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

Hillegers, Ltme Tom; Slot, Jjm Han

doi:10.1002/mats.201600093

Cited by 4 publications

(14 citation statements)

References 29 publications

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“…By means of the ACA algorithm, we find [6,16] CON [ ] ( *) 1 2 The same expression (3.8) is found by some other authors. [4,12,17] Figure 6 shows the graph of ATPL[s], black curve, Equation (3.4), and its asymptotic approximation, straight line, opaque blue, Equation (3.8).…”

Section: The Probabilities P [S]supporting

confidence: 84%

“…These steps: (polynomial equation) → (ordinary differential equation) → (recurrence equation) are well described in Hillegers and Slot. [14] Asymptotic values for P[s], i.e., for s → ∞, are obtained by means of the ACA algorithm (algebraic coefficient asymptotics) [6,15] [ ] 1 2 * ( *) The average total path length, ATPL[s], and some other univariate molecular distributions, can be calculated with these two convolutions of generating function…”

Section: The Probabilities P [S]mentioning

confidence: 99%

“…We have omitted this factor, equivalent to setting b = 1. [6] For a polymeric molecule, the shrink factor is defined as the ratio between its square radius of gyration and that same quantity for a linear polymeric molecule of the same size, which is 1 6 2 s s − .…”

Section: Average Square Radius Of Gyration Asrg[s] and The Average Shrink Factor Asf[s]mentioning

confidence: 99%

“…It is interesting to note that the shrink factor defined by means of the squared radius of gyration, R 2 [s], yields the same value, as we will show in Section 3. [5,6] Figure 2 shows the weighted path length distribution, PLD [s,d], pertaining to the molecule of Figure 1. Each path is weighted by its length d. TPL[s] serves as normalization factor PLD ,…”

Section: Introductionmentioning

confidence: 99%

“…It is interesting to note that the shrink factor defined by means of the squared radius of gyration, R 2 [ s ], yields the same value, as we will show in Section 3. [ 5,6 ]…”

Section: Introductionmentioning

confidence: 99%

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Step‐Growth Polymerized Systems of Type “A1+A2+A3”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

Hillegers¹,

Slot

2021

Macro Theory & Simulations

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might break up later on in the reaction process, and the two now disconnected members of the pair might find other partners. The initial number (molar amounts) of monomers of type A1, A2, and A3 are denoted by n [1], n[2], and n[3], respectively. Then, n tot = n[1] + n[2] + n [3] is the total number of monomeric units, andis the total number (molar amounts) of A groups available to form chemical bonds. The fraction of f tot is denoted by p (named degree of conversion, or extent of reaction) that has formed a bond. It is assumed that after a while, at each value of p, 0 ≤ p ≤ 1, the reaction process has reached a dynamic equilibrium in which the number of bonds does not change over time, and the total number of polymeric molecules remains at the stable value of n tot -1 2 p f tot . When a satisfactory level is reached, the reaction process is frozen by lowering the temperature and/or by removing the condensation by-product. In modeling the reaction process and the formation of polymeric molecules, we make the usual assumptions: i) all reactions occur with equal probability, independent of the size of the molecules involved and independent of the status of other A groups on the same monomeric unit; ii) cycles do not form, i.e., no reaction happens between A groups on the same molecule; and iii) there are no other reactions than between A groups. [1,2] The system "A1+A2+A3," that we study in this paper, is the most simple representative of the more general step-growth polymerizing systems of type "Afi," whereby the functionality (number of A groups) on each monomer species is not limited to three. It is our intention, in future studies, to extend the calculation method developed here for "A1+A2+A3" to step-growth polymerizing systems of type "AfiBgi." Those systems are characterized by monomers bearing A groups and/ or B groups; an A group reacts only with a B group, and vice versa.These step-growth polymerizing systems, "Afi" and "AfiBgi," and the many variations thereof are widely studied in the chemical literature and are of great interest to the polymer industry. Examples of commercially relevant systems "Afi" with AA condensation and water as condensation by-product, are: linear polymers, "A2": POM (polyoxymethylene of paraformaldehyde)Step-growth polymerizing systems of type "A1+A2+A3" are considered. The monomers bear one (A1), two (A2), or three (A3) identical reactive sites. In the reactor vessel, 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 (i.e., are built up by the same number of monomeric units), but differ in number and position of branching points (A3's). 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. The shape and moments of this di...

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Section: The Probabilities P [S]supporting

confidence: 84%

Section: The Probabilities P [S]mentioning

confidence: 99%

Section: Average Square Radius Of Gyration Asrg[s] and The Average Shrink Factor Asf[s]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It is interesting to note that the shrink factor defined by means of the squared radius of gyration, R 2 [ s ], yields the same value, as we will show in Section 3. [ 5,6 ]…”

Section: Introductionmentioning

confidence: 99%

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Step‐Growth Polymerized Systems of Type “A1+A2+A3”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

Hillegers¹,

Slot

2021

Macro Theory & Simulations

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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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Dynamic Networks that Drive the Process of Irreversible Step-Growth Polymerization

Schamboeck

Iedema

Kryven

2019

Sci Rep

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Many research fields, reaching from social networks and epidemiology to biology and physics, have experienced great advance from recent developments in random graphs and network theory. In this paper we propose a generic model of step-growth polymerisation as a promising application of the percolation on a directed random graph. This polymerisation process is used to manufacture a broad range of polymeric materials, including: polyesters, polyurethanes, polyamides, and many others. We link features of step-growth polymerisation to the properties of the directed configuration model. In this way, we obtain new analytical expressions describing the polymeric microstructure and compare them to data from experiments and computer simulations. The molecular weight distribution is related to the sizes of connected components, gelation to the emergence of the giant component, and the molecular gyration radii to the Wiener index of these components. A model on this level of generality is instrumental in accelerating the design of new materials and optimizing their properties, as well as it provides a vital link between network science and experimentally observable physics of polymers.

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

Cited by 4 publications

References 29 publications

Step‐Growth Polymerized Systems of Type “A1+A2+A3”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

Step‐Growth Polymerized Systems of Type “A1+A2+A3”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

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

Dynamic Networks that Drive the Process of Irreversible Step-Growth Polymerization

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