Simultaneous long-chain branching and random scission. II. Analytic expression for the weight-average molecular weights

Abstract

Hetrochain branching theory is applied to develop a matrix formula describing the weight‐average molecular weights for the free‐radical polymerization that involves simultaneous long‐chain branching and random scission. The gel point is simply described as a point at which the largest eigenvalue of the matrix, M, which defines the chain connection statistics, reaches unity. The illustrative calculations show that the calculated results agree satisfactorily with those obtained from the Monte Carlo simulation method developed in the first part of this series. The present matrix formula is superior in the precision of the weight‐average molecular weights and the gel point, while the Monte Carlo simulation technique can provide much more detailed information, and therefore, these two methods are nicely complementary. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 404–414, 2001

Molecular Weight Development in Free-Radical Polymerization with Polyfunctional Chain-Transfer Agents, 1. Equal Reactivity Model

“…A numerical technique to find the gel point can be found elsewhere. [14,16] In ref. [1] , the MC simulation method for the same reaction system was proposed.…”

Molecular Weight Development in Free-Radical Polymerization with Polyfunctional Chain-Transfer Agents, 1. Equal Reactivity Model

“…A numerical technique to find the gel point can be found elsewhere. [14,16] In ref. [1] , the MC simulation method for the same reaction system was proposed.…”

Simultaneous long-chain branching and random scission: I. Monte Carlo simulation

“…Figure 13 shows the obtained weight fractions of gel and their 95% confidence intervals. From the figure, the gel point is determined to be about x = 0.47, which agrees well with that obtained from the analytic solution derived in the second part of this series,17 x = 0.468.…”

Free‐Radical Polymerization with Long‐Chain Branching and Scission: Markovian Solution of the Weight‐Average Molecular Weight

“…In the context of the matrix approach, given by Equation , gelation occurs when the largest eigenvalue of M reaches unity. In the earlier matrix approach, the condition, det( I ‐ M ) = 0 was used . However, it was found that the value of det( I ‐ M ) is too small in the present case, and it was not suitable to determined the point that satisfy det( I ‐ M ) = 0 accurately.…”

Molecular Weight Development in Free-Radical Polymerization with Polyfunctional Chain-Transfer Agents, 1. Equal Reactivity Model

“…A numerical technique to find the gel point can be found elsewhere. [14,16] In ref. [1] , the MC simulation method for the same reaction system was proposed.…”

Simultaneous long-chain branching and random scission: I. Monte Carlo simulation

“…Figure 13 shows the obtained weight fractions of gel and their 95% confidence intervals. From the figure, the gel point is determined to be about x = 0.47, which agrees well with that obtained from the analytic solution derived in the second part of this series,17 x = 0.468.…”

Free‐Radical Polymerization with Long‐Chain Branching and Scission: Markovian Solution of the Weight‐Average Molecular Weight

“…In the context of the matrix approach, given by Equation , gelation occurs when the largest eigenvalue of M reaches unity. In the earlier matrix approach, the condition, det( I ‐ M ) = 0 was used . However, it was found that the value of det( I ‐ M ) is too small in the present case, and it was not suitable to determined the point that satisfy det( I ‐ M ) = 0 accurately.…”

Molecular Weight Development in Free-Radical Polymerization with Polyfunctional Chain-Transfer Agents, 1. Equal Reactivity Model

“…A numerical technique to find the gel point can be found elsewhere. [14,16] In ref. [1] , the MC simulation method for the same reaction system was proposed.…”

Simultaneous long-chain branching and random scission: I. Monte Carlo simulation

“…Figure 13 shows the obtained weight fractions of gel and their 95% confidence intervals. From the figure, the gel point is determined to be about x = 0.47, which agrees well with that obtained from the analytic solution derived in the second part of this series,17 x = 0.468.…”

Free‐Radical Polymerization with Long‐Chain Branching and Scission: Markovian Solution of the Weight‐Average Molecular Weight

“…In the context of the matrix approach, given by Equation , gelation occurs when the largest eigenvalue of M reaches unity. In the earlier matrix approach, the condition, det( I ‐ M ) = 0 was used . However, it was found that the value of det( I ‐ M ) is too small in the present case, and it was not suitable to determined the point that satisfy det( I ‐ M ) = 0 accurately.…”