Entangled strands

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... 

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. 

Me molecular mass of these entanglement strands) and the volume spanned by the entanglement distance is approximately a constant... 

With the number of entanglement strands ne/volume, the desired number of binary contacts per entanglement strand becomes... 

The chain tension arises in a physical way at timescales short enough for the tube constraints to be effectively permanent, each chain end is subject to random Brownian motion at the scale of an entanglement strand such that it may make a random choice of exploration of possible paths into the surrounding melt. One of these choices corresponds to retracing the chain back along its tube (thus shortening the primitive path), but far more choices correspond to extending the primitive path. The net effect is the chain tension sustained by the free ends. 

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... 

Oq total number of entangled strands in the phantom fibril from which the... 

Pc> Pb> Pf cumulative number fraction of grid squares that exhibit craze formation, craze fibril breakdown, and catastrophic fracture, respectively probability that a given entangled strand survives craze fibril formation disentanglement time of i strands in a fibril that survive fibril formation craze interface velocity volume fraction of polymer within craze... 

Hg. 7a-c. A schematic of the advancing craze-interface showing a A pile-up of entangled strands on the void ceiling, b A stretched cross-tie fibril produced by convolution, and c A cross-tie fibril after the polymer strands relax and concomitantly pull the main fibrils out of alignment... 

In view of the fact that the cross-tie fibrils contain some of the entangled strands that were imagined to either break or disentangle in the development of Eqs. (15) and (20), one can ask how accurate these formulae are given the cross-tie fibril microstructure of a typical craze. From the meridonal LAED reflections, Miller estimated that the cross-tie fibrils comprised only at most about 15% of the volume of the main fibrils and therefore that the corrections required to Eqs. (15) and (20) for the cross-tie fibrils are negligible. 

From the microscopic picture for the craze growth it seems clear that one important microscopic variable must be the mean number n of entangled strands within each fibril which survive the geometrically necessary strand loss associated with the interface formation. If the number of such strands is zero, the fibril will fail, since the polymer fluid which flows from the active zone into the fibril has no strain hardening capability and will not be able to support the relatively high tensile stresses necessary to propagate the interface. To obtain n one first estimates n, the total number of strands in the undeformed phantom fibril from which a craze fibril is drawn and which is given by ... 

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...

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. 

The plateau modulus is given by the usual formula for entangled polymers, Gq vksT, where v is the number of entanglement strands per unit volume of melt that is,... 

This tube diameter can be interpreted as the end-to-end distance of an entanglement strand of A e monomers ... 

The entanglement strand has entanglement molar mass M = N Mq. The entanglement strand effectively replaces the network strand in the determination of the modulus for networks made from long strands, and also determines the rubbery plateau modulus of high molar mass polymer -melts ... 

The number of monomers in virtual chains is assumed to change with deformation according to Eq. (7.60), similar to the constrained-junction and diffused-constraints models. If one virtual chain is attached to every entanglement strand of monomers, it contains of order virtual monomers in the undeformed state of the network. The number of monomers in each virtual chain changes as the network is deformed... 

Eq. (7.77) would only serve to estimate the entanglement strand in the preparation state assuming the Edwards tube model is correct (A e ... 

The results in this section were all derived for unentangled networks. The Edwards tube model for entangled networks gives identical results with N replaced by N, the number of Kuhn monomers in an entanglement strand in the preparation state, because both entanglement strands and network strands are assumed to deform affinely in the Edwards tube model. If the Edwards tube model were correct, the universal relations [Eqs (7.91) and (7.92)] would still apply for entangled networks, since they are independent of N. However, the non-affine tube models predict that entangled networks will swell considerably more than the Edwards tube model predicts. 

The tube can be thought of as being composed of NjN sections of size a, with each section containing monomers. The chain can be considered as either a random walk of entanglement strands NjNc strands of size a) or a random walk of monomers (A monomers of size b). 

The average contour length (L) of the primitive path (the centre of the confining tube, see Fig. 7.10) is the product of the entanglement strand length a and the average number of entanglement strands per chain N/N. ... 

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)]. 

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