On the Determination of Gel Points

TL;DR

The paper addresses aligning gelation theory with linear viscoelastic predictions for gel points in cross-linked polysiloxanes. It develops a generalized geometric distribution for prepolymer lengths, accounts for devolatilization via a cutoff $k$, and computes cyclization frequencies and critical dilutions to compare with Scanlan-Winter observations. The results show near-consistency between the two theories under geometric distribution, with the mean-$M_n$ approach coincidentally matching the observed gel point but implying monodispersity contrary to measured polydispersity, underscoring sensitivity to the assumed distribution. The authors advocate monodisperse-precursor experiments to decisively bridge the two theoretical frameworks and clarify the role of distribution and cyclization in gelation behavior.

Abstract

A critical composition of cross-linked polysiloxanes observed by Scanlan and Winter is reinvestigated in comparison with the theory of gelation. We assume, based on the Scott findings, the geometric distribution for one of the monomers, divinyl-terminated poly(dimethylsiloxane). Calculation results show that the two theories are in near-consistency, supporting the Scanlan-Winter estimation based on the linear viscoelastic theory. On the other hand, there is a disturbing result that calculation using the mean molecular weight, $M_{n}$, leads to exact agreement between the two theories, suggesting that the distribution is in effect monodisperse, contrary to the assumed geometric one and also the observed polydispersity, $M_w/M_n=2.1$. Further experimental studies employing monodisperse monomers would be highly valuable to consolidate the bridge between these two fundamental theories.

Figures (6)

• Figure 1: An AB-type polymerization of divinyl-terminated poly(dimethylsiloxane) and tetrakis(dimethyl-siloxy)silane. Z: CH$_3$ or Phenyl.
• Figure 2: Macrocyclization of a linear AB monomer (n $=$ 1, 2, $\dots$). Z: CH$_3$ or Phenyl.
• Figure 3: Probability distribution function, $P(n,x)$, before devolatilization as a function of $n$ and $x$. $\langle n\rangle_n=153\,x$ ($p=\frac{152}{153}$).
• Figure 4: Probability distribution function, $P_k(n,x)$, after devolatilization as a function of $k$, $n$ and $x$. In this example, $k=50$ and $\langle n\rangle_n=153\,x$ ($p=\frac{103}{104}$).
• Figure 5: Critical molar ratios against the reciprocal of the A-functional units concentration $\left([\text{C}=\text{C}]\right)$. Calculated by using the data of $k=50$ in the polymerization of divinyl-terminated poly(dimethylsiloxane) and tetrakis(dimethylsiloxy)silane. The shaded region represents the area where gelation is possible.