Deborah Lacitignola

Spatio-temporal organization in alloy electrodeposition: a morphochemical mathematical model and its experimental validation

This paper proposes a novel mathematical model for the formation of spatio-temporal patterns in electrodeposition. At variance with classical modelling approaches that are based on systems of reaction–diffusion equations just for chemical species, this model accounts for the coupling between surface morphology and surface composition as a means of understanding the formation of morphological patterns found in electroplating. The innovative version of the model described in this work contains an original, flexible and physically straightforward electrochemical source term, able to account for charge transfer and mass transport: adsorbate-induced effects on kinetic parameters are naturally incorporated in the adopted formalism. The relevant non-linear dynamics is investigated from both the analytical and numerical points of view. Mathematical modelling work is accompanied by an extensive, critical review of the literature on spatio-temporal pattern formation in alloy electrodeposition: published morphologies have been used as a benchmark for the validation of our model. Moreover, original experimental data are presented—and simulated with our model—on the formation of broken spiral patterns in Ni–P–W–Bi electrodeposition.

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. ( 2013 J. Solid State Electr. 17 , 467–479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimension-two Turing--Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing–Hopf dynamics.

Analysis of a tuberculosis model with a case study in Uganda

We consider a four-compartment tuberculosis model including exogenous reinfection. We derive sufficient conditions, in terms of the parameters of the system, which guarantee the occurrence of backward bifurcation. We also discuss the global stability of the endemic state by using a generalization of the Poincaré–Bendixson criterion. An application is given for the case of Internally Displaced Peoples Camps in North Uganda. The study suggests how important it is to provide qualitative indications on the threshold value of the population density in the area occupied by the camps, in order to possibly eradicate the disease.

Numerical approximation of Turing patterns in electrodeposition by ADI methods

In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction-diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semi-discretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of time-integrators (IMEX 2-SBDF method, Crank-Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction-diffusion systems with Turing patterns. Eventually, we apply the ADI-ECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached. We validate the model by comparison with experiments on Cu film growth by electrodeposition.

Globally stable endemicity for infectious diseases with information-related changes in contact patterns

Abstract SIR and SIS epidemic models with information—related changes in contact patterns are introduced. The global stability analysis of the endemic equilibrium is performed by means of the Li–Muldowney geometric approach. Biological implications of the stability conditions are given.

Qualitative analysis and optimal control of an epidemic model with vaccination and treatment

We focus on an epidemic model which incorporates a non-linear force of infection and two controls: an imperfect preventive vaccine given to susceptible individuals and therapeutic treatment given to infectious. We study both the cases of constant and non constant controls. In the case of constant controls we perform a qualitative analysis based on Lyapunov stability which allows to integrate the bifurcation analysis performed in a previous paper. The occurrence of a backward bifurcation is discussed in the perspective of disease control. The case of time-dependent controls is studied by means of the optimal control theory. The strategy is to minimize both the disease burden and the intervention costs. We derive the optimality system and solve it numerically. The characterization of the optimal time profile of the controls, together with the qualitative analysis provides a rather complete picture of the possible outcomes of the model.

Travelling waves in a reaction-diffusion model for electrodeposition

In this paper we consider an analytical and numerical study of a reaction-diffusion system for describing the formation of transition front waves in some electrodeposition (ECD) experiments. Towards this aim, a model accounting for the coupling between morphology and composition of one chemical species adsorbed at the surface of the growing cathode is addressed. Through a phase-space analysis we prove the existence of travelling waves, moving with specific wave speed. The numerical approximation of the PDE system is performed by the Method of Lines (MOL) based on high order space semi-discretization by means of the Extended Central Difference Formulae (D2ECDF) introduced in [1]. First of all, to show the advantage of the proposed schemes, we solve the well-known Fisher scalar equation, focusing on the accurate approximation of the wave profile and of its speed. Hence, we provide numerical simulations for the electrochemical reaction-diffusion system and we show that the results obtained are qualitatively in good agreement with experiments for the electrodeposition of Au-Cu alloys.

Investigation into dynamics of Au electrodeposition based on analysis of SERS spectral time series

Abstract A considerable amount of electrochemical and structural information is available on Au electrodeposition, but its dynamic aspects are still largely ignored. In this paper, an investigation of the microscopic Au electrodeposition dynamics is presented. Extensive sets of SERS spectra during prolonged potentiostatic electrodeposition (∼3 h) are recorded. Two different spot sizes have been used, in order to also obtain some information on the coupling of temporal and spatial fluctuations at the cathodic surface. The time evolution of the SERS background and the intensity of the stretching band of adsorbed CN− have been used as markers of surface morphology and chemistry respectively. Statistical analyses of these quantities allowed extraction of detailed information at the coupled dynamics of SERS enhancement and surface concentration of CN−. The spectroelectrochemical investigation is complemented by SEM observations and mathematical modelling of the coupled dynamics of cathode morphology and CN− surface concentration.

Morphological spatial patterns in a reaction diffusion model for metal growth.

In this paper a reaction-diffusion system modelling metal growth processes is considered, to investigate - within the electrodeposition context- the formation of morphological patterns in a finite two-dimensional spatial domain. Nonlinear dynamics of the system is studied from both the analytical and numerical points of view. Phase-space analysis is provided and initiation of spatial patterns induced by diffusion is shown to occur in a suitable region of the parameter space. Investigations aimed at establishing the role of some relevant chemical parameters on stability and selection of solutions are also provided. By the numerical approximation of the equations, simulations are presented which turn out to be in good agreement with experiments for the electrodeposition of Au-Cu and Au-Cu-Cd alloys.

Prediction of Morphological Properties of Smart-Coatings for Cr Replacement, Based on Mathematical Modelling

In this paper we present an extension of a mathematical model for the morphological evolution of metal electrodeposits – recently developed by some of the authors – accounting for mass-transport of electroactive species from the bulk of the bath to the cathode surface. The implementation of mass-transport effects is specially necessary for the quantitative rationalisation of electrodeposition processes from ionic liquids, since these electrolytes exhibit a viscosity that is notably higher than that of cognate aqueous solutions and consequently mass-transport control is active at all practically relevant plating rates. In this work we show that, if mass-transport is coupled to cathodic adsorption of ionic liquid species and surface diffusion of adatoms, it can lead to electrodeposit smoothing. This seemingly paradoxical theoretical result has been validated by a series of Mn electrodeposition experiments from aqueous baths and eutectic ionic liquids. The latter solutions have been shown to be able to form remarkably smoother coatings than the former ones. Mn electroplates have been proposed for Cd replacement and their corrosion protection performance seems comparable, but so far the required surface finish quality has not been achieved with aqueous electrolytes. Ionic liquids thus seem to provide a viable approach to aeronautic-grade Mn electroplating.