Relaxation exponent determination

We also determined the relaxation exponents, n, at the gel point from tan 8 plots and by using equation 7. For an intensity of 0.02 mW cm 2, the values of n were found to be 0.70 0.01 for the continuous mode and 0.71 0.01 for the pulse mode, indicating that the properties of the critical gd do not depend on the mode of UV irradiation. Similar results were obtained for a higher intensity of 0.17 mW cm 2, where n was determined to be 0.67 0.01 for the continuous mode and 0.69 0.02 for the pulse mode. Within experimental error, it seems that values of the relaxation exponents were comparable, regardless of whether pulse or continuous modes were used on the sample. Also, the UV light intensity had little effect on the values of the relaxation exponents determined for the NO A 61 samples. 

Dynamic mechanical data near the gel point allow easy determination of the parameters of the critical gel, Eq. 1-1. Tan 8, as shown in Fig. 26, gives the relaxation exponent n... 

Spin-lattice relaxation time increased continuously as a fimction of time, passing from 4000 ms to 8500 ms. It was possible to see two different evolutions. Ti increased rapidly during the two first days and slowed down after this time. Such a curve can be fitted with a power law model. The evolution was the same as for crystal size during Ostwald Ripening. The power law exponent determined for this system was 0.098, which is lower than the 0.2 to 0.3 which can be found for the evolution of crystal size followed by polarized microscopy. The actual deviation was due to the fact that we measured spin-lattiee relaxation times and not crystal size directly. However, the fact that we retained the power law model led us to expect a relationship between crystal size and spin-lattice relaxation time. 

Into these relations one may introduce specific values (s,z) from percolation theory or from branching theory and determine the corresponding values for Wc- The wide range of values for the relaxation exponent 0 < c < 1 lets us expect that the dynamic exponents s and z are nommiversal. Since s and z can be predicted from theory (47), Uc values can be calculated from equation 13. This result, however, relies on the symmetry hypothesis, which does not seem to be generally valid, at least not for highly entangled polybutadienes (48). 

Sg is called as the gel strength and has an unusual unit of Pa s". n is named the critical relaxation exponent because n determines the stress relaxation rate at the gel point. One may simply understand that Sg is the relaxation modulus at the gel point when the relaxation time f equals Is. The expression of Sg as Sg= G(t)xfn may be of help to understand the physical meaning of Sg. n=0 gives Sg=Go, the elastic modulus, that describes the rigidity of the sample. Also Sg represents the viscosity for w=l. A similar expression can be applied at the gel point for G ((o) and G"((d). 

Fig. 8. Determination of critical gel point and network quality from oscillatory rheology. The critical gel point is the time when the curves of tan(5) at various frequencies coincide. The table shows the relaxation exponent n for gels of various equilibrium moduli.

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

Since the glass transition corresponds to a constant value of the relaxation time [15], dTjdP is just the pressure coefficient of Tg. Comparing Equations 24.10 and 24.13, we see that the scaling exponent is related to quantities—thermal pressure coefficient, thermal expansion coefficient, Tg, and its pressure coefficient—that can all be determined from PVT measurements... 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

The dynamic properties depend strongly on the material composition and structure. This is not included in current theories, which seem much too ideal in view of the complexity of the experimentally found relaxation patterns. Experimental studies involving concurrent determination of the static exponents, df and t, and the dynamic exponent, n, are required to find limiting situations to which one of the theories might apply. 

A unified approach to the glass transition, viscoelastic response and yield behavior of crosslinking systems is presented by extending our statistical mechanical theory of physical aging. We have (1) explained the transition of a WLF dependence to an Arrhenius temperature dependence of the relaxation time in the vicinity of Tg, (2) derived the empirical Nielson equation for Tg, and (3) determined the Chasset and Thirion exponent (m) as a function of cross-link density instead of as a constant reported by others. In addition, the effect of crosslinks on yield stress is analyzed and compared with other kinetic effects — physical aging and strain rate. 

Close to the gel point, in the range AX/X <0.1, the static modulus cannot be measured. Strong relaxation effects are present even at the lowest frequency which could be used, to be consistent with the kinetics (one period of oscillation = 67s). Beyond this range for AX/X >0.1, G (0,015Hz) corresponds to the static relaxed modulus. A critical exponent for the relaxed modulus can be determined by using the equation X - X... 

Pearson and Helfand [12] used a somewhat similar approach to determine the characteristic relaxation time for a branch to disentangle their calculation leads to a similar form, with a different exponent for the front factor ... 

Although glass transition is conventionally defined by the thermodynamics and kinetic properties of the structural a-relaxation, a fundamental role is played by its precursor, the Johari-Goldstein (JG) secondary relaxation. The JG relaxation time, xjg, like the dispersion of the a-relaxation, is invariant to changes in the temperature and pressure combinations while keeping xa constant in the equilibrium liquid state of a glass-former. For any fixed xa, the ratio, T/G/Ta, is exclusively determined by the dispersion of the a-relaxation or by the fractional exponent, 1 — n, of the Kohlrausch function that fits the dispersion. There is remarkable similarity in properties between the JG relaxation time and the a-relaxation time. Conventional theories and models of glass transition do not account for these nontrivial connections between the JG relaxation and the a-relaxation. For completeness, these theories and models have to be extended to address the JG relaxation and its remarkable properties. 

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