Step‐Growth Polymerizing Systems of General Type “AfiBgi”: Calculating the Bivariate (Molecular Size) × (Number of Branch Points) Weight Distribution Using Generating Functions and Recurrences

Hillegers, Ltme Tom; Slot, J.J.M.

doi:10.1002/mats.201500093

Cited by 4 publications

(12 citation statements)

References 22 publications

(77 reference statements)

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“…Furthermore, the criterion for the existence of the giant out-component is identical to (15). For a given directed degree distribution u(n,k), we may associate a one-dimensional degree distribution by disregarding the direction of the edges, d(l) = n+k=l u(n,k).…”

Section: Criterion Of the Phase Transition For An Arbitrary Degrementioning

confidence: 99%

Emergence of the giant weak component in directed random graphs with arbitrary degree distributions

Kryven

2016

Phys. Rev. E

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Emergence of the giant weak component in directed random graphs with arbitrary degree distributions Kryven, I. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for the existence of the giant weak component is derived for directed graphs with arbitrary bivariate degree distributions. In addition, we consider a random process for evolving directed graphs with bounded degrees. The bounds are not the same for different vertices but satisfy a predefined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weak-component criterion to obtain the exact time of the phase transition. The phase-transition time is obtained as a function of the distribution that bounds the degrees. Remarkably, when viewed from the step-polymerization formalism, the new results yield Flory-Stockmayer gelation theory and generalize it to a broader scope.

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Section: Criterion Of the Phase Transition For An Arbitrary Degrementioning

confidence: 99%

Emergence of the giant weak component in directed random graphs with arbitrary degree distributions

Kryven

2016

Phys. Rev. E

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“…This table–the recipe–specifies the “ingredients”, i.e., the monomer species involved, their A functionality ( f ), their B functionality ( g ), the relative amount in moles ( n ), and the extent of reaction ( p A ). System “B1 + B3 + A2 + A1B1” resembles a randomly branched polyamide and will be used here as an illustrative example . Four ( K = 4) monomer species take part in this reaction.…”

Section: The Recipementioning

confidence: 99%

“…The solutions are rational functions, i.e., they can be written as Q ( x ) = F ( x )/ G ( x ), with F ( x ) and G ( x ) polynomials in x with integer coefficients. In this case, explicit expressions exist for q NUM [ s ] and q DEN [ s ], and no further steps are needed to calculate the value of these quantities for large values of s …”

Section: A Recurrence Relation Of Fixed Order For Qnum[s] and Qden[s]mentioning

confidence: 99%

“…System "B1 + B3 + A2 + A1B1" resembles a randomly branched polyamide and will be used here as an illustrative example. [16,17] Four (K = 4) monomer species take part in this reaction.…”

Section: The Recipementioning

confidence: 99%

“…[18] and in the Supporting Information. Note that Equations (14) To proceed with the development of our algorithmic method, we need to eliminate Q A (x) and Q B (x) from Equations (16) to (17) and obtain equations in Q NUM (x) and Q DEN (x) alone. This is achieved by constructing a Gröbner basis for these functions.…”

Section: The Sequences P a [S] P B [S] And R 2 [S] And Associated mentioning

confidence: 99%

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Step‐Growth Polymerizing Systems of General Type “AfiBgi”: Calculating the Radius of Gyration and the g‐Curve Using Generating Functions and Recurrences

Hillegers¹,

Slot

2017

Macro Theory & Simulations

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Step‐growth polymerized systems of general type “AfiBgi” are considered. One or more of the monomer species carries at least three reactive groups and thus can act as a branching point in a polymeric molecule. An algorithmic method is presented to calculate the topology‐averaged square radius of gyration, R 2[s], of the molecules in the class of s‐mers. The degree of polymerization, s, may run through its full range. In addition to R 2[s], the shrinking factor, g[s], is calculated. The method uses integer arithmetic, generating functions, and computer algebra.

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Step‐Growth Polymerized Systems of General Type “AfiBgi”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

Hillegers¹,

Slot

2022

Macro Theory & Simulations

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Step-growth polymerized systems of general type "AfiBgi" are considered. The monomers bear one or more reactive groups of type A and/or B. An A group on one monomeric unit might react with a B group on another unit, thus coupling the two units. This process repeats itself randomly, and at a given degree of conversion, a wide range of polymeric molecules has formed, differing in both molecular size and in branching structure, determined by the laws of probability. In a slice of the molecular size distribution, all polymeric molecules have the same size, but differ in number and position of branching points. A method is presented to calculate the path length distribution for each such slice. Here, path length is the number of chemical bonds in the path connecting two monomeric units in the molecule. From the path length distribution, the light scattering function is calculated.

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Step‐Growth Polymerizing Systems of General Type “AfiBgi”: Calculating the Bivariate (Molecular Size) × (Number of Branch Points) Weight Distribution Using Generating Functions and Recurrences

Cited by 4 publications

References 22 publications

Emergence of the giant weak component in directed random graphs with arbitrary degree distributions

Emergence of the giant weak component in directed random graphs with arbitrary degree distributions

Step‐Growth Polymerizing Systems of General Type “AfiBgi”: Calculating the Radius of Gyration and the g‐Curve Using Generating Functions and Recurrences

Step‐Growth Polymerized Systems of General Type “AfiBgi”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

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