Periodic chain transform

The mean square displacements are calculated in the same spirit as for the simple Rouse model. If for simplicity we consider the periodic chain transform we get ... 

The open-chain transform should be used when dealing with the overall size of the chain and, in dynamics, when separate consideration of the first, most collective modes of motion is required. In the following, the more expedient periodic-chain transform will generally be adopted unless specified otherwise. 

Having found the independent linear combinations of bond vectors [see Eqs. (2.1.17), (2.1.27), and (2.1.28)], through the central-limit theorem it is easy to construct the Gaussian joint distribution and the associated quadratic potential. Adopting for simplicity the periodic-chain transform, we have... 

This same type of modification strategy also can be used to create highly reactive groups from functionalities of rather low reactivity. For instance, carbohydrate chains on glycoproteins can be modified with sodium periodate to transform their rather unreactive hydroxyl groups into highly reactive aldehydes. Similarly, cystine or disulfide residues in proteins can be selectively reduced to form active sulfhydryls, or 5 -phosphate groups of DNA can be transformed to yield modifiable amines. 

With the boundary conditions that the chain ends are free of forces, Eq. (13) is readily solved by cos-Fourier transformation, resulting in a spectrum of normal modes. Such solutions are similar, e.g. to the transverse vibrational modes of a linear chain except that relaxation motions are involved here instead of periodic vibrations. 

PE, the united atom model. We considered a sufficiently long PE chain made up of 5000 united atoms under periodic conditions in each direction. The initial amorphous sample prepared at 600 K was quenched to 100 K and drawn up to 400%. The sample was then quickly heated to various crystallization temperatures, and the molecular processes of fiber formation were monitored in situ via the real-space image and its Fourier transform, the structure function S3d([Pg.79]

Molecules containing polysaccharide chains may be oxidized to possess reactive aldehyde residues by treatment with sodium periodate. Any adjacent carbon atoms containing hydroxyl groups will be affected, cleaving the carbon-carbon bond and transforming the hydroxyls into... 

Under the conditions where the chain oxidation process occurs, this reaction results in chain termination. In the presence of ROOH with which the ions react to form radicals, this reaction is disguised. However, in the systems where hydroperoxide is absent and the initiating function of the catalyst is not manifested, the latter has a retarding effect on the process. It was often observed that the introduction of cobalt, manganese, or copper salts into the initial hydrocarbon did not accelerate the process but on the contrary, resulted in the induction period and elongated it [4-6]. The induction period is caused by chain termination in the reaction of R02 with Mn"+, and cessation of retardation is due to the formation of ROOH, which interacts with the catalyst and thus transforms it from the inhibitor into the component of the initiating system. 

This problem was first approached in the work of Denisov [59] dealing with the autoxidation of hydrocarbon in the presence of an inhibitor, which was able to break chains in reactions with peroxyl radicals, while the radicals produced failed to contribute to chain propagation (see Chapter 5). The kinetics of inhibitor consumption and hydroperoxide accumulation were elucidated by a computer-aided numerical solution of a set of differential equations. In full agreement with the experiment, the induction period increased with the efficiency of the inhibitor characterized by the ratio of rate constants [59], An initiated inhibited reaction (vi = vi0 = const.) transforms into the autoinitiated chain reaction (vi = vio + k3[ROOH] > vi0) if the following condition is satisfied. 

Another important characteristic of inhibitors is the time of their inhibition action. If an inhibitor is consumed only in chain termination reactions, this time is determined by the initial concentration [InH]0, stoichiometric coefficient of inhibition / and Vj. In this case, the rate of inhibitor consumption is vInH = v //. Side reactions of InH with dioxygen and hydroperoxide shorten the inhibitory period and increase the rate of inhibitor consumption. Therefore, an inhibitor is efficient when it provides a minimal chain length v and its own loss in side reactions w is low. Assuming that an efficient inhibitor has w < 0.25, we get the inequality 4k 2[InH][02] < v which can be transformed, by substituting the correlation equation from Table 14.7, into the following equation... 

Some more or less routine transformations with side chains of tetrazolo[l,5-tf]pyridine or its benzologue have also been published during the recent period, and these results are shown in Scheme 19. 

Because of the content of double bonds and a tertiary C-atom, polyisoprene undergoes numerous chemical transformations by the action of oxygen, light, or heat, like chain scission and crosslinking.The addition of suitable stabilizers can suppress these reactions even over a period of years. In the case of the thermo-oxidative crosslinking of polyisoprene this effect can be shown with the following experiment. 

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