Kink, double

Figure 11.6 illustrates the energy that must be supplied by thermal activation. The curve of ab vs. A shows the force that must be applied to the dislocation (per unit length) if it were forced to surmount the Peierls barrier in the manner just described in the absence of thermal activation. The quantity A is the area swept out by the double kink as it surmounts the barrier and is a measure of the forward motion of the double kink. A = 0 corresponds to the dislocation lying along an energy trough (minimum) as in Fig. 11.5a. A2 is the area swept out when maximum force must be supplied to drive the double kink. A4 is the area swept out when the saddle point has been reached and the barrier has been effectively surmounted. The area under the curve is then the total work that must be done by the applied stress to surmount the barrier in the absence of thermal activation. When the applied stress is a a (and too small to force the barrier), the swept-out area is A, and the energy that must be supplied by thermal activation is then the shaded area shown in Fig. 11.6. The activation energy is then...

In many cases molecular displacements depend on other transition processes occurring previously in the neighborhood. Let us consider Fig. 27. The double kink may move from 1 to l Then segment 2 has the possibility to go from 2 to 2. Flowing units 4,5 and 6 may undergo similar processes. At last the molecule may reach the dashed structure. In every case the final position will be reached after some successive place exchange processes each determining the next step. We therefore get a system of simultaneous differential equations. 

Let us calculate the electron affinity (EA) and ionization potential (IP) of clusters of silver species adsorbed to virtual sites near the defect. These levels are shown for the positive kink in Figure 3 relative to their positions in the valence and conduction bands of the model. The EA has a sawtooth behavior but is larger than the AgBr EA for neutral clusters of all sizes up to 8 atoms. Thus, electron trapping will occur at clusters on the positive kink. Corresponding data for the negative and double kink are shown in Figure 4. 

Tram-double bonds cause a double kink of the carbonic chain which maintains the same direction, Fig. (2). Fatty acids with trans-double bonds are very rare in nature and their presence in natural origin extracts is caused in most cases by... 

Figure 2. Modification concepts for para-linked aromatic polymers including typical monomer structures (a) monomer units of different length, (b) kinked comonomers, (c) double kinked comonomers, (d) crankshaft comonomers, (e) flexible lateral substituents, (f) bulky and stiff lateral substituents, and (g) monomers with non-coplanar conformation.

A different structural modification, which is mainly based on lowering the chain stiffness and as a result also on reducing the intermolecular interactions, is the incorporation of kinked and double kinked comonomers (Fig. 2b and c). Typical monomers with kinks are meta-substituted phenylene derivatives or 4,4 -functionalized biphenyl-ethers and 4,4 -functionalized biphenyl-sulfides. Monomers with double kinks are,... 

Polyesters with Kinked or Double Kinked Moieties... 

CR2- -0-, -S-, and -CO-, which are functionalized in the 4,4 -position or the 3,4 -position. Monomers with functional groups in the 3,4 -position, such as 3,4 -di-hydroxy benzophenone (3,4 -DHBP) or 3,4 -dicarboxy-diphenylether (3,4 -CDPE) compensate the kink to a certain extent. These comonomers are also referred to as double kinked comonomers. These types of comonomers can only be incorporated in... 

Figure 6. Examples of kinked and double kinked comonomers for aromatic LC polyesters.

The solution behavior has been significantly enhanced by the same structural modifications as reported previously for aromatic LC polyesters. For example poly-(p-phenylene terephthalamide) has been modified by bulky, stiff substituents [32], flexible alkyl side chains [33], the incorporation of kinked and double kinked comonomers, and comonomers of different lengths [34], as well as the use of noncoplanar bipheny-lene monomers [35]. To develop high performance materials, modifications that increase the solubility while maintaining the rod-like character, high glass transition temperatures, and the temperature stability are of particular interest. The solubility and the chain stiffness are critical factors in achieving lyotropic solutions. 

In another molecular approach Argon (28) has proposed a theory of yielding for glassy polymers based on the concept that deformation at molecular level consists in the formation of a pair of molecular kinks. The resistance to double kink formation is considered to arise from the elastic interactions between chain molecule and its neighbors, ie, from intermolecular forces. This is in contrast to the Robertson theory, where intramolecular forces are of primary consideration. We need to recall that the intramolecular forces are by several orders of magnitude stronger than the intermolecular ones—except for entanglements which operate as if they were primary chemical bonds. 

The quantity is the energy release rate for one of the kinks in the case of double kinking. From (4.28), the stress concentration field phase angle corresponding to (4.86) is... 

Fig. 4.51. Ratio of the energy release rate for incremental crack extension along the interface, as in part (c) or (d) of Figure 4.49, to the energy release rate for incremental crack extension into the substrate, as in part (b) of Figure 4.49, versus the stiffness ratio D. Note that the difference between single and double kinking into the interface is minor.

Yet another useful observation from Figure 4.52 is that the stress concentration phase angles for the cases of single and double kinking depend on Di in much the same way. Thus, even in cases when the interface fracture resistance F(tAkn) depends on the local phase angle, the resistance to a single kink or each of a pair of double kinks can be viewed as being about the same for any practical purposes. 

Taking displacement reactions as the basis and assuming that the adsorption potential at the site of a dislocation (D), double kink, i.e., a pair of neighboring kinks of opposite sign in a ledge (DK), ledge (L), and surface terrace (T) changes in the sequence D>DK>K>L>T, one finds... 

Argon and Bessonov [161, 162] have recently presented a molecular model of shear yielding which involves the rotation of molecular segments against intra- and intermolecular forces and the systematic, pairwise reduction of kinks in a strained polymer. They calculate the free enthalpy of activation of a double kink in a small bundle of collectively acting molecules, AG, as... 

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