Entanglement networks

Under compression or shear most polymers show qualitatively similar behaviour. However, under the application of tensile stress, two different defonnation processes after the yield point are known. Ductile polymers elongate in an irreversible process similar to flow, while brittle systems whiten due the fonnation of microvoids. These voids rapidly grow and lead to sample failure [50, 51]- The reason for these conspicuously different defonnation mechanisms are thought to be related to the local dynamics of the polymer chains and to the entanglement network density. 

N. Nemoto, M. Kishine, T. Inoue, T. Osaki. Tracer diffusion of linear polystyrene in entanglement networks. Macromolecules 22 659-664, 1990. 

In Equation 1, t is a thermal vibration frequency, U and P are, respectively activation energy and volume whereas c is a local stress. The physical significance and values for these parameters are discussed in Reference 1. Processes (a)-(c) are performed with the help of a Monte-Carlo procedure which, at regular short time intervals, also relaxes the entanglement network to its minimum energy configuration (for more details, see Reference 1). 

Many fluids show a decrease in viscosity with increasing shear rate. This behavior is referred to as shear thinning, which means that the resistance of the material to flow decreases and the energy required to sustain flow at high shear rates is reduced. These materials are called pseudoplastic (Fig. 3a and b, curves B). At rest the material forms a network structure, which may be an agglomerate of many molecules attracted to each other or an entangled network of polymer chains. Under shear this structure is broken down, resulting in a shear... 

The mathematical treatment that arises from the dynamic dilution hypothesis is remarkably simple - and very effective in the cases of star polymers and of path length fluctuation contributions to constraint release in Hnear polymers. The physics is equally appealing all relaxed segments on a timescale rare treated in just the same way they do not contribute to the entanglement network as far as the unrelaxed material is concerned. If the volume fraction of unrelaxed chain material is 0, then on this timescale the entanglement molecular weight is renormalised to Mg/0 or, equivalently, the tube diameter to However, such a... 

Reasons have been advanced for both an increase and a decrease of the tube diameter with strain. A justification of the former view might be the retraction process itself [38]. If it acts in a similar way to the dynamic dilution and the effective concentration of entanglement network follows the retraction then Cgjy < E.u > so that a < E.u On the other hand one might guess that at large strains the tube deforms at constant tube volume La. The tube length must increase as < E.u >,so from this effect a < E.u > . Indeed, Marrucci has recently proposed that both these effects exist and remain unnoticed in step strain because they cancel [69] Of course this is far from idle speculation because there is another situation in which such effects would have important consequences. This is in conditions of continuous deformation, to which we now turn. 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

Very thin (average diameter 10 nm) and thin (average diameter 40 nm) H-CNFs were obtained in a highly dispersed form using nanodispersion equipment (T.K. FILM ICS Model 56-50, Primix, Japan) to undo the entangled network with an impeller... 

Chompff and Duiser (232) analyzed the viscoelastic properties of an entanglement network somewhat similar to that envisioned by Parry et al. Theirs is the only molecular theory which predicts a spectrum for the plateau as well as the transition and terminal regions. Earlier Duiser and Staverman (233) had examined a system of four identical Rouse chains, each fixed in space at one end and joined together at the other. They showed that the relaxation times of this system are the same as if two of the chains were fixed in space at both ends and the remaining two were joined to form a single chain with fixed ends of twice the original size. 

The front factor g as defined above5 is unity in all the earlier theories (17). Recently Duiser and Staverman (233) have obtained g = j and Imai and Gordon (259) g — 0.54 with Rouse model theories which make no a priori assumptions about the junction point locations after deformation. Edwards (260) also arrives at and Freed (261) deduces that g= 1 is an upper bound by similar approaches. The front factor usually assumed in the shifted relaxation theory of the plateau modulus is g = 1, although Chompff and Duiser (232) obtain g = j through their extension of the Duiser-Staverman result to entanglement networks. The physical reasons for the different values of g in different treatments are not clear at present. 

Chompff, A.J., Prins.W. Viscoelasticity of networks consisting of crosslinked or entangled macromolecules. II. Verification of the theory for entanglement networks. J. Chem. Phys. 48,235-243 (1968). 

Kramer, O., Carpenter, R. L., Ty, V., Ferry, J. D. Entanglement networks crosslinked in states of strain, paper presented at Society of Rheology Meeting, Montreal, October 1973. Part of this work is described in Entanglement networks of 1,2-poly butadiene crosslinked in states of strain. I. Cross-linking at 0° C. Macromolecules 7,79-84 (1974), by the same authors. 

Chompff, A. J. Linear viscoelasticity of entanglement networks. Thesis, T. H. Delft 1965. 

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