Correlating the Segmental Relaxation Time of Polystyrene

A previous related paper dealing with the density relaxation of polystyrene (PS) has shown that the equilibrium relaxation time (τeq) has a purely exponential temperature dependence (ETD) below ≈100 °C. Such an ETD is now also confirmed based upon available dielectric spectra data for PS. By combining the ETD behavior of τeq (or aT) at low temperatures with a VFTH behavior at higher temperatures (based mainly on available recoverable shear compliance data), a composite correlation for τeq (or aT) is developed, which is continuous with continuous slope at a crossover temperature that is found to be 99.22 °C, where τeq = 92.15 s. This composite representation is shown to describe (without any adjustable parameters) available independent data for the segmental relaxation time over a finite range both above and below Tcrossover (i.e., the glass transition temperature).


Introduction
In a recent paper by Hieber [1], based upon the density relaxation of polystyrene at atmospheric pressure, it was shown that the equilibrium relaxation time is characterized by a purely exponential temperature dependence over the experimental range available in the literature, reaching (under equilibrium) about 16 • C below the nominal glass transition temperature.Such results were shown (in the same paper) to be compatible with the stress relaxation data for polycarbonate of O'Connell and McKenna [2] as well as with the equilibrium dielectric compliance data for PVAc of Zhao and McKenna [3]; in both of these cases, the equilibrium state could again be reached at about 16 or 17 • C below the nominal glass transition temperature.{It is noted that the "T g,nominal " for PS is typically considered to be 373 • K (i.e., essentially 100 • C), as has been done, for example, by Roland and Casalini [4] and He et al. [5]}.
In the present paper, it will be shown for polystyrene that the equilibrium relaxation time τ eq (T) from Hieber [1] can be extended to temperatures above the nominal glass transition temperature by making use of the temperature shift factor (a T ) in the glassrubber transition obtained from available independent data (in terms of recoverable shear creep compliance as well as stress relaxation) from the literature.It will be shown that this composite representation of τ eq (T) describes (without any adjustable parameters) the available data for the segmental correlation time of PS over a temperature range extending both above and below the (nominal) glass transition temperature.

Extending τ eq (T) above T g
Based on the results from fitting the cumulative density relaxation data for PS from Hieber [1], the resulting equilibrium relaxation time (at atmospheric pressure) is given by τ eq (T) = AA exp{−α 3 (T − 100 Combining this with the results from Appendices A and B below, we arrive at the plot in Figure 1 in which the ordinate is a measure of the temperature sensitivity in terms of τ eq (T) or a T , namely, or In particular, based upon the density relaxation data for PS from Hieber [1], we have that Ω = 0.805/ • C (5) over the interval between 83.87 • C and 100 • C. On the other hand, the three curves in Figure 1 are all based upon the VFTH model [6][7][8], namely, In particular, curve 1 in Figure 1 is based upon cumulative data from five sources [9-13] for the "flow regime" reported in Appendix A, with the measured temperatures ranging between 104.
Combining this with the results from Appendices A and B below, we arrive at the plot in Figure 1 in which the ordinate is a measure of the temperature sensitivity in terms of τeq(T) or aT, namely, or In particular, based upon the density relaxation data for PS from Hieber [1], we have that over the interval between 83.87 °C and 100 °C.
On the other hand, the three curves in Figure 1 are all based upon the VFTH model [6][7][8], namely, such that In particular, curve 1 in Figure 1 is based upon cumulative data from five sources [9-13] for the "flow regime" reported in Appendix A, with the measured temperatures ranging between 104.5 °C and 290 °C, and (CC, T∞) = (1793.8°C, 42.27 °C), as given in Equation (A2).On the other hand, curves 2 and 3 are based upon results for the "glass-rubber transition" from Appendix B involving cumulative data from six sources [9,[14][15][16][17][18]   Clearly, all three curves in Figure 1 were extended to temperatures below that of the underlying data (as presented in Figures A1 and A2.Furthermore, it is expected that the present density relaxation results should be directly related to the "glass-rubber transition" results, both reflecting the local molecular behavior, whereas the "flow regime" results reflect long-range molecular motion.In addition, as documented in Appendix B, there is a basis for judging that curve 3 is more representative than curve 2. Accordingly, of special interest in Figure 1 is the intercept of curve 3 with the dashed result given by Equation (5), which occurs at T = 99.22 • C.
These results seem to strongly indicate that the VFTH behavior of the "glass-rubber transition" given by curve 3 in Figure 1 becomes replaced by the constant value given by Equation ( 5) at temperatures below the intersection point at 99.22 • C. In turn, this indicates that the singularity (at T ≡ T ∞ ) in the VFTH equation is only an apparent singularity.
Making use of the results in Figure 1, it seems appropriate to introduce the term "T crossover " to denote where the dashed line and curve 3 intersect.Accordingly, where τ eq (T) is based on Equations ( 1) and (2) for T ≤ T crossover and is extended above T crossover by making use of curve 3 from Figure 1.That is, the resulting composite representation for τ eq (T) is then given by τ eq (T) = 49.18s exp{ −(0.805 for T ≤ T crossover , such that, from Equations ( 8) and ( 9), τ eq (T crossover ) = 92.15s (10) whereas, based upon curve 3 in Figure 1, for T > T crossover we have that It is noted that the composite representation for τ eq (T), given by Equations ( 9) and (11), is continuous with a continuous slope at T crossover {which follows from the definition of Ω in Equation (3)}.Furthermore, the value of T crossover in Equation ( 8) is close to the "nominal T g " of PS, namely, 100 • C. Accordingly, the result in Equation ( 10) is compatible with a convention typically associated with Angell [19], namely, that T g,nominal is where τ eq is on the order of 10 2 s.

Comparison with Experimental Results for the Segmental Relaxation Time
A resulting plot for τ eq (T) based upon Equations ( 9) and ( 11) is shown in Figure 2, together with corresponding data for PS based upon various experimental techniques.Despite the evident scatter, a definite correlation between the data and the composite curve (with no adjustable parameter) seems apparent.
It is worth stressing that the actual level of the τ eq (T) curve in Figure 2 is based upon the density relaxation results obtained in the earlier paper by Hieber [1].On the other hand, the extension of the curve to higher temperatures (i.e., above T crossover ) is based upon fitting cumulative results for a T in the glass-rubber transition region, as presented in Appendix B of the current paper.
In observing Figure 2, it is noted that the experimental results from Roland and Casalini [4] are for two PS samples of significantly different M w , differing by a factor of 43, but the corresponding results for τ eq differ by no more than a half decade.For comparison, if we were dealing with viscous flow, the characteristic time would be proportional to η 0 , which, for these large values of M w , would be proportional to M w raised to the 3.4 power.Accordingly, the respective values of the viscous flow characteristic time for these two polymers would differ by a factor of 43 raised to the 3.4 power, i.e., 3.6 × 10 5 .Clearly, on such a scale, the present results in Figure 2  Accordingly, the respective values of the viscous flow characteristic time for these two polymers would differ by a factor of 43 raised to the 3.4 power, i.e., 3.6 × 10 5 .Clearly, on such a scale, the present results in Figure 2 for the two PS samples are essentially coincident.Stated differently, these results indicate the dramatic difference in behavior of the current results in Figure 2, relating to the local segmental motion of PS, in contrast to the global molecular motion associated with viscous flow.8), ( 9) and ( 11) compared with PS data based on photon correlation spectroscopy [20,21], NMR [22], and dielectric spectroscopy [4,23].The dashed line will be described in Section 6.
As a still further confirmation that the results in Figure 2 are independent of the molecular weight (if sufficiently large), the results for PS in Figure 2b of Hintermeyer et al. [23] are striking; they show that the curves for "lg τα (s) versus T (°K)" are essentially coincident for the three samples with the highest molecular weight (all of NMWD), namely, 96 K, 243 K, and 546 K.In fact, the corresponding data points from Hintermeyer et al. [23] shown in the current Figure 2 were taken from the right-most solid curve in their Figure 2b, which is representative of the high-Mw asymptote.It is noted that Hintermeyer et al. [23] determined "Tg" for each of their polymers as the temperature at which τα equals 100 s.From their Table 2, the corresponding values for the above three samples with high molecular weight were 372.6 °K (99.45 °C), 373.3 °K (100.15°C), and 372.0 °K (98.85 °C), respectively.

Behavior at Higher Temperatures
Whereas Figure 2 extends to only 135 °C (reflecting the underlying related data from Appendix B), Figure 3 extends the plot to higher temperatures.In particular, the solid curve is based upon Equations ( 9) and (11), as in Figure 2, whereas the dashed curve is based upon the empirical fit obtained by He et al. [5], namely, τ , (T) = 0.87 × 10 s exp( °C .°C) (12) for the segmental correlation time.
As a still further confirmation that the results in Figure 2 are independent of the molecular weight (if sufficiently large), the results for PS in Figure 2b of Hintermeyer et al. [23] are striking; they show that the curves for "lg τ α (s) versus T ( • K)" are essentially coincident for the three samples with the highest molecular weight (all of NMWD), namely, 96 K, 243 K, and 546 K.In fact, the corresponding data points from Hintermeyer et al. [23] shown in the current Figure 2 were taken from the right-most solid curve in their Figure 2b, which is representative of the high-Mw asymptote.It is noted that Hintermeyer et al. [23] determined "Tg" for each of their polymers as the temperature at which τα equals 100 s.From their Table 2, the corresponding values for the above three samples with high molecular weight were 372.6 • K (99.45 • C), 373.3 • K (100.15 • C), and 372.0 • K (98.85 • C), respectively.

Behavior at Higher Temperatures
Whereas Figure 2 extends to only 135 • C (reflecting the underlying related data from Appendix B), Figure 3 extends the plot to higher temperatures.In particular, the solid curve is based upon Equations ( 9) and (11), as in Figure 2, whereas the dashed curve is based upon the empirical fit obtained by He et al. [5], namely, for the segmental correlation time.
As noted in Figure 9 of He et al. [5], their data, based upon three PS samples of low M w (namely, 2.05 K, 2.31 K, and 11.45 K), were "horizontally shifted by ∆T g , taking T g = 373 • K for high molecular weight PS".In particular, the highest-temperature data point from He et al. [5], as shown at 274 • C in the current Figure 3, corresponds to M w = 2.05 K and ∆T g = 54 • K/ • C. Similarly, the three data points for M w = 0.59 K from Roland and Casalini [4] were taken from their Figure 4 with a ∆Tg of 119 • C. Furthermore, the five right-most data points from Hintermeyer et al. [23] shown in Figure 3 correspond to M w = 1.350K, as presented in their Figure 11, with a ∆T g of 59 • C.
373 °K for high molecular weight PS".In particular, the highest-temperature data point from He et al. [5], as shown at 274 °C in the current Figure 3, corresponds to Mw = 2.05 K and ΔTg = 54 °K/°C.Similarly, the three data points for Mw = 0.59 K from Roland and Casalini [4] were taken from their Figure 4 with a ΔTg of 119 °C.Furthermore, the five rightmost data points from Hintermeyer et al. [23] shown in Figure 3 correspond to Mw = 1.350K, as presented in their Figure 11, with a ΔTg of 59 °C.8), ( 9) and ( 11)-solid curveor upon Equation ( 12)-dashed curve.Data based on photon correlation spectroscopy, NMR, and dielectric spectroscopy.
Evidently, with the exception of the data from Patterson et al. [24], the dashed curve clearly describes the high-temperature data in Figure 3 quite well.(It should be noted that the data from Lindsey et al. [20] and Patterson et al. [24] are from the same laboratory.)As indicated in Figure 9 of He et al. [5], their higher-temperature NMR data merge well with the lower-temperature NMR data of Pschorn et al. [22].Furthermore (as seen in Figure 3), the correlation given by Equation ( 12) seems to be substantiated by the DS measurements obtained independently by Roland and Cassalini [4] and by Hintermeyer et al. [23].
It should be noted that Equation (12) gives a value of about 10 2 s at 100 °C (often taken as the nominal glass transition temperature for PS).This is consistent with a convention typically associated with Angell [19], which was explicitly employed by Roland and Casalini [4] and by Hintermeyer et al. [23].
In closing this section, one might also consider the limiting behavior of the relaxation time at a hypothetically high temperature (i.e., as T → ∞), namely, "τ∞".In particular, from Equations ( 11) and ( 12) above, we obtain the respective values of 3.546 × 10 −9 s and 0.87 x 10 −12 s.On the other hand, Boyd and Smith [25] noted that the limiting behavior (as T→ ∞) of various modes seem to converge on a time scale of picoseconds (10 −12 s), corresponding to intramolecular and torsional oscillations.Hence, it would seem that Equation ( 12) would be more appropriate than Equation (11).But this is complicated by the generally accepted idea [26,27] that the WLF (or VFTH) model should be replaced by an Arrhenius behavior at sufficiently high temperatures.If that is performed in the case of Equation (11), supposing that the VFTH model in Equation ( 11) is replaced by an Arrhenius behavior at T = T*, with their values and first derivatives being continuous, it can be verified that  8), ( 9) and ( 11)-solid curve-or upon Equation ( 12)-dashed curve.Data based on photon correlation spectroscopy, NMR, and dielectric spectroscopy.
Evidently, with the exception of the data from Patterson et al. [24], the dashed curve clearly describes the high-temperature data in Figure 3 quite well.(It should be noted that the data from Lindsey et al. [20] and Patterson et al. [24] are from the same laboratory.)As indicated in Figure 9 of He et al. [5], their higher-temperature NMR data merge well with the lower-temperature NMR data of Pschorn et al. [22].Furthermore (as seen in Figure 3), the correlation given by Equation ( 12) seems to be substantiated by the DS measurements obtained independently by Roland and Cassalini [4] and by Hintermeyer et al. [23].
It should be noted that Equation (12) gives a value of about 10 2 s at 100 • C (often taken as the nominal glass transition temperature for PS).This is consistent with a convention typically associated with Angell [19], which was explicitly employed by Roland and Casalini [4] and by Hintermeyer et al. [23].
In closing this section, one might also consider the limiting behavior of the relaxation time at a hypothetically high temperature (i.e., as T → ∞), namely, "τ ∞ ".In particular, from Equations ( 11) and ( 12) above, we obtain the respective values of 3.546 × 10 −9 s and 0.87 × 10 −12 s.On the other hand, Boyd and Smith [25] noted that the limiting behavior (as T→ ∞) of various modes seem to converge on a time scale of picoseconds (10 −12 s), corresponding to intramolecular and torsional oscillations.Hence, it would seem that Equation ( 12) would be more appropriate than Equation (11).But this is complicated by the generally accepted idea [26,27] that the WLF (or VFTH) model should be replaced by an Arrhenius behavior at sufficiently high temperatures.If that is performed in the case of Equation (11), supposing that the VFTH model in Equation ( 11) is replaced by an Arrhenius behavior at T = T*, with their values and first derivatives being continuous, it can be verified that τ ∞ ∼ = 10 −13 s if T * ∼ = 222 • C (495 • K), and 10 −12 s if T * ∼ = 242 • C (515 • K).That is, these values indicate that the Arrhenius behavior would decay more rapidly than the VFTH model at these higher temperatures and that the resulting values for τ ∞ based upon such a composite VFTH/Arrhenius model would not be unreasonable.

Further Considerations
Figure 4 compares the results based upon the VFTH fits in Equations ( 12), (A2) and (A5) expressed in terms of Ω(T), a measure of temperature sensitivity, as evaluated via Equation (7).It is noted that curve 1 is based upon the fit obtained in Appendix B based (mainly) upon recoverable shear creep data between 100 and 135 • C. On the other hand, curves 2 and 3 should both be applicable up to 300 • C, based upon the related results in Figures 3 and A1, respectively.It is noted that curve 2 lies consistently above curve 1, that is, with increasing temperature, τ seg,c in Equation ( 12) consistently decays faster (on a fractional basis) than a T in Equation (A5).On the other hand, it is noted from Figure 4 that curve 3, corresponding to a T for the flow regime in Appendix A, intersects curves 1 and 2 at the respective temperatures of 115.9 • C (as can also be seen from Figure 1 above) and 158.5 • C.

𝜏 ≅ 10
s if T * ≅ 222 °C (495 °K), and 10 −12 s if T * ≅ 242 °C (515 °K).That is, these values indicate that the Arrhenius behavior would decay more rapidly than the VFTH model at these higher temperatures and that the resulting values for τ∞ based upon such a composite VFTH/Arrhenius model would not be unreasonable.

Further Considerations
Figure 4 compares the results based upon the VFTH fits in Equations ( 12), (A2) and (A5) expressed in terms of Ω(T), a measure of temperature sensitivity, as evaluated via Equation (7).It is noted that curve 1 is based upon the fit obtained in Appendix B based (mainly) upon recoverable shear creep data between 100 and 135 °C.On the other hand, curves 2 and 3 should both be applicable up to 300 °C, based upon the related results in Figures 3 and A1, respectively.It is noted that curve 2 lies consistently above curve 1, that is, with increasing temperature, τ , in Equation ( 12) consistently decays faster (on a fractional basis) than a in Equation (A5).On the other hand, it is noted from Figure 4 that curve 3, corresponding to a for the flow regime in Appendix A, intersects curves 1 and 2 at the respective temperatures of 115.9 °C (as can also be seen from Figure 1 above) and 158.5 °C.Making use of the three curves in Figure 4, one might address some related results in the literature.For example, Figure 3 of He et al. [5] seems to convincingly indicate that the segmental correlation time and viscosity have the same temperature dependence.In particular, the most extensive viscosity results in their Figure 3 are for the "PS-2" polymer, in terms of eight (solid square) data points that range between 1000/T ≅ 2.046/°K on the left and 2.493/°K on the right.Adjusting these points by ΔTg (=373 °K-331 °K = 42 °K, according to their Figure 9 and Table 1) to account for a small Mw of 2310, we are then dealing (in terms of °C) with the temperatures of about 257 °C and 170 °C, respectively.Based on Equations (A1) and (A2) of Appendix A, it follows that curve 3 in Figure 4 corresponds to Making use of the three curves in Figure 4, one might address some related results in the literature.For example, Figure 3 of He et al. [5] seems to convincingly indicate that the segmental correlation time and viscosity have the same temperature dependence.In particular, the most extensive viscosity results in their Figure 3 are for the "PS-2" polymer, in terms of eight (solid square) data points that range between 1000/T ∼ = 2.046/ • K on the left and 2.493/ • K on the right.Adjusting these points by ∆T g (=373 • K-331 • K = 42 • K, according to their Figure 9 and Table 1) to account for a small M w of 2310, we are then dealing (in terms of • C) with the temperatures of about 257 • C and 170 • C, respectively.Based on Equations (A1) and (A2) of Appendix A, it follows that curve 3 in Figure 4 corresponds to whereas, from Equation ( 12), curve 2 in Figure 4 corresponds to Polymers 2024, 16, 2154 In particular, 2.960 × 10 2 /1.678 × 10 2 = 1.764 = 10 0.246 (15) That is, according to the tight correlation for the viscosity of PS given by Equations (A1) and (A2), the result in Equation (15) indicates that the total variation in a T between the left-most and the right-most solid squares in Figure 3 of He et al. [5] exceeds that of τ seg,c (T), given in Equation (12), by a factor of 1.764.On a logarithmic basis, as in their Figure 3, this becomes a difference of 0.246 decades.{Note: if a T in Equations (A1) and (A2) is replaced by Equations (A1) and (A3), namely, the fit obtained by McKenna et al. [28], the difference becomes 0.263 decades.}Accordingly, such results call into question the close correlation between viscosity and τ seg,c for the polymer d 8 PS-2 shown in Figure 3 of [5].
Another concern relates to the relative behavior of creep compliance and viscosity at higher temperatures.In this regard, one might refer to the work of Ngai [29,30], namely, pp.117/118 from [29] and pp.262/263 from [30].In both cases, Ngai first stated that, above ≈407 • K (134 • C), the (recoverable) creep compliance and viscosity have the same a T .However, Ngai then qualified this by indicating that extrapolating creep compliance data to higher temperatures indicates a weaker temperature dependence than for viscosity.Unlike in [29], Ngai's corresponding plot in [30], namely, Figure 101 (left panel, for PS), explicitly includes a curve for (recoverable) creep compliance that decays more slowly than that for viscosity.In turn, this is in agreement with the present results, as shown in Figure 4, where curve 1 (corresponding to recoverable creep compliance) lies below curve 3 (viscous flow) at higher temperatures, namely, above 115.9• C. Since Ω characterizes the temperature sensitivity, the lower value for curve 1 indicates a slower decay with increasing temperature.
It is noted that Ngai [30] indicated (on p. 263) that the segmental relaxation time (τ α ) has the same temperature dependence as creep compliance up to 384 • K (111 • C); indeed, this agrees with the present results shown in Figure 2, in which there is excellent agreement with the curve, based upon Equations ( 8), ( 9) and (11), up to about 111 • C. On the other hand, there is also evidence that Equations ( 8), ( 9) and ( 11) describe τ α (T) even below T g .This is based upon results from Hintermeyer et al. [23], as documented in the following section.

Unanticipated Corroboration
In Figure 11 of Hintermeyer et al. [23], based on dielectric spectra data for PS, the results are plotted in terms of "lg τ α (s)" versus "z ≡ m (T/T g − 1)", in which "m" is the non-dimensional "fragility index".Of specific interest here is the fact that the data (all of which lie essentially above T g ) coalesce asymptotically onto a straight line as one approaches T g (identified with τ α ≡ 10 2 s) from above.In particular, the straight line in their Figure 11 where τ α is in seconds, and T and T g in • K. From their Figure 6, m ≈ 122 for the three samples with the largest M w and T g ∼ = 373 • K. Hence, Equation (16) becomes τ α (T) = 10 2 s exp{−(0.75/ for the PS polymers of high M w .Indeed, the value of Ω = 0.75/ • C in Equation (17) agrees well (within 7%) with the value of 0.805/ • C in Equation ( 5).This is evidenced by the dashed line in Figure 2, which is based upon Equation (17) with T g = 100 • C. Hence, there is a strong implication that the present correlation, corresponding to Equations ( 8), ( 9) and (11) and plotted as the curve in Figure 2 above, describes τ α (T) for PS not only up to 111 • C but also down to, perhaps, 83 • C, based on the density relaxation results for PS presented by Hieber [1], upon which the value of 0.805/ • C is based.
In a similar manner, based upon the DS data of Hintermeyer et al. [23] for polydimethylsiloxane (PDMS) and polybutadiene (PB), one obtains the respective values for Ω of 1.83/ • C and 1.19/ • C, based upon the higher M w samples.However, since this relates to the segmental relaxation time, the same values for Ω also pertain (essentially) to the polymers of lower M w .

Conclusions
The main results that were obtained in the present paper include the following: (i) It was shown, making use of the extensive DS data of Hintermeyer et al. [23] for PS {as well as for polydimethylsiloxane (PDMS) and 1, 4-polybutadiene (PB)}, that the temperature dependence of the local segmental relaxation time, τ α , is purely exponential below T g , thus confirming previous results for PS obtained by Hieber [1], based on density relaxation considerations.(ii) The fact that the values of 0.805/ • C in Equation ( 5) and 0.75/ • C in Equation ( 17) are in such close agreement strongly suggests that τ eq (T), obtained from density relaxation considerations, and τ α (T), obtained from segmental relaxation considerations, are directly related.(iii) The results shown in Figure 2 indicate that the smooth composite correlation (with no adjustable parameters) given by Equations ( 9) and ( 11) describes the available experimental results for the segmental relaxation time of PS encompassing a definite temperature range both above and below the glass transition temperature.(iv) Based upon the results in Section 5, there is strong evidence that, contrary to some results in the literature, the temperature dependence of τ α (T), given in Equation ( 12), and of a T for viscosity, given in Equations (A1) and (A2), does not become coincident at higher temperatures.

Figure 1 .
Figure 1.Results for Ω versus T. Curves based upon VFTH fits for flow regime (curve 1, from Appendix A) or glass-rubber transition (curves 2 and 3, from Appendix B).Dashed line based upon Equations (1) and (2).
for the two PS samples are essentially coincident.Stated differently, these results indicate the dramatic difference in behavior of the current results in Figure 2, relating to the local segmental motion of PS, in contrast to the global molecular motion associated with viscous flow.