Entanglement effects

The new information necessary to make this approach quantitative is the dependence of the effective entanglement molecular weight on the concentration, (j) of unrelaxed segments. This is known from experiments on dilution of polymer melts by theta-solvents to be approximately which corre-... 

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

From the foregoing it will be clear that whenever entanglements and long chain branching are both present the dynamics in a polymer melt are highly co-operative. The orientational relaxation time of chain segments is exponentially dependent on both the contour distance to the nearest effective free end and on the effective entanglement density of its enviroiunent at all previous timescales. 

For polymers, the chains must reach a critical length before effective entanglement can occur. This number varies from one type of polymer to another. For poly(methyl methacrylate), for example, the critical entanglement degree of polymerization is approximately 300 (M approximately 30,000). 

The basally arranged noncalcareous Polyxenida (Penicillata bristle millipedes) lack defensive glands and instead project hooked bristles against attackers such as ants.12 Similar to modified larval hairs of dermestid beetles, predators are thus effectively entangled. 

The mean number of effectively entangled strands is then given by ... 

Fig. 40a. Craze fibril stability (e — e ) versus n, the mean number of effectively entangled strands per fibril for monodisperse PS (circles) PS molecular weight blends (squares) monodisperse PMMA (diamonds) and monodisperse PotMS (stars) The solid lines are the predictions of the model using the parameters given in the text, b Craze fibril stability (e — e ) vereus 1 / where is the mean force per effectively entangled strand in the fibrils. Same symbols and lines as in a...

Table II contains the values for the number of elastically effective entanglement points both as calculated from Gp (ng(exp)) and as calculated theoretically for an ideal network (n (theo)). The ratio ng(exp)/ng(theo) re-...

Table U. Elastically Effective Entanglement Points as Calculated from Gp and...

Figure 6.1 Log viscosity versus log MW for a polymer showing the critical MW for effective entanglements.

The assumption in deriving this equation is that only polymers with MWs above those for effective entanglements contribute to tensile strength. Thus, O is the fraction of polymer molecules with MWs greater than Mj, the MW for effective entanglements, and is the number-average MW of this fraction. The equation is represented schematically in Figure 6.7, in which O is the shaded area. 

To the naked eye, a craze looks like an extension of the crack, but electron microscopy reveals that load-bearing fibrils about 10 nm In diameter span the gap between the surfaces of the polymer. A network of open holes of similar diameter runs through the craze. Molecular entanglements are essential, since without them there would be little to stabilize the loaded fibrils. If the polymer chains are too short to form effective entanglements, the material is extremely fragile. 

The fracture resistance of a thermoplastic depends critical upon its relative molar mass, essential because Gjc and fall rapidfy when the chains are too short to form effective entanglements. This point is illustrated in Figure 5.19. Very short chains are held tt ther oi by van der Waals forces, and are unable to form stable crazes. 

Brown has shown (Brown 1991b), as will be discussed in much more detail below, that the fracture energy of an interface that fails by the formation and subsequent breakdown of a craze is proportional to the square of the number of effectively entangled chains crossing the interface. Thus we would expect the interfacial fracture energy to vary like... 

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