Entanglement limit

Shi, X Hammond, RW Morris, MD, DNA Conformational Dynamics in Polymer Solutions Above and Below the Entanglement Limit, Analytical Chemistry 67, 1132, 1995. 

A. What would be the viscosity of a sample of polycrud whose (DP)w is 500. [Still above the entanglement limit assume you can use the equation rjm - k(DP) (figure out what the exponent is), even though this... 

Once the chains are long enough to reach the asymptotic characteristic ratio discussed in Chapter 2, it is found that the measured shear viscosity increases linearly with molecular weight for many polymer liquids. This phenomenon can be explained in terms of the dynamics of individual polymer chains as long as the system is below the entanglement limit. The basic theory is due to Rouse and is described in detail by Ferry.i ... 

The role of positional fluctuations in polymer networks is central to some theories of elasticity, and has been investigated with an MC method based on a modified bond-fluctuation model (265). The simple model used in the simulations gave results close to those calculated from theory for a Bethe lattice (also known as a Cayley tree). More extensive results bearing on the role of fluctuations in polymer networks have been reported by Grest and co-workers (225). They find that entanglements limit fluctuations, giving behavior similar to the description provided by the tube model. 

Figure Variation of a glass transition temperature, Tg, for maximally frozen 20 w% solutions against Mn (expressed as a function of DE) for the commercial SHPs in Figure S. DE values are indicated by numbers marked above the x-axis. Areas of specific functional attrftutes, corresponding to three r ons of the diagram, are labeled [Inset plot of Tg vs. l/l (X 10 000) for SHPs with Mn values below entanglement limit, illustrating the theoretically predicted linear dependence.] (Reproduced with permission from ref. 1. Copyright 1986 Elsevier.)...

Figure 10. Schematic plot of Tg (or Tg ) vs. molecular weight (modeled after the data plot for SHPs in Figure 6), which illustrates that, while the segmental Tg remains constant with increasing MW for MWs above the entanglement limit, the network Tg continues to increase monotonically with increasing MW above the entanglement MW limit. (Reproduced with permission from reference 7. Copyright 1991 Plenum.)...

For molecular weights above the entanglement limit, i.e. M > Me, one finds Jg =const and rjo M, therefore... 

For which polymers and under which conditions do crazes occur Crazes form primarily in amorphous polymers, for molecular weights above the entanglement limit. There is no craze formation under compression or under pure shear. The typical situation leading to craze initiation is the imposition of an uniaxial or biaxial tensile stress. If such stresses are applied and fulfill certain threshold conditions, crazes form statistically, preferentially at first at the sample surface. The initiation rate depends on the applied stress, as is shown in Fig. 8.22. The higher the stress imposed, the shorter is the time for the observation of the first crazes. After the initial increase with time, the craze density saturates. Removing the stress, the crazes close their openings somewhat, but survive. They disappear only if the sample is annealed at temperatures above the glass transition. 

A/r c critical number of Rouse-segments at the entanglement limit... 

In linear poly(ester)s, predictions of the end of service time can be based on the relationship between the ductile-brittle transition and the entanglement limit of the molecular weight. However, for crosslinked poly(ester)s, this method is not applicable. The reasons for complications in the degradation behavior arise from the diffusion control of hydrolysis kinetics, heterogeneity of semi-crystalline polymers, and variation of the hydrophilicity with the hydrolysis conversion. 

X. Shi, R. W. Hammond, and M. D. Morris. DNA conformational dynamics in polymer solutions above and below the entanglement limit. Anal. Chem., 67 (1995), 1132-1138. 

We return once again to the frequency-dependent and temperature-dependent measurements of the dielectric function of polyisoprene (PI) presented in Sect. 6.3.2. As shown in Figs. 6.20 and 6.21, two relaxation processes exist. The low frequency process, the normal mode, is the one of interest here. As has already been mentioned, it reflects the movements of the end-to-end distance vector R of the chain. The Rouse model enables these movements to be treated in the case of melts that are not entangled. Earlier, we learned that the motion of the end-to-end distance vector is to a large part due to the superposition of the three lowest order Rouse modes, polarized in the x, y and -directions. Therefore, the dielectric normal mode, when measured for samples with a molar mass below the entanglement limit, may be identified with these primary modes. 

Hence, the effect of the entanglements is two-fold, since both the elastic and the viscous properties are concerned. The observations all indicate the existence of a critical molar mass, introduced earlier as the critical molar mass at the entanglement limit, denoted by Me. Polymers with low molar masses, M < Me, exhibit no entanglement effects, but for M > Me they show up and become dominant. All properties that are founded on motions on length scales corresponding to a molar mass above Me are affected. This holds, in particular, for the viscosity and the dielectric normal mode since these include the whole polymer chain. On the other hand. Rouse dynamics is maintained within the sequences between the entanglement points, as has already been mentioned. 

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