Steady state of a two-species annihilation process with separated reactants.

We describe the steady state of the annihilation process of a one-dimensional system of two initially separated reactants A and B. The parameters that define the dynamical behavior of the system are the diffusion constant, the reaction rate, and the deposition rate. Depending on the ratio between those parameters, the system exhibits a crossover between a diffusion-limited (DL) regime and a reaction-limited (RL) regime. We found that a key quantity to describe the reaction process in the system is the probability p(x_{A},x_{B}) to find the rightmost A (RMA) particle and the leftmost B (LMB) particle at the positions x_{A} and x_{B}, respectively. The statistical behavior of the system in both regimes is described using the density of particles, the gap length distribution x_{B}-x_{A}, the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}), and the reaction kernel. For both regimes, this kernel can be approximated by using p(x_{A},x_{B}). We found an excellent agreement between the numerical and analytical results for all calculated quantities despite the reaction process being quite different in both regimes. In the DL regime, the reaction kernel can be approximated by the probability to find the RMA and LMB particles in adjacent sites. In the RL regime, the kernel depends on the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}).

Colloidal model of two-step protocol for epitaxial growth in one dimension.

We explore the application of a two-step growth protocol to a one-dimensional colloidal model. The evolution of the system is described in terms of the time-dependence of both monomer and island densities,N1andN, while its structure is characterized by using distributions of the gap length, the capture zone, the inter-island distance, and the island length. Analytical results obtained from rate equations are compared with these from molecular dynamics simulations. Since the two-step growth protocol deals with nucleation and aggregation processes in two completely separated time regimes, it makes possible to gain better understanding and control on the island formation mechanism than the standard one-step protocol. The predicted features and advantages of the two-step process could be experimentally tested using deposition of colloidal spheres on pattern substrates.

Two-step unconventional protocol for epitaxial growth in one dimension with hindered reactions.

We study the effect of hindered aggregation and/or nucleation on the island formation process in a two-step growth protocol. In the proposed model, the attachment of monomers to islands and/or other monomers is hindered by additional energy barriers which decrease the hopping rate of the monomers to the occupied sites of the lattice. For zero and weak barriers, the attachment is limited by diffusion while for strong barriers it is limited by reaction. We describe the time evolution of the system in terms of the monomer and island densities, N_{1} and N. We also calculate the gap length, the capture zone and the island distributions. For all the sets of barriers considered, the results given by the proposed analytical model are compared with those from kinetic Monte Carlo simulations. We found that the behavior of the system depends on the ratio of the nucleation barrier to the aggregation barrier. The two-step growth protocol allows more control and understanding on the island formation mechanism because it intrinsically separates the nucleation and aggregation processes in different time regimes.

Island size distribution with hindered aggregation.

We study the effect of hindered aggregation on the island formation processes for a one-dimensional model of epitaxial growth with arbitrary nucleus size i. In the proposed model, the attachment of monomers to islands is hindered by an aggregation barrier, ε_{a}, which decreases the hopping rate of monomers to the islands. As ε_{a} increases, the system exhibits a crossover between two different regimes; namely, from diffusion-limited aggregation to attachment-limited aggregation. The island size distribution, P(s), is calculated for different values of ε_{a} by a self-consistent approach involving the nucleation and aggregation capture kernels. The results given by the analytical model are compared with those from kinetic Monte Carlo simulations, finding a close agreement between both sets of data for all considered values of i and ε_{a}. As the aggregation barrier increases, the spatial effect of fluctuations on the density of monomers can be neglected and P(s) smoothly approximates to the limit distribution P(s)=δ_{s,i+1}. In the crossover regime the system features a complex and rich behavior, which can be explained in terms of the characteristic timescales of different microscopic processes.

Fragmentation approach to the point-island model with hindered aggregation: Accessing the barrier energy.

We study the effect of hindered aggregation on the island formation process in a one- (1D) and two-dimensional (2D) point-island model for epitaxial growth with arbitrary critical nucleus size i. In our model, the attachment of monomers to preexisting islands is hindered by an additional attachment barrier, characterized by length l_{a}. For l_{a}=0 the islands behave as perfect sinks while for l_{a}→∞ they behave as reflecting boundaries. For intermediate values of l_{a}, the system exhibits a crossover between two different kinds of processes, diffusion-limited aggregation and attachment-limited aggregation. We calculate the growth exponents of the density of islands and monomers for the low coverage and aggregation regimes. The capture-zone (CZ) distributions are also calculated for different values of i and l_{a}. In order to obtain a good spatial description of the nucleation process, we propose a fragmentation model, which is based on an approximate description of nucleation inside of the gaps for 1D and the CZs for 2D. In both cases, the nucleation is described by using two different physically rooted probabilities, which are related with the microscopic parameters of the model (i and l_{a}). We test our analytical model with extensive numerical simulations and previously established results. The proposed model describes excellently the statistical behavior of the system for arbitrary values of l_{a} and i=1, 2, and 3.

Interacting steps with finite-range interactions: analytical approximation and numerical results.

We calculate an analytical expression for the terrace-width distribution P(s) for an interacting step system with nearest- and next-nearest-neighbor interactions. Our model is derived by mapping the step system onto a statistically equivalent one-dimensional system of classical particles. The validity of the model is tested with several numerical simulations and experimental results. We explore the effect of the range of interactions q on the functional form of the terrace-width distribution and pair correlation functions. For physically plausible interactions, we find modest changes when next-nearest neighbor interactions are included and generally negligible changes when more distant interactions are allowed. We discuss methods for extracting from simulated experimental data the characteristic scale-setting terms in assumed potential forms.

Statistical behavior of domain systems.

We study the statistical behavior of two out of equilibrium systems. The first one is a quasi-one-dimensional gas with two species of particles under the action of an external field which drives each species in opposite directions. The second one is a one-dimensional spin system with nearest-neighbor interactions also under the influence of an external driving force. Both systems show a dynamical scaling with domain formation. The statistical behavior of these domains is compared with models based on the coalescing random walk and the interacting random walk. We find that the scaling domain size distribution of the gas and the spin systems is well-fitted by the Wigner surmise, which lead us to explore a possible connection between these systems and the circular orthogonal ensemble of random matrices. However, the study of the correlation function of the domain edges shows that the statistical behavior of the domains in both gas and spin systems is not completely well-described by a circular orthogonal ensemble, nor it is by other models proposed such as the coalescing random walk and the interacting random walk. Nevertheless, we find that a simple model of independent intervals describes more closely the statistical behavior of the domains formed in these systems.