Instron model 5543, all the other tests at room temperature were performed with Instron model 5567. PC sheets of dimension 120 mm × 60 mm × 0.5 mm were prepared using the aforementioned technique. PC was melt-stretched using custom-made clamps for the Instron model 5567 that is equipped with an oven (model # 3119-406) accurate to 0.1 at 155 °C. Bluehill Software coupled with the Instron model 5567 allows us to stretch the sample at a constant Hencky rate by increasing the cross-head speed exponentially as the sample lengthens. Samples were then cut perpendicular to the melt-stretching direction using a punch press as shown in the subsequent Figure 4. These samples were then placed in the Instron 5567 at room temperature and extended at 5 mm/min. Aging was performed by placing the PC sheets in a vacuum oven at 147 °C for 10 days under 30 in.Hg vacuum. The samples were then cooled back to room temperature overnight, and samples were cut from the sheets.

terms of a molecular picture and provide our explanations of the experimental results.

II. EXPERIMENTAL SECTION This work studies mechanical behaviors of four different polymer glasses: polystyrene (PS), polycarbonate (PC), poly(methyl methacrylate) (PMMA), and poly(styrene−acrylonitrile) (SAN). The PS is Dow Styron 663. The PC is Lexan TM 141 111 obtained from Sabic (GE Plastic). The PMMA is a Plaskolite brand with the item number CA-86. SAN was graciously donated to us by Diamond Polymers and has an item number SAN 51. This SAN contains 33% acrylonitrile. The chemical and mechanical properties of the materials are described in Table 1.

Table 1 polymer PS PMMA SAN PC PC (fractionated) PC oligomer

Mw (kg/mol) 319 125 116 63 167 6.9

Me (kg/mol)

PDI

τ (s)

13 13 7.2 1.3 1.3

1.44 1.41 1.59 1.58 2.27

2500 2250 400 NA NA

1.3

1.89

NA

temp (°C)

Tg (°C)

130 135 130 149

103 113 107 149 145 NA

In order to prepare PC blends of long and short chains, the asreceived commercial PC was fractionated according to the following procedure. First, the PC was dissolved in chloroform to form a dilute solution of ca. 1% in a beaker under room temperature. Then methanol was added to this dilute solution under rapid stirring. When the solution started to turn cloudy, we slowed down the introduction of methanol. Eventually the solution cloudiness was stable. Subsequently, we heated the solution to a temperature high enough to make the solution clear again. The solution was then allowed to cool under hood toward room temperature, inducing precipitation. We repeated this procedure of precipitation three times and collected the precipitated PC as the fractionated PC listed in Table 1. Table 1 lists the terminal relaxation time τ for each sample that is evaluated from small amplitude oscillatory shear (SAOS) measurements of the storage and loss moduli G′ and G″ as a function of frequency. The SAOS were carried out with an Anton Paar Physica MCR 301 rheometer and is shown in Figure 2a for PS as an example. The molecular weight and PDI were found using GPC, where SEC was done using a Wyatt Dawn Eos multiangle laser light (MALLS) detector in conjunction with Waters Model 2414 differential refractometer concentration detector. This was coupled with Wyatt Astra V 4.73.04 software and three Waters HR styrogel columns using THF at 35 °C flowing at 1 mL/min. Glass transition temperatures

III. RESULTS In the present study, we will systematically explore the effects of melt-stretching on the deformation, yielding, and plastic flow of several brittle polymer glasses. We first present the melt extension measurements of PS, PMMA, SAN, and PC. Then we show how the quenched-melt-stretched glasses differ from the isotropic counterparts in terms of their mechanical behavior as revealed by Instron tensile extension at room temperature. A. Melt Extension of PS, PMMA, SAN, and PC. Uniaxial extension of entangled melts typically involves initial elastic

Figure 2. (a) Small-amplitude oscillatory shear measurements for PS at a reference temperature of 150 °C. The inset shows the WLF shifting factor as a function of temperature. (b) Melt extension data for PS at temperatures from 125 to 150 °C at ε̇ = 0.2 or 0.002 s−1. 6722


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Figure 3. (a) Melt extension of PS at 130 °C at ε̇ = 0.2 s−1 to the various Hencky strains. (b) Melt extension of PMMA at 130 °C at ε̇ = 0.2 s−1 for four different strains. (c) Melt extension of SAN at 130 °C at ε̇ = 0.2 s−1 for five strains. (d) Melt extension of PC at 155 °C at ε̇ = 0.05 s−1 to two strains.

rubber elasticity formula, i.e., σengr = G(λ − 1/λ2), we find that G increases with lowering temperature and its value far exceeds the melt elastic plateau modulus Gpl = 0.2 MPa for PS at 125 °C. Thus, at temperatures 130 °C and lower, nonbonded interchain frictional interactions apparently start to contribute significantly to the initial elastic response. During the meltstretching at 130 °C, sufficient chain entanglements survive so that partial elastic deformation continues until rupture,92 which happens around ε = 2.4 or so. In other words, the molecular extension of the entanglement network increases monotonically with the stretching ratio at these lower temperatures. For the same extensional rate of 0.2 s−1, the lower temperature corresponds to a higher effective stretching rate so that the melt extension is closer to affine deformation and produces higher stress. At 125 °C, the extension of the entanglement network could become non-Gaussian. In a second way, we melt-stretch the PS sample to different degrees at a fixed temperature of 130 °C with the same rate of ε̇ = 0.2 s−1. Similar to the top two curves in Figure 2b, Figure 3a shows a sharp rise in the engineering stress σengr, followed by a smooth increase. Specifically, the last data points in Figure 3a indicate where the melt-stretching was terminated in each of the six samples. A.2. Melt Extension of PMMA and SAN. We also performed similar melt-stretching of the PMMA and SAN samples at relatively low temperatures of 135 and 130 °C, respectively, using a Hencky rate ε̇ = 0.2 s−1. Figures 3b,c show the monotonic rise of the engineering stress σengr with the Hencky strain ε, similar to Figure 3a.

stretching of the entanglement network and subsequent transition to irrecoverable deformation. The transition is a yielding process,91 usually signified by the emergence of a maximum in the tensile force, i.e., the engineering stress σengr. At sufficiently high rates or low enough temperature, the yielding is only partial in the sense that σengr increases monotonically until the point of rupture. 92 Since the entanglement network can be more effectively stretched in this melt rupture regime, the present study primarily involves the high-rate melt extension that can maximally alter the structure of the polymer network. We first present the meltstretching rheological data for PS, followed by those for PMMA, SAN, and PC. A.1. Melt Extension of PS. We have carried out the meltstretching of PS in two ways. First, a fixed Hencky rate ε̇ = 0.2 s−1 was applied to stretch the PS melt to a Hencky strain ε = 2.0 at each of the following temperatures 150, 145, 140, 130, and 125 °C. Since the terminal relaxation time of the PS melt increases by a factor 250 from 150 to 125 °C as shown in the inset of Figure 2a, the temperature change is equivalent to changing the effective applied rate, i.e., the Weissenberg number Wi = τε̇, by the same factor. Figure 2b shows how the engineering stress σengr changes with the degree of melt extension, given by the Hencky strain ε. Since the PS sample has a rather broad molecular weight distribution, the maxima of σengr at 150 and 145 °C are rather broad. At the lower temperatures, σengr increases monotonically because the applied rate ε̇ = 0.2 s−1 is effectively very high, and the stretching enters the non-Gaussian melt rupture regime.92 If we analyze the initial elastic response of the PS melt in terms of the classical 6723


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Figure 4. Schematic showing how the PC is melt-stretched and how a sample is cut perpendicular to the melt-stretching direction.

Figure 5. (a) Engineering stress σengr vs deformation ΔL/L0 (where ΔL = L − L0) in tensile extension of melt-stretched PS (whose initial length is L0) at V = 1 mm/min at room temperature, corresponding to the various stretching conditions depicted in Figure 2b. (b) The same data from (a) for 140 °C and ε̇ = 0.2 s−1, where the yield stress σy is the maximum of σengr, and photos show respectively the original sample, at the onset of necking, after a long neck is drawn, and after failure.

A.3. Melt Extension of PC. The purpose of melt-stretching PC is to demonstrate the anisotropic effect of melt-stretching on deformation, yielding, and failure behavior of a predeformed polymer glass. As illustrated in Figure 4, the meltstretched and quenched PC samples are cut into a dog-boneshaped strip perpendicular to the melt-stretching direction. Subsequently, the predeformed PC are to be subjected to room-temperature tensile extension along the direction perpendicular to the melt-stretching direction, as discussed in section IV.C. Figure 3d shows the melt-stretching data on PC, involving Instron extension at a Hencky rate ε̇ = 0.05 s−1 and temperature 155 °C. As depicted in Figure 4, this extensional testing does not involve the correct sample aspect ratio since the sample is rather wide and short. The sample needs to be wide because it needs to be cut as shown in Figure 4 after the melt-stretching. It needs to be relatively short because the stretching rate V/L0 cannot exceed the upper limit of the crosshead speed V of the Instron 5567. B. Turning Brittle Glasses into Ductile Polymers. For each of these melt-stretched polymer glasses that were rapidly quenched, we carried out an Instron style tensile deformation in the displacement-controlled mode at room temperature (RT). By examining the effect of melt-stretching on deformation, yielding, and failure behavior of several brittle polymer glasses, we show that well-controlled melt-stretching produces a consequence that is universal and independent of the microstructure of the polymer. We first present the Instron room-temperature tensile testing results of the samples that

experienced the melt-stretching shown in Figure 2b. Figure 5a reveals the dramatic effect of melt-stretching, turning the brittle PS into a completely ductile glass. For comparison, we also plot the data for the same PS sample that has undergone a negligible amount of melt deformation at 150 °C because of the low imposed Hencky rate of ε̇ = 0.002 s−1. This slowly meltstretched sample suffered brittle failure just like the isotropic PS sample. On the other hand, after being melt-stretched at the much higher rate of ε̇ = 0.2 s−1 at 150 °C, the same PS is essentially ductile as shown by the down-pointed triangles because the tensile force shows a maximum, and the specimen can now be extended to ΔL/L0 = 0.4. After being meltstretched at progressively lower temperatures, the PS samples show increasing ductility. The PS samples that experienced more efficient melt-stretching (at lower temperatures, i.e., higher Weissenberg number) appear to show higher yield stress and greater degree of extensibility before the final failure that terminates the data points in Figure 5a. It is important to emphasize that such significant extensibility, as an indication of the transition from brittleness to ductility, actually occurs in the form of “necking”, i.e., nonuniform extension. Figure 5b shows, corresponding to the experiment depicted in Figure 5a for 140 °C, the photos at the various stages of the room-temperature extension, including the initial point, the point of macroscopic shear yielding (right after the stress maximum) leading to the subsequent necking, and the final fibril-like failure. Note that we use the symbol σy or σy∥ to distinguish from σy that appears in Figure 1a. Here σy∥ represents the yield stress of the chain network for a melt6724


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Figure 6. (a) Engineering stress σengr vs deformation ΔL/L0 in tensile extension of melt-stretched PS at V = 5 mm/min at room temperature, corresponding to the various stretching conditions depicted in Figure 3a. (b) Engineering stress σengr vs deformation ΔL/L0 for melt-stretched PMMA at V = 5 mm/min at room temperature, corresponding to the various stretching conditions depicted in Figure 3b. (c) Engineering stress σengr vs deformation ΔL/L0 for melt-stretched SAN at V = 5 mm/min at room temperature, corresponding to the various stretching conditions depicted in Figure 3c.

Figure 7. Initial slope of the stress vs extension data at room temperature of (a) PS, (b) PMMA, and (c) SAN that have been melt-stretched to different degrees as shown in Figures 6a−c.

portion of Figures 6a−c as shown in Figures 7a−c. It is clear that the initial slope in the stress versus deformation curves grows with the increasing amount of melt-stretching. In the other words, the Young’s modulus E(λ), as measured by the tensile extension along the direction of the melt-stretching, is found to increase by a factor as high as 2 from its value in the isotropic (nonmelt-stretched) state, E0. This phenomenon does not appear to have been widely recognized in the past. D. Effect of Aging: Ductile PC Turning Brittle. In confirmation with the previous observation that aging can cause a ductile polymer to become brittle, we have examined how an aged PC glass behaves differently in comparison to a quenched PC glass. Figure 8 shows that the aged PC becomes fairly brittle in sharp contrast to the non-aged PC. It is remarkable to note that the yield stress is appreciably higher for the aged PC although the values of yield strain εy are comparable. The photos reveal that the aged PC could barely undergo necking, whereas the non-aged PC undergoes necking until the entire sample is transformed from the initial state to a uniformly strained state, doubling its initial length L0.

stretched polymer in tensile extension parallel to the meltstretching direction. The second photo shows that the first sign of nonuniformity is shear banding. Our discussions on shear yielding will be deferred to section IV.E.2. Corresponding to the melt-stretching histories depicted in Figure 3a, we also show the Instron room-temperature extension data in Figure 6a. It is clear that the effect of meltstretching at a fixed Hencky rate and temperature on the strength and ductility of the melt-stretched PS sample increases with the Hencky melt strain ε, with the exception involving the highest degree of melt-stretching at ε = 2.2 when the sample is close to the point of melt rupture. With ε = 2.2, the sample may have recoiled somewhat before the quenching vitrified the sample. Moreover, some chain disentanglement92,93 may have occurred when stretched from ε = 2.0 to ε = 2.2. In confirmation with the results for PS, we show that other brittle polymer glasses such as PMMA and SAN also systematically turn ductile upon controlled melt-stretching. Figures 6b,c show the Instron extension testing corresponding to the melt-stretching conditions depicted in Figures 3b,c. One common feature in Figures 6a−c is that the yield point where the measured tensile force peaks or levels off involves a degree of extension, (ΔL/L0)y, independent of the melt-stretching condition, which is also somewhat universal numerically, i.e., (ΔL/L0)y = 0.04−0.05, among all the three polymer glasses. C. Influence of Melt-Stretching on Young’s Modulus. The data in the preceding subsection also reveal a second common phenomenon, already evident in Figures 6a−c. To emphasize that this characteristic is universal, we magnify a

IV. DISCUSSION: MECHANISM FOR DUCTILITY IN POLYMER GLASSES A. Geometrical Condensation in Fast Uniaxial Stretching of Entangled Melts. Our melt-stretching in Figures 2b and 3a−d involves a monotonic rise of the engineering stress σengr up to the end point when rapid quenching freezes the effect of melt-stretching. At such high rates, the entanglement network may deform nearly affinely.94 Suppose there are ϕent 6725


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B. Structure and Yielding of the Chain Network. Upon the collapse of the primary (van der Waals) structure during tensile extension, the system should have undergone instant fracture in the absence of another load bearing mechanism. In other words, distances among the nonbond segments should increase without bound under tensile extension in the absence of any chain network. If a polymer chain is not long enough, it can displace large distances without having to hop over many surrounding segments at once. Such short chains as depicted in Figure 10a are ineffective as load-bearing strands and cannot be part of the chain network to support the structural integrity during yielding of the primary structure, which involves energy barrier crossing and segmental rearrangements. In contrast, when a chain can insert itself into others as depicted in Figure 10b at the hairpin-like junctions, its ability to bear load may prevent imminent fracture if there are enough of such chains. Below we explore how the strength of the chain network in the glassy state could be depicted in terms of various structural parameters. First of all, yielding of the chain network occurs when sufficient intrachain forces build up in the chain network during external deformation that exceed the intersegment “vitrifying forces” associated with the pairs of the hairpins. According to a recent report,89 sufficient stress produces greatly enhanced segmental mobility that weakens the intersegment vitrification force and accelerates the approach to the force imbalance. To describe the yield stress for the chain network, we make the following two simplifying assumptions: (a) load bearing occurs only in strands ending in pairs of straggled segments, as depicted in Figure 10b; (b) yielding of the chain network in uniaxial extension requires a stress level proportional to the number of such load-bearing strands (LBS) per unit crosssectional area, ψ, i.e.

Figure 8. Engineering stress σengr vs deformation ΔL/L0 at room temperature of both non-aged and aged PC at V = 5 mm/min, where the photos show brittle failure of the aged PC and long necking of the non-aged PC. For the complete stress vs extension data of the nonaged PC, see Figure 13.

entanglement strands per unit area before stretching. In the limit of affine uniaxial extension, the areal density ϕent of entangled strands increases to λϕent upon reaching a stretching ratio of λ= exp(ε) since the cross-sectional area (XY plane) shrinks by λ in the limit of incompressibility. On the other hand, the areal density of entangled strands in the XZ and YZ planes decreases by a factor of √λ. This situation may be depicted as shown in Figure 9 that illustrates the structural

σ y(T ) = ψfLBS (T )

(3)

where f LBS has the dimension of a force. Since f LBS is assigned to a LBS in eq 3, its maximum value corresponds to the breaking of a covalent bond and is of a similar value for all polymers made of C−C bonds: f LBS(max) = f b ∼ 5 nN. When the system is not far below the glass transition temperature Tg, f LBS might be considerably lower than f b because pullout of a LBS from its surrounding chains is easy. Denoting the average chain length of an LBS by lLBS, we can show how ψ is related to lLBS in two steps. First, the number of strands of chain length lLBS per unit volume is given by

Figure 9. Description of an entangled polymer in both melt and glassy states. In the various cross sections, e.g., the XY plane, the filled circles denotes the entanglement (melt) or load-bearing (glassy) strands. Treating entangled polymers as a network where the straight lines designate Gaussian entanglement strands, if melt-stretching preserves the entanglement (i.e., in the affine extension limit, for notional simplicity), there will still be as many entanglement strands before melt-stretching as there are after melt-stretching in the XY plane. The illustration here demonstrates the geometric condensation of the entanglement strands in the XY plane and geometric dilution in the YZ plane. Also shown schematically is the engineering stress σengr as a function of the melt-stretching ratio λ.

νLBS = ρNa /MLBS = 1/plLBS2

(4)

which is of the same form as eq 1. Then ψ is related to νLBS as, apart from a numerical prefactor, ψ ≈ νLBSlLBS = 1/plLBS

(5)

where use is made of eq 4. A previous study95 has estimated the chain length lc between point A and point B in Figure 10c. Specifically, lc is related to the chain flexibility parameter, i.e., the characteristic ratio C∞, according to

changes of the entanglement network during affine melt extension along the Z direction as well as the geometric condensation and dilation of the resulting chain network. In Figure 9, the lines represent either the entanglement strands or load-bearing strands (LBS), where ψ designates the areal density of the LBS. Even though ψ may not depend on the molecular characteristics in the same manner as ϕent does, it is reasonable to expect that both ψ and ϕent increase in proportion to the stretching ratio λ during affine melt-stretching.

lc ≃ (8/3)1/2 C∞l

(6)

where l is the backbone bond length. Apart from a prefactor, this length lc depends on the molecular characteristics as the Kuhn length lK does. The chain length lhp between two pairs of hairpins, as shown in Figure 10b or 10d, is expected to be 6726


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Figure 10. A depiction of polymer chains in glassy state. (a) When the chain is short, it may not get embedded into other chains in an adequate form to bear sufficient load. Conversely, (b) if a polymer chain is long enough, load-bearing strands (LBS) emerge as shown to build a load-bearing network, i.e., a network of LBS. For the strand to disengage, a great number of segments have to get displaced around at the junctions. (c) Depiction of the minimum chain length for formation of two hairpins in a Gaussian chain, having length lc from A to B that crosses a plane three times. (d) The strand length lhp between two pairs of hairpins at A′ and B′ is expected to be significantly longer than lc, but still proportional to lc of eq 6.

Figure 11. (a) A plot of the distance separating the glass transition temperature Tg and the brittle−ductile transition temperature Tbd as a function of the areal density of subchains that form hairpins: 1/plc. (b) Brittle strength σB = σy(Tbd) of nine glassy polymers obtained by Vincent24 plotted against the areal density of the load-bearing strands, which is assumed here to be proportional to 1/plc, where PPe stands for poly(pentene-1).

higher when p and lLBS ∼ lhp are lower. We know p denotes chain thickness and lhp proportional to the Kuhn length depicts the chain flexibility. Thus, for skinnier and more flexible polymer chains, σy could stay above σ* until a further distance below Tg. In other words, it is reasonable to expect Tg/Tbd to grow with ψ, i.e., Tg/Tbd to be a rising function of ψ:

longer than and proportional to lc. For simplicity, we assume lLBS ≈ lhp and rewrite96 eq 3 along with eq 5 as σ y(T ) ≃ (1/wplc)fLBS (T )

(7)

where the ratio w = lLBS/lc ≈ lhp/lc

(8)

Tg /Tbd = h(ψ )

might be a universal constant for all linear flexible polymers, i.e., for all linear Gaussian chains. Since the presence of two hairpins in Figure 10c does not ensure that a LBS as shown in Figures 10b and 10d will be created in the glassy state, we expect w of eq 8 to be greater than unity. The yield stress σy of eq 7, proportional to the force due to a load-bearing strand, f LBS(T), should depend on temperature and the contour length lLBS depicted in Figure 10b. Around Tg, chain pullout is inevitable upon tensile extension: σy will be low because f LBS is low. Upon lowering T below Tg, f LBS rises, pushing σy to a higher level, and the failing stress corresponding to the breakdown of the primary structure (as denoted by σ* in Figure 1c) also rises. We note that ψ = 1/plLBS of eq 5 in eq 3 is

(9a)

Taking the data from Vincent24 on Tbd for a number of polymer glasses, Figure 11a shows that with increasing ψ ∼ 1/plc Tbd indeed moves down from Tg. Actually, because of eq 6, Figure 11a is consistent with the empirical correlation reported by Wu97 that Tbd/Tg increases linearly with the characteristic ratio C∞. Thus, the origin of Wu’s empirical formula appears to be rooted in how the structure of the load-bearing network depends on the molecular parameters given by p and lc. Similarly, σy is expected to grow as Tg/T increases until it meets σ* at Tbd as shown in Figure 1c. Thus, we could anticipate σB to be a function of Tg/Tbd: 6727


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Figure 12. Our phase diagram to depict the effects on (a) aging and (b) pressurization on the brittle−ductile transition of ductile and brittle polymer glasses, respectively. (c) Analogous to Figure 1c, this phase diagram depicts the room-temperature tensile extension tests of melt-stretched polymers. If the strength of the LBS network can be elevated by melt-stretching by a factor as much as the melt-stretching ratio λ when examined along the direction of melt-stretching (Z axis), and be reduced by √λ when extended along a direction perpendicular to the Z axis, the brittle−ductile transition can be shifted downward or upward along the temperature axis, respectively.

σB = σ y(Tbd) = g (Tg /Tbd)

eq 3. Ideally, the affine melt-stretching increases the yield stress of the chain network in eq 3 as

(9b)

The combination of eqs 9a and 9b indicates that σB might be an increasing function of ψ. We gather the values for σB obtained by Vincent24 and plot them as a function of the areal density of LBS, ψ ∼ 1/plc, for those polymers whose values of p and lc are available,98 as shown in Figure 11b. Figure 11b implies, given eq 9b, that f LBS(Tbd) is constant for the different polymers at the BDT since the data in Figure 11b show σB = σy(Tbd) ∼ ψ. At the present, there is no theory to explain why f LBS(Tbd) should be all the same at the brittle−ductile transition, independent of the chemical microstructure. Since Tg/Tbd is rather different for the different polymers and both the primary structure and the chain network become stronger with increasing Tg/T (>1), it makes sense that a glass with higher areal density ψ of LBS could be cooled further from Tg before it turns brittle, i.e., before the primary structure becomes too stiff. However, any deeper implication of f LBS(Tbd) = constant is beyond the scope of the present study to explore. C. Phase Diagrams: Effects of Aging, Pressure, and Melt-Stretching. At a fixed temperature, upon aging, a ductile glass can turn brittle. The data in Figure 8 imply that the effect of aging on PC is to strength the primary structure and raise the level of the failing stress σ* corresponding to the breakdown of the primary structure. It is convenient to interpret this aging effect in terms of a phase diagram as shown in Figure 12a. Because the strengths of chemical bonds are unaffected by pressure, the pressurization effect affirms that chain pullout, not chain scission, dictates the intrinsic strength of polystyrene at atmospheric pressure and room temperature. Had chain scission dominated the brittle fracture of PS under ambient conditions at σ*, it would be difficult to explain the fact28 that under pressure the ductile PS exhibits a yield stress σy 4 times higher than σ*. When a brittle polymer turns ductile, the chain network must have been enhanced relative to the primary structure. Thus, in our picture, the LBS network must be stronger under high hydrostatic pressure. Since the interchain friction can be greatly enhanced by hydrostatic pressure, the leading effect of pressure is perhaps to make chain pullout more difficult so as to effectively increase ψ in eq 7 for σy. Consequently, we may perhaps depict the pressure effect as shown in Figure 12b. In Figure 9, we proposed that the effect of melt-stretching (along Z axis) is to condense entanglement strands in the XY plane. In other words, the LBS network in the glassy state undergoes the same geometrical condensation so that ψ of eq 5 increases by a factor as high as the stretching ratio λ. In other words, the chain network can be strengthened either by raising ψ (as done here by melt-stretching) or by increasing f LBS(T) in

σ y = λσ y

(10a)

when the tensile extension of the melt-stretched glass turns ψ into ψλ, where σy is the yield stress of an isotropic sample. The conversion from the brittle behavior to ductile yielding, as demonstrated in Figures 5a and 6a−c, suggests that Tbd|| of a melt-stretched polymer is lower than Tbd. To demonstrate this point, i.e., to collect the data along the dashed line in Figure 12c, is beyond the scope of the present study. On the other hand, our analysis of the data in Figures 6a−c indicates that the enhanced yield stress is of the following form: σ y = σ y[A + (1 − A)λ]

(10b)

where A is larger than (1 − A) so that the dependence on λ is only linear instead of linearly proportional. This could reflect either a deviation of melt-stretching from the affine deformation limit or a structural change in the chain network that cannot be accounted for in eq 10a by the simple-minded geometric condensation factor λ. For example, the effect of losing “vertical strands” is not properly depicted in the cartoon of Figure 9. When a melt-stretched polymer glass is extended uniaxially along a direction perpendicular to that of the melt-stretching, the yield stress of the glass is expected to decrease according to σ⊥y = σ y / λ

(10c)

where the reduction factor of 1/√λ appears because the crosssectional area in either XZ or YZ plane could grow by √λ as shown in Figure 9. In other words, we predict that the ductile PC could turn brittle after melt-stretching when examined in tensile extension along a perpendicular direction. Namely, the effect of melt-stretching (ms) is to shift the point of brittle− ductile transition either to an effective lower temperature or to a higher temperature, depending on the extension direction relative to the direction of melt-stretching. In passing, we should also note that melt-stretching does not need to be completely affine to produce significant geometric condensation. In other words, the depiction in Figures 9 and 12c only serves as a guide to anticipate the effects of melt-stretching. To test the prediction of eq 10c, PC was first melt-stretched at 155 °C to a Hencky strain of either ε = 0.56 (i.e., λ = 1.75) or 0.91 (i.e., λ = 2.5), quenched rapidly to room temperature and then cut into a dog-bone-shaped specimen as sketched in Figure 4. The Instron tensile test indeed confirms the anisotropic effect of the melt-stretching as shown in Figure 13. First, the Young’s modulus E decreases with the increasing level of melt-stretching, as shown in the inset, when measured 6728


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Figure 14. Engineering stress σengr vs tensile deformation ΔL/L0 for the isotropic fractionated PC (circles) and its blends with the PC of low molecular weight at weight fractions of the high molecular weight PC equal to 80% (squares), 60% (diamonds), and 40% (triangles). Instead of behaving in a ductile manner, the blends of 60:40 and 40:60 become completely brittle as shown. Inset shows the difference between the as-received commercial PC and its fractionated counterpart.

Figure 13. Engineering stress σengr vs tensile deformation ΔL/L0 for the isotropic PC and two melt-stretched PC samples, for which the extension was carried out in a direction perpendicular to the meltstretching direction as illustrated in Figure 4. After melt-stretching of ratio λ = 2.5, the PC is completely brittle as shown by the photo and the data. The other stretched PC also drew much less as shown. The inset indicates that the initial slope, reflecting the Young’s modulus, systematically decreases from the equilibrium value.

these samples at room temperature. First, all the samples show the same Young’s modulus in the regime of linear response, implying that the short chains can effectively participate in the primary structure. In other words, before yielding of the primary network, the blend is as stiff as the PC made of the pure long chains. But the underlying chain network is apparently considerably weakened upon blending with the short chains. At the weight fraction of 60% long chains, the PC blend exhibits brittleness just like PS at room temperature. At 40% of long chains, the failure stress is even lower as shown in Figure 14. The data of 40% blend in Figure 14 indicate that the primary structure breaks down at a yielding strain of 4%. The entire glass also fails at 4% because the network of LBS is not dense enough to avoid immediate breakup. In contrast, the pure longchain PC can undergo tensile extension well beyond 4% not because the primary structure did not fail at 4% but because the chain network is strong enough to carry the load and further extend until the point of shear yielding. In closing, we note that the stress−strain curves for the two isotropic PC samples are not identical for the same extension condition as shown in the inset of Figure 14. The yield stress σy and overall stress level are considerably higher for the fractionated PC that has a significantly higher Mw. The difference demonstrates the fact that the characteristics associated with yielding of the chain network depend on the molecular weight distribution. On the other hand, the difference in the post-yield regime is due to the difference in the sample geometries. Because of the limited amount of the fractionated PC, the specimen was a strip instead of being dogbone-shaped. Such a specimen tends to involve nearly perfect and stable neck propagation, which gives rise to a constant tensile force. For a dog-bone-shaped specimen, σengr increases slightly with extension when the neck front is near the “end” of the specimen where the specimen’s cross-sectional area is larger. E. Detailed Discussions on Other Important Features. E.1. Young’s Modulus. The systematic anisotropic change in the Young’s modulus E of polymer glasses by melt-stretching

in tensile deformation perpendicular to the melt-stretching direction. Second, the sample with melt-stretching to λ = 1.75 suffers ductile failure at a much lower degree of extension around ΔL/L0 = 0.3 instead of ΔL/L0 = 1.3 seen for the isotropic PC. Third, the sample after melt-stretching to λ = 2.5 shows brittle failure around ΔL/L0 = 0.1. The preceding analysis is evidently rather simple and crude. But the systematic demonstration of the effects of meltstretching suggests that the idea expressed in eqs 7, 10a, and 10c may have captured some physics of polymer glass ductility. Armed with the evidence that all three brittle glasses (PS, PMMA, and SAN) became ductile upon sufficient meltstretching, we are inclined to assert that the effects of meltstretching to affect the polymer glass ductility by either the geometric condensation or dilation can be universally understood in terms of the picture described in eqs 10a and 10c along with Figure 9. D. Ductile-to-Brittle Transition: Reduced Network Density. In our analysis, chain networking in the glassy state requires long chains as depicted in Figure 10b, whereas short chains cannot participate in the chain network as illustrated in Figure 10a. Thus, blending long chains with such short chains should reduce the network density of load-bearing strands, ψ. According to our picture depicted by eq 3, the yield stress σy can be effectively reduced by such blending. To test this prediction, we blend long and short chains of polycarbonate at different weight fraction ϕ of the long chains ϕ:(1 − ϕ) = 100:0, 80:20, 60:40, and 40:60 and study their mechanical behavior in the absence of melt-stretching. The short chain has a molecular weight of 6 kg/mol that could be too low to bridge with other short chains to form a load-bearing network. Thus, the dilution of the LBS network amounts to a reduced σy(ϕ). On the other hand, the blending may not reduce the strength of the primary structure as long as the short chains are above some critical length. Therefore, we can expect some of these PC blends to be brittle when σy(ϕ) < σ*. Figure 14 shows the tensile force (i.e., engineering stress σengr) as a function of the extension up to the point of failure for 6729


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that the drop in the tensile force at the yield point is much smaller percentage-wise. The onset of necking is less obvious; i.e., the sample appears to undergo relatively uniform extension in this case. Moreover, the rate of the rise in σengr is actually similar to that observed after the full conversion, e.g., to the data corresponding to ε = 1.4 (black up-pointed triangles) around and after ΔL/L0 = 0.4 in Figure 6a. Finally, we turn our attention to the rather special case of PMMA. Unlike PC and melt-stretched PS, the room-temperature extension of melt-stretched PMMA could proceed without visible necking. Yet, an isotropic (i.e., non-meltstretched) PMMA does undergo necking during tensile extension at temperatures above the brittle−ductile transition temperature Tbd(PMMA) ∼ 60 °C. Here the only difference is that the melt-stretched PMMA benefits from the geometric condensation effect and can extend considerably even at room temperature < Tbd(PMMA). On the other hand, the meltstretched PS also has a strengthened chain network but still necks. Thus, the condition for shear yielding remains elusive and is currently an open question.

demonstrated in Figures 7a−c (enhanced: parallel to meltstretching) and the inset of Figure 13 (reduced: perpendicular to melt-stretching) indicates that this linear-response property has contributions from both the nonbonded van der Waals interactions and the bonded chain connectivity. In other words, the two separate networks, depicted in Figure 1b, each contributes to the linear mechanical property of the glass. As discussed in Figure 9, one leading effect of melt-stretching is to produce a denser array of load-bearing strands (LBS) per unit area in the XY plane, where the load bearing is in the direction of stretching, Z. When such a melt-stretched glass is under tensile deformation along the Z direction, there is an increased contribution from the chain network. Conversely, the chain network is less loaded under tensile deformation in any direction perpendicular to the Z axis. Thus, the treatment of melt-stretching is also an insightful way to demonstrate the origin of the Young’s modulus E. To our knowledge, the present study is the first systematic observation of the effect of melt-stretching on E and shows how the segmental orientation plays a dominant role to affect the magnitude of the Young’s modulus. Moreover, Figure 14 reveals that the short chains are part of the primary structure, ensuring E of the blend is the same as that of pure long chain PC, but are not part of the chain network that bears load at high deformations. E.2. Shear Yielding in Ductile Glasses and Stable Necking. At T > Tbd, polymer glasses can extend beyond the point of breakdown of the primary structure until the underlying chain network suffers shear yielding. The information in Figure 5b shows that the consequent shear strain localization initiated macroscopic tensile strain localization, i.e., necking.88 As depicted in Figure 1d, the tensile stress σ arising from a significant elastic extension of the glass can be decomposed into two orthogonal components, one of which, σ12 = (σ/2) sin 2θ, acts on a given inclined plane that forms an angle θ with the extension direction. When σ exceeds a critical value so that σ12 at its maximum with θ = π/4 exceeds a critical shear stress σy12, the network of LBS can undergo shear yielding when segments slides around at a subchain level. A microscopic theory should be able to demonstrate that the tensile stress σy required to cause pullout from the hairpins depicted in Figure 10b is higher than 2σy12 that is required for the subchain pullout to occur at a shear plane (e.g., θ = π/4). Conversely, the experimental evidence of shear yielding actually implies that subchain pullout is operative. In other words, if chain scission could take place, the specimen should be breaking in a tensile manner. E.3. Ductile Characteristics as a Function of MeltStretching. All the data in Figures 5a and 6a−c show that apart from the effect on Young’s modulus E the yielding and plastic deformation behaviors are also affected by meltstretching. With increased Young’s modulus E due to meltstretching, the yield stress σy (defined in Figure 5b) is higher, but the point of yield involves a comparable degree of extension. Two more features are worth noting from Figures 6a−c. With increasing melt-stretching ratio λ, (a) the meltstretched polymer shows higher tensile forces at and after the yield point, and (b) the tensile force subsequently can increase more strongly with ΔL/L0. Apparently, the load-bearing chain network is stronger as the amount of the geometric condensation depicted in Figure 9 increases with the degree of melt-stretching. At an optimal level of melt-stretching, as in the case for the top curve in Figure 6a, there is so much geometric condensation to result in such a strong and stiff chain network

V. CONCLUSIONS Polymer glasses are all brittle when tested in tensile extension at sufficiently low temperatures. Conversely, most of them become ductile under tensile deformation at temperatures sufficiently close to their glass transition temperature Tg. Regarding a polymer glass as a hybrid depicted in Figure 1b, its brittle−ductile transition (BDT) may be understood in terms of an interplay between the primary structure due to van der Waals forces and the chain network due to uncrossability illustrated in Figure 10b. In other words, to understand BDT, both the strength of the primary structure and yielding behavior of the chain network should be taken into account. The facts that (a) PMMA can be ductile at Tg/T < 1.2; (b) PC can be brittle at Tg/T > 3.2 as well as upon aging indicate that the entanglement density or molecular characteristics such the packing length p and Kuhn length lK cannot be the only variables in our description of polymer glass mechanics. Conversely, the fact that melt-stretched PS, PMMA, and SAN can turn ductile also suggests that the glass ductility depends on specific characteristics of the chain network. In both brittle and ductile polymer glasses, yielding of the primary structure always precedes the failure of the chain network. How the chain network breaks down depends on how far the glass is below Tg: In its molten state, i.e., well above Tg, the entanglement network of Gaussian chains can undergo a great deal of deformation before displaying strain localizartion. As the polymer vitrifies across the glass transition temperature Tg, in addition to the chain network, and another network emerges from the intersegmental van der Waals interactions. This primary structure must break up before large deformation of the chain network such as shear yielding could take place. When the polymer is further cooled to become a brittle glass, the breakdown of the primary structure involves such a high tensile stress that the resulting load transfer from the primary structure to the chain network produces strain localization. In other words, upon the collapse of the primary structure, the load transfer causes the chain network to undergo structural failure. The weakened network must undergo significant extension because the stress cannot vanish instantly. But a sudden rise in extension is globally impossible at a given external extensional rate. Therefore, strain localization must 6730


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occur. Consequently, we often see crazing as the precursor to brittle fracture. Thus, our approach is to evaluate the strength of the primary structure relative to the ability of the chain network to resist yielding. Opposite to the picture contained in the LDWO hypothesis and depicted in Figure 1a, we assert that when a glass is ductile, i.e., at T > Tbd, the mechanical response is dominated by the chain network. In other words, for T > Tbd, the primary structure is rather weak and falls apart readily before the shear yielding of the chain network takes place to initiate necking at σy. Conversely, as indicated in Figure 1c, at T < Tbd, which is a deeper glassy state, segmental mobility is much lower, and cooperative motions are sufficiently suppressed. Consequently, the inevitable breakdown of the primary structure corresponds a higher stress σ* during tensile extension. Under σ* the chain network has no other option but to suffer immediate collapse through strain localization (i.e., crazing), as asserted in the preceding paragraph. With the pictures of Figures 1b−d in mind, we examined the consequence of melt-stretching on the mechanical behavior of polymer glasses. According to Figure 1c, if melt-stretching alters the structure of the chain network, we can expect it to produce predictable effects as shown schematically in Figure 12c. Specifically, on the basis of three most common brittle polymer glasses, we showed that the effect of melt-stretching in uniaxial extension (along Z direction) is universal and can indeed strengthen the chain network in the polymer glasses along the Z direction. In particular, effective melt extension results in geometric condensation of the load-bearing strands (LBS) in the XY plane and geometric dilation in the other two planes as illustrated in Figure 9. In summary, in this work we propose to treat polymer glasses as a structural hybrid composed of a primary structure made of the van der Waals bonds and a chain network made of chemical bonds. The effect of melt-stretching is to alter the structure of the chain network and shift the relative importance of the double networks. Brittle polymers such as PS, PMMA, and SAN turn into ductile glasses in tensile extension after meltstretching when both cold drawing and melt-stretching are in the same direction. Ductile polymers such as PC can become brittle during uniaxial extension in a direction perpendicular to the melt-stretching direction. Although the rate of the uniaxial extension was limited to a narrow range of 1−5 mm/min with L0 = 15 mm, the same behavior is found to take place when the cross-head speed is increased over 100 times.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We appreciate the constructive remarks by the reviewers. We are grateful to our colleagues Dr. Gary Hamed and Dr. Stephen Cheng for making their Instron testers available to us.

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REFERENCES

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How Melt-Stretching Affects Mechanical Behavior of Polymer Glasses

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