Percolation in an interactive cluster-growth model

Percolation and jamming of linear k-mers on a square lattice with defects: Effect of anisotropy

Using the Monte Carlo simulation, we study the percolation and jamming of oriented linear k-mers on a square lattice that contains defects. The point defects with a concentration d are placed randomly and uniformly on the substrate before deposition of the k-mers. The general case of unequal probabilities for orientation of depositing of k-mers along different directions of the lattice is analyzed. Two different relaxation models of deposition that preserve the predetermined order parameter s are used. In the relaxation random sequential adsorption (RRSA) model, the deposition of k-mers is distributed over different sites on the substrate. In the single-cluster relaxation (RSC) model, the single cluster grows by the random accumulation of k-mers on the boundary of the cluster (Eden-like model). For both models, a suppression of growth of the infinite (percolation) cluster at some critical concentration of defects d(c) is observed. In the zero-defect lattices, the jamming concentration p(j) (RRSA model) and the density of single clusters p(s) (RSC model) decrease with increasing length k-mers and with a decrease in the order parameter. For the RRSA model, the value of d(c) decreases for short k-mers (k<16) as the value of s increases. For k=16 and 32, the value of d(c) is almost independent of s. Moreover, for short k-mers, the percolation threshold is almost insensitive to the defect concentration for all values of s. For the RSC model, the growth of clusters with ellipselike shapes is observed for nonzero values of s. The density of the clusters p(s) at the critical concentration of defects d(c) depends in a complex manner on the values of s and k. An interesting finding for disordered systems (s=0) is that the value of p(s) tends towards zero in the limits of the very long k-mers, k→∞, and very small critical concentrations d(c)→0. In this case, the introduction of defects results in a suppression of k-mer stacking and in the formation of empty or loose clusters with very low density. On the other hand, denser clusters are formed for ordered systems with p(s)≈0.065 at s=0.5 and p(s)≈0.38 at s=1.0.

The ultimate fate of supercooled liquids

In recent years it has become widely accepted that a dynamical length scale ξ(α) plays an important role in supercooled liquids near the glass transition. We examine the implications of the interplay between the growing ξ(α) and the size of the crystal nucleus, ξ(M), which shrinks on cooling. We argue that at low temperatures where ξ(α) > ξ(M) a new crystallization mechanism emerges, enabling rapid development of a large scale web of sparsely connected crystallinity. Though we predict this web percolates the system at too low a temperature to be easily seen in the laboratory, there are noticeable residual effects near the glass transition that can account for several previously observed unexplained phenomena of deeply supercooled liquids including Fischer clusters and anomalous crystal growth near T(g).

Failure process of a bundle of plastic fibers

We present an extension of fiber bundle models considering that failed fibers still carry a fraction 0 < or = alpha < or = 1 of their failure load. The value of alpha interpolates between the perfectly brittle failure (alpha = 0) and perfectly plastic behavior (alpha = 1) of fibers. We show that the finite load bearing capacity of broken fibers has a substantial effect on the failure process of the bundle. In the case of global load sharing it is found that for alpha --> 1 the macroscopic response of the bundle becomes perfectly plastic with a yield stress equal to the average fiber strength. On the microlevel, the size distribution of avalanches has a crossover from a power law of exponent approximately 2.5 to a faster exponential decay. For localized load sharing, computer simulations revealed a sharp transition at a well-defined value alpha(c) from a phase where macroscopic failure occurs due to localization as a consequence of local stress enhancements, to another one where the disordered fiber strength dominates the damage process. Analyzing the microstructure of damage, the transition proved to be analogous to percolation. At the critical point alpha(c), the spanning cluster of damage is found to be compact with a fractal boundary. The distribution of bursts of fiber breakings shows a power-law behavior with a universal exponent approximately 1.5 equal to the mean-field exponent of fiber bundles of critical strength distributions. The model can be relevant to understand the shear failure of glued interfaces where failed regions can still transmit load by remaining in contact.

Computer simulation of electrical conductivity of colloidal dispersions during aggregation

The computation approach to the simulation of electrical conductivity of colloidal dispersions during aggregation is considered. We use the two-dimensional diffusion-limited aggregation model with multiple-seed growth. The particles execute a random walk, but lose their mobility after contact with the growing clusters or seeds. The two parameters that control the aggregation are the initial concentration of free particles in the system p and the concentration of seeds psi. The case of psi=1, when all the particles are the immobile seeds, corresponds with the usual random percolation problem. The other limiting case of psi=0, when all the particles walk randomly, corresponds to the dynamical percolation problem. The calculation of electrical conductivity and cluster analysis were done with the help of the algorithms of Frank-Lobb and Hoshen-Kopelman. It is shown that the percolation concentration phi c decreases from 0.5927 at psi=1 to 0 at psi --> 0. Scaling analysis was applied to study exponents of correlation length v and of conductivity t. For all psi>0 this model shows universal behavior of classical 2d random percolation with v approximately t approximately 4/3. The electrical conductivity sigma of the system increases during aggregation reaching up to a maximum at the final stage. The concentration dependence of conductivity sigma(phi) obeys the general effective medium equation with apparent exponent ta(psi) that exceeds t. The kinetics of electrical conductivity changes during the aggregation is discussed. In the range of concentration Pc(phi)<p<0.9527 the time of percolation cluster formation pi c decreases with increasing phi.

Susceptibility and percolation in two-dimensional random field Ising magnets

The ground-state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations. First we show that the ferromagnetism, which exists for small system sizes, vanishes with a large excitation at a random field strength-dependent length scale. This breakup length scale L(b) scales exponentially with the squared random field, exp(A/delta(2)). By adding an external field H, we then study the susceptibility in the ground state. If L>L(b), domains melt continuously and the magnetization has a smooth behavior, independent of system size, and the susceptibility decays as L-2. We define a random field strength-dependent critical external field value +/-H(c)(delta) for the up and down spins to form a percolation type of spanning cluster. The percolation transition is in the standard short-range correlated percolation universality class. The mass of the spanning cluster increases with decreasing Delta and the critical external field approaches zero for vanishing random field strength, implying the critical field scaling (for Gaussian disorder) H(c) approximately (delta-delta(c))(delta), where delta(c)=1.65+/-0.05 and delta=2.05+/-0.10. Below Delta(c) the systems should percolate even when H=0. This implies that even for H=0 above L(b) the domains can be fractal at low random fields, such that the largest domain spans the system at low random field strength values and its mass has the fractal dimension of standard percolation D(f)=91/48. The structure of the spanning clusters is studied by defining red clusters, in analogy to the "red sites" of ordinary site percolation. The sizes of red clusters define an extra length scale, independent of L.