rheo-optical properties of polyethylene films in the nonlinear viscoelastic region

Rheo-optical Properties of Polyethylene Films in the Nonlinear Viscoelastic Region

AKIRA TAN=,* MITSUHIRO FUKUDA, HIROYUKI NAGAI, MASAYUKI SHINOHARA, and SHIGEHARU ONOGI? Department of Polymer Chmistry, Faculty of Engineering,

Kyoto Uniuersity, Kyoto, Japan.

Synopsis

A new apparatus was designed to investigate the dynamic viscoelastic properties of solid polymer materials in the nonlinear viscoelastic region. The apparatus was combined with a birefringence apparatus in such a way that birefringence could be measured simultaneously with stress under oscillatory deformation. The nonlinear viscoelastic behavior of bulk-crystallized high-density polyethylene films was examined. Nonlinearity of mechanical properties became evident around 30°C, while optical properties became markedly nonlinear around 50°C. The nonlinearity of viscoelastic properties changes very little when the films are swollen with tetrachloroethane. It is proposed that disruption of lamellae to crystallites in the drawing process is one of the most important causes of the nonlinear behavior of high-density polyethylene films.

INTRODUCTION

It is well known that crystalline polymers exhibit nonlinear viscoelasticity even under small strains. A number of engineering and phenomenological studies on nonlinear viscoelasticity'-8 have been carried out, but no system- atic study has been done, and the precise nature of the nonlinear viscoelasticity is unclear. In order to clarify the nature of the nonlinearity, it is necessary to examine the time-dependence of deformation mechanisms in the nonlinear viscoelastic region.

Rheo-optical studies involve observing the time dependence of one or more optical quantities, each of which is sensitive to a particular aspect of morphology. Rheo-optical techniques have been utilized for resolving the relatively complex deformation process into constituent mechanisms. Thus, the rheo- optical technique should be very helpful in improving understanding of the molecular basis for the nonlinear viscoelastic behavior. There have been a few studies such as the stress relaxation and creep experiments by Tanaka et al.9-'1 and Yannas et a1.12 on the static nonlinear viscoelasticity.

In the present work, the dynamic viscoelasticity and dynamic birefringence were examined for high-density polyethylene (HDPE) films in the nonlinear viscoelastic region. In this paper the nonlinearity is restricted to that observed at rather small strains, under which macroscopic necking cannot be recog- nized, and above room temperature, that is, in the alpha dispersion region.

*To whom correspondence should be addressed. 'Present address: Matsue Technical College, Matsue 690, Japan.

Journal of Polymer Science: Part B: Polymer Physics, Vol. 27, 2283-2293 (1989) Q 1989 John Wiley & Sons, Inc. CCC 0887-6266/89/112283-11$04.00

2284 TANAKA ET AL.

Many morphological studies have been carried out on polyethylene films, and the relation between the viscoelastic properties and deformation mechanisms has also been examined at small strains (i.e., in the linear viscoelastic region).13 Hence, we chose polyethylene for this study.

First, we describe briefly the morphology of the bulk-crystallized polyethylene, the superstructure of which is usually ~pherulitic'~ with the b axis directed along the radius and with the a and c axes helicoidally oriented perpendicular to the b axis with more-or-less radial order.15 The b axis is believed to lie in the direction of the extension of the lamellae. The stretching of polyethylene is accompanied by deformation of spherulites.I6 The deformation is not simply affine. I t involves crystal reorientation within the deformed spherulite. The reorientation process is not the same throughout the spherulite. The principal mechanism at small deformation in the equatorial part of the spherulite has been postulated to be a twisting of the lamellae about their b axis so as to turn to the c axis (chain direction) toward the stretching directi~n.'~-'' In the 45" to polar parts of the spherulite, shearing of the lamellae leads to chain tilt with respect to the lamellar plane.

The application of a large deformation effects a transformation from the spherulitic to fibrous structures. The transformation may be accompanied by a disruption of spherulites. The disruption of spherulites seems to be particu- larly significant for nonlinear viscoelastic behavior. The disruption may mainly depend upon the perfection of alignment of crystallites and/or disorder within spherulites and their mobility. The perfection and mobility may also vary with temperature. The raising of the temperature brings about not only a partial melting of crystals but also a substantial reduction of the deformation rate. Thus, the nonlinear behavior of polyethylene films can be expected to be influenced by the nature of spherulitic structure, temperature, and the deformation rate.

, I t is our purpose to clarify these hypotheses experimentally. For this purpose, a "nonlinear dynamic viscoelasticity apparatus" in which one can measure the dynamic viscoelasticity under large amplitude static and dynamic strains, was designed. A dynamic birefringence apparatus was combined with the apparatus to enable the simultaneous measurements of dynamic birefringence with dynamic mechanical properties in the nonlinear viscoelastic region.

EXPERIMENTAL

Apparatus

The apparatus is designed to measure the dynamic viscoelasticity in the nonlinear region. A block diagram of the mechanical part of the apparatus is shown in Figure la. A Rheovibron Model DDVIII, (Toyo Baldwin Co., LTD.) allows a large amplitude of deformation (up to about 16 mm in gage length), which is derived by oil pressure. A servo system adjusts the strain to balance the signal of an electric oscillator. Therefore, strains with different deformation patterns such as sinwidal, square, and triangular can be applied by selecting a signal available on an oscillator. The stress as well as strain waves can be recorded on a pen recorder (or computer through the AD convertor) at low frequencies and on a transient computer memory at high frequencies.

RHEO-OPTICAL PROPERTIES

I

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2285

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(b)

Fig. 1. Block diagram of apparatus for measuring nonlinear dynamic viscoelastic properties (a) and nonlinear dynamic birefringence (b).

Furthermore, the apparatus is combined with a birefringence apparatus so that one can measure the biref'ringence simultaneously with mechanical quantities. The block diagram of the birefringence apparatus is shown in Figure lb. The laser beam is polarized by a polarizer at 45" to the stretching direction, passes through the sample and an analyzer oriented perpendicular to the polarizer, and passes into a photomultiplier. The transmitted intensity thus obtained is recorded by a pen recorder or a transient computer memory and is then converted to birefringence by the procedure described below.

Viscoelastic and Optical Functions

When the total strain y comprises an oscillatory strain superimposed on a static strain y,;

y( t) = y, + yd sin wt (1)

where y = 2 r f ( f is frequency in Hz), and yd is the dynamic strain amplitude, the stress u may be e x p r d by

u(t) =a , /2+a ,s inwt+azs in2wt+ ...

2286 TANAKA ET AL.

Then, the coefficients, a,, a,, b, can be calculated by eqs. (3), (4), and (5), respectively.

a , = ( w/77)/2n'a( 0 t) dt (3)

a, = (W/7i)/2n'~u(t)sinnwtdt 0 ( n = 1,2,3, . . . ) (4)

b, = (~/lr)/~"'oo(t)cosnwtdt * o ( n = 1,2,3,.. .) (5)

For the fundamental components (n = l), the amplitude of the complex modulus I E: I , dynamic modulus E;, loss modulus E r , and the phase difference 6, are defined by the following equations:

(9) 6, = cos-l[ a, / ( a," + bp) 1/2 ]

Positive 6, means that the strain lags the stress by a phase angle 6,. The birefringence An is converted from the transmitted intensity T by eqs. (10) and (11).

T = A sin2( r/2) (10)

where r is the retardation, and A is an arbitrary constant.

An = (Xr)/(27rd) (11)

where X is the wavelength of light, and d is the thickness of the specimen. The birefringence may also be written as a Fourier expansion for a nonlinear body as follows:

An(t) = a6/2 + aisinwt + a;sin2wt + - . .

From the coefficients of the fundamental components, in a similar way, the amplitude of the complex strain optical coefficient (SOC) ( K : I , in-phase SOC K[, out-of-phase SOC Kf', and phase difference a, (positive a, means that the birefringence is behind the strain by a phase angle a,) are defined by the

RHEO-OPTICAL PROPERTIES 2287

following equations:

a1 = cos-1[ u; / ( + b ; 2 y 2 ] (16)

Furthermore, the mechanical and optical nonlinear parameters Pa and Pa, as defined by eqs. (17) and (18), were introduced as a measure of mechanical and optical nonlinearities, respectively.

Pa = (AS)/( u," + b f ) 1 / 2 (17)

and

pa = (AS')/( + bi2)'l2 (18)

where A S (or A S ) is determined as the total deviation of the experimental stress (or birefringence) wave from the fundamental wave of the stress (or birefringence) over one cycle, as indicated by the shaded area in Figure 2. The thick, thin, and chain lines indicate, respectively, the experimental stress (or birefringence) wave, the fundamental stress (or birefringence) wave, and the strain wave. The response is always nonlinear when the nonlinear parameter is not zero. However, the response is not always linear when the nonlinear parameter is zero, because the latter becomes zero whenever the time-dependent and strain-dependent response terms are separable and at least one term is linear.

Fig. 2. Illustration of mechanical and optical nonlinear parameters. The thick solid line indicates observed stress (or birefringence) wave, the thin solid line indicates a fundamental of stress (or birefringence) wave, and the chain line indicates a strain wave. A S is given by the shaded area.

2288 TANAKA ET AL.

Sample Specimens

The high-density polyethylene used in this study was Hizex 5000S, pro- duced by Mitsui Petrochem. Co. The number-average and weight-average molecular weights M, and M, are 1.88 X lo4 and 2.51 X lo5, respectively. Film specimens (HDPE) were prepared as follows. Pellets of the material were melt-pressed a t 220°C and 100 kg/cm2 and quenched in an ice-water bath (0°C). The quenched films were then heat-treated in boiling water (about 98°C) for 1 h.

The density of the films thus prepared was 0.940 g/cm3 a t 30"C, giving the degree of the crystallinity of 62.5 wt%. The H , light-scattering pattern for the films was the so-called "four-leaf clover" pattern, which clearly evidences the existence of spherulitic structure. The radius of the spherulites was estimated to be about 6 pm by the method of Stein and Rhodes2'

Experimental Procedure

In the static strain dependence measurements, a static strain of 10% was first applied to the film specimen at a given temperature. After the stress had almost completely relaxed, a dynamic strain of 2% was superimposed on the static strain. The static strain was increased stepwise by 5% from 10% to 40%. This procedure made uniform drawing possible without causing macroscopic necking. In the frequency dependence measurements, the static strain was fixed a t 10% and the dynamic strain was fixed at 1%. The frequency was varied from 0.005 to 0.5Hz.

Generally speaking, the Lissajous figure of stress (or birefringence) versus strain does not close in one cycle. After many cycles (usually about 20) the Lissajous figure does close, showing that a stationary state has been attained. In the present study the viscoelastic and optical functions were determined in such a stationary state.

RESULTS AND DISCUSSION

In Figure 3 mechanical and optical functions are plotted against temperature a t several static strain levels. The dynamic strain was fixed at 2.0%. As shown in Figure 3, I E: I decreases with increasing temperature a t all levels of static strain. On the other hand, lK:I increases with increasing temperature a t all static strains, although it decreases with increasing temperature above 60°C. At all levels of static strain tan 6 , is not zero but shows a positive value, indicating that strain lags behind the stress. The temperature dependence of tan 6 , is quite similar among different levels of static strains. Tan 6, increases with increasing temperature below 40"C, reaches a maximum between 40 and 60"C, and then decreases above 60OC. Also, tan a, a t all levels of static strain is not zero, but positive. However, in this case, positive tana, indicates that strain leads the birefringence in phase as defined in eqs. (12) and (16). Tan a, a t all levels of static strain is almost constant or increases a little with increasing temperature below 50"C, and then decreases with increasing temperature. However, the magnitude is weakly dependent upon the level of static strain in the temperature region below 50OC. That is, tana, decreases with increasing static strain.


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At all static strain levels on the plot of the mechanical nonlinear parameter 1’, against temperature, a maximum is seen at 30°C. The maximum is very small at 10% of y,, shows a stepwise increase between 10 and 15%, and then gradually increases with increasing y,. Also at all static strain levels for the optical nonlinear parameter PA, a maximum is seen near 50°C. A t 15% of y,, the maximum becomes especially high. Thus, it was found that there are two kinds of nonlinearities in the process of oscillatory deformation, that is, mechanical nonlinearity at low temperatures (20-40°C) and optical nonlinearity at high temperatures (40-60°C).

In the Lissajous figures for stress versus strain and birefringence versus strain, characteristic features were also found, as represented schematically in Figure 4. At low temperatures, a marked skew from elliptical shape is observed in the Lissajous figure for stress versus strain but not in that of birefringence versus strain. The skew results from a small growth in the stress a t the high strain end. A t high temperatures, on the other hand, such a skew is not observed in the Lissajous figure for stress versus strain, but a different type of skew is observed in the Lissajous figure for birefringence versus strain. The Lissajous figure shows a parallelogramlike shape. This shape is due to a small change in the birefringence near both the high and low strain limits, that is, at low strain rate.

The frequency dependence of the nonlinear viscoelastic and optical quantities were measured at 30,50, and 70°C. The static strain was fixed at 10% and the dynamic strain at 1%. The value of I E: I increased and 1 K : I decreased with increasing frequency. These frequency dependences were quite similar to those of I E * I and I K * I in small dynamic strains (i.e., in the linear viscoelas-

2290 TANAKA ET AL.

temp st ress -strain biref . -strain

Fig. 4. Schematic representation of typical Lissajous figures for high-density polyethylene films.

tic region), which were previously ~btained.'~ In the previous study, the Arrhenius activation energy was obtained at 30 kcal/mol. With shift factors estimated from the activation energy of 30 kcal/mol, superposition was carried out. The master curves thus obtained are shown in Figure 5. The reference temperature was 50°C. To obtain smooth master curves, small vertical shifts were also necessary. As shown in Figure 5, the superposition of I E: I and I is very satisfactory. The master curve for 1 E: I increases with increasing reduced frequency and that for IK: I decreases. The master curves for tan6, and tancw, showed a maximum at about 0.01 and 0.1 Hz, respectively. The master curves were very similar to those previously obtained in a linear viscoelastic region.13 This similarity suggests that deformation in the nonlinear viscoelastic region is caused mainly by the same mechanisms as in the linear viscoelastic region.

In the master curves of P, and Pa, the following tendencies can be seen, although the superposition is not so satisfactory. The master curve of P, shows lower values at low frequencies and higher values at high frequencies, although the difference is quite small. The master curve of PA shows a maximum at about 0.01 Hz. Here again, one can recognize two kinds of nonlinearities: the mechanical nonlinearity at higher frequencies and the optical nonlinearity at lower frequencies.

The Lissajous figures for stress versus strain and birefringence versus strain were examined at different temperatures and frequencies. The mechanical Lissajous figure changed with increasing reduced frequency from a smooth ellipse to a distorted ellipse. The distortion of the mechanical Lissajous figure was highest in the temperature-frequency region denoted by letters c and d in the figure. The distortion is similar to that observed at low temperatures (see Fig. 4). On the other hand, the distortion of the optical Lissajous figure is highest in an intermediate temperature-frequency region denoted by letter f.


-4 -3 -2 - 1 0 1 l o g f (Hz.)

Fig. 5. Master curves of I E r 1 , I K: 1 , tan a,, tan al, Po, and Pa at reference temperature 50°C. f is reduced frequency.

The distortion is similar to that observed at high temperatures (see Fig. 4). Thus deviation mechanisms are also time-dependent.

According to previous rheo-optical studiesYg* l3 in the linear viscoelastic region of polyethylene films having spherulitic structure there exist three deformation mechanisms in the a1 dispersion region; i.e., deformation of spherulites as a whole ( d n e deformation), orientation of crystallites, and orientation of amorphous chains. Of these processes, &ne deformation and orientation of amorphous chains respond quickly to strain (i.e., they are elastic, and tan 6 and tan a are nearly zero). The orientation of crystallites responds to strain with a time or phase lag (ie., the process is viscoelastic, and tan S and tan a are not near zero).

The fundamental components of mechanical and optical quantities appar- ently can be interpreted by the same deformation mechanisms in the linear viscoelastic region; i.e., by the a1 loss mechanisms, as far as the static and dynamic strains covered in this study are concerned. The high IE: I and low

2291

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2292 TANAKA ET AL.

IK:I a t low temperatures are mainly due to deformation of stiff spherulites. (The orientation of crystallites hardly occurs a t low temperatures because of its long relaxation time.) The high I K: I at higher temperatures (below 60°C) is attributed to the orientation of crystallites within the spherulites. The decrease in I K: I above 60°C indicates the reduction of both the Orientation of crystallites and amorphous chains. It is supposedly due to partial melting of the crystal and/or substantial reduction of the deformation rate.

We believe that the nonlinear viscoelasticity is caused by deviation from the a1 loss deformation mechanism. The deviation is seen as a small growth of stress a t high strain, which was marked a t low temperatures and high frequencies. Consequently, we propose that the deviation is caused by deformation mechanisms other than those which cause the increase in stress with the strain. The pertinent deformation mechanisms are supposedly plastic deformation such as void opening or slippage of crystallites. However, in the same temperature-frequency region, the nonlinearity in optical quantities was not as great. This suggests that the effect of plastic deformation on the change in birefringence is very small. Birefringence is known to be insensitive to plastic deformation. We will seek more direct evidence for plastic deformation by other methods such as ultrasonic measurements in future.

The skew of the Lissajous figure for birefringence versus strain is attributed to a reduction in orientation a t low strain rate. The reduction in orientation was marked a t high temperatures and low frequencies (i.e., a t low stresses). Thus, it is suggested that the increase in birefringence is small below a critical stress and then becomes high beyond the critical stress. This result can be interpreted as follows. The total birefringence is composed of the birefringence of crystalline and amorphous components and form birefringence. The contri- bution of form birefringence can be neglected in this study, because it is unlikely that alignment of morphological entities with different refractive indices, which causes form birefringence, changes under the mild conditions of strain and strain rate we have used. Accordingly, the change in total birefringence simply is the sum of the changes in birefringence of the crystalline and amorphous components. The orientation of amorphous chains is very elastic, but that of crystallites is viscoelastic and depends upon strain rate. Hence, the ratio of the orientations of amorphous chains and crystallites varies during oscillatory deformation. Moreover, the increase in birefringence per unit strain caused by the orientation of the amorphous chains is smaller than that due to the orientation of crystallites. These situations may cause a nonlinearity in optical quantities. However, it is expected that these situations do not affect the relationship between stress and strain. As expected, the nonlinearity in the mechanical quantities is small in the same temperature-frequency region.

It is expected that swelling of the film specimens causes some structural changes in the amorphous region. It is very interesting to elucidate the effect of the structural change in the amorphous region on the nonlinear viscoelastic behavior of polyethylene films. For this purpose, the nonlinear viscoelastic and optical functions were measured for swollen PE films. The swelling was achieved by immersing film specimens in tetrachloraethane (CC1,) a t room temperature for 5 days. The increase in volume of the films thus obtained was about 4%. Before measurements CC1, on the surface of the film specimens was removed gently with filter paper. The dynamic measurements were carried out at relatively low temperatures (20 and 30°C). The static strains were varied


from 5 to 25%, and the dynamic strain was fixed at 2%. The measurements were carried out as quickly as possible to avoid evaporation of CC1,. However, the nonlinear viscoelastic and optical functions for the swollen films were almost the same as those obtained for the dry PE films. I t was found that the structural change in the amorphous region caused by the swelling of CC1, hardly affects the nonlinear viscoelasticity.

CONCLUSION

It was observed that the nonlinear behavior of the mechanical properties becomes pronounced around 30°C, while that of the optical properties becomes pronounced around 50°C. However, it was found that the fundamental components of mechanical and optical quantities are due to the same deformation mechanism as in the linear viscoelastic region (the a1 loss mechanism). I t is supposed that the nonlinear viscoelasticity is caused by small deviation from the a1 loss deformation mechanism. The deformation mechanisms which cause the deviation are presumably plastic deformations, such as void opening or slippage of crystallites in the process of the transformation from spherulitic to fibrous structures. Also, the deviation showed a time dependence very similar to that of the a1 loss mechanism. This may be understood by taking into account the fact that the nonlinearity appears at the onset of the transformation. The nonlinearity in viscoelastic properties changes very little when the films are swollen with tetrachloroethane.

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Polymers, Textile Foundation, Washington, D.C., 1943. 6. T. Yamaguchi, Doctoral Dissertation, Kyoto University, 1976. 7. 0. Nakada, J . Phys. SOC. Japan, 15, 2280 (1960). 8. K. Okano and 0. Nakada, J . Phys. SOC. Japan, 16, 2071 (1961). 9. S. Onogi, A. Tanaka, Y. Ishikawa, and T. Igarashi, Polymer J., 7 , 467 (1975).

10. A. Tanaka, A. Yamamoto, and S Onogi, Rep. Progr. Polym. Phys. Japan, 21, 385 (1978). 11. A. Tanaka, K. Tanai, and S. Onogi, BuU. Znst. Chem. Res., Kyoto Uniu., 55, 177 (1977). 12. I. V. Yannas and M. J. Doyle, J . Polym. Sci., A-2, 10, 159 (1972). 13. A. Tanaka, E. P. Chang, B. Delf, I. Kimura, and R. S. Stein, J . Polym. Sci., Polym. Phys.

14. P. H. Geil, Polymer Single Crystals, John Wiley & Sons, New York, 1963. 15. I. L. Hay and Keller, Kolloid-Z., 204, 43 (1965). 16. K. Kobayashi and T. Nagasawa, in US.-Japan Symposium on Polymer Physics, R. S. Stein

and S. Onogi, Eds., Interscience, New York, 1967, p. 163. 17. K. Sasaguri, S. Hashino, and R. S. Stein, J. Appl. Phys., 35, 47 (1964). 18. K. Sasaguri, R. Yamada, and R. S. Stein, J. Appl. Phys., 35, 3188 (1964). 19. K. Fujino, H. Kawai, T. Oda, and H. Maeda, Proceedings of the Fourth Zntermtioml

20. T. Oda, S. Nomura, and H. Kawai, J. Polym. Sci. A , 3,1993 (1965). 21. T. Oda, N. Sakaguchi, and H. Kawai, in U.S.-Japan Symposium in Polymer Physics,

F L S. Stein and S. Onogi, Eds., Interscience, New York, 1966, p. 223. 22. R. S. Stein and M. B. Rhodes, J. Appl. Phys., 31, 1873 (1960).

Received November 20, 1987 Accepted March 2, 1989

Ed., 11, 1891 (1973).

Congress on Rheology, Interscience, New York, 1965, Part 3, p. 501.

rheo-optical properties of polyethylene films in the nonlinear viscoelastic region

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