Ion transport in thin cell electrodeposition: modelling three-ion electrolytes in dense branched morphology under constant voltage and current conditions

Electrochimica Acta 50 (2005) 3436–3445

Ion transport in thin cell electrodeposition: modelling three-ion electrolytes in dense branched morphology under constant voltage and current conditions G. Marshalla,b,∗ , F.V. Molinac , A. Sobab a Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Laboratorio de Sistemas Complejos, Departamento de Computaci´on, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina INQUIMAE, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina b

c

Received 11 July 2004; received in revised form 2 November 2004; accepted 20 December 2004 Available online 21 January 2005

Abstract Electrochemical deposition (ECD) and spatially coupled bipolar electrochemistry (SCBE) experiments in thin-layer cells are known to produce complex ion transport patterns concomitantly with the growth of dendrite-like structures. Here we present a macroscopic model of ECD and SCBE with a three-ion electrolyte in conditions of dense branched morphology. The model describes ion transport and deposit growth through the one-dimensional Nernst–Planck equations for ion transport, the Poisson equation for the electric field and, for ECD, a growth law for deposit evolution. We present numerical simulations for typical electrochemical deposition experiments: dense branched morphology in ECD and the incubation period in SCBE. In ECD the model predicts cation, anion and proton concentration profiles, electric field variations and deposit growth speed, that are in qualitative agreement with experiments; the predicted evolution and collision of the deposit and proton fronts reveal a time scaling close to those observed in experiments. In SCBE, the model predicts that the inverse of the incubation time scales linearly with the applied voltage. Such behaviour was observed in experiments. © 2004 Elsevier Ltd. All rights reserved. Keywords: Electrodeposition; Thin cells; Ion transport; Numerical simulations; Migration

1. Introduction The electrochemical deposition of metals is a topic intensely studied for more than one century, and the conditions needed to obtain a smooth morphology are well known [1–4]. The development of rough or dendritic structures, on the other hand, is less known. In particular, when metal cations are reduced under far from equilibrium conditions, in the absence of support electrolyte, branched structures develop, with dendrite-like growth. This form of electrodeposition, termed electrochemical deposition in thin cells (ECD), has

∗

Corresponding author. Tel.: +54 11 48268181; fax: +54 11 45763359. E-mail address: [email protected] (G. Marshall).

0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2004.12.018

become a paradigmatic model for studying growth pattern formation (GPF) [5–39]. In an ECD experiment, the electrolytic cell consists essentially of two glass plates sandwiching a binary electrolyte and two parallel electrodes of the same metal as the electrolyte cation (typically, a CuSO4 solution with copper electrodes or ZnSO4 with Zn electrodes). The thickness of the solution layer in the z-direction (the cell setup is shown in Fig. 1) is usually below 1 mm, thus yielding a quasi-two dimensional growth. Depending on cell geometry, electrolyte concentration, applied voltage and other parameters, the deposit can be fractal, dendritic or densely branched [5–31]. When the branches or dendrites grow as a uniform front rather than with a hierarchy of branch sizes, as illustrated in Fig. 2, the deposit is termed densely branched [10].

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

Fig. 1. Scheme of an ECD cell with the definition of coordinates. The deposit grows in the y-direction.

Issues of ECD are particularly pertinent in the case of macrowiring using bipolar electrochemistry (spatially coupled bipolar electrochemistry, SCBE, [40]), in which electrodissolution and electrodeposition in an applied electric field can be exploited to create directional growth of copper deposits between copper particles that are not connected to an external circuit. In analogy with ECD, in SCBE experiments, the cell consists of two microscope slides holding two platinum electrodes, the electrolyte solution (a diluted H2 SO4 solution) and two copper discs aligned in the field direction, with the distance between discs being much smaller than that between electrodes (the cell setup is shown in Fig. 3). When the circuit is closed a Cu2+ front propagates in the electrolyte solution from one copper particle (acting as anode and dissolving) to the next (acting as cathode) during the incubation period. When that front reaches the cathodic particle the incubation period ends and the electrodeposition starts. The incubation time is thus determined by the transport of metal cations, and is relevant for the characteristics and performance of the SCBE process. The experimental study of the SCBE evolution in the region between discs show that

Fig. 2. Schlieren snapshots (from [42]) showing two examples of morphology under galvanostatic conditions (0.1 M ZnSO4 solution, zinc electrodes): (a) hierarchical dendritic morphology (j = 40 mA cm−2 ), and (b) dense branched morphology (j = 80 mA cm−2 ).

3437

Fig. 3. Scheme of a SCBE cell with the definition of coordinates. The deposit grows in the y-direction between the copper discs. A and C are the platinum anode and cathode, respectively.

the inverse of the incubation time scales linearly with the electric field, indicating that during that period migration is the most relevant transport mode. After branching develops, convection plays a relevant role as well, the deposit front growing linearly in time, with a transition point at half the inter disc gap [41]. Fig. 4 shows the deposit morphology of an SCBE experiment when contact between the two discs has been made. The parallel between SCBE and ECD experiments runs as follows. SCBE consists in two different stages: during the first one, the so called incubation time, the electrodes are fixed since no electrodeposition is taking place, while a cloud of dissolved metal ions moves from anode to cathode; during the second stage, termed growth time, electrodeposition takes place. Clearly, the first stage is similar to an ECD experiment with a constant voltage, except that the initial electrolyte solution differs and there is absence of electrodeposition. The second stage is similar to an ECD experiment in the sense that there are already metal ions dissolved in the electrolyte. The transport of ions in ECD is due to a combination of migration, diffusion and convection and plays a crucial role in the growth of the deposit. In fact, dendrite growth induces a complex physicochemical hydrodynamic ion transport process, in which all transport modes (diffusion, migration and convection) are relevant. Convection is triggered by the

Fig. 4. A snapshot showing an SCBE connection between copper discs (reproduced from [42]) (1 × 10−4 M H2 SO4 , mean field applied 10 V cm−1 , distance between discs: 1 mm).

3438

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

presence of buoyancy and coulombic forces as follows. When a high current density flows through the cell, strong density gradients develop at both electrodes (due to the electrochemical reactions at the electrodes coupled to migratory transport) which lead to the development of gravity-driven convection rolls propagating from the electrodes [9–13]. After a few seconds, an instability develops, triggering the growth of a ramified deposit at the cathode. The deposit develops as a three-dimensional (3D) array of thin porous metallic filaments. Coulombic forces concentrate at the tips causing the development of electrically driven vortex rings, according to a model developed in [7,10–12,17,25,27]. The influence of convection was studied by Huth et al. [28] who used a particle tracking technique to measure the convective velocity field. Thus, the full picture of ion transport in these phenomena is rather complex. Some experimental results point to a key role of migration in the development of the ECD phenomenon: several researchers have shown that in many cases the deposit growth speed is equal or similar to the anion drift speed in the relatively unperturbed solution ahead of the growing tip (see for instance [10]). Also, in cells with thickness less than 50 ␮m, convection is strongly damped so that diffusion and mainly migration, are the dominant modes in ion transport [29]. Considerable theoretical efforts have been dedicated to predict the observed experimental behaviour in ECD. In a pioneering work, Chazalviel [7] (see also [8]), introduced a onedimensional (1D) model for ion transport in ECD. This model included only diffusion and migration as transport modes, and consisted essentially of the Nernst–Planck equations for ion transport and the Poisson equation for the electric field. The equations were solved considering a growthless regime (i.e., fixed boundaries), potentiostatic conditions (constant anode–cathode potential difference), and low concentrations (due to numerical constraints). Even with all these simplifications, numerical simulations gave useful results, allowing Chazalviel to predict the existence and behaviour of a depletion layer next to the cathode. This model gave great insight into the problem but was limited to steady state regimes. Marshall et al. [32] and Marshall and Mocskos [33] introduced the first two-dimensional macroscopic model describing ion transport in the presence of diffusion, migration and convection in an electro convective limited regime. The model also featured a stochastic growth model based on a Dielectrical Breakdown Model (DBM) [34]. The model was further extended in [35] to include gravity currents. The prediction of these models regarding ion transport were remarkable: the existence of vortex pairs near fixed filaments and of convection rolls near each electrode and their evolution following the same scaling as in the experiments, was corroborated. However, the growth law of the model was of limited value due to the fluctuations inherent in the DBM model giving rise to non realistic growth morphologies. Further comparison of simulations using those models with experimental results at different viscosities [37,38] demonstrated their ability for capturing the

most relevant aspects of the physical phenomena. Recently, the first results using a 3D numerical simulation, showing the interplay between gravity-driven and electrically driven convection, were presented by Marshall et al. [39]. All the numerical simulations thus far mentioned assumed a binary electrolyte and (excepting [33]) used fixed boundary conditions, either without deposit or with a static one (one or a few fingers of fixed dimensions) under potentiostatic conditions. Because of these restrictions, present theoretical modelling still is a rather limited approximation of reality, particularly in the question related to growth pattern formation and evolution. As said above, a binary electrolyte was always assumed. The experimental evidence, however, indicates that this is not the case. Changes in the growth rate and in the morphology of the deposit have been reported when fronts emanating from the anode encounters the deposit. The first pattern transition or Hecker effect [14–23] is associated with a migratory pH front (as it has been shown by the use of acid–base indicators), whereas a second pattern transition is due to the anodic convective front [28,36]. The evolution of the interaction of all the transport modes was presented in Fig. 9 of Huth et al. [28], here reproduced as Fig. 5. The figure is constructed from monochromatic video images taken with the camera focusing the electrodeposition cell from above. Measurements made each 10 s are averaged over 10% of the cell width (the x-direction) to reduce each image to a line; then these lines are stacked to yield the space–time diagram shown in the picture. This remarkable diagram shows in Fig. 5(a) the evolution of the deposit front, the migratory pH front, and the cathodic and anodic convection rolls. The point of intersection between the pH and the deposit fronts reveals the Hecker effect that induces a change in the deposit growth rate (associated with a morphology change). The figure also reveals the second transition in the deposit growth rate when the deposit meets the anodic concentration front. Fig. 5(b) shows concentration of zinc ions as a function of the distance from the cathode for different times. These results were obtained by titration of aliquotes of the cell solution at different times. We note that in these measurements the pH front detection was enhanced through the addition of H2 SO4 at a low concentration, thus the visualized front is actually the depletion of H+ ions; however, in other cases, as in the work of Otero et al. [23], protonic fronts coming from the anode, without any acid/base addition, have been observed. The origin of this front is due to the hydrolysis of the concentrated metallic cations produced by anode dissolution: Cu2+ + H2 O = CuOH+ + H+

(1)

The thermodynamic equilibrium constant of Eq. (1) is 1 × 10−8 , so that in Cu2+ solutions above about 0.01 M an acidic medium is present. The copper ions concentration is higher in the vicinity of the anode, thus the protons in excess migrate towards the cathode. The case of Zn is similar, but

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

3439

Fig. 5. Full space–time evolution of the ion transport modes (reproduced from [28]). Visualization of the different transport modes is done through the addition of tracer particles, sulphuric acid and a colour pH indicator. In (a) are shown the deposit front (darker pixels), the migratory pH or protonic front and the cathodic and anodic concentration fronts. In (b) it is shown the concentration of zinc ions as a function of the cathode–anode distance, for different times. Cell dimensions: 100 mm × 50 mm × 1.0 mm; 0.1 M ZnSO4 , 1 × 10−4 M H2 SO4 , j = 100 mA cm−2 .

the effect should be less pronounced, because the equilibrium constant is about 4 × 10−10 . The above considerations show that, for a more precise description, even in the simplest cases a three-ion electrolyte (typically Zn2+ or Cu2+ , H+ , and SO4 2− ) should be considered. A simplified model describing the Hecker effect was presented by Fleury et al. [17] assuming a pure migration process. Furthermore, Fleury et al. [10] presented calculations of ternary electrolytes under a galvanostatic and steady state growth regime describing a dense branched morphology. Trigueros et al. [24] presented similar calculations but in

the presence of sodium ions. In these models the simulations were done under potentiostatic conditions and the ternary electrolyte and the moving boundary were only considered in a simplified steady state regime. It is clear from previous discussions, the need for a fully three-dimensional model of ion transport taking into account a three-ion electrolyte, convective effects and a realistic aggregation model, for describing ECD problems. But this is, indeed, a formidable task. Here, as a first step towards that goal, we present a simplified 1D model that we claim, possesses many interesting

3440

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

features. The simplified macroscopic model for ECD and SCBE presented here is based in the following considerations. The ECD problem can be schematically viewed, from an electrical point of view, as a closed ohmic circuit containing the power source and the ‘one-dimensional’ cell. The circuit sees the electrolytic cell as a resistance, which is mainly a function of the deposit front-anode length (initially the cathode–anode length), decreasing in time as the deposit grows. In this context, assuming that ion transport is solely governed by diffusion and migration and a dense branched regime, when the circuit is closed, two concentration fronts emerge from each electrode advancing in opposite directions. Concomitantly, the deposit front advances towards the anode with a velocity proportional to the anion mobility times the electric field and a pH front emanating from the anode moves towards the cathode with a velocity proportional to the proton mobility. In SCBE we distinguish two well-defined stages: an incubation period with no deposit growth and a growth period where the deposit grows and eventually contact is made. In the incubation period, as in ECD, two concentration fronts emerge from each electrode advancing in opposite directions: a proton front and an anion front. In addition, a copper front advances towards the cathode; when it reaches the cathode electrodeposition starts and thus the growth period. Here, we concentrate in the study of the incubation period (the growth period is similar to ECD). Naturally, the model proposed here for ECD and SCBE is a first step towards a full 3D ion transport model in which convection and an aggregation model will be incorporated into the model.

The ion transport problem discussed above can be described in one dimension using the Nernst–Planck equations for ion transport [43–46] and the Poisson equation for the electric potential. This should be combined with an aggregation model describing the advance of the deposit. Here we assume, following [10] that the deposit front advances with velocity equal to the anion mobility times the electric field, which is consistent with a densely branched morphology. Morphology transitions can only occur when incoming fronts collide with the deposit front: in the present model, their space location can only be predicted in a crude way, tracking ion concentration fronts and their collision with the deposit front. Clearly, we neglect the influence of convection. The one-dimensional system of equations in dimensionless form can be written as (see Fig. 1 for the reference frame)

1 ji = −Mi Ci ∇φ − ∇Ci Pei
∇ 2 φ = Po zi Ci i

φ(y0 ) =

kT ln(CC (y0 , t)) zeφ0

 ∂CC  =0 ∂y y0  ∂CH  =0 ∂y y0 jA (y0 ) = 0

2. Theoretical analysis

∂Ci = −∇ · ji ∂t

Here φ is the dimensionless electrostatic potential, Ci and ji the dimensionless concentration and flux of the ionic species i, where i = C, A and H, standing for Me2+ (a divalent cation), SO4 2− and H+ ions, respectively. Hydrogen ions are considered to be present due to either H2 SO4 addition or to the hydrolysis of the salt cation (Eq. (1)). The MeOH+ ions produced in this process, having a lower concentration and roughly half the mobility than Me2+ , will be neglected. The quantities Mi = µi φ0 /x0 u0 , Pei = x0 u0 /Di , and Po = x02 C0 e/εφ0 , stand for the dimensionless numbers Migration, Peclet, and Electric Poisson, respectively (see details in [39]). The quantities zi , µi , and Di are respectively the charge number, mobility and diffusion coefficient of an ionic species i, µi and zi are signed quantities, being positive for cations and negative for anions, e is the electronic charge, and ε is the permittivity of the medium. x0 , u0 , φ0 , and C0 , are reference values of the length, velocity, electrostatic potential, and concentration, respectively (later discussed). Under a constant voltage ECD simulation, the boundary conditions for the cathode and the deposit (y = y0 , where y0 is the deposit position; in the absence of deposit y0 = 0), following Chazalviel [7] are:

(2) (3) (4)

(5) (6) (7) (8)

where jA is the anion flux. In addition to the usual conditions [7], it is assumed that the H+ ions behave at the cathode like the metal cations. The boundary conditions for the anode (y = 1) are: φ(1) = 1 −

kT ln(CC (1, t)) zeφ0

(9)

zC CC (1) = θzA CA (1)

(10)

zH CH (1) = (1 − θ)zA CA (1)

(11)

jA (1) = 0

(12)

where θ is a parameter varying between 0 and 1, its meaning later explained. In a galvanostatic ECD simulation, the boundary conditions for y = y0 are: φ(y0 ) =

kT ln(CC (y0 , t)) zeφ0

 ∂CC  =0 ∂y y0  ∂CH  =0 ∂y y0

(13) (14) (15)

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

jA (y0 ) = 0

(16)

where jC is the cation flux, and J the current density. The boundary conditions for the anode (y = 1) are: 1

dy z i i Ci µ i



φ(1) = J 0

(17)

jC (1) =

θJ zC e

(18)

jH (1) =

(1 − θ)J zH e

(19)

jA (1) = 0

(20)

where in Eq. (17) the cell resistivity it is assumed to be of migratory nature, thus neglecting diffusion effects in calculating the total voltage drop in the cell. As discussed in Section 1, the high concentration of metal cations generated by anode dissolution causes a local acidification due to a reaction described in Eq. (1), which is expected to be much faster than the transport processes. From the equilibrium constant and appropriate activity coefficients, the local H+ concentration could be computed. However, as discussed below, due to numerical constraints we are forced to use values for the dimensionless numbers which correspond to concentrations considerably lower than typical experimental values. Because of this, and also because we are mainly interested in the study of front evolution, a parameter θ is introduced in Eqs. (10), (11), (18) and (19) to simulate the presence of protons in the vicinity of the anode. θ is taken as the fraction of the anodic charge effectively resulting in the introduction of metal cations (θ represents the fraction of current spent in generating metal cations). Thus, θ = 1 means that only Me2+ cations are produced (so that no anodic proton wave will be present), whereas θ = 0, corresponds to the (hypothetical) case where only H+ ions appear at the anode. For convenience, a second parameter θ 0 representing the initial cation concentration in the cell, is introduced. Clearly, θ 0 is used to specify the initial MeSO4 /H2 SO4 ratio: θ 0 is the fraction of MeSO4 and (1 − θ 0 ) is the fraction of H2 SO4 . Thus, at t = 0, the dimensionless anion concentration CA is uniform and equal to 1, the Me2+ concentration CC is equal to θ 0 and the H+ concentration CH equals 2(1 − θ 0 ). Also, at t = 0 the electrostatic potential is a ramp function. Typical experimental values of the dimensionless numbers in a cell with L = 1.5 cm, d = 0.05 cm, C = 0.1 M and I = 10 mA, result in: MC = 0.66, MA = 1.0, MH = 4.4, PeC = 18, PeA = 12, PeH = 2.7, Po = 18 × 108 (the reference values used here were: x0 = d and u0 = µA E0 = µA φ0 /x0 [35]). Also, the limiting ionic mobilities are used, thus neglecting non-ideal corrections. The range of values of the dimensionless numbers reflects the disparity of scales of the processes involved in ECD ([35]) yielding a stiff numerical problem. The one-dimensional system is solved, for each time step in a fixed or moving domain on a two-dimensional uni-

3441

form lattice using finite differences and deterministic relaxation techniques, under potentiostatic and galvanostatic conditions. These conditions are reflected by the boundary conditions imposed at the end of the deposit-anode distance (see Eqs. (8)–(20)). The numerical simulation of a potentiostatic ECD experiment for a fixed cathode is straightforward and has already been described in the literature [7]. For a moving cathode, in each time step, given the voltage V at the boundaries, ion concentrations and the potential are calculated inside the cell; the front is then advanced with a velocity proportional to the anion mobility times the electric field. Here, for advancing the front, we impose an empirical functional relation between the front velocity and the electric field, in which the mobility is considered constant and the electric field is calculated at each time step, as the average of the local electric field in front of the deposit. This value emerge, at each time step, from the numerical solution of the complete system of Eqs. (2)–(20). In the simulation of a galvanostatic experiment, in each time step, given the electric current density J, the global voltage drop is calculated through Ohm’s law using the migratory cell resistance Eq. (17). The calculated voltage is used as boundary condition and the calculation proceeds as in the potentiostatic case, except for the remaining boundary conditions that now must satisfy Eqs. (13)–(16) and (18)–(20).

3. Numerical results and discussion Here we present numerical simulation results with the aim of describing ECD and SCBE experiments, under potentiostatic and galvanostatic conditions, respectively. The ECD experiments presented in Huth et al. [28] is mimicked with a ternary electrolyte and moving boundaries (dense branched morphology assumption) under galvanostatic conditions. The SCBE experiment is mimicked in its initial phase (before aggregation starts) with fixed boundaries under constant voltage conditions. The disparity of scales of the processes being simulated precludes the use of some of the real dimensionless numbers; in particular, the value of the Poisson number employed in the simulations (Po = 500) differ considerably from those found in typical experiments. This restriction is tantamount, in the framework of a 1D model, to a low C0 value. The computational model is written in the C language and implemented on a Pentium class computer. All the results are plotted in dimensionless form. 3.1. The ECD numerical simulation In the zinc electrodeposition experiment presented in Huth et al. [28] shown in Fig. 5, initially there is a uniform MeSO4 solution concentration to which a small amount of H2 SO4 has been added. As time evolves, as seen in Fig. 5(a), two concentration fronts emerge from each electrode advancing in opposite directions: a higher concentration front from the anode, and a depletion front from the cathode. Meanwhile,

3442

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

Fig. 7. Deposit and pH front trajectories for a MeSO4 + H2 SO4 solution (θ 0 = 0.9), with no proton production at the anode (θ = 1), under galvanostatic conditions.

Fig. 6. Concentration and potential profiles for a MeSO4 + H2 SO4 solution (θ 0 = 0.9), with no proton production at the anode (θ = 1), and deposit advancing as anion mobility times the electric field under galvanostatic conditions: (a) t = 150; (b) t = 250; (c) t = 450.

a deposit front advances towards the anode with a velocity proportional to the anion mobility times the electric field, and a pH front emanating from the anode moves towards the cathode with a velocity that is proportional to the proton mobility times the electric field. The pH front is visualized by the addition of a pH indicator to the solution. The measured velocity of the pH front coincides with the proton migration. The deposit and the pH front meet at about 35% of the cathode–anode distance, at which place, a morphology transition occurs. Here, we present numerical simulations of that experiment in which the initial and boundary conditions for the protons are mimicked with the values θ 0 = 0.9 and θ = 1 (proton production at the anode is considered negligible in comparison with the acid already present), and the deposit front is advanced proportional to the electric field times the anion mobility. Fig. 6 shows the potential and concentration profiles at different times predicted by our model. Initially, there is a uniform ion concentration (cations and ions). As time evolves, two concentration fronts emerge from each electrode advancing in opposite directions, a deposit front advances towards the anode with a velocity proportional to the anion mobility times the electric field, and a high pH front (that is, a depletion of H+ ions) emanating from the anode moves towards the cathode with a speed of a migration front. Cation and anion concentrations profiles are in

qualitative agreement with the experimental measurements of Fig. 5(b). Fig. 7 shows simulated deposit and pH front trajectories under galvanostatic conditions (the characterization of the proton trajectory has been obtained numerically as follows: a time stack formed with the successive values of the proton concentration for different time steps results in a space–time surface CH (y, t), a particular contour of which is plotted in the figure). As seen in the figure, the deposit reaches the proton depletion front at about t = 250 at a position y ≈ 0.3, in reasonable agreement with the experimental results of Fig. 5 (y ≈ 0.35). In our model, no change in the velocity of the front is observed in its encounter with the protonic front. This is because we used constant values for the mobilities rather than using, as it was done in Ref. [10], values of the mobilities as a function of ionic concentration, which indeed, would be a refinement of the model. The deposit front speed is simulated as the anion mobility times the average electric field in front of the deposit. The use of an averaged field approximation to simulate the deposit front speed appears to be reasonable, since this trajectory limits the evolution of a depletion zone with a high potential drop concentrated in a small spatial range. In the experiments, a delay (or induction time) is usually observed before the branched ECD starts to develop. Presumably, during this time a depletion layer effectively develops, triggering the branched pattern formation, but is prevented from increasing by the deposit growth. On the other hand, in the experiments, changes in the deposit growth speed (and morphology) are usually observed when the deposit front collides with a protonic front. Again, in our simulations little or no change in deposit growth speed is observed, because our model is invariant to changes in the morphology produced by anodic roll and front collision; however, their spatial location is qualitatively well predicted. We are presently extending the model to 2D and 3D with the inclusion of convective terms that will indeed affect the deposit trajectory. In summary, an appropriate election of E and an appropriate functional relation between the mobility and the concentration, would reveal transitions in the velocity slope due to concentration fronts and the Hecker effect. We will be able

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

3443

to see these transitions when we include convective effects and use variable mobilities in our 3D extension of the present model (work in progress). 3.2. The SCBE numerical simulation In the SCBE experiments presented in Bradley et al. [40,41], in the gap between discs, initially there is a uniform H2 SO4 concentration within the copper discs. Here we focus our simulations in the inter disc gap, with the simulated anode and cathode corresponding to the copper discs nearest to the platinum anode and cathode, respectively (see Fig. 3 and [40,41]). With the passage of an electric current, the protons migrate towards the cathode faster than the copper cations, generating a cationic depletion (anion excess) zone. Electroneutrality thus imposes the same velocity for both cations, with a boundary between them. Experiments show that the inverse of the incubation time (time needed for dissolved anodic copper to arrive to the cathode [40]) scales linearly with the electric field, indicating that during this period migration is the most relevant transport mode. This is shown in Fig. 11 from [41] where a plot of the inverse of the incubation time versus the electric field yields a linear law with the slope related to µCu2+ . After branching develops, as shown in [41], convection plays a relevant role as well, the deposit front growing linearly in time. Here we present numerical simulations under potentiostatic conditions with fixed boundaries in a mimic of the first part of the SCBE experiment, that is, during incubation time. The fixed boundaries of the domain are determined by the gap between the copper discs, and the electrostatic potential difference between the boundaries of the domain is taken constant in time. The experimental conditions are mimicked by θ 0 = 0 and θ = 1. Fig. 8 presents simulations of the concentration and potential profiles for different times with a fixed cathode–anode distance and under potentiostatic conditions. As shown in the figure, our model predicts a depletion proton front and a copper front moving from anode towards the cathode and an anion front moving towards the anode. This yields a boundary or limit between H+ and Me2+ zones that is well defined (a consequence of the different mobilities involved), although as earlier mentioned, the computed gradients are considerably less pronounced than the real ones. The charge unbalance in this boundary region is not apparent from the concentration profiles, but is revealed in Fig. 8(a) and (b), by the change in slope of the potential profile, as dictated by Eq. (4) (this profile has a less pronounced slope in the protonic region, in accordance with the lower resistance presented by an acidic solution). Fig. 8(c) corresponds to the time when protons have almost vanished; at a later time, the profiles are similar to those found in final stages of ECD simulations in the absence of protons [7]. Fig. 9 presents the time evolution of the cell resistivity under galvanostatic and potentiostatic conditions. The cell resistance increases with time, due to the replacement of the

Fig. 8. Concentration and potential profiles for an acid solution with a metallic anode, showing the replacement of H+ by Me2+ , with a fixed cathode–anode distance under potentiostatic conditions for different times: (a) t = 300; (b) t = 450; (c) t = 700.

fast protons with slower metal cations. When protons disappear, the resistance reaches a constant value, because the overall solution composition remains invariant. The cell resistance under galvanostatic conditions (Fig. 9, solid line) increases at a faster rate than under potentiostatic conditions, because in the former the boundary advances more rapidly. Under constant applied voltage, a resistance increase results in a decrease of the current and, as a consequence, a decrease in the electric field in the vicinity of the deposit. Fig. 10 shows the space–time trajectories of the boundary between the metal cation and the proton zones previously mentioned, here called yb , under galvanostatic and potentiostatic conditions. We observe that at constant voltage, the velocity of trajectories decreases with time due to a decrease in the current; the decrease in current is caused by the increase in resistance shown in Fig. 9. The velocity increase

Fig. 9. Time dependence of the resistance (assumed migratory) for an acid solution (θ 0 = 0) with a metallic anode (θ = 1), with a fixed cathode–anode distance under galvanostatic (solid line) and potentiostatic (dotted line) conditions.

3444

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

Fig. 10. Time dependence of the H+ and Me2+ boundary position, for an acid solution with a metallic anode (θ = 1, θ 0 = 0), with a fixed cathode–anode distance: (a) potentiostatic conditions, and (b) galvanostatic (the continuous line indicates the straight portion of the curve).

found at the end of the curve corresponds to the extinction of the proton rich zone. In the galvanostatic regime, the velocity is nearly constant. It is worth mentioning that this case resembles the wellknown moving boundary method for measuring transport numbers [47]. In a classic application, an anodically generated Cu2+ solution replaces an acidic one under galvanostatic conditions, with a sharp boundary between both solutions. The plot of the boundary position as a function of time is linear. The proton transport number is obtained from the slope of this plot. In the moving boundary method, the evolution of the concentration and electrical potential profiles is similar to those shown in Fig. 8. In our simulations, the boundary trajectory under galvanostatic conditions (Fig. 10, curve b) shows, in its central section, a straight line as expected. From the slope of this section, the limiting transport number of H+ ions in sulphuric acid is recovered within 1%, thus demonstrating the consistency of our model simulations. The initial part of curve b in Fig. 10, with a somewhat higher slope, corresponds to the establishment of the boundary. The final part, as discussed above, is due to the vanishing of the proton zone. The features of the limiting boundary method were qualitatively predicted a long time ago, but the actual concentration and electric potential profiles are shown here for the first time. In the SCBE experiments, as previously mentioned, the inverse of the incubation time (the time needed by the metal cation wave to reach the cathodic disc) versus the mean electric field yields a linear law with a slope related to the mobility of Cu2+ , µCu2+ : 1/tinc = µCu2+ E/L

Fig. 11. Inverse of the incubation time vs. applied voltage in an SCBE simulated experiment.

different voltages. The plot yields a slope having the same order of magnitude as the metal limiting mobility, about 10−4 cm2 V−1 s−1 . This result demonstrates the migratory character of the simulated metal front coming from the anode, and therefore, the good predicting capabilities of our model.

4. Conclusions We presented a macroscopic model of ECD and SCBE with a three-ion electrolyte and densely branched morphology and numerical simulations for typical electrochemical deposition experiments: densely branched morphology in ECD and the incubation period in SCBE. We conclude that the general qualitative behaviour of the ECD and SCBE experiments is well represented by our model. In particular, the model predicts, for ECD experiments, ionic concentration profiles, electric field variations and deposit growth speed that are in qualitative agreement with experiments; moreover, the evolution and collision of the deposit and proton fronts reveal a time scaling close to those observed in experiments. Likewise, in SCBE the predicted evolution of concentration fronts and boundary zones between H+ and Me2+ appears to be reasonable. In particular, the model predicts that the applied voltage scales linearly with the inverse of the incubation time, which is in agreement with experiments. At present, we are extending a 3D version of the model presented here to include the effects of convection on ion transport.

(21)

as it was experimentally verified (Fig. 11 from [41]). Here, tinc is the incubation time, L the cathode–anode distance (actually, interdisc distance in a real experiment) and E the mean electric field. We carried out constant voltage simulations, for the SCBE case (θ 0 = 0 and θ = 1), using different values for the boundary φ(1); from these simulations, the time needed for the H+ /Me2+ boundary to reach the cathode was obtained. Our model predictions, presented in Fig. 11, show a linear graph of the inverse of the incubation time for

Acknowledgements We thank anonymous referees for their valuable comments. GM and FVM are investigators at the National Research Council of Argentina (CONICET). AS is supported by CNEA and University of Buenos Aires (UBA). This work was partially supported by UBA Grant No. TX187/02, FOMEC Grant No. 376/98 from the Department of Computer Sciences, FCEyN, UBA and CONICET Grant No. PIP379/98.

G. Marshall et al. / Electrochimica Acta 50 (2005) 3436–3445

References [1] M. Thompson, Theoretical and Applied Electrochemistry, MacMillan, New York, 1939. [2] A.R. West, Solid State Chemistry and its Applications, Wiley, New York, 1992. [3] D.M. Kolb, in: R.C. Alkire, D.M. Kolb (Eds.), Advances in Electrochemical Science and Engineering, vol. 7, Wiley/VCH, Weinheim, 2002, p. 107. [4] G.R. Stafford, C.L. Hussey, in: R.C. Alkire, D.M. Kolb (Eds.), Advances in Electrochemical Science and Engineering, vol. 7, Wiley/VCH, Weinheim, 2002, p. 275. [5] T. Vicsek, Fractal Growth Phenomena, 2nd ed., World Scientific, Singapore, 1992. [6] F. Argoul, J. Huth, P. Merzeau, A. Arneodo, H.L. Swinney, Physica D 62 (1993) 170. [7] J.N. Chazalviel, Phys. Rev. A 42 (1990) 7355. [8] J.N. Chazalviel, Coulomb Screening by Mobile Charges Applications to Material Science, Chemistry and Biology, Birkhauser Verlag, 1999. [9] R.M. Brady, Ball, Nature (London) 309 (1984) 225. [10] V. Fleury, J.N. Chazalviel, M. Rosso, B. Sapoval, Phys. Rev. A 44 (1991) 6693. [11] V. Fleury, J.N. Chazalviel, M. Rosso, Phys. Rev. Lett. 68 (1992) 2492. [12] V. Fleury, J.N. Chazalviel, M. Rosso, Phys. Rev. E 48 (1993) 1279. [13] D. Barkey, P. Garik, E. Ben-Jacob, B. Millar, B. Orr, J. Electrochem. Soc. 139 (1992) 1044. [14] N. Hecker, D.J. Grier, L.M. Sander, in: R.B. Laibowitz, B.B. Mandelbrot, D.E. Passoja (Eds.), Fractal Aspects of Materials, Materials Research Society, University Park, PA, 1985. [15] P. Garik, D. Barkey, E. Ben-Jacob, E. Bochner, N. Broxholm, B. Miller, B. Orr, R. Zamir, Phys. Rev. Lett. 62 (1989) 2703. [16] J.R. Melrose, D.B. Hibbert, R.C. Ball, Phys. Rev. Lett. 65 (1990) 3009. [17] V. Fleury, M. Rosso, J.N. Chazalviel, Phys. Rev. A 43 (1991) 6908. [18] P. Trigueros, J. Claret, F. Mas, F. Sagues, J. Electroanal. Chem. 312 (1991) 219. [19] A. Kuhn, F. Argoul, Fractals 1 (1993) 451. [20] P. Carro, S.L. Marchiano, A. Hernandez Creus, S. Gonzalez, R.C. Salvarezza, A.J. Arvia, Phys. Rev. E 48 (1993) 2374. [21] A. Kuhn, F. Argoul, J. Electroanal. Chem. 371 (1994) 93. [22] A. Kuhn, F. Argoul, Phys. Rev. E 49 (1994) 4298. [23] D. Otero, G. Marshall, S. Tagtachian, Fractals 4 (1996) 7. [24] P.P. Trigueros, F. Sagues, J. Claret, Phys. Rev. E 49 (1994) 4328.

3445

[25] V. Fleury, J. Kaufman, B. Hibbert, Nature 367 (1994) 435. [26] K.A. Linehan, J.R. de Bruyn, Can. J. Phys. 73 (1995) 177. [27] V. Fleury, M. Rosso, J.N. Chazalviel, in: R.A. Lowden, J.R. Hellmann, M.K. Ferber, S.G. DiPietro, K.K. Chawla (Eds.), Ceramic Matrix Composites: Advanced High-Temperature Structured Materials, MRS Symposia Proc 367, Material Research Society, Pittsburg, 1995, p. 183. [28] J. Huth, H. Swinney, W. McCormick, A. Kuhn, F. Argoul, Phys. Rev. E 51 (1995) 3444. [29] C. Leger, J. Elezgaray, F. Argoul, Phys. Rev. Lett. 78 (1997) 5010. [30] M.Q. Lopez-Salvans, F. Sagues, J. Claret, J. Bassas, Phys. Rev. E 56 (1997) 6869. [31] D. Barkey, in: Alkire, Kolb (Eds.), Advances in Electrochemical Science and Engineering, vol. 7, Wiley/VCH, 2000. [32] G. Marshall, E. Perone, P. Tarela, P. Mocskos, Chaos Solitons Fract. 6 (1995) 315. [33] G. Marshall, P. Mocskos, Phys. Rev. E 55 (1997) 549. [34] L. Pietronero, H.J. Weismann, J. Stat. Phys. 36 (1984) 909. [35] G. Marshall, P. Mocskos, H.L. Swinney, J.M. Huth, Phys. Rev. E 59 (1999) 2157. [36] S. Dengra, G. Marshall, F. Molina, J. Phys. Soc. Jpn. 69 (2000) 963. [37] G. Gonzalez, G. Marshall, F.V. Molina, S. Dengra, M. Rosso, J. Electrochem. Soc. 148 (2001) C479. [38] G. Gonzalez, G. Marshall, F.V. Molina, S. Dengra, Phys. Rev. E 65 (2002) 051607. [39] G. Marshall, E. Mocskos, F.V. Molina, S. Dengra, Phys. Rev. E 68 (2003) 021607. [40] J.C. Bradley, H.M. Chen, J. Crawford, J. Eckert, K. Ernazarova, T. Kurzeja, M. Lin, M. McGee, W. Nadler, S.M. Stephens, Nature 389 (1997) 268. [41] J.C. Bradley, S. Dengra, G.A. Gonzalez, G. Marshall, F.V. Molina, J. Electroanal. Chem. 478 (1999) 128. [42] G. Gonzalez, Transporte ionico y patrones de crecimiento en electrodeposicion ramificada, PhD Thesis, University of Buenos Aires, 2003. [43] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Wiley, New York, 1980. [44] J.S. Newman, Electrochemical Systems, Prentice-Hall, New Jersey, 1973. [45] R.F. Probstein, Physicochemical Hydrodynamics: An Introduction, Wiley, New York, 1994. [46] V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, New Jersey, 1962. [47] M. Spiro, in: A. Weissberger, B.W. Rossiter (Eds.), Techniques of Chemistry, vol. IIA, Wiley/Interscience, New York, 1971, p. 205.