Entanglement strand density

The combined effects of a divalent Ca counterion and thermal treatment can be seen from studies of PMMA-based ionomers [16]. In thin films of Ca-salts of this ionomer cast from methylene chloride, and having an ion content of only 0.8 mol%, the only observed deformation was a series of long, localized crazes, similar to those seen in the PMMA homopolymer. When the ionomer samples were subject to an additional heat treatment (8 h at 100°C), the induced crazes were shorter in length and shear deformation zones were present. This behavior implies that the heat treatment enhanced the formation of ionic aggregates and increased the entanglement strand density. The deformation pattern attained is rather similar to that of Na salts having an ion content of about 6 mol% hence, substitution of divalent Ca for monovalent Na permits comparable deformation modes, including some shear, to be obtained at much lower ion contents. 

The mechanical properties of ionomers are generally superior to those of the homopolymer or copolymer from which the ionomer has been synthesized. This is particularly so when the ion content is near to or above the critical value at which the ionic cluster phase becomes dominant over the multiplet-containing matrix phase. The greater strength and stability of such ionomers is a result of efficient ionic-type crosslinking and an enhanced entanglement strand density. 

From X and X, together with stress-strain measurements in extension from the state of ease, or simply from the equilibrium stress at X, the concentration of trapped entanglement strands can be calculated and compared with the entanglement strand density estimated from transient measurements on the uncrosslinked polymer. To obtain consistent results, especially for stress-strain relations in large extensions of the dual network from its state of ease, it is necessary to attribute deviations from neo-Hookean elasticity to the trapped entanglement network, as described by the... 

The different experimental systems all yield a similar pattern of variation of toughness with interface width. The toughness initially increases slowly with width at low interface width, and then increases rapidly with width and saturates at high width at a value close to the bulk toughness. If the density of entangled strands controlled the toughness, then the interface width at which the toughness... 

Finally we require an expression for the relaxation modulus consistent with the dilution hypothesis in which each tube segment relaxing its stress typically at a time r(x) does so in a background whose effective density of entangled strands is 0 The appropriate general form is... 

Fig. 9. Plot of the true suain ratio in craze and deformation zones showing the transition from crazing to shear deformation as a function of network strand (entangled + crosslinked) density v. The open squares and open diamonds represent uncrossiinked homopolymers and copolymers, the open triangles and hexagons represent uncrossiinked blends of PS and PPO and the filled triangles and circles represent crosslinked PS (After Ref. courtesy of J. Polym. Sd.-Polym. Phys. (Wiley))...

It has been previously suggested that fibril stability can be correlated uniquely with n, the mean number of entangled strands within each fibril which survive fibril formation. The present analysis does not quantitatively bear out this claim as demonstrated by the plot of fibril stability versus n shown in Fig. 40a. While the fibril stability certainly increases with n, not even the data for the monodisperse PS s and the PS blends fall on the same curve. In particular the use of the incorrect formula for the entanglement density of the diluted blends (v = [v] % instead of the correct v = [v] x ) caused a fortuitous superposition of the data in the paper by Yang et al. 

Since monomers are space-filling in the melt, the number density of entanglement strands is just the reciprocal of the entanglement strand volume, leading to a simple expression for the plateau modulus of an entangled polymer melt [Eq. (7.47)]. 

The plateau modulus in this regime is G VefeeT, where Ve is the number density of entanglement strands. The volume per entanglement strand is =fa. The scaling model... 

The idea described above for glassy amorphous homopolymers can be extended to include miscible amorphous polymer blends, such as PS/PPO. Furthermore, a low degree of covalent cross-links can be considered as equivalent to entanglements for controlling the deformation mode. The strand density of cross-linked polymers is defined as the sum of the entanglement density and the covalent cross-link density [18] as... 

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