spatio-temporal organization in a morphochemical electrodeposition model: analysis and numerical...

Acta Appl MathDOI 10.1007/s10440-014-9910-3

Spatio-Temporal Organization in a MorphochemicalElectrodeposition Model: Analysis and NumericalSimulation of Spiral Waves

Deborah Lacitignola · Benedetto Bozzini · Ivonne Sgura

Received: 20 November 2013 / Accepted: 25 January 2014© Springer Science+Business Media Dordrecht 2014

Abstract In this paper we derive Hopf instability conditions for the morphochemicalmathematical model for alloy electrodeposition introduced and experimentally validated inBozzini et al. (J. Solid State Electrochem., 17:467–479, 2013). Using normal form theorywe show that in the neighborhood of the Hopf bifurcation, essential features of the sys-tem dynamics are captured by a specific Complex Ginzburg-Landau Equation (CGLE). Thederived CGLE yields analytical results on the existence and stability of spiral waves. More-over, the arising of spiral instability is discussed in terms of the relevant system parametersand the related phenomenology is investigated numerically. To face with the numerical ap-proximation of the spiral structures and of their longtime oscillating behavior we apply anAlternating Direction Implicit (ADI) method based on high order finite differences in space.

Keywords Reaction diffusion models · Hopf bifurcation · Normal forms · ComplexGinzburg-Landau equation · Spiral instability · High order finite differences · ADI method

1 Introduction

Electrochemistry has been found to be a fertile ground for the controlled generation and un-derstanding of spatial organization phenomena. Many experiments have shown that complex

D. Lacitignola (B)Dipartimento di Ingegneria Elettrica e dell’Informazione, Università di Cassino e del LazioMeridionale, Via di Biasio, 43, 03043 Cassino, Italye-mail: [email protected]

B. BozziniDipartimento di Ingegneria dell’Innovazione, Università del Salento, Via per Monteroni, 73100 Lecce,Italye-mail: [email protected]

I. SguraDipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Via per Arnesano,73100 Lecce, Italye-mail: [email protected]

mailto:[email protected]



D. Lacitignola et al.

spatio-temporal patterns can result from simple reaction mechanisms and that even smallchanges in the reaction conditions can give rise to radical pattern reorganizations. Withinthis framework, appropriate manipulation and control of specific chemical patterns can onlyrely on a deep theoretical understanding of the involved physico-chemical phenomena.

Along these lines, we aim at rationalizing the formation of morphological patterns inelectrodeposition, through a reaction-diffusion modeling approach in which the couplingof morphology (surface profile) and surface chemistry (composition) is the key ingredient[4–8]. In a previous publication [8], we have developed, from physico-chemical fundamen-tal information, a 2D reaction-diffusion system which proved to be interestingly capable ofaccounting for all the spatio-temporal organization types that, to the best of our efforts, wewere able to find in the experimental literature. In [8] the mathematical study of the modelwas limited to numerical computations and no analytical results were presented. A first moreinsightful mathematical analysis of this model was offered in [18], where spatial pattern ini-tiation has been investigated from analytical and numerical points of view, based on theinterplay between Turing and Hopf instabilities. In the present paper we advance the math-ematical analysis of the reaction-diffusion model proposed in [8], explicitly showing that itcan also exhibit spiral wave behavior. In fact—in addition to steady spatial patterns—spiralwaves have also been widely documented in the experimental electrochemical literature[9, 10, 17, 19]. In order to rigorously prove the existence of spiral wave behavior, we firstdetect a region in the parameter space where the system exhibits oscillatory dynamics be-cause of the occurrence of a supercritical Hopf bifurcation. Then, by using the approachdeveloped in [14–16], we show that in the neighborhood of the Hopf bifurcation value, es-sential features of system dynamics are captured by a specific Complex Ginzburg-LandauEquation (CGLE), whose properties have been extensively studied in literature and whichhas been shown to admit a rich variety of coherent 2D structures such as target and spiralwaves [2]. The derived CGLE allows to derive analytical results on existence and stabilityproperties of spiral waves.

We tested these findings by numerical approximation of the full model and found in-teresting scenarios leading to spiral breaking for suitable variations of appropriate systemparameters. To capture the spiral structures, careful space discretization and longtime inte-gration until the asymptotic state is reached are needed. To face with this challenging task,we apply the ADI-ECDF scheme described in [22] that yields efficient numerical solutionsat moderate computational cost.

2 The Model and Remarks on the Spatially Uniform Case

We consider the following reaction diffusion system in adimensional form:

ηt = �η + f (η, θ),

θt = d�θ + g(η, θ),(1)

where � is the bidimensional Laplacian operator. Nonlinear source terms which take intoaccount generation and loss of the relevant material are

f (η, θ) = A1(1 − θ)η − A2η3 − B(θ − α1),

g(η, θ) = C(1 + k2η)(1 − θ)[1 − γ (1 − θ)

] − Dθ(1 + γ θ) − Dk3ηθ(1 + γ θ).(2)

Here η ∈ R is the adimensional morphological variable, measuring the displacement fromthe average plane of the deposit and 0 ≤ θ ≤ 1 is the surface coverage with the adsorbate.

Spiral Waves in a Morphochemical Electrodeposition Model

The source term f includes (i) the charge-transfer rate at sites free from adsorbatesthrough the term A1(1 − θ)η; (ii) mass-transport limitations to the electrodeposition pro-cess through the term A2η

3; (iii) the effect of adsorbates on the electrodeposition rate bythe term −B(θ − α1). Moreover the source term g features adsorption (parameter C) anddesorption (parameter D) terms including both chemical and electrochemical contributions.We refer to [8] for full details. All the involved constants are meant as real positive or equalto zero, with 0 < α1 ≤ 1 and 0 ≤ γ ≤ 1. We restrict our mathematical discussion to themanifold

D = C(1 − α1)(1 − γ + γα1)[α1(1 + γα1)

]−1, (3)

and assume k3 < k2 so that the effect of electrochemistry on desorption is weaker than onadsorption. We also require (1)–(2) to be equipped with zero-flux boundary conditions andthe following initial conditions:

η(x, y,0) = η0(x, y), θ(x, y,0) = θ0(x, y), (x, y) ∈ [0,L] × [0,L],where L is a characteristic length of the electrode. In the following, we focus on Pe =(ηe, θe) = (0, α1) which is easily shown to be a spatially uniform equilibrium for (1)–(2).The way Pe looses stability is particularly notable from the physical point of view since, be-ing characterized by ηe = 0, it corresponds to a flat electrode surface from which corrugationand outgrowth morphologies can develop.

As a first step to investigate the occurrence of spiral waves behavior for system (1)–(2),we show that in the spatially uniform case a Hopf bifurcation can occur, destabilizing Pe insuch a way that an oscillatory regime takes over.

The Jacobian matrix J (η, θ) evaluated at the homogeneous steady state Pe is given by:

J (ηe, θe) =[

A1(1 − α1) −B

C(k2 − k3)F1(α1, γ ) −CF2(α1, γ )

], (4)

where

F1(α1, γ ) = (1 − α1)(1 − γ + α1γ ); F2(α1, γ ) = 2α1γ (1 + α1γ − γ ) + 1 − γ

α1(1 + α1γ ). (5)

Let be τe = trace(J (ηe, θe)) and δe = det(J (ηe, θe)). Linear stability analysis shows thatPe can change its stability properties because of a Hopf bifurcation. This bifurcation occursin the system when a couple of complex conjugate eigenvalues of J (ηe, θe) crosses the imag-inary axis. The Hopf bifurcation can hence be detected by requiring τe = 0 and δe > 0, that is

C = A1(1 − α1)F2(α1, γ )−1, B > A1(1 − α1)F2(α1, γ )[(k2 − k3)F1(α1, γ )

]−1.

A further bifurcation, involved in the spatially uniform case, is the transcritical bifurcation,where the equilibrium Pe exchanges its stability properties with another equilibrium. Atthe bifurcation value, Pe becomes non hyperbolic since one real eigenvalue of the relatedJacobian matrix vanishes. Hence, the transcritical bifurcation can be detected by requiringδe = 0, i.e. B = A1(1 − α1)F2(α1, γ )[(k2 − k3)F1(α1, γ )]−1.

Figure 1 shows such bifurcation lines in the parameter space (C,B) for a specific choiceof the other parameter values: the vertical and horizontal lines are related to the Hopf andtranscritical bifurcations, respectively. The regions to the right of the Hopf line and abovethe transcritical line are characterized by τe < 0 and δe > 0 so that, in the spatially uniformcase, the homogeneous equilibrium Pe is unconditionally stable.


Fig. 1 The spatially uniformcase. By fixing the parametersvalues α1 = 0.5; γ = 0.2;k2 = 2.5; k3 = 1.5; A1 = 10;A2 = 30, bifurcation diagram inthe parameter space (C,B) isshown. The bifurcation point TBwhere transcritical and Hopf linesmeet is (CTB,BTB) = (2.8061,

19.7979)

Fig. 2 The spatially uniform case. Simulations near the supercritical Hopf bifurcation at CHopf = 2.8061with other parameters as in the caption of Fig. 1. Left: Phase portraits for B = 22 and increasing C values.Right: Time dependent behaviors of η(t) and θ(t)

By choosing parameters below the transcritical bifurcation line, δe < 0 holds, so that Pe

can be destabilized by small homogeneous perturbations and system trajectories tend towarda different stable steady state. Moreover, in the region to the left of the Hopf line and abovethe transcritical one, Pe is unstable and we can expect homogeneous oscillations, owing tothe presence of a stable limit cycle caused by a supercritical Hopf bifurcation, as shown inFig. 2. A rigorous analytical proof of the supercritical nature of the Hopf bifurcation hasbeen provided in a separate paper [18].

3 In the Neighborhood of the Hopf Bifurcation: Derivation of CGLE

Numerical investigations on system (1)–(2) show that—in the presence of oscillatorydynamics—this reaction-diffusion system can exhibit bound states of spiral waves as wellas interesting spirals breaking when specific system parameters are varied. In this sectionwe give analytical evidence of both circumstances.


The key requirement for the existence of spiral waves in model (1)–(2) is the occurrenceof a supercritical Hopf bifurcation: spiral waves are in fact very common in many of thesystems exhibiting an oscillatory dynamics. By using classical tools from dynamical systemtheory, it is possible to show that, when a reaction-diffusion model is near a supercriticalHopf bifurcation, the essential features of its dynamics are captured by a suitable CGLE.

To obtain the proper CGLE for our system, we employ a systematic approach for generaldynamical systems developed in [15] and extended in [14, 16] to reaction-diffusion models.This method is essentially based on the idea that system dynamics near a bifurcation can besuitably described in a low-dimensional space of amplitudes through the so-called ampli-tude equations which are a very important tool to capture the qualitative behavior of manydifferent systems [11, 12, 15].

We briefly summarize the bases of this approach, as developed in [14–16], for a dynami-cal system in the neighborhood of a local bifurcation. The motion of its state variables in ther-dimensional center manifold Wc can be described by r amplitudes yi , i = 1, . . . , r , whichare coordinates of a point y in the center subspace Ec with respect to a basis of eigenvectors(related to the critical eigenvalues) of the linearized vector field. It is then possible to derivea set of differential equations for the amplitudes yi and a transformation h(y) from Ec toWc such that h(y(t)) describes the dynamical evolution of the state variables on Wc .

The function h(y) is chosen in a way that allows to successively eliminate from the ki-netics equations as many nonlinear terms as possible, starting from the lowest order ones.As in the normal form theory, each type of bifurcation is characterized by the nonlineari-ties that cannot be removed. Once the general amplitude equation is obtained, the relevantcoefficients can be obtained directly from the original vector field.

As far as the amplitude equation for our system is concerned, it is worth noting that atthe Hopf bifurcation the Jacobian matrix (evaluated at the steady state of interest) has twocomplex conjugate right eigenvectors, u and u, and two complex conjugate left eigenvectorsu+ and u+, corresponding to the critical eigenvalues λ1 = λ2 = iω0 (here the overbar standsfor complex conjugation). The left and right eigenvectors are taken as normalized such that:

u · u+ = 1, u+ · u = 0, u+ · u = 0, u+ · u = 1. (6)

The generic amplitude equation for the Hopf bifurcation can be shown take on the followingform [15]:

y = (iω0 + σC)y − g|y|2y,

where y = dy/dt , y and y are the complex coordinates of y = yu + yu ∈ Ec; i.e. in thebasis spanned by u and u: (i) C is the bifurcation parameter and (ii) σ and g are coefficientsto be determined from the original vector field.

In the spatial case, when diffusion is considered, the amplitudes are a function of bothspace s and time t . One can then write y(s, t) = eiω0tw(s, t), where w(s, t), w(s, t) arescaled amplitudes that describe the modulation of local oscillations of a frequency ω0. In theneighborhood of the Hopf bifurcation, system dynamics is thus described by the followingamplitude equation [14, 16]:

wt = σCw − g|w|2w + d�sw, (7)

where

d = u+ · D · u, σ = u+ · FxC · u + u+ · Fxx(u,h001),

g = −[

u+ · Fxx(u,h110) + u+ · Fxx(u,h200) + 1

2u+ · Fxxx(u,u, u)

].

(8)


We recall that in this notation F = (f, g) is the vector field whose components are given

in (2); x = (x1, x2) = (η, θ) is the vector of the state variables and D = ( 1 00 d

).

The other quantities in (8) are defined as follows [15]:

FxC =(

∂2F∂xi∂C

)

|Pe

, Fxx(u,hpqr ) =2∑

i,j=1

∂2F(Pe,Chopf )

∂xi∂xj

ui(hpqr )j ,

Fxxx(u,u, u) =2∑

i,j,k=1

∂3F(Pe,Chopf )

∂xi∂xj ∂xk

uiujuk.

(9)

Moreover h001, h110 and h200 can be determined by the following relations:

[J (Pe,Chopf ) − γ0I

] · h001 = −FC


] · h110 = −Fxx(u, u)


] · h200 = −1

2Fxx(u,u)

(10)

with γk = kiω0 and I the 2 × 2 identity matrix.For model (1)–(2), we fix the parameter values as in Fig. 1 and set B = 22. Accordingly,

the homogeneous steady state Pe = (0,0.5) experiences a supercritical Hopf bifurcation atC = Chopf = 2.8061. The Jacobian matrix J (Pe,Chopf ) has two complex conjugate eigen-values λ1 = λ2 = iω0 = i1.6675. The right and left eigenvectors associated with λ1 andnormalized according to (6) are given by:

u = (3.9595 + i3205,1)T , u+ = (−0.3786i,0.4999 + i1.4992). (11)

By straightforward calculations we obtain the CGLE (7) with coefficients (8) specialized as

σ = σr + iσi = −0.8909 + i0.29712,

g = gr + igi = 2292.6790 + i6970.0742,

d = dr + idi = 0.5(1 + d) + i1.4992(d − 1).

Moreover, by the change of variables

w =√

σrC

gr

w′ exp

[iσi

σr

t ′], t = t ′

Cσr

, s =√

dr

σrCs′

and by removing the primes, one obtains the CGLE (7) in the more compact form:

wt = w − (1 + iα)|w|2w + (1 + iβ)�sw, (12)

with β = di/dr and α = gi/gr . For the chosen set of parameter values, we get

α = 3.0401, β = 2.9984

(d − 1

d + 1

), (13)

so that, in our case, the properties of (12) are only determined by the control parameter d .


4 Spiral Waves

For the system under consideration, the CGLE (12) predicts the emergence of spiral waveswhose velocity and frequency can be derived analytically. Toward this end, we can use forthe solution of (12)—as in [20]—a traveling wave Ansatz, holding in arbitrary dimensionsand therefore suitable for 1D traveling waves, 2D spiral waves and 3D scroll waves. In-deed, for the investigated patterns, properties as wavelength and frequency still agree withthose of traveling waves. The linear spreading velocity v∗ of the propagating fronts can beobtained by linearizing (12) around the steady state w = 0 and using the traveling waveAnsatz w(s, t) = w exp [−iΩt − ik · s]. This leads to the following dispersion relation:

Ω(k) = βk2 + i(1 − k2

),

with k = |k|. We observe that the state w = 0 is linearly unstable for wavenumbers k2 < 1since in this range Im(Ω(k)) > 0. As insightfully discussed in [21], v∗ can be obtained byfinding a point k∗—the so called linear spreading point—such that

dΩ(k)

dk

∣∣∣∣k∗

= ImΩ(k∗)Imk∗

= v∗.

Taking into account of (13), it follows

v∗ = 2

√β2 + 1

3= 2√

3

√

8.9904

(d − 1

d + 1

)2

+ 1.

The frequency Ω of the spiral waves are instead obtained by considering the full nonlinearequation (12). By making a traveling-wave Ansatz, w(s, t) = Z exp [−iq · s − iΩt], oneobtains the dispersion relation:

Ω(q) = α + q2(β − α), (14)

with q = |q|. As observed before in this Section, traveling wave solutions can exist in therange of wavenumbers q2 < 1. The first term in (14) is related to nonlinear system dynamics,whereas the second one is due to the spatial character of the system. In order to sustain avelocity v∗, the dynamics selects a unique wavenumber qs(α,β), such that [21]:

Ω(qs) = α + v∗qs. (15)

The geometry and the dynamical features of the spiral waves are thus fully defined by α

and β . From (13), (14) and (15) it follows:

qs = v∗

(β − α)= 2√

3

√β2 + 1

(β − α)=

0.3798√

8.9904( d−1d+1 )2 + 1

[0.9862( d−1d+1 ) − 1] .

The frequency of the spirals is then given by:

Ωs := Ω(qs) = α + 4

3

β2 + 1

β − α= 3.0401 + 0.4385[8.9904( d−1

d+1 )2 + 1][0.9862( d−1

d+1 ) − 1] . (16)


In the polar coordinates (r,ϑ), a spiral wave solution of the 2D CGLE can be represented as

w(r,ϑ, t) = A(r)ei[ϑ+ψ(r,t)].

As r → +∞, A(r) → √1 − q2

s and ψ(r, t) → qsr − Ωst = qs(r − vf t), where the phasevelocity vf is given by vf = Ωs/qs . By (13) it follows that, for the case under study,α > β such that qs < 0. The group velocity vg = dΩs/dqs = 2(β − α)qs is therefore a posi-tive quantity, but the phase velocity vf can change its sign according to the sign of Ωs givingrise to spiral waves if vf > 0 and to antispiral ones in the other case. In this study, however,we are not interested in such distinction and generically we speak of spiral waves behavior.

Since spiral waves tend to plane wave solutions as r → +∞, it is reasonable to expectthat stability criteria for plane waves can also be applied to spiral waves. Plane waves arestable against long wavelength perturbation in the range q2 < q2

E = 1+αβ

3+αβ+2α2 and 1+αβ > 0

(Benjamin-Feir-Newell criterion), where qE defines the Eckhaus threshold [2]. Eckhaus in-stability arises from a longitudinal perturbation and the corresponding destabilization modeshave a group velocity vg . This kind of instability is of convective nature and it has beenshown that it does not necessarily cause the loss of pattern stability. In fact, in the convec-tively unstable range, localized fluctuations do grow, but it is not warranted that they growat fixed location, since they can drift away fast enough. In the region of convective insta-bility, the phenomenon of spiral wave nucleation can be observed: spiral wave solutionsbreak down but, after a certain transient time, two or more stable spirals appear [13]. Re-calling (13), we observe that, for positive d , the function f (d) = q2

s (d) − q2E(d) is positive

with a minimum at dm = 1.0608, such that f (dm) = .0959. Hence, for d = dm it results thatq2

s ≈ q2E . In the following we shall investigate the full system behavior in this special case.

5 Numerical Simulations

We consider our reaction-diffusion model (1)–(2) and choose the value of the diffusion coef-ficient d in a small neighborhood of d = 1.0608, i.e. d ∈ [0.9,1.2]; the other parameter val-ues are the same as for the spatially uniform case as given in Fig. 1, with C ∈ [1.6,Chopf ].Wesolve the PDE system (1)–(2) on the domain Ω = [0,100] × [0,100] and for t ∈ [0, T ] un-til asymptotic oscillating surface profile is attained. The numerical approximation of spiralwaves is a challenging task for several reasons: (i) in order to identify the solution structure,highly accurate discretization in space is needed, that is very fine meshes on large domainof integration must be used; (ii) long-time integration is required; (iii) oscillatory solutionsare expected.

We perform the semi-discretization in space by high order finite differences given by theExtended Central Difference Formulae (ECDFs) that approximate Neumann BCs with thesame accuracy. As far as the discretization in time is concerned, we apply the Peaceman-Rachford Alternating Direction Implicit (ADI) scheme to reduce the computational costs,for more details on this numerical approach refer to [22]. In particular, for the followingsimulations we apply the ADI-ECDF of order p = 4 in space and with explicit approxima-tion of the reaction terms. We fix Nx = Ny = 120 meshpoints in the space and ht = 0.004for time stepsize. The initial conditions are given by η0(x, y) = ηe + c · sη(x, y), θ0(x, y) =θe + c · sθ (x, y) with c = 10−5 and sη(x, y) = cos(Θ + R), sθ (x, y) = sin(Θ + R), whereΘ = arctan((y − L/2)/(x − L/2)),R = √

(x − L/2)2 + (y − L/2)2, archimedean spiralscentered in (x, y) = (50,50). In many cases, along the time evolution a restart technique ona macroscopic grid Tk = k�T, k = 1, . . . , n has been used such that �T = 20, Tn = Tf and


Fig. 3 Case d = 1, C = 2.2. Numerical simulations for the model (1)–(2) illustrating the emergence ofa bound state of spiral waves. Snapshots show different stages of the bound state formation during timeevolution

at the stage k + 1 we set η0(x, y) = η(x, y,Tk), θ0(x, y) = θ(x, y,Tk), that is the approxi-mations obtained at the previous stage k.

Figure 3 is the result of numerical simulations performed in the case d = 1, C = 2.2,near the border of the convectively unstable region: the emergence of a bound state of spiralwaves is shown. This turns out to be consistent with studies showing the existence of boundstates in the convective region of the CGLE with β = 0 (see [1, 2]).

In Fig. 4(a), we also note that, for fixed d and by decreasing C from 2.2, the aggregationof bound states increases in complexity at C = 2 resulting in an even more irregular struc-ture at C = 1.8 and 1.6. This same feature can be observed by considering an approximationof the space integral 〈η(t)〉 = 1

|Ω|∫

Ωη(x, y, t)dxdy which exhibits periodic oscillations for

the two higher values of C and behaviors for C = 1.8 and C = 1.6 that are aperiodic inthe investigated time interval, as shown in Fig. 4(b). However, in all the cases, the presenceof standing oscillations indicates the oscillatory nature of the asymptotic pattern. Numeri-cal investigations suggest that the above described phenomenology remains unchanged ford <≈ 1. In particular, Fig. 5 shows bound states of spiral waves obtained for d = 0.9 andC = 2.2. Also in this case, the time-dependent behavior of 〈η(t)〉 in Fig. 5(a) highlights anoscillating pattern. Moreover, snapshots in Fig. 5(b) also suggests a kind of breathing behav-ior [3]. This phenomenology is particularly interesting since it has been observed in variousoscillatory reaction diffusion systems, e.g. the oscillatory chlorine dioxide-iodinemalonicacid. Moreover, it has been reported in the Belousov-Zhabotinsky model as an intermediatebehavior between regular spiral waves and spiral turbulence [3]. In this respect, Fig. 5(c)shows a space-time plot along the line y = 32.5 (crossing the bottom spiral branch in itscore (xc, yc)) that reveals slight oscillations in the speed and shape of the waves, resultingin a weak breathing behavior. In addition, Fig. 5(d) shows that the shape of time dependentoscillations at different points outside the spiral core is constant in time and does not de-pend on the position. We finally wish to stress that, for d moving away from dm, a notablymore irregular phenomenology develops, that can be connected with the onset of absoluteinstability [23]. In particular, the case d = 1.2 and C = 2.2 highlights this aspect both in the


Fig. 4 (a) Spatial system behavior of η(x, y,T ) at T = 720 for decreasing values of the parameter C.(b) Time-dependent behavior of the space integral 〈η(t)〉 corresponding to spiral wave solutions in (a)

pattern spatial organization and, even more clearly in the complex 〈η(t)〉 time-dependentbehavior as depicted in Fig. 6.

6 Concluding Remarks

The varied phenomenology highlighted in this study suggests further comments on the roleof the relevant system parameters in order to further clarify their relationship to the exper-


Fig. 5 Case d = 0.9, C = 2.2. (a) 〈η(t)〉 in the interval t ∈ [720,743]. (b) Snapshots of numerical solu-tion corresponding to typical minima (t = 723), blue circle) and maxima (t = 743, red square) of 〈η(t)〉in the breathing motion shown in (a). (c) Space-time plot along the line y = 32.5 drawn through the core(xc, yc) of the bottom spiral; time increases vertically from t = 720 to t = 743. (d) Oscillations of the vari-able η(x0,32.5, t) calculated at three different points x0. Solid line (blue in the color version) corresponds tothe spiral core motion in (xc,32.5); dashed line (green in the color version) and dashed-dotted line (red inthe color version) correspond to the points (50,32.5) and (70,32.5), respectively (Color figure online)

Fig. 6 Case d = 1.2, C = 2.2. Left: Snapshot at t = 900. Right: 〈η(t)〉 time-dependent behavior

iments. As far as the spiral waves behavior is concerned, we have stressed the key role ofthe parameters C and d . In particular, the parameter C turns out to be strictly related to theinception of oscillatory behavior in the local kinetics, that occurs in the system when C isset below the supercritical Hopf bifurcation threshold Chopf . In the oscillatory regime forthe local kinetic, developing close to the Hopf bifurcation, we have shown that our systemcan exhibit spiral wave behavior. In this study, the case d ≈ 1 has been taken specificallyinto account. Recalling that the parameter d is defined as the ratio between the diffusioncoefficients of the two master equations, d = dθ/dη [8], we have focused on the case dθ

close enough to dη . When dθ <≈ dη and C ≈ Chopf we have found a regular spiral wave


behavior characterized by bound spiral wave states. The aggregation of these bound spiralstates increases in complexity when C becomes much lower than the bifurcation value. Onthe contrary, the case dθ >≈ dη leads to a kind of spiral break-up, characterized by irreg-ular spatio-temporal dynamics. The presented phenomenology turns out to be particularlyconsistent with the experiments. In fact, under the physico-chemical point of view, the pa-rameter C describes the rate of release of adsorbable species during electrodeposition; as aconsequence, if this quantity is lower than a critical threshold, the self-inhibiting effect of thegrowth process is lost and formation of dendritic structures—corresponding experimentallyto the oscillating behavior—and the development of irregular free outgrowth features are fa-vored. Moreover, the case dθ <≈ dη corresponds to a relatively low surface mobility of ad-sorbates, tending to stabilize growth patterns: this type of behavior is specially emphasizedby the oscillating growth, related—as recalled above—to the minimization of self-inhibitingelectrodeposition conditions. The results of the analysis described in this paper are a notablestep forward in the link between the reaction-diffusion modeling of electrodeposition mor-phochemistry and practical metal-plating, for the following reasons: (i) 3D growth shapeswith undercuts can be modeled and (ii) the dynamic scenarios brought about by changes inparameters can be easily compared with in situ video-rate microspectroscopy experimentstracking stepwise changes in electrodeposition conditions, such as additive injection. Thepossibility of designing dynamic experiments and comparing them with computed morpho-chemical transients is expected to yield a more insightful and robust approach to parameteridentification and sensitivity analysis, especially in view of disentangling combined effectsof experimental variables on model parameters. We finally stress that the capability to gen-erate spiral wave behavior, exhibited by this system, is a considerable further confirmationof the flexibility and generality of our metal electrodeposition model.

Acknowledgements The authors thank the anonymous Referees for their helpful comments and remarks.

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