Topological interactions

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

The obtained values of "CRT differ from 2C by only 10% or less, whereas other investigators (13,18,34) have found that topological interactions contribute significantly to 2Cj. To account for this different finding, additional data on polyurethane elastomers would be required. 

It has long been reaUsed that the key physics determining the rheology of high molecular weight polymers in the melt state arises from the topological interactions between the molecules [1,2]. This is deduced from observations on many different monodisperse materials that ... 

From a conceptual point of view the Rouse model is limited towards large scales by the onset of topological interactions (the associated confinement and rep-... 

When the solution is dilute, the three diffusion coefficients in Eq. (40a, b) may be calculated only by taking the intramolecular hydrodynamic interaction into account. In what follows, the diffusion coefficients at infinite dilution are signified by the subscript 0 (i.e, D, 0, D10> and Dr0). As the polymer concentration increases, the intermolecular interaction starts to become important to polymer dynamics. The chain incrossability or topological interaction hinders the translational and rotational motions of chains, and slows down the three diffusion processes. These are usually called the entanglement effect on the rotational and transverse diffusions and the jamming effect on the longitudinal diffusion. In solving Eq. (39), these effects are taken into account by use of effective diffusion coefficients as will be discussed in Sect. 6.3. 

Fig.3 Computer simulation of a disordered densely packed amorphous polymer (BPAPC, M= 19800) the density of the simulated structure (1.2 x 103kg/m3) agrees with experimental data. Note that the shown cube of 3 nm edge length contains segments of about 30 different chains thus giving rise to a large number of topological interactions or entanglements (Courtesy U. Suter [7])...

Ivanov D, Nasmyth K. A topological interaction between cohesin rings and a circular minichromosome. Cell 2005 122 849-860. 

The fact that two chains cannot pass through one another creates topological interactions known as entanglements that raise the network modulus. 

Virtual chains represent topological interactions between strands, as opposed to effective chains that model the effects of strand connections. 

To rjsk jkT (b jkT. These relations are very general, and can be applied whenever the topological interactions, called entanglements (discussed in detail in Chapter 9), can be ignored. 

The reptation ideas discussed above will now be combined with the relaxation ideas discussed in Chapter 8 to describe the stress relaxation modiihis G t) for an entangled polymer melt. On length scales smaller than the tube diameter a, topological interactions are unimportant and the dynamics are similar to those in unentangled polymer melts and are described by the Rouse model. The entanglement strand of monomers relaxes by Rouse motion with relaxation time Tg [Eq. (9.10)] ... 

On length scales larger than the correlation length but smaller than the tube diameter a, hydrodynamic interactions are screened, and topological interactions are unimportant. Polymer motion on these length scales is described by the Rouse model. The relaxation time Tg of an entanglement strand of monomers is that of a Rouse chain of N jg correlation volumes [Eg. (8.76)] ... 

On length scales larger than the tube diameter, topological interactions are important and the motion is described by the reptation model with the chain relaxation time given by the reptation time ... 

Comparison of Schemes 77 and 79 reveals that the topological interaction of the two menthyl ester groups is fundamentally different in [4 + 2] additions of fumarates (308) compared with those of methylenemalonates (318). In the latter, both ester auxiliaries cooperate in terms of activating the dienophiile through formation of a stable chelate (319) but their individual stereoface biases in (319) are reversed. However, the observed ir-face discrimination can be rationalized by assuming that the C-2,C-3 moiety of cyclopentadiene is more bulky than the 5-CH2 group so that orientations c and d are favored over orientations a and b. Consistent with this hypothesis, no asymmetric induction could be achieved on addition of (318 R = H) to cyclohexadiene, which has comparably bulky C-2.C-3 and C-5.C-6 moieties. 

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