Ismael S. S. Carrasco

The interplay between surface relaxation over a finite time window and diffusive cation flux produces compact, randomly branched, or hierarchically branched (dendritic) growth. Lattice symmetry and electrode orientation select dendrite shapes.

In a recent work [Phys. Rev. E 109, L042102 (2024)2470-004510.1103/PhysRevE.109.L042102], interesting dimensional crossovers [from two- to one-dimensional (2D to 1D) scaling] were found in the growth of Kardar-Parisi-Zhang (KPZ) interfaces on rectangular substrates, with lateral sizes L_{y}>L_{x}. Here, we extend this study to other universality classes for interface growth-specifically, the Edwards-Wilkinson (EW), the Mullins-Herring (MH), and the Villain-Lai-Das Sarma (VLDS) classes. From extensive simulations, we demonstrate that, in all systems with sufficiently large aspect ratio R=L_{y}/L_{x}, the roughness W scales with time t in the growth regime as W∼t^{β_{2D}} for t≪t_{c} and W∼t^{β_{1D}} for t≫t_{c}, where t_{c}∼L_{x}^{z_{2D}} in most cases. For the VLDS class, this crossover is also observed in the height distribution (HD), which approaches its characteristic probability density function for the 2D case at short times (t≪t_{c}) and then crosses over to the asymptotic 1D HD. Dimensional crossovers are also found in the steady-state regime, both in the roughness scaling and in the VLDS HD, which interpolate between the 2D and 1D ones as R increases. The particular case L_{x}=L_{y}^{δ}, with 0<δδ^{*}. Moreover, this leads the saturation roughness to display a nonuniversal scaling, W_{s}∼L_{y}^{Λ}, with Λ=(1-δ)α_{1D}+δα_{2D}.

Thin film deposition on weakly interacting substrates exhibits a unique growth mode characterized by initially strong island formation and rapidly increasing roughness, which reaches a maximum and subsequently decreases as the film returns to a smooth morphology. Here we show this rough-to-smooth growth mode experimentally for two molecular systems with substantially different geometries, namely, the effectively spherical buckminsterfullerene (C_{60}) and the disk-like 1,4,5,8,9,11-hexaazatriphenylenehexacarbonitrile. This growth mode is explained by a geometrical model that captures the basic mechanisms of multilayer island growth, island coalescence, and formation of a continuous film. Additionally, kinetic Monte Carlo simulations with minimal ingredients demonstrate that this mode generally occurs for weakly interacting substrates, providing quantitative estimates of parameters that characterize adsorbate-adsorbate and adsorbate-substrate interactions. Both the model and simulations accurately describe the experimental data and highlight the generic nature of the phenomenon, independently of the details of the interactions and the molecular flux, which opens up a path for controlling nanoscale film roughness.

Moving beyond simple associations, researchers need tools to quantify how variables influence each other in space and time. Correlation functions provide a mathematical framework for characterizing these essential dependencies, revealing insights into causality, structure, and hidden patterns within complex systems. In physical systems with many degrees of freedom, such as gases, liquids, and solids, a statistical analysis of these correlations is essential. For a field Ψ(x,t) that depends on spatial position x and time t, it is often necessary to understand the correlation with itself at another position and time Ψ(x_{0},t_{0}). This specific function is called the autocorrelation function. In this context, the autocorrelation function for order-parameter fluctuations, introduced by Fisher [J. Math. Phys. 5, 944 (1964)0022-248810.1063/1.1704197], provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent η is introduced to correct the spatial behavior of the correlation function at T=T_{c}. In recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)2470-004510.1103/PhysRevE.110.L062107] demonstrated that at T_{c} a fractal analysis is necessary for a complete description of the correlation function. In this study, we investigate the fundamental physics and mathematics underlying phase transitions, emphasizing the deep interplay between scaling behavior, critical exponents, and fractal geometry. In particular, we show that the application of modern fractional differentials allows us to write an equation for the correlation function that recovers the correct exponents below the upper critical dimension. We obtain the exact expression for the Fisher exponent η. Furthermore, we examine the Rushbrooke scaling relation, which has been questioned in certain magnetic systems, and, drawing on results from the Ising model, we confirm that both our relations and the Rushbrooke scaling law hold even when d is not an integer.

Island formation in strain-free heteroepitaxial deposition of thin films is analyzed using kinetic Monte Carlo simulations of two minimal lattice models and scaling approaches. The transition from layer-by-layer (LBL) to island (ISL) growth is driven by a weaker binding strength of the substrate, which, in the kinetic model, is equivalent to an increased diffusivity of particles on the substrate compared to particles on the film. The LBL-ISL transition region is characterized by particle fluxes between layers 1 and 2 significantly exceeding the net flux between them, which sets a quasiequilibrium condition. Deposition on top of monolayer islands weakly contributes to the second-layer nucleation, in contrast with the homoepitaxial growth case. A thermodynamic approach for compact islands with one or two layers predicts the minimum size in which the second layer is stable. When this is linked to scaling expressions for submonolayer island deposition, the dependence of the ISL-LBL transition point on the kinetic parameters qualitatively matches the simulation results, with quantitative agreement in some parameter ranges. The transition occurs in the equilibrium regime of partial wetting, and the convergence of the transition point upon reducing the deposition rate is very slow and practically unattainable in experiments.

We develop a hypothesis that the dynamics of equilibrium systems at criticality have their dynamics constricted to a fractal subspace. We relate the correlation fractal dimension associated with this subspace to the Fisher critical exponent controlling the singularity associated with the correlation function. This fractal subspace is different from that associated with the order parameter. We propose a relation between the correlation fractal dimension and the order parameter fractal dimension. The fractal subspace we identify has as a defining property that the correlation function is restored at the critical point by restricting the dynamics this way. We determine the correlation fractal dimension of the two-dimensional Ising model and validate it by computer simulations. We discuss growth models briefly in this context.

Modeling carbonate growth in fractures and pores is important for understanding carbon sequestration in the environment or when supersaturated solutions are injected into rocks. Here, we study the simple but nontrivial problem of calcite growth on fractures with rough walls of the same mineral using kinetic Monte Carlo simulations of attachment and detachment of molecules and scaling approaches. First, we consider wedge-shaped fracture walls whose upper terraces are in the same low-energy planes and show that the valleys are slowly filled by the propagation of parallel monolayer steps in the wedge sides. The growth ceases when the walls reach these low-energy configurations so that a gap between the walls may not be filled. Second, we consider fracture walls with equally separated monolayer steps (vicinal surfaces with roughness below 1 nm) and show that growth by step propagation will eventually clog the fracture gap. In both cases, scaling approaches predict the times to attain the final configurations as a function of the initial geometry and the step-propagation velocity, which is set by the saturation index. The same reasoning applied to a random wall geometry shows that step propagation leads to lateral filling of surface valleys until the wall reaches the low-energy crystalline plane that has the smallest initial density of molecules. Thus, the final configurations of the fracture walls are much more sensitive to the crystallography than to the roughness or the local curvature. The framework developed here may be used to determine those configurations, the times to reach them, and the mass of deposited mineral. Effects of transport limitations are discussed when the fracture gap is significantly narrowed.