Structural interpretation of the topological index. 2. The molecular connectivity index, the Kappa index, and the atom-type E-State index

The structural interpretation is extended to the topological indices describing cyclic structures. Three representatives of the topological index, such as the molecular connectivity index, the Kappa index, and the atom-type E-State index, are interpreted by mining out, through projection pursuit combining with a number theory method generating uniformly distributed directions on unit sphere, the structural features hidden in the spaces spanned by the three series of indices individually. Some interesting results, which can hardly be found by individual index, are obtained from the multidimensional spaces by several topological indices. The results support quantitatively the former studies on the topological indices, and some new insights are obtained during the analysis. The combinations of several molecular connectivity indices describe mainly three general categories of molecular structure information, which include degree of branching, size, and degree of cyclicity. The cyclicity can also be coded by the combination of chi cluster and path/cluster indices. The Kappa shape indices encode, in combination, significant information on size, the degree of cyclicity, and the degree of centralization/separation in branching. The size, branch number, and cyclicity information has also been mined out to interpret atom-type E-State indices. The structural feature such as the number of quaternary atoms is searched out to be an important factor. The results indicate that the collinearity might be a serious problem in the applications of the topological indices.     

Effect of fullerene C60 as a branching center on molecular and polarization properties of star-shaped polystyrenes

The behavior of molecules of a star-shaped six-arm polystyrene with a covalently bound fullerene C₆₀ as a branching center was studied by viscometry and by measuring the electrooptical Kerr effect and the dielectric polarization in solutions. It was shown that polarization and electrooptical characteristics of a fullerene-containing polystyrene (C₆₀ ～ 3 wt %) differ by an order of magnitude or even greater from the corresponding characteristics of the parent polymer. A comparison of the above properties with the analogous characteristics of the model hexaadduct (the products of reaction between octyllithium and fullerene C₆₀) demonstrated that a difference in the behavior of the star-shaped polystyrene and its parent analog is associated with the structural features of the branching center, among which is the occurrence of six proton addends that are bonded rather weakly to the fullerene cage in the hexaadducts under study.     

Diffusion-limited hyperbranched polymers with substitution effect

Highly branched structure has the essential influence on macromolecular property and functionality in physics and chemistry. In this work, we proposed a diffusion-limited reaction model with the consideration of macromolecular unit relaxations and substitution effect of monomers to study the structure of hyperbranched polymers prepared by slow monomer addition to a core molecule. The exponential relationship (R(g) ～ N(λ)) between the radius of gyration R(g) and the degree of polymerization N, was systematically analyzed at various branching degrees. It is shown that the effective exponent λ(eff) decreases at lower N and but increases toward that of diffusion-limited aggregation (DLA) clusters (λ(DLA) = 0.4) with the degree of polymerization increasing. The substitution effect of monomers in reaction strongly influences the evolution pathway of λ(eff). With the static light scattering technique, the fractal property of internal chains was further calculated. A general law about the radial distribution of the units of diffusion-limited hyperbranched polymers was found that, at smaller reactivity ratio k(12), the radial density of all monomer units D(A) declines from the center region to the peripheral layer revealing the dense core structure; however, at larger k(12), the density distribution shows a loose-dense-loose structure. These structural characteristics are helpful to deeply understand the property of hyperbranched polymers.     

橡胶相具有交联结构的新型抗冲聚丙烯合金

利用具有"颗粒反应器技术(RGT)"特征的Ziegler-Natta催化剂进行丙烯多相共聚(丙烯均聚+乙烯/丙烯无规共聚),通过在乙丙共聚阶段引入双烯烃单体1,9-癸二烯,使乙丙共聚物在聚合的同时实现交联,制备了新型抗冲聚丙烯合金.聚合反应结果表明,1,9-癸二烯可参与乙丙共聚,同时对聚合反应速率和共聚物组成影响较小;1,9-癸二烯使乙丙共聚物发生支化/部分交联,合金聚合物的熔体流动速率在引入1,9-癸二烯后显著降低,且凝胶含量随1,9-癸二烯用量的增加而增大.形态研究结果表明,乙丙共聚物的交联显著降低了其在聚丙烯基体中的分散尺度,提高了分散均匀性,分散相粒径随支化/交联程度提高而减小.力学性能测试结果表明,乙丙共聚物的交联使合金聚合物在保持较高韧性的同时显著提升了刚性,有利于实现抗冲聚丙烯合金的刚韧平衡.     

Exhaustive enumeration of molecular substructures

This article addresses the systematic and complete enumeration of all the substructures of any size present in a given molecule. The study is not restricted to features which could be defined a priori such as rings or chains. Contrary to prior expectation the exhaustive enumeration is tractable with current computational tools. Results are presented for several families of skeletons which are widespread in chemistry. It is shown that the numbers of constituent substructures of each size are related to the molecular topology, in particular the degree of branching. The number substructures which are distinct depends additionally on the number of different atom and bond types present. The overall shapes of the distribution of substructure counts as a function of substructure size are found to be similar within particular classes of molecules. These distributions are compared and found to be characteristic of certain topologies. For several simple classes of molecule, analytic expressions are provided for the numbers of substructures as a function of fragment and molecule size. These results hold promise for identifying potentially useful scaffolds for use in combinatorial chemistry. © 1997 by John Wiley & Sons, Inc.     

Relationship between degree-rank function and degree distribution of protein-protein interaction networks

It is argued that both the degree-rank function r=f(d), which describes the relationship between the degree d and the rank r of a degree sequence, and the degree distribution P(k), which describes the probability that a randomly chosen vertex has degree k, are important statistical properties to characterize protein-protein interaction (PPI) networks, both rank-degree plot and frequency-degree plot are reliable tools to analyze PPI networks. An exact mathematical relationship between degree-rank functions and degree distributions of PPI networks is derived. It is demonstrated that a power law degree distribution is equivalent to a power law degree-rank function only if scaling exponent is greater than 2. The puzzle that the degree distributions of some PPI networks follow a power law using frequency-degree plots, whereas the degree sequences do not follow a power law using rank-degree plots is explained using the mathematical relationship.     

Structural features of star-shaped fullerene (C60)-containing polystyrenes: Neutron scattering experiments

Structural features of star-shaped polyprotostyrene and polydeuterostyrene containing fullerene C₆₀ as a branching center have been studied by small-angle neutron scattering in benzene solutions. The results are compared with the corresponding characteristics of linear PSs, the molecular mass of which is equal to the molecular mass of one star arm in star-shaped macromolecules. The molecular masses of star-shaped polymers are estimated, and their branching center is shown to be hexafunctional. At relatively low concentrations of starshaped polymers in solutions, one can observe excluded volume effects, which are related by the presence of regions with higher densities at the center of a macromolecule. Using the Fourier transform of the scattering cross section, three-dimensional correlation functions are obtained, and the regular structure of stars is proved. Conclusions about the local correlations of units within one star arm and averaged correlations between units of neighboring arms within a given star are derived. An analysis of three-dimensional correlations shows that the centers of mass of all star arms are directed along orthogonal axes passing through the C₆₀ branching center of a star-shaped macromolecule.     

A quasichemical model for comb-like aggregation of monohydric alcohols: supramolecular structure and macroscopic properties

A new quasichemical model of the supramolecular structure of the liquid consisting of chain-like and comb-like hydrogen-bonded aggregates with branches of unit length has been developed. Analytical expressions for structural characteristics (size and structure distributions of aggregates), dielectric (permittivity, dipole correlation factor), optic (mean molecular anisotropy in liquid, optic correlation factor, anisotropic Rayleigh light scattering ratio) and thermodynamic (energy of intermolecular interactions in liquid, vaporisation enthalpy) properties of the liquid as functions of structural and thermodynamic aggregation parameters have been derived. The analytical model developed creates the foundation for studying the role of branched aggregates in the supramolecular structure of liquids and various macroscopic properties determined by different molecular parameters. The model is applied to pure methanol at ambient conditions. The dependence of structural characteristics of liquid, dipole correlation factor, permittivity, mean molecular anisotropy in liquid, vaporisation enthalpy on the equilibrium constants of chain-like and branched aggregation is studied. The results are compared with experimental data and computer simulation. The influence of branching degree of aggregates on different physicochemical properties of liquid is revealed and discussed.     

A Conditional Monte Carlo Method to Determine the Architectures of Metallocene Catalyzed Polyethylene

Two polymer molecules of the same length (n) and the same number of branch points (N) can have different properties, since they may possess distinct architectures. In this paper we present a conditional Monte Carlo algorithm for the virtual synthesis of metallocene‐catalyzed polyethylene (PE) in a continuous stirred tank reactor (CSTR). The condition for the Monte Carlo method consists of a fixed chain length distribution (CLD) and a degree of branching distribution (DBD). These distributions are calculated with a Galerkin finite element method. The synthesis method is a recursive algorithm that subsequently creates insertions of sub‐structures containing numbers of branch points according to a certain probability density function. This provides an adjacency matrix describing the connectivity between the branch points, while separately a vector containing the length of segments between branch points and terminal segments is generated. Characterization of the architectures proceeds by rheological features, seniorities and priorities, and molecular properties like the radius of gyration. Comparing the radii of gyration of metallocene polyethylene and low density PE (ldPE) shows the former to possess a more comb‐like structure on average. This is confirmed by the rheological characterization. The found bivariate seniority/priority distribution agrees well to the results of an analytical study of the same chemical system.

Constituting elements of the algorithm.     

A structural characteristic of hyperbranched polymers

The structural characterization of hyperbranched polymers is discussed. Current approaches describing the structure of these polymers in terms of such parameters as the degree of branching and the average number of branches are shown to be inadequate. We propose characterizing the structure of polymers using the distribution over the degree of branching and the $ \bar D_w /\bar D_n $ ratio as the quantitative measure of distribution width. Using the example of AB₂ type monomer polycondensation, it is shown how this value varies in the course of the process along with other structural parameters.     

The photodissociation dynamics of tetrachloroethylene

We present a direct current slice imaging study of tetrachloroethylene (C(2)Cl(4)) photodissociation, probing the resulting ground state Cl ((2)P(3/2)) and spin-orbit excited state Cl* ((2)P(1/2)) products. We report photofragment images, total translational energy distributions and the product branching ratio of Cl*/Cl following dissociation at 235 and 202 nm, obtained using a two-color reduced-Doppler dissociation/probe. Near 235 nm, the Cl translational energy distribution shows a peak at the limit of the available energy, indicating a direct dissociation through a σ*(C-Cl) ← π (C=C) transition, which is superimposed on a broader underlying distribution. The ground state Cl image and associated translational energy distribution at 202 nm is broad and peaked at lower energy, suggesting either internal conversion to the ground state or a lower excited state prior to dissociation. The Cl* images are similarly broad at both wavelengths. The branching ratio is presented as a function of recoil energy, but after integration shows a near-statistical average of Cl:Cl* as 70:30 at both wavelengths. All the images are largely isotropic, with anisotropy parameters (β) of 0.05 ± 0.03.     

Topological characterization of cyclic structures

The sum of the topological distances in the molecular graph (the Wiener number) is used for a topological characterization of the condensed polycyclic molecular systems. This topological index discriminates well the isomeric cyclic molecules. In addition, it also properly reflects their structural features. On this basis the principal points of molecular cyclicity are formulated in 15 rules.     

Kinetic Evaluation of Hyperbranched A2 + B3 Polycondensation Reactions

The kinetics of hyperbranched A₂ + B₃ systems is discussed theoretically with respect to the development of the 7 different structural units, the degree of branching, DB, and the monomer sequences considering the adjacent groups of a structural unit. For A₂ + B₃ systems, the comonomer ratio, the relative rate constants and the process conditions have an influence on the resulting structure as shown by numerical simulations. With increasing A:B ratios f_A/B, the degree of branching will be increased. Also the relative reaction rate constants have a strong impact on the distribution of structural units, especially when the reaction rate constants for the pathway of the B₃ monomer are changed. On the other hand, differences in the reaction rate constants for the pathway of the A₂ monomer do not have any influence on the degree of branching. The simulation indicates that slow addition of either both monomers or just the B₃ monomer has the strongest effect on the resulting DB. In all cases, the conversion is a critical issue to obtain high molecular weight products.

Degree of branching (DB) versus conversion of A‐functionalities (p_A) for various monomer compositions.     

Analyse topologique du comportement d'esters aliphatiques saturés et insaturés en chromatographie gaz-liquide

(Topological analysis of the gas-liquid chromatographic behaviour of saturated and unsaturated aliphatic esters.) Topological analysis is applied to tje Kovats retention indices for a collection of aliphatic esters on six stationary phases. Three subclasses are treated separately: (1) 100 saturated esters; (2) 45 esters unsaturated in the acidic part and 53 saturated but topologically similar esters; (3) 24 esters unsaturated in the alcohol moiety and 6 saturated but topologically similar esters. A detailed study is reported for the correlation of structural effects with numerical data for the specific solute/stationary phase interactions at the level of carbon atoms and ethylenic double bonds. The roles of the carboxylic ester nucleus, the branching and lengthening of the chain and the presence of ethylenic double bonds is elucidated. The selectivity of trifluoropropyl- and phenyl-bonded stationary phases is clarified.     

Modeling molecular weight distribution of comb-branched graft copolymers

Grafting one type polymer onto a different polymer type may yield a comb-branched copolymer. The branching density has a significant effect on its overall molecular weight distribution. A general model is derived to describe the bivariate distribution of molecular weight and branching density for such comb copolymers. The model is applicable for various grafting mechanisms provided the side chains are randomly grafted onto the backbone. The determining parameters are the molecular weight distributions of backbone and side chains, and the branching density. Analytical expressions are obtained for the cases of the side chains having uniform and Schulz–Zimm distributions. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 705–714, 1998     

Information theory for experiments with known conditional probabilities: Application to the prediction of branching ratios

Information theory is used, under conditions of incomplete data, to provide a maximally conservative prior expectation of a population distribution. This is done by systematically refining the estimates of the full distribution via the incorporation of fragmentary data in the form of conditional probabilities. The net effect is to reduce the available phase space volume of the distribution as more data are incorporated. The above refinements are then applied to calculate estimates of branching ratios for chemical reactions with more than one possible product. Two new branching ratio expressions are derived: one for the case where each product's internal state distribution is known for reactions originating from various specific reactant state?, and when the joint state-to-state reactive probability is known.