Step-Growth Polymerized Systems of General Type “AfiBgi”: Generating Functions and Recurrences to Compute the MSD

Abstract

General step‐growth polymerization systems of order 2 are considered, i.e. systems of type “AfiBgi”. We describe an algorithmic method to calculate the molecular size distribution (MSD). Input to the algorithm is the “recipe”: a list of the monomers involved stating their A and B functionalities and their molar amounts, and the degree of conversion. Output is the MSD and its moments. Three main steps lead from input to output: (i) setting up a polynomial equation for the generating function that generates the MSD, (ii) transforming this polynomial equation to a differential equation, and (iii) transforming the latter one further to a recurrence equation. The recurrence yields the MSD and is of constant order.

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

“…This restriction is implemented as a single upper bound on the * i.kryven@uva.nl degrees. At the same time, the algorithmic study on directed graphs (inspired by the polymerization structures) [16] does allow one to impose the degree bounds as a distribution, yet the algorithm is applicable only prior to the phase transition.…”

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

“…[ 17 ] and [ 18 ] , we have derived that the three generating functions have to satisfy the following set of equations, for all values of x and y . …”

“…[ 15 ] In the recent past, Hillegers et al have successfully applied univariate generating functions to calculate molecular size and molecular weight distributions of step-growth systems of general type. [16][17][18] We now use bivariate generating functions to tackle the problem of computing the statistical distribution of weight over the two molecular characteristics size and number of branch points . To that end, defi ne three bivariate generating functions, namely…”

“…The transformation leads to a linear recurrence of the form. [ 15,18 ] For the system specifi ed by the recipe of Table 1 …”

“…The computational procedure to transform the polynomial equation to a differential equation is described elsewhere. [ 15,18,20 ] The next step is to transform the differential Equation ( 6.1) for the bivariate generating function Q ( x , y ) to a recurrence relation for the coeffi cients Q HOR [ v ]( x ) of Q ( x , y ). These coeffi cients are themselves univariate generating functions-the horizontal ones, cf.…”