A stochastic model, simulation, and application to aggregation of cadmium sulfide nanocrystals upon evaporation of the Langmuir–Blodgett matrix

A stochastic model of nanocrystals clusters formation is developed and applied to simulate an aggregation of cadmium sulfide nanocrystals upon evaporation of the Langmuir–Blodgett matrix. Simulations are compared with our experimental results. The stochastic model suggested governs mobilities both of individual nanocrystals and its clusters (arrays). We give a comprehensive analysis of the patterns simulated by the model, and study an influence of the surrounding medium (solvent) on the aggregation processes. In our model, monomers have a finite probability of separation from the cluster which depends on the temperature and binding energy between nanocrystals, and can also be redistributed in the composition of the cluster, leading to its compaction. The simulation results obtained in this work are compared with the experimental data on the aggregation of CdS nanocrystals upon evaporation of the Langmuir–Blodgett matrix. This system is a typical example from real life and is noteworthy in that the morphology of nanocrystals after evaporation of the matrix cannot be described exactly by a model based only on the motion of individual nanocrystals or by a cluster-cluster aggregation model.

Random walk on spheres algorithm for solving steady-state and transient diffusion-recombination problems

We further develop in this study the Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion-recombination equations first suggested in our recent article [K. Sabelfeld, First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations, Appl. Math. Lett. 88 2019, 141–148]. The random walk on spheres process mimics the isotropic diffusion of two types of particles which may recombine to each other. Our motivation comes from the transport problems of free and bound exciton recombination. The algorithm is based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. Therefore, the method is mesh free both in space and time. In this paper we implement the RWS algorithm for solving the diffusion-recombination problems both in a steady-state and transient settings. Simulations are compared against the exact solutions. We show also how the RWS algorithm can be applied to calculate exciton flux to the boundary which provides the electron beam-induced current, the concentration of the survived excitons, and the cathodoluminescence intensity which are all integral characteristics of the solution to diffusion-recombination problem.

A diffusion Monte Carlo method for charge density on a conducting surface at non-constant potentials

We develop a last-passage Monte Carlo algorithm on a conducting surface at non-constant potentials. In the previous researches, last-passage Monte Carlo algorithms on conducting surfaces with a constant potential have been developed for charge density at a specific point or on a finite region and a hybrid BIE-WOS algorithm for charge density on a conducting surface at non-constant potentials. In the hybrid BIE-WOS algorithm, they used a deterministic method for the contribution from the lower non-constant potential surface. In this paper, we modify the hybrid BIE-WOS algorithm to a last-passage Monte Carlo algorithm on a conducting surface at non-constant potentials, where we can avoid the singularities on the non-constant potential surface very naturally. We demonstrate the last-passage Monte Carlo algorithm for charge densities on a circular disk and the four rectangle plates with a simple voltage distribution, and update the corner singularities on the unit square plate and cube.

Global sensitivity analysis of statistical models by double randomization method

The paper addresses a global sensitivity analysis of complex models. The work presents a generalization of the hierarchical statistical models where uncertain parameters determine the distribution of statistical models. The double randomization method is applied to increase the efficiency of the Monte Carlo estimation of Sobol indices. Numerical computations are provided to study the accuracy and efficiency of the proposed technique. The issue of optimization of the suggested approach is considered.

A global random walk on grid algorithm for second order elliptic equations

In this paper we develop stochastic simulation methods for solving large systems of linear equations, and focus on two issues: (1) construction of global random walk algorithms (GRW), in particular, for solving systems of elliptic equations on a grid, and (2) development of local stochastic algorithms based on transforms to balanced transition matrix. The GRW method calculates the solution in any desired family of prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula. The use in local random walk methods of balanced transition matrices considerably decreases the variance of the random estimators and hence decreases the computational cost in comparison with the conventional random walk on grids algorithms.

A fully backward representation of semilinear PDEs applied to the control of thermostatic loads in power systems

We propose a fully backward representation of semilinear PDEs with application to stochastic control. Based on this, we develop a fully backward Monte-Carlo scheme allowing to generate the regression grid, backwardly in time, as the value function is computed. This offers two key advantages in terms of computational efficiency and memory. First, the grid is generated adaptively in the areas of interest, and second, there is no need to store the entire grid. The performances of this technique are compared in simulations to the traditional Monte-Carlo forward-backward approach on a control problem of thermostatic loads.

3D sputtering simulations of the CZTS, Si and CIGS thin films using Monte-Carlo method

The future of the industry development depends greatly on the permanently ensured energy needs and can be achieved only through the use of a variety of sustainable energy sources where the solar energy, which gains its optimal exploitation directly by linking it to the properties of solar cells and in particular to the crystallographic quality of the used semiconductor substrates, is one of them. Many growth processes are used to obtain a high quality of semiconductor formation and deposition, among them the DC sputtering. In this work, based on the Monte-Carlo method, a 3D DC sputtering simulation of the CZTS {\mathrm{CZTS}} , Si {\mathrm{Si}} and CIGS {\mathrm{CIGS}} semiconductors thin film formation is proposed by considering Argon as vacuum chamber bombardment gas. We extrapolate firstly the best sputtering yield possible of the semiconductors CZTS {\mathrm{CZTS}} and Silicon represented by their chemical formulas Cu 2 ⁢ ZnSnS 4 {\mathrm{Cu}_{2}\mathrm{Zn}\mathrm{Sn}\mathrm{S}_{4}} and Si {\mathrm{Si}} , respectively, by the application of different energies and incidence angles. From the obtained results, firstly we deduce that the best sputtering angle is 85 ∘ {85^{\circ}} ; in the same time, CZTS {\mathrm{CZTS}} is more efficient comparing to the Si {\mathrm{Si}} . Secondly, with the application of this angle ( 85 ∘ {85^{\circ}} ) in the sputtering process for the CZTS {\mathrm{CZTS}} ( Cu 2 ⁢ ZnSnS 4 {\mathrm{Cu}_{2}\mathrm{Zn}\mathrm{Sn}\mathrm{S}_{4}} ) and CIGS {\mathrm{CIGS}} represented by its chemical formula CuIn x ⁢ Ga ( 1 - x ) ⁢ Se 2 {\mathrm{Cu}\mathrm{In}_{x}\mathrm{Ga}_{(1-x)}\mathrm{Se}_{2}} , and the variation of the bombardment energy in order to find the total ejected atoms from each element of these two materials, we deduce that the sulfide ( S 4 {\mathrm{S}_{4}} ) and selenide ( Se 2 {\mathrm{Se}_{2}} ) elements give the majority of the sputtering yield amount obtained.

A global random walk on grid algorithm for second order elliptic equations

We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.