Dynamic Networks that Drive the Process of Irreversible Step-Growth Polymerization

Abstract

Many research fields, reaching from social networks and epidemiology to biology and physics, have experienced great advance from recent developments in random graphs and network theory. In this paper we propose a generic model of step-growth polymerisation as a promising application of the percolation on a directed random graph. This polymerisation process is used to manufacture a broad range of polymeric materials, including: polyesters, polyurethanes, polyamides, and many others. We link features of step-growth polymerisation to the properties of the directed configuration model. In this way, we obtain new analytical expressions describing the polymeric microstructure and compare them to data from experiments and computer simulations. The molecular weight distribution is related to the sizes of connected components, gelation to the emergence of the giant component, and the molecular gyration radii to the Wiener index of these components. A model on this level of generality is instrumental in accelerating the design of new materials and optimizing their properties, as well as it provides a vital link between network science and experimentally observable physics of polymers.

Coloured random graphs explain the structure and dynamics of cross-linked polymer networks

“…Step-growth polymerisation is one example of a network formation process that only constrains the degree distribution, beyond that, the graph is fully random and history-independent. Such networks can be model by a random graph that satisfies a fixed degree distribution at each time point 21,23,24 . However, if the emerging structure depends on history of its formation, it cannot be represented by degree distribution alone, as shown in the following example: Consider a system with linear polymer chains having a narrow length distributions.…”

“…If , the observed network is not history-dependent and can be sufficiently described by the uncoloured random graph, i.e., colours are redundant. One example of such a process is the conventional step-growth polymerisation 24 .…”

“…Following the work by Flory and Stockmayer, branching and coagulation processes were used to derive the size of gel molecules and predict the time when the gel starts to form, see Gordon 17 , 18 and Ziff 19 , 20 . In our recent works 21 – 24 , we continued this avenue by identifying the entire polymer network as a random graph with a given degree distribution 25 , which provides a new theory and algorithms of general step-growth polymer systems irrespective of the number of monomers and their functionality. However successful such a technique proves to be for step-growth polymer networks, it cannot explain the networks produced by chain-growth procedure due to history dependence of the latter.…”

“…One striking feature of history-dependent processes is that the moment in time when an extensive cluster emerges, that is the gel point 13 , differs from the percolation threshold found when randomly removing bonds in the final network. This provides a strong contrast to network formation by step-growth polymerisation, where the polymerisation process is equivalent to random percolation on the final network topology 21,23,24 . In some cases, however, the uncoloured random graph gives a surprisingly good estimate of the phase transition and the gel fraction 13 , which suggests that under certain conditions chain growth polymerisation may not induce a significant history dependency.…”

Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution

“…One therefore may speak of a ball of radius r centered at a chosen node, where r is a positive integer. Studying how the expected volume of such a ball grows with r gives a good proxy characterizing the interplay of the space and structure of spatial random networks [1][2][3][4][5]: If this quantity has a simple monomial shape r d , then one says that d is the network's dimension or volume scaling [3,4,6,7]. Dimensions of many real-life and artificial networks tend to range from approximately 1.25 for the New York City subway to 8.1 for YouTube [3].…”