# A model for the zero shear viscosity.

INTRODUCTION

Zero shear viscosity, [[Eta].sub.0], being a fundamental property of polymeric materials, has been the subject of intensive studies aimed at elucidating its relationship with the polymeric structure (1-5). From experimental and theoretical studies the presence of a double regime of viscosity was quickly ascertained, depending on the molecular weight. At low molecular weights the relationship of viscosity with molecular weight was found to be substantially linear (1, 6), while at molecular weights higher than a critical value, [M.sub.c], the relationship was found to follow the 3.4 power of molecular weight. The 3.4 power law equation was first proposed in 1951 by Fox and Flory (3) on the basis of measurements on narrow distribution fractions of polystyrene and polyisobutylene. It has since been shown to apply to both melts and concentrated solutions for many species of polymers (1, 4).

The presence of a sharp transition between the two regimes at M = [M.sub.c] has been the subject of particular attention since it turned out to be a characteristic constant of the species in the melt state; [M.sub.c] was found equal to about 2-3 [M.sub.e], where [M.sub.e] is the average entanglement molecular weight (1). The sharpness of the transition, however, remained quite puzzling.

The nature of the entanglements has often been discussed and criticized in literature; perhaps, a sounder representation is to consider them as being time fluctuating and rather de-localized (1). More recently a different description of the macromolecular structure has been proposed, the so-called reptation model, introduced by De Gennes (7) and developed by Doi and Edwards (8, 9), where an entanglement spacing is not specified but an equivalent concept, the tube diameter, has been introduced, which allows an alternative description of the polymeric structure. The entanglements, however, are considered a key factor controlling not only the melt rheology but also the solid mechanical (10-12) and adhesive properties of polymers (13). For instance, according to Kramer, their relative values determine crazing versus shear yielding behavior (11, 12).

The increasing evidence of a correlation of material properties to entanglement junctions has stimulated new detailed studies on the correlation between entanglements and molecular structures, resulting, aside basic considerations, in a deep reexamination of experimental data and the ensuing extensive compilations of [M.sub.e] for a broad variety of polymeric species (14, 15).

In this paper we have reexamined the definition of the average number of entanglements per molecule in order to provide a quantitative ground for the zero shear viscosity. It will be shown that a suitable expression for the average number of entanglements per molecule may allow a simple new model for viscosity, which, by the way, quite naturally predicts the features of the sharp transition observed for the relationship viscosity versus molecular weight.

THE NUMBER OF ENTANGLEMENTS PER MOLECULE

For the moment, we shall restrict our attention to monodisperse polymers.

Let us consider the average number of entanglements per molecule, [n.sub.e]. It can be easily seen from geometric considerations, Fig. 1, that [n.sub.e] increases according to the equation

[n.sub.e] = (M/[M.sub.e] - 1) (1)

which, in the limit of very high molecular weights, tends to M/[M.sub.e]. This function predicts that the number of entanglements per molecule is 1 when M = 2[M.sub.e]; at this level, indeed, it is considered that the entanglement-dominated behavior begins for the viscosity, characterized by the well-known power 3.4.

Equation 1, while adequate to represent [n.sub.e] at high molecular weights, is not fully satisfactory for molecular weights comparable to [M.sub.e]. Indeed, according to this formula, we should expect [n.sub.e] = 0 for M = [M.sub.e] and even negative values for still lower molecular weights, a nonsensical expectation from the physical point of view.

A possible way out for this may be found along these lines:

a) Macromolecules may obstruct themselves, when flowing, even at molecular weights equal to or lower than [M.sub.e]. Since the nature of these "effective contacts" or "flow restrictions" is very similar to that defined by the entanglements, we may count as entanglements all the flow restrictions, even those experienced by short molecules;

b) The entanglements do not show up abruptly at some well-defined molecular weight but appear gradually as a function of molecular weight. Accordingly, we require that the equation defining [n.sub.e] as a function of M must be continuous, starting from zero for M = 0 and converging to formula (1) at high M.

A tentative empirical equation satisfying such requirements is the following one,

[n.sub.e] = M/[M.sub.e] [e.sup.-[M.sub.e]/M (2)

This equation may possibly find a theoretical ground; however, without trying to further pursue this goal, we note that it certainly satisfies the above-mentioned requirements, as can be easily seen. Indeed it tends to zero when M tends to zero and to Eq 1 at high M, as can be easily seen by a series expansion of the exponential factor

[n.sub.e] = [M/[M.sub.e] exp(-[M.sub.e]/M) [approaches] M/[M.sub.e] (1 - [M.sub.e]/M) = M/[M.sub.e] - 1. (3)

The use of Eq 2 to estimate the number of entanglements per molecule, here introduced on purely mathematical grounds, entails some significant differences and novelties over the usually accepted estimations of entanglements accounts, such as Eq 1.

First, it describes a continuous development of entanglements over the whole molecular weight range. Second, it satisfies our requirement of avoiding negative values for M [less than] [M.sub.e], which cannot be avoided when using Eq 1. Third, it predicts a number of entanglements differing from zero for M [less than] [M.sub.e]. At this molecular weight, Eq 2 predicts 1/e = 0.37 entanglements per molecule. For M = 2[M.sub.e] Eq 2 gives an estimation of [n.sub.e] of 1.21, a value 21% higher than the classical value of 1 obtained from Eq 2. Finally, as mentioned above, [n.sub.e] estimated by Eq 2 converges to the values estimated by Eq 1 for molecular weights higher than n[M.sub.e], when n is higher than about 4.

In conclusion, we can summarize the above considerations stating that Eq 2, which does not introduce new molecular parameters other than [M.sub.e], looks formally adequate to estimate the number of entanglements per molecule over the full range of molecular weights. From the physical point of view, it may be useful to stress again the fact that the above analysis simply means that short macromolecules experience some restriction to their movements even if they do not form a fluctuating network, as implied in the concept of entanglements.

THE VISCOSITY OF POLYMER MELTS

Having a formula for estimating the number of entanglements/molecule, we are now ready to reconsider the dependence of zero shear viscosity, [[Eta].sub.0], on molecular weight for monodisperse polymers. Let's suppose that the viscosity could be written as the sum of two terms: a) the first one, describing the friction between small molecules, depending on the monomeric friction factor, [[Zeta].sub.0], and on the total number of chain monomers, M/[m.sub.0]: b) the second one, dealing with the difficulties found by flowing entangled macromolecules, depending on the entanglement friction factor, [[Zeta].sub.e], and on the number of entanglements per molecule, [n.sub.e]; this last term is taken with the 3.4 power to take into account all the available experimental information, which points to such power in the entanglements region. Then we write:

[Mathematical Expression Omitted] (4)

where [[Eta].sub.0] is the zero shear viscosity, [[Eta].sub.0] = M/[m.sub.0] is the number of monomeric units in a chain, i.e., the polymerization degree, and [n.sub.e] the number of entanglements per molecule.

The friction coefficients [[Zeta].sub.0] and [[Zeta].sub.e].sup.3.4], having the dimension of the viscosity, contain, respectively, all the relevant information about the restraints experienced by small molecules moving over the others and the additional restraints to the movement caused by the entanglements. Some comments on them will be made later.

A virtue of Eq 4, Fig. 2, is to be able to describe by a single continuous function the behavior of viscosity of low and polymeric systems over a full range of molecular weights. At low M, i.e., M [less than] [M.sub.e], it converges toward the classical linear relationship, known for low molecular liquids, since the first term is prevailing

log [[Eta].sub.0] = log ([[Zeta].sub.0] M/[m.sub.0]) (M [less than] [M.sub.e]) (5)

and at high M, i.e. M [greater than] 4[M.sub.e], toward the law

log [[Eta].sub.0] = 3.4 log [[Zeta].sub.e] + 3.4 log (M/[M.sub.e] exp (-[M.sub.e]/M)) (6)

which, for M [greater than] 10 Me, further simplifies, reducing to the classical 3.4 power law of high polymeric materials

log [[Eta].sub.0] [Congruent] 3.4 log [[Zeta].sub.e] + 3.4 log M/[M.sub.e] (7)

In the molecular weight region, [M.sub.e] [less than] M [less than] 4[M.sub.e], the two contributions are comparable, so they both contribute significantly to the viscosity.

Accordingly, we may speak of three flow regimes: a) the monomeric regime, up to M [less than] [M.sub.e], where Eq 5 applies; b) the transition regime, for [M.sub.e] [less than] M [less than] 4 [M.sub.e], where the monomeric flow behavior coexists with an incipient entanglement-like behavior; and finally c) the high polymer or entanglement regime, M [greater than] 4 [M.sub.e], where Eq 6 applies. Within this region, we can use the approximate law expressed by Eq 7 only when the molecular weight is sufficiently high, i.e., higher than 10 [M.sub.e].

A second characteristic of Eq 4, which clearly shows up from Fig. 3, is that the viscosity shows a rather abrupt slope variation in a narrow molecular weight range, centered around 2[M.sub.e]. This behavior, well known in literature (1), comes out naturally from Eq 4, indicating, as expected, that the entanglement regime becomes really effective on viscosity when, on the average, one entanglement per molecule is present. On this ground, [M.sub.c] can be suitably defined on the basis of the number of entanglements per molecule.

The friction coefficients [[Zeta].sub.0] and [[Zeta].sub.e].sup.3.4], which are characteristic of a given polymeric species and depend on temperature and pressure, can be easily estimated for each polymeric species by Eqs 5 and 7 by testing on low and high molecular weight polymers respectively, as shown in the next section.

As for the estimation of [[Zeta].sub.0], a particular point has to be considered to get meaningful values of it, which was deeply investigated by Graessley and co-workers (16). Since the monomeric friction factor reflects the local chain dynamics, it depends on the concentration of chain ends, the effect of which vanishes with increasing molecular weight. This entails that experimental viscosity values must be corrected in order to establish the real dependence of viscosity from M in the low molecular weight range. Graessley's paper (16) suggests a method to estimate the correction, based on free volume concepts, which results in the following correcting equation

[Mathematical Expression Omitted] (8)

where [[Eta].sub.corr] and [[Eta].sub.exp] are the corrected and experimental viscosities for a low polymer of molecular weight M; [C.sub.1] is the WLF constant for the molecular weight M, and [[C.sub.1].sup.varies] is the value of C at high molecular weight. The correction, requiring a preliminary analysis on a set of low molecular weight polymers, is important because only in this way can low polymers be shown to follow the familiar power low pattern, i.e., near to unity, that we have taken for granted.

COMPARISON OF THE MODEL WITH EXPERIMENTAL DATA

In order to check the applicability of Eq 4 and to estimate the constants of the model, we have taken into account experimental data from literature on selected well-characterized polymers. Details of characterization, not reported here, may be found in the original papers.

a. The Case of Polybutadiene

We have used the remarkably accurate set of data from Colby, Fetters, and Graessley (17) covering a very extended range of molecular weights, from 1 x [10.sup.3] up to 1.65 x [10.sup.7]. Our analysis, however, was limited to molecular weights up to 350,000, where the 3.4 power was found to apply; above this value, experimental results were found consistent with the lower power of M, about 3, as suggested by the reptation theory. Samples obtained by anionic polymerization were nearly monodisperse and with similar microstructure. [T.sub.g] was found to be -99 [degrees] C and [M.sub.e] = 1850, based on a plateau modulus [[G.sub.N].sup.0] = 1.20 X [10.sup.7] dyn/[cm.sup.2].

In Table 1, zero shear viscosity data are reported at 25 [degrees] C for the molecular weight range where the entanglement regime is supposed to hold, i.e., 10,000 [less than] M [less than] 350,000. In the two last columns are reported the number of entanglements per molecule, [n.sub.e], and the entanglements friction factor, [[Zeta].sub.e], estimated respectively by Eqs 2 and 6. As can be seen, the values of [[Zeta].sub.e] are rather similar, ranging from 0.49 to 0.43. The range could be even a little more narrowed, considering only samples having a well-formed entanglement network, i.e., n [greater than] 20, and excluding sample B3, [TABULAR DATA FOR TABLE 1 OMITTED] which is in the borderline of the validity of the 3.4 power law. Accordingly, for polybutadiene at 25 [degrees] C, [[Zeta].sub.e] = 0.445 [+ or -] 0.02 [(Pa.s).sup.1/3.4]. This indicates that a proper choice of [n.sub.e] in the model results in a unique estimation of the entanglements friction factor.

In order to estimate [[Zeta].sub.0] we have taken into account four low polymer samples, Table 2, with molecular weights ranging from 1030 to 1420. Using for the viscosity the already chain ends corrected values, as mentioned at the end of the above paragraph, we got for polybutadiene at 25 [degrees] C: [[Zeta].sub.0] = 0.031 [+ or -] 0.03 Pa.s. Finally, in order to check the validity of Eq 4 we have compared experimental and calculated viscosity data for intermediate molecular weights 1420 [less than] M [less than] 10500, see Table 3 and Fig. 4, which is the transition region from low to high polymer behavior. This region, where the two flow regimes mix together, appears the most suitable for a critical test of Eq 4. As can be seen, estimated viscosity data appear in rather good agreement with the experimental data.

[TABULAR DATA FOR TABLE 3 OMITTED]

To conclude, we have shown in Fig. 5 the comparison of experimental values of viscosity with calculated ones over the whole range of molecular weights.

b. Other Remarks

The example of polybutadiene developed in detail has shown us how to deal with experimental data in order to extract the parameters of the model. For [[Zeta].sub.e] it may be sometimes more straightforward to use the relationship [[Eta].sub.0] versus molecular weight on monodisperse polymers, [[Eta].sub.0] = k [M.sup.3.4], provided that viscosity data were obtained on samples of sufficiently high molecular weight, i.e., M [greater than] 10 [M.sub.e]. Under this condition

[[Zeta].sub.e] = [M.sub.e] [k.sup.(1/3.4)] (9)

where k is the coefficient of the viscosity-molecular weight relationship in the high entanglement range.

Table 4 shows examples of such calculations, based on literature data (16-22). The [[Zeta].sub.e] values were estimated, considering both the M powers reported by the [TABULAR DATA FOR TABLE 4 OMITTED] authors, the [Alpha] values of column 6, and the standard 3.4 power, last column. Usually the resulting values are not too different. In case of too big a discrepancy, we think it preferable to rely on values obtained from the power estimated from experimenters, since the coefficient k, from which we get the friction factor, and [Alpha] are somehow related. In any case, [[Zeta].sub.e] values for a very broad set of polymers span over two decades with the average value centering around 1 [(Pa.s).sup.1/3.4].

In order to improve our knowledge about [[Zeta].sub.e], it proved fruitful to examine its dependence on temperature. A preliminary analysis, Fig. 6, shows that:

a) [[Zeta].sub.e] has a very similar temperature dependence for a number of polymeric species;

b) The temperature dependence is a WLF-type, i.e., it can be rationalized by T-[T.sub.g].

These statements may help to predict viscosity, since [[Zeta].sub.e] can be easily estimated at any given temperature from plot 6.

As for [[Zeta].sub.0], the above analysis on polybutadiene has indicated that, before getting acceptable results, it is necessary to make relevant corrections to experimental data. This would require a more specific study and will be postponed to a later time. For the moment we simply note that, at the same temperature, [[Zeta].sub.0] values for polybutadiene turn out to be about half the value of [[Zeta].sub.3.4].

It may be interesting to observe that the present model considers the friction factors as constants, independent from the molecular weight. This is in contrast to what is stated in the literature (Ref. 1, Ch. 10, Eq 14), where the monomeric friction coefficient is assumed to rise from the value [[Zeta].sub.00], for the monomer to an equilibrium value [[Zeta].sub.0] at high molecular weights, as a consequence of the additional free volume associated with molecular ends. Our approach, Eq 2, explains the small deviations from linearity of the viscosity of low polymers as due to additional entanglement-type constraints to flow found from small molecules even below 2 [M.sub.e].

CONCLUSIONS

We can summarize the above considerations along these lines:

The average number of entanglements per molecule, [n.sub.e], was critically reexamined, resulting in a new equation, Eq 2, which applies to monodisperse polymers. The modifications introduced for [n.sub.e], although apparently minor from the quantitative point of view, offer in principle a means to describe by a unique continuous function the evolution with the molecular weight of the number of entanglements per molecule.

The analytical equation for [n.sub.e] was then used to get a description of the zero shear viscosity over an extended range of molecular weights, from very low polymer up to the molecular weights where the 3.4 power law applies. Two well-known experimental findings turned out to be properly described: the continuous evolution of the viscosity as a function of molecular weight, and the sharp slope variation when changing from the monomer friction regime to the high polymer, i.e., to the entanglement regime.

The parameters appearing in the new viscosity equation, Eq 4, are [m.sub.0]. the monomeric molecular weight, [M.sub.e], the average molecular weight between entanglements, [[Zeta].sub.0], the monomeric friction coefficient, and [[Zeta].sub.e], the entanglement friction factor. In particular we underline here the novelty of the introduction of the entanglement friction coefficient, which accounts for additional restraints experienced by a macromolecule when moving within the network of randomly distributed macromolecules. The friction factors depend on temperature and pressure. Eqs 5 and 7 may be used for a quantitative evaluation.

Whereas [[Zeta].sub.0] was estimated only in the particular case of polybutadiene, leaving a deeper analysis to further studies, Se values for many polymers were shown to follow a WLF-type equation, i.e., with quite similar values when referred to temperatures equally distant from [T.sub.g].

Finally, using the friction factors extracted from experimental data at low and high molecular weights, the validity of the general viscosity equation, Eq 4, was tested in the most critical molecular weight region, i.e., for [M.sub.e] [less than] M [less than] 4[M.sub.e]. finding good agreement with the experimental data.

REFERENCES

1. D. J. Ferry, Viscoelastic Properties of Polymers, 3rd Ed., John Wiley & Sons, New York, (1980).

2. T. G. Fox and P. J. Flory, J. Am. Chem. Soc., 70, 2384 (1948).

3. T. G. Fox and P. J. Flory, Phys. Colloid Chem, 55, 221 (1951).

4. G. C. Berry and T. G. Fox, Adv. Polym. Sci, 5, 261 (1968).

5. W. W. Graessley, Adv Polym. Sci, 16, 1 (1974).

6. P. E. Rouse, J. Chem. Phys., 21, 1272 (1953).

7. P. G. De Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, N.Y. (1979).

8. M. Doi and S. F. Edwards, J. Chem Soc., Faraday Trans. II, 74, 1789, 1802, 1818 (1978).

9. M. Doi and S. F. Edwards, J. Chem Soc., Faraday Trans. II, 75, 38 (1979).

10. H. H. Kausch, Polymer Fracture, 2nd Ed., Springer Verlag, Berlin (1987).

11. A.M. Donald, and E. J. Kramer, J. Polym. Sci., Polym Phys. Ed., 20, 899 (1982).

12. A.M. Donald, and E. J. Kramer, J. Mater. Sci., 17, 1871 (1982).

13. S. Wu, Polymer Interfaces and Adhesion, Marcel Dekker, New York (1982).

14. S. M. Aharoni, Macromolecules, 19, 426 (1986).

15. S. Wu, J. Polym. Sci., Polym. Phys. Ed., 27, 723 (1989).

16. L. J. Fetters, W. W. Graessley, and A.D. Kiss, Macromolecules, 24, 3136 (1991).

17. R. H. Colby, L. J. Fetters, and W. W. Graessley, Macromolecules, 20, 2226 (1987).

18. J. T. Gotro and W. W. Graessley, Macromolecules, 17, 2767 (1984).

19. D. J. Plazek and V. M. O'Rourke, J. Polym. Sci.: Pt A-2, 9, 209 (1971); D. J. Plazek and P. Agarwal, J. Appl. Polym. Sci., 22, 2127 (1978).

20. W. M. Prest Jr. and R. S. Porter, Polymers, Polym. J., 4, 154 (1973).

21. S. H. Wasserman and W. W. Graessley, J Rheol., 36 (4), 543 (1992).

22. G. Locati and L. Gargani, J. Polym. Sci., Polym. Lett. Ed., 11, 95 (1973).

Zero shear viscosity, [[Eta].sub.0], being a fundamental property of polymeric materials, has been the subject of intensive studies aimed at elucidating its relationship with the polymeric structure (1-5). From experimental and theoretical studies the presence of a double regime of viscosity was quickly ascertained, depending on the molecular weight. At low molecular weights the relationship of viscosity with molecular weight was found to be substantially linear (1, 6), while at molecular weights higher than a critical value, [M.sub.c], the relationship was found to follow the 3.4 power of molecular weight. The 3.4 power law equation was first proposed in 1951 by Fox and Flory (3) on the basis of measurements on narrow distribution fractions of polystyrene and polyisobutylene. It has since been shown to apply to both melts and concentrated solutions for many species of polymers (1, 4).

The presence of a sharp transition between the two regimes at M = [M.sub.c] has been the subject of particular attention since it turned out to be a characteristic constant of the species in the melt state; [M.sub.c] was found equal to about 2-3 [M.sub.e], where [M.sub.e] is the average entanglement molecular weight (1). The sharpness of the transition, however, remained quite puzzling.

The nature of the entanglements has often been discussed and criticized in literature; perhaps, a sounder representation is to consider them as being time fluctuating and rather de-localized (1). More recently a different description of the macromolecular structure has been proposed, the so-called reptation model, introduced by De Gennes (7) and developed by Doi and Edwards (8, 9), where an entanglement spacing is not specified but an equivalent concept, the tube diameter, has been introduced, which allows an alternative description of the polymeric structure. The entanglements, however, are considered a key factor controlling not only the melt rheology but also the solid mechanical (10-12) and adhesive properties of polymers (13). For instance, according to Kramer, their relative values determine crazing versus shear yielding behavior (11, 12).

The increasing evidence of a correlation of material properties to entanglement junctions has stimulated new detailed studies on the correlation between entanglements and molecular structures, resulting, aside basic considerations, in a deep reexamination of experimental data and the ensuing extensive compilations of [M.sub.e] for a broad variety of polymeric species (14, 15).

In this paper we have reexamined the definition of the average number of entanglements per molecule in order to provide a quantitative ground for the zero shear viscosity. It will be shown that a suitable expression for the average number of entanglements per molecule may allow a simple new model for viscosity, which, by the way, quite naturally predicts the features of the sharp transition observed for the relationship viscosity versus molecular weight.

THE NUMBER OF ENTANGLEMENTS PER MOLECULE

For the moment, we shall restrict our attention to monodisperse polymers.

Let us consider the average number of entanglements per molecule, [n.sub.e]. It can be easily seen from geometric considerations, Fig. 1, that [n.sub.e] increases according to the equation

[n.sub.e] = (M/[M.sub.e] - 1) (1)

which, in the limit of very high molecular weights, tends to M/[M.sub.e]. This function predicts that the number of entanglements per molecule is 1 when M = 2[M.sub.e]; at this level, indeed, it is considered that the entanglement-dominated behavior begins for the viscosity, characterized by the well-known power 3.4.

Equation 1, while adequate to represent [n.sub.e] at high molecular weights, is not fully satisfactory for molecular weights comparable to [M.sub.e]. Indeed, according to this formula, we should expect [n.sub.e] = 0 for M = [M.sub.e] and even negative values for still lower molecular weights, a nonsensical expectation from the physical point of view.

A possible way out for this may be found along these lines:

a) Macromolecules may obstruct themselves, when flowing, even at molecular weights equal to or lower than [M.sub.e]. Since the nature of these "effective contacts" or "flow restrictions" is very similar to that defined by the entanglements, we may count as entanglements all the flow restrictions, even those experienced by short molecules;

b) The entanglements do not show up abruptly at some well-defined molecular weight but appear gradually as a function of molecular weight. Accordingly, we require that the equation defining [n.sub.e] as a function of M must be continuous, starting from zero for M = 0 and converging to formula (1) at high M.

A tentative empirical equation satisfying such requirements is the following one,

[n.sub.e] = M/[M.sub.e] [e.sup.-[M.sub.e]/M (2)

This equation may possibly find a theoretical ground; however, without trying to further pursue this goal, we note that it certainly satisfies the above-mentioned requirements, as can be easily seen. Indeed it tends to zero when M tends to zero and to Eq 1 at high M, as can be easily seen by a series expansion of the exponential factor

[n.sub.e] = [M/[M.sub.e] exp(-[M.sub.e]/M) [approaches] M/[M.sub.e] (1 - [M.sub.e]/M) = M/[M.sub.e] - 1. (3)

The use of Eq 2 to estimate the number of entanglements per molecule, here introduced on purely mathematical grounds, entails some significant differences and novelties over the usually accepted estimations of entanglements accounts, such as Eq 1.

First, it describes a continuous development of entanglements over the whole molecular weight range. Second, it satisfies our requirement of avoiding negative values for M [less than] [M.sub.e], which cannot be avoided when using Eq 1. Third, it predicts a number of entanglements differing from zero for M [less than] [M.sub.e]. At this molecular weight, Eq 2 predicts 1/e = 0.37 entanglements per molecule. For M = 2[M.sub.e] Eq 2 gives an estimation of [n.sub.e] of 1.21, a value 21% higher than the classical value of 1 obtained from Eq 2. Finally, as mentioned above, [n.sub.e] estimated by Eq 2 converges to the values estimated by Eq 1 for molecular weights higher than n[M.sub.e], when n is higher than about 4.

In conclusion, we can summarize the above considerations stating that Eq 2, which does not introduce new molecular parameters other than [M.sub.e], looks formally adequate to estimate the number of entanglements per molecule over the full range of molecular weights. From the physical point of view, it may be useful to stress again the fact that the above analysis simply means that short macromolecules experience some restriction to their movements even if they do not form a fluctuating network, as implied in the concept of entanglements.

THE VISCOSITY OF POLYMER MELTS

Having a formula for estimating the number of entanglements/molecule, we are now ready to reconsider the dependence of zero shear viscosity, [[Eta].sub.0], on molecular weight for monodisperse polymers. Let's suppose that the viscosity could be written as the sum of two terms: a) the first one, describing the friction between small molecules, depending on the monomeric friction factor, [[Zeta].sub.0], and on the total number of chain monomers, M/[m.sub.0]: b) the second one, dealing with the difficulties found by flowing entangled macromolecules, depending on the entanglement friction factor, [[Zeta].sub.e], and on the number of entanglements per molecule, [n.sub.e]; this last term is taken with the 3.4 power to take into account all the available experimental information, which points to such power in the entanglements region. Then we write:

[Mathematical Expression Omitted] (4)

where [[Eta].sub.0] is the zero shear viscosity, [[Eta].sub.0] = M/[m.sub.0] is the number of monomeric units in a chain, i.e., the polymerization degree, and [n.sub.e] the number of entanglements per molecule.

The friction coefficients [[Zeta].sub.0] and [[Zeta].sub.e].sup.3.4], having the dimension of the viscosity, contain, respectively, all the relevant information about the restraints experienced by small molecules moving over the others and the additional restraints to the movement caused by the entanglements. Some comments on them will be made later.

A virtue of Eq 4, Fig. 2, is to be able to describe by a single continuous function the behavior of viscosity of low and polymeric systems over a full range of molecular weights. At low M, i.e., M [less than] [M.sub.e], it converges toward the classical linear relationship, known for low molecular liquids, since the first term is prevailing

log [[Eta].sub.0] = log ([[Zeta].sub.0] M/[m.sub.0]) (M [less than] [M.sub.e]) (5)

and at high M, i.e. M [greater than] 4[M.sub.e], toward the law

log [[Eta].sub.0] = 3.4 log [[Zeta].sub.e] + 3.4 log (M/[M.sub.e] exp (-[M.sub.e]/M)) (6)

which, for M [greater than] 10 Me, further simplifies, reducing to the classical 3.4 power law of high polymeric materials

log [[Eta].sub.0] [Congruent] 3.4 log [[Zeta].sub.e] + 3.4 log M/[M.sub.e] (7)

In the molecular weight region, [M.sub.e] [less than] M [less than] 4[M.sub.e], the two contributions are comparable, so they both contribute significantly to the viscosity.

Accordingly, we may speak of three flow regimes: a) the monomeric regime, up to M [less than] [M.sub.e], where Eq 5 applies; b) the transition regime, for [M.sub.e] [less than] M [less than] 4 [M.sub.e], where the monomeric flow behavior coexists with an incipient entanglement-like behavior; and finally c) the high polymer or entanglement regime, M [greater than] 4 [M.sub.e], where Eq 6 applies. Within this region, we can use the approximate law expressed by Eq 7 only when the molecular weight is sufficiently high, i.e., higher than 10 [M.sub.e].

A second characteristic of Eq 4, which clearly shows up from Fig. 3, is that the viscosity shows a rather abrupt slope variation in a narrow molecular weight range, centered around 2[M.sub.e]. This behavior, well known in literature (1), comes out naturally from Eq 4, indicating, as expected, that the entanglement regime becomes really effective on viscosity when, on the average, one entanglement per molecule is present. On this ground, [M.sub.c] can be suitably defined on the basis of the number of entanglements per molecule.

The friction coefficients [[Zeta].sub.0] and [[Zeta].sub.e].sup.3.4], which are characteristic of a given polymeric species and depend on temperature and pressure, can be easily estimated for each polymeric species by Eqs 5 and 7 by testing on low and high molecular weight polymers respectively, as shown in the next section.

As for the estimation of [[Zeta].sub.0], a particular point has to be considered to get meaningful values of it, which was deeply investigated by Graessley and co-workers (16). Since the monomeric friction factor reflects the local chain dynamics, it depends on the concentration of chain ends, the effect of which vanishes with increasing molecular weight. This entails that experimental viscosity values must be corrected in order to establish the real dependence of viscosity from M in the low molecular weight range. Graessley's paper (16) suggests a method to estimate the correction, based on free volume concepts, which results in the following correcting equation

[Mathematical Expression Omitted] (8)

where [[Eta].sub.corr] and [[Eta].sub.exp] are the corrected and experimental viscosities for a low polymer of molecular weight M; [C.sub.1] is the WLF constant for the molecular weight M, and [[C.sub.1].sup.varies] is the value of C at high molecular weight. The correction, requiring a preliminary analysis on a set of low molecular weight polymers, is important because only in this way can low polymers be shown to follow the familiar power low pattern, i.e., near to unity, that we have taken for granted.

COMPARISON OF THE MODEL WITH EXPERIMENTAL DATA

In order to check the applicability of Eq 4 and to estimate the constants of the model, we have taken into account experimental data from literature on selected well-characterized polymers. Details of characterization, not reported here, may be found in the original papers.

a. The Case of Polybutadiene

We have used the remarkably accurate set of data from Colby, Fetters, and Graessley (17) covering a very extended range of molecular weights, from 1 x [10.sup.3] up to 1.65 x [10.sup.7]. Our analysis, however, was limited to molecular weights up to 350,000, where the 3.4 power was found to apply; above this value, experimental results were found consistent with the lower power of M, about 3, as suggested by the reptation theory. Samples obtained by anionic polymerization were nearly monodisperse and with similar microstructure. [T.sub.g] was found to be -99 [degrees] C and [M.sub.e] = 1850, based on a plateau modulus [[G.sub.N].sup.0] = 1.20 X [10.sup.7] dyn/[cm.sup.2].

In Table 1, zero shear viscosity data are reported at 25 [degrees] C for the molecular weight range where the entanglement regime is supposed to hold, i.e., 10,000 [less than] M [less than] 350,000. In the two last columns are reported the number of entanglements per molecule, [n.sub.e], and the entanglements friction factor, [[Zeta].sub.e], estimated respectively by Eqs 2 and 6. As can be seen, the values of [[Zeta].sub.e] are rather similar, ranging from 0.49 to 0.43. The range could be even a little more narrowed, considering only samples having a well-formed entanglement network, i.e., n [greater than] 20, and excluding sample B3, [TABULAR DATA FOR TABLE 1 OMITTED] which is in the borderline of the validity of the 3.4 power law. Accordingly, for polybutadiene at 25 [degrees] C, [[Zeta].sub.e] = 0.445 [+ or -] 0.02 [(Pa.s).sup.1/3.4]. This indicates that a proper choice of [n.sub.e] in the model results in a unique estimation of the entanglements friction factor.

In order to estimate [[Zeta].sub.0] we have taken into account four low polymer samples, Table 2, with molecular weights ranging from 1030 to 1420. Using for the viscosity the already chain ends corrected values, as mentioned at the end of the above paragraph, we got for polybutadiene at 25 [degrees] C: [[Zeta].sub.0] = 0.031 [+ or -] 0.03 Pa.s. Finally, in order to check the validity of Eq 4 we have compared experimental and calculated viscosity data for intermediate molecular weights 1420 [less than] M [less than] 10500, see Table 3 and Fig. 4, which is the transition region from low to high polymer behavior. This region, where the two flow regimes mix together, appears the most suitable for a critical test of Eq 4. As can be seen, estimated viscosity data appear in rather good agreement with the experimental data.

Table 2. Estimation of [[Zeta].sub.0] for Polybutadiene. T: 25 [degrees] C. Sample M corr.[Eta](Pa.s) [[Zeta].sub.0] (Pa.s) CDS-B2 1030 0.6 0.0315 C1 1130 0.7 0.0335 C2 1190 0.61 0.0277 C3 1420 0.85 0.0323 Ref. as in Tab. 1.

[TABULAR DATA FOR TABLE 3 OMITTED]

To conclude, we have shown in Fig. 5 the comparison of experimental values of viscosity with calculated ones over the whole range of molecular weights.

b. Other Remarks

The example of polybutadiene developed in detail has shown us how to deal with experimental data in order to extract the parameters of the model. For [[Zeta].sub.e] it may be sometimes more straightforward to use the relationship [[Eta].sub.0] versus molecular weight on monodisperse polymers, [[Eta].sub.0] = k [M.sup.3.4], provided that viscosity data were obtained on samples of sufficiently high molecular weight, i.e., M [greater than] 10 [M.sub.e]. Under this condition

[[Zeta].sub.e] = [M.sub.e] [k.sup.(1/3.4)] (9)

where k is the coefficient of the viscosity-molecular weight relationship in the high entanglement range.

Table 4 shows examples of such calculations, based on literature data (16-22). The [[Zeta].sub.e] values were estimated, considering both the M powers reported by the [TABULAR DATA FOR TABLE 4 OMITTED] authors, the [Alpha] values of column 6, and the standard 3.4 power, last column. Usually the resulting values are not too different. In case of too big a discrepancy, we think it preferable to rely on values obtained from the power estimated from experimenters, since the coefficient k, from which we get the friction factor, and [Alpha] are somehow related. In any case, [[Zeta].sub.e] values for a very broad set of polymers span over two decades with the average value centering around 1 [(Pa.s).sup.1/3.4].

In order to improve our knowledge about [[Zeta].sub.e], it proved fruitful to examine its dependence on temperature. A preliminary analysis, Fig. 6, shows that:

a) [[Zeta].sub.e] has a very similar temperature dependence for a number of polymeric species;

b) The temperature dependence is a WLF-type, i.e., it can be rationalized by T-[T.sub.g].

These statements may help to predict viscosity, since [[Zeta].sub.e] can be easily estimated at any given temperature from plot 6.

As for [[Zeta].sub.0], the above analysis on polybutadiene has indicated that, before getting acceptable results, it is necessary to make relevant corrections to experimental data. This would require a more specific study and will be postponed to a later time. For the moment we simply note that, at the same temperature, [[Zeta].sub.0] values for polybutadiene turn out to be about half the value of [[Zeta].sub.3.4].

It may be interesting to observe that the present model considers the friction factors as constants, independent from the molecular weight. This is in contrast to what is stated in the literature (Ref. 1, Ch. 10, Eq 14), where the monomeric friction coefficient is assumed to rise from the value [[Zeta].sub.00], for the monomer to an equilibrium value [[Zeta].sub.0] at high molecular weights, as a consequence of the additional free volume associated with molecular ends. Our approach, Eq 2, explains the small deviations from linearity of the viscosity of low polymers as due to additional entanglement-type constraints to flow found from small molecules even below 2 [M.sub.e].

CONCLUSIONS

We can summarize the above considerations along these lines:

The average number of entanglements per molecule, [n.sub.e], was critically reexamined, resulting in a new equation, Eq 2, which applies to monodisperse polymers. The modifications introduced for [n.sub.e], although apparently minor from the quantitative point of view, offer in principle a means to describe by a unique continuous function the evolution with the molecular weight of the number of entanglements per molecule.

The analytical equation for [n.sub.e] was then used to get a description of the zero shear viscosity over an extended range of molecular weights, from very low polymer up to the molecular weights where the 3.4 power law applies. Two well-known experimental findings turned out to be properly described: the continuous evolution of the viscosity as a function of molecular weight, and the sharp slope variation when changing from the monomer friction regime to the high polymer, i.e., to the entanglement regime.

The parameters appearing in the new viscosity equation, Eq 4, are [m.sub.0]. the monomeric molecular weight, [M.sub.e], the average molecular weight between entanglements, [[Zeta].sub.0], the monomeric friction coefficient, and [[Zeta].sub.e], the entanglement friction factor. In particular we underline here the novelty of the introduction of the entanglement friction coefficient, which accounts for additional restraints experienced by a macromolecule when moving within the network of randomly distributed macromolecules. The friction factors depend on temperature and pressure. Eqs 5 and 7 may be used for a quantitative evaluation.

Whereas [[Zeta].sub.0] was estimated only in the particular case of polybutadiene, leaving a deeper analysis to further studies, Se values for many polymers were shown to follow a WLF-type equation, i.e., with quite similar values when referred to temperatures equally distant from [T.sub.g].

Finally, using the friction factors extracted from experimental data at low and high molecular weights, the validity of the general viscosity equation, Eq 4, was tested in the most critical molecular weight region, i.e., for [M.sub.e] [less than] M [less than] 4[M.sub.e]. finding good agreement with the experimental data.

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Author: | Locati, G.; Pegoraro, M.; Nichetti, D. |
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Publication: | Polymer Engineering and Science |

Date: | Apr 1, 1999 |

Words: | 3791 |

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