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Molecular network plays a critical role in determining dynamic elasticity of soft condensed polymers, e.g., their varied cross-linking densities and end-to-end distances with changed Young's moduli. Although it has been studied for decades, the coupling relationship between vertices and edges of molecular networks is not well understood, mainly because the degree of the crosslinking points and asymmetry molecular networks have not been considered in the previously models. In this study, a graph theory was employed to formulate an undirected graphical model and describe the coupling between vertices and edges in molecular networks, based on which the dynamic elasticity of polyelectrolyte hydrogel was modeled. From the Kirchhoff graph theory and bead-spring model, the coupling relationship between vertices and edges was obtained using end-to-end distance and viscosity parameters. Combining the Watts–Strogatz model and kinetic probability, the coupling between vertices and edges for the polyelectrolyte network was studied. Furthermore, an adjacency matrix with eigenvalue, number of vertices and mean degree was proposed to formulate constitutive relationships including dynamic elasticity and stress-strain, according to rubber elasticity theory and Mooney-Rivlin model, respectively. The linking between the vertices and edges determines the network structure and dynamic elasticity of the polyelectrolyte hydrogel. Based on the graph theory, the vertices and edges are encoded by adjacency matrix, which is proposed to describe the dynamic elasticity of symmetric and asymmetric network structures using the crosslinking density and end-to-end distance. Finally, effectiveness of the undirected graphical model was verified using both finite element analysis and experimental results of polyelectrolyte hydrogels reported in literature. Constitutive stress-elongation ratio curves for the polyelectrolyte hydrogel, of which the graph network is determined by the mean degree of d = 2, d = 3, d = 4, d = 5, d = 6 and d = 7, at the same vertex number of N = 8 in molecular network. [Display omitted] • An undirected graphical model and describe the coupling between vertices and edges in molecular networks. • Dynamic elasticity of polyelectrolyte hydrogel has been modeled using the adjacency matrix. • A constitutive relationship is developed to understand the molecular networks and their dynamic elasticities. [ABSTRACT FROM AUTHOR]

Mechanical robustness is one of the challenges for soft hydrogels, which are difficult to repair once fractured, mainly because of their chemical bonds and large binding energies required to heal the fractured surfaces within a reasonable time scale. In this study, an extended Maxwell model is proposed to describe intra-chain ionic bonds and inter-chain hydrogen bonds, coupled in physical hydrogels undergoing self-assembly and mechanical toughening, during both of which the intra-chain and inter-chain bonds are working as sub-entanglement and physical crosslink, respectively. According to the rubber elastic theory, a topology model is formulated to identify the working principle of intra-chain and inter-chain coupled binding in the physical hydrogels. Furthermore, a constitutive stress-strain relationship is developed to understand their self-assembling topology signatures and topological transitions. Finally, effectiveness of the proposed model is verified using molecular dynamic simulations and experimental results of physical hydrogels reported in literature. Illustration of dependence of mechanical behaviors on topology signature in terms of intra-chain and inter-chain coupled binding in a hydrogel, where topological geometry is determined by the number of chains and the number of crosslinks. [Display omitted] • An extended Maxwell model is proposed for the physical hydrogels. • A topology model is formulated to identify the working principle of coupled binding. • A constitutive relationship is developed to understand the self-assembly and topological transition. [ABSTRACT FROM AUTHOR]

In this study, a cooperative model has been proposed for the double network (DN) hydrogel, which synchronously undergoes heuristic swelling and inhibitive micellization by the ionic dissociation of polyelectrolyte. Flory-Huggins solution theory is initially employed to identify the working mechanism of dielectric constants on swelling behavior of the DN hydrogel. Then a free-energy function is introduced to formulate the constitutive relationship of the DN hydrogels, in which the first hydrotropic network undergoes a heuristic swelling and the second hydrophobic network undergoes an inhibitive micellization. Finally, the proposed model has been verified using the experimental results reported in the literature. A good agreement between the theoretical results and experimental ones has been achieved. This study provides a fundamental approach to formulate the constitutive relationship and to understand the cooperative dynamics of two types of networks in DN hydrogels induced by the polyelectrolyte. A new cooperative model was proposed to explore the working mechanism of double network (DN) hydrogel undergoes heuristic swelling and inhibitive micellization, synchronously, by the ionic dissociation of polyelectrolyte. Image 10787 • A cooperative model has been proposed for the heuristic swelling and inhibitive micellization. • A free-energy function is introduced to formulate the constitutive relationship of the DN hydrogels. • This study provides a fundamental approach to formulate the cooperative dynamics in DN hydrogels. [ABSTRACT FROM AUTHOR]