General expressions of average molecular weights in condensation polymerisation of polyfunctional monomers

Abstract

A new method, using the propagation expectation defined by Case, is presented to compute directly and simply the general expressions of average molecular weights for condensation polymerisation of polyfunctional monomers. In particular, the weight average molecular weight in systems composed of monomers endowed with both A and B coreactive functional groups is derived.

A Fast Method to Compute the MSD and MWD of Polymer Populations Formed by Step‐Growth Polymerization of Polyfunctional Monomers With Identical Reactive Groups

“…We have programmed both methods in the programming language Mathematica and applied them to the system “A1 + A2 + A3 + A4” with specification ( n 1 , n 2 , n 3 , n 4 ) = (11, 600, 3, 5) and p = 615/639 = 0.962441. For this system, the degree of conversion at which gelation occurs, p gel , is p gel = 620/639 = 0.970266 [Durand and Bruneau,10 expression (11), p gel = 1/( f f 0 –1)].…”

“…While simple expressions, i.e., without multiple summations or integrals, for the MSD and MWD are not available, such expressions for the first few moments of the distributions have been derived, notably by Macosko and Miller9 and by Durand and Bruneau 10. These authors published expressions for the average properties of the population of polymer molecules, like, for example, the number‐average molecular weight ($\overline {M} _{{\rm n}} $ ) or the weight‐average molecular weight ($\overline {M} _{{\rm w}} $ ).…”

Monte Carlo Simulation of Randomly Branched Step‐Growth Polymers: Generation and Analysis of Representative Molecular Ensembles

“…The analytical methods developed in the past, to calculate the first few moments of the distribution provide a validation test. Durand and Bruneau [8] give explicit formulas for the number-, weight-and z-average molecular weights of step-polymerization systems with reactive groups of type A and B. The comparison between analytical and numerical results for a typical, moderately high conversion is shown in Table 2.…”

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

“…Section ). Both systems lead to an equation Ψ ( x , Q ) = 0 for Q ( x ), whereby Ψ is of degree 1 in Q , d = 1, and neither can gelate for feasible values of p A …”

“…In some computer programming systems, this integral can be calculated by the “residue” method. The MSD curve thus calculated is shown in Figure .Once the sequence { S (1), S (2), S (3), …} has been obtained, molecular averages can be calculated: If wished for, these values can be compared to those of Macosko and Miller and/or of Durand and Bruneau, after applying their direct formulae for these averages. Also, the numerical values of the sums and can be found by differentiating the generating function Q ( x ), Equation (16), with respect to x and evaluating Q ′( x ) and Q ″( x ) in x...…”