Glass Transition and Ultrasonic Relaxation in Polystyrene
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influence of molecular weight on the relaxation time and the fragility of polystyrene. EXPERIMENT The polymers were purchased from Pressure Chemicals Co. and were used as received. All samples, about 3.5 grams each, were melted into disks under a pressure of 22.5 MPa and K. Then, while maintaining pressure, each sample was T, +200 + annealed for 15 min at quenched to well below T. The measurements of the specific volume, V, the longitudinal sound velocity, v, and the ultrasonic attenuation, a, were then immediately taken at a heating rate of 2K /min from 220 to 520K. Details of the experimental setup were given elsewhere [3]. All samples were prepared following the same procedure so that the effect of molecular weight could be studied. From the quantities, V, v and a we compute the complex longitudinal modulus, 1L* = L' + iL", with the storage, L', and loss, L", moduli given by [4]: L~=--- 1j-Jj-
1l+
-J
and
L",=a
+
where cw= 2 I-rf is the angular frequency; here we used f = 2.7 MHz. 183
Mat. Res. Soc. Symp. Proc. Vol. 455 @1997 Materials Research Society
(1)
RESULTS
Representative data of L' and L" as a function of temperature are shown in Fig. 1. The symbols represent the experimental data and the solid line is a fit to the theoretical expressions, as explained below. With increasing temperature, the storage modulus decreases first slowly then more rapidly. The break point from slow to rapid change in the modulus is the glass transition temperature, T,, where the glass-like structure of the polymer starts to collapse and the system goes into the molten state. In the low- and high-temperature regions, L' tends toward the same limiting values indicating the independence of the modulus on the molecular weight in the completely glassy state as well as in the liquid state. On the other hand, with increasing M, the loss peak shifts to higher temperatures and its height increases steadily before saturating at high molecular weights.
The viscoelastic modulus, L*, can also be written in terms of a relaxed modulus,
LR,
and an
unrelaxed modulus, LU, as: L* = LR +(Lu
(2)
- LR)L(wc) ,
LU and LR are the low and high temperature limits of the storage modulus. L( w) is the HN relaxation function, L( a)= (1 + (i aon)'•)-"[5], where r is the relaxation time and a•, r are shape
parameters of the loss peak. The temperature dependence of r is given by the empirical VTF equation [6]: (3)
r= ziexp[B/(T-To)]
where z;ois a microscopic time scale which relates to energy barrier crossing which oppose chain rearrangements during structural relaxation and is the same for all samples (-104 s) [7]. The Vogel temperature, To, and B are parameters specific to the material. The experimental data are fitted to the theoretical expression of the modulus from eq. 2. Thus, the unknown parameters are B, TO, a, and y. The results of the fit are the solid lines in Fig. 1. The agreement is very good 6
5
0
803
o
1586
0 0
803 1586
,7
8490
v
8490
z
545000
z
545000
0.4
0,3
3
0,
0. 300
400
300
500
400
500
T (K) T (
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