X(t),
B(t), and
C(t) processes by the molecular weight-dependent factorR
K(M)(see eq 5 of ref 32 or eq 11.6 of ref 33),1,2,29,3133which has a plateau value 3.3 in the high-molecular-weight region; starts to decline with decreasing molecular weight at around 10Me or 10
M
e' (M
e' M
eW
2 with W2 being the weight faction of the entangled component in a blend solution); and reaches 1 as the molecular weight reaches MeorM
e'.(3) The K value in the
X(t),
B(t), and
C(t) processes of ERT is in quantitative100 Ferry, J. D. Viscoelastic Properties of Polymers, 3rded.; Wiley: New York, 1980.
101 See Table 1 and Appendix B of ref 1 and note at ref 38 of ref 2.
agreement with the frictional factor K in the Rouse theory (as appearing in the Rouse relaxation time
; see Figure 20 of ref 31 or Figure 11.1 of ref 33),
1R meaning that both theories have the same footing at the Rouse-segmental level.2,31,33As indicated by the observation
R
K(M
)1asM
M
e orM
e', K’being greater than K by the factor RK(M) is due to the topological constraint of entanglement. At temperatures close to Tg, in the relatively short yet macroscopic time scales of and
G S (non-ergodic times; see Table 2 of paper 1) the strong energetic interactions among segments keep many configurations from being explored, while in the long time scales of,
1A X,
B, and
Cor (ergodic times) there is enough time to
1R explore the configurational space effectively, leading to entropy-derived modulus (as represented by the entropy force constant; see the note at ref 39)102 and dynamics (as described by the Langevin equation; see the note at ref 40).103 Thus, except for the factor RK(M) due to the topological constraint effect of entanglement, there is not a fundamental difference between K’and K. The expression for K (eq 2) is equally102 This is indicated by the fact that no shift along the compliance coordinate is required in the quantitative analyses of J(t) line shapes of polystyrene melts in the expected entropic region using ERT (in paper 1)1or the Rouse theory (in paper 3)2 as the reference frame, even when the temperature is very close to Tg.
103 The successful quantitative analyses of the J(t) line shapes of polystyrene melts in the expected entropic region indicate that the functional forms for the relaxation processes in the stress-relaxation modulus and the structural factors of the relaxation times as given in ERT or in the Rouse theory remain valid, even when the
temperature is very close to Tg.1,2
applied to K’; namely
where the diffusion constant D’is smaller than D by the factor RK(M).
As detailed in section 4 of paper 3,2for more directly reflecting the close relation between the
G(t) and
A(t) processes)the two processes occurring next to each other in time, the normalized glassy-relaxation time s has been modified by introducing s’as defined by
Then the structural-relaxation time defined by
S 18
G
(see ref 41 (paper 3) for adetailed study of the structural-relaxation time)104 can be expressed by
'
R
K indicates that, when the tube (of the reptational model) is disappearing and the Rouse theory becoming applicable, the dynamics in the system has only one K and becomes isotropic as it should. As detailed in section 4 of paper 3,2 in an entanglement-free system (sample C), s can be regarded as s’and K as K’.104 Lin, Y.-H. J. Phys. Chem. B 2005, 109, 17670.
As far as sample C is concerned in the discussions below; s and s’, and K and K’will be used interchangeably without further explanation. As s or s
’decreases with
increasing temperature, the entropic region shifts more away from the glassy region as can be observed in Figures 6 and 7 of ref 1 and in comparing Figures 8 and 9 of ref 2.In the literature,1113,105,
S reaching 100～1000 sec has often been used as the criterion for defining Tg. Following this practice, the glass transition point Tg has been defined as the temperature where
S=1000 sec for all the three samples whose J(t) results have been analyzed, two entangled and one entanglement-free. The thus defined Tg provides a common reference point equivalent for all samples, with respected to which the structural and dynamic quantities
S, s’and K’obtained from analyzing the J(t) results may be compared in a perspective way; the effect of the Tgdifference among samples on
S, s’and K’can be accounted for by expressing them as a function of the temperature difference from Tg, T
T
T
g. It has been foundthat
S, s’and K’values of the three samples plotted as a function of individual T fall on a common curve, indicating that TRC as closely related with the glass transition behaves in a universal way within the polystyrene system, entangled or not.When the temperature is significantly closer to Tgthan 127.5oC, the formation of structure as related to the increase in s’starts to affect K through K’; in other words, K behaves in such a way that K’values as a function of T fall on a universal curve)K’
differs from K by the factor RK(M). As a result, below ～120oC, K begins to deviate from the behavior purely controlled by the topological constraint of
105 Angell, C. A. J. Non-Cryst. Solids 1991, 131-133, 13.
entanglement)namely, being independent of molecular weight and thus the difference in Tg)which holds at higher temperatures. However, below ～ 120oC, ERT remains valid in describing the topological effect on the line shape of the viscoelastic responses as explained in section 8 of paper 3 even though K becomes gradually dependent on molecular weight as the temperature approaches Tg (see Figure 11 of paper 3)2. Because the effect of the formation of structure becomes dominant in the close proximity of Tg, we should use K’and its conjugate structural parameter s’to discuss the Tg-related effects instead of K and s as used in paper 1. However, the basic mechanism for the TRC developed in terms of K and s in paper 1 remains equally valid as the only difference is a proportional constant. In an entangled case, as the RouseMooney process occurs right after the glassy-relaxation process
G(t), using K’and s’instead of K and s also indicates the need to shift our focus to the shorter-time region for studying the Tg-related effects.Being Brownian motion, the diffusion constant of a Rouse segment can generally be expressed as
t l D kT
2
' '
(6)where l is the step length that the Rouse segment has moved in a time intervalt. The only criterion for choosing t and l is that the steps are independent of one another;
then after sufficiently large number of steps of movement have taken place, the central limit theorem ensures that the dynamic process becomes Gaussian as required
by the Langevin equation being applicable.28,33,106,107
At high temperatures, there is a wide range down to very small values to choose l andt to satisfy eq 6; the dynamic process is often referred to as the continuous (small-step) or “free”diffusion.11,108 At a temperature close to Tg, the structure is formed with a certain lifetime
S which has increased greatly with s’; then, the smallest independent time step that can be chosen is of the order of the lifetime of the structure
S 18
G. We can choose as
the time step because it is still much shorter than,
1A X,
B, and
C or (see Table 2
1R of paper 1 and Figure 8 of paper 3).1,2 Corresponding to being longer at lower
temperatures, a larger length-scale denoted by d is expected for the step length as explained in the following: As K’(or K ) has been determined from the quantitative line-shape analysis of J(t), so D’(or D) is defined. Due to eq 3, the following constraint is imposed on the system
where eq 5 has been used. To maintain D’
K’being constant, d has to increase by
about 5 times as s’increases from about 1500 at temperatures higher than ～T
g+40oto about 40000 at Tg(see Figure 5 of paper 3).106 Mooney, M. J. Polym. Sci. 1959, 34, 599.
107 Doi, M. J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 1005.
108 Thirumalai, D.; Mountain, R. D. Phys. Rev. E 1993, 47, 479.
The relaxation times in the long-time region (
,
1A X,
B, and
C or ) are
1R proportional to K /kT /d
2 or K’ ’ /kT
/d
2 while the structural relaxation time is
S
G. With decreasing temperature, increases more
than
/d
2 as the structure is formed following the increase in s’. This difference in temperature dependence is the basic mechanism for the TRC as concluded from the analyses of the polystyrene J(t) curves reported in papers 1 and 3.It was pointed out in paper 1 that this basic mechanism should be also the reason for the breakdown of the StokesEinstein equation in relating the translational diffusion constant Dg with the shear viscosity
or rotational relaxation time
rot as observed in OTP, when Tg is approached from above. Without the entropy-derived modes of motion)as described by ERT or by the Rouse theory)in OTP,
0 G(t
)dt
G S rot (8)What is explained above for the diffusion of the Rouse segment in polystyrene melts can similarly be applied to the molecular diffusion Dgin OTP;
2
g
D d
(9)Thus, from eqs 8 and 9, one sees that Dg
rot increases with increasing d as Tg is approached from above, meaning BSE.To characterize the BSE, a translational diffusion enhancement parameter
has
been defined by
where DSE is the translational-diffusion constant expected when the StokesEinstein relation holds. As the StokesEinstein relation holds at temperatures far above Tgin OTP, its
value at a temperature T close to T
gmay be calculated from)
where Thigh stands for a high temperature in the region where the StokesEinstein relation holds.
As s’for the polystyrene system reaches a plateau value of about 1500 as the temperature is more than 40oabove Tg, under the constraint imposed by eq 7, d should reach a lower limiting value at high temperatures, which is denoted by d0. Denoting the plateau value of s’at high temperatures by
s (1500 for polystyrene), then the s’
0' value at a temperature T close to Tgmay be expressed by2
Applying the same idea to the translational diffusion enhancement parameter in OTP and substituting eqs 8 and 9 into eq 11, the
value at a temperature T close to T
gmaybe expressed by
2
0
) ) (
(
d
T T d
(13)On the basis of comparing eqs 12 and 13, we conclude that
(T) for OTP is equivalent
tos
'(T
)s
0'for polystyrene melts, reflecting the same mechanism.3. Comparison of the Results of (T) and s’ (T)s
0’
The data of