Continuum percolation of polydisperse rods in quadrupole fields: Theory and simulations

Re-entrant percolation in active Brownian hard disks

Non-equilibrium clustering and percolation are investigated in an archetypal model of two-dimensional active matter using dynamic simulations of self-propelled Brownian repulsive particles. We concentrate on the single-phase region up to moderate levels of activity, before motility-induced phase separation (MIPS) sets in. Weak activity promotes cluster formation and lowers the percolation threshold. However, driving the system further out of equilibrium partly reverses this effect, resulting in a minimum in the critical density for the formation of system-spanning clusters and introducing re-entrant percolation as a function of activity in the pre-MIPS regime. This non-monotonic behaviour arises from competition between activity-induced effective attraction (which eventually leads to MIPS) and activity-driven cluster breakup. Using an adapted iterative Boltzmann inversion method, we derive effective potentials to map weakly active cases onto a passive (equilibrium) model with conservative attraction, which can be characterised by Monte Carlo simulations. While the active and passive systems have practically identical radial distribution functions, we find decisive differences in higher-order structural correlations, to which the percolation threshold is highly sensitive. For sufficiently strong activity, no passive pairwise potential can reproduce the radial distribution function of the active system.

Synergistic effect in improving the electrical conductivity in polymer nanocomposites by mixing spherical and rod-shaped fillers

In this work, coarse-grained molecular dynamics simulation is adopted to investigate the effect of hybrid fillers [nanospheres (NSs) and nanorods (NRs)] on the conductive probability of polymer nanocomposites (PNCs) in the quiescent state and under the shear field. The percolation threshold gradually rises as the volume fraction ratio (α) of NSs to all the fillers increases in the quiescent state. Compared to the NSs, the greater number of beads in the NRs help them connect to other NRs to form the conductive network. Meanwhile, compared to NSs, more NRs participate in building the conductive network. A transition from the synergistic effect to the antagonistic effect occurs as the NS-NR tunneling distance is reduced. Furthermore, the shear field induces a more direct aggregation structure of NSs, which act as linkers between fillers to protect the conductive network. This result is confirmed by the fact that more NSs occupy the conductive network under the shear field. As a result, the percolation threshold declines with increasing shear rate. Finally, compared to in the quiescent state, the percolation threshold increases at α = 0.0 and remains nearly unchanged for α = 0.25 under the shear field, while it gradually decreases for α≥ 0.5. In total, the results further our understanding of how to realize the synergistic effect between NSs and NRs when forming a conductive network of PNCs.

Irreversible bilayer adsorption of straight semirigid rods on two-dimensional square lattices: Jamming and percolation properties

Numerical simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of straight semirigid rods adsorbed on two-dimensional square lattices. The depositing objects can be adsorbed on the surface forming two layers. The filling of the lattice is carried out following a generalized random sequential adsorption (RSA) mechanism. In each elementary step, (i) a set of k consecutive nearest-neighbor sites (aligned along one of two lattice axes) is randomly chosen and (ii) if each selected site is either empty or occupied by a k-mer unit in the first layer, then a new k-mer is then deposited onto the lattice. Otherwise, the attempt is rejected. The process starts with an initially empty lattice and continues until the jamming state is reached and no more objects can be deposited due to the absence of empty site clusters of appropriate size and shape. A wide range of values of k (2≤k≤64) is investigated. The study of the kinetic properties of the system shows that (1) the jamming coverage θ_{j,k} is a decreasing function with increasing k, with θ_{j,k→∞}=0.7299(21) the limit value for infinitely long k-mers and (2) the jamming exponent ν_{j} remains close to 1, regardless of the size k considered. These findings are discussed in terms of the lattice dimensionality and number of sites available for adsorption. The dependence of the percolation threshold θ_{c,k} as a function of k is also determined, with θ_{c,k}=A+Bexp(-k/C), where A=θ_{c,k→∞}=0.0457(68) is the value of the percolation threshold by infinitely long k-mers, B=0.276(25), and C=14(2). This monotonic decreasing behavior is completely different from that observed for the standard problem of straight rods on square lattices, where the percolation threshold shows a nonmonotonic k-mer size dependence. The differences between the results obtained from bilayer and monolayer phases are explained on the basis of the transversal overlaps between rods occurring in the bilayer problem. This effect (which we call a "cross-linking effect"), its consequences on the filling kinetics, and its implications in the field of conductivity of composites filled with elongated particles (or fibers) are discussed in detail. Finally, the precise determination of the critical exponents ν, β, and γ indicates that, although the increasing in the width of the deposited layer drastically affects the behavior of the percolation threshold with k and other critical properties (such as the crossing points of the percolation probability functions), it does not alter the nature of the percolation transition occurring in the system. Accordingly, the bilayer model belongs to the same universality class as two-dimensional standard percolation model.

Shape, geometric percolation, and electrical conductivity of clusters in suspensions of hard platelets

Using Monte Carlo simulations, we investigate how geometric percolation and electrical conductivity in suspensions of hard conducting platelets are affected by the addition of platelets and their degree of spontaneous alignment. In our simulation results for aspect ratios 10, 25, and 50, we consistently observe a monotonically decreasing percolation threshold as a function of volume fraction, i.e., the addition of particles always aids percolation. In the nematic phase, the distribution of particles inside the percolating clusters becomes less spherically symmetric and the aspect ratio of the clusters increases. However, the clusters are also anisotropically shaped in the isotropic phase, although their aspect ratio remains constant as a function of volume fraction and is only weakly dependent on the particle aspect ratio. Mapping the percolating clusters of platelets to linear resistor networks, and assigning unit conductance to all connections, we find a constant conductivity both across the isotropic-nematic transition and in the respective stable phases. This behavior is consistent with the other observed topological properties of the networks, namely, the average path length, average number of contacts per particle, and the Kirchhoff index, which all remain constant and unaffected by both the addition of particles and the degree of alignment of their suspension. In contrast, using an anisotropic conductance model that explicitly accounts for the relative orientation of the particles, the network conductivity decreases with increasing volume fraction in the isotropic, and further diminishes at the onset of the nematic while preserving the same trend deep in the nematic. Hence, our observations consistently suggest that, unlike for rodlike fillers, the network structures that arise from platelet suspensions are not very sensitive to the particle aspect ratio or to alignment. Hence platelets are not as versatile as fillers for dispersion in conductive composite materials as rods.

Geometric percolation of hard nanorods: The interplay of spontaneous and externally induced uniaxial particle alignment

We present a numerical study on geometric percolation in liquid dispersions of hard slender colloidal particles subject to an external orienting field. In the formulation and liquid-state processing of nanocomposite materials, particle alignment by external fields such as electric, magnetic, or flow fields is practically inevitable and often works against the emergence of large nanoparticle networks. Using continuum percolation theory in conjunction with Onsager theory, we investigate how the interplay between externally induced alignment and the spontaneous symmetry breaking of the uniaxial nematic phase affects cluster formation in nanoparticle dispersions. It is known that particle alignment by means of a density increase or by an external field may result in a breakdown of an already percolating network. As a result, percolation can be limited to a small region of the phase diagram only. Here, we demonstrate that the existence and shape of such a "percolation island" in the phase diagram crucially depends on the connectivity length-a critical distance defining direct connections between neighboring particles. For some values of the connectivity range, we observe unusual re-entrance effects, in which a system-spanning network forms and breaks down multiple times with increasing particle density.

Unusual geometric percolation of hard nanorods in the uniaxial nematic liquid crystalline phase

We investigate by means of continuum percolation theory and Monte Carlo simulations how spontaneous uniaxial symmetry breaking affects geometric percolation in dispersions of hard rodlike particles. If the particle aspect ratio exceeds about 20, percolation in the nematic phase can be lost upon adding particles to the dispersion. This contrasts with percolation in the isotropic phase, where a minimum particle loading is always required to obtain system-spanning clusters. For sufficiently short rods, percolation in the uniaxial nematic mimics that of the isotropic phase, where the addition of particles always aids percolation. For aspect ratios between 20 and infinity, but not including infinity, we find reentrance behavior: percolation in the low-density nematic may be lost upon increasing the amount of nanofillers but can be regained by the addition of even more particles to the suspension. Our simulation results for aspect ratios of 5, 10, 20, 50, and 100 strongly support our theoretical predictions, with almost quantitative agreement. We show that a different closure of the connectedness Ornstein-Zernike equation, inspired by scaled particle theory, is as least as accurate in predicting the percolation threshold as the Parsons-Lee closure, which effectively describes the impact of many-body direct contacts.

Percolation of aligned rigid rods on two-dimensional triangular lattices

The percolation behavior of aligned rigid rods of length k (k-mers) on two-dimensional triangular lattices has been studied by numerical simulations and finite-size scaling analysis. The k-mers, containing k identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The connectivity analysis was carried out by following the probability R_{L,k}(p) that a lattice composed of L×L sites percolates at a concentration p of sites occupied by particles of size k. The results, obtained for k ranging from 2 to 80, showed that the percolation threshold p_{c}(k) exhibits a increasing function when it is plotted as a function of the k-mer size. The dependence of p_{c}(k) was determined, being p_{c}(k)=A+B/(C+sqrt[k]), where A=p_{c}(k→∞)=0.582(9) is the value of the percolation threshold by infinitely long k-mers, B=-0.47(0.21), and C=5.79(2.18). This behavior is completely different from that observed for square lattices, where the percolation threshold decreases with k. In addition, the effect of the anisotropy on the properties of the percolating phase was investigated. The results revealed that, while for finite systems the anisotropy of the deposited layer favors the percolation along the parallel direction to the alignment axis, in the thermodynamic limit, the value of the percolation threshold is the same in both parallel and transversal directions. Finally, an exhaustive study of critical exponents and universality was carried out, showing that the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered.

Clustering and anisotropic correlated percolation in polar flocks

We study clustering and percolation phenomena in the Vicsek model, taken here in its capacity of prototypical model for dry aligning active matter. Our results show that the order-disorder transition is not related in any way to a percolation transition, contrary to some earlier claims. We study geometric percolation in each of the phases at play, but we mostly focus on the ordered Toner-Tu phase, where we find that the long-range correlations of density fluctuations give rise to an anisotropic percolation transition.

Benefits of Fractal Approaches in Solid Dosage Form Development

Pharmaceutical formulations are complex systems consisting of active pharmaceutical ingredient(s) and a number of excipients selected to provide the intended performance of the product. The understanding of materials' properties and technological processes is a requirement for building quality into pharmaceutical products. Such understanding is gained mostly from empirical correlations of material and process factors with quality attributes of the final product. However, it seems also important to gain knowledge based on mechanistic considerations. Promising is here to study morphological and/or topological characteristics of particles and their aggregates. These geometric aspects must be taken into account to better understand how product attributes emerge from raw materials, which includes, for example, mechanical tablet properties, disintegration or dissolution behavior. Regulatory agencies worldwide are promoting the use of physical models in pharmaceutics to design quality into a final product. This review deals with pharmaceutical applications of theoretical models, focusing on percolation theory, fractal, and multifractal geometry. The use of these so-called fractal approaches improves the understanding of different aspects in the development of solid dosage forms, for example by identifying critical drug and excipient concentrations, as well as to study effects of heterogeneity on dosage form performance. The aim is to link micro- and macrostructure to the emerging quality attributes of the pharmaceutical solid dosage forms as a strategy to enhance mechanistic understanding and to advance pharmaceutical development and manufacturing processes.

Connectivity, Not Density, Dictates Percolation in Nematic Liquid Crystals of Slender Nanoparticles

We show by means of continuum theory and simulations that geometric percolation in uniaxial nematics of hard slender particles is fundamentally different from that in isotropic dispersions. In the nematic, percolation depends only very weakly on the density and is, in essence, determined by a distance criterion that defines connectivity. This unexpected finding has its roots in the nontrivial coupling between the density and the degree of orientational order that dictate the mean number of particle contacts. Clusters in the nematic are much longer than wide, suggesting the use of nematics for nanocomposites with strongly anisotropic transport properties.