Structure-property relationships--linear and star-branched macrostructures.

Rheological characterization

The theories of entanglement coupling are central to the discussion of the viscoelastic behavior of linear polymeric materials when measured in the solid state. The addition of long chain branching further alters the flow characteristics of elastomers. It has been reported that at low overall molecular weights, the effect of introducing branching alters the linear viscoelastic behavior of many polymers by actually lowering the viscosity (refs. 99, 62 and 100). However, when the branch molecular weight is longer than the entanglement molecular weight ([M.sub.e]), an increase in the low shear viscosity is seen. Kraus and Gruver (ref. 61) demonstrated that, for alkyllithium initiated PBds with three-and four-arm star macrostructures, [[eta].sub.0] became higher than the linear control PBd at an overall molecular weight of approximately 1.00E+05 g/mol. The same was seen by Graessley, et al. (ref. 101) for poly(isoprenes), where it was also shown that at similar molecular weights the recoverable compliance ([J.sub.e.sub.0]) of the star macrostructure is much higher than the linear analog of equivalent molar mass. Similar results were also reported by Masuda (ref. 102) for poly(styrene) systems. In terms of branch molecular weight, this was about 3[M.sub.e] and 4[M.sub.e] for the tri-chain and tetra-chain poly(butadiene) structures, respectively. It was concluded that the smaller hydrodynamic size of the branched structures dominated the flow properties until the branch length was long enough to entangle with other molecules. At this point, the additional constraint of having a fixed branch point at the center of the star molecules that is incapable of slippage increases the zero shear viscosity of the star samples to well above that of the linear control.

Again, molecular theory has been applied to describe a subset of entanglement theory that accounts for long chain branching. In the references cited above, Graessley has most thoroughly developed this model. The conventional entanglement theory used for linear chains does not adequately explain the behavior of chains containing branch points. Consider the case of a polymer system with tertiary branch points where all three branches that emanate from the node are long enough to further entangle. The branch nodes cannot easily slip through the matrix of entanglements. For this to occur, all the chains bound by the branch node would have to completely rearrange their conformations in order for the parent chain to pass the points of restriction. The translational diffusion coefficient, as defined by reptation theory, for branched chains is significantly lower than for the linear case. Using this type of argument, it is clear that mobility along the contour of the chain or, equivalently, inside the tube defined by entanglement loops, is subsequently reduced as branch nodes are introduced. In the absence of the permanent network structure typical of crosslinked materials, the reduction in chain mobility will result in longer relaxation times and higher zero shear viscosities. Entanglement theory corrections for the case of branched structures utilize an enhancement factor that accounts for the diffusion of the branches in concert with the center of gravity of the translating molecule. The side branches must diffuse into entirely new tunnels for each incremental displacement of the center of gravity. In the above reference, Graessley proved that when plotting the viscosity enhancement due to branches as a function of a parameter that describes the topological impedance due to side tunnels, the data fall on a single line regardless of branch molecular weight as long as it is above [M.sub.e]. Also, the data suggest that the presence of one branch point (idealized by a three-arm star) contributes the majority of the viscosity increase; the viscosity-enhancing effect diminishes as more arms are added about the node.

Subsequent work by Doi and Kuzuu (ref. 103) has accurately modeled the experimental data generated in the linear viscoelastic regime from the references cited above. The authors accurately reproduced the differences in trends between linear and star samples seen for [[eta].sub.0] plotted as a function of molar mass, and G' and G" as a function of frequency in the terminal region of the relaxation spectra.

Pearson and Helfand (ref. 104) have taken the reptation concept and applied it to the specific case of star-branched polymers and reduced the disentanglement process to diffusion of the ends of an arm in a free energy potential field. Several researchers have shown that the viscosity of stars does not depend on the overall molecular weight, but only that of the arms. The viscosity of star polymers with 4 to 33 arms can be plotted on a single reduced curve if the independent variable was arm molecular weight (refs. 105 and 106). This concept has been summarized by Fetters, et al. (ref. 107) in work with poly(isoprene) stars where the arm functionality equals 3 and 4. The authors have chosen the span molecular weight ([M.sub.s]) as their reduced variable. The span molecular weight corresponds to the longest, continuous span that can be traced along the molecule, and is taken as twice the arm molecular weight of the star. It is shown that the zero shear viscosities of the star samples do not increase above the linear counterparts until the Ms is approximately 1.00E+05 g/mol.

The non-linear viscoelastic behavior of elastomers is also affected by branching. Under higher shear rates, the viscosity of a branched sample enters the non-linear regime of viscoelasticity at lower shear rates (longer relaxation times) than a linear counterpart at the same molar mass (refs. 1 and 75). The star-branched samples also have lower viscosities at the higher shear rates (ref. 101). The characteristic shear rate ([[gamma].sub.0]), [[eta].sub.0] and [J.sub.e.sup.0 ]are closely related. For linear polymer with low polydispersity, equation 6 holds true (ref. 54).

(6) [[eta].sub.0] [J.sub.e.sup.0][[gamma].sub.0] = 0.6 [+ or -] 0.2

The higher viscosities of branched materials in the linear regime and their increased dependence of viscosity as a function of shear rate can result in products which display a higher resistance to cold flow, yet also display improved processing characteristics. The viscoelastic properties of the experimental samples were studied below. The flow behavior of the linear and star materials was evaluated in both the linear and non-linear regimes.

In industry, often the goal of rheological measurements on both raw gum elastomers and filled compound stock is to correlate the behavior of the materials under shear deformation to the processing characteristics that will be displayed when the material is used on industrial processing equipment. That the macrostructure of the material plays a dominant role in determining the processing behavior is not lost on anyone who has attempted to substitute an emulsion-prepared elastomer with its solution-prepared counterpart. As outlined above, there are several methods used to characterize the processing behavior of elastomeric materials with varied macrostructure. In this study, oscillatory shear measurements under various testing conditions are evaluated to better understand the effect of long chain branching.

Oscillatory shear measurements were used to reconstruct the plateau and terminal zones of the relaxation spectra for selected experimental samples. From the relaxation spectra, qualitative information regarding the shape of the shear moduli curves as a function of frequency can be observed. Quantitative information such as the plateau moduli ([G.sub.N.sup.0]) and the frequency at which the storage and loss shear moduli crossover (equivalent to tangent [delta] = 1.0, indicating the beginning of the terminal zone) can also be taken from the measurements. This information can help discern between macrostructural differences in the samples, such as the overall molecular weight and the presence or absence of branching, since the chain mobility is affected by such changes.

A parallel plate rheometer was used for the oscillatory shear measurements of the PBds. The time-temperature superposition principle was employed to reconstruct the plateau and terminal zones of the relaxation spectra for each sample by combining the frequency sweep data at several different temperatures using calculated shift factors. Table 4 summarizes the temperatures and corresponding shift factors used for the analysis. The shift factors were calculated using the reported empirical constants for lithium PBd of 3.64 and 186.5 degrees for [C.sub.1.sup.0] and [C.sub.2.sup.0], respectively (ref. 108). The reference temperature was 323[degrees]K. It has been established that the shift factors for linear and simple star polymers are equivalent, regardless of macrostructure, given that the molecular weight is above [M.sub.e] (refs. 102 and 107). This equivalence does not hold for more complex branched structures, including long chain branching and dendritic structures. That the shift factors are universal for a given microstructure indicates that the tractional free volume and thermal expansion coefficient do not change significantly with the number of chain ends in high molecular weight polymers.

Figure 8 contains a log-log plot of shifted shear moduli for the lowest and highest molecular weight linear samples (samples A and F). It is apparent that the plateau modulus seen at high frequencies was equivalent for the two samples, as was expected. The magnitude of G' and G" are independent of molecular weight in the plateau zone, as the moduli values here are governed by the segmental motion of chain segments between entanglements. The onset of the terminal zone, however, was extended as the molecular weight of the sample increases. This is clearly evident in figure 9, displaying the data for the lowest and highest [M.sub.w] star samples (samples G and L). It is apparent that the delay in the translational flow of the polymer molecules is governed by molecular weight in both the macrostructures studied.

[FIGURES 8-9 OMITTED]

Figure 10 compares the relaxation spectra tot the linear and branched samples at equivalent molecular weights (nominal 250,000 g/mol, samples D and J). Looking at the high frequency area of the plot, it was determined that the plateau moduli are roughly equivalent for each pair of samples. The onset of the terminal zone has been extended to lower frequencies for the star sample, however, indicating that the rigid branch node represents an additional hurdle for reptational flow. Table 5 contains the crossover frequency data as a function of sample molecular weight and macrostructure. The crossover frequency delineating the transition between the plateau and terminal zones has been equated to approximately the reciprocal of the longest relaxation time of the sample. Within a certain macrostructure, the crossover frequency decreased with molecular weight. When comparing the branched and linear samples at a given molecular weight, the star macrostructure consistently had much lower crossover frequencies.

The shape of the loss moduli curve above the crossover frequency also has been used to indicate that there is another flow constraint inherent to the star-branched macrostructure. Above the crossover frequency, G" for a linear sample decreases to a minimum before increasing again towards the transition zone.

The G" curves of star polymers continue to increase past the crossover frequency and reaches a maximum before the onset of the transition zone. The G" curves in figure 10 demonstrate this behavior. The maximum in the G" curve within the plateau zone for star macrostructure has been correlated to the relaxation time of the arm, which is proportional to the arm molecular weight (ref. 109).

[FIGURE 10 OMITTED]

The shape of the relaxation spectra can also be influenced by polydispersity. It has been demonstrated through blending studies for several polymer types that an increase in the polydispersity of the sample can alter the crossover frequency and make the transition between the zones less distinct (refs. 110 and 111). Since the star samples do contain a small amount of uncoupled linear material, the shorter relaxation times associated with this material may have had a slight effect on the crossover frequencies. Kasehagen, et al. (ref. 112), has shown how altering the percentage of branched structures in an otherwise linear system affects the rheological properties of PBd in both oscillatory and creep recovery testing.

Overall, the samples proved to be rheologically simple, such that when the calculated shift factors were applied, there was a good fit between the data collected at the different temperatures. However, the highest molecular weight star sample (sample L) did not produce as clean a spectra as the other samples.

The dependence of G' and G" on frequency and strain was also measured using a rubber process analyzer. The RPA is a morn robust piece of equipment commonly used in industrial applications where ease of sample preparation, simplicity of operation and minimal maintenance are required. Unfortunately, the advantages in operation also come with the disadvantage of limitations on the frequencies and strains attainable. At angular strains below one degree, the RPA can provide frequencies between 0.20 to 200.0 rad/s, but at larger strains, the frequency range becomes more limited (ref. 113). The RPA was used over the limited flequency range available to reproduce the relaxation spectra at low strain (2%). Figures 11 and 12 display the results tot the linear and star samples, respectively. Again, the plateau moduli were equivalent for the different samples, and the presence and value of the crossover frequency (in addition to the shape of the curves) was dependent on the macrostructure. The relaxation spectra are not as well developed as in the parallel plate-derived data, owing to the limited frequency range of the RPA. This limitation also results in a loss of differentiation between the star samples, since the terminal zone is never reached under these test conditions, and experiments at higher temperatures did not produce significantly different results. Shear moduli data was simplified by plotting tangent 6 as a function of frequency on a semi-log plot in figure 13. In this type of representation of the data, it is easy to see how far into the terminal zone the samples progress under the test conditions. Also, this type of representation allows for the comparison of samples with very different absolute values of shear moduli, since the data are essentially normalized in the tangent [delta] values. The higher the tangent [delta] at low frequency, the more flow the sample exhibits. The star samples do not attain tangent [delta] values above unity, which is the value that marks the onset of flow.

[FIGURES 11-13 OMITTED]

To determine shear rate dependence, the RPA frequency sweep collected at low strain (2.0%) can be utilized. Cox and Merz (ref. 114). found thai the complex viscosity |[eta]*| measured as a function of angular frequency under small deformation correlates very closely to the non-linear shear viscosity measured as a function of shear rate. This close correspondence has been confirmed by a number of studies (refs. 54 and 55), and is outlined in equation 7.

(7) |[eta]*| ([omega]) [approximately equal to] ([eta][gamma])

Figure 14 contains a log-log plot of the complex viscosity data as a function of frequency. It is evident from the data that the linear sample exhibited linear viscoelastic behavior at the lower frequencies (corresponding to lower shear rates) and entered the power-law regime at the higher frequencies. The star sample, however, exhibits a strong shear rate dependence at all frequencies. Although the curves are not complete, it is obvious that the star samples will have higher low shear viscosities (low cold flow properties) and lower high shear viscosities (improved processing) than the samples with linear macrostructure. Typical of linear PBds, the low frequency (zero shear) viscosity data for these samples is proportional to the molecular weights.

[FIGURE 14 OMITTED]

The dynamic storage moduli collected at 8.0 rad/s were also compared for the selected samples. Figure 15 summarizes the G' data as a function of overall molecular weight and macro-structure at low strain. Closer examination of the slopes of the trendlines for the different macrostructures allowed for further analysis of the flow mechanism. Table 6 summarizes the slope data as a function of macrostructure using both overall molecular weight and span molecular weight ([M.sub.s]. When overall [M.sub.w] was used for comparison. there were large differences in the slopes for the two macrostructures at low strain. Upon shearing, the conformation of the polymer chain does not obey the random coil dimensions assumed in solution theory, but becomes elongated in the shear field. The conditions governing the alignment of molecules under strong flows has been addressed by Tanner and Huilgo (ref. 115) and Larson (ref. 116). Because the arms of the star are tethered at the branch node, perhaps the Ms is a more appropriate dimension to use in the analysis. Figure 16 uses Ms when comparing G' by macrostructure. By comparing the slope data with [M.sub.s] as the variable, the magnitude of the dependence of G' at low strain was shown to be similar for the two macrostructures. Again, entanglement theory and the additional restraints that the branch node imparts to molecular translational movement can explain the differences in the flow behaviors.

[FIGURES 15-16 OMITTED]

Table 4--test temperature and
corresponding calculated shift factors

Test temp.
(K)          [a.sub.T]

293            4.890
323            1.000
353            0.313
373            0.170
393            0.102
413            0.065
453            0.032
473            0.024

Table 5--crossover frequencies as a function of
macrostructure and [M.sub.w]

                  Peak [M.sub.w]   Shifted crossover
Sample    Type       (kg/mol)      frequency (rad/s)

A        Linear        208             2.92E+00
D        Linear        246             1.35E+00
F        Linear        296             9.24E-01

G         Star         199             1.90E-02
J         Star         243             2.95E-03
L         Star         292            <5,00E-03

Table 6--slopes of the storage moduli plots as
a function of weight average molecular weight,
span molecular weight and macrostructure

Slope @ 8.0 rads/

            Linear   Star

[M.sub.w]            1.22
[M.sub.s]    2.70    2.45

Table 6--slopes of the
storage moduli plots as
a function of weight
average molecular
weight, span molecular
weight and
macrostructure

Slope @ 8.0 rad/s

            Linear   Star

[M.sub.w]            1.22
[M.sub.s]    2.70    2.45

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Steven K. Henning, Goodyear (This is the third of a four-part series)