Chemical combinatorics. Part I. Chemical kinetics, graph theory, and combinatorial entropy

Gordon, M.; Temple, William B.

doi:10.1039/j19700000729

Cited by 19 publications

(9 citation statements)

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“…The partition functions may be decomposed into a product of contributions from translational, rotational, vibrational, electronic and nuclear modes . The contribution to the free energy of formation from the combinatorial entropy change, Δ S comb, n A , n B , k , contains the contribution from the inverse dependence of the rotational partition functions on their symmetry numbers, while the remaining energetic and entropic contributions are contained in the term Δ G AA 0 l AA + Δ G BB 0 l BB + Δ G AB 0 l AB . − , The combinatorial entropy of formation is defined by the expression − normalΔ S comb , n normalA , n normalB , k = k normalB .25em ln ( | G A | n A | G B | n B | G n A , n B , k | ) where | G A | = r A ! and | G B | = r B !…”

Section: Theoretical Methodsmentioning

confidence: 99%

“…are the symmetry numbers of single A and B monomeric units, respectively, assumed to be symmetric to all permutations of the arms. The symmetry number of the cluster, denoted by | G n A , n B , k |, is related to the number of distinct rooted, ordered trees , obtained from the graph of the polymer cluster. − This relationship enables the simplification of eq and, consequently, eq . We present the final results below, while the details are relegated to Appendix A.…”

Section: Theoretical Methodsmentioning

confidence: 99%

“…Gordon and co-workers − provided a sound statistical mechanical basis for Flory–Stockmayer theory by deriving the condition for gelation via statistical mechanics and graph theory. This work relies on the relation of the reaction equilibria to the symmetries of the reactants and products, where the latter are quantified via the enumeration of trees from a graphical representation of the polymers.…”

Section: Introductionmentioning

confidence: 99%

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Macro- and Microphase Separation in Multifunctional Supramolecular Polymer Networks

2011

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We present a field-based model for the phase separation and gelation of a binary melt of multifunctional monomeric units that can reversibly bond to form copolymer networks. The mean-field phase separation behavior of several model networks with heterogeneous bonding is calculated via the random phase approximation (RPA). By this technique, the spinodal phase boundary (stability limit of the homogeneous disordered phase) is obtained by a combination of analytical and numerical methods. It is demonstrated that higher functionality and more favorable bonding energy suppresses macroscopic phase separation due to greater connectivity between unlike species. Gelation occurs with sufficiently high connectivity of trior higher functional monomeric units and microphase separation of the copolymer network can occur either preceding or after the gel point. Eutectic-like behavior of the spinodal is seen in highly connected networks due to excess loosely connected material in systems having nonstoichiometric ratios of the two components.

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Section: Theoretical Methodsmentioning

confidence: 99%

Section: Theoretical Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Macro- and Microphase Separation in Multifunctional Supramolecular Polymer Networks

2011

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“…If these / orientations represent all The factor l is defined here to be identical with the factor / described by Bishop and Laidler8 as being the number of differently labeled sets of given products that can be formed if all identical atoms in reactants are labeled. Factoring of both the partition functions and the reaction cross section in eq 15 leads to an overall statistical factor, Sc, given by (17) For the usual case of achiral reactants, this expression reduces to a simple product of l and the corresponding symmetry numbers.…”

Section: Effects Of Chirality On Statistical Factorsmentioning

confidence: 99%

Statistical factors in reaction rate theories

Coulson¹

1978

J. Am. Chem. Soc.

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The use of various statistical factors in absolute rate theory (ART) is examined and criticized. It is concluded that of the various procedures proposed for incorporating statistical factors into ART rate constant expressions, only the conventional one employing a symmetry number ratio is consistent with the classical formulation of ART. A generalization of this latter factor to include the effects of chirality is also given. Finally, the use of statistical factors in other statistical rate theories is examined.Among reaction rate theories those based on a statistical approach have enjoyed widespread use over the years, in large part because of the enormous popularity of absolute rate theory' (ART). Other important statistical theories contributing to this use include RRKM,* phase space,3 and collision4 theories. I n the descriptions of rate constants given by these theories it has been found necessary to use statistical factors to correct for inadequacies in the basic theory. In their original formulation,' as part of ART, such factors took the form of a ratio of the product of symmetry numbers of the reactants to that of the activated complex. These symmetry numbers arose naturally from the use of classical rotational partition functions in the derivations and were viewed as purely quantum mechanical correction factors. However, the contemporary view of statistical factors in A R T and related theories includes a correction that accounts for the presence of multiple reaction pathways5 Schlag6 for example, has advocated replacing the conventional symmetry number ratio by a "reaction path degeneracy factor", n , derived from an elegant group-theoretical treatment of reaction pathways. To simplify the computation Schlag and Hailer' have described a modified direct-count method, not involving group theory, for determining the value of n . A variant of this method, described by Bishop and Laidler, uses the notation I* for n and describes it simply as a "statistical factor". In addition, these workers defined a "statistical factor", r*, for the reverse process of an activated complex returning to reactants. To further complicate matters, other statistical factors designated L* and U,-h have been proposed by Elliott and Frey9 and Johnston,'O respectively. These latter factors appear to be identical with both l* and n .In a study predating the development of this contemporary view of statistical factors, Rapp and Weston' concluded, from an application of A R T to some elementary exchange reactions, that the usual symmetry number ratio is an inadequate statistical factor in certain cases. They went on to describe a basis for modifying the symmetry number ratio which yielded statistical factors comparable to those obtained using the direct-count methods.

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“…Recently, click chemistry has emerged as a powerful technique to design HPs via step polymerization . On the topological front, use of combinatorics and graph theory to determine structural characteristics of synthesized polymer molecules has been ongoing; − it is primarily used in conjunction with the polymerization kinetics to obtain the branching information. , Moreover, the identification of linear/branched structures and their isomers has been largely missing from these studies. Our approach, on the other hand, leverages the correspondence between polymer molecules to rooted full m -ary trees (FMTs) to obtain the topological information of polymer chains, which are consistent with the statistics of step-growth polymerization.…”

Section: Introductionmentioning

confidence: 99%

On the Role of Half-Catalan Numbers and Pathwidth in Hyperbranched Polymers Synthesized by AB_m Step Polymerization

et al. 2021

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Topological features of polymer chains have been used as the key controlling mechanism for the physicochemical properties of hyperbranched polymers (HPs) and, therefore, provide a significant impetus to determine their branching characteristics. Single monomer methodology (SMM) involving AB m step polymerization has been one of the routes to synthesize both compact and segmented HPs. Here, we explore Catalan and half-Catalan numbers in the context of AB m step polymerization to deduce the structural information of HPs. Our approach harnesses the concepts of combinatorics and graph theory to calculate the exact numbers of isomeric, branched and linear, structures of polymer chains. We also demonstrate that the extent of branching of a polymer chain can be measured via pathwidth and establish its bounds as a function of its length. We believe that our findings can be leveraged to design and control the architecture of HPs synthesized via SMM or any other chemistries in a straightforward manner.

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Chemical combinatorics. Part I. Chemical kinetics, graph theory, and combinatorial entropy

Cited by 19 publications

References 0 publications

Macro- and Microphase Separation in Multifunctional Supramolecular Polymer Networks

Macro- and Microphase Separation in Multifunctional Supramolecular Polymer Networks

Statistical factors in reaction rate theories

On the Role of Half-Catalan Numbers and Pathwidth in Hyperbranched Polymers Synthesized by AB_m Step Polymerization

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Chemical combinatorics. Part I. Chemical kinetics, graph theory, and combinatorial entropy

Cited by 19 publications

References 0 publications

Macro- and Microphase Separation in Multifunctional Supramolecular Polymer Networks

Macro- and Microphase Separation in Multifunctional Supramolecular Polymer Networks

Statistical factors in reaction rate theories

On the Role of Half-Catalan Numbers and Pathwidth in Hyperbranched Polymers Synthesized by ABm Step Polymerization

Contact Info

Product

Resources

About

On the Role of Half-Catalan Numbers and Pathwidth in Hyperbranched Polymers Synthesized by AB_m Step Polymerization