The application of stress relaxation to quality control of polyisobutylene.

Polyisobutylene rubber (IM) is produced at molecular weights ranging into several millions resulting in extremely high polymer viscosities. These high viscosities make it impossible to apply simple rheological tests for measuring common MW parameters, such as Mooney viscosity or melt flow rate. An additional concern with high MW polymers is shear degradation. Dynamic tests at low strains can yield MW information, but for high MW very low test frequencies are required which imply impractical slow tests and require highly sensitive instruments. MW of IM is therefore commonly determined by dilute polymer solutions methods such as GPC and intrinsic viscosity. The later test is the one used for QC and specification of IM rubber in terms of a viscosity average MW (VAMW). Solution viscosity is also a very slow test primarily due to low shear dissolving over a period of two to three days required to avoid molecular degradation. Such a slow test is not acceptable for production control and it imposes great difficulties in quality control.

Stress relaxation can provide a practical and quick basis for testing MW dependence (ref. 1). There are, however, a few possible problems to overcome for a stress relaxation test to be successful. The instrument should be reasonably sensitive at low stresses; a sharp step strain mode is desirable for initiating the stress; and stresses introduced during sample preparation should be relaxed before the testing stress is imposed. There are a few instruments, however, that can satisfy most of the above. This article describes the development of a method for a MW test based on one such instrument with stress relaxation capabilities. Test conditions, including sample preparation, and the proper choice of test parameters and their analysis are described.

Stress relaxation and linear viscoelasticity

Stress relaxation after a small constant step in strain is an established technique for the determination of linear viscoelastic properties. Linearity requires that in response to a step strain, the relaxation modulus is only a function of time and can be written as:

G(t) = [sigma]/[gamma] (1)

where [sigma] is the shear stress, [gamma] is the shear strain imposed at time zero and t is the time.

For demonstration purposes, the simplest model that can describe stress relaxation, as well as dynamic experiments, is the Maxwell element which is composed of a spring attached to a dashpot in series. If both components in the model display linear responses between force or stress to displacement (a Hookian spring) or displacement rate or shear rate (a dashpot containing a Newtonian, fluid), the relaxation modulus for this model can be shown to decay exponentially

[Mathematical Expression Omitted]

where G is the spring constant and [tau] is the relaxation time constant for the Maxwell element which is a combination of the spring constant G and the dashpot viscosity [eta]

[tau] = [eta]/G (3)

The area under the relaxation modulus vs. time curve from time zero to infinity is the viscosity il as can be derived by integrating equation (2)

[Mathematical Expression Omitted]

A number of Maxwell elements in parallel are needed to represent the relaxation mechanisms associated with a distribution of molecular weights. The relaxation modulus for such an array of M Maxwell elements can be shown to be

[Mathematical Expression Omitted]

where G and [tau] are the spring constant and relaxation time for each of the M Maxwell elements. The entire array of [G.sub.i] and [[tau.sub.i]] pairs is known as the relaxation spectrum where the value of [tau] is increasing from i = 1 to i = M. If the long time tail of the relaxation curve is fitted with a single exponential function, the relaxation time for this exponential decay, [[tau.sub.M]], is known as the longest or maximum relaxation time for the polymer. The area under the relaxation modulus vs. time curve as described by equation 4 is the zero shear viscosity, [[eta.sub.0]], for any linear viscoelastic material (ref. 1), regardless of which model may describe the relaxation data.

The dependence of the zero shear viscosity or the longest relaxation time on the weight average molecular weight for various polymers, including IM, was shown in earlier experimental work to follow a power law relationship (refs. 2-5) with a power of about 3 to 3.4. A power dependence of about 3.4 for zero shear viscosity was confirmed in a recent paper by Fetters et al (ref. 6).

[Mathematical Expression Omitted]

These dependencies were later derived by applying molecular modeling such as scaling and the reptation model (refs. 7 and 8). A scaling factor of three was originally predicted, but with some modifications (ref. 9) it may be possible to predict the observed 3.4 power factor.

Equations 6 and 7 suggest that parameters which can be directly measured by stress relaxation, such as the area under the relaxation modulus curve, or a characteristic time for complete relaxation of stress, have a very strong dependence on molecular weight. Stress relaxation is expected therefore to be a sensitive method for measuring MW if the proportionality constants in equations 6 or 7 are known. The constants may be obtained by calibrating stress relaxation results for a given family of polymers with known MW. However, stress relaxation parameters can be useful as relative MW indicators in the absence of a calibration.

Stress relaxation data also contains information on the molecular weight distribution (MWD) of a polymer. A polymer with a very narrow MWD is expected to display a relaxation behavior similar to a single Maxwell element while a polymer with a wider MWD has fast and slow relaxation modes which may be approximated by the generalized Maxwell model. A qualitative description of stress relaxation vs. time for polymers with various MW and MWD is shown in figure 1 for polymers which are well above their glass transition temperature. Since stress decays over a long time period it is customary to display stress relaxation data using logarithmic scales. The initial decay represents a transition from a glassy to a rubbery state which can not practically be measured at higher temperatures since it occurs over extremely short times. Stress decay of polymers with higher MW is significantly slowed down as shown by equation 7 and as shown schematically in figure 1. A wider MWD polymer contains elements of low and high MW and the resulting stress decay spans rates and times corresponding to the various chain lengths present. Therefore, measuring stress relaxation parameters at various times can potentially reveal information on MWD (ref. 10). Accordingly, attempts were made to reproduce a complete MWD from dynamic or stress relaxation data (ref. 11).

While the total area under the relaxation curve is a function of the weight average MW as shown in equations 6 and 7, relative decay rates or areas at partial times should contain information on MWD at least for relative comparisons. It should be pointed out that Mooney relaxation was proposed as a practical tool for evaluation of processability and quality control of rubbers (ref. 12). Mooney relaxation produces similar information to step strain stress relaxation but its analysis is complicated by two major factors. First, Mooney relaxation is a cessation of flow experiment and as such it masks some of the shorter relaxation mechanisms which relax during the steady flow portion of the experiment. A second complication arises from the large strains experienced in a steady flow experiment which take it beyond linear viscoelasticity. Mooney relaxation is advantageous, however, in that it provides an established MW parameter, Mooney viscosity, in the first phase of the test and analysis of the following relaxation portion provides additional information on structure such as MWD, branching, etc. A Mooney viscometer on the other hand can not be used with very high MW polymers and is not applicable to the analysis of IM rubbers.

Experimental

Polymers

A series of commercially available IM rubbers was used (table 1). The polymers varied in viscosity average MW (VAMW) from about 1-4 million. The polymers were obtained from two commercial sources that vary in MWD width. One family of polymers had a relatively narrow MVD with Mw/Mn about 2.1 to 2.5 while the other family had a wider distribution with Mw/Mn about 4.0. A few polymers were special polymers with a Mw/Mn in the 3.5 to 4.0 range.

Testing

Stress relaxation tests were performed with a rubber process analyzer, equipped with stress relaxation capability (ref. 13). The following test procedure provided for relaxing the initial stresses which result from molding the sample into the test cavity. The conditions used were also optimized to provide for maximum test sensitivity and reproducibility while staying as close as possible to the linear viscoelastic region.

A 5 g polymer sample was loaded into the test cavity after it was heated to 195[degrees]C. The analyzer was programmed to keep the sample for four minutes at 195[degrees]C followed by cooling down to 125[degrees]C over a four minute period. Since cooling is fast, the sample equilibrated at 125[degrees]C for about one minute. Stress relaxation was started by imposing a 5[degrees] (or about 70%) strain. The test was terminated when the stress dropped down to a value of I dNm which is a safe value above the zero noise limit of the instrument. The total area under the linear relaxation stress vs. linear time from one second up to the cut-off time was computed by the computer which is supplied with the analyzer for test control and data analysis. The area up to one second is negligibly small relative to total area measured for high MW polymers. A typical relaxation stress vs. time response in a dual logarithmic scale was duplicated from the instrument printout and is shown in figure 2.

Results and discussion

Stress relaxation test results for the IM rubbers listed in table 1 are shown in figure 3 in terms of RPA area vs. the respective VAMW values. The solid lines in figure 3 were obtained by a power law regression analysis of the data corresponding to polymers listed as series "A" polymers in table 1 and shown in the figure as circles. The resulting power dependence of RPA area on VAMW for the measured data up to the cut-off point of 1 dNm was 2.6. This result differs from the expected result of 3.4 shown in equation 6. It can also be seen that the points for polymers of series "B" and "C" resulted in measured RPA areas lower than that predicted by the polymers in series "A".

It was assumed that a large contribution to both types of discrepancies discussed above stems from the missing area fraction from cut-off time to infinite time required to describe zero shear viscosity. Although the cut-off stress of 1 dNm represents a low stress, the high MW fraction which is responsible for the terminal stress may relax over many decades of time and the missing area can therefore be significant. This missing area should be even more significant for polymers with a wide MWD having a longer high MW tail. We attempted to calculate the missing terminal area by assuming that the last part of relaxation measured on the RPA can be represented by a single exponential element such as was assumed by Andrews and Tobolsky (refs. 2 and 4) in their "Procedure X" for the analysis of stress relaxation. If a single exponential element from the series described by equation 5, where i = M, is fitted to the measured relaxation data from time [t.sub.2] to the cut-off time [t.sub.1], then,

[Mathematical Expression Omitted]

and the missing area from time [t.sub.1] to infinite time for a single terminal element is G([t.sub.1])[[tau.sub.M]]. If the values for G([t.sub.1]) and G([t.sub.2]) are set to 1 and 2 dNm respectively for times [t.sub.1] and [t.sub.2], then the missing area is 1.44([t.sub.1-[t.sub.2]). Correcting the relaxation area for all the data points shown in figure 3 as empty symbols resulted in the full symbols shown in this figure. Correcting for total relaxation area with this model brought most of the points corresponding to the wider MVM polymers of series "B" and "C" much closer to the regression line obtained for the narrower polymers of series "A". However, the slope of the corrected line moved up slightly, from 2.6 to 2.7. This may indicate that the area was not sufficiently corrected by the assumed model and that there are additional modes corresponding to the high MW tail that were not accounted for. Another possible reason for obtaining a lower power law fit to the data in figure 3 is the possible deviation from linear viscoelasticity at the approximate 70% strain used for the data in this article. A more sensitive instrument that can measure much lower stresses may be able to operate at lower strain and capture the missing area. Such tests are in progress using a mechanical spectrometer. Some preliminary data (ref. 14) indicated that deviations from linear viscoelasticity start at about 15% strain. However, tests in the linear viscoelastic regime as well as at 70% strain produced similar power dependencies which are close to those shown with the RPA. It is believed that these preliminary experiments have not been conducted for a long enough relaxation time to capture the total contribution to zero shear viscosity.

It should be noted that the initial evaluation of the instrument used in this work for MW determination involved dynamic test evaluation. It was found that in the dynamic test mode the instrument was not able to test at low enough frequencies to enter the flow region for the high MW IM rubbers of interest. Dynamic data at higher frequencies may be sensitive to MW at certain narrow regions of frequency but there is a possibility of entering the plateau modulus region which is insensitive to MW as is the initial data measured in stress relaxation. If, on the other hand, the instrument was capable of operating at the low frequencies required for the flow region, the test would have been extremely slow which would have made it unacceptable for QC.

To further evaluate the potential of the stress relaxation test used with the polymers in table 1, we applied the test to a production campaign of IM rubber. The results for this evaluation are shown in figure 4. The data in figure 4 show measured, uncorrected RPA area vs. measured VAMW similar to the lower curve in figure 3. Over 900 samples from the single campaign plus most of the samples from table 1 were incorporated in figure 4. The approximate slope of the line in figure 4 is 2.9.

We believe that stress relaxation measured with the RPA is suitable for quick QC tests relative to the more highly sensitive instruments currently used for research. Applying an area correction such as discussed above should improve the indication of an unknown MW. We also believe that a simple MWM index can be constructed from stress relaxation data. We plan to pursue such an index, perhaps by comparing relative areas at different time or stress portions of the measured stress relaxation. This should be complementary to the practical Mooney relaxation test which we discussed in the introduction and in a previous publication by Friedersdorf and Duvdevanil (ref. 2) and which is not applicable to high MW polymers such as the IM rubbers tested in this work.

A discussion of attributes necessary for a QC test can also be found in the Mooney relaxation paper by Friedersdorf and Duvdevani. To quantify the repeatability of the test with the procedure prescribed for this work, we conducted multiple repeats of a control polymer similar to sample A3 of table 1. We found that the standard deviation of the uncorrected RPA area was 43.7 dNm for a mean area of 957 dNm. Using the regression fit it translates to a VAMW of 1.79 with a standard deviation of 0.031, or 1.7%, in terms of MW in millions. This repeatability is similar to what was obtained for the MLR area parameter of Friedersdorf and Duvdevani. It also represents an acceptable repeatability for a QC test relative to the desired specification range in production but a further improvement in repeatability will be desirable. In an effort to improve the test for QC purposes, we plan to further investigate other potential parameters and other possible corrections. For example, running the test at two or more temperatures depending on the MW of the product may speed up the test and improve its accuracy. A relaxation time parameter, rather than an area parameter, will also be investigated as suggested by equation 7. Another important improvement which will be investigated is a MWD index derived from other test parameters as mentioned above.

     Table 1 - IM rubbers used in the study

[Polymer.sup.a]           [VAMW.sup.b]             [Mw/Mn.sup.c]
                        (MW in millions)
      A1                     0.93                       2.4
      A2                     1.21                       ---
      A3                     1.77                       2.1
      A4                     2.03                       2.2
      A5                     2.29                       ---
      A6                     2.60                       2.5
      B1                     3.29                       3.6
      B2                     4.10                       4.1
      C1                     2.60                       3.4
      C2                     2.81                       ---
      C3                     2.83                       3.4
      C4                     2.86                       ---

a. Polymers of series "A" are commercial Exxon Vistanex polyisobutylene polymers (Exxon Chemical Co.); polymers of series "B" are commercial BASF Oppanol polyisobutylenes; and polymer series "C" are special polymers with wider MWD. b. VAMW refers to viscosity average molecular weight expressed in millions of MW units and determined by dilute solution viscosity. c. MWD width index expressed as the ratio of weight average MW to number average MW obtained by GPC with a differential refractive index detector.

Summary

This article demonstrated a quick QC test based on stress relaxation for MW of high MW polymers such as IM rubbers. The test can potentially deliver MWD information, as well. The current test uses the area under the relaxation curve as the leading parameter since die area is theoretically related to zero shear viscosity. Improvements to the test accuracy are desired in order to account for MWM effects. Test repeatability may be acceptable but it can possibly be further improved by refining the analysis or by utilizing other test parameters.

References

[1.] J.L. White, in "Science and technology of rubber," 2nd ed., J.E. Mark, B. Erman and F.R. Eirich, Eds., Academic Press, San Diego, CA, 1994, ch. 6.

[2.] R.D. Andrews, N. Hofman-Bang and A. V. Tobolsky, J. Polym. Sci. 3, 669 (1948).

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

[4.] R.D. Andrews and A. V. Tobolsky, J. Polym. Sci. 7, 221 (1952).

[5.] G. Berry and T.G. Fox, Adv. Polym. Sci. 5, 261 (1968).

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

[7.] P.G. de Gennes, "Scaling concepts in polymer physics," Cornell Univ. Press, Ithaca, NY, 1979.

[8.] M. Doi and S.F. Edwards, "The theory of polymer dynamics," Clarendon Press, Oxford, 1986.

[9.] D.S. Pearson, Rubber Chem. Technol. 60, 439 (1987).

[10.] S.H. Wasserman and W.W. Graessley, J. Rheol. 36, 543 (1992).

[11.] W.J. McGrory and W.H. Tuminello, J. Rheol. 34, 867 (1990).

[12.] C.B. Friedersdorf and I. Duvdevani, Rubber World, 211, No. 4, 30 (January 1995).

[13.] H.A. Pawlowski and J.S. Dick, paper No. 50 presented at a meeting of the Rubber Division, American Chemical Society, Philadelphia, Pennsylvania, May 2-5, 1995.

[14.] R.A. Mendelson, private communication.