Universal features of cluster numbers in percolation

Void Connectivity and Criticality in the Compression-Induced Gel-to-Glass Transition of Short-Range Attractive Colloids

Gels and attractive glasses-both dynamically arrested states formed through short-range attraction-are commonly found in a range of soft matter systems, including colloids, emulsions, proteins, and wet granular materials. Previous studies have revealed intriguing similarities and distinctions in their structural, dynamic, vibrational, and mechanical properties. However, the microstructural mechanisms underlying the gel-to-glass transition remain elusive. To address this, we investigate uniaxial compression-induced gel collapse using confocal microscopy, which provides experimental access to the relationship between mechanical stress and microstructure. Together with Brownian dynamics simulations, our study reveals two sequential transitions in void structure: from gels with percolating voids to isolated voids, and ultimately to voidless glasses. These transitions are closely linked to a shift from superlinear power-law scaling to explosive divergence toward the packing limit in both normal and deviatoric stresses as a function of volume fraction, particularly at lower temperatures. Understanding these mechanically self-organized transitions and their associated criticality deepens our insight into disordered solids, enabling better control over mechanical properties, interfacial characteristics, and transport behavior in porous materials.

A Concept of Local Coordination Number for the Characterization of Solute Clusters within Atom Probe Tomography Data

Identifying clusters of solute atoms in a matrix of solvent atoms helps to understand precipitation phenomena in alloys, for example, during the age hardening of certain aluminum alloys. Atom probe tomography datasets can deliver such information, provided that appropriate cluster identification routines are available. We investigate algorithms based on the local composition of the neighborhood of solute atoms and compare them with traditional approaches based on the local solute number density, such as the maximum separation distance method. For an ideal solid solution, the pair correlation functions of the kth nearest solute atom in the coordination number representation are derived, and the percolation threshold and the size distribution of clusters are studied. A criterion for selecting optimal control parameters based on maximizing the phase separation by the degree of clustering is proposed for a two-phase system. A map of phase compositions accessible for cluster analysis is constructed. The coordination number approach reduces the influence of density variations commonly observed in atom probe tomography data. Finally, a practical cluster analysis technique applied to the early stages of aluminum alloy aging is described.

Percolation and dissolution of Borromean networks

Inspired by experiments on topologically linked DNA networks, we consider the connectivity of Borromean networks, in which no two rings share a pairwise-link, but groups of three rings form inseparable triplets. Specifically, we focus on square lattices at which each node is embedded a loop which forms a Borromean link with pairs of its nearest neighbors. By mapping the Borromean link network onto a lattice representation, we investigate the percolation threshold of these networks (the fraction of occupied nodes required for a giant component), as well as the dissolution properties: the spectrum of topological links that would be released if the network were dissolved to varying degrees. We find that the percolation threshold of the Borromean square lattice occurs when approximately 60.75% of nodes are occupied, slightly higher than the 59.27% typical of a square lattice. Compared to the dissolution of Hopf-linked networks, a dissolved Borromean network will yield more isolated loops, and fewer isolated triplets per single loop. Our simulation results may be used to predict experiments from Borromean structures produced by synthetic chemistry.

Exact results for average cluster numbers in bond percolation on infinite-length lattice strips

We calculate exact analytic expressions for the average cluster numbers 〈k〉_{Λ_{s}} on infinite-length strips Λ_{s}, with various widths, of several different lattices, as functions of the bond occupation probability p. It is proved that these expressions are rational functions of p. As special cases of our results, we obtain exact values of 〈k〉_{Λ_{s}} and derivatives of 〈k〉_{Λ_{s}} with respect to p, evaluated at the critical percolation probabilities p_{c,Λ} for the corresponding infinite two-dimensional lattices Λ. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in 〈k〉_{Λ_{s}} determine the radii of convergence of series expansions for small p and for p near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.