A Fast Method to Compute the MSD and MWD of Polymer Populations Formed by Step‐Growth Polymerization of Polyfunctional Monomers Bearing A and B Coreactive Groups

Abstract

General step‐growth polymerization systems of order 2 are considered, i.e., systems of type “AfiBgi”, and a fast algorithmic method is presented to compute, at a given degree of conversion, the MSD and the MWD. The complete distribution is calculated; not just statistical averages of the polymer population such as $\overline {M} _{{\rm n}} $ or $\overline {M} _{{\rm w}} $. For the computation of the low‐ and intermediate size/weight parts of the distribution curves, a set of recurrence relations is used. The high‐molecular size/weight parts of the curves (right tails) are computed using an accurate approximation derived from generating functions. In a previous paper, we applied our method to general order‐1 systems, i.e., systems of type “Afi”. magnified image

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…”

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

“…In ref . it is shown that the three generating functions { Q A ( x ), Q B ( x ), Q ( x )} satisfy a set of simultaneous equations for all values of x .…”

“…As we are more interested in Q ( x ), we wish to eliminate Q A ( x ) and Q B ( x ) from Equation (9) to (11). This is done by calculating the resultant polynomial from the set of polynomials { Ψ A ( Q A , Q B ), Ψ B ( Q A , Q B ), Ψ S ( Q A , Q B , Q )} and involves the calculation of the Sylvester determinant . The programming language Mathematica —capable of performing symbolic algebra—has a separate command to compute a resultant polynomial.…”

“…To validate the calculated MSD, two checks are quite easily made. These two identities should hold: An approximation to the high‐molecular tail part of the MSD curve offers a third check: an explicit formula for this part of the curve can be derived . It is given by: with, for system “A1B1 + B1 + A2 + B4”, p A = 94/100, …”