# Calculation of the relaxation spectrum as a log-normal distribution and influence of the molecular parameters.

INTRODUCTIONThe fiber-spinning process is important among polypropylene applications. In recent years, consistent with the increase in the industrial processing speeds, polypropylenes intended for spinning operations have exhibited greater and greater fluidity. This property is obtained by controlled peroxide degradation after polymerization which modifies the molecular weight distribution (MWD).

However, the influence of MWD, and especially its breadth, on the processability and the ultimate properties of fibers is not yet totally elucidated. In this area of investigation melt rheological characterization is very helpful since it constitutes the link between molecular structure and strain behavior. It was previously established that there are strong connections between melt spinning behavior, fiber properties, and the mean relaxation time of melt material (1). Nevertheless, the breadth of the MWD affects not only the mean relaxation time but all the relaxation time spectrum. Therefore, the aim of the present study is to determine how this spectrum influences spinnability.

Recent molecular models tend to explain relation processes by considering more and more details of molecular motion. Unfortunately, these models are difficult to apply to polypropylenes due to two main reasons. First, the molecular characterization of polyolefins is known to be difficult, and second, broad MWDs are not yet correctly accounted for by the models.

So, it seems to be easier to characterize the material directly by considering its linear viscoelastic behavior. Moreover, this characterization is very helpful when that non-linear viscoelastic behavior is analyzed which is essential to assess processability.

In this work, we propose a method to calculate the relaxation spectrum of commercial polypropylenes from shear dynamic experiments. Then, the influence of the method of synthesis and, consequently, of the molecular weight distribution on the relaxation spectrum is studied.

A second paper will follow, which will study the influence of the relaxation spectrum obtained on the spinnability of polypropylene.

THEORETICAL BACKGROUND

The linear viscoelastic behavior of polymer melt can be described using the parallel association of N Maxwell elements (2, 3). Each of them is defined by the rigidity of the spring ([G.sub.i]) and the relaxation time which is the ratio between the viscosity of the dash-pot and the rigidity of the spring ([[Tau].sub.i] = [[Eta].sub.i]/[G.sub.i]). The behavior is entirely characterized by the knowledge of the discrete relaxation spectrum which is represented by the number N and the different values of [G.sub.i] and [[Tau].sub.i]. If the number N is increased without limit, the relaxation spectrum becomes continuous (H([Tau])). In that case, each infinitesimal contribution to rigidity is associated with relaxation times lying in the range between [Tau] and [Tau] + d[Tau].

The relaxation modulus is related to the relaxation spectrum by:

G(t) = [summation of] [G.sub.i][multiplied by][e.sup.-t/[Tau]i] where i = 1 to N with a discrete spectrum (1.a)

G(t) = [integral of] H([Tau])[multiplied by][e.sup.-t/[Tau]] between limits +[infinity] and -[infinity] dLn[Tau]

with a continuous spectrum (1.b)

The dynamic modulus becomes:

[G.sup.*] = j[Omega] [summation of][G.sub.i][[Tau].sub.i]/1 + j[Omega][[Tau].sub.i] where i = 1 to N with a discrete spectrum (2.a)

[G.sup.*]([Omega]) = j[Omega] [integral of][Tau]H([Tau])/1 + j[Omega][Tau] between limits +[infinity] and -[infinity]

with a continuous spectrum (2.b)

Considering a continuous relaxation spectrum for the terminal zone and without accounting for the rubber-glass transition, limiting values are obtained by the following expressions:

Newtonian viscosity:

[[Eta].sub.0] = [integral of] [Tau]H([Tau]) between limits +[infinity] and -[infinity] dLn[Tau] (3)

Plateau modulus:

[Mathematical Expression Omitted]

Limiting compliance:

[Mathematical Expression Omitted]

The number average ([Tau]n) and weight average ([Tau]w) relaxation times are also defined from the relaxation spectrum:

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Many methods have been developed to obtain the relaxation spectrum from the experimental dynamic modulus. One can mention different approximation methods (4-7). However, these methods are not very efficient in the case of broad distributions as for the samples presented below. When the relaxation times of the material span a large domain compared with the experimental frequency range, the relaxation spectrum obtained by these methods may not be entirely defined. Therefore, the use of this spectrum does not lead to correct calculations of the rheological functions. Moreover, in the case of polypropylene, it is not easy to enlarge the frequency domain using the time-temperature principle because the activation energy is not very important (about 40 kJ/mol).

Recently, various methods have been published to obtain a discrete relaxation spectrum (8-10) but the problem mentioned above persists. However, when data are incomplete, these methods may be improved by introducing asymptotic behavior governed by the material parameters as the plateau modulus value [Mathematical Expression Omitted] (11). Unfortunately, this value is not available for polypropylene. Indeed, the experimental plateau modulus is not pronounced because of the broad time distribution which leads to a coupling phenomenon between the flow regime and the glass transition.

In other words, for polypropylene, it is difficult to obtain experimental data of [G.sup.*]([Omega]) lying on a sufficiently large frequency domain to obtain the relaxation spectrum with these methods. For low frequencies, the limiting slopes of log(G[prime]([Omega])) and log(G[double prime]([Omega])) vs. log([Omega]) (respectively 2 and 1) are not experimentally reached. On the other hand, for high frequencies, the plateau modulus is not reached either.

In the case of broad molecular distribution, a continuous relaxation spectrum is more suitable. It is often obtained from a fit of an experimental function which leads to the relaxation spectrum using the complex algebraic inversion formula (12):

H([Tau] = 1/[Omega]) = [+ or -] 1/[Pi] Im[[G.sup.*]([Omega] exp([+ or -] i[Pi]))] (8)

The experimental curve fitting is often achieved using a power law of the frequency (13-16). Actually, this fitting can be considered as an extrapolation of the experimental data to obtain an entire relaxation spectrum, but this law may not always be convenient. For example, the real part of complex viscosity may be fitted using (17):

[Eta][prime]([Omega]) = 1/a[[Omega].sup.b] + c (9)

From such an expression, H([Tau])can be calculated using Eq 8. One obtains:

H([Tau] = 1/[Omega]) = 2/[Pi] a[[Omega].sup.b+1] sin(b[Pi]/2)/2 ac[[Omega].sub.b] cos(b[Pi]/2) + [c.sup.2] + [a.sup.2][[Omega].sup.2b] (10)

But, this expression of H([Tau]) is not convenient in all cases. It can be easily shown that when b is smaller than 1, the functions H([Tau]) and [[Tau].sup.2][multiplied by]H([Tau]) tend to infinity when [Tau] tends respectively to 0 and infinity. That is typically the case when the polydispersity index is greater than 2 (as for the samples studied). It follows that the plateau modulus and the limiting compliance cannot be obtained because the integrals which take place in Eqs 4 and 5 do not converge. This result disagrees with the linear viscoelastic theory.

THE LOG-NORMAL LAW

All these considerations lead us to choose another way to assess a continuous relaxation spectrum in order to describe correctly the linear viscoelastic behavior and to be in accordance with the theory. By analogy with the expressions often applied to describe the molecular weight distributions (18), we assume that the relaxation spectrum can take the shape of a Log-Normal law as was proposed by Wiechert (19) for glasses. The relaxation spectrum is then expressed by (20, 21):

H([Tau]) = [[Eta].sub.0]/[square root of [Sigma][Tau]] exp[-[(1/[Sigma] Ln[([Tau]/[Tau]m)).sup.2]] (11)

with [Tau]m = [square root of [Tau]w[multiplied by][Tau]n] and [Sigma] = [square root of 2 Ln([Tau]w/[Tau]n)]

This relaxation spectrum is described by three material characteristic parameters: the Newtonian viscosity ([[Eta].sub.0]), the number ([Tau]n), and weight ([Tau]w) average relaxation times. The reduced spectrum [Tau]H([Tau])/[[Eta].sub.0] vs. Ln([Tau]) is bell-shaped. The abscissa of the maximum is Ln([Tau]m) and the area under this curve equals 1. The breadth of the relaxation spectrum is characterized by the ratio [Tau]w/[Tau]n.

Furthermore, all the functions as [[Tau].sup.n] [multiplied by] H([Tau]) are also Log-Normal laws so they exhibit finite integrals. Consequently, all the limiting parameters in Eqs 3, 4 and 5 can be calculated.

The relaxation spectrum is calculated from an experimental rheological function. In the present case, the modulus of the complex viscosity is fitted using:

[Mathematical Expression Omitted]

The three characteristic parameters of H([Tau]) in Eq 11 are adjusted using a nonlinear least-squares regression procedure. The initial values can be graphically estimated. Then, the procedure converges towards the solution.

EXPERIMENTAL RESULTS

The samples studied are commercial polypropylenes intended for spinning, so their viscosity is relatively low. The Melt Flow Index (MFI: ASTM D 1238) is the useful characteristic of commercial thermoplastics defined by the mass flow rate of a polymer through an orifice of specified dimensions under prescribed conditions of load and temperature. Obviously, the greater the MFI, the lower the viscosity. However, information provided by the MFI is inadequate since the measured flow rates are not steady-state values.

Moreover, the samples have been obtained either directly from polyrnerization or after a controlled degradation. In the first case, their molecular weight distribution is broad. In the second case, the degradation increases the MFI of the control polymer. From a molecular characteristics point of view, degradation leads not only to decreased molecular weight but also to a narrowed distribution. However, a polydispersity index lower than 2 cannot be achieved using this method of synthesis.

The molecular weight distributions of the samples have been studied using GPC (gel permeation chromatography) measurements. Nevertheless, it is known that GPC analysis is difficult for polyolefins due to their poor solubility. So, the experiments must be achieved at high temperature (150 [degrees] C). Moreover with broad molecular weight distributions, the elution of high molecular weight material may be doubtful when they are not previously filtered. Consequently, the accuracy of the measured molecular weight distributions is poor.

The dynamic shear experiments have been carried out using a Rheometrics mechanical spectrometer (RMS 800) with a parallel plate geometry. The experimental temperature was 180 [degrees] C except for sample A, for which five experiments have been carried at various temperatures from 140 (overmelting, Ref. 22) to 280 [degrees] C. These five experiments allowed us to obtain a master curve for which the reference temperature is 180 [degrees] C. The temperature domain is limited at low values by the crystallization of the polymer and at high values by its degradation. Moreover, the activation energy being small, the resulting master curve spans only five decades.

After rheological experiments, the three characteristic parameters of the relaxation spectrum of each sample were calculated following the procedure defined above.

All results are reported in Table 1. The column heading "Control" shows the values of the MFI before degradation. If the polymer has been obtained directly from polymerization, it is labeled "Dir."

DISCUSSION

Figures 1 and 2 allow characterization of the proposed shape of the relaxation spectrum as a Log-Normal [TABULAR DATA FOR TABLE 1 OMITTED] law (Eq 2b). Indeed, the experimental curves of log(G[prime]) and log(G[double prime]) vs. log([Omega]) are in accordance with the calculated curves (Eq 11 with parameters in Table 1).

As to the results in Table 1, it appears clearly that the MFI measurement alone is not sufficient to provide a good rheological characterization. Indeed, some samples can exhibit the same MFI when their relaxation spectra are different. For example, samples D and E show the same MFI of 25 but their Newtonian viscosities and especially their relaxation times distributions are different. This is due to the fact that sample D did not undergo any degradation. So, the molecular weight distribution is broad. On the contrary, sample E was degraded from a control MFI of 2; its molecular weight distribution is then narrowed. The relaxation spectrum breadth of both samples, characterized by the ratio [Tau]w/[Tau]n, well accounts for the two methods of synthesis. The same conclusion can be applied to samples F and G (MFI: 35). They both underwent a degradation but not from the same control. So, sample G was more degraded than sample F, and its ratio [Tau]w/[Tau]n is lower.

On the other hand three samples present the same Newtonian viscosity without the same MFI (samples B, C, and D), again, due to the different degradation rates. Sample D was obtained directly from polymerization whereas the degradation of sample B was important. Sample C is intermediate. The ratios [Tau]w/[Tau]n are in the same order.

The comparison between Figs. 3 and 4 show directly the influence of molecular weight distribution on the reduced relaxation spectrum. Obviously, it is possible to compare the ratios [Tau]w/[Tau]n when the shapes of the molecular weight distributions are similar as in the present case of commercial polymers with relatively broad molecular weight distributions (in any case Mw/Mn [greater than] 2). Indeed, the simple Log-Normal law assumes that the relaxation spectrum is symmetric. However, easy modifications of this law can be considered to adapt the relaxation spectrum expression to other cases and then, for example, to put forward the influence of dissymmetric or multiple molecular weight distributions on the linear viscoelastic behavior.

CONCLUSION

This study puts forward the influence of molecular parameters on the relaxation spectrum of commercial polypropylenes obtained by different routes. The chosen shape of the relaxation spectrum (Log-Normal law) is remarkably suitable to characterize the linear viscoelastic behavior of the materials. Furthermore it provides a good extrapolation of the experimental results when the frequency domain is too short with respect to the breadth of the relaxation times distribution.

Moreover, contrary to often used continuous models, the Log-Normal law is in accord with the linear viscoelastic theory since it provides all the limiting values.

Furthermore, the analysis of the relaxation spectrum parameters (especially the ratio [Tau]w/[Tau]n) allows one to distinguish the methods of synthesis of the polymers whereas the MFI cannot. Particularly, the influence of the controlled peroxide degradation appears clearly on rheological behavior.

Finally, it is well known that the study of polymer processing cannot be achieved without considering the nonlinear viscoelastic behavior, for which the knowledge of the relaxation spectrum is very helpful. In a companion paper, we will show how it can be applied to spinning operations.

ACKNOWLEDGMENT

The authors gratefully acknowledge the society APPRYL (Lavera France) for the supply of samples, GPC analyses, and financial support.

REFERENCES

1. C. Prost, G. Nemoz, and A. Michel, Makromol. Chem. Macromol. Symp., 23, 161 (1989).

2. J. D. Ferry, in Viscoelastic Properties of Polymers, 2nd Ed., J. Wiley & Sons, New York (1971).

3. N. W. Tschoegl, in The Phenomenological Theory of Linear Viscoelastic Behavior - An Introduction, Springer-Verlag, Berlin (1989).

4. J. D. Ferry and M. L. Williams, J. Colloid. Sci., 7, 347 (1952).

5. M. L. Williams and J. D. Ferry, J. Polym. Sci., 11, 169 (1953).

6. F. Schwarlz and A. J. Staverman, Appl. Sci. Res., A4, 127 (1953).

7. N. W. Tschoegl, Rheol. Acta. 10, 582 (1971).

8. A. K. Livesey, P. Licinio, and M. Delaye, J. Chem. Physics, 84, 5102 (1986).

9. J. Honerkamp and J. Weese, Macromolecules, 22, 4372 (1989).

10. M. Baumgaertel and H. H. Winter. Rheol. Acta. 28, 511 (1989).

11. J. Honerkamp, Rheol. Acta. 28, 363 (1989).

12. B. Gross, in Mathematical Structure of the Theories of Viscoelasticity, Hermann, Paris (1953).

13. K. S. Cole and R. H. Cole, J. Chem. Phys., 9, 341 (1947).

14. P. J. Carreau, Trans. Soc. Rheol. 16, 99 (1972).

15. S. Wu, Polym. Eng. Sci., 25. 122 (1985).

16. I. P. Briedis, Rheol. Acta, 24, 357 (1989).

17. R. Fulchiron, V. Verney, and A. Michel, Sixth annual meeting of the Polymer Processing Society, Nice, France (April 1990).

18. L. H. Peebles, in Molecular Weight Distributions in Polymers, J. Wiley & Sons, New York (1971).

19. E. Wiechert, Ann. Phys., 50, 335 (1893).

20. R. Fulchiron, V. Verney, P. Cassagnau, and A. Michel, XIth Congress on Rheology, Brussels (August 1992).

21. R. Fulchiron, V. Verney, and G. Marin, J. Non-Newt. Fluid Mech., 48, 49 (1993).

22. V. Verney, results to be published.

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Title Annotation: | Correlations Between Relaxation Time Spectrum and Melt Spinning Behavior of Polypropylene, part 1 |
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Author: | Fulchiron, R.; Verney, V.; Michel, A. |

Publication: | Polymer Engineering and Science |

Date: | Mar 1, 1995 |

Words: | 2734 |

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