# Long Chain Branching and Polydispersity Effects on the Rheological Properties of Polyethylenes.

The rheological behavior of linear, and branched polyethylenes is
studied as a function of the weight average molecular weight ([M.sub.w])
and its distribution (MWD) as well as the level of long chain branching
in an attempt to identify correlations between long chain branching,
polydispersity and rheological properties. It is found that a need for
vertical shift of the viscoelastic moduli data to obtain the master
curves using the time-temperature superposition principle is associated
with the existence of long chain branching in the structure of the
polymer. The degree of vertical shift is found to correlate with the
level of long chain branching. This correlation corroborates with the
observation that long chain branching correlates with the horizontal
flow energy of activation. Plots of atan(G"/G') vs. G* (known
as Van Gurp plots) also reveal some important features that can be used
as signs of specific features in the structure of polymers. More
specifically, the area included below the Van Gurp curves c orrelates
with the level of long chain branching and polydispersity index. The
correlations are presented in graphical form and they can be used to
associate rheological properties with the presence of long chain
branching and/or polydispersity.

INTRODUCTION

The melt rheology of entangled polymers is strongly influenced by molecular weight, its distribution (polydispersity) and the level of long-chain branching [1-3]. Although the effects of polydispersity and long chain branching are pointing in the same direction, the latter are much stronger and more evident (4-8). It would be desirable to have a way in separating these effects and if possible to assess whether or not possible effects in rheology or processing are due to long chain branching or polydispersity.

There is lately a considerable interest in detecting long chain branching in the structure of polymers using rheological techniques (9-17). These techniques aim at detecting small level of long chain branching in metallocene polyethylenes. This family of resins is a major recent technological advancement in the polyolefins industry, i.e., the development of homogeneous metallocene catalysts for polymerization reactions. Metallocene catalysts produce polymers with narrow molecular weight and comonomer distributions, which, combined with controlled amounts of long chain branching, is claimed to lead to both excellent processability and superior mechanical properties (18-20).

The presence of long branches in the structure of polymers affects the rheological properties in different ways. The zero-shear viscosity of polymer melts with long branches has been found to scale exponentially with [M.sub.w] [21]. This makes their zero-shear viscosity much higher than those of their linear counterparts. In addition, branched polymers exhibit higher degrees of shear thinning [1], the onset of shear thinning occurs at smaller shear rates, finally they exhibit higher extrudate swell, and extensional viscosities [22, 23]. Similar effects to these of long chain branching are also exhibited by polydispersity, although to a milder degree [6, 7, 9, 23, 24-26]. The effects of number of arms and the arm molecular weight on the rheological properties in the case of star-polymers have been studied in detail elsewhere (27-29).

Various authors have developed and proposed correlations between long chain branching and rheological properties. These include the relationships between the long chain branching and (i) flow energy of activation [4, 25, 26], (ii) a normalized elevation of the flow energy of activation [16], [iii] the area in the Van Gurp plot [atan{G"/G') vs G*) included between the curve corresponding to the branched polymer and its linear counterpart, and finally (iv) the shift in the [M.sub.w]-location that corresponds to the maximum in the predicted molecular weight distribution from rheological data with respect to the [M.sub.w]-location of the maximum from GPC data. It is worthwhile to mention that Janzen and Colby [13] have developed a model to theoretically predict the effect of molecular characteristics on the zero-shear viscosity of the polymers. Their method is beyond from a simple correlation.

In the present work, we use the viscoelastic moduli of a large number of polyethylenes in order to evaluate/test some of the previously proposed correlations between levels of long chain branching and polydispersity with the rheological properties. These correlations together with some new ones in a graphical form can be used to correct for the effects of polydispersity or long chain branching in order to assess the effect of these two molecular features on the rheological properties independently.

EXPERIMENTAL

Polymers Samples

The viscoelastic moduli data of a large number of polyethylenes were used in this work. Some of the data are already published elsewhere [9, 25], while some others were measured as part of this work. All polyethylenes whose data are used in the present work are summarized in Tables 1 and 2 along with molecular and structural characteristics. The molecular data were provided by the manufacturers along with the resins. Molecular weight and its distribution were determined by GPC, while Long Chain Branching (LCB) by means of the spectroscopic technique of [[blank].sup.13]C NMR.

Table 1 includes two series of linear low-density polyethylenes (LLDPEs). The first series includes ten LLDPEs (Dow's Dowlex 2049 and LLDPE A-I) having about the same molecular weight and various polydispersities (defined as PD [equivalent] [M.sub.w]/[M.sub.n] ranging from 3.3 to 12.7. These polymers are essentially blends of LLDPE A and I [9]. The second series includes six LLDPE's (LLDPE J-O) having about the same polydispersity and various molecular weights ranging from 51 k to 119 k. The data for these resins have been published by Kazatchkov [9] and Hatzikiriakos et al. [25].

Table 2 includes a series of LDPE's (randomly branched polyethylenes) of various melt indexes ranging from 2 to 7.11. There are also two samples with a small degree of long chain branching, labeled as mLLDPE A and B, which are used to assess the effect of LCB on the rheological properties of polymers. The viscoelastic moduli data of all eight branched samples were obtained as part of the present work.

Rheological Measurements

Linear dynamic oscillatory measurements were carried out for all resins at temperatures in the range of 160-220[degrees]C using: (a) a Rheometrics System IV equipped with 25 mm parallel plates and (b) an ARES (Rheometrics Scientific) in the parallel plate geometry having a plate diameter of 25 mm. From the experiments the complex moduli, G' (storage modulus) and G" loss modulus), the complex viscosity, [[eta].sup.*] [equivalent] [square root][G'.sup.2] + [G".sup.2]/[omega] and the flow energy of activation (using the time-temperature superposition) were determined for all polymers listed in Tables 1 and 2.

An Interlaken sliding plate rheometer with a 0.5-mm gap was used to determine the true shear viscosities, which were also compared with Bagley- and Rabinowitsch-corrected capillary data [1]. An Instron capillary rheometer was used with dies having the same diameter and various length-to-diameter, L/D, ratios (0, 10, 20 and 40) in order to calculate the true shear viscosities of the polymers as well as the end pressure, [delta][P.sub.ends], known as Bagley correction. The pressure drop associated with changes in velocity profiles at the entrance and exit regions (end or Bagley correction) was calculated using the standard Bagley technique [6].

RESULTS AND DISCUSSION

Zero-Shear Viscosity

Figure 1 depicts the complex viscosity of a linear metallocene polyethylene (Exxon Chemical's Exact [R] 3128), two metallocene LLDPE's and LDPE A listed in Table 2. The effects of long chain branching (LCB) on the shape of the complex viscosity are evident. The zero-shear viscosity increases with increase of LCB, although at the frequency of 0.01 rad/s the Newtonian viscosity has not been reached for resins mLLDPE B and LDPE A. In addition, it is clear from Fig. 1 that with increase of LCB, the degree of shear thinning increases, while its onset occurs at smaller frequencies. The continuous lines in Fig. 1 have been drawn to guide the eye to see easily the effects of LCB on the shape of the complex viscosity curve.

Figure 2 plots the zero-shear viscosities of a number of resins listed in Tables 1 and 2 as a function of molecular weight, [M.sub.w] at 200[degrees]C. These were calculated either directly from the experimental data or by fitting a Carreau-Yasuda model that gives the zero-shear viscosity as a parameter. Two important issues arise from Fig. 2 that must be discussed. The zero-shear viscosities of resins J to 0 define a line with slope of 4.1, which means that the zero shear viscosity for this series of resins scales as, [[eta].sub.0] [alpha] [[M.sup.4.1].sub.w]. The two branched polyethylenes (mLLDPE A and B) possess viscosities much greater than those defined by the continuous line, an observation expected. In addition, it can be seen that the viscosities of resins LLDPE A to I, are also greater than those defined by the continuous line. The higher the polydispersity, the higher the deviation from the scaling [[eta].sub.0] [alpha] [[M.sup.4.1].sub.w]. It clearly appears that polydispersity has a significant effect on the zero-shear viscosity of LLDPE. It is worth mentioning that resins LLDPE A to I have the same percent of comonomer and therefore the increase in viscosity is not due to the higher level of short chain branching (SOB). Obviously, more systematic studies are needed in order to confirm the observed effect of polydispersity on the zero-shear viscosity of linear polyethylenes and the deviation from the well-established scaling of scaling [[eta].sub.0] [alpha] [[M.sup.3.4].sub.w] that is true for monodisperse and nearly monodisperse polymers.

Therefore, the effects of polydispersity and LCB on the shape of the complex viscosity curve as well as on the Newtonian viscosity are similar, It can also be concluded that these effects are the same on the absolute viscosity curve in view of the validity of the Cox-Merz rule. It would be desirable to have a method to separate these two effects, i.e., to know, for example, if a certain effect is due to polydispersity or presence of LOB.

Linear Viscoelastic Moduli

Mavridis and Shroff [2] studied the viscoelastic behavior of a large number of polyethylenes in an attempt to identify the effects of long branches on the rheological behavior of polyethylenes. It was reported that in the case of linear polyethylenes (high-density and linear low density) a vertical shift to the viscoelastic moduli data is not necessary in order to obtain single master curves, i.e. when the time-temperature superposition is applied. On the other hand, it was shown theoretically and demonstrated experimentally that neglect of the vertical shift cause failure to the applicability of the time-temperature superposition principle. This observation also leads to a stress-dependent flow energy of activation.

For temperatures greater than [T.sub.g] + 100, the horizontal shift factor resulting from application of the time-temperature superposition to the viscoelastic moduli data can be interpreted in terms of a flow activation energy defined through an Arrhenius type equation, as follows:

[[alpha].sub.T] = exp[[E.sub.H]/R(1/T - 1/[T.sub.ref])] (1)

where [[alpha].sub.T] represents the horizontal shift factor, [E.sub.H] the horizontal flow energy of activation, R is the gas constant, R is the absolute temperature, and [T.sub.ref] is the reference temperature. Similarly, Mavridis and Schroff (4) interpreted the vertical shift in terms of a vertical flow energy of activation using an Arrhenius type equation as follows:

[b.sub.T] = exp[[E.sub.V]/R(1/T - 1/[T.sub.ref])] (2)

where [b.sub.T] represents the vertical shift factor, and [E.sub.V] the vertical flow energy of activation.

To see if a vertical shift is necessary a Cole-Cole plot (G" versus G') may be constructed. If this is independent of temperature, a vertical shift is not necessary and the master curves for the viscoelastic moduli can be obtained by applying the time-temperature superposition in the horizontal direction only. Such Cole-Cole plots were constructed for all resins listed in Tables 1 and 2. It was found that no vertical shift was necessary to superpose the data for all polyethylenes listed in Table 1 (linear with no LCB). This was not true for those listed in Table 2, which include polymers havings various levels of LCB. This observation can obviously be used to distinguish between the effects of LCB and PD in view of the fact that Cole-Cole plots of polydisperse polyethylenes are temperature independent. Therefore, failure of superposition in a Cole-Cole plot is a sign of the presence of LCB in the molecular structure of a polymer. Note that up to this point LCB and PD exhibited similar effects on the rheologi cal properties of the resins.

Figure 3 is a Cole-Cole plot for three different LLDPEs having about the same [M.sub.w] but different polydispersities. The data for three temperatures were plotted. It can be seen that the Cole-Cole plot for these linear polymers of various polydispersities (3.3, 6.6, and 12.7) is independent of temperature. Therefore, a vertical shift is not necessary to obtain the master curves of viscoelastic moduli. However, this is not the case for branched polyethylenes. Figure 4 is a Cole-Cole plot for two branched polyethylenes, namely mLLDPE A and LDPE A. It can clearly be seen that a vertical shift is necessary to superpose the data. Calculating the vertical shift factors by minimizing the standard deviation of all data from the data corresponding to the reference temperature, and referring all data to this reference temperature, Fig. 5 is obtained. In fact, these shift factors were calculated using an alternative form of the Cole-Cole plot, that is the shift angle, [delta] [equivalent] [alpha]tan (G"/G') versus G* [equivalent] [square root][G'.sup.2] + [G".sub.2], a plot known as the Van Gurp plot [10, 17]. Note that in a Cole-Cole plot, the data shift should be performed in both directions, whereas in a Van Gurp plot, the data shift should be done in only along the G* axis. Therefore, by comparing Figs. 4 and 5, the need for a vertical shift to superpose the data for branched polymers becomes clear.

The vertical shift factors were calculated for all branched polymers listed in Table 2. These were used to fit Eq 2 in order to calculate the vertical flow energy of activation, [E.sub.V]. Once the linear viscoelastic moduli data were corrected by taking into account the vertical shift factors, [b.sub.T], the time-temperature superposition was applied in the horizontal direction in order to obtain the master curves of the viscoelastic moduli for all resins listed in Tables 1 and 2. Moreover, the horizontal flow energy of activation, [E.sub.H], was calculated by means of fitting Eq 1. The values of [E.sub.V], and [E.sub.H] are listed in Table 3 and are plotted in Fig. 6 as a function of PD and LCB. First, from an inspection of the values of [E.sub.V], and [E.sub.H], it can be seen that these correlate well with each other. A higher value of [E.sub.V] implies a higher value of [E.sub.H] and vice versa. This is somehow expectable, since the need of vertical shift to the data and consequently application of such a correction before the application of the time-temperature superposition principle, imposes a higher temperature sensitivity to the viscoelastic data and therefore a higher [E.sub.H] value is calculated. The values of [E.sub.H] calculated in this work agree well with those reported previously in the literature [4, 16, 26]. However, the [E.sub.V] values are higher than those calculated by Mavridis and Shroff (4). They have reported values in the range of up to 2.5 kcal/ mol, whereas in the present work we have calculated [E.sub.V] values in the range from 3.2 to 5.26 kcal/mol.

It can be seen from Fig. 6 that [E.sub.V], and [E.sub.H] correlate well with LCB and that [E.sub.H] correlates with PD as well. The values for [E.sub.V], and [E.sub.H] of LDPE (LCB [greater than or equal to] 3) were calculated by averaging all values listed in Table 2. The continuous lines on Fig. 6 are drawn to guide the eye. Once again, it was seen that the effects of PD and LCB are similar as far as the horizontal flow energy of activation concerns. The differences between the effects of LCB and PD on the rheological properties of polyethylenes are two. First, the values of the horizontal flow energy of activation for the case of polydisperse polymers (up to polydispersities of 12.7) are smaller than those of branched polymers with 0.01 branches per 1000 C atoms or more. Second, for the case of branched polymers a vertical shift should be applied to the viscoelastic moduli to obtain the master curves. These two differences may be used to differentiate between the effects of PD and LCB. For example, a valu e of [E.sub.v] greater than 2 kcal/mol and a value of [E.sub.H] greater than 9 kcal/mol would imply the existence of LCB. Then, the correlations plotted in Fig. 6 may be used to determine the level of LCB. Smaller values for [E.sub.H] (smaller than 9 kcal/mol) as well as calculation of insignificant values of [E.sub.v] (smaller than 1 kcal/mol) would imply a polydisperse polyethylene with polydispersity that depends on [E.sub.H] and according to the correlation plotted in Fig. 6.

Van Gurp versus Cole-Cole Plot

Besides the Cole-Cole plot that is used to verify whether or not a fluid is thermorheologically simple (applicability of the time-temperature superposition principle), another way is to use the Van Gurp plot (10). This is a plot of the loss phase angle, [delta], defined as [delta] [equivalent] atan (G"/G'), versus the magnitude of the complex modulus, G* defined as G* [equivalent] [square root][G'.sup.2] + [G".sup.2]. Such plots have been used in the past (4) to determine the vertical shift needed to superpose the viscoelastic moduli of the low-density polyethylenes (see a previous section). Friedrich et al. (17) have used such plots in an attempt to identify a rheological correlation with the level of long chain branching (30). The same method is also used in the present work for assessing not only the effect of long chain branching on the rheological properties but those of polydispersity as well. It is noted that the Cole-Cole plot might give the same information with the corresponding Van Gurp ones, but t he latter show the effects more clearly.

Figure 7 is a Van Gurp plot for the polyethylene resins listed in Table 1 that have about the same polydispersity (3.4-3.9) and various molecular weights in the range of 51k to 114k. It can be seen that all the data define about a single line. It seems that molecular weight has no effect on the Van Gurp plot at least in the molecular weight range examined in the present work. This range of molecular weight covers mostly the range of interest from the industrial point of view. On the other hand, the effect of polydispersity on the Van Gurp plot is more obvious. Figure 8 is a Van Gurp plot for some polyethylenes listed in Table 1 that have about the same molecular weight (90k-119k) and various polydispersities in the range of 3.3 to 12.7. The effect is clear. Increasing the polydispersity, the Van Gurp curves shifts to smaller values of the phase angle, [delta]. One may use the area included under the curve (called Van Gurp Area) as a parameter to correlate with polydispersity. This is done and discussed in det ail below.

The effect of long chain branching on the Van Gurp plot is similar to that of polydispersity. Figure 9 is a Van Gurp plot of a series of polyethylenes listed in Table 2 that have different levels of long chain branching. It can be seen that by increasing the level of long chain branching, the Van Gurp curve shifts to smaller values of the phase angle, [delta]. Again the area included under the curve (called Van Gurp Area) is used below as a parameter to correlate with long chain branching.

Determining the viscoelastic moduli of various polymers at various temperatures, and consequently applying the time-temperature superposition principle, one may end up with data covering a different range of frequency and complex modulus values for different resins. Therefore, in order to calculate the area included under the van Gurp plot, one has to refer the data of all resins under the same range of complex modulus. In this case we have selected as an accessible range of complex modulus values for all polymers to be from [10.sup.2] to [10.sup.5] Pa. This range of complex modulus values covers the accessible by most rheometers' range of frequencies. In some cases, data extrapolation was used for some resins in order to cover the whole range, as appears evident by the continuous lines plotted in Figs. 8 and 9. Having defined the range of complex modulus, the areas under each curve in Figs. 8 and 9 were calculated, and these are plotted in Fig. 10 as a function of polydispersity and long chain branching. The existence of correlations between the Van Gurp area ([10.sup.2]-[10.sup.5] Pa) with long chain branching and polydispersity are clear. It is once more evident that the effects of long chain branching and polydispersity are similar.

The significance of Fig. 10 can be defined as follows. In many cases, long chain branching and polydispersity are present in the structure of a single polymer. Therefore, before using Fig. 10 to detect the level of long chain branching, a correction may be applied to account first for the effect of polydispersity. Consequently, having the new "corrected" value of the Van Gurp area, one may proceed to use Fig. 9 in order to calculate the level of long chain branching. Clearly, such an approach excludes synergism between LOB and polydispersity. Such synergism might be possible.

It has been discussed in the past that crossover frequency, where the storage (G') and loss (G") moduli are equal, can serve as an indicator of the resin viscoelasticily (polydispersity and long chain branching effects). In other words, its inverse may represent a single characteristic relaxation time of a given resin (21). Zeichner and Patel (31) have noticed a correlation between polydispersity of a linear resin and its crossover modulus ([G.sub.c] = G' = G"). They have found that if a rheological polydispersity index defined as PI [equivalent] [10.sup.5]/[G.sub.c], where [G.sub.c] is in Pa, is plotted versus [M.sub.w]/[M.sub.n] on a double logarithmic plot, the data points would fall on a straight line (6). These types of correlation as well as some others discussed in (7) are based on a single point and they are not always successful (9). The correlations proposed in this work are based on the use of Van Gurp plot that utilizes the whole viscoelastic moduli curves of the resins and therefore might be more appropria te to use. Most important, the proposed correlations are temperature independent ones. Finally, it might be argued that the shapes of the Van Gurp plots mirror to a large extent those of the complex viscosity versus frequency and therefore they show the expected dependence on polydispersity and branching.

Ends Pressure In Capillary Flow

Figure 11 plots the end pressure correction known also as Bagley correction, of representative polyethylenes listed in Table 1 as a function of the apparent shear rate. These were calculated by using a conventional Instron capillaxy rheometer with an orifice die having a contraction angle of 90 degrees and a final diameter of 0.5 mm. It is interesting to note that the Bagley correction function shifts to higher values with increase of polydispersity. Similar effects can also be observed for the effect of long chain branching on the Bagley correction. Figure 12 plots the ends pressure correction of representative polymers having various degrees of long chain branching listed in Table 2 as a function of the apparent shear rate. The end pressure correction (Bagley correction) increases with increase of the long chain branching.

These results are not surprising since it is known that the elasticity of polyethylenes increases with polydispersity and long chain branching. Since the end pressure (Bagley correction) can serve as an indication of the level of extensional viscosity of resins [32] and since this increases with polydispersity and long chain branching [22-24], this observation (increase of Bagley correction with polydispersity and long chain branching) is in order. These results also corroborate our previous discussion that the polydispersity and long chain effects on the rheological properties are similar. The only difference is that in the case of branched polymers, a vertical shift of the viscoelastic moduli is needed in order to obtain master curves with the application of the time-temperature superposition.

CONCLUSIONS

Several experimental techniques were employed to study the rheological behavior of a large number of polyethylenes in order to assess independently the effects of polydispersity (PD) and long chain branching (LCB). It was found that most of the effects of LCB and PD are similar. In particular (i) PD and LCB increase the zero-shear viscosity more than that predicted by the scaling relationship, [[eta].sub.0] [alpha] [[M.sup.3.4].sub.w], where the exponent of 3.4 might be slightly higher, i.e., 4.1 in the present work, (ii) PD and LCB decrease the critical shear rate for the onset of shear thinning as well as increase the degree of shear thinning, (iii) PD and LCB increase the horizontal flow energy of activation, [E.sub.H], in other words both of these parameters increase the sensitivity of the rheological properties to temperature, (iv) PD and LCB increase the Van Gurp area (area included by the curve in a phase angle, [delta], versus the magnitude of the complex modulus, G*, curve), (v) PD and LCB increase the ends pressure (Bagley correction) and as a result they increase the extensional viscosity of polymers. The only difference in the effects of PD and LCB on the rheological properties detected in this work, was found to be in terms of a vertical flow energy of activation that is needed to shift the viscoelastic moduli data in order to obtain single master curves for G' and G" in the case of branched polymers.

Based on the experimental measurements, several correlations are proposed in this work. Namely, it was found that LCB and PD correlate well with the horizontal flow energy of activation and the Van Gurp area. In addition, LCB was found to also correlate well with the vertical flow energy of activation. Finally, these correlations were discussed in terms of using them as tools in separating the effects of LCB and PD as well as tools in detecting PD and LCB from rheological measurements.

ACKNOWLEDGMENTS

Financial assistance from the Natural Sciences and Engineering Research Council of Canada (NSERC) is also gratefully acknowledged. Most of the resins used in this work were supplied by NOVA Chemicals.

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INTRODUCTION

The melt rheology of entangled polymers is strongly influenced by molecular weight, its distribution (polydispersity) and the level of long-chain branching [1-3]. Although the effects of polydispersity and long chain branching are pointing in the same direction, the latter are much stronger and more evident (4-8). It would be desirable to have a way in separating these effects and if possible to assess whether or not possible effects in rheology or processing are due to long chain branching or polydispersity.

There is lately a considerable interest in detecting long chain branching in the structure of polymers using rheological techniques (9-17). These techniques aim at detecting small level of long chain branching in metallocene polyethylenes. This family of resins is a major recent technological advancement in the polyolefins industry, i.e., the development of homogeneous metallocene catalysts for polymerization reactions. Metallocene catalysts produce polymers with narrow molecular weight and comonomer distributions, which, combined with controlled amounts of long chain branching, is claimed to lead to both excellent processability and superior mechanical properties (18-20).

The presence of long branches in the structure of polymers affects the rheological properties in different ways. The zero-shear viscosity of polymer melts with long branches has been found to scale exponentially with [M.sub.w] [21]. This makes their zero-shear viscosity much higher than those of their linear counterparts. In addition, branched polymers exhibit higher degrees of shear thinning [1], the onset of shear thinning occurs at smaller shear rates, finally they exhibit higher extrudate swell, and extensional viscosities [22, 23]. Similar effects to these of long chain branching are also exhibited by polydispersity, although to a milder degree [6, 7, 9, 23, 24-26]. The effects of number of arms and the arm molecular weight on the rheological properties in the case of star-polymers have been studied in detail elsewhere (27-29).

Various authors have developed and proposed correlations between long chain branching and rheological properties. These include the relationships between the long chain branching and (i) flow energy of activation [4, 25, 26], (ii) a normalized elevation of the flow energy of activation [16], [iii] the area in the Van Gurp plot [atan{G"/G') vs G*) included between the curve corresponding to the branched polymer and its linear counterpart, and finally (iv) the shift in the [M.sub.w]-location that corresponds to the maximum in the predicted molecular weight distribution from rheological data with respect to the [M.sub.w]-location of the maximum from GPC data. It is worthwhile to mention that Janzen and Colby [13] have developed a model to theoretically predict the effect of molecular characteristics on the zero-shear viscosity of the polymers. Their method is beyond from a simple correlation.

In the present work, we use the viscoelastic moduli of a large number of polyethylenes in order to evaluate/test some of the previously proposed correlations between levels of long chain branching and polydispersity with the rheological properties. These correlations together with some new ones in a graphical form can be used to correct for the effects of polydispersity or long chain branching in order to assess the effect of these two molecular features on the rheological properties independently.

EXPERIMENTAL

Polymers Samples

The viscoelastic moduli data of a large number of polyethylenes were used in this work. Some of the data are already published elsewhere [9, 25], while some others were measured as part of this work. All polyethylenes whose data are used in the present work are summarized in Tables 1 and 2 along with molecular and structural characteristics. The molecular data were provided by the manufacturers along with the resins. Molecular weight and its distribution were determined by GPC, while Long Chain Branching (LCB) by means of the spectroscopic technique of [[blank].sup.13]C NMR.

Table 1 includes two series of linear low-density polyethylenes (LLDPEs). The first series includes ten LLDPEs (Dow's Dowlex 2049 and LLDPE A-I) having about the same molecular weight and various polydispersities (defined as PD [equivalent] [M.sub.w]/[M.sub.n] ranging from 3.3 to 12.7. These polymers are essentially blends of LLDPE A and I [9]. The second series includes six LLDPE's (LLDPE J-O) having about the same polydispersity and various molecular weights ranging from 51 k to 119 k. The data for these resins have been published by Kazatchkov [9] and Hatzikiriakos et al. [25].

Table 2 includes a series of LDPE's (randomly branched polyethylenes) of various melt indexes ranging from 2 to 7.11. There are also two samples with a small degree of long chain branching, labeled as mLLDPE A and B, which are used to assess the effect of LCB on the rheological properties of polymers. The viscoelastic moduli data of all eight branched samples were obtained as part of the present work.

Rheological Measurements

Linear dynamic oscillatory measurements were carried out for all resins at temperatures in the range of 160-220[degrees]C using: (a) a Rheometrics System IV equipped with 25 mm parallel plates and (b) an ARES (Rheometrics Scientific) in the parallel plate geometry having a plate diameter of 25 mm. From the experiments the complex moduli, G' (storage modulus) and G" loss modulus), the complex viscosity, [[eta].sup.*] [equivalent] [square root][G'.sup.2] + [G".sup.2]/[omega] and the flow energy of activation (using the time-temperature superposition) were determined for all polymers listed in Tables 1 and 2.

An Interlaken sliding plate rheometer with a 0.5-mm gap was used to determine the true shear viscosities, which were also compared with Bagley- and Rabinowitsch-corrected capillary data [1]. An Instron capillary rheometer was used with dies having the same diameter and various length-to-diameter, L/D, ratios (0, 10, 20 and 40) in order to calculate the true shear viscosities of the polymers as well as the end pressure, [delta][P.sub.ends], known as Bagley correction. The pressure drop associated with changes in velocity profiles at the entrance and exit regions (end or Bagley correction) was calculated using the standard Bagley technique [6].

RESULTS AND DISCUSSION

Zero-Shear Viscosity

Figure 1 depicts the complex viscosity of a linear metallocene polyethylene (Exxon Chemical's Exact [R] 3128), two metallocene LLDPE's and LDPE A listed in Table 2. The effects of long chain branching (LCB) on the shape of the complex viscosity are evident. The zero-shear viscosity increases with increase of LCB, although at the frequency of 0.01 rad/s the Newtonian viscosity has not been reached for resins mLLDPE B and LDPE A. In addition, it is clear from Fig. 1 that with increase of LCB, the degree of shear thinning increases, while its onset occurs at smaller frequencies. The continuous lines in Fig. 1 have been drawn to guide the eye to see easily the effects of LCB on the shape of the complex viscosity curve.

Figure 2 plots the zero-shear viscosities of a number of resins listed in Tables 1 and 2 as a function of molecular weight, [M.sub.w] at 200[degrees]C. These were calculated either directly from the experimental data or by fitting a Carreau-Yasuda model that gives the zero-shear viscosity as a parameter. Two important issues arise from Fig. 2 that must be discussed. The zero-shear viscosities of resins J to 0 define a line with slope of 4.1, which means that the zero shear viscosity for this series of resins scales as, [[eta].sub.0] [alpha] [[M.sup.4.1].sub.w]. The two branched polyethylenes (mLLDPE A and B) possess viscosities much greater than those defined by the continuous line, an observation expected. In addition, it can be seen that the viscosities of resins LLDPE A to I, are also greater than those defined by the continuous line. The higher the polydispersity, the higher the deviation from the scaling [[eta].sub.0] [alpha] [[M.sup.4.1].sub.w]. It clearly appears that polydispersity has a significant effect on the zero-shear viscosity of LLDPE. It is worth mentioning that resins LLDPE A to I have the same percent of comonomer and therefore the increase in viscosity is not due to the higher level of short chain branching (SOB). Obviously, more systematic studies are needed in order to confirm the observed effect of polydispersity on the zero-shear viscosity of linear polyethylenes and the deviation from the well-established scaling of scaling [[eta].sub.0] [alpha] [[M.sup.3.4].sub.w] that is true for monodisperse and nearly monodisperse polymers.

Therefore, the effects of polydispersity and LCB on the shape of the complex viscosity curve as well as on the Newtonian viscosity are similar, It can also be concluded that these effects are the same on the absolute viscosity curve in view of the validity of the Cox-Merz rule. It would be desirable to have a method to separate these two effects, i.e., to know, for example, if a certain effect is due to polydispersity or presence of LOB.

Linear Viscoelastic Moduli

Mavridis and Shroff [2] studied the viscoelastic behavior of a large number of polyethylenes in an attempt to identify the effects of long branches on the rheological behavior of polyethylenes. It was reported that in the case of linear polyethylenes (high-density and linear low density) a vertical shift to the viscoelastic moduli data is not necessary in order to obtain single master curves, i.e. when the time-temperature superposition is applied. On the other hand, it was shown theoretically and demonstrated experimentally that neglect of the vertical shift cause failure to the applicability of the time-temperature superposition principle. This observation also leads to a stress-dependent flow energy of activation.

For temperatures greater than [T.sub.g] + 100, the horizontal shift factor resulting from application of the time-temperature superposition to the viscoelastic moduli data can be interpreted in terms of a flow activation energy defined through an Arrhenius type equation, as follows:

[[alpha].sub.T] = exp[[E.sub.H]/R(1/T - 1/[T.sub.ref])] (1)

where [[alpha].sub.T] represents the horizontal shift factor, [E.sub.H] the horizontal flow energy of activation, R is the gas constant, R is the absolute temperature, and [T.sub.ref] is the reference temperature. Similarly, Mavridis and Schroff (4) interpreted the vertical shift in terms of a vertical flow energy of activation using an Arrhenius type equation as follows:

[b.sub.T] = exp[[E.sub.V]/R(1/T - 1/[T.sub.ref])] (2)

where [b.sub.T] represents the vertical shift factor, and [E.sub.V] the vertical flow energy of activation.

To see if a vertical shift is necessary a Cole-Cole plot (G" versus G') may be constructed. If this is independent of temperature, a vertical shift is not necessary and the master curves for the viscoelastic moduli can be obtained by applying the time-temperature superposition in the horizontal direction only. Such Cole-Cole plots were constructed for all resins listed in Tables 1 and 2. It was found that no vertical shift was necessary to superpose the data for all polyethylenes listed in Table 1 (linear with no LCB). This was not true for those listed in Table 2, which include polymers havings various levels of LCB. This observation can obviously be used to distinguish between the effects of LCB and PD in view of the fact that Cole-Cole plots of polydisperse polyethylenes are temperature independent. Therefore, failure of superposition in a Cole-Cole plot is a sign of the presence of LCB in the molecular structure of a polymer. Note that up to this point LCB and PD exhibited similar effects on the rheologi cal properties of the resins.

Figure 3 is a Cole-Cole plot for three different LLDPEs having about the same [M.sub.w] but different polydispersities. The data for three temperatures were plotted. It can be seen that the Cole-Cole plot for these linear polymers of various polydispersities (3.3, 6.6, and 12.7) is independent of temperature. Therefore, a vertical shift is not necessary to obtain the master curves of viscoelastic moduli. However, this is not the case for branched polyethylenes. Figure 4 is a Cole-Cole plot for two branched polyethylenes, namely mLLDPE A and LDPE A. It can clearly be seen that a vertical shift is necessary to superpose the data. Calculating the vertical shift factors by minimizing the standard deviation of all data from the data corresponding to the reference temperature, and referring all data to this reference temperature, Fig. 5 is obtained. In fact, these shift factors were calculated using an alternative form of the Cole-Cole plot, that is the shift angle, [delta] [equivalent] [alpha]tan (G"/G') versus G* [equivalent] [square root][G'.sup.2] + [G".sub.2], a plot known as the Van Gurp plot [10, 17]. Note that in a Cole-Cole plot, the data shift should be performed in both directions, whereas in a Van Gurp plot, the data shift should be done in only along the G* axis. Therefore, by comparing Figs. 4 and 5, the need for a vertical shift to superpose the data for branched polymers becomes clear.

The vertical shift factors were calculated for all branched polymers listed in Table 2. These were used to fit Eq 2 in order to calculate the vertical flow energy of activation, [E.sub.V]. Once the linear viscoelastic moduli data were corrected by taking into account the vertical shift factors, [b.sub.T], the time-temperature superposition was applied in the horizontal direction in order to obtain the master curves of the viscoelastic moduli for all resins listed in Tables 1 and 2. Moreover, the horizontal flow energy of activation, [E.sub.H], was calculated by means of fitting Eq 1. The values of [E.sub.V], and [E.sub.H] are listed in Table 3 and are plotted in Fig. 6 as a function of PD and LCB. First, from an inspection of the values of [E.sub.V], and [E.sub.H], it can be seen that these correlate well with each other. A higher value of [E.sub.V] implies a higher value of [E.sub.H] and vice versa. This is somehow expectable, since the need of vertical shift to the data and consequently application of such a correction before the application of the time-temperature superposition principle, imposes a higher temperature sensitivity to the viscoelastic data and therefore a higher [E.sub.H] value is calculated. The values of [E.sub.H] calculated in this work agree well with those reported previously in the literature [4, 16, 26]. However, the [E.sub.V] values are higher than those calculated by Mavridis and Shroff (4). They have reported values in the range of up to 2.5 kcal/ mol, whereas in the present work we have calculated [E.sub.V] values in the range from 3.2 to 5.26 kcal/mol.

It can be seen from Fig. 6 that [E.sub.V], and [E.sub.H] correlate well with LCB and that [E.sub.H] correlates with PD as well. The values for [E.sub.V], and [E.sub.H] of LDPE (LCB [greater than or equal to] 3) were calculated by averaging all values listed in Table 2. The continuous lines on Fig. 6 are drawn to guide the eye. Once again, it was seen that the effects of PD and LCB are similar as far as the horizontal flow energy of activation concerns. The differences between the effects of LCB and PD on the rheological properties of polyethylenes are two. First, the values of the horizontal flow energy of activation for the case of polydisperse polymers (up to polydispersities of 12.7) are smaller than those of branched polymers with 0.01 branches per 1000 C atoms or more. Second, for the case of branched polymers a vertical shift should be applied to the viscoelastic moduli to obtain the master curves. These two differences may be used to differentiate between the effects of PD and LCB. For example, a valu e of [E.sub.v] greater than 2 kcal/mol and a value of [E.sub.H] greater than 9 kcal/mol would imply the existence of LCB. Then, the correlations plotted in Fig. 6 may be used to determine the level of LCB. Smaller values for [E.sub.H] (smaller than 9 kcal/mol) as well as calculation of insignificant values of [E.sub.v] (smaller than 1 kcal/mol) would imply a polydisperse polyethylene with polydispersity that depends on [E.sub.H] and according to the correlation plotted in Fig. 6.

Van Gurp versus Cole-Cole Plot

Besides the Cole-Cole plot that is used to verify whether or not a fluid is thermorheologically simple (applicability of the time-temperature superposition principle), another way is to use the Van Gurp plot (10). This is a plot of the loss phase angle, [delta], defined as [delta] [equivalent] atan (G"/G'), versus the magnitude of the complex modulus, G* defined as G* [equivalent] [square root][G'.sup.2] + [G".sup.2]. Such plots have been used in the past (4) to determine the vertical shift needed to superpose the viscoelastic moduli of the low-density polyethylenes (see a previous section). Friedrich et al. (17) have used such plots in an attempt to identify a rheological correlation with the level of long chain branching (30). The same method is also used in the present work for assessing not only the effect of long chain branching on the rheological properties but those of polydispersity as well. It is noted that the Cole-Cole plot might give the same information with the corresponding Van Gurp ones, but t he latter show the effects more clearly.

Figure 7 is a Van Gurp plot for the polyethylene resins listed in Table 1 that have about the same polydispersity (3.4-3.9) and various molecular weights in the range of 51k to 114k. It can be seen that all the data define about a single line. It seems that molecular weight has no effect on the Van Gurp plot at least in the molecular weight range examined in the present work. This range of molecular weight covers mostly the range of interest from the industrial point of view. On the other hand, the effect of polydispersity on the Van Gurp plot is more obvious. Figure 8 is a Van Gurp plot for some polyethylenes listed in Table 1 that have about the same molecular weight (90k-119k) and various polydispersities in the range of 3.3 to 12.7. The effect is clear. Increasing the polydispersity, the Van Gurp curves shifts to smaller values of the phase angle, [delta]. One may use the area included under the curve (called Van Gurp Area) as a parameter to correlate with polydispersity. This is done and discussed in det ail below.

The effect of long chain branching on the Van Gurp plot is similar to that of polydispersity. Figure 9 is a Van Gurp plot of a series of polyethylenes listed in Table 2 that have different levels of long chain branching. It can be seen that by increasing the level of long chain branching, the Van Gurp curve shifts to smaller values of the phase angle, [delta]. Again the area included under the curve (called Van Gurp Area) is used below as a parameter to correlate with long chain branching.

Determining the viscoelastic moduli of various polymers at various temperatures, and consequently applying the time-temperature superposition principle, one may end up with data covering a different range of frequency and complex modulus values for different resins. Therefore, in order to calculate the area included under the van Gurp plot, one has to refer the data of all resins under the same range of complex modulus. In this case we have selected as an accessible range of complex modulus values for all polymers to be from [10.sup.2] to [10.sup.5] Pa. This range of complex modulus values covers the accessible by most rheometers' range of frequencies. In some cases, data extrapolation was used for some resins in order to cover the whole range, as appears evident by the continuous lines plotted in Figs. 8 and 9. Having defined the range of complex modulus, the areas under each curve in Figs. 8 and 9 were calculated, and these are plotted in Fig. 10 as a function of polydispersity and long chain branching. The existence of correlations between the Van Gurp area ([10.sup.2]-[10.sup.5] Pa) with long chain branching and polydispersity are clear. It is once more evident that the effects of long chain branching and polydispersity are similar.

The significance of Fig. 10 can be defined as follows. In many cases, long chain branching and polydispersity are present in the structure of a single polymer. Therefore, before using Fig. 10 to detect the level of long chain branching, a correction may be applied to account first for the effect of polydispersity. Consequently, having the new "corrected" value of the Van Gurp area, one may proceed to use Fig. 9 in order to calculate the level of long chain branching. Clearly, such an approach excludes synergism between LOB and polydispersity. Such synergism might be possible.

It has been discussed in the past that crossover frequency, where the storage (G') and loss (G") moduli are equal, can serve as an indicator of the resin viscoelasticily (polydispersity and long chain branching effects). In other words, its inverse may represent a single characteristic relaxation time of a given resin (21). Zeichner and Patel (31) have noticed a correlation between polydispersity of a linear resin and its crossover modulus ([G.sub.c] = G' = G"). They have found that if a rheological polydispersity index defined as PI [equivalent] [10.sup.5]/[G.sub.c], where [G.sub.c] is in Pa, is plotted versus [M.sub.w]/[M.sub.n] on a double logarithmic plot, the data points would fall on a straight line (6). These types of correlation as well as some others discussed in (7) are based on a single point and they are not always successful (9). The correlations proposed in this work are based on the use of Van Gurp plot that utilizes the whole viscoelastic moduli curves of the resins and therefore might be more appropria te to use. Most important, the proposed correlations are temperature independent ones. Finally, it might be argued that the shapes of the Van Gurp plots mirror to a large extent those of the complex viscosity versus frequency and therefore they show the expected dependence on polydispersity and branching.

Ends Pressure In Capillary Flow

Figure 11 plots the end pressure correction known also as Bagley correction, of representative polyethylenes listed in Table 1 as a function of the apparent shear rate. These were calculated by using a conventional Instron capillaxy rheometer with an orifice die having a contraction angle of 90 degrees and a final diameter of 0.5 mm. It is interesting to note that the Bagley correction function shifts to higher values with increase of polydispersity. Similar effects can also be observed for the effect of long chain branching on the Bagley correction. Figure 12 plots the ends pressure correction of representative polymers having various degrees of long chain branching listed in Table 2 as a function of the apparent shear rate. The end pressure correction (Bagley correction) increases with increase of the long chain branching.

These results are not surprising since it is known that the elasticity of polyethylenes increases with polydispersity and long chain branching. Since the end pressure (Bagley correction) can serve as an indication of the level of extensional viscosity of resins [32] and since this increases with polydispersity and long chain branching [22-24], this observation (increase of Bagley correction with polydispersity and long chain branching) is in order. These results also corroborate our previous discussion that the polydispersity and long chain effects on the rheological properties are similar. The only difference is that in the case of branched polymers, a vertical shift of the viscoelastic moduli is needed in order to obtain master curves with the application of the time-temperature superposition.

CONCLUSIONS

Several experimental techniques were employed to study the rheological behavior of a large number of polyethylenes in order to assess independently the effects of polydispersity (PD) and long chain branching (LCB). It was found that most of the effects of LCB and PD are similar. In particular (i) PD and LCB increase the zero-shear viscosity more than that predicted by the scaling relationship, [[eta].sub.0] [alpha] [[M.sup.3.4].sub.w], where the exponent of 3.4 might be slightly higher, i.e., 4.1 in the present work, (ii) PD and LCB decrease the critical shear rate for the onset of shear thinning as well as increase the degree of shear thinning, (iii) PD and LCB increase the horizontal flow energy of activation, [E.sub.H], in other words both of these parameters increase the sensitivity of the rheological properties to temperature, (iv) PD and LCB increase the Van Gurp area (area included by the curve in a phase angle, [delta], versus the magnitude of the complex modulus, G*, curve), (v) PD and LCB increase the ends pressure (Bagley correction) and as a result they increase the extensional viscosity of polymers. The only difference in the effects of PD and LCB on the rheological properties detected in this work, was found to be in terms of a vertical flow energy of activation that is needed to shift the viscoelastic moduli data in order to obtain single master curves for G' and G" in the case of branched polymers.

Based on the experimental measurements, several correlations are proposed in this work. Namely, it was found that LCB and PD correlate well with the horizontal flow energy of activation and the Van Gurp area. In addition, LCB was found to also correlate well with the vertical flow energy of activation. Finally, these correlations were discussed in terms of using them as tools in separating the effects of LCB and PD as well as tools in detecting PD and LCB from rheological measurements.

ACKNOWLEDGMENTS

Financial assistance from the Natural Sciences and Engineering Research Council of Canada (NSERC) is also gratefully acknowledged. Most of the resins used in this work were supplied by NOVA Chemicals.

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Author: | HATZIKIRIAKOS, SAVVAS G. |
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Publication: | Polymer Engineering and Science |

Date: | Nov 1, 2000 |

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