s’and K’among the three samples as well as the implied universality is very clear
7. Information in G(t) as Extracted from the Analysis 1. Length-Scale at T g
8.3. K Values in the Close Neighborhood of T g
As pointed out above, the common length-scale at
S=1000 or 1sec (at T=0 or 9.7) as shown in Figures 8 and 9, respectively, and the same relative position of
Swith respect to the Rouse-segmental motional time
v as shown in Figure 10 for the three samples are direct results of the universal behavior of s’and K’as a function ofT. As Figures 810 are all displayed in real time, the positions or magnitudes of the relaxation times are ultimately determined by the K values. Thus, although, as expected, the K values as a function ofT for the three samples as shown in Figure 7 do not fall on a universal line, the K value of a sample at a certainT does not occur without following a certain “rule”as the somewhat “chaotic”look of the collective display may suggest. As the temperature is approaching Tg (T 20; in this temperature region
S12), because K’=KR
K(M) (K’=1.61K for s-A; K’=3.16K for s-B; and K’=K for s-C) K has to change with
T in such a way that the corresponding53 Fujara, F.; Geil, B.; Sillescu, H.; Fleischer, G.
Z. Phys. B: Condens. Matter 1992, 88, 195.
54 Cicerone, M. T.; Ediger, M. D. J. Phys. Chem. 1993, 97, 10489.
55 Kind, R.; Liechti, N.; Korner, N.; Hulliger, J. Phys. Rev. B 1992, 45, 7697.
56 Swallen, S. F.; Bonvallet, P. A.; McMahon, R. J.; Ediger, M. D. Phys. Rev. Lett.
2003, 90, 015901.
K’
values will behave in the universal way as shown in Figure 6. As the temperature is very close to Tg, this effect becomes dominant; K becomes influenced through K’by the Tg value, which declines with decreasing molecular weight below ～10Me for polystyrene. This has to be reconciled with the fact that K is independent of molecular weight at and above 127.5oC. As shown in Figure 11, the comparison of the
K values as a function of temperature between s-A, -B and -C illustrates such a
transition.As s-C and E167 have very similar Mw values, their K values as a function of temperature should not be very different. The physical difference in K between s-C and E167 should be quite small even though the viscoelastic response in the entropic region of the former is described by the Rouse theory while that of the latter by ERT.
See the Appendix for a detailed discussion of the viscoelastic difference between s-C and E167. As described by the Rouse theory, the dynamics in s-C has only one frictional factor K)dynamically isotropic. In the case of E167, because its molecular weight is so close to Me, it has been found that K=K’within a small experimental error)virtually isotropic dynamically. Thus, as far as the frictional factor K is concerned, it is basically the same in both samples. The pattern that the K values of s-A and s-C diverge as the temperature approaching Tg and merge at high temperatures, 130oC, as shown in Figure 11, should similarly occur between s-A and E167. The divergence in approaching Tgin the latter case should be smaller as E167 has a slightly higher
M
w value and a much narrower molecular-weight distribution)factors favoring a higher Tg. At 127.5oC the K values (Table 1) for s-C and for E167 are both about 17～20% smaller than the average value K=4.9×10910% over the molecular weight range from 3.4×104 to 6.0×105.3,7,10 The 17%
smaller in K for s-C may be due to the effect of a substantial Tg difference, while the 20% smaller in K for E167 is at least partly due to the likely effect that very small amounts of components with molecular weight below Me in the sample system)as its
M
w is only 1.24Me)will reduce the obtained K value somewhat. In any case, as these differences in K are so small, these results actually confirm that K at 127.5oC is independent of molecular weight to a molecular weight virtually as low as Meand that the Rouse theory and ERT have the same footing at the Rouse-segmental level.Because s-B is contaminated by residual plasticizers, the K values of s-B cannot be directly compared with those of s-A.2,3 To illustrate the point made above, we show the curve calculated from the FTH equation that has been obtained from least-squares fitting to the K values of s-B and shifted to the higher temperature side by 1.5o. The temperature shift is to account for the decrease in Tg by the contamination of residual plasticizers; after the shift, the curve superposes on the FTH curve of s-A over the region of 118140oC very closely, including at 127.5oC, where the K values of s-A and the “uncontaminated s-B”are expected to be in close agreement. After such a shift, the FTH curve of s-B begins to rise above that of s-A below～115oC, illustrating the divergence similar to, but smaller than, that between s-A and s-C. If we use the value K’=1.35×103expected atT=0 (see Figure 6), the
K value (=K’ /3.16) at T
g should be around 4.3×104, which occurs at 99.55oC on the shifted FTH curve of s-B. In other words, from such a “restoration”of s-B to its uncontaminated state, its Tg is estimated to be about 99.5oC. Thus, we have the Tgvalues for s-A, hypothetically uncontaminated s-B, and s-C to be 97, 99.5 and 93.8oC,
respectively; these values are consistent with what may be expected from calorimetric measurements (see the note at ref 56).57
The above discussion of the results shown in Figure 11 suggests that the glass transition is a nonlinear effect not only in form as expressed by the product of s’and
K’(eq 17) but also in the interplay between the two variables: an increase in s’can
enhance a further increase in K’and vice versa. The sharper rise in both s’and K’as the temperature is getting closer to Tg may be the manifestation of such an effect.Such an effect is imposed through K’on the frictional factor K; consequently, as the temperature decreases below ～120oC, K gradually deviates from its purely topologically controlled behavior)namely, being independent of Tg and molecular weight)which holds at higher temperatures. Below ～120oC, K becoming dependent on molecular weight does not mean ERT ceases to be valid in describing the topological effect on viscoelasticity; as shown in paper 1 (ref 3), in this low-temperature range the line shapes of J(t) in the rubber(like)-to-fluid region remains quantitatively described by ERT. Clearly, only the value of the frictional factor K is affected; the functional forms of the entropy-derived dynamic modes and the structural factors of their relaxation times as given in ERT are not affected. Thus, using ERT as the reference frame in analyzing the J(t) line shapes remains valid below ～120oC. On such a basis, it is no accident that the T dependences of s’and
K’are found to be universal; the result of analysis represents a real advancement in
57 Note: From Figure 3 of ref 36, one may obtain Tg=93.4, 97 and 100oC at Mw=16400, 46900 and 122000, respectively.
understanding the glass transition of polystyrene, which may be generalized to polymers in general.
9. Summary
The quantitative success of the Rouse theory and ERT in describing the entropic region of relaxation modulus G(t) or creep compliance J(t) allows them to be used as a reference frame)the former for an entanglement-free system (s-C) and the latter for an entangled system (s-A and s-B))with respect to which the glassy-relaxation process that occurs in the short-time region can be studied in a perspective way.
From the analyses of the J(t) results of three polystyrene samples: s-A, -B and -C, the structural-growth parameter s’, the frictional factor K’for the RouseMooney process
A(t) in the entangled case or the frictional factor K (equivalent to K’; and also treated as K’in notation as explained in section 6) for the Rouse process
R(t) in the entanglement-free case, and the structural-relaxation time
S 18s
'K
' are extracted.It has been shown that the thermorheological complexity occurring in J(t) of polystyrene is due to the temperature dependence of the glassy-relaxation process
G(t) being stronger than that of the entropy-derived ones. The uneven thermorheological complexity in J(t) is fully characterized by a simple increase of the structural-growth parameter s’with decreasing temperature in both the entangled and entanglement-free systems.For all the three studied samples, Tg is defined by the temperature at which the structural-relaxation time
S=1000 sec. Thus defined Tgprovides a common reference point equivalent to all the samples, with respect to which the obtained
S, s’and K’results may be compared in a proper perspective. When these values of s-A, -B and -C are displayed as a function ofT=TTg, universal behavior related to Tgis revealed.
A significant point of this study is that the universal T-dependence of
S is separated into two decoupled effects: structural-growth as represented by s’and frictional-slowdown as represented by K’)by a clean-cut process: s’being first determined entirely by the J(t) line shape; then K’determined from the time-scale shifting factor)both changing with T individually in a universal way. Because of the universality of s’and K’as a function of T, the
G(t) process and the
A(t) or
R(t) process in both their absolute positions and positions relative to each other in time depend onT in the same way for all the three studied samples. In other words, the thermorheological complexity is a universal effect within the polystyrene system,entangled or not.
Ultimately this conclusion supports the result from the study of the blend-solution systems that ERT and the Rouse theory have the same footing at the Rouse-segmental level.9,10 It also strongly indicates the importance of the roles which the obtained time-scale and length-scale may play.It has been shown that the relaxation times of the RouseMooney normal modes (for s-A and s-B) or the Rouse normal modes (for s-C) may be used as the
“graduations”of an internal yardstick for estimating the extent of the influence of the glassy relaxation at Tg. As a logical consequence of s’and K’depending onT in a universal way, the length-scales at Tg obtained this way for the three samples agree with each other closely)about 3nm for all the three samples. As pointed out previously,3 this kind of analysis represents a new methodology for studying the length-scale at Tg. As the characteristic ratio and entanglement molecular weight of
various polymers have been well documented,10,48,49 this method would be of wide application.
Key concepts: decoupling of the structural-growth and frictional-slowdown effects; universality in the thermorheological complexity; and the time-scale and length-scale in respect of the motion and size associated with the Rouse segment, as developed, revealed and obtained in the previous and present studies represent a new way to see and study the glass transition of a polymer.
Acknowledgement
This work is supported by the National Science Council (NSC 93-2113-M-009-015, NSC94-2113-M-009-002). The author thanks one of the reviewers for drawing his attention to ref 50.