Connectedness percolation of hard convex polygonal rods and platelets

Continuum percolation in colloidal dispersions of hard nanorods in external axial and planar fields

We present a theoretical study on continuum percolation of rod-like colloidal particles in the presence of axial and planar quadrupole fields. Our work is based on a self-consistent numerical treatment of the connectedness Ornstein-Zernike equation, in conjunction with the Onsager equation that describes the orientational distribution function of particles interacting via a hard-core repulsive potential. Our results show that axial and planar quadrupole fields both in principle increase the percolation threshold. By how much depends on a combination of the field strength, the concentration, the aspect ratio of the particles, and percolation criterion. We find that the percolated state can form and break down multiple times with increasing concentration, i.e., it exhibits re-entrance behaviour. Finally, we show that planar fields may induce a high degree of triaxiality in the shape of particle clusters that in actual materials may give rise to highly anisotropic conductivity properties.

Effect of Nanorod Physical Roughness on the Aggregation and Percolation of Nanorods in Polymer Nanocomposites

Using molecular dynamics simulations, we elucidate the effect of nanorod roughness on nanorod aggregation, dispersion, and percolation in polymer nanocomposites (PNCs). By choosing coarse-grained models that enable systematic variation of the nanorod roughness and by selecting purely repulsive pairwise interactions for nanorods and polymer chains, we show how nanorod roughness affects the entropic driving forces for various PNC morphologies. At this entropically driven limit, we find that increasing nanorod roughness hinders nanorod aggregation and promotes nanorod percolation in the polymer melt. As nanorod roughness increases, the nanorod volume fraction needed to induce nanorod aggregation also increases. Increasing nanorod roughness increases the configurational entropy of the polymer chains and lowers the entropically induced depletion attraction between nanorods.

Percolation of rigid fractal carbon black aggregates

We examine network formation and percolation of carbon black by means of Monte Carlo simulations and experiments. In the simulation, we model carbon black by rigid aggregates of impenetrable spheres, which we obtain by diffusion-limited aggregation. To determine the input parameters for the simulation, we experimentally characterize the micro-structure and size distribution of carbon black aggregates. We then simulate suspensions of aggregates and determine the percolation threshold as a function of the aggregate size distribution. We observe a quasi-universal relation between the percolation threshold and a weighted average radius of gyration of the aggregate ensemble. Higher order moments of the size distribution do not have an effect on the percolation threshold. We conclude further that the concentration of large carbon black aggregates has a stronger influence on the percolation threshold than the concentration of small aggregates. In the experiment, we disperse the carbon black in a polymer matrix and measure the conductivity of the composite. We successfully test the hypotheses drawn from simulation by comparing composites prepared with the same type of carbon black before and after ball milling, i.e., on changing only the distribution of aggregate sizes in the composites.

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.

Percolation thresholds for discorectangles: Numerical estimation for a range of aspect ratios

Using Monte Carlo simulation, we have studied the percolation of discorectangles. Also known as stadiums or two-dimensional spherocylinders, a discorectangle is a rectangle with semicircles at a pair of opposite sides. Scaling analysis was performed to obtain the percolation thresholds in the thermodynamic limits. We found that (i) for the two marginal aspect ratios ɛ=1 (disc) and ɛ→∞ (stick) the percolation thresholds coincide with known values within the statistical error and (ii) for intermediate values of ɛ the percolation threshold lies between the percolation thresholds for ellipses and rectangles and approaches the latter as the aspect ratio increases.