Recoverable compliance function

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.)...

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 compliance function measured in a creep experiment can be decomposed into recoverable and viscous flow components... 

Fig. 20. Double-logarithmic plot of the recoverable compliance JAt) of the National Bureau of Standards (NBS) PIB sample measured as a function of time at six temperatures as indicated.

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).

Fig. 11. The dashed curves give the linear recoverable compliance (T/To)Ro(0=i< ) as a function of the recovery time 0.

Fig. 2.27. Recoverable-compliance, Ji(t), data of PPMS 5000 at temperatures -32.2°C ( ), -35.0°C U),-38.6°C (t),-40.0°C ( ), -41.1 °C (x), -42.6 °C n, -44.5 °C ( ), -45.2 °C (V), -46.9 °C (A), and -50 °C (o). The data taken at different temperatures have been shifted horizontally along the log t axis by a temperature-dependent shift factor log ut in order to superpose the curves at the short-time end with the data for -35.0 °C. The inset shows the retardation spectrum, L, as a function of the reduced retardation time X with reference temperature To = -35.0 °C, which was obtained numerically from J lf) data.

The variation of recoverable compliance with molecular weight also differs for linear and nonlinear polymers. In contrast to the behavior of nearly monodisperse linear polymers, for which becomes a constant 2/ G ) beyond about 5Mg, for stars simply continues to increase in direct proportion to Mb, which is exemplified by the comparison of data for linear polystyrene [71] and four-arm polystyrene stars [66] in Fig. 3.47. Experimentally, the behavior of for nearly monodisperse stars, irrespective of branch-point functionality, is described well by [52]... 

Berry, et al. report studies of shear thinning and recoverable compliance of poly-a-methylstyrene in a-chloronaphthalene(32). The normalized viscosity r/(K)/tio was reported as a function of the dimensionless shear tioRok, Ro being the low-shear limit of the recoverable compliance. Figure 13.20b, c, and d shows these data. For all molecular weights and concentrations, t] k) is described to within the experimental accuracy by a simple exponential in k. 

Jd in equation (6) is equal to Jg - Jg, where Jg is the steady state compliance which is equal to the long-time limiting value of the recoverable compliance for a non-cross-linked system. i/(t) is the normalized memory function for the compliance, and it goes from i/ (0) = 0.0 to (oo) = 1.0. The normalized memory function can often be described using the generalized Kohlrausch-William-Watts (KWW) (37,38) or stretched exponential function ... 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

An analytic expression for the reduced creep function, obtained from Eq. (3.11) with Eq. (3.6) for the linear array, is shown in Table 4 Eq. (T4). Je is the elastic equilibrium compliance and y (1/2, w) is the incomplete gamma function of order 1/2, extensively tabulated and available in most computer subroutine libraries. The last term accounts for contribution of plastic flow, where present. Reduced creep functions (recoverable and including flow contributions) are plotted in Fig. 4 as a function of Z. The... 

FIGURE 5.17 Logarithmic presentation of the recoverable shear compliance, Jr(t), of Epon 100 IF as a function of the logarithm of time I at nine temperatures as indicated. Dramatic loss of long-time viscoelastic mechanisms is evident when temperature is decreased toward Tg. 

With all of the viscoelastic functions it is important to note the limiting values or forms which are qualitatively independent of the molecular structure. For a viscoelastic liquid, lini, /(f) = Jg, lim, /(f) = tlr], and lim,, Jr t) = J t) -thr] = Jg + Jd = /j.The last Umiting value Js is called the steady-state recoverable shear compliance. It is the maximum recoverable strain per unit stress, which reflects the maximum configurational orientation achievable at the present stress. 

Equation 24 indicates that the strains arising from different molecular mechanisms add simply in the compliances and in principle can be separated. On the other hand, the stresses do not add and the different mechanisms cannot be easily resolved in modulus fimctions. In solution the solvent and the polymer contributions are also seen to be additive, as shown below. To understand the somewhat esoteric viscoelastic fimctions this additivity is helpful. It should also be noted that there are only two functions that do not contain contributions involving viscous flow, which obscure the recoverable responses which are largely, if not completely. 

Fig. 30. The local segmental relaxation correlation time Tc of high molecular weight PS as a function of temperature obtained by two-dimensional exchange nmr (154,155) up to long times exceeding 100 s, and compared with the shift factor ot,s from time-temperature superposition of recoverable creep compliance Jr( ) curves of another high molecular weight...

Figure 6.10 Creep compliance (/) as a function time (t). Calculation of zero-shear-rate viscosity and steady-state recoverable shear compliance.

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