Dispersive Linear Chain

Figure 34. Surface phonon dispersion for 2H-TaSe2. The HAS data are shown as solid circles except for weak points which appear in the TOP spectra as hybridized longitudinal modes that are shown as crosses. All the data were obtained at 60 K, well into the low-temperature phase. The calculated striped and shaded regions, corresponding to transverse and longitudinal polarizations respectively, are the slab-adapted bulk phonon bands, while the solid line is a calculation for the Rayleigh wave based on the Dispersive Linear Chain Model (shown schematically in Fig. 35). The open circles at g = 0 are from Raman scattering experiments. (This figure has been corrected from Fig. 23 in Ref. 54.)...

Figure 35. Schematic drawing of the Dispersive Linear Chain Model applied to 2H-TaSe . The force constants /, and A describe the Ta-Se and Se-Se forces within a TaSej subunit, and / gives the Se-Se force between layers, rf, and are the spring constants between Se atoms and between Ta atoms, respectively, for neighboring subunits in the same layer.

Let us now study the effect of including a perturbation //per to the zero-order Hamiltonian Hq, so that the new Hamilton operator H will be given by // = Hq + //per. We further proceed with our example of the linear chain of period a with //at = 1, for the sake of simplicity, and continue to extract fundamental information. Figure 1.32 shows the band dispersion of this half-filled system. 

Mention has already been made of the numerous effects attendant upon chemical substitutions on the polysaccharide linear chain. Natural branches impart a dispersion stability to amylopectin that is not afforded amylose. One only has to compare cellulose ethers, deesterified chitin, and the lysis product of protopectin with the underivatized parent compound to appreciate the impact of chemical substituents on functionality. The loosening of compact, parallel structures with alkyl, hydroxyalkyl, and alkoxyl groups facilitates hydration and transforms insoluble, refractory polysaccharides to soluble, reactive polysaccharides. Not only do these substituents obstruct the crystallization tendency, they almost always confer secondary functionalities like q enhancement and foam, suspension, and freeze-thaw stabilization. 

For the dispersion relation given above for the linear chain, we have... 

Fig. 12. Schematic phonon energy versus reciprocal lattice wavevector for a linear chain. The actual dispersion curve depends upon the masses of and forces between the constituent...

So far, activation of mechanophores in the cyclic chain has not been reported. One important issue is the location of the mechanophore in the cyclic polymer. If only one mechanophore is incorporated into a ring chain (Fig. 18a), it is unlikely to experience the maximum hydrodynamic force (red dots) because the ring has no definitive midpoint in the flow field. Even if the ring chain breaks, the positimi of the mechanophore is unlikely to locate just at the midpoint of the linear product To improve the chance of activation, it is better to have multiple mechanophores incorporated into the cyclic chain, such as random, alternative, or block cyclic copolymers. For example, in Fig. 18b the mechanophores are randomly dispersed into a cyclic macromolecule to increase the activation probability. If the cyclic chain breaks, the mechanophores still have the chance to be located near the midpoint of the linear chain. The linear fragment then undergoes CST and activates the mechanophores. 

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