Entangled system transition point

For the weakly entangled systems, one can expect, that the ratio E/B, that is the parameter of internal viscosity is small. It can be demonstrated in Section 4.2.3, that transition point from weakly to strongly entangled systems occurs at E B. To describe these facts, one can use any convenient approximate function for the measure of internal resistance, for example, the simple formula... 

The transition point can be different, if one uses different modes, but only the transition point for the first mode is considered here. For the strongly entangled systems, according to relation (5.17), ijj = 7t2/x, so that, at %l> 1,... 

One can consider the parameter B to be a function of x and, taking equations (3.17), (3.25) and empirical value 6 = 2.4 into account, finds a solution of the equation, and estimate the value of the transition point between weakly and strongly entangled systems as... 

The results of estimation of coefficient of self-diffusion due to simulation for macromolecules with different lengths are shown in Fig. 12. The introduction of local anisotropy practically does not affect the coefficient of diffusion below the transition point M, the position of which depends on the coefficient of local anisotropy. For strongly entangled systems (M > M ), the value of the index —2 in the reptation law is connected only with the fact of confinement of macromolecule, and does not depend on the value of the coefficient of local anisotropy. At the particular value ae = 0.3, the simulation reproduces the results of the conventional reptation-tube model (see equation (5.21)) and corresponds to the typical empirical situation (M = 10Me). 

The critical value of molecular weight can be identified with the transition point between weakly and strongly entangled systems, the position of which was estimated in Sections 4.2.3 and 5.1.2 as... 

For a loosely cross-linked network, polymerized above the glass transition temperature, R should approach the flexible limit, i.e., R, because the distance from the last attachment to the network is significantly larger and the overall mobility of the system is increased by polymerization above the Tg, As the polymerization temperature is lowered below Tg, the distance back to the last network attachment point (or entanglement) becomes less important, and the mobility of the radical chain end is reduced to the point where it is virtually immobile on the time scale of propagation. In this case, the rigid limit should be applicable, and R should approach R m, just as it would for a highly cross-linked network. 

The mean-field nature of our entanglement model is perhaps its weakest point. It has long been recognized that mean-field theories are rather rough approximations near the critical points for systems with d <4 dimensions. For example, the experimentally measured exponent for the liquid-gas critical point is (J= 0.33 and not S = 0.5. Mean-field theories generally predict more abrupt transitions than are actually observed. The discrepency between mean-field theories and experiments stems from the absence of fluctuation phenomena which are often more important near the transition. 

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