Finite dimension and particle heterogeneous DLAs.

We study heterogeneous Diffusion Limited Aggregates (DLAs) i.e. those formed by a mixture, in different proportions, of 4-legged and 2-legged particles. We fixed the total number of particles, let the proportions vary, and computed their finite dimension, a recent addition to the list of "fractal" dimensions. At one extreme, when all particles are 4-legged, the corresponding DLAs are complex, fractal structures whose appearance resembles very much that of the DLAs that occur in Nature. At the other extreme, when almost all particles are 2-legged, the DLAs lose much of their complexity and acquire long rectilinear stretches so that their appearance resembles more and more the structure of the underlying lattice. We expected the complexity in between would decrease monotonically, and this would be reflected in the finite dimension of the corresponding DLAs. However, the finite dimension first increases and then, when the proportion of 4-legged to 2-legged particles is about 30 to 70, starts decreasing. In the paper, we study and explain the mechanisms behind this unexpected, counter-intuitive behaviour.

Finite dimension unravels the structural features at the glass transition.

Physics Nobel Prize winner P.W. Anderson famously wrote in 1995: "The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of the glass and the glass transition". Despite much effort in the intervening years, the problem is still unsolved. We contribute a novel mathematical approach to this problem. The main new ingredient is finite dimension, a recently introduced "fractal" dimension defined only for finite sets. Our methods sharply distinguish the glass transition temperature and give hints as to the structural changes that occur in the transition.

Order and disorder in the adsorption model of repulsively interacting binary mixtures on triangular lattices: theory and Monte Carlo simulations.

The problem of interacting binary mixtures adsorbed on triangular lattices has been studied by means of ground-state (GS) calculations, Monte Carlo (MC) simulations and exact counting of configurations on finite cells [the so-called cluster-exact approximation (CA)]. We focus on the case of repulsive intraspecies couplings and null interspecies interaction, where a rich variety of ordered phases is found in the adsorbed layer. Each surface structure is separated from a disordered state by a phase transition occurring at a finite temperature. The ordered phases are identified by means of GS analysis, and their dependence on temperature is studied by MC simulations and CA. Total and partial adsorption isotherms are obtained for values of the lateral interactions in the different regions of the phase diagram. MC results are compared with CA calculations. A very good coincidence is obtained between both methods, supporting the validity of the exact counting of states on finite cells as an adequate approach to describe the behavior of multicomponent competitive adsorption with adsorbate-adsorbate interactions.

Adsorption of interacting particles on bivariate diffusion-limited aggregates.

The adsorption of pairwise interacting particles on fractal surfaces has been studied by grand canonical Monte Carlo simulations. The substrate is built from a mixture of two types of objects: (i) objects with two bonds and (ii) objects with four bonds. These objects move on a square lattice, according to diffusion-limited aggregation (DLA) rules, and stick to each other only if they have a free bond pointing at each other and, of course, are first neighbors of each other. The resulting substrate, which is named as bivariate diffusion-limited aggregate (BDLA), is a fractal structure composed by two bonds units with fraction [Formula: see text] and four bonds units with concentration [Formula: see text]. Different surface morphologies are obtained by varying [Formula: see text] and [Formula: see text]. In the limit case of [Formula: see text] and [Formula: see text], the standard DLA model is recovered. In addition, repulsive lateral interactions between adsorbed particles are considered. Adsorption isotherms and differential heats of adsorption are calculated for different values of the parameters of the system. In the case of high repulsive couplings, a wide variety of structural orderings are observed in the adlayer. The main characteristics of these ordered phases are discussed in terms of the topological properties of the bivariate aggregates.