Equivalent chain

The foregoing discussion of equivalent chains requires merely that its root-mean-square end-to-end distance shall equal that of the real chain. In order to define completely the equivalent chain, its contour lengths may also be required to coincide with that of the real chain. 

Thus the fully extended length of the equivalent chain will be set equal to that of the real chain with all valence angle B and ) restrictions removed. Then n and V (and therefore m) for the equivalent chain are completely defined by... 

This hypothetical equivalent chain should resemble in statistical behavior the actual chain of given size n and mean extension v. ... 

Exact Treatment for the Freely Jointed Chain (or Equivalent Chain).4 >5— Consider one of the bonds of a freely jointed chain acted upon by a tensile force r in the x direction. Letting xpi represent the angle between the bond and the o -axis, its component on the x-axis is Xi = l cos pi. The orientation energy of the bond is —rXi, and the probability that its x component has a value between Xi and Xi- -dxi therefore is proportional to... 

Considering the large variation of / for the poly[2]catenand 51b, it is expected that little correlation will exist between the spatial orientation of neighboring monomer segments and that it will represent the closest synthetic equivalent of the freely jointed chain model [63]. In this model, a real polymer chain is replaced by an equivalent chain consisting of N rectilinear segments of length Z, the spatial orientations of which are mutually independent (Scheme 24) [63]. 

A similar method was described by Rezanka and Podojil (100) and by Wiley et al. (101), including choline chloride as the source of the quaternary ammonium counterion. The molecular species of PE, PC, and PI were separated by a mixture of methanol, 286 mM aqueous choline chloride, and acetonitrile in a 90.5/7/2.5 ratio. Graphs of the logarithm of the relative retention time versus the equivalent chain length of the fatty acid found in the 1-position yields a straight line for a given fatty acid in the 2-position. For different fatty acids in the 2-position, a series of parallel straight lines is obtained. 

The equivalent chain corresponding to a CMIMx copolymer chain has the following characteristics ... 

The quantitative approach accounts well for the observed increase of Tu when the CMI content increases. Furthermore, considering the uncertainties of experimental determination of both Tu and Me (which controls Ne and Coo of the equivalent chain) the agreement seems reasonable. 

In the case of the xTyh-y copolyamides, another difficulty arises from the determination of characteristics of the equivalent chain (Sect. 2.1), namely... 

The structures deduced from the mass spectra must account for the observed equivalent chain length (ECL) or Kovats indices (KI) (Carlson et al., 1998 Katritzky and Chen, 2000 Zarei and Atabati, 2005). The values for monomethylalkanes (Mold et al., 1966 Szafranek et al., 1982) can be used to estimate the expected ECL or KI for a di-, tri- or tetramethyl-alkane structure proposed from a mass spectrum. For example, a dimethylalkane, such as 3,11 -dimethylnonacosane, with 31 carbons, would have its elution time decreased by about 0.3 carbons for the 3-methyl group and about 0.7 carbons for the 11-methyl group. Thus, the predicted ECL is approximately 30 and this is the ECL observed. 

By carrying out the above procedure from time 0 to time /,Mm, we evidently obtain only one possible realization of the stochastic process. In order to get a statistically complete picture of the temporal evolution of the system, we must actually carry out several independent realizations or runs. These runs must use the same initial conditions of the problem but different starting numbers for the uniform random number generator in order for the algorithm to result in different but statistically equivalent chains. If we make K runs in all, and record the population sizes (k, t) in run k at time t (i = 1,..., m and k = 1,...,K), then we may assert that the average number of particles at time t is... 

The square planar [Pt(CN)4]2- complex ions are not always strictly parallel to each other and not always perpendicular to the c axis. In NaCP 3 H20 they are parallel to each other within every chain, but form a dihedral angle of 14° with those of the next crystallographically non-equivalent chain. In KCP 3 H20 the normals of the complex planes deviate by 3° from the c axis, in RbCP H20 7°, in CsCP H20 15°, in CaCP 5 H20 6°, in SrCP 5 H20 5°, in BaCP 4 H20 3°. In KNaCP 3 H20 the dihedral angle between two neighbouring complexes within the chain is 2.3°. In RbCP 1.5 H20 the planes of the complexes form alternatively dihedral angles of 8.1° and 13.7° with the ab plane. 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

The flexibility of the chain is caused by a rotation about the O-C and 0-C bonds between neighboring glucose rings. If a real chain is replaced by an equivalent chain each unit of which consists of two parallel A/2 bonds about which rotation is possible and one 6 bond (normal to the two first bonds) about which no rotation takes place, it can be shown that the number of monomer units Sf in the Kuhn segment of a cellulose chain with unhindered rotation is given by... 

Figure 1.16 Schematic representation of the equivalent chain comprising N hypothetical bonds of length b connected by free joint.

The hetcrocyclc component of this general structure is most commonly a 4-monosubstituted imida/.ole or bioisast-cric equivalent. Chains A and B can be of various structures and lengths, and there is al.so wide latitude in the structural icquircmcnls for the polar group. Halogenated phenyl, cycloalkyl, and heteroaryl structures are usually found for ihc lipophilic moiety (Hig. 21-19). 

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