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.

A phenomenological model based on the three-dimensional theory of nonlinear elasticity is developed to describe the phenomenon of overstretching in the force-extension curve for double-stranded DNA (dsDNA). By using the concept of a material with multiple reference configurations a single formula is obtained to fit the force-extension curve.

This paper concerns the numerical approximation of Boundary Value ODEs (BVPs) with non-smooth coefficients and solutions. Different strategies are presented to tackle the cases of known and unknown singularity locations. In the former, the original problem is transformed in a multipoint BVP and high order Extended Central Difference Formulas (ECDFs) are used to approximate the smooth branches of the solution and the Neumann boundary conditions (BCs) with same accuracy. In the latter, an iterative Hybrid method coupling ECDFs and the shooting technique has been introduced to approximate both the discontinuity point and the solution. Convergence analysis and numerical comparisons with other approaches from literature are also presented. Good performances in terms of errors and convergence order are reported by applying ECDFs and the Hybrid method to linear test BVPs and to a nonlinear bio-mechanical model, in both cases of mixed and Dirichlet BCs.

We analyze the effects of cross-diffusion on pattern formation in a PDE reaction-diffusion system introduced in Bozzini et al. 2013 to describe metal growth in an electrodeposition process. For this morphochemical model - which refers to the physico-chemical problem of coupling of growth morphology and surface chemistry - we have found that negative cross-diffusion in the morphological elements as well as positive cross-diffusion in the sur- face chemistry produce larger Turing parameter spaces and favor a tendency to stripeness that is not found in the case without cross-diffusion. The impact of cross-diffusion on pat- tern selection has been also discussed by the means of a stripeness index. Our theoretical findings are validated by an extensive gallery of numerical simulations that allow to better clarify the role of cross-diffusion both on Turing parameter spaces and on pattern selec- tion. Experimental evidence of cross-diffusion in electrodeposition as well as a physico- chemical discussion of the expected impact of cross diffusion-controlled pattern formation in alloy electrodeposition processes complete the study.

In this paper, we consider the numerical approximation of a reaction-diffusion system 2D in space whose solutions are patterns oscillating in time or both in time and space. We present a stability analysis for a linear test heat equation in terms of the diffusion d and of the reaction timescales given by the real and imaginary parts $alpha" and $beta$ of the eigenvalues of J(Pe), the Jacobian of the reaction part at the equilibrium point Pe. Focusing on the case $alpha = 0,beta neq 0$, we obtain stability regions in the plane $(xi ,nu )$, where $xi =lambda (h;d)h_t$ , $nU =beta h_t$ , $h_t$ time stepsize, $lambda$ lumped diffusion scale depending also from the space stepsize h and from the spectral properties of the discrete Laplace operator arising from the semi-discretization in space. In space we apply the Extended Central Difference Formulas (ECDFs) of order p = 2,4,6. In time we approximate the diffusion part in implicit way and the reaction part by a selection of integrators: the Explicit Euler and ADI methods, the symplectic Euler and a partitioned Runge-Kutta method that are symplectic in absence of diffusion. Hence, by estimating l , for each method we derive stepsize restrictions $h_t <≈ F_met(h;d,beta , p) $ in terms of the stability curve $F_met$ depending on diffusion and reaction timescales and from the approximation order in space. For the same schemes, we provide also a dispersion error analysis. We present numerical simulations for the test heat equation and for the Lotka- Volterra PDE system with solutions oscillating only in time for the presence of a center-type dynamics. In these cases, the implicit-symplectic schemes provide the best choice. We solve also the Schnakenberg model with spatial patterns oscillating in space and time in presence of an attractive limit cycle due to the Turing-Hopf instability. In this case, all schemes attain closed orbits in the phase space, but the Explicit ADI method is the best choice from the computational point of view.

Synchrotron-based soft-X-ray imaging and microspectroscopy with sub-micrometer spatial resolution are used to investigate the electrodeposition of Mn-Cu-ZnO, a prospective active material for supercapacitors. This study is focused on the correlation of the local current density and the spatial distribution of the composition and chemical-state in electrodeposits grown potentiostatically at -0.7V (vs a saturated calomel electrode), as well as the optimal potential for the achievement of high specific capacitance and cycling stability. The morphology, elemental distribution, and the local chemical state of both the electrode deposits and the grown dendrite structures are followed by using X-ray imaging, X-ray fluorescence mapping, and X-ray absorption microspectroscopy. For transmission soft-X-ray measurements, a thin-layer Hull-type microcell is developed and fabricated by using electron-beam lithography. The information obtained for the spatial distribution of the deposit is complemented by using electrochemical measurements and numerical simulations.

This paper reports on spiral pattern formation in In–Co electrodeposition. We propose an approach to the understanding of this process based on: (i) compositional and chemicalstate distribution analysis by high-resolution photoelectron microspectroscopy and (ii) a mathematical model able to capture the morphological features highlighted in the experiments. Microspectroscopy—complemented by electrochemical, structural and morphological characterisations—combined with mathematical modelling, analytical and numerical investigations, converge in pointing out the key role played by intermetallic electrodeposition in spiral formation.

We propose and analyse a novel surface finite element method that preserves the invariant regions of systems of semilinear parabolic equations on closed compact surfaces in R3 under discretisation. We also provide a fully-discrete scheme by applying the implicit-explicit (IMEX) Euler method in time. We prove the preservation of the invariant rectangles of the continuous problem under spatial and full discretizations. For scalar equations, these results reduce to the well-known discrete maximum principle. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. In particular we provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up due to the nature of the kinetics.

Weconsider a lumped surface finite element method (LSFEM) for the spatial approximation of reaction–diffusion equations on closed compact surfaces in R3 in the presence of crossdiffusion. We provide a fully-discrete scheme by applying the Implicit–Explicit (IMEX) Euler method. We provide sufficient conditions for the existence of polytopal invariant regions for the numerical solution after spatial and full discretisations. Furthermore,weprove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. To support our theoretical findings, we provide two numerical tests. The first test confirms that in the absence of lumping numerical solutions violate the invariant region leading to blow-up due to the nature of the kinetics. The second experiment is an example of Turing pattern formation in the presence of cross-diffusion on the sphere.

Contamination of fuel-cell membranes with metal ions, resulting from the use of ferrous alloys, leads to corrosion and ultimately failure of the device. By using a combination of X-ray spectroscopy techniques, the corrosion processes of Ni and Fe electrodes in contact with a hydrated Nafion film are investigated. The results show diffusion of corrosion products within the film only in the case of the Fe electrodes, whereas Ni electrodes appear corrosion resistant.

A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an indepth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics.We apply our method to Treloar’s classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.

This paper offers an overview of morphogenetic processes going on in metal electrodeposition processes and provides a systematisation of the morphology classes identified experimentally in terms of an electrokinetic theory accounting for charge-transfer and masstransport rates. In addition, it provides a review of the modelling work by the authors, based on a reaction-diffusion system coupling morphology with surface chemistry of the growing metal and briefly describes the experimental validation of the model.

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.

This paper describes the numerical modelling of a key material-stability issue within the realm of Molten Carbonate Fuel Cells (MCFC). Differential models have been developed for the 2D and 3D distributions of current density as well as peroxide and carbon dioxide concentrations. By suitable variations of the integration domain - based on the agglomerate concept - one can describe the morphological and attending electrocatalytic evolution of porous NiO electrodes. On the basis of electrochemical data recorded during the operation of a laboratory MCFC, we have shown that this model is able to rationalise the evolution of cathode conditions leading to both improvements of electrocatalytic performance - such as lithiation - and degradation - such as agglomeration

In this paper we investigate from the numerical point of view the discrete DNA model proposed in Lacitignola and Saccomandi (Bull. Math. Biol., 2014) in order to test the robustness of the parametric resonance condition found in the limit of the continuum approximation. To describe more realistically the binding of RNA polymerase to the DNA macromolecule during the first stage of the transcription process, we here consider a localized DNA-RNA polymerase interaction and a relatively high number of base-pairs. Even with these more realistic assumptions, our findings confirm the ones found in the continuum limit and indicate that the parametric resonance phenomenon can be an intrinsic property of the discrete DNA model.

In this paper we consider a parameter identification problem (PIP) for data oscillating in time, that can be described in terms of the dynamics of some ordinary differential equation (ODE) model, resulting in an optimization problem constrained by the ODEs. In problems with this type of data structure, simple application of the direct method of control theory (discretize-thenoptimize) yields a least-squares cost function exhibiting multiple ‘low’ minima. Since in this situation any optimization algorithm is liable to fail in the approximation of a good solution, here we propose a Fourier regularization approach that is able to identify an iso-frequency manifold S of codimensionone in the parameter space Rm, such that for all parameters in S the ODE solutions have the same frequency of the assigned data. Further to the identification of S, we propose to minimize on this manifold the least squares, the phase (or time lag) and infinity norm errors between data and simulations. Hence, the Fourier-PIP can be regarded as a new constrained optimization problem, where the iso-frequency sub-manifold represents a further constraint. First we describe our approach for simulated oscillatory data obtained with the two-parameter Schnakenberg model, in the Hopf regime. Finally, we apply Fourier-PIP regularization to follow original experimental data with the morphochemical model for electrodeposition (Lacitignola et al 2015 Eur. J. Appl. Math. 26 143–73) in the case of two and three parameters .

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.

We propose and analyse a lumped surface finite element method for the numerical approximation of reaction–diffusion systems on stationary compact surfaces in R3. The proposed method preserves the invariant regions of the continuous problem under discretization and, in the special case of scalar equations, it preserves the maximum principle. On the application of a fully discrete scheme using the implicit–explicit Euler method in time, we prove that invariant regions of the continuous problem are preserved (i) at the spatially discrete level with no restriction on the meshsize and (ii) at the fully discrete level under a timestep restriction. We further prove optimal error bounds for the semidiscrete and fully discrete methods, that is, the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings.We provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up.

This paper reports on the electrodeposition of Mn-Cu-ZnO for hybrid supercapacitors. This material exhibits a dual structure consisting of Mn-rich highly active, but poorly electronically conducting, grains, which are locked by a Cu-rich highly conductive network that also possesses some degree of charge-storage capacity. This work focuses on morphological, compositional, and chemical-state distributions with submicrometer lateral resolution. This information, which is crucial because doping distribution controls supercapacitor performance, has been obtained by combining electrochemical and in situ Raman measurements with synchrotron-based X-ray fluorescence and absorption microspectroscopy. Using a microfabricated thin-layer three-electrode microcell, we followed the morphochemical changes at different electrodeposition stages and found that pulse-plating allows the growth of Mn-and Cu-doped ZnO as self-organized structures with a consistent spatially stable composition distribution.

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 nonlinear 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 abenchmark 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.

One of the least studied universal deformations of incompressible nonlinear elasticity, namely the straightening of a sector of a circular cylinder into a rectangular block, is revisited here and, in particular, issues of existence and stability are addressed. Particular attention is paid to the system of forces required to sustain the large static deformation, including by the application of end couples. The influence of geometric parameters and constitutive models on the appearance of wrinkles on the compressed face of the block is also studied. Different numerical methods for solving the incremental stability problem are compared and it is found that the impedance matrix method, based on the resolution of a matrix Riccati differential equation, is the more precise.

We consider the elastic deformation of a circular cylindrical sector composed of an incompressible isotropic soft solid when it is straightened into a rectangular block. In this process, the circumferential line elements on the original inner face of the sector are stretched while those on the original outer face are contracted. We investigate the geometrical and physical conditions under which the latter line elements can be contracted to the point where a localized incremental instability develops. We provide a robust algorithm to solve the corresponding two-point boundary value problem, which is stiff numerically. We illustrate the results with full incremental displacement fields in the case of Mooney–Rivlin materials and also perform an asymptotic analysis for thin sectors.

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 phasespace 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.

The present paper deals with the pattern formation properties of a specific morpho- electrochemical reaction-diffusion model on a sphere. The physico-chemical background to this study is the morphological control of material electrodeposited onto spherical parti- cles. The particular experimental case of interest refers to the optimization of novel metal- air flow batteries and addresses the electrodeposition of zinc onto inert spherical supports. Morphological control in this step of the high-energy battery operation is crucial to the energetic efficiency of the recharge process and to the durability of the whole energy- storage device. To rationalise this technological challenge within a mathematical modeling perspective, we consider the reaction-diffusion system for metal electrodeposition intro- duced in [Bozzini et al., J. Solid State Electr.17, 467–479 (2013)] and extend its study to spherical domains. Conditions are derived for the occurrence of the Turing instability phe- nomenon and the steady patterns emerging at the onset of Turing instability are investi- gated. The reaction-diffusion system on spherical domains is solved numerically by means of the Lumped Surface Finite Element Method (LSFEM) in space combined with the IMEX Euler method in time. The effect on pattern formation of variations in the domain size is investigated both qualitatively, by means of systematic numerical simulations, and quan- titatively by introducing suitable indicators that allow to assign each pattern to a given morphological class. An experimental validation of the obtained results is finally presented for the case of zinc electrodeposition from alkaline zincate solutions onto copper spheres.

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equa- tion on a surface in R3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) [Dziuk, Eliott, Finite element methods for surface PDEs, 2013] and the recent VEM [Beirao da Veiga et al, Basic principles of virtual element methods, 2013] in order to allow general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

We focus on the morphochemical reaction–diffusion model introduced in Bozzini et al. (2013) and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns arising as the result of Turing instability of the physically relevant equilibrium. We perform a weakly nonlinear multiple scales analysis, and derive the normal form equations governing the amplitude of the patterns. These amplitude equations allow us to construct relevant solutions of the model equations and reveal the presence of multiple branches of stable solutions arising as the result of subcritical bifurcations. Hysteretic type phenomena are highlighted also through numerical simulations. We show the occurrence of spatial pattern propagation and derive the Ginzburg–Landau equation describing the envelope of the traveling wavefront.

In this paper we focus on the following map identification problem (MIP): given a morphochemical reaction–diffusion (RD) PDE system modeling an electrodepostion process, we look for a time t*, belonging to the transient dynamics and a set of parameters p, such that the PDE solution, for the morphology h(x, y, t^*; p) and for the chemistry theta(x, y, t^*; p) approximates a given experimental map M*. Towards this aim, we introduce a numerical algorithm using singular value decomposition (SVD) and Frobenius norm to give a measure of error distance between experimental maps for h and θ and simulated solutions of the RD-PDE system on a fixed time integration interval. The technique proposed allows quantitative use of microspectroscopy images, such as XRF maps. Specifically, in this work we have modelled the morphology and manganese distributions of nanostructured components of innovative batteries and we have followed their changes resulting from ageing under operating conditions. The availability of quantitative information on space-time evolution of active materials in terms of model parameters will allow dramatic improvements in knowledge-based optimization of battery fabrication and operation.