Branch index

The Randic connectivity index, X, is also called the connectivity index or branching index, and is defined by Eq. (18) [7], where b runs over the bonds i-j of the molecule, and and dj are the vertex degrees of the atoms incident with the considered bond. 

The molecular uniformity of constituting components of a nb/lcb glucan fraction of potato starch was investigated with Sepharose CL 2B (Fig. 16.16) as well as with Sephacryl S-1000 (Fig. 16.17). Therefore, each of the subsequently eluted 3-ml fractions was analyzed on their potential to form inclusion complexes with iodine, a sensitive test for the presence of nb/lcb glucans. Results are shown in Fig. 16.17 in terms of branching index, the ratio of extinction of pure iodine solution and of nb/lcb glucan/iodine complex the higher the index, the more pronounced the nb/lcb characteristics. 

In accordance with expectations, the branching index classifies the major partition of the investigated sample as nb/lcb glucans. On CL 2B, as well as on S-1000, the major percentage of components was eluted in the selective separation range, and a minor amount of high dp scb glucans in both systems was eluted at the exclusion limit. 

FIGURE 16.17 Nonbranched/long chain branched glucans of potato starch dissolved in hot water-steam and 0.1 M NaOH 1.2 ml of the 18-mg/ml solution was separated on Sephacryl S-1000 (95 X 1.6 cm) 3-ml fractions were collected for further analysis normalized (area = 1.0) eluogram profiles (ev) constructed from an off-line determined mass of carbohydrates for each of the fractions branching index ( ) determined from iodine-complexing potential of individual 3-ml fractions flow rate 0.40 ml/ min V ,i = 75 ml, Vtot = 162 ml eluent 0.005 tA NaOH. 

Both LALLS and MALLS can also be used to estimate polymer branching. The degree of polymer branching can be determined using the Zimm-Stockmeyer branching index, which is given by 190... 

Figure 10 On-line determination of molecular weight and branching by light scattering. (A) Molecular weight as a function of elution volume. (B) Dependence of branching index, gM, on molecular weight. (From Yu, L. P. and Rollings, J. E., /. Appl. Polym. Sci., 33, 1909, 1987. With permission.)...

Two analytical methods, size exclusion chromatography and rheology, provide a long chain branching index. The application of size exclusion chromatography to long chain branching analysis was described in Section 5.2.3.1. 

Branching index, 79 839 branch-poly-(n-b ityl acrylate), 7 610t Brass, 7 753... 

The concept of the molecular connectivity index (originally called branching index) was introduced by Randic [266]. The information used in the calculation of molecular connectivity indices is the number and type of atoms and bonds as well as the numbers of total and valence electrons [176,178,181,267,268]. These data are readily available for all compounds, synthetic or hypothetical, from their structural formulas. All molecular connectivity indices are calculated only for the non-hydrogen part of the molecule [269-271]. Each non-hydrogen atom is described by its atomic 6 value, which is equal to the number of adjacent nonhydrogen atoms. For example, the first-order Oy) molecular connectivity index is calculated from the atomic S values using Eq. (38) ... 

The elements of D represent the sum over all unit cells of the interaction between a pair of atoms. D has 3n x 3n elements for a specific q and j, though the numerical value of the elements will rapidly decrease as pairs of atoms at greater distances are considered. Its eigenvectors, labeled e ( fcq), where k is the branch index, represent the directions and relative size of the displacements of the atoms for each of the normal modes of the crystal. Eigenvector ejj Icq) is a column matrix with three rows for each of the n atoms in the unit cell. Because the dynamical matrix is Hermitian, the eigenvectors obey the orthonormality condition... 

Vibrations in a real crystal are described by the lattice dynamical theory, discussed in section 2.1, rather than by the atomic oscillator model. Each harmonic phonon mode with branch index k and wavevector q then has, analogous to Eq. 

The Mark-Houwlnk parameters K and a, where appropriate for the sample of interest, are calculated from a plot of log [h] vs. log M(ld) for linear polymers. For branched polymers, the branching index, g, is calculated from... 

The branching index, g, calculated from the Intrinsic viscosity measurement is related to the g value calculated from the light scattering experiment by the following equations... 

Figure 15. Plot of Branching Index as a Function of Molecular Weight for a Randomly Branched Polystyrene.

Berger and Shultz (26) have computed the weight-average of products of the type gbz° over the distribution of species produced by random trifunctional branching, where gz is the value of g0 corresponding to DP z, and a and b are constants, for various values of a branching index. 

X Randic branching index extension of the Randic approach by Kier and Hall [19]... 

Kinetic studies of primary and higher order star formation concluded that well-defined first order stars with narrow molecular weigth distribution could be prepared with [SiH]/[C=C] = 1.25 at room temperature whereas higher order stars were obtained with [SiH]/[C=C]=4.0 at 120 °C. While primary star formation was very slow and could require up to a week to complete at room temperature, higher order star formation was essentially complete in 24 h. Higher order stars with up to 28 arms have been prepared by this method. Intrinsic viscosities and branching index g were also studied. The intrinsic viscosities of stars were much lower than those of linear PIBs of the same MW. As expected, it was found that g values of stars depend on the number of arms and not on the MW of the arms. The stars were found to be resistant to acids and bases suggesting that the PIB corona protects the vulnerable core. 

Randic branching index, or molecular connectivity Kappa shape index Hammett constant Dipole moment... 

Milan Randic (1975) proposed a branching index ( ), commonly referred to now as the connectivity index ... 

The participation of individual bonds in the branching index is shown in Figure 5.10. 

Figure 5.10 Bond distribution of the branching index (1D) in 2,3-dimethylhexane.

The skeletal branching index, published by Randic, motivated significant research to enlarge its applicability to chemical systems other than alkanes and to predict properties other than boiling point. Kier and coworkers developed the molecular connectivity idea into a full paradigm for the representation of molecular structure (Hall and Kier, 2001). 

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