Steady-state recoverable compliance

Doi and Edwards were able to develop an equation for the stress relaxation modulus G(t) of monodisperse entangled linear chain liquids in the terminal region without resorting to the independent alignment approximation From G(t), expressions can be obtained for the plateau modulus, the steady-state viscosity and steady-state recoverable compliance. The following dependences on chain length are obtained ... 

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...

While all relaxation times depend on temperature and pressure, only the global motions (viscosity, terminal relaxation time, steady-state recoverable compliance) are functions of Af , (and to a lesser extent MWD). The glass transition temperature of rubbers is independent of molecular weight because chain ends for high polymers are too sparse to affect this bulk property (Figure 3.14 Bogoslovov et al., 2010). The behavior can be described by the empirical Fox-Hory equation (Fox and Flory, 1954) ... 

The steady-state recoverable compliance, which is a measure of the elastic strain, can be calculated using... 

The steady-state recoverable compliance can be measured by fitting Eq. (6.13) to creep data at long times, or more accurately, jf is determined from the recovery after removal of the stress subsequent to attainment of steady-state flow. 

Alternatively, from steady shearing experiments, which yield rio directly in the limit of low shear rate, the steady-state recoverable compliance can be obtained from the first normal stress coefficient, which is the ratio of the first normal stress difference to the square of the shear rate, measured at low shear rate... 

From Eq. (6.12) the steady-state recoverable compliance is also given by... 

The steady-state recoverable compliance is an important rheological parameter because it is very sensitive to the high molecular tail of the molecular weight distribution, and thus can be correlated with elastic properties of a rubber. 

FIGURE 6.14 Steady-state recoverable compliance, which becomes constant at high molecular weight, for 1,4-polyisoprene (circles) and 1,4-polybutadiene (squares). The vertical arrows denote the Mu) for entanglements to affect the viscosity, which is smaller than the value of the molecular weight at which 7 become constant (Roland, 2011). 

Thus the steady state recoverable compliance Js is the sum of the glassy compliance and the limiting delayed compliance. 

For the viscoelastic liquids Jg is the steady-state recoverable compliance and for the viscoelastic solids, beyond the gel point it is the equilibrium compliance. 

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

In addition to the large molecular weight dependence of the viscosity, another aspect of the chain dynamics of polymers in the terminal region is the observation that the relaxation spectrum is very broad. One measure of the breadth of the relaxation spectrum is the product of the rubbery plateau value G°n and the steady-state recoverable compliance J°e- The predicted and observed values for polymer melts are... 

Fig. 42. Double logarithmic representation of the steady-state (recoverable) compliance as a function of molecular weight for linear (squares and crosses) and cyclic (circles) polystyrenes melts. After McKenna et al. (122).

This phenomenon originates from the relaxation time (r) of molecular chains and the steady-state recoverable compliance (/ o) in a polymer solution. With increasing concentration, the polymer chains are more strongly entangled with each other and get more and more... 

Steady-State viscosity, t/o, and the steady-state recoverable compliance, were found to be related to the molecular weight as follows ... 

In addition to timescale shifts with temperature, the magnitude of the compliance or modulus can change. The kinetic theory of rubber-like elasticity suggests that the entropically based contribution of the modulus to the viscoelastic response should increase in direct proportion to the absolute temperature. Correspondingly, the reciprocal of the steady-state recoverable compliance should be directly proportional to the absolute temperature. This is true at temperatures that are greater than 2Tg, but, between 1.2Tg and 2Tg, the steady-state recoverable compliance Js is essentially independent of temperature. At still lower temperatures a strong decrease of Js, is seen [51]. 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

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... 

Similarly, the steady-state recoverable compliance /e = yt/rioy (with rigf being the steady-state shear stress), which characterizes a strain recovered (recoiled) on removal of the stress from a steady fiow state, is determined from the G and G" data at a f/rj as... 

The corresponding relationships are noted for the storage and loss moduli, zero-shear viscosity, steady-state recoverable compliance, and average relaxation times (cf. eqns [32]-[35]) ... 

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