Stress and birefringence measurements during the uniaxial elongation ...

Stress and birefringence measurements during the uniaxial elongation of polystyrene melts ¨ ttinger D. C. Venerus,a) S.-H. Zhu, and H. C. O Department of Materials, ETH Zu¨rich, Institute of Polymers, CH-8092 Zu¨rich, Switzerland (Received 4 November 1998; final revision received 2 February 1999)

Synopsis A rheometer for generating uniaxial elongations in molten polymers has been modified to allow for the simultaneous measurement of stress and flow-induced birefringence. Tensile stress s and birefringence Dn 8 data in flows at constant strain rates up to 1 s21 were collected on a polydisperse polystyrene melt at temperatures of 160 and 170 °C. From these data, the stress-optic rule was followed for stresses below roughly 1 MPa. For stresses less than 1 MPa, the stress-optic coefficient u C u 5 u Dn 8 u / s was found to have a value of 4.831029 Pa21, which was independent of strain, strain rate, and temperature. At stress levels higher than 1 MPa, uCu decreased indicating a failure of the stress-optic rule. A criteria for failure of the stress-optic rule was formulated using simple arguments from network models and characteristic times suggested by the tube model. This criteria, which is based on the hypothesis that failure of the stress-optic rule is the result of significant chain stretching, was found to be consistent with the data reported in this study and with data from previous studies on polystyrene melts. © 1999 The Society of Rheology. @S0148-6055~99!01003-2#

I. INTRODUCTION The importance of studying the rheological behavior of polymeric liquids subjected to elongational, or shear-free, deformations is well known. Techniques and devices for generating controlled uniaxial elongations of molten polymers have been advanced significantly by Meissner and his co-workers over the last 30 years @Meissner ~1969!; Meissner and Hostettler ~1994!#. In the latest version of Meissner’s rheometer, known as the Rheometrics Melt Elongational ~RME! rheometer, elongational deformations are imposed on a molten sample that floats on a gas cushion by two pairs of opposed rotating belt clamps. These features of the RME make it possible to generate large, homogeneous deformations in polymer melts over a range of elongation rates and temperatures @Meissner and Hostettler ~1994!#. Uniaxial elongational flows are believed to be strongly aligning flows in that a high degree of orientational anisotropy in polymer chain segments can be induced. Uniaxial elongation at constant strain rate is also classified as a strong flow because this deformation history has a tendency to stretch polymer chain segments. The high degree of chain segment orientation and stretch induced by uniaxial elongation can result in rather draa!

Author to whom all correspondence should be addressed. Current address: Department of Chemical and Environmental Engineering and Center of Excellence in Polymer Science and Engineering, Illinois Institute of Technology, Chicago, IL 60616. Electronic mail: [email protected]

© 1999 by The Society of Rheology, Inc. J. Rheol. 43~3!, May/June 1999

0148-6055/99/43~3!/795/19/$20.00

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¨ TTINGER VENERUS, ZHU, AND O

matic rheological responses including the phenomenon known as strain hardening, where the elongational viscosity increases well above the linear viscoelastic prediction. Polymeric liquids undergoing deformation also exhibit optical anisotropy in the form of birefringence, which can be measured to gain insight into the relationship between molecular orientation and stress in deforming polymers @Fuller ~1995!#. Here we focus on the behavior of amorphous, molten polymers in uniaxial elongation. The Hencky strain is defined as e 5 ln(l), where the elongation ratio l 5 L/L 0 is the ratio of the sample length L to initial length L 0 . In the uniaxial elongation of constant density liquids, there is a single material function that can be measured called the tensile ~principal! stress difference: s 5 t 112 t 33 . The difference in the principal components of the refractive index tensor is called the flow-induced birefringence: Dn 8 5 n 11 2n 33 . The stress-optic rule ~SOR! states that the refractive index tensor n and the extra stress tensor t are linearly related through the stress-optical coefficient C. For uniaxial elongations, the SOR can be written as: C 5 Dn 8 / s . Arguments for the validity of the SOR are often based on the theory of rubber elasticity @Treloar ~1958!#, but can be easily generalized to entangled polymer liquids @Lodge ~1955!; Janeschitz-Kriegl ~1983!#. Let Q be the polymer chain segment connector vector for a chain segment between entanglement points. By considering the polarizability of chain segments in directions both parallel and perpendicular to Q and using the relationship between refractive index and polarizability, the refractive index tensor can be shown to be proportional to the second moment of Q: n } ^ QQ & . Here, ^...& indicates an average weighted by the distribution function for Q. The Kramers form @Bird et al. ~1987!# for the extra stress tensor t suggests a proportionality to the average of the force along the connector vector F and Q: t } ^ FQ & . For moderate levels of deformation, where stretching of chain segments is minimal, F is proportional to Q, t } ^ QQ & , and, hence, the SOR follows. For deformations that induce a high degree of stretch in chain segments, the linear relationship between F and Q is lost. In such cases the birefringence saturates while the stress continues to increase with increasing deformation causing C to decrease @Treloar ~1958!#. The SOR has been tested extensively on a variety of polymeric liquids and appears to have a rather wide range of applicability for amorphous polymer melts at temperatures well above the glass transition temperature T g . Studies involving shear deformations are numerous and indicate C is independent of stress and relatively insensitive to temperature, molecular weight, and molecular weight distribution @Janeschitz-Kriegl ~1983!#. Far fewer studies have been carried out in elongational flows and the general validity of the SOR is not as clear as for shear flows. However, violations of the SOR have been reported in uniaxial elongation flows when the stress exceeds roughly 1 MPa @JaneschitzKriegl ~1983!#. Violations of the SOR have also been reported for amorphous systems in the rubberto-glass transition region, which can be probed by experiments conducted at temperatures near the glass transition temperature ( , T g 120 °C), or at large frequencies or small times @Read ~1964!; Retting ~1979!; Read ~1983!#. A review of these phenomena is available @Osaki et al. ~1994!# where deviations from the SOR are attributed to localized perturbations of the polymer chain and are sometimes described as nonequilibrium effects @Janeschitz-Kriegl ~1983!#. More recent elongational flow measurements on polymers at temperatures near T g have been reported where rather large deviations from the SOR were observed @Muller and Pesce ~1994!; Kro¨ger et al. ~1997!#. The deviations from the SOR observed in systems near T g appear to be of a different nature than those resulting from the stretching of chain segments since the former have been reported for small strain deformations @Osaki et al. ~1994!# while the latter would occur only in large

STRESS AND BIREFRINGENCE DURING ELONGATION

797

strain deformations. In other words, stress-optical behavior at temperatures near T g cannot be simply related to behavior well above T g by invoking the time–temperature superposition principle. Here we briefly summarize previous studies in which stress and birefringence were measured during uniaxial elongational flow of amorphous polymer melts at temperatures above T g . The first study appears to be that of Matsumoto and Bogue ~1977! who conducted constant elongation rate experiments on a molten polystyrene over a temperature range of 120–157 °C. The rheometer used in this often cited study had a combination of a fixed clamp and a rotating gear clamp, and Hencky strains of 2.5 were achieved. The absolute value for the stress-optic coefficient uCu reported was 6.131029 Pa21; for s * 1 MPa, uCu was found to decrease slightly indicating a violation of the SOR @Matsumoto and Bogue ~1977!#. Muller and Froelich ~1985! used a device consisting of two fixed clamps for the simultaneous measurement of stress and birefringence during elongational flows. From constant strain rate experiments up to Hencky strains of 2.5 conducted on a molten polystyrene at temperatures ranging from 126 to 146 °C, uCu was found to be 4.731029 Pa21 with deviations from the SOR for s * 2 MPa. Note that the stress-optic coefficient for polystyrene measured in shear flows is negative with an absolute value in the range 4.0– 5.031029 Pa21 @Janeschitz-Kriegl ~1983!#. Koyama and Ishizuka ~1989! studied low-density polyethylene at 120 °C using a rheometer configured with a pair of rotating gear clamps. These experiments were conducted to a Hencky strain of 3.0 and the SOR was satisfied to the maximum stress achieved, which was below 1 MPa. Hoppler et al. ~1995! measured stress and birefringence in extensional flows of a polystyrene melt and reported a stress-optic coefficient that depended on time and strain rate. However, these authors noted it was necessary to employ time- and strain ratedependent corrections to their data to account for significant deviations from ideal uniaxial deformation. In a more recent study, Kotaka et al. ~1997! modified an RME device to allow for simultaneous stress and birefringence measurements. Results were reported on a polystyrene melt at 150 °C and a low-density polyethylene melt at 140 °C over a range of strain rates to Hencky strains of 3.0. For the low-density polyethylene melt, stresses of roughly 1 MPa were achieved and the SOR was found to hold. Results for the polystyrene melt are somewhat ambiguous since C was found to be a function of strain rate and to increase with increasing stress at a given strain rate. Both of these observations deviate from the findings of other investigators and the expected departure from the SOR; and were attributed in part to difficulties with the optical system @Kotaka et al. ~1997!#. It appears that relatively few studies have reported simultaneous measurement of the stress and birefringence in polymer melts undergoing uniaxial elongations. In several previous studies the test temperature was rather close to T g ~or the polymer melting temperature! and modest Hencky strains, corresponding to a l of 10–20, were achieved. Also, for the most frequently studied material ~polystyrene!, there are inconsistencies in reported deviations from the SOR. Hence, it is difficult to ascertain if reported failures of the stress-optic rule are the result of chain segment stretching, the result of tests being conducted close to T g , or due to some other factor. Given the widespread use of flow birefringence measurements to determine stress fields in complex polymer flows @Baaijens et al. ~1995!; Quinzani et al. ~1994!; Li et al. ~1998!#, which often have a significant elongational content, a better understanding of conditions leading to failures of the SOR would seem essential. The objective of this investigation is to develop an apparatus and method to measure the tensile stress s and birefringence Dn 8 during the uniaxial elongation of molten polymers over a range of strain rates and to large Hencky strains. This apparatus will be


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FIG. 1. Schematic diagram of modification of RME for optical experiments.

used to collect data on an amorphous polymer melt ~polystyrene! at temperatures well above T g ~100 °C!. With this new data set, and previously published data on polystyrene melts, we will examine the SOR and its failure paying particular attention to flows with large Hencky strains generated by high strain rates. II. EXPERIMENT Experiments were conducted on an RME rheometer modified for optical measurements. A detailed description of the RME can be found elsewhere @Meissner and Hostettler ~1994!#. Deformation of samples having rectangular cross section is imposed by rotating belt clamps, one of which is mounted on a force transducer. In the most recent version of the RME, a check of deformation homogeneity can be made from a video recording of the elongating sample taken through a large glass window on the top of the instrument. The video image is digitized and analyzed so that the time-dependent position of tracer particles sprinkled on the molten sample just before the test can be determined @Rohr ~1996!#. Modifications of the RME rheometer for optical measurements were made so that a laser beam could pass through the elongating sample. The most significant alteration involved the sample support table, through which a beam tube was inserted, as shown in Fig. 1. One end of the beam tube, having an inside diameter of 2.5 mm, was flush with the top surface of the porous metal frit; the other end, with an inside diameter of 4.0 mm, extended below the lower wall of the RME. The beam tube allowed the beam to pass


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FIG. 2. Schematic diagram of optical setup for birefringence experiments.

through the sample and out of the rheometer while maintaining the integrity of the gas cushion sample support system. To minimize heat loss and undesirable gas flow through the beam tube, a small window of good quality optical glass was attached to the end of the beam tube below the RME. A small optical glass window was also integrated with the large window on the top of the RME. Removable pinholes at the ends of the beam tube were used to ensure the beam passed through the center of the beam tube and normal to the sample. Birefringence measurements were made using a simple optical system shown in Fig. 2. A rigid steel frame, mounted on a heavy table, was built such that the optical components and the RME ~elevated relative to the table! could be securely mounted to it. The overall system consisted of optical elements above and below the RME whose function will be described below. On the upper optical train, mirrors M1 and M2 were used to steer the beam produced by a 2 mW HeNe laser through pinholes ~PH1 and PH2! that ensured the beam passed through a convex lens ~CL! ~focal length of 30 cm! and ~Glan–Thompson! polarizer P1. The convex lens focused the beam so that its diameter at the sample was approximately

800


0.5 mm. The orientation angle of the linearly polarized beam produced by P1 was arbitrarily specified as zero. This beam entered the RME and passed through the sample S, having retardance d8 given by

d8 5

2pdDn8 lHeNe

,

~1!

where d is the sample thickness, Dn 8 is the birefringence, and l HeNe is the wavelength of the HeNe laser (l HeNe 5 633 nm). The orientation of the extinction angle of sample S with respect to the probe beam polarization was fixed at p/4. After passing through the sample, the beam exited the RME and was redirected by mirror M3 so that it would pass through the optical elements below the RME. After passing through PH3, the beam was split by a 50/50 nonpolarizing beamsplitter ~BS!. One beam passed through a second polarizer P2 oriented at an angle of a 5 p /2 ~or zero for calibration of the system! with respect to P1, then to photodiode D1 ~Hamamatsu, model S2281!. The second beam produced by the BS passed directly to a second photodiode D2. The output current from the photodiodes were sent to photosensor amplifiers ~Hamamatsu, model C2719! producing voltages proportional to the intensities of the beams striking D1 and D2: V 1 5 C 1 I D1 ; V 2 5 C 2 I D2 , where C 1 and C 2 are constants. Voltages V 1 and V 2 were sent to a 12 bit analog to digital ~A/D! converter in a personal computer. Relationships between beam intensities at various locations along the optical train can be found using Jones or Mueller calculus @Fuller ~1995!#. For the optical train shown in Fig. 2, it is easy to show that ID

1

I0

5

f 4

@11cos~2a!cos~d8!#,

~2!

where I 0 is the output intensity of the HeNe laser and f is the fraction of the beam intensity passing through the BS. In principle, it would be possible to determine C 1 , I 0 , and f and use Eq. ~2! with a 5 p /2 to find a relationship between the measured voltage V 1 and the retardance d8. Measurement of the absolute intensity ~voltage! to extract retardance would not, however, be successful for the rheometer used in this study. Unlike previous studies where the sample was surrounded by an oil bath, in this study the sample is surrounded by a gas. This is significant because the refractive index of a typical oil is much closer to the refractive index of a molten polymer sample than is the refractive index of a gas. Consequently, refraction of the probe beam at polymer-melt/ambient-fluid interfaces was a more serious problem in this study. Ideally, the probe beam is centered and has normal incidence with respect to the sample so that deviations of the beam should be minimal. However, the sample can undergo small lateral displacements during elongation. This problem becomes particularly acute at large strains where the width of the sample approaches the diameter of the probe beam. Hence, refractive effects cause an artificial time dependence of the transmitted beam intensity not due to the retardance of the sample. It is interesting to note that the flowing gas sample support system used in the RME, which is advantageous for mechanical measurements relative to an oil bath, actually creates a problem for optical measurements. A solution to the beam refraction problem described above was to split off and measure a portion of the transmitted beam. This second beam created by the BS was used as a reference signal to normalize the spurious fluctuations in the intensity. It was also necessary to use a pinhole just before the BS ~PH3 in Fig. 2! to maximize the fraction of


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the beam passing through the remainder of the optical system that had passed cleanly through the sample. For the intensity at D2, it is easy to see that I D2 /I 0 5 (12 f )/2. Hence, Eq. ~2!, with a 5 p /2 can be written as V1 V2

K

5

2

@12cos~d8!#,

~3!

where K 5 f C 1 /(12 f )C 2 . K was determined by measuring V 1 /V 2 with the polarizers aligned ( a 5 0) and no sample ( d 8 5 0) and found to have an approximate value of 1.2. We note that with the polarizers crossed ( a 5 p /2) and no sample ( d 8 5 0), the voltage ratio V 1 /V 2 was approximately 0.02 making the maximum extinction ratio for the optical system roughly 60. This rather small extinction ratio is most likely the result of parasitic birefringence in the windows and mirrors of the optical system. Because the focus of this work was on situations where Dn 8 was rather large, no attempts were made to compensate for imperfections in the optical elements. The tensile stress is the tensile force F divided by the cross-sectional area of the sample

s5

F Wd

~4!

,

where W is the width and d the thickness, respectively, of the sample. For constant volume, uniaxial elongational flows at constant Hencky strain rate e˙ 0 , we have W W0

5

d d0

5 exp~2e˙ 0t/2! ,

~5!

where W 0 and d 0 are the width and thickness, respectively, of the sample before deformation. The ratio of the birefringence to tensile stress difference defines the stress-optic coefficient uCu 5

uDn8u

s

.

~6!

The absolute value is taken since the technique described above is unable to resolve the sign of Dn 8 . Finally, we note that for high stresses leading to large birefringences, the retardance d8 will pass through values of p,2p,3p,... or multiple orders, causing V 1 /V 2 to go through values of K, zero, K,... as required by Eq. ~3!. It is also evident from Eqs. ~1!, ~5!, and ~6! that if C is constant, it is possible for d8 to go through a maximum. Thus, care must be taken in analyzing the evolution of V 1 /V 2 with time to track the number of maxima and minima so that d8 can be determined correctly from Eq. ~3!. In this study, a commercial polystyrene supplied by BASF ~Ludwigshafen, Germany! with the designation PS 158 K was used. For PS 158 K, the weight–average molecular weight M w 5 335 000 and M w /M n 5 2.85. Linear viscoelastic characterization of PS 158 K was made by performing small-shear-strain oscillatory tests over a range of frequencies at temperatures of 150, 170, and 190 °C. Shear storage G 8 and loss G 9 moduli data were shifted to a temperature of 170 °C using the principle of time–temperature superposition. A discrete relaxation spectrum ~with 13 relaxation times shown in Table I! was fit to the shifted G 8 and G 9 data and is represented by the solid lines in Fig. 3. From


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TABLE I. Relaxation spectra for PS 158 K at 170 °C. i

t i ~s!

g i ~Pa!

1 2 3 4 5 6 7 8 9 10 11 12 13

0.001 580 0.004 810 0.014 64 0.044 56 0.1356 0.4129 1.257 3.826 11.65 35.45 107.9 328.5 1000

199 300 371 15 171 20 405 40 327 20 242 10 183 90 100 10 3598 1263 147.6 45.88 6.372

these data we estimate the zero-shear-rate viscosity ( 5 S i g i t i ) to be 2.03105 Pa s and the average relaxation time ( 5 S i g i t 2i /S i g i t i ) to be 75 s for PS 158 K at 170 °C. Samples were prepared by extruding a small quantity of polymer through a die and into the shape of a blob of molten polymer. The blob was then placed between heated glass plates on a laboratory press, slowly pressed into a plate, and allowed to cool slowly under pressure. The smooth surface of samples pressed between glass plates was found to

FIG. 3. Shear storage G 8 (h) and loss G 9 (s) moduli for PS 158 K polystyrene melt at 170 °C. Solid lines are fit of linear viscoelastic spectrum given in Table I.


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FIG. 4. Particle tracking results for elongation tests at a constant rate of 0.1 s21 on PS 158 K at 170 °C made on standard ~right! and modified ~left! RME devices.

make a considerable difference in optical experiments due to a reduction in scattered light compared with samples pressed using metal plates. Solid polymer plates produced by this method had a nominal thickness of 1.5 mm and were machined into samples having widths of either 7.0 or 10.0 mm and nominal lengths of 56 mm. As mentioned earlier, a check of the deformation homogeneity could be made using a particle tracking analysis of video images of the deforming sample. Particle pathlines from a typical experiment performed on a standard RME are shown in Fig. 4 for a strain rate of 0.1 s21. From Fig. 4 it appears as though the deformation is indeed a homogeneous uniaxial elongation. Also shown in Fig. 4 are particle pathlines from an experiment performed on an RME modified as described above. Comparison of these images indicates that the homogeneity of the flow is not compromised by the modifications made to the sample support system. Strain rates as functions of time calculated from the particle tracking analysis in three different zones of the sample are plotted in Fig. 5 for an imposed strain rate of 0.1 s21. From Fig. 5 it appears that the actual deformation rate in the sample is uniform and within 3% of the imposed value. ~Note that the high frequency fluctuations in the calculated strain rate are an artifact of the tracking procedure used.! This means the sample thickness d can be determined accurately using Eq. ~5!.

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FIG. 5. Actual Hencky strain rate and strain from particle tracking analysis for elongation test at a constant strain rate of 0.1 s21 on PS 158 K at 170 °C made on a modified RME. Strain rate data are shown by symbols connected by lines for left ~n!, middle ~h!, and right ~s! thirds of sample and solid line is strain based on average strain rate. Dashed lines indicate specified strain rate and strain.

The procedure used to carry out tests on the RME was similar to that described in the article by Meissner and Hostettler ~1994!. In this study, alignment of the optical elements was performed before each test. Extra care was used in loading the samples to ensure the HeNe beam passed through the center of the sample. Samples were allowed to melt for 10 min during which time the intensities at detectors D1 and D2 were monitored to ensure the sample was of good optical quality and did not become distorted during melting. Tests were conducted at temperatures of 160 and 170 °C and reported data are the average of at least-two repeat experiments. III. RESULTS AND DISCUSSION Constant strain rate experiments were carried out to a Hencky strain of approximately four. Figure 6 shows the growth of the viscosity in uniaxial extension h u 5 s / e˙ 0 at 170 °C for five values of e˙ 0 ranging from 0.01 to 1.0 s21. Also shown in Fig. 6 ~solid line! is the linear viscoelastic prediction calculated using the relaxation spectrum in Table I. It appears that the data and linear viscoelastic prediction coincide at small strains for each strain rate. At large strains, rather significant deviations between the data and solid line are apparent and appear to occur at strain values that decrease slightly with increasing e˙ 0 . The behavior of the PS 158 K material shown in Fig. 6 is consistent with previously reported results on other polystyrene materials. We also note that repeat runs, which numbered from three to five in this case, were within 2.5% of the reported average values for the tensile stress difference s. As discussed in Sec. II the intensity of the beam passing through the second polarizer and striking the detector I D1 was normalized to compensate for anomalous intensity


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FIG. 6. Growth of elongation viscosity for PS 158 K at 170 °C at constant strain rates ~s21! of 0.01 ~n!; 0.03 ~L!; 0.1 ~h!, 0.3 ~,!; 1.0 ~s!. Solid lines are linear viscoelastic prediction using spectrum given in Table I.

fluctuations. To demonstrate how this approach works, Figs. 7 and 8 show raw data for runs with e˙ 0 equal to 0.1 and 1.0 s21, respectively, in the form of voltages V 1 and V 2 . Ideally, V 2 should be independent of time, but is not as a result of imperfect interactions of the probe beam and the sample. Clearly, the ratio V 1 /KV 2 ~shown by the dasheddotted lines in Figs. 7 and 8! is much smoother than V 1 because the irregular fluctuations have been normalized by the reference signal V 2 . We note in passing that a similar technique called variance reduction is used to reduce noise in Brownian dynamics simulation results. The evolution of the retardance d8 and birefringence Dn 8 for tests at three strain rates is shown in Fig. 9. It should be noted that the optical data shown in this plot, like the stress data in Fig. 6, are collected to a Hencky strain of approximately four, which corresponds to l ' 50. For each strain rate, the retardance goes through a maximum while the birefringence appears to increase for all times. From Eq. ~1! it is clear that the maximum in d8 is explained by Dn 8 initially increasing faster and then later increasing slower than the rate of decrease in sample thickness d. For the highest strain rate ~1.0 s21!, the retardance reaches a maximum value of approximately 10 before decreasing. Hence, d8 goes through three orders ~3p! and this results in the first three maxima/ minima shown in Fig. 8. The shallow minimum corresponds to the point where d8 goes through its maximum value; the remaining maximum and minimum correspond to d8 passing back through 3p and 2p. Finally, we note that repeat runs were within 5% of the reported average values for the birefringence Dn 8 . Figure 10 shows a log–log plot of the birefringence Dn 8 versus the tensile stress s for data collected at five e˙ 0 values ranging from 0.01 to 1.0 s21 at 170 °C. The error bars

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FIG. 7. Photodetector voltages vs time for signal V 1 ~solid!, reference V 2 ~dashed!, and normalized ratio V/KV 2 ~dashed-dot! for elongation test at a constant rate of 0.1 s21 on PS 158 K at 170 °C.

FIG. 8. Photodetector voltages vs time for signal V 1 ~solid!, reference V 2 ~dashed!, and normalized ratio V 1 /KV 2 ~dashed-dot! for elongation test at a constant rate of 1.0 s21 on PS 158 K at 170 °C.


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FIG. 9. Evolution of retardance and birefringence for elongation tests on PS 158 K at 170 °C at constant strain rates ~s21! of 0.01 ~n!, 0.1 ~h! and 1.0 ~s!. Symbols with dots inside are retardance d8 and open symbols are birefringence Dn 8 .

shown in Fig. 10 were estimated in the following manner. From the value of the extinction ratio given above as 60, the minimum retardance d8 that can be resolved is approximately 0.25; this corresponds to an uncertainty in the birefringence Dn 8 of approximately 231025 . According to the stress-optic rule, Eq. ~6!, the plot shown in Fig. 10 should have a slope of one, which is the slope of the straight line in this figure. The vertical position of the line in Fig. 10 was determined from data at stress levels below 5 3105 Pa. From Fig. 10 it appears that for s & 53105 Pa, the SOR is valid and from these data uCu is found to be 4.831029 Pa21. This value is in good agreement with those reported previously for polystyrene in both shear @Janeschitz-Kriegl ~1983!# and uniaxial elongation @Muller and Froelich ~1985!#, which fall in the range (4 – 5)31029 Pa21. We note that the systematic deviations of the data from the straight line shown in Fig. 10 for Dn 8 & 1024 are most likely caused by parasitic birefringence in various components of the optical train, but are consistent with the SOR within the estimated error in the data. A parallel set of experiments was conducted at a temperature of 160 °C. At this temperature the zero-shear-rate viscosity is roughly 7.03105 Pa s and the average relaxation time roughly 260 s. At 160 °C, the growth of uniaxial elongational viscosity h u and birefringence Dn 8 are qualitatively similar to those shown in Figs. 6 and 9, respectively. Figure 11 shows a log–log plot of Dn 8 vs s for data collected over the same range of e˙ 0 values at 160 °C. The results shown in Fig. 11 are similar to those obtained at 170 °C shown in Fig. 10. From Fig. 11 it appears that for s & 13106 Pa, the SOR is valid with the same absolute value for the stress-optic coefficient (4.831029 Pa21) as that found for 170 °C. The independence of C with temperature is consistent with results found in previous studies on polystyrene melts @Janeschitz-Kriegl ~1983!#.

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FIG. 10. Birefringence vs stress for constant rate elongation tests on PS 158 K at 170 °C. Different symbols are for different strain rates as given in caption to Fig. 6. Solid line has a slope of one and gives a stress-optic coefficient u C u 5 4.831029 Pa21.

At high stress levels ( s ; 1 MPa) the data shown in Figs. 10 and 11 show deviations from the SOR that are not within the estimated error. To see this more clearly, the data of Figs. 10 and 11 are replotted on a linear scale in Fig. 12. The straight line in Fig. 12 has as slope of 4.831029 Pa21; the error bars reflect the uncertainty in Dn 8 measurements from repeat experiments, which is 5%. The gradual deviation from the SOR at a stress level of 0.5–1.0 MPa shown by the data is consistent with the observations of Matsumoto and Bogue ~1977! and Muller and Froelich ~1985!, but at variance with the more recent results of Kotaka et al. ~1997!. It should also be noted that these deviations cannot be explained by error in the sample thickness d, since these would, according to the results shown in Fig. 5, result in an error of roughly 5% in Dn 8 values. From the results reported in this study it seems fair to conclude that the failure of the SOR for molten polystyrene is not caused by factors associated with the polymer being near its glass transition temperature. Rather, it would seem likely that the observed deviation from the SOR at high elongation rates is the result of the high degree of chain segment stretch induced by the flow. IV. CRITERIA FOR FAILURE OF THE STRESS-OPTIC RULE A. Theoretical development Here we develop an expression to estimate the critical stress s c above which the SOR is no longer valid. The development is based on concepts from the network theory for rubber elasticity, which are discussed at length by Treloar ~1958!. The application of ideas for systems composed of permanent entanglements to systems of temporary en-


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FIG. 11. Birefringence vs stress for constant rate elongation tests on PS 158 K at 160 °C. Different symbols are for different strain rates as given in caption to Fig. 6. Solid line has a slope of one and gives a stress-optic coefficient u C u 5 4.831029 Pa21.

tanglements ~polymer melts! is only possible in the high-strain-rate limit. Hence, to quantify the permanence of the entanglement points, we use time scales for chain dynamics suggested by the tube model. According to the tube model @Doi and Edwards ~1986!#, the most important features of polymer chain dynamics are described by two times constants. Reptative motion corresponding to the relaxation of flow-induced orientation of chain segments occurs on a time scale t d ; and the relaxation of flow-induced chain stretching occurs on the much shorter time scale t R . The reptation picture suggests that the ratio of relaxation times is given by t d / t R 5 3Z, where Z is the number of entanglements per chain @Doi and Edwards ~1986!#. Z can be estimated from the ratio M w /M e , where M e is the molecular weight between entanglement points. At high strain rates ( e˙ 0 t R . 1) the entanglement points are effectively permanent and chain stretching can occur. In the affine deformation limit we assume the fluid behavior is neo-Hookean: s c . G Nl 2c , where G N is the plateau modulus and l c is the critical elongation. The length of an affinely deformed chain segment can be approximated by Q . a KAN Kl, where there are N K Kuhn steps of length a K between entanglement points @Treloar ~1958!#. We now assume that the linear relationship between F and Q is lost when a chain segment is extended to a length comparable to its fully extended length of a KN K so that l c . AN K . For the plateau modulus we use the result @Treloar ~1958!#: G N 5 r RT/M e , which is also used frequently for monodisperse polymer melts @Onogi et al. ~1970!; Ferry ~1980!#. Here r is the mass density, R is the gas constant and T is the absolute temperature. Combining these ideas, the criteria for s c is given by

810


FIG. 12. Composite linear plot of birefringence vs stress for constant rate elongation tests on PS 158 K. Different symbols are for different strain rates as given in caption to Fig. 6 with open and filled symbols for 170 and 160 °C, respectively. Solid line has a slope equal to the stress-optic coefficient u C u 5 4.8 31023 MPa21.

sc 5

S D Mw Mn

2p

rRT M Kuhn

,

~7!

where M Kuhn is the molecular weight of a Kuhn step. The front factor involving the ratio of molecular weights in Eq. ~7! is included to take into account the effect of polydispersity on G N @Ferry ~1980!# where p . 0. The expression given in Eq. ~7! suggests s c is relatively insensitive to changes in temperature and depends on chain stiffness through M Kuhn and on polydispersity. B. Experimental evaluation Table II summarizes the results of studies where stress and birefringence were measured during constant rate uniaxial elongation flows of molten polystyrene. Estimation of t d for polydisperse systems is difficult, but a reasonable value is the average relaxation time, which is 75 s for PS 158 K at 170 °C. Estimation of t R is made from t R 5 t d/3Z, where Z 5 M w /M e and M e 5 18 000 for polystyrene @Onogi et al. ~1970!; Ferry ~1980!#. Values of t d and t R for the systems considered in other investigations were estimated using values for PS 158 K at 170 °C and established correlations to account for differences in temperature and molecular weight. Also, we have chosen to report data from previous studies at temperatures greater than 130 °C to ensure the data are not affected by factors associated with T g and to limit extrapolation errors associated with estimating relaxation times.


811

TABLE II. Summary of stress-optical experiments for elongational flows of polystyrene. M w ~g/g mol!

M w /M n

T ~°C!

t d ~s!a

t R ~s!

e˙ 0 t R

lcb

SOR failure?

This study

335 000

2.85

Matsumoto and Bogue Mueller and Froelich Kotaka et al.

283 000

4.6

250 000

2.44

275 000

1.15c

170 170 160 160 157 140 146 136 150 150

75 75 260 260 215 3700 800 5600 450 450

1.3 1.3 4.6 4.6 4.6 78 19 130 10 10

1.3 0.4 4.6 1.4 0.3 5.9 1.9 13 10 1.0

33 ~55! 10 33 ~12! 12 ~12! 8 — —

yes no yes yes no yes no yes yesd yesd

Ref.

Estimated using t d 5 75 s and scaling for molecular weight using t d } M 3.4 w and temperature using the WLF equation @Ferry ~1980!#: log@td(T)/ t d(100) # 5 214(T2100)/(501T2100). b Extension at which SOR failed; ~..! indicates maximum extension for test if SOR did not fail. c Because of the low M w /M n of this material, t d was estimated from published data @Onogi et al. ~1970!# on a similar system at 160 °C. d Deviations from SOR showed trend opposite those reported in other studies listed in table.

a

The results shown in Table II suggest that @in all but the study of Kotaka et al. ~1997!# conditions for the failure of the SOR can be consistently described using the simple argument discussed above. For significant chain stretching to occur, the strain rate must be greater than the reciprocal of the Rouse time for the chain, or e˙ 0 t R . 1; relatively large stretches must be achieved, or l @ 1. Together, these conditions indicate that failure of the SOR will occur in polymer melts when large strains are generated by high-strain-rate extensional flows. We now evaluate the expression in Eq. ~7! for s c . For narrow molecular weight polystyrene, the value reported by Onogi et al. ~1970! for G N is 0.2 MPa. For PS 158 K with M w /M n 5 2.85, we assume G N . G 8 ( v ) 5 G 9 ( v ) 5 0.025 MPa ~see Fig. 3!, which corresponds to p . 2. For polystyrene, there are roughly 180 monomers between entanglement points and we use an estimate of 10 monomers per Kuhn step @Ferry ~1980!# so that N K . 18. The molecular weight of a Kuhn step is given by M Kuhn 5 M e /N K 5 1000. Using these values, we estimate for PS 158 K that s c 5 0.4 MPa, which is consistent with the results shown in Fig. 12. The dependence of s c ~through G N) on molecular weight distribution might also explain the difference in s c values reported in other studies. For the study of Muller and Froelich ~1985! where M w /M n 5 2.44, s c 5 0.6 MPa. In fact, for the nearly monodisperse polystyrene (M w /M n 5 1.15) used in the study of Kotaka et al. ~1997!, we find using Eq. ~7! that s c 5 3 MPa, which is above the stress level achieved in their experiments. As noted above the results of Kotaka et al. ~1997! are qualitatively different than those reported in this and other studies. The reported increase of uCu at high strains for several strain rates could be explained by the following scenario. Just before the sample ruptures, necking causes a reduction of cross-sectional area and, consequently, measured force. If this failure point is not near the laser beam, birefringence measurement would be possible up to the point in time when the sample ruptures. Necking would therefore cause the measured stress s to decrease while the birefringence Dn 8 in the sample has not relaxed appreciably causing an anomalous increase in uCu. This argument is speculative, but consistent with the fact that more tenable results were obtained @Kotaka et al. ~1997!# on a low-density polyethylene melt, which is more stable in elongational flows. The same device has been used recently to examine the mechanical and optical behavior of linear

812


low-density polyethylene @Okamoto et al. ~1998a!#, and blends of low-density polyethylene with high molecular-weight polyethylene @Okamoto et al. ~1998b!#. V. CONCLUSIONS An existing rheometer for generating uniaxial elongations in polymer melts has been modified to allow for simultaneous measurement of the tensile stress s and birefringence Dn 8 . A simple optical arrangement in which the deforming sample was located between crossed polarizers was used. To overcome anomalous intensity fluctuations caused by imperfect interaction of the probe beam with the sample, a reference beam was used to normalize the transmitted intensity. This study differs in several important ways from previous studies. The simple but novel optical system used in this work made interpretation of raw optical data rather straightforward and unambiguous. Second, we have carefully checked the homogeneity of the deformations so that data collected at large strains were reliable. Finally, the experiments conducted in this study were made at temperatures well above the glass transition and in the range encountered during processing. The new apparatus was tested on a polydisperse polystyrene material over a range of strain rates up to 1.0 s21 at temperatures of 160 and 170 °C. Moderate levels of strain hardening in the uniaxial extensional viscosity were observed. Plots of birefringence Dn 8 versus tensile stress s were used to examine the stress-optic rule, which was found to be valid for stress levels below 1 MPa. From data below this stress, the ~absolute value of the! stress-optic coefficient uCu was estimated to be 4.831029 Pa21, a value in good agreement with previous investigations on polystyrene melts. For s greater than 1 MPa, uCu decreased indicating a violation of the stress-optic rule. An expression for the critical stress s c above which deviations from the SOR are observed was formulated using the network model picture. This expression indicates that s c is inversely proportional to the molecular weight of a Kuhn step, M Kuhn , which is related to the stiffness of a polymer chain. The expression for s c is based on the argument that pronounced chain stretching is responsible for failure of the SOR, which will occur at large strains in high strain rate extensional flows, or, l @ 1 and e˙ 0 t R . 1, where t R is the relaxation time of an equivalent Rouse chain. The proposed criteria were found to be consistent with the new data set and with previously published data for molten polystyrenes. As a final remark, we mention that aside from polystyrene, stress and birefringence data in uniaxial elongation are available for only a few other polymer melts. Two data sets have been obtained on low-density polyethylene melts @Koyama and Ishizuka ~1989!; Kotaka et al. ~1997!# and the SOR was found to hold for stresses up to 1 MPa. Muller and Pesce ~1994! conducted experiments on a polycarbonate melt and found the SOR failed at a stress of 4 MPa. This limited body of data seems to suggest that for molten polymers chain architecture ~i.e., chain branching and stiffness! plays a significant role is the extensibility of chains. ACKNOWLEDGMENTS One of the authors, D.C.V., is grateful to the Institut fu¨r Polymere at the ETH, Zu¨rich for support during his sabbatical. The authors are grateful to Dr. T. Schweizer, J. Hostettler, W. Schmidheiny, and F. Mettler for their assistance in conducting this study and to Professors W. R. Burghardt, J. Meissner, J. D. Schieber, and P. Schurtenberger for helpful discussions.


813

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