Effect of dimensionality on the percolation threshold of overlapping nonspherical hyperparticles

Geometric measures of uniaxial solids of revolution in higher-dimensional Euclidean spaces and their relation to the second virial coefficient

We provide analytical expressions for the second virial coefficients of various hard, convex, monoaxial solids of revolution in higher-dimensional Euclidean spaces. Therefore, the rotation-averaged mutual excluded volume per particle is calculated employing the Brunn-Minkowski theorem using quermassintegrals of the respective shape. In addition to geometries without singularities in their surface curvature, so far unknown quermassintegrals for geometries with singularities in their surface curvature, such as hyperlenses and hypercones, are calculated. Studying the influence of the detailed particle shape, the second virial coefficients are analyzed in four dimensions depending on the aspect ratio ν. These analytical expressions are extended to arbitrary-dimensional Euclidean spaces. The resulting virial coefficients are compared to available data for analogs in two and three dimensions. For hard hyperspheroids, the universal parity B_{2}^{*}(ν)=B_{2}^{*}(ν^{-1}) of the reduced second virial coefficient with respect to the aspect ratio ν is proven. Unlike other geometric shapes, the excluded volume of hyperspherocylinders in any dimension solely depends on at most three quermassintegrals.

Chemically reactive and aging macromolecular mixtures. II. Phase separation and coarsening

In a companion paper, we put forth a thermodynamic model for complex formation via a chemical reaction involving multiple macromolecular species, which may subsequently undergo liquid-liquid phase separation and a further transition into a gel-like state. In the present work, we formulate a thermodynamically consistent kinetic framework to study the interplay between phase separation, chemical reaction, and aging in spatially inhomogeneous macromolecular mixtures. A numerical algorithm is also proposed to simulate domain growth from collisions of liquid and gel domains via passive Brownian motion in both two and three spatial dimensions. Our results show that the coarsening behavior is significantly influenced by the degree of gelation and Brownian motion. The presence of a gel phase inside condensates strongly limits the diffusive transport processes, and Brownian motion coalescence controls the coarsening process in systems with high area/volume fractions of gel-like condensates, leading to the formation of interconnected domains with atypical domain growth rates controlled by size-dependent translational and rotational diffusivities.

Numerical modeling of the effects of the shape and aspect ratio of 3D curved fiber on the percolation threshold and electrical conductivity of conductive polymer composites

For designing conductive polymer composites (CPCs), understanding how the fiber curvature affects the percolation behavior of curved conductive fibers is essential for determining the effective electrical conductivity σeff of the CPCs. In this work, CPCs were considered as a polymer matrix filled with the random packing of overlapped curved spherocylinders. The geometries of the curved spherocylinders were defined, and inter-curved spherocylinder contact-detecting and system-spanning fiber cluster searching algorithms were developed. The finite-size-scaling method was used to explore how the aspect ratio α and bending central angle θ of a curved spherocylinder affect the percolation threshold ϕc of an overlapped curved spherocylinder system in 3D space. The findings suggest that ϕc decreases as α increases and increases initially before declining as θ increases. An empirical approximation formula was proposed to quantify the effect of the curved spherocylinder's morphology, characterized by the dimensionless excluded volume Vdex of the curved spherocylinder, on ϕc. The new rigorous bound for ϕc of the soft-curved spherocylinder system was further proposed. A random resistor network model was constructed, and the reliability of this model was validated by comparing the simulations and published data. Finally, a fitting formula was developed to assess the impacts of the normalized reduced density (η - ηc)/ηc and Vdex on the σeff of CPCs. A distinct linear correlation between σeff and (η - ηc)/ηc was constructed, denoted as σeff ∼ [(η - ηc)/ηc]t(α,θ). An empirical approximation model was proposed to establish the relationship between the fiber shape and conductivity exponent t. Our study may provide a theoretical hint for the design of CPCs.

Percolation threshold and electrical conductivity of conductive polymer composites filled with curved fibers in two-dimensional space

Quantifying the influence of fiber curvature on the percolation behavior of flexible conductive fiber and further on the electrical conductivity of conductive polymer composites (CPCs) is crucial for the design of CPCs. This study considers CPCs as a random packing of soft curved discorectangles (CDCRs) in a polymer matrix. The geometry of CDCR is developed, and an inter-CDCR contact detection algorithm is used to generate a random packing structure of CDCRs. The effects of aspect ratio α and bending central angles θ of CDCR on the percolation threshold ϕc of the overlapped CDCR system in a two-dimensional plane are then investigated using the finite-size scaling method. The result reveals that ϕc decreases monotonically as α grows and increases monotonically as θ rises. A shape-independent power law formula, denoted as ϕc = 2.2015 A-0.8172dex is developed to quantify the relationship between the Adex and ϕc. A comparison of our numerical simulations, published data, and predictions verifies the reliability and universality of the fitting model. Subsequently, a resistor network searching algorithm (RNSA) is proposed to construct the random resistor network model (RRNM). A power law model, denoted as is developed to evaluate the effects of the normalized reduced density (η - ηc)/ηc on the effective conductivity σeff of CPC. Comparing our predictions with data from the literature and our simulation verifies the reliability of our RNSA and the fitting model. This paper's methodology and findings may provide a theoretical hint for the CPC's design.

Virial coefficients of hard hyperspherocylinders in R^{4}: Influence of the aspect ratio

We provide second- to sixth-order virial coefficients of hard hyperspherocylinders in dependence on their aspect ratio ν. Virial coefficients of an anisotropic geometry in four dimensions are calculated employing an optimized Mayer-sampling algorithm. As the second virial coefficient of a hard particle is identical to its excluded hypervolume, the numerically obtained second virial coefficients can be compared to analytical relations for the excluded hypervolume based on geometric measures of the respective, convex geometry in dependence on its aspect ratio ν.

Elliptic percolation model for predicting the electrical conductivity of graphene-polymer composites

Graphene-based polymers exhibit a conductive microstructure formed by aggregates in a matrix which drastically enhances their transmitting properties. We develop a new numerical framework for predicting the electrical conductivity based on continuum percolation theory in a two dimensional stochastically-generated medium. We analyze the role of the flake shape and its aspect ratio and consequently predict the onset of percolation based on the particle density and the domain scale. Simultaneously, we have performed experiments and have achieved very high electrical conductivity for such composites compared to other film fabrication techniques, which have verified the results of computing the homogenized electrical conductivity. As well, the proximity to and a comparison with other analytical models and other experimental techniques are presented. The numerical model can predict the composite transmitting conductivity in a larger range of particle geometry. Such quantification is exceedingly useful for effective utilization and optimization of graphene filler densities and their spatial distribution during manufacturing.

Modeling multicellular dynamics regulated by extracellular-matrix-mediated mechanical communication via active particles with polarized effective attraction

Collective cell migration is crucial to many physiological and pathological processes such as embryo development, wound healing, and cancer invasion. Recent experimental studies have indicated that the active traction forces generated by migrating cells in a fibrous extracellular matrix (ECM) can mechanically remodel the ECM, giving rise to bundlelike mesostructures bridging individual cells. Such fiber bundles also enable long-range propagation of cellular forces, leading to correlated migration dynamics regulated by the mechanical communication among the cells. Motivated by these experimental discoveries, we develop an active-particle model with polarized effective attractions (APPA) to investigate emergent multicellular migration dynamics resulting from ECM-mediated mechanical communications. In particular, the APPA model generalizes the classic active-Brownian-particle (ABP) model by imposing a pairwise polarized attractive force between the particles, which depends on the instantaneous dynamic states of the particles and mimics the effective mutual pulling between the cells via the fiber bundle bridge. The APPA system exhibits enhanced aggregation behaviors compared to the classic ABP system, and the contrast is more apparent at lower particle densities and higher rotational diffusivities. Importantly, in contrast to the classic ABP system where the particle velocities are not correlated for all particle densities, the high-density phase of the APPA system exhibits strong dynamic correlations, which are characterized by the slowly decaying velocity correlation functions with a correlation length comparable to the linear size of the high-density phase domain (i.e., the cluster of particles). The strongly correlated multicellular dynamics predicted by the APPA model is subsequently verified in in vitro experiments using MCF-10A cells. Our studies indicate the importance of incorporating ECM-mediated mechanical coupling among the migrating cells for appropriately modeling emergent multicellular dynamics in complex microenvironments.

Calculating the Electrical Conductivity of Graphene Nanoplatelet Polymer Composites by a Monte Carlo Method

Electrical conductivity is one of several outstanding features of graphene–polymer nanocomposites, but calculations of this property require the intricate features of the underlying conduction processes to be accounted for. To this end, a novel Monte Carlo method was developed. We first established a randomly distributed graphene nanoplatelet (GNP) network. Then, based on the tunneling effect, the contact conductance between the GNPs was calculated. Coated surfaces (CSs) were next set up to calculate the current flow from the GNPs to the polymer. Using the equipotential approximation, the potentials of the GNPs and CSs met Kirchhoff’s current law, and, based on Laplace equation, the potential of the CSs was obtained from the potential of the GNP by the walk-on-spheres (WoS) method. As such, the potentials of all GNPs were obtained, and the electrical conductivity of the GNP polymer composites was calculated. The barrier heights, polymer conductivity, diameter and thickness of the GNP determining the electrical conductivity of composites were studied in this model. The calculated conductivity and percolation threshold were shown to agree with experimental data.

Scalable Fabrication of Porous Microchannel Nerve Guidance Scaffolds with Complex Geometries

Microchannel scaffolds accelerate nerve repair by guiding growing neuronal processes across injury sites. Although geometry, materials chemistry, stiffness, and porosity have been shown to influence nerve growth within nerve guidance scaffolds, independent tuning of these properties in a high-throughput manner remains a challenge. Here, fiber drawing is combined with salt leaching to produce microchannels with tunable cross sections and porosity. This technique is applicable to an array of biochemically inert polymers, and it delivers hundreds of meters of porous microchannel fibers. Employing these fibers as filaments during 3D printing enables the production of microchannel scaffolds with geometries matching those of biological nerves, including branched topographies. Applied to sensory neurons, fiber-based porous microchannels enhance growth as compared to non-porous channels with matching materials and geometries. The combinatorial scaffold fabrication approach may advance the studies of neural regeneration and accelerate the development of nerve repair devices.

Continuum percolation of congruent overlapping polyhedral particles: Finite-size-scaling analysis and renormalization-group method

The continuum percolation of randomly orientated overlapping polyhedral particles, including tetrahedron, cube, octahedron, dodecahedron, and icosahedron, was analyzed by Monte Carlo simulations. Two numerical strategies, (1) a Monte Carlo finite-size-scaling analysis and (2) a real-space Monte Carlo renormalization-group method, were, respectively, presented in order to determine the percolation threshold (e.g., the critical volume fraction ϕ_{c} or the critical reduced number density η_{c}), percolation transition width Δ, and correlation-length exponent ν of the polyhedral particles. The results showed that ϕ_{c} (or η_{c}) and Δ increase in the following order: tetrahedron < cube < octahedron < dodecahedron < icosahedron. In other words, both the percolation threshold and percolation transition width increase with the number of faces of the polyhedral particles as the shape becomes more "spherical." We obtained the statistical values of ν for the five polyhedral shapes and analyzed possible errors resulting in the present numerical values ν deviated from the universal value of ν=0.88 reported in literature. To validate the simulations, the corresponding excluded-volume bounds on the percolation threshold were obtained and compared with the numerical results. This paper has practical applications in predicting effective transport and mechanical properties of porous media and composites.

Continuum percolation-based tortuosity and thermal conductivity of soft superball systems: shape dependence from octahedra via spheres to cubes

Understanding the effect of particle shape on the percolation threshold, tortuosity and thermal conductivity of soft (geometrical overlapping) particle systems is very crucial for the design and optimization of such materials, including colloids, polymers, and porous and fracture media. In this work, we first combine the excluded-volume theory with the Monte Carlo simulations to determine the percolation threshold for a family of soft superballs, the shape of which interpolates between octahedra and cubes via spheres. Then, we propose two continuum percolation-based models to respectively obtain the tortuosity and effective thermal conductivity of soft superball systems considering their percolation behavior, where monodisperse overlapping superballs are uniformly distributed in a homogeneous solid matrix. Specifically, both models cover the whole feasible range of superball volume fractions, including near the percolation threshold. Comparison with extensive experimental, numerical and theoretical results confirms that the present models are capable of precisely predicting the percolation threshold, tortuosity and thermal conductivity of such systems. Furthermore, we apply the proposed models to probe the influence of particle shape on these important parameters. Our results show that the decreasing percolation threshold and tortuosity as soft particles become more anisotropic is consistent with increasing conductivity. It suggests that the anisotropic-shaped inclusion phase is more conducting than the spherical inclusion phase. The present theoretical strategies and conclusions may provide sound guidance for the synthesis of colloidal and polymer superballs.

Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems

With the advances in artificial particle synthesis, it is possible to create particles with unique shapes. Particle shape becomes a feasible parameter for tuning the percolation behavior. How to accurately predict the percolation threshold by particle characteristics for arbitrary particles has aroused great interest. Towards this end, a versatile family of cuboidlike particles and a numerical contact detection algorithm for these particles are presented here. Then, combining with percolation theory, the continuum percolation of randomly distributed overlapping cuboidlike particles is studied. The global percolation threshold ϕ_{c} of overlapping particles with broad ranges of the shape parameter m in [1.0,+∞) and aspect ratio a/b in [0.1, 10.0] is computed via a finite-size scaling technique. Using the generalized excluded-volume approximation, an analytical formula is proposed to quantify the dependence of ϕ_{c} on the parameters m and a/b, and its reliability is verified. The results reveal that the percolation threshold ϕ_{c} of overlapping cuboidlike particles is heavily dependent on the shapes of particles, and much more sensitive to a/b than m. As the cuboidlike particles become spherical (i.e., m=1.0 and a/b=1.0), the maximum threshold ϕ_{c,max} can be obtained.

Continuum percolation of congruent overlapping spherocylinders

Continuum percolation of randomly orientated congruent overlapping spherocylinders (composed of cylinder of height H with semispheres of diameter D at the ends) with aspect ratio α=H/D in [0,∞) is studied. The percolation threshold ϕ_{c}, percolation transition width Δ, and correlation-length critical exponent ν for spherocylinders with α in [0, 200] are determined with a high degree of accuracy via extensive finite-size scaling analysis. A generalized excluded-volume approximation for percolation threshold with an exponent explicitly depending on both aspect ratio and excluded volume for arbitrary α values in [0,∞) is proposed and shown to yield accurate predictions of ϕ_{c} for an extremely wide range of α in [0, 2000] based on available numerical and experimental data. We find ϕ_{c} is a universal monotonic decreasing function of α and is independent of the effective particle size. Our study has implications in percolation theory for nonspherical particles and composite material design.

Third-order thermo-mechanical properties for packs of Platonic solids using statistical micromechanics

Obtaining an accurate higher order statistical description of heterogeneous materials and using this information to predict effective material behaviour with high fidelity has remained an outstanding problem for many years. In a recent letter, Gillman & Matouš (2014 Phys. Lett. A 378, 3070-3073. ()) accurately evaluated the three-point microstructural parameter that arises in third-order theories and predicted with high accuracy the effective thermal conductivity of highly packed material systems. Expanding this work here, we predict for the first time effective thermo-mechanical properties of granular Platonic solid packs using third-order statistical micromechanics. Systems of impenetrable and penetrable spheres are considered to verify adaptive methods for computing n-point probability functions directly from three-dimensional microstructures, and excellent agreement is shown with simulation. Moreover, a significant shape effect is discovered for the effective thermal conductivity of highly packed composites, whereas a moderate shape effect is exhibited for the elastic constants.

Characterization of maximally random jammed sphere packings: Voronoi correlation functions

We characterize the structure of maximally random jammed (MRJ) sphere packings by computing the Minkowski functionals (volume, surface area, and integrated mean curvature) of their associated Voronoi cells. The probability distribution functions of these functionals of Voronoi cells in MRJ sphere packings are qualitatively similar to those of an equilibrium hard-sphere liquid and partly even to the uncorrelated Poisson point process, implying that such local statistics are relatively structurally insensitive. This is not surprising because the Minkowski functionals of a single Voronoi cell incorporate only local information and are insensitive to global structural information. To improve upon this, we introduce descriptors that incorporate nonlocal information via the correlation functions of the Minkowski functionals of two cells at a given distance as well as certain cell-cell probability density functions. We evaluate these higher-order functions for our MRJ packings as well as equilibrium hard spheres and the Poisson point process. It is shown that these Minkowski correlation and density functions contain visibly more information than the corresponding standard pair-correlation functions. We find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ packings, consistent with previous findings for other pair correlations [A. Donev et al., Phys. Rev. Lett. 95, 090604 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.090604], indicating that large-scale volume fluctuations are suppressed by accompanying large Voronoi cells with small cells, and vice versa. In contrast to the aforementioned local Voronoi statistics, the correlation functions of the Voronoi cells qualitatively distinguish the structure of MRJ sphere packings (prototypical glasses) from that of not only the Poisson point process but also the correlated equilibrium hard-sphere liquids. Moreover, while we did not find any perfect icosahedra (the locally densest possible structure in which a central sphere contacts 12 neighbors) in the MRJ packings, a preliminary Voronoi topology analysis indicates the presence of strongly distorted icosahedra.