Recoverable compliance

The recoverable compliance Je° is very sensitive to molecular weight distribution, especially to the tail of the distribution at high molecular weights. According to the Rouse model [Eq.(4.28)], when samples with the same Mw are compared, their compliances should vary as MZMZ+ JM. Based on the success of the Rouse model mixing law for viscosity, one might hope for correlations of the form ... 

In addition to the primary glass-rubber relaxation which follows the empirical shifts determined by Eq. (26), part of the recoverable compliance does not obey time-temperature superposition. The shortest time data at the lowest temperatures has a component which shifts according to the Arrhenius temperature dependence... 

The strong effect of molecular chains on the viscoelastic behavior of polymeric solutions, even in the most dilute ones, is shown in Figure 8.24 (37). Here the recoverable compliance of a very dilute solution of polystyrene of weight-average molecular weight 860,000 in tri-m-tolyl phosphate is compared with that of the solvent. It is noteworthy that the value of the steady-state compliance for the solvent is 10 cm /dyn while that of the very dilute solution (Wpoi = 0.001) is nearly 10 cm /dyn. In other words, a very small fraction of the molecular chains are responsible for the fact that the steady-state compliance of the solution is more than 10 times that of the solvent. 

Figure 8.24 Double logarithmic plots of the recoverable compliance function in the time domain for (a) a dilute (0.1%, w/w) solution of polystyrene in tri-m-tolyl phosphate (b) tri-m-tolyl phosphate. (From Ref. 37.)...

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 ... 

The ejqpression obtained in this way for the diffusion < efficient appears to work very well, providing not only the observed dependence on molecular weight but also close numerical agreement with measured values A universal expression is obtained for recoverable compliance ... 

With these equations the steady state viscority and recoverable compliance for sufh-dently long diains (so that relaxaticm jmK es for t te make negligible contributions to the integrals) can be calculated ... 

Fig. 6. Comparisons of viscosity and recoverable compliance predictions by the Doi-Edwards theory with experimental observations. The predicted tio h too large, but its chmn length dependence is slightly weaker than observed. The predicted J is too small, but independent of chain length as observed. The dashed lines indicate predictions of the Rouse model...

Constraint release is likely to be very important in the relaxation of branched polymer liquids. However, if we ignore that complication, the stress relaxation modulus for a liquid of highly entan ed stars is given simply by Eq. 69 with v replaced by vi. The viscosity and recoverable compliance can then be calculated from Eq. 69 with Eqs. 25 and 26. 

Expressions for viscosity and recoverable compliance are readily obtained with Eqs. 25 and 26 ... 

Expressions for Viscosity and Recoverable Compliance of Entangled Star Polymers... 

Values of storage modulus G (a)) and loss modulus G"(m) can then be obtained by separating Eq. V-8 into its real and imaginary parts (Eq. V-2). Viscosity and recoverable compliance in the large N limit can be obtained from ... 

It was originally assumed that Gj should be proportional to T, and hence have a temperature dependence small compared to that for Xi, and further that all the Xi have a common dependence on temperatxure (7). Thus, it is assumed here that the temperature dependence of rj reflects that of only the T(. Recent investigators have suggested that for simple liquids, the Gf can have an appreciable temperatxure dependence xmder some conditions, so that it may be unreasonable to expect a simple fimction to correlate iq T) over extended temperature intervals 142a, 143b). Similar considerations may be necessary for a complete analysis of C as defined by Eq. (2.1) with temperature. Recent results of Plazek (777) showed that the temperature dependences of the viscous and recoverable contributions to the creep compliance of polystyrene are different from each other (after allowing for an expected proportionality of G,- to T). This can be interpreted as an indication of a significant temp>erature dependence of the G. Since our primary concern here is the temperature dependence of C for polymeric materials, for which the requisite data on deviation of Gj from proportionality to T do not exist, we will henceforth assume that there is only one contribution to the temperatiue dependence of (or of rj for simple liquids). Careful comparison of the temperature dependence of rj and of viscoelastic properties such as the recoverable compliance, may eventually provide an assessment of this assumption. 

Strain from a creep experiment with constant applied stress a for a viscoelastic solid (lower curve) and a viscoelastic liquid (upper curve). The slope at long times is the steady shear rate 7, from which the viscosity is calculated as = cr/7 (the viscosity of any solid is infinite, corresponding to zero slope). The extrapolation of this straight line to zero time (dotted line) gives the elastic part of the strain, from which the recoverable compliance is determined. 

Two steady states are recognized for the long-time creep compliance of materials. Either the sample is a solid and the compliance becomes time independent or the sample is a liquid and the compliance becomes linear in -time. Once steady state has been achieved in creep, the stress can be removed (a = 0) and the elastic recoil, called creep recovery, can be measured. Recovery strain is defined as 7r(0 s 7(0) — 7(0 for t > 0, where t is defined to be zero at the start of recovery. The recoverable compliance is defined as the ratio of the time-dependent recovery strain 7r(0 and the initially applied stress a, where both 7r and t are now defined to be zero at the start of recovery ... 

Boltzmann superposition relates the recoverable compliance after steady state has been achieved in creep to the creep compliance ... 

At low frequencies G of a liquid contains information on viscosity 77, while G of a liquid contains information on recoverable compliance /gq. At low frequencies G of a solid contains information on its modulus Ggq. What does G" tell us about the network at low frequency ... 

Find the relation between creep compliance J(t) and recoverable compliance /R(f) using the Boltzmann superposition principle. Dielectric spectroscopy indicates that water molecules respond to an oscillating electric field at a frequency of 17 GHz at room temperature. Is water still a Newtonian liquid at this high a frequency or is it viscoelastic If... 

Creep, Swelling, and Extraction Studies. Additional indications of component interaction were found in the results of creep experiments with the xl series shown in Tables VI and VII. Creep in the xl series is fully recoverable, i. e. there is no permanent set, consistent with a PUx continuous phase. The increase in recoverable compliance, however, indicates a reduction in apparent crosslink density with increasing delay time before irradiation. This result is reinforced by the data in Table VII Ci and Cg are the material constants in the Mooney-Rivlin equation. The rubbery plateau modulus and the crosslink density of PUx prepared in BHA, which mimics xl formation, is less than that of PU prepared neat. 

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...

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