The Steady State Compliance Je

Now as t -> 0 the spectral function L reduces to the area under the distribution. This is the steady state compliance Je ... 

The low-frequency limiting viscoelastic behavior is thus governed by the sums S1 and 82- Values of these, together with 1S2/5 and the first two relaxation time ratios, are given in Table 9-II for several choices of h and other parameters. The greatest differences are seen in the ratio 82/8 to which the steady-state compliance Je is proportional. The table also includes some data for branched polymers which will be discussed in Section 8. 

System relaxation times have been determined from the relaxation of the stress after abrupt cessation of shear flow. Representative applications of the approach are found in Takahashi, etal. 9), who examined 355-3840 kDa polystyrenes in benzyl- -butylphthalate, at concentrations identified as showing dilute-solution behavior for the steady-state compliance Je and semidilute behavior for the zero shear viscosity. The stress relaxation after shear cessation, identified as the transient viscosity, decreased exponentially with time except at the shortest times studied, leading to an identification of an observed longest relaxation time Xm, whose c and M dependences were determined. 

From equation 26, setting to = 0, we have for the equilibrium compliance Je in a viscoelastic solid or the steady-state compliance 7° in a viscoelastic liquid ... 

For M M c, if there were really a single terminal relaxation time as implied by equation 14, combination of that equation with equation 34 of Chapter 3 would make the steady-state compliance the same as the plateau compliance 7 = 7/v = /G%. with J% given by equation 2. Actually, as shown by Graessley, the ratio Je/J% is the ratio of what may be termed the weight- and number-average relaxation times in the terminal zone ... 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

This means that for a linear elastico-viscous liquid, the steady-state shear compliance Je must be equal to the constrained shear recovery sx which follows on a steady shear flow at unity shear stress. From this one deduces the following relation between and Je ... 

Figure 4 Variations with temperature of the steady state recoverable compliance Je° (a), and of the contribution (Jf — JN°) of slow retardation mechanisms to Je° (b). Gluten from Olympic x Gabo cross line -jl7 + 18/-) in water (filled symbols) and in deuterium dioxide (empty symbols). Je° was obtained from the recovery curves...

The renormalized horizontal and vertical factors used to build up the master curve follows the longest relaxation time, Xz, and the steady-state creep compliance Je, respectively (d, 19, 4S). Therefore, the scaling relationships expressed by eqs.4 and 5 can be used to calculate the static exponents s and t. Figure 10 shows the log-log plot of ah and ay versus log e, in case of sample G1 (Table III), for which t=1.95 and s=0.69. The values of s, t and A are listed in Table III not for all the samples, but only when enough experimental data were available in the sol-gel transition so as to construct accurately the master curves. 

The two material parameters which characterize polymeric fluids at low strain rates, the viscosity, j, and the recoverable shear compliance, Je, can be directly determined, rj follows from the measurement of the torque under steady state conditions, Jq shows up in the reverse angular displacement subsequent to an unloading, caused by the retraction of the melt. From the discussion of the properties of rubbers we know already that simple shear is associated with the building-up of normal stresses. More specifically, one finds a non-vanishing... 

The estimate ((t) g(t) w exp(— t/x ) used earlier is known to be inaccurate, as g(t) is not a simple exponential frmction [114]. On the other hand, if g(t) is expressed as a sum of exponentials, as would be expected [96,111-114], then Tl for a pseudo-exponenti behavior might lie between Tn and x = qJJ, where Je is the steady-state recoverable compliancy for linear flexible diain polymers... 

Fig. 23. Normalized reciprocal steady-state recoverable compliance Je,max/Je for three polymers, poly(dimethyl siloxane), PIB, and PS, versus the reduced temperature T/Tg Tg is the glass temperature and the normalized compliance

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