Correlating the Segmental-Relaxation Time of Polystyrene

1. Introduction

In a recent paper by Hieber [1], based upon density relaxation of polystyrene at atmospheric pressure, it has been shown that the equilibrium relaxation time is characterized by a purely exponential temperature dependence over the experimental range available in the literature, reaching down (under equilibrium) to about 16ºC 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 the equilibrium dielectric-compliance data for PVAc of Zhao and McKenna [3]; in both of these cases, the equilibrium state could again be reached down to about 16 or 17ºC below the nominal glass-transition temperature. {It is noted that “T_g,nominal” for PS is typically taken as 373ºK (i.e., essentially 100ºC), as has been done, for example, by Roland & 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 glass-rubber 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 for τ_eq(T) describes (without any adjustable parameters) available data for the segmental-correlation time of PS over a temperature range extending both above and below the (nominal) glass-transition temperature.

2. Extending τ_eq(T) to 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

$${\mathsf{\tau }}_{\mathrm{e}\mathrm{q}}\left(\mathrm{T}\right)=\mathrm{A}\mathrm{A}\mathrm{}\exp \{-{\mathsf{\alpha }}_3\left(\mathrm{T}\mathrm{}-100°\mathrm{C}\right)\}$$

(1)

where

$$\mathrm{A}\mathrm{A}=49.18\mathrm{}\mathrm{s}\mathrm{e}\mathrm{c},\mathrm{}{\mathsf{\alpha }}_3\mathrm{}=0.805/°\mathrm{C}.$$

(2)

Combining this with the results from Appendix A & Appendix 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 $\mathrm{a}$_T, namely

$$\mathsf{\Omega }\equiv -\frac{\mathrm{d}\ln {\tau }_{\mathrm{e}\mathrm{q}}\left(\mathrm{T}\right)}{\mathrm{d}\mathrm{T}}$$

(3)

or

$$\mathsf{\Omega }\equiv -\frac{\mathrm{d}\ln {\mathrm{a}}_{\mathrm{T}}}{\mathrm{d}\mathrm{T}}$$

(4)

In particular, based upon the density-relaxation data for PS from Hieber [1], we have that

$$\mathsf{\Omega }\mathrm{}=0.805/°\mathrm{C}$$

(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,

$${\mathrm{a}}_{\mathrm{T}}=\mathrm{B}\mathrm{B}\mathrm{}\exp \left(\frac{\mathrm{CC}}{\left(\mathrm{T}-{\mathrm{T}}_{\infty }\right)}\right)$$

(6)

such that

$$\mathsf{\Omega }=\frac{\mathrm{CC}}{(\mathrm{T}-{\mathrm{T}}_{\infty }{)}^2}.$$

(7)

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).

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).

In particular, curve 1 in Figure 1 is based upon cumulative data from 5 sources [9,10,11,12,13] for the “flow regime” reported in Appendix 1, 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 (A.2). On the other hand, curves 2 and 3 are based upon results for the “glass-rubber transition” from Appendix B involving cumulative data from 6 sources [9,14,15,16,17,18] in the temperature range from 100°C to 135°C, with curve 2 corresponding to (CC, T_∞) = (669.8°C, 71.60°C) from Eqn. (B.1) of Appendix B and curve 3 to (CC, T_∞) = (714.4°C, 69.43°C) from Equation (B.2).

Clearly, all three curves in Figure 1 have been extended to temperatures below that of the underlying data (as presented in Figures A and B). Furthermore, it is expected that the present density-relaxation results should be directly related to the “glass-rubber transition” results, both reflecting 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 especial interest in Figure 1 is the intercept of curve 3 with the dashed result given by Eqn. (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 gets replaced by the constant value, given by Eqn. (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,

T_crossover = 99.22°C

(8)

where τ_eq(T) is based on Eqns. (1, 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) = 49.18 sec exp{−(0.805/°C)(T−100°C)}

(9)

for T ≤ T_crossover, such that, from Eqns. (8, 9),

τ_eq(T_crossover) = 92.15 sec

(10)

whereas, based upon curve 3 in Figure 1, for T > T_crossover we have that

$${\mathsf{\tau }}_{\mathrm{eq}}\left(\mathrm{T}\right)=92.15\sec \exp \left\{\frac{714.4\text{℃}}{\mathrm{T}\mathrm{}-69.43\text{℃}}-\frac{714.4\text{℃}}{99.22\text{℃}-69.43\text{℃}}\right\}.$$

(11)

It is noted that the composite representation for τ_eq(T), given by Eqns. (9) & (11), is continuous with a continuous slope at T_crossover {which follows from the definition of Ω in Eqn. (3)}. Furthermore, the value of T_crossover in Eqn. (8) is close to the “nominal T_g” of PS, namely 100º C. Accordingly, the result in Eqn. (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 10² sec.

3. Comparison with Experimental Results for the Segmental-Relaxation Time

A resulting plot for τ_eq(T) based upon Eqns. (9) and (11) is shown plotted in Figure 2, together with corresponding data for PS based upon various experimental techniques. Despite 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, 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 & Casalini [4] are for two PS of significantly different M_w, differing by a factor of 43, but that 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 η₀ which, for these large values of M_w, would be proportional to M_w raised to the 3.4 power. Accordingly, the respective values for 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 × 10⁵. Clearly, on such a scale, the present results in Figure 2 for the two PS 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.

As a still further confirmation that the results in Figure 2 are independent of molecular weight (if sufficiently large), the results for PS in Figure 2(b) of Hintermeyer, et al. [23] are striking, in which the curves for “lg τ_α (sec) versus T(ºK)” are essentially coincident for the three highest molecular weights (all of NMWD), namely 96K, 243K and 546K. In fact, the corresponding data points from Hintermeyer, et al. [23] shown in the current Figure 2 have been taken from the right-most solid curve in their Figure 2(b), which is representative of the high-M_w asymptote. It is noted that Hintermeyer, et al. (23) have determined “T_g” for each of their polymers as the temperature at which τ_α equals 100 seconds. From their Table 2, the corresponding values for the above three high molecular weights were 372.6ºK (99.45ºC), 373.3ºK (100.15ºC) and 372.0ºK (98.85ºC), respectively.

4. Behavior at Higher Temperatures

Whereas Figure 2 extends to only 135ºC (reflecting the underlying related data from Appendix B), Figure 3 extends the plot to higher temperatures. In particular, the solid curve is based upon Eqns. (9) and (11), as in Figure 2, whereas the dashed curve is based upon the empirical fit obtained by He, et al. [5], namely

$${\mathsf{\tau }}_{\mathrm{s}\mathrm{e}\mathrm{g},\mathrm{c}}\left(\mathrm{T}\right)=\mathrm{}0.87\mathrm{}\times \mathrm{}{10}^{-12}\mathrm{}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{1248°\mathrm{C}}{\mathrm{T}\mathrm{}-\mathrm{}61.55°\mathrm{C}}\right)$$

(12)

for the segmental-correlation time.

As noted in Figure 9 of He, et al. [5], their data, based upon three PS of low M_w (namely 2.05K, 2.31K and 11.45K), have been “horizontally shifted by ΔT_g, taking T_g = 373ºK for high molecular weight PS”. In particular, the highest-temperature data point from He, et al. [5], as shown at 274ºC in the current Figure 3, corresponds to M_w = 2.05K and ΔT_g = 54ºK/ºC. Similarly, the three data points for M_w = 0.59K from Roland & Casalini [4] are taken from their Figure 4 with a ΔT_g of 119ºC. 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 ΔT_g of 59ºC.

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 Eqn. (12) seems to be substantiated by the DS measurements obtained independently by Roland & Cassalini [4] and by Hintermeyer et al. [23].

It should be noted that Eqn. (12) gives a value of about 10² sec at 100ºC (often taken as the nominal glass-transition temperature for PS). This is consistent with a convention typically associated with Angell [19] which has been explicitly employed by Roland & 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 Eqns. (11) and (12) above we get respective values of 3.546 x 10^-9 sec and 0.87 x 10^-12 sec. On the other hand, Boyd & Smith [25] note that the limiting behavior (as T→ ∞) of various modes all seem to converge on a time scale of picoseconds (10^-12 sec), corresponding to intramolecular and torsional oscillations. Hence, it would seem that Eqn. (12) would be more appropriate than Eqn. (11). But this is complicated by the generally accepted idea [26,27] that the WLF (or VFTH) model should get replaced by an Arrhenius behavior at sufficiently high temperatures. If that is done in the case of Eqn. (11), supposing that the VFTH in Eqn. (11) gets replaced by an Arrhenius at T = T^*, with their values & first derivatives being continuous, it can be verified that ${\tau }_{\infty }\cong {10}^{-13}\mathrm{s}\mathrm{e}\mathrm{c}$ if ${\mathrm{T}}^{*}\cong 222°\mathrm{C}$ (495ºK) and 10^-12 sec if ${\mathrm{T}}^{*}\cong 242°\mathrm{C}$ (515ºK). That is, these values reflect that the Arrhenius would decay more rapidly than the VFTH at these higher temperatures and indicate that the resulting values for τ_∞ based upon such a composite VFTH/Arrhenius model would not be unreasonable.

Further consideration of the temperature dependence in Eqn. (12), compared with the correlations for a_T in Appendix A and Appendix B, is given in Appendix C.

5. An Unanticipated Corroboration

It is noted that Ngai [28] indicates (on p. 263) that the segmental-relaxation time (τ_α) has the same temperature dependence as the creep compliance up to 384ºK (111ºC); indeed, this agrees with the present results shown in Figure 2, in which there is excellent agreement with the curve, based upon Eqns. (8, 9, 11), up to about 111ºC. On the other hand, there is also evidence that Eqn. (8, 9, 11) describes τ_α (T) even below T_g. This is based upon results from Hintermeyer, et al. [23], as follows...

In Figure 11 of Hintermeyer, et al. [23], based on dielectric-spectra data for PS, results are plotted in terms of “lg τ_α (sec)”, 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 where τ_α ≡ 10² sec) from above. In particular, the straight line in their Figure 11 corresponds to

$$\log \mathsf{\tau }{\mathsf{\alpha }}_{}=2-\frac{m}{T_g}(\mathrm{T}-{\mathrm{T}}_{\mathrm{g}})$$

(13)

where τ_α is in seconds and T & T_g in ºK. From their Figure 6, m ≈ 122 for the three largest M_w and T_g ≅ 373ºK. Hence, Eqn. (13) becomes

τ_α (T) = 10² sec exp{ − (0.75/ºC) (T − T_g)}

(14)

for the PS polymers of high M_w. Indeed, the value of Ω = 0.75/ºC in Eqn. (14) agrees well (within 7%) with the value of 0.805/ºC in Equation (5). This is evidenced by the dashed line in Figure 2, which is based upon Eqn. (14) with T_g = 100ºC.

Hence, there is the strong implication that the present correlation, corresponding to Eqns. (8, 9, 11) and plotted as the curve in Figure 2 above, describes τ_α (T) for PS not only up to 111ºC but also down to perhaps 83ºC, based on the density-relaxation results for PS presented by Hieber [1], upon which the value of 0805/ºC is based.

In a similar manner, based upon the DS data of Hintermeyer, et al. [23] for polydimethylsiloxane (PDMS) and polybutadiene (PB), one obtains respective values for Ω of 1.83/ºC and 1.19/ºC, 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, as is demonstrated in Appendix D.

6. Conclusions

Main results that have been obtained in the present paper include the following:

(i) It has been 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/ºC in Eqn. (5) and 0.75/ºC in Eqn. (14) 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 Eqns. (9) and (11) describes 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 Appendix C, there is strong evidence that, contrary to some results in the literature, the temperature dependence of τ_α(T), as given in Eqn. (12), and of a_T for the viscosity, as given in Eqns. (A.1) and (A.2), do not become coincident at higher temperatures.