The Relaxation in Creep Compliance

With G(t) known)for instance as given by eqs 1, 4 and 5 of ref. 11)J(t) can be calculated numerically by the method of Hopkins and Hamming.^132,133 It has been shown in detail in ref. 11 that the rubber(like)-fluid region of Plazeks J(t) results of two nearly monodisperse polystyrene samples^134,135can be well described by ERT and that the dynamic information of the glassy-relaxation process as contained in the small-compliance/short-time region of J(t) can be meaningfully extracted by using the successful description of the rubber(like)-fluid region in terms of ERT as the reference frame. The glassy-relaxation process is found to be well described by the stretched exponential form

as incorporated into eq 4 of ref 11. In the whole relaxation-time distribution, the glassy-relaxation region is situated in a certain position relative to the rubber(like)-fluid region, where all the relaxation times are proportional to the frictional factor K. The relative position has been expressed by

G sK

 (10)

where s is a proportional constant and has the unit of Dalton square. The parameter s represents the glassy-relaxation time with K fixed at 1 or any constant; it is regarded as a normalized glassy-relaxation time. In the vicinity of T_g, the parameter s increases with decreasing

temperature, reflecting the thermorheological complexity between the glassy-relaxation process, AGG(t), and the ERT processes:A(t),X(t),B(t), andC(t), in the rubber-fluid region and indicating the existence of a structural length scale as discussed in detail in ref. 11.

Sample B whose J(t) results was analyzed in ref. 11 is contaminated by residual

plasticizers; the K value extracted from it cannot be used for comparing with studies on normal (uncontaminated) samples. Thus, in this report, we only discuss the results of sample A. It has been found for sample A that AG(=5482) and the stretching parameter(=0.41) are very much independent of temperature, while s increases with decreasing temperature significantly)by about an order of magnitude over the covered temperature range. The obtained K and s values at different temperatures for sample A are listed in Table 1. Using the obtained K and s values in theAp

equation (i.e. eq 5 with K replaced by K=1.61K as calculated from eq 8 of ref. 11 for M=4.69x10⁴; M replaced by M_e; and N_rreplaced by N_e=16) and eq 10, thevA15

and_G values at different temperatures can be, respectively, calculated, as also shown in Table 1.

One may calculate the J(t) curves at different temperatures in real time with the K and s values shown in Table 1. Instead of doing this way, the comparisons of the J(t) curves of sample A measured at different temperatures to those calculated with K fixed at 5x10^9and the s values listed in Table 1 are shown in Figure 1 of ref. 11. This illustrates using the description of the rubber(like)-fluid region of J(t) in terms of ERT as the reference frame to show the effect of temperature on the glassy-relaxation process; such a comparison serves the purpose of reflecting and characterizing in perspective the thermorheological complexity occurring in J(t) as the temperature is near T_g. As also shown in ref. 11, unlike the extensive overlapping of the

different processes in J(t), the individual processes can be clearly shown in the G(t) form. Based on the G(t) results, a structural relaxation timeSwas defined as the time when G/R has declined to 3 as described in detail in ref. 11. The thus defined structural relaxation time becomes greater thanvjust before the temperature reaches Tg, indicating vitrification at the Rouse-segmental level.

As will be shown below, the structural relaxation time defined by G/R =3 can be considered as basically equivalent to the so-calledrelaxation time. In the literature, the -relaxation time has been "defined" in different ways,^5,136such as the reciprocal of the frequency at the peak of tanand the reciprocal of the frequency at which the storage modulus G⁽) is at 10⁸dyn/cm². The relaxation time defined in any of these ways can in principle be determined clearly by experiment. However, it does not really characterizes a relaxation process in a simple and clear manner; with a temperature change, it is affected not only by the intrinsic temperature dependence of the relaxation process that matters but also by the change in the line shape of the viscoelastic spectrum)namely, the thermorheological complexity. The structural relaxation time defined as the time when G/R=3 has a similar defect.

To further illustrate the physical effect on the bulk mechanical property by the glassy relaxation, another analysis will be made below. This analysis confirms the basic physical uniqueness ofSas defined by the time when G/R=3. Based on these findings, an optimum definition forSis chosen, which has an unambiguous meaning in its temperature dependence and at the same time properly reflects the effect on the bulk property by the glassy relaxation.

And it will be shown below that the thus definedSis very close to the-relaxation time defined by one of the traditional ways.

We consider that the time when the absolute value of the slope d(log G(t))/d(log t), denoted by H, reaches its first maximum reflects a unique physical meaning associated with the glassy-relaxation process as explained in the following: As shown previously and mentioned above, theG(t) process can be well described by the stretched exponential form with=0.41,

which is very much independent of temperature. In the high-modulus/short-time region where

As shown in Figure 2, initially following eq 11, log H increases with log t with a slope of=0.41, indicating a gradually steeper decline of log G(t) with log t. At the time, denoted by tm, when H reaches its first maximum, while the rate of the glassy-relaxation process has the greatest influence, its modulus magnitude is losing its dominance as deviation from eq 11 begins taking place. As it turns out, the location of the H maximum occurs in the neighbourhood of the structural relaxation time defined as the time when G/R=3. The obtained tmvalues at different temperatures are listed in Table 1, which occur in the range of 1525 _G, depending on the temperature. The obtained_Gvalues occur in the too short-time region to clearly reflect the dynamic effect of the glassy-relaxation process on the bulk mechanical property; however, they carry the intrinsic temperature dependence of the glassy-relaxation process. To have the benefits of both t_mand_G, we redefine the structural relaxation time arbitrarily asS=18_G, whose values at different temperatures are also listed in Table 1. Allowing a 20% deviation from this somewhat arbitrarily chosenS)for instance one may as well chooseS=22G)the main point that will be explained in terms of the definedSremains the same.

For comparing the above-definedS with the-relaxation time defined in the literature, the storage-, loss-modulus and tanspectra of sample A are shown in Figures 3 and 4, all the spectra being “normalized”with respect to K=5x10^-9(see the Appendix for the calculations of the spectra). As, being basically a mirror image, the G⁽) spectrum has a close match to G(t) if

=0.7/t is used in the conversion between time and frequency, we defineS=0.7/S.¹³⁷ The thus definedSvalues at 114.5, 104.5 and 97^oC are compared in Figure 4 with what have been

used traditionally: at the peak of tanand at G()=10⁸dyn/cm². It can be seen that theS

values at the three shown temperatures occur in the close neighbourhood of the frequencies where the respective storage-modulus has the value 10⁸dyn/cm²; however, they deviate considerably from the respective frequencies at the tanmaximum. In a case where a careful analysis as done in this study is not feasible, using G()=10⁸dyn/cm²as the criterion for deciding the-relaxation time may be a good choice except bearing that the thus determined relaxation time does not follow exactly the temperature dependence ofGas the above defined

Sdoes.

In Figure 4, the frequency corresponding to the highest RouseMooney normal mode,

v=0.7/v, is also indicated. One can see that at a temperature between 104.5 and 97^oC,S

becomes smaller thanv, signalling the initiation of vitrification at the Rouse-segmental level, a prelude to the glass transition. This was pointed out in terms of the previously defined

structural-relaxation time,¹¹which reflected the similar effect of the glassy-relaxation process. In fact, as values of the previously definedSat different temperatures are very close to the values based on the present definition (see Table 2 of ref. 11), the discussion of the physical role of the structural relaxation in terms ofSdefined by G/R=3 remains essentially the same as in terms of the above definedS, which has the additional advantage that, as shown below, its temperature dependence can be unambiguously compared with those of other dynamic quantities.

5. Comparison of the Temperature Dependences of Various Dynamic Quantities