“…AB-, CD- reaction), the master equation becomes more complex due to a combinatorial number of monomer species of defined type and state. In that case, automated reaction networks can be applied to algorithmically construct the corresponding master equation 23 .…”
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.
“…AB-, CD- reaction), the master equation becomes more complex due to a combinatorial number of monomer species of defined type and state. In that case, automated reaction networks can be applied to algorithmically construct the corresponding master equation 23 .…”
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.
“…Since molecular structures derived from triolein can bear several functional groups, manually writing down all of the reaction steps and intermediate products is infeasible due to the large number of combinations in which multifunctional molecules may interact. To set up an automated exploration process yielding such reaction steps, 34 we will first formalize the structures of irregular macromolecules so that they become palatable to a computer algorithm. As illustrated in Figure 2 a, we distinguish two hierarchical levels: the macromolecule level (Level 1) and the fragment level (Level 2).…”
Section: Resultsmentioning
confidence: 99%
“…However, using this algorithmic discovery of reaction steps on large molecules becomes prohibitive because the algorithm is NP-hard. 40 To overcome this problem, we have formulated ARNG for random graphs in Orlova et al, 34 and, in this paper, extended this method further to many cross-link types. Because of our approach, a large irregular macromolecule is viewed as an ensemble of randomly interconnected monomers, which have precise structures and are represented as molecular graphs.…”
The autoxidation
of triglyceride (or triacylglycerol, TAG) is a
poorly understood complex system. It is known from mass spectrometry
measurements that, although initiated by a single molecule, this system
involves an abundance of intermediate species and a complex network
of reactions. For this reason, the attribution of the mass peaks to
exact molecular structures is difficult without additional information
about the system. We provide such information using a graph theory-based
algorithm. Our algorithm performs an automatic discovery of the chemical
reaction network that is responsible for the complexity of the mass
spectra in drying oils. This knowledge is then applied to match experimentally
measured mass spectra with computationally predicted molecular graphs.
We demonstrate this methodology on the autoxidation of triolein as
measured by electrospray ionization-mass spectrometry (ESI-MS). Our
protocol can be readily applied to investigate other oils and their
mixtures.
“…For instance, cascade events my occur under strong notion of connectivity 22,34–36 . There are many studies that focus on important special cases, and apply the multiplex formalism to explain phenomena in empirical data 25,35,37–40 . However, the theoretical aspects are less developed in these studies as they scatter between special cases to explain empirical observations rather than to develop a single universal theory.Fig.…”
Percolation in complex networks is a process that mimics network degradation and a tool that reveals peculiarities of the network structure. During the course of percolation, the emergent properties of networks undergo non-trivial transformations, which include a phase transition in the connectivity, and in some special cases, multiple phase transitions. Such global transformations are caused by only subtle changes in the degree distribution, which locally describe the network. Here we establish a generic analytic theory that describes how structure and sizes of all connected components in the network are affected by simple and colour-dependent bond percolations. This theory predicts locations of the phase transitions, existence of wide critical regimes that do not vanish in the thermodynamic limit, and a phenomenon of colour switching in small components. These results may be used to design percolation-like processes, optimise network response to percolation, and detect subtle signals preceding network collapse.
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