Continuum Percolation Model Research Articles

Percolation of nonequilibrium assemblies of colloidal particles in active chiral liquids.

The growing interest in the non-equilibrium assembly of colloidal particles in active liquids is driven by the motivation to create novel structures endowed with tunable properties unattainable within the confines of equilibrium systems. Here, we present an experimental investigation of the structural features of colloidal assemblies in active liquids of chiral E. coli. The colloidal particles form dynamic clusters due to the effective interaction mediated by active media. The activity and chirality of the swimmers strongly influence the dynamics and local ordering of colloidal particles, resulting in clusters with persistent rotation, whose structure differs significantly from those in equilibrium systems with attractive interactions, such as colloid-polymer mixtures. Our colloid-bacteria mixture displays several hallmark features of a percolation transition at a critical density, where the clusters span the system size. A closer examination of the critical exponents associated with cluster size distribution, the average cluster size, and the correlation length in the vicinity of the critical density shows deviations from the prediction of the standard continuum percolation model. Therefore, our experiments reveal a richer phase behavior of colloidal assemblies in active liquids.

Mean-field bounds for Poisson-Boolean percolation

We establish the mean-field bounds γ≥1, δ≥2 and △≥2 on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of γ), or not at all (in the case of δ and △). We deduce these as consequences of the mean-field bound β≤1, recently established under the same moment condition [8], using a relative entropy method introduced by the authors in previous work [7].

Percolative Channels for Superionic Conduction in an Amorphous Conductor.

All-solid-state batteries greatly rely on high-performance solid electrolytes. However, the bottlenecks in solid electrolytes are their low ionic conductivity and stability. Here we report a new series of amorphous xAgI·(1-x)Ag3PS4 (x = 0∼0.8 with interval of 0.1) conductors, among which the sample with x = 0.8 exhibits the highest ionic conductivity (about 1.1 × 10-2 S cm-1) and ultrahigh chemical stability. We discovered the existence of mixed disordered Ag3PS4 and AgI clusters in the amorphous conductors using solid-state nuclear magnetic resonance spectroscopy. The high ionic conductivity was ascribed to the formation of the interconnecting AgI clusters, i.e., the percolative channels for superionic conduction. The composition dependence of the ionic conductivity for this series of amorphous conductors was clarified by a continuum percolation model. These findings provide fundamental guidance for designing and fabricating high-performance amorphous solid electrolytes for all-solid-state batteries.

The direct-connectedness function in the random connection model

AbstractWe investigate expansions for connectedness functions in the random connection model of continuum percolation in powers of the intensity. Precisely, we study the pair-connectedness and the direct-connectedness functions, related to each other via the Ornstein–Zernike equation. We exhibit the fact that the coefficients of the expansions consist of sums over connected and 2-connected graphs. In the physics literature, this is known to be the case more generally for percolation models based on Gibbs point processes and stands in analogy to the formalism developed for correlation functions in liquid-state statistical mechanics.We find a representation of the direct-connectedness function and bounds on the intensity which allow us to pass to the thermodynamic limit. In some cases (e.g., in high dimensions), the results are valid in almost the entire subcritical regime. Moreover, we relate these expansions to the physics literature and we show how they coincide with the expression provided by the lace expansion.

Two-dimensional continuum percolation models with disks under the generalized Achlioptas process

We investigate the disk percolation phase transitions in continuum models in two dimensions under four kinds of generalized product rule and four kinds of generalized sum rule, respectively. We study the critical behaviors of the largest, second largest clusters and their size ratio to characterize the universality class of percolation transition. Using the finite size scaling analysis and the Monte Carlo simulation, we calculate the critical exponents of the percolation transitions under the above rules. We find that the universality classes of continuum percolation under the generalized product rule, under the generalized sum rule, and the lattice percolation are different.

Phase transitions for the Boolean model of continuum percolation for Cox point processes

We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

Sharp phase transition for Cox percolation

We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.

A Biodegradable Copolyester, Poly(butylene succinate-co-ε-caprolactone), as a High Efficiency Matrix Former for Controlled Release of Drugs.

A biodegradable copolyester, poly(butylene succinate-co-ε-caprolactone) (PBS_CL), was used for first time as an excipient for pharmaceutical dosage forms using direct compression and hot processing techniques (ultrasound-assisted compression (USAC) and hot melt extrusion (HME)). Robust binary systems were achieved with hot processing techniques, allowing a controlled release of the drug. With only 12% v/v of PBS_CL, controlled release forms were obtained using USAC whereas in HME over 34% v/v of excipient is necessary. Amounts over 23% v/v allowed a long-extended release for more than 72 h following diffusional kinetic. Thanks to the high melting point of theophylline and the physicochemical properties of PBS_CL selected and synthesized, the structure of the excipient inside the USAC tablets and HME filaments corresponds to a continuum medium. A percolation threshold around 23% v/v was estimated, which agrees with a continuum percolation model. The polymer shows a high excipient efficiency value using HME and USAC. A nanostructured matrix with wall thicknesses lower than 0.1 µm was obtained. This leads to a very effective coating of the drug particles by the excipient, providing a slow and reproducible release. The present study therefore supports the use of PBS_CL, for the preparation of controlled release dosage forms using hot processing techniques.

Continuum percolation and stochastic epidemic models on Poisson and Ginibre point processes

The most studied continuum percolation model in two dimensions is the Boolean model consisting of disks with the same radius whose centers are randomly distributed on the Poisson point process (PPP). We also consider the Boolean percolation model on the Ginibre point process (GPP) which is a typical repelling point process realizing hyperuniformity. We think that the PPP approximates a disordered configuration of individuals, while the GPP does a configuration of citizens adopting a strategy to keep social distancing in a city in order to avoid contagion. We consider the SIR models with contagious infection on supercritical percolation clusters formed on the PPP and the GPP. By numerical simulations, we studied dependence of the percolation phenomena and the infection processes on the PPP- and the GPP-underlying graphs. We show that in a subcritical regime of infection rate the PPP-based models show emergence of infection clusters on clumping of points which is formed by fluctuation of uncorrelated Poissonian statistics. On the other hand, the cumulative numbers of infected individuals in processes are suppressed in the GPP-based models.

Statistical analysis of the critical percolation of ITZ around polygonal aggregates in three-phase concrete materials

Concrete is generally regarded as a kind of three-phase granular material composed of aggregates, interfacial transition zone (ITZ) and bulk cement paste, among which ITZ is an inescapable transport path of aggressive agents due to its high porosity. As the weakest part in concrete, the existence of ITZ usually triggers an unfavorable change of the holistic properties of concrete structures. In particular, the percolation behavior of ITZ is likely to be an important factor that accelerates the penetration of aggressive agents. Previous studies about the percolation of ITZ around aggregates are mainly carried out based on the models of circles, spheres and ellipsoids, which are very different from the real features of aggregates. In this work, the regular polygons which are more similar to the geometrical shapes of fine and coarse aggregates are applied to represent the aggregate particles. The ITZ around these aggregates is simplified to the soft interphase layer with identical thickness and the models of concrete structures can be modeled by the random sequential addtion (RSA) algorithm. By combining with the continuum percolation models of hard core/soft shell, the percolation properties of ITZ around different 2D multi-sized aggregates are studied in detail and the effect of both aggregates and ITZ on the percolation thresholds ϕagg,c of ITZ is quantified. Based on the obtained values of ϕagg,c here, an approximately numerical formula of ϕagg,c is proposed and its reliability is further verified by comparing with the results in the literature. The comparison confirms that the proposed model can be utilized to predict the critical thresholds of ITZ around various polygonal particles with the broad particle size range and ITZ thickness. The research findings may provide a helpful guidance for the design and optimization of concrete materials.

Critical polynomials in the nonplanar and continuum percolation models.

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial P_{B}(p,L) was introduced for planar-lattice percolation models, where p is the occupation probability and L is the linear system size. The solution of P_{B}=0 can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of P_{B}, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, P_{B} suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds p_{c}(z) as a function of coordination number z for equivalent-neighbor percolation with z up to O(10^{5}) and clearly confirm the asymptotic behavior zp_{c}-1∼1/sqrt[z] for z→∞. For the continuum percolation model, we surprisingly observe that the finite-size correction in P_{B} is unobservable within uncertainty O(10^{-5}) as long as L≥3. The estimated threshold number density of disks is ρ_{c}=1.43632505(10), slightly below the most recent result ρ_{c}=1.43632545(8) of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.

Homological Percolation: The Formation of Giant k-Cycles

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

Investigation of flow boiling heat transfer and boiling crisis on a rough surface using infrared thermometry

We design and build a special heater to enable infrared investigations of boiling heat transfer on surfaces featuring the typical roughness and scratch pattern of commercial-grade heat transfer surfaces (in this case a zirconium alloy typically used as fuel cladding material in nuclear reactors). We use high-speed infrared thermometry to investigate surface effects on the boiling process for both the rough infrared heater and a reference more conventional, nano-smooth infrared heater. Compared to the nano-smooth surface, the rough surface has larger nucleation sites, which require a lower nucleation temperature. The rough surface has a much smaller bubble departure volume. However, it has a much higher nucleation site density, and, overall, a higher heat transfer coefficient. We capture this behavior with a stochastic heat flux partitioning model. Notably, while the two surfaces have very different boiling dynamics, the boiling crisis has a common “signature”. For both surfaces, the probability density functions of bubble footprint areas follow a power law with a negative exponent smaller than 3, also known as a scale-free distribution. We predict these observations and the onset of the boiling crisis using a continuum percolation model. These results corroborate the hypothesis of the boiling crisis as a percolative critical phase transition of the bubble interaction process.

Synergy effect in hybrid nanocomposites based on carbon nanotubes and graphene nanoplatelets

Hybrid nanocomposites reinforced with a mixture of graphene nanoplatelets (GNPs) and carbon nanotubes (CNTs) have shown improvement in filler dispersion while providing a cost-effective alternative to CNT monofiller composites. Depending on their composition, hybrid composites can exhibit electrical performance superior to either of the constituent monofiller composites due to synergistic effects. In this work, we develop a three-dimensional tunneling-based continuum percolation model for hybrid nanocomposites filled with hardcore particles of elliptical GNPs and cylindrical CNTs. Using Monte Carlo simulations, parametric studies of the filler content, composition and morphology are carried out to analyze the conditions required for synergy in percolation onset and electrical conductivity. Our results suggest that for hybrid systems with well-dispersed fillers, the electrical performance is linked to the number of tunneling junctions per filler inside the percolated network of the nanocomposites. More importantly, hybrid composites filled with specific morphology of GNP and CNT, exhibit synergy in their electrical performance when the monofiller composites of each of those exact fillers have similar percolation onset values. The simulations results are in agreement with relevant experimental data on hybrid nanocomposites.

Numerical study for the percolation threshold and transport properties of porous composites comprising non-centrosymmetrical superovoidal pores

The aggressive media can be transported into the composite through the connected pores, which will damage the microstructure and then lead to a subsequent reduction in the strength and serviceability of composite. In the statistical physics, the continuum percolation model is usually used to study the system-spanning connectivity in the porous composite. It is widely believed that the dimensionless excluded volume Vdex of discrete pore is of great importance in the evaluation of the global percolation threshold ϕc of porous network. The porous networks of composites are normally composed of overlapping symmetrical spheres. However, the actual shape of pores in composite is asymmetrical. In this work, the dimensionless Vdex of discrete superovoidal pores are theoretically and numerically calculated. Then, the quantitative relationship between components (pore shape) and microstructures (global percolation threshold ϕc) of porous networks comprising superovoids is established by a dimensionless excluded volume based empirical approximation. It is observed that the porous composites comprising symmetrical pores are more difficult to form the system-spanning connected porous path than the composites consisting of non-centrosymmetrical pores. Moreover, the transport properties of composites considering their microstructures (i.e., the thresholds ϕc) are theoretically predicted by the continuum percolation-based generalized effective medium theory (CP-GEMT) and the theoretical values are validated by comparing with the experimental results.

Percolative Scale-Free Behavior in the Boiling Crisis.

We present the first experimental observations of scale-free behavior in the bubble footprint distribution during the boiling crisis of water, in pool and flow boiling conditions. We formulate a continuum percolation model that elucidates how the scale-free behavior emerges from the near-wall stochastic interaction of bubbles and provides a criterion to predict the boiling crisis. It also offers useful insights on how to engineer surfaces that enhance the critical heat flux limit.

A 3D percolation model for multicomponent nanocarbon composites: the critical role of nematic transition

A three-dimensional (3D) continuum percolation model has been developed on the basis of Monte Carlo simulation to investigate the percolation behavior of an electrically insulating matrix reinforced with multiple conductive fillers of different dimensionalities. Impenetrable fillers of large aspect ratio are found to preferentially align with each other to maximize the packing entropy rather than forming randomly oriented clusters. This entropy-driven transition from isotropic to nematic phase is shown to critically affect the percolation threshold. It suggests that an isotropic phase with a smaller nematic order parameter leads to a reduction in percolation threshold. In addition, a combination of two fillers with different dimensionalities can achieve a working concentration below the percolation threshold of single component system, which is further validated by the experiments of electrical conductivity in multicomponent multidimensional nanocarbon composites.

Sharp phase transition for the random-cluster and Potts models via decision trees

We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $\beta<\beta_c$. 2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. 3. For the random-cluster models with cluster weight $q\geq1$ on planar quasi-transitive graphs $\mathbb{G}$, $$\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]). These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.

Annealed scaling relations for Voronoi percolation

We prove annealed scaling relations for planar Voronoi percolation. To our knowledge, this is the first result of this kind for a continuum percolation model. We are mostly inspired by the proof of scaling relations for Bernoulli percolation by Kesten [22]. Along the way, we show an annealed quasi-multiplicativity property by relying on the quenched box-crossing property proved by Ahlberg, Griffiths, Morris and Tassion [3]. Intermediate results also include the study of quenched and annealed notions of pivotal events and the extension of the quenched box-crossing property of [3] to the near-critical regime.

Topology, connectivity and percolation in fracture networks

The connectivity of a fracture network depends on both its geometry and topology, which may be characterized in terms of the types of nodes and branches in the system. A number of dimensionless parameters (dimensionless intensity (P22, B22), connections per line (CL) and connections per branch (CB), are used to characterize and assess network connectivity based on well-established lattice and continuum percolation models. These models result in networks with only I- and X-nodes, whereas most natural fracture networks have many Y-nodes. A simple scheme to transform IX systems into IYX systems is developed that allows the expected value of dimensionless branch intensity (B22C) at the percolation threshold to be derived for any network topology. The method is used to assess if a network is likely to be connected or not, and demonstrated by analysing the development of a network of desiccation cracks.