Universal Behavior of Connectivity Properties in Fractal Percolation Models

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of lambda that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions d greater than or equal to 2. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for d=2 we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.Comment: 29 pages, 4 figure

A grid of chemical evolution models along the Hubble Sequence

We have computed a grid of multiphase chemical evolution models whose results are valid for any spiral galaxy, using as input the maximum rotation velocity and the morphological type or index T.Comment: 2 pag., contribution to the conference Cosmic Evolution (Paris, Nov. 2000

The Quantum Compass Model on the Square Lattice

Using exact diagonalizations, Green's function Monte Carlo simulations and high-order perturbation theory, we study the low-energy properties of the two-dimensional spin-1/2 compass model on the square lattice defined by the Hamiltonian $H = - \sum_{\bm{r}} (J_x \sigma_{\bm{r}}^x \sigma_{\bm{r} + \bm{e}_x}^x + J_z \sigma_{\bm{r}}^z \sigma_{\bm{r} + \bm{e}_z}^z)$. When $J_x\ne J_z$, we show that, on clusters of dimension $L\times L$, the low-energy spectrum consists of $2^L$ states which collapse onto each other exponentially fast with $L$, a conclusion that remains true arbitrarily close to $J_x=J_z$. At that point, we show that an even larger number of states collapse exponentially fast with $L$ onto the ground state, and we present numerical evidence that this number is precisely $2\times 2^L$. We also extend the symmetry analysis of the model to arbitrary spins and show that the two-fold degeneracy of all eigenstates remains true for arbitrary half-integer spins but does not apply to integer spins, in which cases eigenstates are generically non degenerate, a result confirmed by exact diagonalizations in the spin-1 case. Implications for Mott insulators and Josephson junction arrays are briefly discussed.Comment: 8 pages, 8 figure

Identification of an RVB liquid phase in a quantum dimer model with competing kinetic terms

Starting from the mean-field solution of a spin-orbital model of LiNiO$_2$, we derive an effective quantum dimer model (QDM) that lives on the triangular lattice and contains kinetic terms acting on 4-site plaquettes and 6-site loops. Using numerical exact diagonalizations and Green's function Monte Carlo simulations, we show that the competition between these kinetic terms leads to a resonating valence bond (RVB) state for a finite range of parameters. We also show that this RVB phase is connected to the RVB phase identified in the Rokhsar-Kivelson model on the same lattice in the context of a generalized model that contains both the 6--site loops and a nearest-neighbor dimer repulsion. These results suggest that the occurrence of an RVB phase is a generic feature of QDM with competing interactions.Comment: 8 pages, 12 figure

Fat fractal percolation and k-fractal percolation

We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s.Comment: 27 pages, 3 figure

Dissolution and phosphate-induced transformation of ZnO nanoparticles in synthetic saliva probed by AGNES without previous solid-liquid separation. Comparison with UF-ICP-MS

The variation over time of free Zn2+ ion concentration in stirred dispersions of ZnO nanoparticles (ZnO NPs) prepared in synthetic saliva at pH 6.80 and 37 degrees C was followed in situ (without solid liquid separation step) with the electroanalytical technique AGNES (Absence of Gradients and Nernstian Equilibrium Stripping). Under these conditions, ZnO NPs are chemically unstable due to their reaction with phosphates. The initial stage of transformation (around 5-10 h) involves the formation of a metastable solid (presumably ZnHPO4), which later evolves into the more stable hopeite phase. The overall decay rate of ZnO NPs is significantly reduced in comparison with phosphate-free background solutions of the same ionic strength and pH. The effective equilibrium solubilities of ZnO (0.29-0.47 mg.L-1), as well as conditional excess-ligand stability constants and fractional distributions of soluble Zn species, were determined in the absence and presence of organic components. The results were compared with the conventional ultrafiltration and inductively coupled plasma-mass spectrometry (UF-ICP-MS) methodology. AGNES proves to be advantageous in terms of speed, reproducibility, and access to speciation information. KeywordsThis work was supported by the Spanish Ministry MINECOunder Grant No. CTM2016-78798 and European UnionSeventh Framework Programme FP7-NMP.2012.1.3-3 underGrant No. 310584 (NANoREG). FQ gratefully acknowledgesa grant from AGAUR

Dynamics of trace metal sorption by an ion-exchange chelating resin described by a mixed intraparticle/film diffusion transport model. The Cd/Chelex case

The time-evolution of Cd2+ ion sorption by Chelex 100 resin was studied in batch experiments as a function of time, pH, ionic strength, stirring rate, mass of resin and initial metal ion concentration. In the experimental conditions, the amount of resin sites are in excess with respect to the amount of metal ion, leading to extensive depletion of metal in bulk solution when equilibrium is reached. The data were described using a mixed control mass transport model in finite volume conditions (MCM) that includes explicitly both intraparticle and film diffusion steps. Exact numerical computations and a new approximate analytical expression of this model are reported here. MCM successfully predicts the influence of pH and ionic strength on the experimental Cd(II)/Chelex kinetic profiles (which cannot be justified by a pure film diffusion controlled mechanism) with a minimum number of fitting parameters. The overall diffusion coefficient inside the resin was modelled in terms of the Donnan factor and the resin/cation binding stability constant. The values of the latter coefficient as a function of pH and ionic strength were estimated from the Gibbs-Donnan model. Even though MCM is numerically more involved than models exclusively restricted to film or intraparticle diffusion control, it proves to be accurate in a wider range of values of the mass transfer Biot number and solution/resin metal ratios.The authors gratefully acknowledge support for this research from the Spanish Ministry MINECO (Projects CTM2013-48967 and CTM2016-78798) and by the “Comissionat d'Universitats i Recerca de la Generalitat de Catalunya” (2014SGR1132). FQ acknowledges a grant from AGAUR