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JOURNAL OF POLYMER SCIENCE Polymer Physics Edition VOL. 15, 1663-1674 (1977) Stress Birefringence in Amorphous Polymers under Nonisothermal Conditions TAKAYOSHI MATSUMOTO and DONALD C. BOGUE, Department of Chemical, Metallurgical, and Polymer Engineering, The University of Tennessee, Knoxville, Tennessee 37916 Synopsis The relationship of birefringence to stress in an amorphous polymer was studied, with emphasis on conditions of high stress and rapid cooling. The latter (nonisothermal) conditions are important in connection with studies of polymer processing operations. Polystyrene was pulled at a constant elongation rate (0.075 to 2 sec-l in the present and related work) under both isothermal conditions (in the range 120 to 157OC) and nonisothermal conditions (with cooling rates in the range 0.6 to 1.7OC/sec). Generally we conclude that stress is proportional to birefringence under a wide range of conditions, except that a nonlinear regime appears at stresses higher than about lo’ dyn/cm2. In this regime, stress increases more rapidly with deformation than does birefringence. INTRODUCTION The present studies, comprising the measurement of birefringence in amor- phous polymers under conditions of rapidly changing temperature, accompany similar studies in nonisothermal rheological response.1*2 We are interested in nonisothermal response both as a basic problem in rheology and as an applied problem in polymer processing. The idea of time-temperature superposition is, of course, well developed in rheology; however, simultaneous deformations and rapid temperature changes have been studied only quite recently. Therefore in the models now available for actual processing operations one turns to a very simple rheological description-that of a temperature-dependent viscosity function multiplied by a deformation rate-and neglects elastic effects; this is done, for example, in analyses of melt spinning314and injection rn~lding.~ Clearly, however, elastic effects cannot be neglected when the material is de- formed rapidly. . In other papers we have dealt with transient stress responses while simulta- neously pulling the samples and cooling them, in the one case pulling them at constant crosshead speed1and in another case pulling them at a constant elon- gation rate.2 The former is an experiment at dL/dt = const and the latter is an experiment a t (l/L)(dL/dt) = const, where L(t) is the instantaneous length of the sample. To deal with the changing temperature history we proposed a generalized viscoelastic theory of the form 0 1977 by John Wiley & Sons, Inc.

Stress birefringence in amorphous polymers under nonisothermal conditions

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JOURNAL OF POLYMER SCIENCE Polymer Physics Edition VOL. 15, 1663-1674 (1977)

Stress Birefringence in Amorphous Polymers under Nonisothermal Conditions

TAKAYOSHI MATSUMOTO and DONALD C. BOGUE, Department of Chemical, Metallurgical, and Polymer Engineering, The University of

Tennessee, Knoxville, Tennessee 37916

Synopsis

The relationship of birefringence to stress in an amorphous polymer was studied, with emphasis on conditions of high stress and rapid cooling. The latter (nonisothermal) conditions are important in connection with studies of polymer processing operations. Polystyrene was pulled at a constant elongation rate (0.075 to 2 sec-l in the present and related work) under both isothermal conditions (in the range 120 to 157OC) and nonisothermal conditions (with cooling rates in the range 0.6 to 1.7OC/sec). Generally we conclude that stress is proportional to birefringence under a wide range of conditions, except that a nonlinear regime appears at stresses higher than about lo’ dyn/cm2. In this regime, stress increases more rapidly with deformation than does birefringence.

INTRODUCTION

The present studies, comprising the measurement of birefringence in amor- phous polymers under conditions of rapidly changing temperature, accompany similar studies in nonisothermal rheological response.1*2 We are interested in nonisothermal response both as a basic problem in rheology and as an applied problem in polymer processing. The idea of time-temperature superposition is, of course, well developed in rheology; however, simultaneous deformations and rapid temperature changes have been studied only quite recently. Therefore in the models now available for actual processing operations one turns to a very simple rheological description-that of a temperature-dependent viscosity function multiplied by a deformation rate-and neglects elastic effects; this is done, for example, in analyses of melt spinning314 and injection rn~lding.~ Clearly, however, elastic effects cannot be neglected when the material is de- formed rapidly. .

In other papers we have dealt with transient stress responses while simulta- neously pulling the samples and cooling them, in the one case pulling them at constant crosshead speed1 and in another case pulling them at a constant elon- gation rate.2 The former is an experiment at dL/dt = const and the latter is an experiment at ( l /L) (dL/d t ) = const, where L ( t ) is the instantaneous length of the sample. To deal with the changing temperature history we proposed a generalized viscoelastic theory of the form

0 1977 by John Wiley & Sons, Inc.

1664 MATSUMOTO AND BOGUE

where aij is the stress, p is the isotropic pressure, c;' is the Finger strain tensor, and the G: and T~ are, respectively, the moduli and time constants in a discrete relaxation spectrum. A shift of the reference modulus G: from TO to some ar- bitrary temperature T(t ) is shown in eq. (1) although this small correction is often omitted in the calculations. The important effect of temperature occurs, of course, through the dependence of T~ on temperature, which is taken to be of the form

with designating the time constants at the reference temperature and UT being the shift factor, stated in the form of a Williams-Landel-Ferry (WLF) equation.6 It is assumed that one shift factor can be used over all time scales which are rel- evant to the experiments being done. The generalized form of eq. (1) follows from earlier suggestions of Morland and Lee? Bernstein et al.? and Abbott and White.g A nonlinear form for the time constants T~ was also used in some of the w0rk.l

The general conclusion is that, while there are some nonlinear discrepancies in both the isothermal and nonisothermal elongations, eq. (1) is satisfactory if one is far removed from the glass transition temperature Tg. However, as one approaches Tg to within 20 to 30°C, and particularly at high cooling rates (in the range 1 to 10°C/sec), some systematic deviations between theory and experiment occur. More rapid than expected immobilization of the molecule seems to take place when the cooling rate is high. Corrections to the UT factor of eq. (2) were suggested by introducing a rate-dependent Tg into the WLF equation.2

In accompanying studies we have also examined published melt spinning data for high density polyethylene and find that the nonisothermal theory and the data are in good agreement.1° In this case the temperatures are far from Tg and the above effects were not considered.

In the present work transient birefringence as well as transient tensile stresses have been measured during simultaneous pulling and cooling. The fact that stress, reflecting molecular orientation, is a key variable in understanding pro- cessing operations has been recognized in several contexts. Spruiell and White and co-workers (at the University of Tennessee) have shown that the spin-line stress in melt spinning is the important unifying variable in understanding the subsequent structure and mechanical properties.llJ2 Similar ideas appear in injection molding where a distinct skin-core distribution of orientation and properties has been observed in both crystalline and amorphous polymer,13J4 which are presumably caused by the elongational stresses in the melt front and the shear stresses in the center of the mold.15 In connection with these studies Oda et a1.16 have recently considered the stress-birefringence relationship in a general way for a variety of experiments and several polymers. They emphasize the essential linearity of this relationship.

The work of the present paper is connected to that work, but here we em- phasize the nonisothermal aspects of the problem. As distinct from the melt spinning type of experiment, the deformation and the temperature histories of the present work are controlled variables. We focus on how the orientation (birefringence) develops, as compared with the stress, as the sample is cooled rapidly during the deformation.

STRESS BIREFRINGENCE 1665

T

"1 I I I I I

i I I - --

1666 MATSUMOTO AND BOGUE

~

0 E O 40 60 80

t , sec Fig. 2. Variation of sample thickness and width during elongation. (0) Thickness, (v) width.

dido = e-*t/2; E = 0.0750 sec-l.

EXPERIMENTAL

The basic experimental apparatus is that reported by Matsui and Boguel and later modified by Matsumoto and Bogue.2 It now consists of a pulling apparatus in which one end of a small cylindrical sample or a film sample is held in a clamp attached to an Instron load cell and the other end is driven by two rollers. It is a constant elongational rate experiment: & = & = (l/L)(dL/dt) = const where L ( t ) is the instantaneous length of the sample. During the pulling, a cooling history is applied to the sample by means of radiant heaters and circulating air. The effects of temperature variation in the sample and volume contraction were considered in an earlier paper.2

The material was commercial polystyrene Shell TC-3-30, as _ - was used ear1ier.l The molecular weight parameters are Vw = 2.83 X lo5 and Mw/Mn = 4.6.

In the present studies film samples, rather than cylindrical samples, were used. A two roller drive was used in all of the measurements. A schematic diagram of the mechanical and optical system is shown in Figure 1. The light source was a helium-neon gas laser (wavelength, 632.8 nm). The light beam passed through small holes in the heating chamber, then the film sample, and finally impinged on a photocell. The temperature (as measured with a bare thermocouple in the vicinity of the sample), the stress (as measured on the load cell), and the light intensity were recorded simultaneously as a function of time. The original di- mensions of the samples, before pulling, were 1-1.5 X 0.015 X 13.8 cm.

In order to relate the measured light intensity to the birefringence, several calibrating measurements were necessary. First, it was necessary to establish that the sample thinned down in a predictable way during the pulling. This was verified in a series of runs in which the pulling was stopped at various points, the

STRESS BIREFRINGENCE 1667

70

sa

sa

4 c

I 3 c

P C

1 1

I

I \ I p"'

r , mlr Fig. 3. Optical calibration: transmitted intensity vs. retardation. Z = 50.5 sin2 (TI'h).

samples removed, and their thicknesses and widths measured. The results are shown in Figure 2. The data are in good agreement with the predicted expo- nential change shown there. This equation was used subsequently to determine the sample thickness at any instant of time.

Next the light intensity I for a sample of given thickness was related to the retardation r by means of the relation:17

I = c sin2 (arb) (3) where X is the wavelength of light, c is a constant depending on the intensity of the entering beam and the angle between the principal axis of the sample and the optic axis of the polarizer. Actually this constant also depends on the ab- sorption and the scattering by the sample and on the optical system.

The constant in eq. (3) was determined by calibration with samples whose birefringence had previously been measured in a separate optical system. The calibration of the test optical system, described by eq. (3), is shown in Figure 3 with the constant being taken as 50.5. The numbers on the points indicate the thickness of the samples. As one expects, the correlation is independent of thickness.

1668 MATSUMOTO AND BOGUE

-1

-2

a 8 -3 -

-4

-5 -

TC3-30 E ,40750 Sez'

0 1 log t , sac

Fig. 4. Birefringence as a function of time during various (0) isothermal and (0, 'I) nonisothermal experiments. (0 ) N1, (v) N2.

Finally, then, the retardation is obtained from the measured intensity by means of eq. (3) and the thickness of the sample d is calculated from the ex- pression in Figure 1. These are combined by

A = r / d (4) to obtain the birefringence A at each instant of time. The sign of the birefrin- gence for polystyrene is known to be negative2l but here we use the symbol A to mean the absolute value of the birefringence.

PRESENTATION OF RESULTS

The birefringence data are first shown directly in the form of a plot of bire- fringence as a function of time at an elongation rate of 0.075 sec-l (Fig. 4); it has much the same shape as the corresponding stress plot (Fig. 5). In a rheological sense we are in the elastic (stress buildup) regime; for much lower values of 8, one expects that the stress would approach an asymptotic value and be related to the applied elongational rate through a Trouton viscosity relationship.18 In Figures 4 and 5 the open data points are the isothermal runs at the temperatures shown there and the closed data points, marked N1 and N2, are nonisothermal (cooling) runs.

In the film samples of the present work (Fig. 5) it was not possible to completely eliminate time lags between the sample and the measuring thermocouple; the cooling rates, as measured with a bare thermocouple, varied in the range 0.65- 3.3-1.5'Chec for run N1 and in the range 0.75-1.75"C/sec for run N2. In pre- vious work with fiber samples 2 it was possible to use samples small enough

STRESS BIREFRINGENCE 1669

- 1 TC3-30 ' I

4 1 -1 0 1 2

log t , sec

Fig. 5, Stress as a function of time during various (0) isothermal and nonisothermal (0, V) film-pulling experiments. (0 ) N1, (v) N2.

(300-400p initial diam) to eliminate time lags and to impose almost linear cooling rates. A portion of these results is reported in Figure 6, plotted as stress divided by the constant elongation rate 8, and compared with the theoretical predictions from eq. (1). The agreement is generally good although, as noted previously, we found systematic differences as one approached Tg, particularly at high cooling rates. From these studies we are satisfied that we have a good under- standing of the relationships between stress, strain history, and temperature history. We return now to the stress birefringence data on the film samples.

To determine the stress-birefringence relationship, these variables are plotted directly against each other in Figure 7. All of the data, including both the iso- thermal and nonisothermal runs, can be superposed on one curve, although clearly the function is not linear over the entire range. In the lower (linear)range, the stress-optical coefficient K defined by

K = A/g ( 5 ) has the value 6.1 X and 4.1 X cm21dyn reported by Han and Dexlerlg and Wales,2O respectively. Some comments on the nonlinear part of Figure 7 will be made later.

Other aspects of the data are considered in Figures 8 and 9. Consideration of a possible strain-optical coefficient [A/(X2 - X-')] is shown in Figure 8 in both the isothermal and nonisothermal runs, where the symbol X is the elongational ratio LILO. There have been many papers describing the strain-optical coeffi-

cm2/dyn, which compares with values of 4.95 X

1670 MATSUMOTO AND BOGUE

TC3-30 I 0.0520 sec-' I I

-1 0 1 2 log t , sec

Fig. 6. Stress as a function of time during various isothermal ( R ) = 0) and nonisothermal (R # 0) fiber-pulling experiments (the lines are theoretical results from eq. (1)). R: (-, Om), 0 "C/sec; (-,v) 1.25"Chec; (- - -,O) 1.60°C/sec; (--,O), 2.18"C/sec.

-1

-2

Q a 0 -

-3

-4

TC3-30

E, ao75oeed

4 5 6 7 8

l o g 6 , dynes/cm2

Fig. 7. Summary of stress-birefringence data. (0) 120°C; (v) 130°C; ( 0 ) 140°C; (0) 157°C; (0) N1; (v) N2.

TC3 - 30

1671

k, 0.0750 set’

-- 3

41: I

0 - -4

-5 -1 0 1 2

log t , sec

Fig. 8. Strain-optical coefficient as a function of time during various (0) isothermal and (0 , V) nonisothermal experiments. (0) N1, (‘I) N2.

cient of crystalline polymers,21,22 but there are only a few concerning amorphous polymers above Tg.23 As shown in Figure 8, the strain-optical coefficient de- creases rapidly with increasing time or strain. The shape of the strain-optical curve, however, depends on the measure of strain. In general, one does not suppose that strain and birefringence are simply related unless one is in an elastic regime, where stress can be easily related to strain.

CONSIDERATION OF THE STRESS-BIREFRINGENCE RELATION

According to the theory of rubber elasticity for small deformations, the product of the stress-optical coefficient (A/a) and absolute temperature T is given by24

where n is the average refractive index of the material, k is the Boltzmann con- stant, and a1 and a2 are the principal polarizabilities of the statistical segment. The generalization to a temporary network (polymer melt) was made by Lodge.25 In essence one finds a relation between stress and a statistically averaged mo- lecular strain and between birefringence and this same strain. The relation is the same in both cases, however; so that the strain can be eliminated between them to yield eq. (6). The constancy of the stress-optical coefficient has been noted in many systems under time varying (sinusoidal) deformations (Stein and Tobolsky26), conditions of high shear rate (PhilippofP7), and under noniso- thermal conditions in a melt spinning apparatus (Oda et a1.I6). Contrary ex- amples have also been reported, however, for low density polyethylene in melt spinning experiments (Katayama et a1.28). These authors separate the stress into viscous and elastic (birefringent) portions.

We have already implicitly shown the stress-optical coefficient in Figure 7 by

1672 MATSUMOTO AND BOGUE

log6 , dynes/cm2

Fig. 9. Temperature-multiplied stress-optical coefficient as a function of stress. (0) 120OC; (V) 130OC; (0) 140OC; (0) 157OC; (0) N1; (v) N2.

plotting birefringence against stress. We now replot these results in Figure 9, also introducing the factor of the absolute temperature (the instantaneous temperature in the case of the nonisothermal runs). The small variation of the absolute temperature is such that this factor does not appreciably affect the results. As previously seen both the isothermal and nonisothermal data su- perpose on a single curve. Up to a stress level of about lo7 dyn/cm2 the tem- perature-multiplied stress-optical coefficient is nearly constant, having an ab- solute value of about 2.5 X OKcm2/dyn. This agrees exactly with the value a/(AT) = 0.40 X lo7 reported by K ~ h n . ~ ~ To make some molecular interpre- tation of Figure 9, let us consider numerical values for (a1 - a2) calculated for the styrene monomer by Stein and T o b o l ~ k y . ~ ~ These values are of the order -6.2 to -62.8 X cm3, depending on whether one considers the benzene ring free to rotate or perpendicular to the axis of the chain. Estimating a value of (a1 - a2) = -180 X cm3 from our data (and Kuhn's data), one concludes that there are about 3-30 monomer units in the statistical segment. This is consistent, then, with the usually assumed size of the statistical segment30 and it seems reasonable to conclude therefore that the observed birefringence is as- sociated with near molecular level optical anisotropies.

For the high stress portion of Figure 9 the picture becomes more complicated. A t large stresses there is some nonlinear effect which contributes to the stress but not the birefringence. Intuitively it seems reasonable that molecules will not continue to show increases in their optical anisotropy as they become almost fully aligned, whereas their ability to support stress will be enhanced by the alignment. This idea is noted explicitly in connection with non-Gaussian rubber elasticity theory and the corresponding optical theory.24 The optical theory of Kuhn and Griin3l and later modifications are reviewed by Tre10ar;~~ a plot of optical anisotropy versus stress shows a linear relationship for the Gaussian theories but various departures, tending to an asymptote in the optical aniso- tropy, for the non-Gaussian theories.

STRESS BIREFRINGENCE 1673

To within the accuracy of our data, the extra stress seems to be uniquely though nonlinearly related to the birefringence; that is, we can superpose all of the data on a single curve. With so few data points in the nonlinear regime, however, we would hesitate to make a general case for superposition under all circum- stances.

For the broader problem of using birefringence as a means of measuring stress in processing operations, we conclude, along with Oda et a1.,I6 that there is a simple relation between these quantities over a wide range of deformation and cooling rates. They emphasized the essential linearity of the relationship for many polymers in several experiments (in isothermal shear flow, in nonisothermal melt spinning elongational flow, and in samples vitrified from various defor- mation states). For precise determinations, however, one should note the small but systemic departures from linearity in their polystyrene data and the more pronounced departures in our data at stresses above lo7 dyn/cm2.

The authors acknowledge with thanks the support of the Polymer Consortium of the University of Tennessee for a major portion of this work and the National Science Foundation (Grant No. ENG76-15644) for the final stages.

References

1. M. Matsui and D. C. Bogue, Trans. SOC. Rheol., 21,133 (1977). 2. T. Matsumoto and D. C. Bogue, Trans. SOC. Rheol., 21, (1977). 3. S. Kase and T. Matsuo, J. Polym. Sci., A3,2451 (1966); J. Appl. Poly. Sci., 11,251 (1967). 4. A. Ziabicki, in Man-Made Fibers, H. Mark, S. Atlas, and E. Cernia, Eds., Wiley, New York,

5. M. R. Kamal and S. Kenig, Poly. Eng. Sci., 12,294 (1972). 6. M. L. Williams, R. F. Landel, and J. D. Ferry, J. Amer. Chem. SOC., 77,3701 (1955). 7. L. W. Morland and E. H. Lee, Trans. SOC. Rheol., 4,233 (1960). 8. B. Bernstein, E. A. Kearsley, and L. J. Zapas, Trans. SOC. Rheol., 7,391 (1963); J. Res. Nut.

9. L. E. Abbott and J. L. White, Appl. Polym. Symp., 20,247 (1973).

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Bur. Stds., 68B. 103 (1964).

10. M. Matsui and D. C. Bogue, Poly. Eng. Sci., 16,735 (1976). 11. J. E. Spuiell and J. L. White, Appl. Poly. Symp., 27,121 (1975). 12. V. G. Bankar, J. E. Spruiell, and J. L. White, J. Appl. Poly. Sci. (in press). 13. E. S. Clark, Appl. Poly. Symp., 24,45 (1974). 14. G. Menges and G. Wubken, Ann. Tech. Conf. SOC. Plastics Engrs. Preprints, Montreal, Canada,

1973; G. Wubken, Dr.-Ing. Dissertation, Institut fur Kunststoffverarbeitung, Techniichen Hochschule Aachen, 1974.

15. Z. Tadmor, J. Appl. Poly. Sci., 18,1753 (1974). 16. K. Oda, J. L. White, and E. S. Clark, Poly. Eng. Sci. (in press). 17. H. T. Jessop and F. C. Harris, Photoelasticity: Principles and Methods, Cleaver-Hume,

18. H. Chang and A. S. Lodge, Rheol. Acta. 11,127 (1972). 19. C. D. Han and I. H. Dexler, J. Appl. Poly. Sci., 17,2329 (1973). 20. J. L. S. Wales, Rheol. Acta, 8,38 (1969). 21. R. S. Stein, S. Onogi, and D. A. Keedy, J. Poly. Sci., 57,801 (1962). 22. S. Onogi, Y. Fukui, T. Asada, and Y. Naganuma, Proceedings of the 5th International Congress

23. A. Ziabicki and K. Kedzierska, J. Appl. Poly. Sci., 6,361 (1962). 24. L. R. G. Treloar, The Physics of Rubber Elasticity, 2nd ed., Oxford U. P., London, 1958. 25. A. S. Lodge, Kolloid2., 171,46 (1960). 26. R. S. Stein, and A. V. Tobolsky, Textile Res. J., 18,302 (1948). 27. W. Philippoff, Trans. SOC. Rheol., 1,95 (1958); 4,159,169 (1960); 7,45 (1963).

London, 1949.

on Rheology, S. Onogi, Ed., University of Tokyo Press, Tokyo, 1970, Vol. 4, p. 87.

1674 MATSUMOTO AND BOGUE

28. K. Katayama, T. Amano, and K. Nakamura, Appl. Poly. Symp., 20,237 (1973). 29. W. Kuhn, J. Poly. Sci., 1,380 (1946). 30. R. S. Stein and A. V. Tobolsky, J. Poly. Sci., 11,285 (1953). 31. W. Kuhn and F. Grun, Kolloid Z., 101,248 (1942).

Received March 1,1977 Revised April 21,1977