Entanglement concentration

The concentration at which the correlation length is of the order of the coil size R bN is the overlap concentration (f , given by Eq. (5.19)  

The concentration at which the tube diameter a [from Eqs (9.30) or (9.31)] equals the coil size R [Eq. (9.29)] is the entanglement concentration (p  

Chain size R, tube diameter a, and correlation length in a good solvent. The semidilute unentangled regime is 

For (p (pe, entanglement effects control chain dynamics and the reptation model must be used as described below. Between the overlap concentration and the entanglement concentration (cp p p, the solution is semidilute hut not entangled, and the unentangled solution model of Section 8.5 describes dynamics. The width of this semidilute unentangled regime is given by the ratio of Eqs (9.33) and (9.32)  

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Langley, N.R. and Polmanteer, K.E., Relation of elastic modulus to crosslink and entanglement concentrations in rubber networks. J. Polym. Sci. Polym. Phys. Ed., 12(6), 1023-1034 (1974). 

These expectations are indicated as regime III in Figs. 1 and 2 for polyelectrolyte concentrations greater than the entanglement concentration c. ... 

Zero shear viscosities have been determined in solution over a wide range of concentrations with a cone-plate Rheometrics Stress Rheometer. For linear macromolecules, the viscosity is proportional to c below the so called "entanglement concentration", c above c, is proportional to c. However, the viscosity will rise steeply at some concentration below c in the case where particular interconnections are formed at the concentration at which the molecules come into contact with one another. Ideally this will be the overlap threshold c. Below c, the molecules may associate partially but cannot form a network continuous over the entire sample space. Above c, plastic flow will require separation and... 

Table 9.1 shows that the number of Kuhn monomers in an entanglement strand in the melt state varies over a wide range (7 < A e(l) < 80) making 4 < 0e/0 < 30 for solutions in an athermal solvent. Since the entanglement concentration

overlap concentration (p, the expressions for a -solvent [Eqs (9.31), (9.33), and (9.34)] are valid for [A e(l)] - This condition is not very restrictive and it is satisfied for all experimental studies to date. 

Polymer solutions can be classified into five regimes (dilute, semidilute not entangled, semidilute entangled, concentrated not entangled, and concentrated entangled) [22] according to the polymer concentration and molar mass however, it is much common to classify them into only... 

Experimentally, as the solvent quality approaches the theta condition, the exponent ai approaches the Mark-Houwink-Sakurada exponent, a, which, in turn, approaches the limiting value 0.5. The applicability of Eqs. (1.100) and (1.101) was found to extend to concentrations in excess of the entanglement concentration. 

FIGURE 1.16 Determination of the chain overlap concentration c, the entanglement concentration Cg, the electrostatic hlob overlap concentration c from the concentration dependence of specific viscosity for a 17%-quaternized P2VP copolymer (17PMVP-C1) in solution in ethylene glycol at 25°C. Symbols are experimental data and solid lines represent the power laws predicted from scaling theory. (Adapted from Don and Colby [2006].)... 

As described above, NMR can be used to measure chemical crosslink concentrations however, the spectra are unaffected by physical entanglements. Since these entanglements contribute to the elastic response of cured rubber (see Chapter 4), the difference between crosslink determinations from NMR and modulus measurements can be used to quantify the entanglement concentration. Most applications of this method have been directed toward assessing the role of filler-polymer interaction on the stiffness of elastomers [95]. 

Celluloses are described by the type of substituent carried by the cellulose backbone and the viscosity. They can separate DNAs at low concentrations (<2%). Hydroxyethylcellulose (HEC) is a linear derivative with bulky ethylene oxide side chains terminating in hydroxyl groups. In aqueous solutions, it has a stiff, extended conformation. It has relatively high viscosities at the entanglement concentration. A mixture of HEC of different sizes offers a wide range of DNA separation from 100 bp to 23 kbp. Hydroxypropylcellulose (HPC) solution provides similar separation ability, and it has 2-3 times lower viscosity than HEC solution. 

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