Dimension of heterogeneous network architecture formed in emulsion cross‐linking copolymerization

Abstract

The mean‐square radius of gyration Rg2 and the graph diameter D of highly heterogeneous network polymers formed in emulsion vinyl/divinyl copolymerization are investigated to find a linear relationship, Rg2 = a D. The proportionality coefficient, a, is dominated by the cycle (circuit) rank, kc, or the number of intramolecular cross‐links. The magnitude of a is slightly larger than that for the random cross‐linked network polymers, but the functional form of a (kc) is simply proportional to that for the random networks proposed earlier. This relationship makes it possible to determine Rg2 based on D and kc, enabling quick estimation of Rg2 for large network polymers. It is found that the contraction factor g of the present heterogeneous network polymer is larger than that for the homogeneous networks formed through random cross‐linking of polymer chains. The ratio of these two contraction factors is kept constant, irrespective of the magnitude of kc.

Dimensions of Network Polymers: Universal Relationship for the Ratio between Mean‐Square Radius of Gyration and Graph Diameter

“…The secondary fraction polymers are shown by the blue dots. The size of the secondary fraction polymers are rather small and the number fraction of such polymers are small enough, [21,22] and therefore, the analyses are conducted only for the major fraction polymers. The red circles show the expected (average) values for given r's.…”

Archie E. Hamielec memorial issue

“…The authors of the articles in this issue join us from many countries and institutions across the globe (Figure 1 and Table 1) to thank Archie for his enduring legacy in polymerization reaction engineering, a field of academic and industrial interest that he established practically single‐handedly. [ 1–45 ]…”

Dimensions of Large‐Sized Network Polymers Formed in Miniemulsion Polymerization

“…With the development of graph theory [ 6–8 ] and of the mathematical software to conduct the graph calculations, it has become possible to determine Rg 2 of network polymer whose structure is generated in the Monte Carlo (MC) simulation in a straightforward manner, as reported in a series of articles. [ 9–12 ]…”

“…In a recent series of articles [ 10–12 ] that investigate the relationship between Rg 2 and D of the random networks, it was found that the cycle rank r dominates the magnitude of ratio ( Rg 2 / D ), with negligible effects of the primary chain length distribution and the number of primary chains in the network. An empirical equation for the relationship between the ratio a r ,ran = ( Rg 2 / D ) r of the random network proposed [ 12 ] is given by: $$$$\begin{equation}{a}_{r{\mathrm{,ran}}} = {a}_{0,{\mathrm{ran}}}\ {\left[ {{{\left( {1 + r} \right)}}^{ - 2/3} + r/2} \right]}^{ - 0.25}\end{equation}$$$$where a 0,ran is the ratio for the ring‐free random crosslinked polymers, which is given by a 0,ran = 0.178, irrespective of the primary chain length distribution.…”

“…For various types of nonrandom, heterogeneous networks investigated so far, [ 11,12 ] the ratio a r = ( Rg 2 / D ) r is simply proportional to that for the random, homogeneous networks. $$$$\begin{equation}{a}_r = f{a}_{r{\mathrm{,ran}}}\ \end{equation}$$$$…”