Emergence of the giant weak component in directed random graphs with arbitrary degree distributions

Thermokinetic and Chemorheology of the Geopolymerization of an Alumina-Rich Alkaline-Activated Metakaolin in Isothermal and Dynamic Thermal Scans

Alkaline sodium hydroxide/sodium silicate-activating high-purity metakaolin geopolymerization is described in terms of metakaolin deconstruction in tetrahedral hydrate silicate [O[Si(OH)3]]- and aluminate [Al(OH)4]- ionic precursors followed by their reassembling in linear and branched sialates monomers that randomly copolymerize into an irregular crosslinked aluminosilicate network. The novelty of the approach resides in the concurrent thermo-calorimetric (differential scanning calorimetry, DSC) and rheological (dynamic mechanical analysis, DMA) characterizations of the liquid slurry during the transformation into a gel and a structural glassy solid. Tests were run either in temperature scan (1 °C/min) or isothermal (20 °C, 30 °C, 40 °C) cure conditions. A Gaussian functions deconvolution method has been applied to the DSC multi-peak thermograms to separate the kinetic contributions of the oligomer's concurrent reactions. DSC thermograms of all tested materials are well-fitted by a combination of three overlapping Gaussian curves that are associated with the initial linear low-molecular-weight (Mw) oligomers (P1) formation, oligomers branching into alumina-rich and silica-rich gels (P2), and inter- and intra-molecular crosslinking (P3). The loss factor has been used to define viscoelastic behavioral zones for each DMA rheo-thermogram operated in the same DSC thermal conditions. Macromolecular evolution and viscoelastic properties have been obtained by pairing the deconvoluted DSC thermograms with the viscoelastic behavioral zones of the DMA rheo-thermograms. Two main chemorheological behaviors have been identified relative to pre- and post-gelation separation of the viscoelastic liquid from the viscoelastic solid. Each comprises three behavioral zones, accounting for the concurrently occurring linear and branching oligomerization, aluminate-rich and silica-rich gel nucleations, crosslinking, and vitrification. A "rubbery plateau" in the loss factor path, observed for all the testing conditions, identifies a large behavioral transition zone dividing the incipient gelling liquid slurry from the material hard setting and vitrification.

Chemorheology of a Si/Al > 3 Alkali Activated Metakaolin Paste through Parallel Differential Scanning Calorimetry (DSC) and Dynamic Mechanical Analysis (DMA)

Although geopolymers, as structural materials, should have superior engineering properties than traditional cementitious materials, they often need to improve their final characteristics' reproducibility due to the need for more control of the complex silico-aluminate decomposition and polymerisation stages. Thermosetting of a reactive geopolymeric paste involves tetrahedral Silicate and Aluminate precursor condensation into polyfunctional oligomers of progressively higher molecular weight, transforming the initial liquid into a gel and a structural solid. Viscosity and gelation control become particularly critical when the geopolymer is processed with 3D printing additive technology. Its physical state modification kinetics should match the flow and setting characteristics required by the deposition process. The reaction kinetics and the elastic and viscous characteristics preceding gelation and hardening have been investigated for an alkali-activated Metakaolin/Sodium Silicate-Sodium Hydroxide paste with a Si/Al ratio > 3. A chemoreological approach has been extended to these inorganic polymerisable systems, as already utilised for organic thermosetting polymers. Differential scanning calorimetry and Oscillatory DMA were carried out to monitor the advancement of the polymerisation reaction and the associated variations of the rheological viscoelastic properties. Dynamic thermal scans were run at 1 °C/min and a frequency of 10 Hz for the dynamic mechanical tests. The observed kinetics of polymerisation and variations of the elastic and viscous components of the complex viscosities and shear moduli are described in terms of polycondensation of linear and branched chains of oligomeric macromolecules of increasing complexity and molecular weight up to gelation (Gel1) and cross-linking of the gelled macrostructure (Gel2) and final glassy state. Geopolymerization can be allocated into two main behavioural zones: a viscoelastic liquid paste below 32.5% of reaction advancement and a viscoelastic solid above. Initial complex viscosities range from 2.3 ± 0.9 × 10-5 MPa*s to 6.8 ± 0.9 × 10-2 in the liquid-like state and from 1.9 ± 0.1 MPa to 9.6 ± 2.1 × 102 MPa in the solid-like state.

Learning heterogenous reaction rates from stochastic simulations

Reaction rate equations are ordinary differential equations that are frequently used to describe deterministic chemical kinetics at the macroscopic scale. At the microscopic scale, the chemical kinetics is stochastic and can be captured by complex dynamical systems reproducing spatial movements of molecules and their collisions. Such molecular dynamics systems may implicitly capture intricate phenomena that affect reaction rates but are not accounted for in the macroscopic models. In this work we present a data assimilation procedure for learning nonhomogeneous kinetic parameters from molecular simulations with many simultaneously reacting species. The learned parameters can then be plugged into the deterministic reaction rate equations to predict long time evolution of the macroscopic system. In this way, our procedure discovers an effective differential equation for reaction kinetics. To demonstrate the procedure, we upscale the kinetics of a molecular system that forms a complex covalently bonded network severely interfering with the reaction rates. Incidentally, we report that the kinetic parameters of this system feature peculiar time and temperature dependences, whereas the probability of a network strand to close a cycle follows a universal distribution.

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

Step-growth and chain-growth are two major families of chemical reactions that result in polymer networks with drastically different physical properties, often referred to as hyper-branched and cross-linked networks. In contrast to step-growth polymerisation, chain-growth forms networks that are history-dependent. Such networks are defined not just by the degree distribution, but also by their entire formation history, which entails a modelling and conceptual challenges. We show that the structure of chain-growth polymer networks corresponds to an edge-coloured random graph with a defined multivariate degree distribution, where the colour labels represent the formation times of chemical bonds. The theory quantifies and explains the gelation in free-radical polymerisation of cross-linked polymers and predicts conditions when history dependance has the most significant effect on the global properties of a polymer network. As such, the edge colouring is identified as the key driver behind the difference in the physical properties of step-growth and chain-growth networks. We expect that this findings will stimulate usage of network science tools for discovery and design of cross-linked polymers.

Enhancing the robustness of a multiplex network leads to multiple discontinuous percolation transitions

Determining design principles that boost the robustness of interdependent networks is a fundamental question of engineering, economics, and biology. It is known that maximizing the degree correlation between replicas of the same node leads to optimal robustness. Here we show that increased robustness might also come at the expense of introducing multiple phase transitions. These results reveal yet another possible source of fragility of multiplex networks that has to be taken into the account during network optimization and design.

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

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.

Bond percolation in coloured and multiplex networks

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.

Percolation is a tool used to investigate a network’s response as random links are removed. Here the author presents a generic analytic theory to describe how percolation properties are affected in coloured networks, where the colour can represent a network feature such as multiplexity or the belonging to a community.

Modeling the free-radical polymerization of hexanediol diacrylate (HDDA): a molecular dynamics and graph theory approach

In the printing, coating and ink industries, photocurable systems are becoming increasingly popular and multi-functional acrylates are one of the most commonly used monomers due to their high reactivity (fast curing). In this paper, we use molecular dynamics and graph theory tools to investigate the thermo-mechanical properties and topology of hexanediol diacrylate (HDDA) polymer networks. The gel point was determined as the point where a giant component was formed. For the conditions of our simulations, we found the gel point to be around 0.18 bond conversion. A detailed analysis of the network topology showed, unexpectedly, that the flexibility of the HDDA molecules plays an important role in increasing the conversion of double bonds, while delaying the gel point. This is due to a back-biting type of reaction mechanism that promotes the formation of small cycles. The glass transition temperature for several degrees of curing was obtained from the change in the thermal expansion coefficient. For a bond conversion close to experimental values we obtained a glass transition temperature around 400 K. For the same bond conversion we estimate a Young's modulus of 3 GPa. Both of these values are in good agreement with experiments.

Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions

This work presents exact expressions for size distributions of weak and multilayer connected components in two generalizations of the configuration model: networks with directed edges and multiplex networks with an arbitrary number of layers. The expressions are computable in a polynomial time and, under some restrictions, are tractable from the asymptotic theory point of view. If first partial moments of the degree distribution are finite, the size distribution for two-layer connected components in multiplex networks exhibits an exponent -3/2 in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents -1/2 and -3/2.

General expression for the component size distribution in infinite configuration networks

In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary degree distribution the corresponding size distribution of connected components. This equation is suitable for fast and stable numerical computations up to the machine precision. The analytical analysis reveals that the asymptote of the component size distribution is completely defined by only a few parameters of the degree distribution: the first three moments, scale, and exponent (if applicable). When the degree distribution features a heavy tail, multiple asymptotic modes are observed in the component size distribution that, in turn, may or may not feature a heavy tail.