Delineating nature of stress responses during ductile uniaxial extension of polycarbonate glass

Polymer 89 (2016) 143e153

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Delineating nature of stress responses during ductile uniaxial extension of polycarbonate glass Panpan Lin, Jianning Liu, Shi-Qing Wang* Department of Polymer Science, University of Akron, Akron, OH 44325-3909, United States

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 December 2015 Received in revised form 18 February 2016 Accepted 21 February 2016 Available online 24 February 2016

We carry out simultaneous mechanical and IR-thermal-imaging-based temperature measurements of tensile extension on untreated, milled (mechanically “rejuvenated”) and melt-stretched bisphenol Apolycarbonate (PC). The extension is found to cause significant buildup of both excess internal energy u2 and plastic dissipation. The magnitude of u2 is one to two orders of magnitude higher than the energy involved in rubbery elastic deformation. While the ratio of u2 to the mechanical work w decreases with increasing rate of extension for untreated PC, milled PC is found to be more dissipative at lower rates. Homogeneous extension of melt-stretched PC in the post-yield regime including strain hardening behavior reveals largely non-dissipative responses, emphasizing the plastic deformation of glassy polymer may not be fully dissipative. The experimental results clearly indicate that a significant component of stress can be intrasegmental leading to the observed buildup of internal energy by distortions of covalent bonds. The glassy polymer physics at the chain level complements the more familiar idea of inter-segmental dissipation as the dominant event during plastic deformation. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Glassy polymers Internal energy Plastic dissipation Strain hardening

1. Introduction Mechanical properties are an important characteristic of polymer materials. A core question concerns the nature of mechanical stress arising from polymers of high molecular weight under large deformation, either well above or below the glass transition temperature Tg or crystallization temperature Tc. To delineate the origin of stress in either liquid or solid state, both phenomenological and molecular-level viewpoints have been adopted. The task belongs to the realm of polymer rheology when T > Tg and Tm. In the past many decades a disproportionally large number of studies have concentrated on nonlinear polymer melt rheology [1e4]. Although the explicit molecular picture concerning how chain disentanglement takes place in large deformation is still under active development [5e12], we have achieved a satisfactory level of phenomenological understanding [13,14]. In our view [15], it is key to recognize that such strongly viscoelastic liquids as entangled polymers necessarily undergo yielding, i.e., a transition from elastic deformation to flow at high rates of deformation, and it is crucial to explore the condition for chain disentanglement leading to macroscopic yielding. Stress prior to the yield point is largely rubbery like, arising from stretching of the entanglement network.

* Corresponding author. E-mail address: [email protected] (S.-Q. Wang). http://dx.doi.org/10.1016/j.polymer.2016.02.051 0032-3861/© 2016 Elsevier Ltd. All rights reserved.

The challenge is to describe when the elastic deformation ceases and irreversible deformation begins to dominate. An appreciable amount of research has been carried out to investigate mechanical responses of amorphous polymers in their glassy state at T < Tg [16e40]. Strain hardening is an important phenomenon [21,24,32e35,41e55], typical of ductile deformation of polymer glasses, involving growing stress with strain in the post-yield (plastic flow) regime. Kramer [56] indicated that the microscopic origin of stress had remained elusive, e.g., the cause of “strain hardening” was unclear. More recent activities searched for and identified a dissipative mechanism to explain strain hardening as predominantly arising from plastic flow associated with intersegmental processes, based on computer simulations [22e26,50e52], and theoretical formulations [55,57e60]. On the other hand, experiments, based on calorimetric detections [61,62], DSC measurements [63,64] and direct temperature readings [65e67], have revealed appreciable buildup of internal energy in the post-yield regime. Simulations also found a small portion of energetic stress at large deformation while the majority of stress was related to plastic arrangements [22,23,50,51,68]. The present study aims to explore further the nature of stress responses as a function of the deformation rate during either inhomogeneous or homogeneous uniaxial extension of ductile bisphenol A-polycarbonate (PC). In contrast to the suggestion that strain hardening is either associated with reduction of conformational entropy [19] or plastic dissipation [50,55,69], we

P. Lin et al. / Polymer 89 (2016) 143e153

2. Experimental Bisphenol A-polycarbonate (PC) under study is Lexan TM 141 111, received from Sabic (GE Plastic). The average weight molecular weight is 63 kg/mol, with a polydispersity of 1.58. Its glass transition temperature Tg is 145  C, measured by a TA Q2000 DSC at a ramping rate of 10  C/min. In the present work, we study three “types” of PC that are either untreated, mechanical “rejuvenated” by milling, or melt-stretched. To prepare untreated PC sheets, PC pellets were placed into a 100 mm  100 mm  0.8 mm compression mold with two sheets of Kapton HN polyimide film at the surface of both sides in a 50-ton Dake hydraulic press. The pressing temperature was set at 200  C. PC was first pre-heated between two plates for 30 min, and then pressed with 25 tons of force for another 30 min. The sample was then removed from the press to allow cooling down to room temperature. Dog-bone shaped samples were obtained by a punch press at room temperature with a dog-bone mold (ASTM D-412) involving an effective length of L0 ¼ 39 mm and a width of W0 ¼ 3.30 mm. The milled samples were prepared by placing an untreated PC sheet with a dimension of 100 mm  100 mm  2 mm into a gap between two identical counter-rotating (at 10 rpm) rolls of 150 mm in diameter. Such a milling procedure involves a thickness reduction of 0.1 mm in each passing. After many passes, we achieved an accumulated thickness decrease of 30%. Such milled sheets were then cut into dog-bone (ASTM D-412) shaped with a thickness of H0 ¼ ca. 1.4 mm, a length of L0 ¼ 39 mm and a width of W0 ¼ 3.30 mm. To obtain melt-stretched samples, dog-bone shaped PC sheets with middle dimensions of 80 mm  35 mm  0.5 mm were first pressed using the same procedure as that adopted to prepare untreated PC. We then fixed the sample onto Instron 5567 and allowed it to relax at 160  C for 15 min to reach the thermal equilibrium. Uniaxial extension of PC was then carried out at a crosshead speed V0 ¼ 6 mm/s to a draw ratio of 2.5. At the end of melt stretching, icy water was sprayed onto the sample. Melt-stretched samples were obtained by cutting such samples in the stretching direction by a dog-bone mold (ASTM D-638) with an effective length of L0 ¼ 13.7 mm, a width of W0 ¼ 3.30 mm and a thickness of 0.3 mm. All the uniaxial extension tests of three different PC were carried out using Instron 5543 at around 23  C. In situ measurements of the specimens' temperature were performed by an infrared camera

(FLIR SC325) operating at 60 Hz to record the whole timedependent temperature profile on a video clip. To determine the convective heat transfer coefficient h, we preheated a comparable dog-bone shaped sample with a thickness of 1.2 mm and used the IR camera to record the temperature decrease as a function of time due to the air cooling. Fig. 1 shows the temperature drop DT¼(T  Ta) as a function of time t. From the energy balance equation, we have

rcp

dT 2h ¼  DT; dt H

(1a)

where H represents the thickness of sample. Approximating the specimen as a thin sheet, we have


 2h DT ¼ ðTi  Ta Þexp  t Hrcp

(1b)

where Ti is the initial temperature and Ta is the ambient temperature. For PC, the density is r ¼ 1200 kg/m3, and the specific heat capacity is cp ¼ 1200 J/(kg,K). By fitting the temperature data to the exponential function of eq (1), we obtain the decay time constant to be 33 s, from which h is estimated to be h ¼ 25 W/(m2,K). 3. Results 3.1. Untreated sample The elastic yielding phenomenon [71] suggests to us that significant energy storage (i.e., U2) takes place during neck formation. The structure of the neck front is sketched in Fig. 2. To characterize the nature of the stress response during uniaxial extension of untreated PC, we carried out theoretical analysis in Appendix A. As shown in Appendix A.1, we can gain more insight about the nature of ductile deformation of polymer glasses by determining the ratio U2/W according to either eq (A.16) or eq (A.9), depending on whether the extension speed is high or low and whether air cooling contribution needs to be incorporated or not. Fig. 3a shows the IR thermal imaging of shear yielding upon startup extension at a low speed of V0 ¼ 3 mm/min. See supporting information for two movies with V0 ¼ 3 and 300 mm/min respectively. Fig. 3b displays the temperature profile at the neck front during its steady

80 70 60 o

demonstrate that ductile extension of polymer glasses could result in a significant buildup of excess internal (potential) energy u2 of both inter-segmental and intra-segmental origins that well exceeds contributions due to rubbery elastic stretching of the entanglement network. Specifically, we apply IR thermal imaging to measure the internal (kinetic) energy u1 increase associated with the temperature rise against the mechanical work density w and find u2 ¼ (w  u1) to vary with the applied rate in opposing ways depending on whether or not the polymer glass has undergone pre-treatment, i.e., mechanical “rejuvenation”. Based on the various pieces of evidence from elastic retraction of a necked sample to stress relaxation, a microscopic picture is proposed here to explain how plastic deformation can take place while u2 builds up during large deformation at different rates. The paper is organized as follows. After the Experimental Section 2 to describe the material characteristics, we present the experimental results in Section 3 and discuss the microscopic origins of u2 according to the detailed theoretical analyses that evaluate u2 based on the simultaneous mechanical and thermal measurements. The paper ends with a conclusion in Section 5.

ΔT ( C)

144

50 40 30 20

0

50

100

150 t (s)

200

250

300

Fig. 1. IR camera measurement of the surface temperature of a heated PC that drops due to air cooling according to eq 1a-b.

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logarithmically as shown in Fig. 4. Over this range of speed, there is significant increase in Tnk, as shown in Fig. 5, although the stress level sengr(nk) stays nearly constant, only increasing logarithmically with V0 at high speeds, which we might anticipate. Moreover, it is perhaps unsurprising that sengr(nk) does not continue to decrease with lowering V0 because a substantial stress is required to propagate the neck front regardless of the speed. At the two low speeds of 3 and 24 mm/min, air cooling and lateral heat conduction are non-negligible. In other words, according to eq (A.16), in absence of air cooling and lateral heat conduction, the temperature reading at the neck front would have been given by

  2Snf hðlnk  1Þ kðlnk  1Þ Tnk  Ta ¼ ðTnk  Ta ÞIR 1 þ þ rcp D0 V0 rcp A0 V0

(2)

i.e., higher than the actual IR measurement by a factor indicated in the squared brackets. In Fig. 5, the open squares are (Tnk  Ta)IR. After the correction, all data points fall onto a straight line, revealing (Tnk  Ta) ~ (V0)1/5. We should also notice that the temperature readings in Fig. 5 are maximum values, which involves around 2.5% of experimental error. Given the information in Figs. 4 and 5, we can estimate u1, u2 and w according to eqs (A.5b), (A.6b) and (A.8) as a function of V0,

1.85 1.8

nk

1.75

λ

Fig. 2. Illustration of the neck front propagation at a crosshead speed of V0 for a specimen of initial dimensions of A0 ¼ D0H0 in cross-section and L0 in length. During the neck front propagation, over Dt, a fresh amount A0DLiso is converted from the unnecked part to increase the neck by a length of DLnk.

propagation. The corresponding stress vs. strain curve is given in Fig. 3c, where also plotted is the temperature measurement of one fixed spot on the specimen. The temperature at this arbitrarily chosen point rises to the maximum when the neck front passes through. As estimated in eq A.10 through A.13, at this low speed there are non-negligible air cooling and lateral heat conduction that affect our measurement of the true temperature rise. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.polymer.2016.02.051. We have carried out similar experiments at higher values of V0. With increasing V0 there is more shape change, i.e., lnk increases

1.7 1.65

L = 39 mm 0

1.6

1

10 100 V0 (mm/min)

1000

Fig. 4. Draw ratio lnk, characterizing the shape change associated with necking, as a function of the crosshead speed V0.

Fig. 3. Temperature profiles during uniaxial extension at V0 ¼ 3 mm/min at 23  C. (a) Shear yielding occurs at the yield point. (b) Temperature mapping at a subsequent moment, showing the propagation of the neck front at the top whereas the bottom neck front slows down. Color bar listed on right refers to temperatures in the unit of Celsius (oC). (c) Corresponding stressestrain curve (red circles) of untreated PC and temperature (blue squares) measured from a fixed spatial point on the extending specimen, as a function of either draw ratio (top X axis) or time (bottom X axis) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

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100

60

3.2. Milled sample

engr(nk)

40 30

1/5

o

σ

nk

[T − Ta]max ( C)

(MPa)

50

20

10

10 0

1

10 100 V (mm/min)

1000

0

Fig. 5. The maximum temperature rise (Tnk  Ta) (open squares and right Y axis), as measured by the IR camera, at the various values of the crosshead speed V0, and the “plateau” stress level sengr(nk) (red circles and left Y axis) during the neck front propagation at different values of V0 for L0 ¼ 39 mm. Filled diamonds are the hypothetical temperature readings that contain corrections from the air cooling and lateral heat conduction according to Eq. (2) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

80

40

12 10

σengr

8

60

ΔT

6

40

4

20 0

σ 1

1.1

1.2

1.3

λ

1.4

2 engr(E)

1.5

0 -2 1.6

Fig. 7. Stressestrain curve (red circles) of milled PC during uniaxial extension with V0/ L0 ¼ 1.2 min1 at 23  C, and simultaneous temperature measurement (blue squares) for L0 ¼ 39 mm. Also plotted is the elastic component of stress decomposition sengr(E) (green diamonds) according to eq (A.23) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

λ

w (MPa) u (MPa) 1 u (MPa)

14

V0/L0= 1.2 min -1

o

50

100

ΔT ( C)

taking (Tnk  Ta)max as (Tnk  Ta) that appears in eqs A.7, A.9 and A.16. Fig. 6a shows how the work density w, internal energy change associated with the temperature rise u1 and excess energy storage u2 respectively change with V0. The process of neck formation becomes increasingly plastic with increasing V0, in agreement with the finding of an earlier similar study [73]. It is more insightful to evaluate the normalized changes. Fig. 6b shows that u2/w actually decreases with V0 logarithmically in the explored range of speed. Since lnk also has logarithmic dependence on V0, u2/w is a linear decreasing function of lnk as shown in Fig. 6b. We see this correlation to mean that u2/w depends on the severity of the spatial rearrangement associated with necking, depicted by lnk. Intriguing implications of the data in Fig. 6aeb will be discussed in Section 4. It is interesting to remark that Fig. 6b also shows the nature of the plateau stress during the neck propagation at different speeds according to eq (A.25).

Necking can be avoided in extension of PC if we pre-treat PC with mechanical rejuvenation. After two-rolling milling, the engineering stress (red circles) monotonically grows and shows yielding around 50 MPa at 5% extension at 1.2 min1. The specimen uniformly heats up as indicated by the in situ IR thermal imaging. For homogeneous extension, the theoretical analysis is much simpler as shown in Appendix A.2. Fig. 7 uses double Y axis to display both the engineering stress sengr and temperature rise DT as a function of the stretching ratio l. The monotonic increase of sengr with l can be regarded as “strain hardening” [74]. Based on the stress vs. strain curve and eq (A.23), we also show in Fig. 7 that the elastic component of the tensile stress sengr(E) is more than measurable and is responsible for the observed stress increase. A second way to depict the nature of the mechanical response is to evaluate over the course of the continuous extension how the excess internal energy u2 stores up relative to the mechanical work w. Fig. 8 shows that during extension u2, estimated using eq (A.21), peaks at the early stage because of the initial temperature drop [75] while w increases monotonically. At this low rate of 0.077 min1, u2 stays constant for the rest of extension, suggesting that the

σengr (MPa)

146

(a)

1

1.65

1.7

neck

1.75

1.8

1.85

(b)

2

0.8

u /w

30

0.4

10 0

0.6

2

20

0.2

1

10 100 V (mm/min) 0

1000

0

1

10 100 V (mm/min)

1000

0

Fig. 6. (a) Mechanical work density w (red circles), internal (kinetic) energy change u1 (blue squares) and internal (potential) energy u2 (green diamonds) as a function of the crosshead speed V0 for L0 ¼ 39 mm. (b) the ratio of u2/w as a function of V0 on logarithmic scale (bottom X axis) and draw ratio lnk on linear scale (top X axis) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

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by melt stretching and elevation of the energy landscape by milling. Fig. 9b also shows that u2 grows much more strongly with l in melt-stretched PC, i.e., there is a much higher component of the elastic stress sengr(E). Because of a slight spatial variation in the temperature, we present measurements in triangles from two spots on the specimen in Fig. 11. The diamonds are based on the curve of up-pointed triangles, thus an under-estimate of the elastic component. Because of the less temperature rise, there is more excess internal energy storage u2, as represented by the half-filled squares in Fig. 9aeb. 4. Discussions 4.1. Evidence of internal energy storage

Fig. 8. Mechanical work density w (red circles) and excess energy u2 (green diamonds) as a function of extensional ratio l with V0/L0 ¼ 0.077 min1 of milled PC for L0 ¼ 39 mm. The inset shows the elastic component of stress decomposition sengr(E) (green diamonds) and the total engineering stress sengr (red circles) as a function of stretching ration l (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

subsequent extension is entirely irreversible deformation, i.e., totally plastic flow. We have also carried out tensile extension of milled PC at other rates. To summarize, we present the ratio u2/w as a function of the stretching ratio l as shown in Fig. 9a. Several remarks are in order. First, u2/w can indeed be greater than unity initially because the extending polymer glass cools down to build internal energy in the first stage of startup extension [75]. Second, the trend of u2/w indicates the extension becomes increasingly plastic. Third, since w increases monotonically with strain, the leveling-off of u2/w means that u2 still grows with strain, as shown in Fig. 9b. Fourth, unlike the rate dependence of u2 for untreated PC involving necking, milled PC shows the opposite trend. At any given draw ratio, u2 is lower at a lower rate. Finally, this internal energy buildup has little to do with the conformational entropy change [19] that produces rubber elasticity above Tg. The work wre required to affinely stretch the entanglement network in PC is given by the small dots in Fig. 9b, i.e., wre << u2. 3.3. Melt-stretched sample Another way to have near-homogeneous extension of PC is by a treatment of melt stretching that produces a geometric condensation of load-bearing strands (LBSs) in the chain network [76]. The strengthened chain network is much more effective in activating the primary structure during extension. Fig. 10 shows a comparison of stressestrain curve for three different PC samples: (i) untreated; (ii) milled; and (iii) melt-stretched. Because the extension of meltstretched PC is essentially uniform, it provides an interesting comparison with the preceding case of milled PC. Fig. 11 is plotted in the same way as Fig. 7, involving a rate of 3.5 min1. Here “strain hardening” is also evident. Without mechanical rejuvenation, the melt stretched PC is less dissipative, leading to a higher ratio of u2/ w in Fig. 9a. Another difference is also immediately clear: the temperature increase in the melt stretched PC is lower than the milled PC although the stress level is actually higher, as shown in Fig. 9c. The remarkable contrast in u2 between melt-stretched and milled PC in Fig. 9b stems from both geometric condensation of LBS

Our experiments of combined mechanical and thermal measurements enabled in situ time-resolved temperature determination and indicated that extension builds up internal energy. One form of energy storage may be through covalent bond distortions [77], leading to chain tension in the chain network [71,78,79]. Important supporting evidence emerges when an elastic retractive stress shows upon warming up a pre-necked PC specimen above the storage temperature. We have carried out a series of elastic yielding experiments based on pre-necked PC produced at the various speeds. The emergent retractive stress 65  C below Tg in Fig. 12 is a clear indication of the residual elastic stress of an intrasegmental origin trapped at room temperature, plausibly associated with covalent-bond level distortions. Moreover, the comparable magnitude of the stress sEY suggests that there is little correlation between sEY and u2 since Fig. 6a reveals the overall u2 to decrease with increasing rate. This is possible only if u2 has both inter-segmental and intra-segmental components. We may take Fig. 12 to suggest that the intra-segmental component of u2 is less rate sensitive and more internal energy of inter-segmental origin is stored at lower rates. In fact, the combination of Figs. 6a and 12 indicates that u2 is dominantly inter-segmental at low rate since u2 drops to a rather low common level at the highest rates. 4.2. Nature of excess internal energy u2: a more detailed picture of segmental activation Fig. 6b explicitly suggests that the level of excess internal (potential) energy u2 is rather high at low speeds. On the other hand, the elastic yielding data in Fig. 12 do not reveal higher sEY at a lower rate. To reiterate, this implies that there must be a substantial intersegmental component in u2. Since necking involves significant shape change, we face the challenge to explain why a great deal of inter-segmental u2 can build up during necking. To answer the question, we need to understand the other extreme. According to Fig. 6b, u2 diminishes at high rates where additional deformation mechanism (b motions) had been proposed by Garg et al. [66] as the origin of plasticity at high rates. With growing rate, the necking process becomes largely dissipative, involving segments overcoming the potential barrier and hopping past one another. In contrast, the large u2 at low speeds corresponds to insufficient plastic events of segmental hopping. Perhaps we can consider a mesoscopic picture as shown in Fig. 13. Here the dark dots denote segments stuck in a high potential energy state and the light dots represent segments that have overcome the potential barrier and are mobile. In other words, during the severe plastic deformation at the necking front, segments either become jammed again one another or slide past one another after climbing over the potential barrier and converting their potential energy into kinetic energy in the form of heat. The ratio of these two population varies with the applied rate. The data

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Fig. 9. (a) Fraction of excess internal energy storage u2/w; (b) excess energy u2; and (c) temperature increase DT as a function of the extensional ratio l at different values of V0/L0 for milled PC (open) with L0 ¼ 39 mm and melt-stretched PC (filled) with L0 ¼ 13.7 mm.

Fig. 10. Stressestrain curves of three different PC during uniaxial extension at V0 ¼ 48 mm/min at 23  C: (i) untreated (circles); (ii) milled (diamonds); and (iii) meltstretched (triangles).

suggests that at low rates much more segments remain localized, sitting in a high energy state. It is important to emphasize that these domains of high energy state may not be as small as implied in (a) through (c) of Fig. 13. These domains may be of a much greater scale. The depiction given in (d) through (f) of Fig. 13 represents them on some unspecified scales because we do not have a theory to describe the relationship between the domain size and crosshead speed V0. The cartoons in Fig. 13 are proposed to envision how global plastic deformation may occur without turning the majority of segments into a mobilized state of low energy. Fig. 13a and d show presence of large immobilized domains (dark dots or areas) that coexist with the mobilized regions during global plastic deformation. Since mobilized segments (light dots or regions) have hopped over the confining barriers, the fully plastic state depicted by Fig. 13c and f should possess much less internal energy u2. In other words, the phenomenological cartoons are prompted by the trend clearly shown in Fig. 6(a)-(b). Fig. 13 also implies that during extension there is less structural and dynamical heterogeneity at higher values of V0. The more plastic state depicted by either (c) or (f) of Fig. 13 should indeed show less heterogeneity. This is consistent with the report based on optical photobleaching measurements [80,81] that b in the

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of higher mobility produced by the higher rate, the initial stress relaxation is faster [86]. Consistently, the measurable shrinkage occurs at different rates, as revealed in Fig. 15. Clearly, the sample is hardly glassy in the initial moments after the clamp release. We note the necked part of the sample can retract 9% within a second for 300 mm/min and 8% in over 100 s for 3 mm/min. Instead of using a common value of t0 ¼ 0.03 s to mark to the initial time, we choose t0 ¼ 0.01 s for the shrinkage data involving 300 mm/min and t0 ¼ 1 s for the data involving 3 mm/min and re-plot Fig. 15 as the inset in Fig. 15. This figure indicates that the sample shrinkage clearly takes place on a much longer time scale when released from the extension at 3 mm/min. Actually, the initial sharp retraction is roughly consistent with the initial stress relaxation data in Fig. 14. It is entirely plausible that the state of (a) may undergo significantly slower retraction than the state of (c) in Fig. 13 in presence of so many “roadblocks”, i.e., the “dark patches”. In other words, the data in Figs. 14 and 15 are consistent with the picture depicted in Fig. 13.

Fig. 11. Stressestrain curve (circles) of melt-stretched PC during uniaxial extension at V0/L0 ¼ 3.5 min1 at 23  C, and simultaneous temperature measurement from two spots on the specimen (two types of triangles). Also plotted is the elastic component of stress decomposition sengr(E) (diamonds). This melt-stretched PC involves a meltstretching ratio of lms ¼ 2.5.

stretched exponential of the KWW characterization for relaxation dynamics increases with increasing rate. Moreover, a glass is inherently heterogeneous, perceived by Long and Lequeux [82] to be made of a percolating set of glassy domains. Other studies have also suggested glasses to contain soft and hard spots [83e85]. In such a context, our picture of Fig. 13 for plasticity depicts the patches on a different length scale from those envisioned by these previous studies. To further explore the implication of Fig. 13, we carry out one more experiment to learn about the glassy state after necking. Fig. 14 shows extension behavior of untreated PC at two speeds in terms of stress vs. strain curves. We can either unload to capture how the necked portion of the PC specimen retracts as a function of time or conduct stress relaxation tests as shown in Fig. 14. Because

4.3. Difference between milled and melt-stretched PC: dissipative vs. elastic The difference between milled and melt-stretched PC is rather remarkable, as shown in Fig. 9 (a)-(b) between open and half-filled squares. The rejuvenation by milling decreases the confining energy barrier and thus has obvious effects, e.g., making it easier to activate the glassy state (so that the yield stress is notably lower as shown in Fig. 10) and making it harder to build chain tension. Thus, u2 is expected to be noticeably lower than that of the meltstretched PC, which has the additional enhancement arising from the geometric condensation [76] of LBSs. If we assume that after yielding around l ¼ 1.1 the glassy states of milled and meltstretched PC are comparable, then at any given draw ratio the ability for the post-yield state to retain chain tension in LBSs should be the same. Then, the growing difference in u2 between the halffilled squares and open diamonds or squares is due to the increased density of LBSs because of the geometric condensation. It is conceivable that apart from inter-segmental contributions more intrasegmental energy storage also occurs due to the enhanced network by melt-stretching. The state of milled PC during the postyield extension can be close to Fig. 13c and f. In contrast, a meltstretched PC has many of its segments stuck in a high energy state during cold drawing. Significantly higher u2 arises during the ductile extension so that the state is perhaps close to that depicted by 13a or 13d. 5. Conclusion

Fig. 12. Retractive stress as a function of annealing time for pre-necked untreated PC at 80  C. These prenecked specimens were obtained by cold-drawing at room temperature with various crosshead speed V0, ranging from 3 to 600 mm/min and initial length of L0 ¼ 39 mm and final length of 1.7  39 ¼ 66 mm. A middle portion (ca. 30 mm) of such specimens were heated up to 80  C to observe the elastic yielding behavior.

The origin of stresses in large deformation of polymer glasses has been elusive because they appear to be both plastic and anelastic in the post-yield regime. Even today, a decade after Kramer's viewpoint article [56], entropic elastic force, commonly identified as the origin of rubber elasticity and important for polymer melt rheology, is still sometimes regarded [19,35] to be relevant during post-yield deformation of glassy polymers. There are two elusive issues. When the large deformation is not entirely dissipative, how much of this non-dissipative (elastic) component is inter-segmental vs. intra-segmental in origin? When the elastic component is intra-segmental in nature, is it due to the conformational entropic change that produces rubber elasticity in deformation of elastomers? Although it is beyond the scope of the present study to answer these questions, we have obtained some helpful hints. Based on homogeneous ductile extension of treated (milled and melt-stretched) polycarbonate samples, we demonstrate that the observed level of energy storage actually far exceeds that associated

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Fig. 13. Microscopic pictures of untreated PC during extension as a function of speed V0, indicated by the arrow. Dark dots or areas represent immobilized segments or domains stuck in high potential energy, and light dots or areas are mobilized segments or domains that have hopped over the potential barrier. As a function of speed V0, we can expect the number of immobilized segments (from (a) to (c)) to shrink, i.e., immobilized regions to shrink as shown in (d) to (f). Correspondingly the excess energy u2 decreases. Note we omit any structural and dynamic heterogeneities that may present inherently. The dark patches in Fig. 13(d)e(f) may contain the heterogeneities.

In terms of a recent molecular model [79] for yielding and brittleeductile transition of polymer glasses, along with the detailed theoretical analysis given in Appendix A, we have drawn significant conclusions. Ductile deformation of polymer glasses in post-yield regime is accompanied by molecular processes leading to buildup of internal (potential) energy u2 of both inter-segmental and intra-segmental origins. There is not only distortion of covalent bonds in the backbones. Large extension also cause some segments to hop over one another, contributing to the dissipative fraction of the mechanical work w, other segments to sit high in the potential well, which contribute to the non-dissipative fraction of w that we collectively term u2. This inter-segmental fraction corresponds to distortion of van der Waals bonds. During ductile extension, for both plastic deformation and internal energy buildup to take place, we envision a microscopic picture to depict the state of the primary structure. In this picture, there exist non-activated domains where high inter-segmental internal (potential) energy is stored. Such immobilized regions can co-exist with plastic flow. This explicit

1.02 l(t)/l(t )

with rubbery stretching of an embedded entanglement network, as shown in Fig. 9b. The significant internal-energy buildup can be more than 50% the mechanical work as shown in Fig. 6b, contrasting the literature consensus that ductile deformation of polymer glasses in the post-yield regime is largely dissipative [69]. In this work, we explored how the decomposition of dissipative and energetic components changes with the applied rate. For untreated PC that undergoes shear yielding and subsequent necking, the u2/w decreases logarithmically with increasing crosshead speed V0. At the lowest rate of V0/L0 ¼ 0.077 min1 (i.e., V0 ¼ 3 mm/min), u2 > 10 MPa. For milled PC under extension, u2 is the lowest at 0.077 min1, only ca. 2 MPa, about a fifth of that of the untreated PC. Moreover, uniform drawing produces much higher u2/w and u2 for melt-stretched PC than for milled PC. This difference explicitly indicates that the chain network in a more vitreous state drives a larger buildup of the internal energy storage. In other words, the observed difference between milled and melt-stretched PC supports the recently proposed picture of a structural hybrid for polymer glasses under large deformation. In this model, it is postulated that chain networking is necessary to permit post-yield plastic deformation in polymer glasses.

0

l(t)/l(t =0.03 s)

0

1

1.02 t =1s 1 t = 0.01 0s 0

0.98

0.96 0.92

0.96

-2

0

10

10

0

10

t (s)

0.94

10

2

4

10

0.92 0.9 -2 10

Fig. 14. Stress response of untreated PC at two different rates up to a stretching ratio of L/L0 ¼ 1.5 at 23  C. Two different crosshead speeds are 0.05 (circles) and 0.5 mm/min (squares) for an initial length of L0 ¼ 13.7 mm.

-1

10

10

1

t (s)

2

10

3

10

Fig. 15. Length shrinkage of pre-necked PC as a function of time, prepared with two different crosshead speeds of V0 ¼ 3 (circles) and 300 mm/min (squares) at 23  C and L0 ¼ 39 mm to extend to L/L0 ¼ 1.6 < lnk before the necking is completed. Instead of a common value for t0, the inset replot the data using two different starting points: t0 ¼ 0.01 s for 3 mm/min and t0 ¼ 1 s for 300 mm/min.

P. Lin et al. / Polymer 89 (2016) 143e153

picture is supported by or at least consistent with both stress relaxation behavior and elastic recovery characteristic, as shown in sub-section 4.2. Although there is an agreement that a significant component of stress in the strain hardening regime is dissipative, there exists confusion about the nature of the non-dissipative component, which has often been labeled as entropic [19,35]. Whenever a nonnegligible component of the stress is intrasegmental in nature, it could be mistaken as entropic. Surely, stress of intrasegmental origin can occur during large deformation in the glassy state. The elastic yielding phenomenon shown in Fig. 12 and studied in detail recently [71,78,79,87] reveals the existence of such stress. But this retractive stress is not related to entropic forces produced during melt extension above Tg because one can show that melt-stretched polymers do not show elastic yielding at any temperature below Tg [78]. The entropic intramolecular force is too weak to cause any yielding of the glassy state. Conversely, the chain tension produced during cold drawing to result in intrasegmental stress is entirely a different concept and can be sufficiently high to drive the polymer out of its glassy state upon annealing above the storage temperature. We assert that the origin of chain tension produced below Tg is not entropic, but rather enthalpic, associated with distortions of covalent bonds [77].

151

V0 ¼ Vnk  Viso ;

(A.1)

where Viso is the speed at which the un-necked portion shrinks because of its conversion to the necked region. The mass and volume conservations dictate

A0 Viso ¼ AVnk ;

(A.2)

where A0 and A is the cross sectional area of un-necked and necked regions respectively, as shown in Fig. 2. From eqs A.1 and A.2 we have

Vnk ¼ V0 =ð1  1=lnk Þ;

(A.3)

where lnk is a ratio depicting the shape change due to necking, given by

lnk ¼ A0 =A:

(A.4)

When V0 is relatively high, the mechanical power can be considerably greater than the rate of heat loss due to the air convection. In other words, the tensile extension test can be essentially adiabatic. In this limit during the steady neck front propagation, the measured temperature rise at the neck front can be used to estimate a portion of the internal energy change rate,

Acknowledgments

dU1 =dt ¼ rcp AVnk ðTnk  Ta Þ ¼ rcp A0 V0 ðTnk  Ta Þ=ðlnk e1Þ; This work is, in part, supported by NSF-DMR (EAGER-1444859) as well as ACS-PRF (54047-ND7). Appendix A. Theoretical analysis We need to carry out elementary yet necessary theoretical analysis that will be used to draw conclusions from the experimental data. The experiments involve simultaneous stress measurement and temperature profiling as a function of time at various extensional rates for three different types of specimens. For the untreated PC, the post-yield regime is characterized by stable neck front propagation, and there is considerable temperature rise in the neck front region. For milled and melt-stretched PC, uniform extension beyond the yield point is accompanied by a spatially homogeneous temperature increase. The analysis to evaluate the concurrent irreversible plastic and reversible elastic processes is different depending on whether the extension involves the strain localization or not. Thus, the following two subsections deal with each case separately. 1. Neck front propagation in untreated PC Tensile extension of ductile PC at room temperature amounts to performing mechanical work on the specimen, where one end of the specimen is held fixed and the other is displaced at the crosshead speed V0. For thermally quenched (untreated) PC that undergoes necking in post-yield regime, we need to carry out temperature profiling measurements using an IR camera for thermal imaging to determine the internal kinetic energy increase dU1 during ductile drawing when the neck fronts propagate steadily. After shear yielding to initiate necking, the neck front propagates at the expense of the un-necked parts of the specimen. Although there are elastic extension of a few percent away from the neck, we treat the un-necked parts as un-deformed to simplify the description. The conversion from the un-necked to the necked can be depicted in terms of the speed with which the neck grows in length. Assuming the neck length increases at a speed of Vnk at one front, then it is related to V0 as

(A.5a) where (Tnk  Ta) represents the temperature rise across the specimen during necking, with Ta denoting the ambient temperature. We can arrive at the same expression as eq (A.5a) by acknowledging that after extension the entire sample would have undergone a temperature rise of (Tnk  Ta) within a period of tneck, which is the time it takes for the neck front propagation to complete at an overall draw ratio of lnk, given by tneck¼ (L0/V0)(lnk  1). In other words, dU1/dt ¼ rcpA0L0[(Tnk  Ta)/tneck], which is eq (A.5a). Over the course of the extension until the completion of the necking, increase in the internal energy density u1 is simply

u1 ¼ U1 =A0 L0 ¼ rcp ðTnk  Ta Þ:

(A.5b)

The increase of U1 over time is to be compared with the mechanical power approximately given by

dW=dt ¼ sengrðnkÞ A0 V0 ;

(A.6a)

so that the work density at the end of necking completion is approximately

w ¼ sengrðnkÞ ðlnk  1Þ:

(A.6b)

During neck front propagation, the plateau engineering stress

sengr(nk) is nearly constant. This means that these two rates in eqs A.5 and A.6 are essentially constant. Thus, taking the ratio of eq (A.5a) to eq (A.6a) or eq (A.5b) to eq (A.6b), we have

U1 =W ¼ u1 =w ¼ rcp ðTnk  Ta Þ=ðlnk e 1ÞsengrðnkÞ:

(A.7a)

This expression gives a crude estimate of the ratio. A more actual one is given by

, Zlnk u1 =w ¼ rcp ðTnk  Ta Þ

sengr ðlÞdl;

(A.7b)

1

where the mechanical work density w is evaluated from the stress

152

P. Lin et al. / Polymer 89 (2016) 143e153

vs. strain data. Under adiabatic condition, the first law of thermodynamics states, in terms of the total internal energy u ¼ u1 þ u2,

  2Snf hðlnk  1Þ kðlnk  1Þ rcp ðTnk  Ta Þ U2 u2 ¼ ¼1 Z l 1þ þ nk rcp D0 V0 rcp A0 V0 W w sengr dl 1

u1 þ u2 ¼ w:

(A.8)

(A.16)

Therefore, the excess internal (potential) energy density u2 resulting from tensile extension can be evaluated from a combination of eqs A.7b and A.8 as

At high rates where the air cooling and lateral heat conduction corrections are negligible in eq (A.16) so that eq (A.16) reduces to eq (A.9), the maximum temperature rise is experimentally observed to occur away from the neck front. In applying eq (A.16), we will take Tnk to represent the observed maximum temperature in the extending specimen.

, Zlnk u2 =w ¼ 1  u1 =w ¼ 1  rcp ðTnk  Ta Þ

sengr ðlÞdl

(A.9)

1

In principle, u2/w is finite because ductile extension of polymer glasses may not be entirely plastic. Elastic yielding reported in the literature [70,71] is evidence that there is elastic stress in colddrawn polymer glasses. In general, it is necessary to compare the mechanical power with the air cooling rate that can be significant at low crosshead speeds. The heat loss at the neck front involves a surface area of 2Snf that is illustrated in Fig. 2. Air cooling is important when

2Snf hðTnk eTa Þ  dU1 =dt ¼ rcp A0 V0 ðTnk  Ta Þ=ðlnk e1Þ; (A.10) which can be rewritten to determine the threshold crosshead speed V0(ac) as

  . V0ðacÞ z2 h=rcp Snf A0 ðlnk  1Þ:

dw=dt ¼ sengr l_

(A.17)

where l_ ¼ V0/L0 is time-independent for a fixed V0. When V0 is relatively low, we also need to consider the air cooling effect. In terms of the rate of internal kinetic energy change per unit volume, du1/dt, given by

(A.18)

the first law of thermodynamics is of the following form,

rcp dT=dt þ du2 =dt ¼ sengr l_  ð2h=HÞðTs  Ta Þ (A.12)

2

Taking Snf ~ (D0) and A0 ¼ D0H0, where D0 and H0 is the width and thickness of the undeformed region, we have

V0ðacÞ zðD0 =H0 Þ mm=min:

Mechanically “rejuvenated” and sufficiently melt-stretched PC specimens both undergo homogeneous extension. For a uniformly extended sample, we can describe the mechanical work per unit volume per unit time as

du1 =dt ¼ rcp dT=dt (A.11)

Given h ¼ 25 J/(s,m2,K) and lnk ¼ 1.7, we have

 . V0ðacÞ z Snf A0 mm=min:

2. Uniform extension of milled and melt-stretched PC

(A.13)

Given typical values of our specimens, i.e., D0 ¼ 3.0 mm, and H0 ¼ 0.8 mm, we have V0(ac) ¼ 3.8 mm/min. Thus, for air cooling to have little effect, i.e., for the extension test to be adiabatic, we need V0 >> 4 mm/min. Similarly, we can estimate the effect of lateral heat conduction across the neck front, which is important when

(A.19)

where T can differ from Ts slightly [72]. Defining the work density w(l) as a function of the draw ratio l ¼ L/L0

Zl wðlÞ ¼

sengr ðl0 Þdl0

(A.20)

1

we can once again estimate the ratio of the excess energy increase u2 to the mechanical work as

8 9 Zt < =. 0 0 0 u2 =w ¼ 1  rcp ½TðtÞ  Ta  þ 2h dt ½Ts ðt Þ  Ta =Hðt Þ wðlÞ : ; 0

T  Ta T  Ta kA0 nk  rcp A0 V0 nk D0 lnk  1

Given k ¼ 0.20 W/(m,K), D0 ¼ 3.0 mm, and lnk ¼ 1.7, we have V0 ¼ 2.0 mm/min. Thus, this heat conduction along the stretching direction can be negligible when V0 >> 2.0 mm/min. Incorporating the contribution of air cooling and heat conduction along stretching direction, the first law of thermodynamics is given by, in the rate form,

dU1 dU2 dW T  Ta  2Snf hðTnk  Ta Þ  kA0 nk þ ¼ dt dt dt D0

(A.21)

(A.14)

(A.15)

where the first term on either side of eq (A.15) has been expressed in eq (A.5) and eq (A.6) respectively [72]. We can integrate eq (A.15) to estimate the fraction of the mechanical work W that is converted into U2. Specifically, from the combination of eqs A.5, A.6, and A.15, we have

where T can be set equal to Ts without causing a significant error [72]. Finally, it is insightful to divide the stress into two components, one plastic in nature, associated with change of the internal kinetic energy u1 characterized by the rising sample temperature and the other elastic in origin, corresponding to change of u2 that can be either inter-segmental or intra-segmental in origin. In either untreated PC that undergoes necking or milled PC that experiences homogeneous extension, we are interested in the ratio of the elastic component sengr(E) to the total stress sengr. For milled PC, we can rewrite eq (A.19) as

sengr ¼ sengrðEÞ þ sengrðVÞ

(A.22)

following the practice of Hoy and Robbins [50], where first term is given by

P. Lin et al. / Polymer 89 (2016) 143e153

sengrðEÞ ¼ du2 =dl ¼ sengr  rcp dT=dl  2hðT  Ta Þ



 Hl_ (A.23)

which can be estimated by measuring sengr and monitoring the temperature rise according to eq (A.23). 2 For the untreated PC, the necking results in a finite dU in eq dt (A.15). In steady state of neck front propagation, we have

dU2 =dt U2 u2 ¼ ¼ dW=dt W w

(A.24)

so that the engineering stress sengr(neck) resulting from the neck front propagation has its elastic component given by

sengrðEÞ ¼


dU2 1 U dW 1 u ¼ 2 ¼ 2 sengrðnkÞ A0 V0 dt A0 V0 W dt w

(A.25)

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