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Growth and forms in quasi-two-dimensional electrocrystallization
Physics Reports 337 (2000) 97}115
Growth and forms in quasi-two-dimensional electrocrystallization Francesc SagueH s*, Marta Queralt LoH pez-Salvans, Josep Claret Departament de Qun& mica Fn& sica, Universitat de Barcelona, Martn& i Franque% s 1, 08028 Barcelona, Catalonia, Spain
Abstract A review on the morphologies and corresponding growth dynamics observed in quasi-two-dimensional electrocrystallization is presented. Emphasis is devoted to the understanding of the speci"c and growth mode-dependent role played by the di!erent transport mechanisms, i.e. di!usion, migration and convection, as well as the chemical processes. The role of chemical reactions is decissive for the understanding of the well-known morphological transitions and is at the origin of the "nger-like texture. 2000 Elsevier Science B.V. All rights reserved. PACS: 05.45.!a
1. Introduction Quasi-two-dimensional electrocrystallization, or simply electro(chemical) deposition, ECD, as it will be referred to on most of what follows, constitutes a fascinating example of a growth phenomenon operated under nonequilibrium conditions [1]. In its simplest form, it consists in the aggregation of a metallic deposit from the electrochemical reduction of its cations dissolved in a thin layer of an electrolyte solution. Although standard electrodeposition, i.e. operated in three-dimensional cells, has been a particularly familiar subject to electrochemists for many years, it has been only during these last two decades, when conducted in thin geometries, that has been widely recognized as a genuine example of growth process, similar and even phenomenologically richer as compared to other interfacial phenomena studied exhaustively within the physics community [2]. In relation with the just mentioned wide scenario of phenomena, ECD o!ers the opportunity to investigate very di!erent morphologies, from largely rami"ed to well-ordered anisotropic dendritic * Corresponding author. E-mail address: [email protected] (F. SagueH s). 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 5 7 - 0
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structures and even comprising quite unexpected digitation forms, very similar indeed to those found in viscous #uid displacement. Such a diversity is easily observed by appropriately varying a few basic paramenters of the experimental system, mainly the composition of the electrodepositing solution, and/or the driving force for the growth process, either the current intensity or the applied electric potential. The richness of the phenomenon indicates that beyond its apparent experimental and even conceptual simplicity, many times somewhat misleadingly invoked, several di!erent physicochemical ingredients are delicately coupled in any ECD experiment. Actually, this forces us to adopt an enlarged perspective with respect to the predominant position at the mid-1980s, when research on ECD was initiated [3}5]. According to the dominant idea at those times, ECD would be essentially a paradigmatic example of a Laplacian, DLA-like, pattern-forming system. In fact, during these last 15 years, much more complete, diverse and systematically oriented experiments conducted by our group and many others have enabled us to start to have a more complete understanding of the process. What is more remarkable is that such a deeper comprehension of the phenomenon at hand has developed mainly when widening our attention and redirecting it from the morphological and scale invariance properties of the observed deposits, towards the basic chemical and transport mechanisms behind the whole electrodeposition process, i.e. di!usion, electric migration and convection, and the essential role they play in selecting the observed morphologies. The present contribution intends to brie#y summarize the work of our group following this aim. Our research, and in this way it is sketched on what follows, adopted from the very beginning a multiperspective and as a consequence has followed di!erent strategies and methodologies. It is worth recalling them at this point as a way to comment on our choice of the selected material and how it appears organized in this contribution. After Introduction, Section 2 is devoted to give a very schematic presentation of the standard experimental protocol in ECD. Section 3 reports on our starting experiments on ECD, which essentially were aimed at classifying the di!erent morphologies observed when electrodepositing zinc sulphate solutions in parallel cells under constant applied potentials. The general idea expressed in the paragraph above and concerning the di!erent mechanisms involved in ECD is shortly ellaborated in Section 4, where a sort of minimal model, based on the "rst-principle equations governing the process, is also proposed. Purely di!usive-like morphological features of tree-like ECD deposits, obtained in the limit of very low applied potentials and high electrolyte concentrations, are examined in Section 5. It is there where aboundant scaling arguments borrowed from the by now standard fractal and multifractal methodology, are employed to resolve statically and dynamically the electrodeposited structures. Changing considerably the experimental conditions, by applying large potentials to very low concentrations, we observe a completely di!erent morphological pattern. It consists in arrays of metallic "laments enclosed with a well-de"ned advancing front. As explained in Section 6, such a texture, commonly referred to in the literature as dense parallel, or homogeneous as we sometimes use it, turns out to be the best candidate to be examined in terms of an ohmic model of the growth process, i.e. by invoking simple electric migration concepts. One of the most peculiar experimental observations in ECD refers to the morphological transitions experienced by the deposits as they advance through the electrolized solution. As it is discussed in Section 7, the understanding of such morphological changes needs for a precise characterization of the most chemically oriented aspects of the ECD process. Evenmore, it turned out that when we were investigating those transitions we encountered direct evidences for the last, and for a long time
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overlooked, transport ingredient involved in ECD experiments, i.e. buoyancy driven convection. Its role is emphasized in Section 8, fully devoted to describe the recently observed "ngered-like morphology as it appears when electrolyzing copper sulphate solutions containing small amounts of an inert electrolyte. Finally, this contribution ends with a summary.
2. Experimental system In a typical experiment [6], electrodeposits are grown from a thin layer of an aqueous electrolyte solution sandwiched between two 5 mm thick glass plates enclosing two parallel metallic electrodes separated by a distance of 4 cm. The cell gap is determined by the electrodes, typically copper wires of ca. 100 lm diameter. Since the composition and morphology of electrodeposits are very sensitive to the solution purity, they are prepared with ultra pure water and p.a. quality chemicals. Most of our experiments are conducted applying a constant potential di!erence between the cathode and the anode. The conductivity and pH of the initial electrolyte solution are measured under the same conditions as for the ECD experiment. The temporal evolution of current passing through the cell and in some cases, that of cathode potential are conveniently monitored and processed in a personal computer. In this last kind of measurements, an additional quasi-reference electrode (typically a copper wire) must be included in the electrochemical cell [7]. Experiments are carried out at room temperature. In order to quantitatively analyse the morphology of the electrodeposits, they are recorded using a stereo zoom microscope connected to a standard video camera, transferred to a PC-based image analysis system and conveniently digitised with a 512;512 resolution. The fractal features of the deposit structure are evaluated by means of the MFRAC software [8]. The chemical composition of the deposits is investigated by means of X-ray powder di!raction experiments using a di!ractometer with Cu Ka radiation (j"1.5418 As ) at 40 kV and 30 mA [9]. In order to prepare the samples, electrodeposits are collected, dried and stored under an argon atmosphere before the measurement and then, mounted over a Si(001) single crystal.
3. Diagram of ECD morphologies Electrodeposition of zinc from a thin layer of a non-deaerated aqueous zinc sulphate solution is a clear example of the wide morphological diversity observed in ECD experiments. These morphologies depend on the applied potential and the electrolyte concentration and are represented in a sort of morphological diagram in Fig. 1 [6,10]. Five typical patterns can be identi"ed: (a) Compact: a non-structured deposit with a rough interface is obtained at the lowest applied potential values (up to 2 V). In this case, the growth velocity is extremely low and the deposit texture is characterized by the absence of rami"cation at least at cell length scales. (b) Homogeneous or dense parallel: an array of regularly spaced trees is formed at low zinc sulphate concentrations (up to 0.04 M). The main characteristics of this morphology are: (i) the tips of the trees de"ne a growing front parallel to the cathode and retaining in this way the symmetry of the electrochemical cell; (ii) the growth velocity de"ned by the advancing front increases with applied potential; and (iii) the spacing between trees and the branching e!ects both decrease when
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Fig. 1. Diagram of morphologies for zinc electrodeposition from a non-deaerated ZnSO aqueous solution in a parallel cell. Cell thickness: 70 lm. Electrode separation: 3 cm.
increasing the applied voltage. In the limit of high potential values, a columnar (channel like) and non-rami"ed deposit is formed. (c) Dendritic: typical dendrites are obtained at intermediate concentration values ranging from 0.05 to 0.1 M. These patterns are de"ned by straight and anisotropic backbones with a considerable amount of side branching. Unlike the dense parallel morphology, these dendrites do not de"ne any sort of advancing front but rather some of them grow faster and remain as the only growing structures at high development stages. Although a growth velocity for the whole deposit cannot be de"ned, the individual growth rate of these growing dendrites is higher than that of homogeneous patterns at similar potential values. (d) Open fractal: at high concentration and low applied potential values (up to 10 V), a new morphology is formed displaying continuous tip splitting of the growing trees that leads to the formation of highly rami"ed and self-similar structures. Three characteristics of these deposits are
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worth mentioning: (i) the absence of a well-de"ned growing front; (ii) their growth that occur at very low velocities; and (iii) screening e!ects that prevent the further development of some initial trees or of the less exposed regions of a growing tree. (e) Mixed: this pattern does not correspond to a pure morphology but rather shows features characteristic of both dendritic and open fractal textures. The trees consist of main dendritic backbones, but the lateral branches are typical open fractal structures. Increasing the applied potential the lateral branching decreases in such a way that at very high values (550 V), the deposit is formed by non-rami"ed stringy patterns with a very high growth velocity. On all what follows we leave apart both the compact morphology, which is likely to be essentially controlled by the electrochemical reduction process itself [11], and the pure and mixed dendritic forms. Actually, dendritic structures would merit a detailed study on its own which was not pursued in our group.
4. Electrocrystallization mechanisms As in any non-equilibrium growth process, two di!erent categories of elementary processes are involved in even the simplest realization of an electrodeposition experiment. The "rst category refers to the transport mechanisms that bring the individual components to be deposited close to the advancing interface, where they are going to aggregate. The second class of processes concern the interfacial processes themselves. Let's comment a little bit more extensively on the di!erent mechanisms belonging to these two categories. On what concerns the transport mechanisms, one could in principle refer to two basic components: di!usion and migration. The di!usive mode, genuinous to most of the non-equilibrium growth process one can imagine, appears as a consequence of the concentration gradients built inside the electrolized solution as it is being invaded by the advancing interface. Transport by di!usion is associated with a typical length scale, the standard di!usion length scale, given in terms of the ratio between the di!usion coe$cient of the electrodeposited cations and the growth velocity of the interface. On the other hand, what is singular of ECD is the fact that the electrodepositing species, being electrically charged, experience typical migration processes under the applied potential di!erence that drive the growth process. In principle, the migration transport acts on a much larger length scale, typical of the cell dimension, as de"ned by the distance between both electrodes. A third transport mechanism, largely overlooked for many years in the context of ECD, refers to convection. Without being too precise, we could even distinguish between two di!erent modes of convective phenomena in ECD experiments. The "rst one, generically referred to as electroconvection [12,13], involve small scale #uid motion con"ned to the very close proximity of the metallic deposit. The second one, much more interesting on what follows, is known as buoyancy-driven convection [14}16]. For cells whose thickness is about or exceeds around a hundred of microns, the concentration gradients induced by the reduction and oxidation processes on the respective electrodes result in density gradients. These inhomogeneities, in turn, triger the propagation of convective rotating rolls extending from each electrode until they merge and encompass, in large covective currents, the whole cell. Naturally to say, such convective motions not only modify the transport of the depositing cations, but may markedly in#uence the physicochemical properties of the solution close to the metallic aggregate.
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Now, let's turn to the second family of mechanisms, i.e. those related to the electrochemical reactions taking place at the growing cathode. Under the usual standard conditions to which our experiments refer, the interfacial kinetics is assumed to be fast as compared to the previously mentioned transport mechanisms. However, this does not mean that we can completely forget the electrochemical processes that occur at the cathode. The most important role is played by possible concurrent reactions [7,17}21] accompanying the reduction process itself of the depositing cations. Several reactions could be considered depending on the composition of the solution or even the presence of impurities, the most important, however, being hydrogen evolution from water reduction. The implications of such a reaction will be emphasized later on in Sections 7 and 8, when referring to morphological transitions and singularly to the "ngering growth mode. Let's convert the picture just given of a typical electrodeposition process into a sort of simpli"ed and comprehensive model. We will build it for the simplest situation corresponding to the electrodeposition of a binary electrolyte, i.e., composed simply of the cations to be deposited and the corresponding anions. Furthermore and for the sake of simplicity we will neglect the presence of any concurrent reaction di!erent from the pure reduction of the depositing cations. We start by writing down the basic transport equation for the depositing cations in terms of the three previously mentioned ingredients, i.e. di!usion, migration and convection (Nernst}Planck equation [22]). Thus, the #ux of cations, J , is expressed in terms of the concentration gradient, C , the electric "eld, E, and the advection velocity, V, zFD C E#C V , J "!DC # R¹ being such a relation readily transformed into a bulk equation zF ) (DC )# ) C V . * C " ) (DC )# R R¹
(1)
Now, we have to supplement such a bulk relation with boundary conditions for the #ux of cations at the interface. Actually this is formulated as a conservation equation ov ) n"!MJ ) n , (2) relating the growth velocity, v, to the #ux of cations. In turn such a #ux is related to the current density, j j"nFJ , (3) where n is the number of electrons involved in the cation reduction and F is the Faraday constant. Finally, experimental constraints "x this last quantity directly if we work under what is called galvanostatic control (prescribed current intensity) or indirectly in terms of the applied potential di!erence, *<, j"iE(*<), if we take this last one, *<, as a control parameter as we mostly do in our experiments. In this last equation, i is the conductivity of the solution. Just by looking at the simpli"ed model we have just proposed, we can advance a sort of general reasoning that will enable us to interpret, at least in an approximate way, the principle of morphological selection we need for in view of the morphological diversity observed in our experimental system.
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Let's forget at this point about convective e!ects (actually one could always imagine himself working with thin enough cells to supress them, while still one would observe the basic morphologies in Fig. 1), and concentrate on two particularly di!erent growth textures: the so-called open fractal and the homogeneous, or dense parallel, one. They, respectively, correspond to very di!erent growth modes and we assume that they both result from a di!erent balance between the otherwise unique set of ingredients participating in the whole process. As already mentioned, the open fractal morphology develops with very small growth velocities, typically a few microns per second. This in turn result in large di!usion lengths of the order of hundreds of microns. It seems then natural to consider such a growth mode as the natural candidate to be examined when looking for morphological features typical of a Laplacian growth mode. Actually, this conclusion is supported by the evident signatures of self-similarity appearing for those highly branched structures. Contrarily, the dense parallel morphology appears with large growth velocities, at least one order of magnitude larger than in the previous case, resulting in very small di!usion lengths, an order of magnitude smaller, i.e. several tenths of microns. Consequenly, di!usive-like features are going to be restricted to such small length scale in this case and thus will be rather unobservable, but contrarily such a growth mode seems particularly suited to be examined based on pure migration e!ects. The arguments just stated are going to be exploited to resolve separately in the next two sections these two signi"cative ECD morphologies. 5. Fractal characteristics of the open ECD morphology Right after the "rst experiments were done on quasi-two-dimensional electrocrystallization, fractal concepts were applied to quantitatively characterize the highly rami"ed open morphology obtained in ECD [3,23]. More precisely, de"nitive signatures of DLA [24] scaling were evidenced with a fractal dimension close to . Our aim was to extend such an analysis in two ways: (i) "rst we wanted to check the true self-similar nature of the geometrical mass measure of the grown branched aggregates by applying to them a multifractal analysis, and (ii) we aimed at analysing such deposits, not only from a static point of view but also from a dynamic perspective, by resolving their distribution of local growth velocities in terms of multifractal concepts. Let's go "rst with the static analysis. It was performed using a box-counting algorithm [8], restricted purely to the contour of typical tree-like aggregates grown from zinc sulphate solutions. A digitized image of the examined pattern, as that presented in Fig. 2, is covered with a mobile grid composed of square boxes of "xed size l (alternatively "xed mass), taken as the scaling variable. By de"ning e"l/¸ (¸ linear size of the grid), the set of scaling exponents q(q) is obtained from the scaling relation ,C Z(q)" pOJ eOO , (4) G C G where p is the relative measure of the cluster's contour contained in the ith cell. In the case of the G box-counting "xed-mass algorithm, one has 1!q(q)"lim N
log[1/N(p) ,N e\O(p)] G * , log p
(5)
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Fig. 2. Quasi-two-dimensional open fractal deposit (512 pixels"6.5 mm) obtained in zinc electrodeposition from a non-deaerated ZnSO solution. Cell thickness: 70 lm. Electrode separation: 3 cm. [ZnSO ]"0.4 M. *<"5 V.
where N(p) is the number of boxes with measure p. Due to the unavoidable existence of large and small experimental cut-o!s, one is compelled in practice to check any fractal or multifractal scaling limited to a range of length scales. We optimized such a choice by the computation of local slopes and gain the necessary statistics by moving the grid over 50 positions. Di!erent patterns corresponding to slightly di!erent experimental conditions, together with di!erent magni"cations of the same pattern, were examined in the same way [6,25]. Typical results for the pattern in Fig. 2 are shown in Fig. 3. There we plot q(q), a convenient rede"nition of the set of fractal dimensions D , O i.e. q(q)"(q!1)D . On what follows we use indistinctely q or q(q). The linear behavior there O O evidenced is a clear indication of a true self-similar measure, the di!erent values of D practically O collapsing around a value of 1.61$0.02 in agreement with previous results [23,26] and DLA simulations [23,24]. More interesting is the dynamic analysis [27}29]. Here a multifractal analysis was applied to the growth probability distribution (GPD) of an electrodeposited pattern (this measure is called the harmonic measure in a DLA-like context [30,31]). This means, in short, that the measure p now refers rather than to the cluster's contour mass to the local growth probability distribution. Actually such analysis was done in two di!erent and complementary ways. The "rst procedure is completely experimental. It consists in taking successive pictures of a growing branched-like pattern. The length of segments drawn between two successive perimeters, taken on perpendicular directions to any point in the surface contour, give us an estimation of the distribution of local growth velocities. In order to obtain a large number of growth sites, this procedure is repeated by comparing a set of successive images with a "xed original one, "nally averaging the computed growth velocities over time. The second related procedure of evaluating the GPD is in fact partially
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Fig. 3. Indices q versus q for the open-fractal morphology when the mass-distribution measure is analyzed. Two algorithms are used: "xed size (䢇) and "xed mass (*). The regression range for the "xed-size algorithm is 68 lm4l41104 lm. The regression range for the "xed-mass algorithm is the standard one.
experimental since it relies mostly on a computer resolution. It amounts to solve numerically the Laplace equation in the bulk region limited by two Dirichlet conditions taken, respectively, at the contour of the experimental aggregate (to which we applied the previous analysis) and at an outer contour far away from it. The GPD is then computed in terms of the normal gradient of the Laplacian "eld. The results of such a double-way analysis, applied to a similar pattern to that in Fig. 2, are summarized in Fig. 4 and Table 1. In spite of the experimental limitations, mainly related to the crucial acquisition and digitization techniques necessary to resolve the smaller GPD values, i.e. those characterizing the negative q indices, evidences of a Laplacian multifractality exist for the GPD of open-fractal ECD deposits, at least taking the GPD measure as indicator and given the spatial resolution and corresponding length scale chosen for such an analysis.
6. A simpli5ed model for homogeneous (dense parallel) growth As anticipated, the main features of the growth dynamics of dense parallel or homogeneous deposits can be analysed considering that migration is the main contribution to the transport of electroactive metal cations to the electrode surface [17]. This leads us to propose a drastic simpli"cation of the general set of equations formulated in Section 4. Later on this will be checked
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Fig. 4. Indices q versus q for the open fractal morphology when the GPD measure is assumed Laplacian. The discrete points stand for the values computed with "xed size (䢇) and "xed mass (*). The regression ranges are the same as in Fig. 3.
Table 1 Some representative multifractal indices in the q-representation for the open fractal morphology when the GPD measure is either assumed Laplacian or calculated from experimentally determined growth velocities. The regressions in the box-counting "xed-size algorithm are achieved in the range 60 lm4l41100 lm. (a ,d/dq(q ); f ,qa !q ). O O O O O q inf
0 1 3 sup
Singularity exponent
Open fractal (ECD) Laplacian GPD
Open fractal (ECD) Growth velocities distribution
q q a f (a ) D f (a )"D D a D "f (a )"a q q q a f (a ) D
!3 !7.5$ 0.8 6.7$0.2 0.10$0.01 1.9$0.2 1.53$0.01 3.41$0.09 0.94$0.01 1.51$0.02 15 8.8$0.2 0.61$0.01 0.07$0.01 0.63$0.01
!1 !3.9$0.2 3.02$0.07 0.40$0.07 1.96$0.08 1.43$0.02 1.82$0.02 1.05$0.02 1.53$0.06 15 8.3$0.4 0.53$0.03 !0.05$0.03 0.59$0.03
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Fig. 5. Diagram of morphologies for copper electrodeposition from a 0.05 M CuSO aqueous solution containing Na SO in a parallel cell. Cell thickness: 100 lm. Electrode separation: 4 cm.
for the experimental system CuSO /Na SO (Fig. 5). A sort of ideal ohmic model can thus be built taking into account the following assumptions: (i) metal electroreduction is the only reaction on the electrode surface and only takes place on the growing front of the deposit; (ii) concentration of metal cation on the electrode surface and in the solution embedding the deposit vanishes; (iii) ionic migration towards the electrode surface is the limiting step of the whole process; (iv) contributions due to the concentration layer are neglected and thus, the conductivity and the electric "eld are considered uniform throughout the whole cell; (v) convection e!ects are neglected; (vi) electroneutrality is ful"lled in any point of the solution; and (vii) the zone between the cathode and the advancing front is considered to be an equipotential region. This model has been developed for a solution containing an inert electrolyte with the same anion as the binary electroactive salt [17]. Our aim in choosing this system is to be able to gain an extra parameter to check the e!ect of conductivity on the growth velocity. Simple manipulations with the transport and balance equations which follow from the assumptions stated above lead to a simple expression for the growth velocity, v (u #u )E > C , v"u E# \ GL \ C
(6)
where u are the ionic mobilities of the ions of the inert electrolyte, E is the electric "eld, and C and C are the concentrations of electroactive and inert electrolyte, respectively. This equation shows GL
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Fig. 6. Growth velocities of homogeneous electrodeposits versus Na SO concentration. Cell thickness: 100 lm. Electrode separation: 4 cm. [CuSO ]"0.05 M. triangles: 7 V. circles: 5 V. crosses: 4 V.
that the growth velocity is proportional to the electric "eld, and that, for a binary electrolyte solution, it is purely determined by the anion mobility. Actually, this last result had been previously derived by di!erent groups [32}34], but applied to binary electrolyte solutions. This approach has been experimentally checked with the CuSO /Na SO system. The addition of sodium sulphate to a 0.05 M copper sulphate solution leads to a change in the morphology of the deposit as shown in the diagram of Fig. 5. At low sodium sulphate concentrations, open fractal patterns are formed resulting in smaller growth velocities than those expected for homogeneous deposits. At high sodium sulphate concentrations, hydrogen evolution coexists with a new ECD "nger-like morphology (see Section 8), and only at intermediate values of this parameter, homogeneous structures are formed on the cathode surface. For this last morphology, we test the model proposed above by plotting the growth velocity in front of sodium sulphate concentration (Fig. 6). In spite of the limited range of experimental conditions where our model can be checked, growth velocity values reasonably agree with the model predictions. Actually, slopes (0.22 and 0.14 lm s\ mM\ at 7 and 5 V, respectively) fall quite close to the theoretical values (0.20 and 0.15) [17]. Finally, it is worth remarking at this point that this model reasonably predicts the kinetics of this growth mode but not its "lamentary structure. One could conceive that the spacing between "laments and their own existence is related to electroconvection e!ects that take place in front of and between the tips of the growing "laments [12,13].
7. Dynamical morphological transitions A common feature in the thin layer electrodeposition experiments is the occurrence of abrupt changes in the morphology of the deposit reproducing the anode symmetry and appearing at
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Fig. 7. Morphological transitions in a copper electrodeposition experiments from a 0.05 M CuSO aqueous solution. Electrode separation: 4 cm. Cell thickness: 300 lm. Picture width: 16 mm. (A) *<"30 V. (B) *<"60 V.
a reproducible stage of its development (Fig. 7) [6,35}37]. These changes can a!ect the rami"cation degree of the deposit and its colour and texture, and can appear for any initial morphology of the deposit. In spite that they have been mainly evidenced and examined for the homogeneous growth mode, they have been also reported for open fractal and "nger-like deposits. The main characteristics of these transitions suggest that they are due to cationic fronts coming from the anode. When these fronts meet the growing deposit, the chemical environment of the metal cation reduction process is modi"ed, inducing a change in the composition and therefore, in the observable features of the deposit. With the primary aim of investigating such morphological transitions, we conducted a systematic series of experiments for copper deposition from an aqueous copper sulphate solution in the concentration range 0.05}0.1 M [38]. In these experiments, anode dissolution increases the copper ion concentration in the anodic region together with the proton concentration due to copper ion hydrolysis. This prompts the formation of an acid front travelling from the anode towards the cathode (i.e. contrary to the advancing deposit) which is at the origin of the transition. The previous considerations clearly lead us to consider in more detail the chemical features of ECD. For the experiments just mentioned we investigated the composition of the initial deposit. The result was that instead of a pure copper deposit it is actually a mixture of copper and cuprous oxide. This "nding can be interpreted as originated from a competition between the two electrode reactions 2Cu>#2OH\#2e\PCu O#H O , Cu>#2e\PCu .
(7) (8)
At the pH values of the initial 0.05 M CuSO solution (pH&4.5), the contribution of the "rst reaction is not negligible at all, but when the acid front meets the growing deposit, this concurrent
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process is hindered and the direct reduction of copper ions to metallic copper is enhanced. This has been also veri"ed by phase analysis of the deposit before and after the transition using X-ray powder di!raction. These measurements allow us to conclude that the content of cuprous oxide in the deposit obtained from a 0.05 M copper sulphate solution is higher before (&40%) than after (&20%) the transition has occurred. This is in complete accordance with: (i) the in situ observation that the initial deposit is reddish and dull and turns to a red and bright structure with a metallic appearance when the transition has been completed; and (ii) the formation of this metallic deposit at high CuSO concentrations (i.e. lower pH values) where the transition is not observed ([CuSO ]'0.2 M). The second point to discuss is the position of the transition. It is clear that this will depend on the growth velocity of the deposit and that of migration of the acid front. If we accept the previously proposed ohmic model, the position of the transition would depend on the mobility of the ions responsible for the anodic front and the growth velocity of the deposit, proton and sulphate respectively. If we de"ne the position of the transition, j, as the distance from the transition to the cathode relative to the electrode separation, an straightforward calculation leads to u \ . (9) j" u #u \ & The experimental values of j are close to 0.25 at low potential values (<430 V), but increase with potential up to 0.43 at 100 V. However, when gravity-induced convection e!ects are completely suppressed placing the cell in a vertical con"guration with the cathode facing downwards, the position of the transition is not potential dependent with values very close to those obtained at low potential values (0.25) (Fig. 8). Thus, the potential dependence of j must be attributed to gravity
Fig. 8. Morphological transitions in a copper electrodeposition experiment from a 0.1 M CuSO aqueous solution in a vertical cell with the cathode facing downwards. Electrode separation: 4 cm. Cell thickness: 100 lm. Picture width: 16 mm. (A) *"60 V. (B) *"105 V.
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driven convection, and the value of 0.25 has to be considered as the genuinous value for pure ionic migration, i.e. in absence of any convection e!ect. Taking as sulphate mobility the value given by the growth velocity of the deposit, proton mobilities derived from this value of j are in reasonable agreement with those obtained with di!erent experimental techniques for similar systems. Let's "nish the section by stressing the role of the buoyancy-driven convection. As mentioned in Section 4, concentration gradients developing at both electrodes create rotating rolls that propagate through the solution. Looking in particular at the anodic region, where such a propagation is not a!ected by the growing deposit, it is found that the width of the propagating roll, l, depends on time as [14,15] lJt? with a(1. The rate of this motion will increase with the density gradient in the anodic region and therefore with the applied potential. Thus, it is reasonable to believe that at high potential values the protons originated in the anodic region will be at least transiently somehow trapped, decreasing their transport rate, and then delaying the position of the transition.
8. Finger-like morphology When small amounts of an alkaline sulphate are added to a 0.05 M copper sulphate solution, a deposit with new and totally unusual morphology in the "eld of ECD [17] is formed on the cathode (see Fig. 5). These aggregates consists of an array of rounded-like patterns that enclose a soft and very highly rami"ed microstructure (Fig. 9). The main morphological characteristics of this growth mode are: (i) a grey and dull appearance with a non-metallic aspect; (ii) the coexistence with hydrogen bubbles, and (iii) their regular pro"le, with rounded tips at the centimeter scale,
Fig. 9. Finger-like deposits obtained in a ECD experiment from a 0.05 M CuSO #3 mM Na SO aqueous solution. Cell thickness: 100 lm. Electrode separation: 4 cm. *<"25 V. Picture width: 16 mm.
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recalling those of viscous "ngering commonly ascribed to hydrodynamic phenomena [39,40]. For this reason, this texture is hereafter referred as the "nger morphology. Finger-like electrodeposits are an excellent example to stress the interplay between some of the ECD mechanisms mentioned before. In particular, migration, convection and chemical reactions play a crucial role. On what follows, we propose a sort of qualitative intrepretation of the origin of such a morphology. The addition of the inert electrolyte increases the conductivity of the solution and in turn the electric current. This prompts water reduction and hydrogen evolution as a concurrent electrochemical reaction. This reaction originates a basic medium around the growing interface that, in turn, explains the formation of copper hydroxylated species accumulating in a sort of hydrogel boundary layer surrounding the deposit, as is clearly observed in Fig. 10. These species are reduced to cuprous oxide, which is the predominant component of "nger-like deposits (see Eq. (7)). In contrast, common copper aggregates (i.e. open or homogeneous) are richer in copper than in cuprous oxide. This fact explains the di!erent color and texture of both type of microstructures. The special physico-chemical properties of the boundary layer are responsible for the enhancement of the e!ective surface tension associated with the interfacial region separating the depleted solution embedding the growing deposit, from the bulk. Such a surface tension acts as a restoring force opposite to the driving "eld, which is nothing but the electric force genuinous to any ECD experiment. Note that such a picture leads us directly to look for similarities between the phenomenon at hand and the usual "ngering instabilities in #uid displacement. Actually, we can even take for granted from such similar scenario the standard linear dispersion relation, to analyze the initial growth dynamics in our "ngering experiments [18]. One important remark is worth
Fig. 10. An optical image showing the hydrogel boundary layer surrounding a "nger-like deposit obtained from a 0.05 M CuSO #7 mM Na SO aqueous solution. Cell thickness: 250 lm. Electrode separation: 4 cm. *<"12 V. Picture width: 4 mm.
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making at this point. Since the value of the e!ective surface tension can not be calculated by any reasonable means, we assume that it is going to be inversely proportional to the mixing layer width, l [41]. According to parallel experimental observations, this last quantity is in turn related to the @J growth velocity according to a power law as v?. In such a way one can formally predict a value for the parameter b, de"ned in the adapted dispersion relation, NJk J(vl ) Jv@, as
@J b"(a!1)/2a, where k represents the most unstable wave number. Taking a"1.5 as determined
previously, bK0.16. This was experimentally checked by conducting a series of experiments varying the applied potential di!erence, and recording the number of incipient "ngers, N, at di!erent growth velocities, v. These results "tted a power law, N&v@ with bK0.18. Another experimental observation reinforcing the particular role of the mixing layer in the origin of this morphology in ECD, concerns the singular e!ect of convection. It has been mentioned before (see Section 7) that gravity-induced convection can be largely suppressed by turning the cell into a vertical con"guration with the cathode facing downwards. Under this arrangement, the deposits display their typical "ne texture but enclosed this time by a planar front which is free of any long wavelength modulation (Fig. 11). This behaviour can be understood by considering that in absence of convection e!ects the boundary layer would be largely compressed and its endowed e!ective surface tension increased up to a value such that the "nger structure cannot be developed, in spite of the fact that the microstructural texture is largely preserved. The "nal aspect we want to refer to in relation with the "nger growth mode concerns the morphological transitions they experience [7]. Complementary to what has been said in the last section one observes a richer scenario when examining such transitions in ECD "ngers. Actually, there are two transitions appearing at di!erent potential ranges. Both di!er in their position and in the changes they cause to the initial deposit (Fig. 12). At low potential values, the transition simply
Fig. 11. Deposit obtained in an ECD experiment from a 0.05 M CuSO #2 mM Na SO aqueous solution in a vertical cell with growth directed downwards. Cell thickness: 100 lm. Electrode separation: 4 cm. *<"25 V. Picture width: 16 mm.
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Fig. 12. Morphological transitions experimented by "nger-like deposits obtained from a 0.05 M CuSO # 2 mM K SO aqueous solution in a parallel cell. Cell thickness: 100 lm. Electrode separation: 4 cm. (A) *"20 V. (B) 10 V.
re#ects the depletion of alkaline ions due to their migration to the cathodic region and a typical copper deposit is recovered after the morphological change. This is in good agreement with its position relative to the distance between the electrodes. Furthermore this would explain the observed #at boundary separating the two morphologies. At high potential values, the proton concentration in the anodic region increases prompting the formation of an acid front which neutralizes the basic layer around the deposit and, in many cases, even leading the penetration inside the "nger microstructure (Fig. 12). In fact, this transition is the same as that discussed in Section 7.
9. Summary A review of di!erent aspects of the phenomenon of quasi-two-dimensional electrocrystallization has been presented. Emphasis has been devoted to the understanding of the delicate coupling between the di!erent transport mechanisms and chemical processes, which "nally results in the diverse growth modes observed in the experiments. In particular fractal and multifractal features associated to di!usive-like growth processes have been examined for the open and highly rami"ed patterns, obtained at very low applied potentials and high concentrations from a binary electrolyte. By invoking pure ionic migration concepts, we were able to predict the growth velocity, and its dependence on the conductivity properties of the solution, for the so-called dense parallel or homogeneous morphology, obtained contrarily at low concentrations when high enough potentials are applied to these systems. Buoyancy-driven convection e!ects and the role of chemical reactions are stressed in relation to both, morphological transitions and the appearance of a particularly interesting and unexpected
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"nger-like morphology obtained by electrolizing electrolytes containing a small amount of an added inert ionic substance.
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Growth and forms in quasi-two-dimensional electrocrystallization Francesc SagueH s*, Marta Queralt LoH pez-Salvans, Josep Claret Departament de Qun& mica Fn& sica, Universitat de Barcelona, Martn& i Franque% s 1, 08028 Barcelona, Catalonia, Spain
Abstract A review on the morphologies and corresponding growth dynamics observed in quasi-two-dimensional electrocrystallization is presented. Emphasis is devoted to the understanding of the speci"c and growth mode-dependent role played by the di!erent transport mechanisms, i.e. di!usion, migration and convection, as well as the chemical processes. The role of chemical reactions is decissive for the understanding of the well-known morphological transitions and is at the origin of the "nger-like texture. 2000 Elsevier Science B.V. All rights reserved. PACS: 05.45.!a
1. Introduction Quasi-two-dimensional electrocrystallization, or simply electro(chemical) deposition, ECD, as it will be referred to on most of what follows, constitutes a fascinating example of a growth phenomenon operated under nonequilibrium conditions [1]. In its simplest form, it consists in the aggregation of a metallic deposit from the electrochemical reduction of its cations dissolved in a thin layer of an electrolyte solution. Although standard electrodeposition, i.e. operated in three-dimensional cells, has been a particularly familiar subject to electrochemists for many years, it has been only during these last two decades, when conducted in thin geometries, that has been widely recognized as a genuine example of growth process, similar and even phenomenologically richer as compared to other interfacial phenomena studied exhaustively within the physics community [2]. In relation with the just mentioned wide scenario of phenomena, ECD o!ers the opportunity to investigate very di!erent morphologies, from largely rami"ed to well-ordered anisotropic dendritic * Corresponding author. E-mail address: [email protected] (F. SagueH s). 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 5 7 - 0
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structures and even comprising quite unexpected digitation forms, very similar indeed to those found in viscous #uid displacement. Such a diversity is easily observed by appropriately varying a few basic paramenters of the experimental system, mainly the composition of the electrodepositing solution, and/or the driving force for the growth process, either the current intensity or the applied electric potential. The richness of the phenomenon indicates that beyond its apparent experimental and even conceptual simplicity, many times somewhat misleadingly invoked, several di!erent physicochemical ingredients are delicately coupled in any ECD experiment. Actually, this forces us to adopt an enlarged perspective with respect to the predominant position at the mid-1980s, when research on ECD was initiated [3}5]. According to the dominant idea at those times, ECD would be essentially a paradigmatic example of a Laplacian, DLA-like, pattern-forming system. In fact, during these last 15 years, much more complete, diverse and systematically oriented experiments conducted by our group and many others have enabled us to start to have a more complete understanding of the process. What is more remarkable is that such a deeper comprehension of the phenomenon at hand has developed mainly when widening our attention and redirecting it from the morphological and scale invariance properties of the observed deposits, towards the basic chemical and transport mechanisms behind the whole electrodeposition process, i.e. di!usion, electric migration and convection, and the essential role they play in selecting the observed morphologies. The present contribution intends to brie#y summarize the work of our group following this aim. Our research, and in this way it is sketched on what follows, adopted from the very beginning a multiperspective and as a consequence has followed di!erent strategies and methodologies. It is worth recalling them at this point as a way to comment on our choice of the selected material and how it appears organized in this contribution. After Introduction, Section 2 is devoted to give a very schematic presentation of the standard experimental protocol in ECD. Section 3 reports on our starting experiments on ECD, which essentially were aimed at classifying the di!erent morphologies observed when electrodepositing zinc sulphate solutions in parallel cells under constant applied potentials. The general idea expressed in the paragraph above and concerning the di!erent mechanisms involved in ECD is shortly ellaborated in Section 4, where a sort of minimal model, based on the "rst-principle equations governing the process, is also proposed. Purely di!usive-like morphological features of tree-like ECD deposits, obtained in the limit of very low applied potentials and high electrolyte concentrations, are examined in Section 5. It is there where aboundant scaling arguments borrowed from the by now standard fractal and multifractal methodology, are employed to resolve statically and dynamically the electrodeposited structures. Changing considerably the experimental conditions, by applying large potentials to very low concentrations, we observe a completely di!erent morphological pattern. It consists in arrays of metallic "laments enclosed with a well-de"ned advancing front. As explained in Section 6, such a texture, commonly referred to in the literature as dense parallel, or homogeneous as we sometimes use it, turns out to be the best candidate to be examined in terms of an ohmic model of the growth process, i.e. by invoking simple electric migration concepts. One of the most peculiar experimental observations in ECD refers to the morphological transitions experienced by the deposits as they advance through the electrolized solution. As it is discussed in Section 7, the understanding of such morphological changes needs for a precise characterization of the most chemically oriented aspects of the ECD process. Evenmore, it turned out that when we were investigating those transitions we encountered direct evidences for the last, and for a long time
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overlooked, transport ingredient involved in ECD experiments, i.e. buoyancy driven convection. Its role is emphasized in Section 8, fully devoted to describe the recently observed "ngered-like morphology as it appears when electrolyzing copper sulphate solutions containing small amounts of an inert electrolyte. Finally, this contribution ends with a summary.
2. Experimental system In a typical experiment [6], electrodeposits are grown from a thin layer of an aqueous electrolyte solution sandwiched between two 5 mm thick glass plates enclosing two parallel metallic electrodes separated by a distance of 4 cm. The cell gap is determined by the electrodes, typically copper wires of ca. 100 lm diameter. Since the composition and morphology of electrodeposits are very sensitive to the solution purity, they are prepared with ultra pure water and p.a. quality chemicals. Most of our experiments are conducted applying a constant potential di!erence between the cathode and the anode. The conductivity and pH of the initial electrolyte solution are measured under the same conditions as for the ECD experiment. The temporal evolution of current passing through the cell and in some cases, that of cathode potential are conveniently monitored and processed in a personal computer. In this last kind of measurements, an additional quasi-reference electrode (typically a copper wire) must be included in the electrochemical cell [7]. Experiments are carried out at room temperature. In order to quantitatively analyse the morphology of the electrodeposits, they are recorded using a stereo zoom microscope connected to a standard video camera, transferred to a PC-based image analysis system and conveniently digitised with a 512;512 resolution. The fractal features of the deposit structure are evaluated by means of the MFRAC software [8]. The chemical composition of the deposits is investigated by means of X-ray powder di!raction experiments using a di!ractometer with Cu Ka radiation (j"1.5418 As ) at 40 kV and 30 mA [9]. In order to prepare the samples, electrodeposits are collected, dried and stored under an argon atmosphere before the measurement and then, mounted over a Si(001) single crystal.
3. Diagram of ECD morphologies Electrodeposition of zinc from a thin layer of a non-deaerated aqueous zinc sulphate solution is a clear example of the wide morphological diversity observed in ECD experiments. These morphologies depend on the applied potential and the electrolyte concentration and are represented in a sort of morphological diagram in Fig. 1 [6,10]. Five typical patterns can be identi"ed: (a) Compact: a non-structured deposit with a rough interface is obtained at the lowest applied potential values (up to 2 V). In this case, the growth velocity is extremely low and the deposit texture is characterized by the absence of rami"cation at least at cell length scales. (b) Homogeneous or dense parallel: an array of regularly spaced trees is formed at low zinc sulphate concentrations (up to 0.04 M). The main characteristics of this morphology are: (i) the tips of the trees de"ne a growing front parallel to the cathode and retaining in this way the symmetry of the electrochemical cell; (ii) the growth velocity de"ned by the advancing front increases with applied potential; and (iii) the spacing between trees and the branching e!ects both decrease when
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Fig. 1. Diagram of morphologies for zinc electrodeposition from a non-deaerated ZnSO aqueous solution in a parallel cell. Cell thickness: 70 lm. Electrode separation: 3 cm.
increasing the applied voltage. In the limit of high potential values, a columnar (channel like) and non-rami"ed deposit is formed. (c) Dendritic: typical dendrites are obtained at intermediate concentration values ranging from 0.05 to 0.1 M. These patterns are de"ned by straight and anisotropic backbones with a considerable amount of side branching. Unlike the dense parallel morphology, these dendrites do not de"ne any sort of advancing front but rather some of them grow faster and remain as the only growing structures at high development stages. Although a growth velocity for the whole deposit cannot be de"ned, the individual growth rate of these growing dendrites is higher than that of homogeneous patterns at similar potential values. (d) Open fractal: at high concentration and low applied potential values (up to 10 V), a new morphology is formed displaying continuous tip splitting of the growing trees that leads to the formation of highly rami"ed and self-similar structures. Three characteristics of these deposits are
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worth mentioning: (i) the absence of a well-de"ned growing front; (ii) their growth that occur at very low velocities; and (iii) screening e!ects that prevent the further development of some initial trees or of the less exposed regions of a growing tree. (e) Mixed: this pattern does not correspond to a pure morphology but rather shows features characteristic of both dendritic and open fractal textures. The trees consist of main dendritic backbones, but the lateral branches are typical open fractal structures. Increasing the applied potential the lateral branching decreases in such a way that at very high values (550 V), the deposit is formed by non-rami"ed stringy patterns with a very high growth velocity. On all what follows we leave apart both the compact morphology, which is likely to be essentially controlled by the electrochemical reduction process itself [11], and the pure and mixed dendritic forms. Actually, dendritic structures would merit a detailed study on its own which was not pursued in our group.
4. Electrocrystallization mechanisms As in any non-equilibrium growth process, two di!erent categories of elementary processes are involved in even the simplest realization of an electrodeposition experiment. The "rst category refers to the transport mechanisms that bring the individual components to be deposited close to the advancing interface, where they are going to aggregate. The second class of processes concern the interfacial processes themselves. Let's comment a little bit more extensively on the di!erent mechanisms belonging to these two categories. On what concerns the transport mechanisms, one could in principle refer to two basic components: di!usion and migration. The di!usive mode, genuinous to most of the non-equilibrium growth process one can imagine, appears as a consequence of the concentration gradients built inside the electrolized solution as it is being invaded by the advancing interface. Transport by di!usion is associated with a typical length scale, the standard di!usion length scale, given in terms of the ratio between the di!usion coe$cient of the electrodeposited cations and the growth velocity of the interface. On the other hand, what is singular of ECD is the fact that the electrodepositing species, being electrically charged, experience typical migration processes under the applied potential di!erence that drive the growth process. In principle, the migration transport acts on a much larger length scale, typical of the cell dimension, as de"ned by the distance between both electrodes. A third transport mechanism, largely overlooked for many years in the context of ECD, refers to convection. Without being too precise, we could even distinguish between two di!erent modes of convective phenomena in ECD experiments. The "rst one, generically referred to as electroconvection [12,13], involve small scale #uid motion con"ned to the very close proximity of the metallic deposit. The second one, much more interesting on what follows, is known as buoyancy-driven convection [14}16]. For cells whose thickness is about or exceeds around a hundred of microns, the concentration gradients induced by the reduction and oxidation processes on the respective electrodes result in density gradients. These inhomogeneities, in turn, triger the propagation of convective rotating rolls extending from each electrode until they merge and encompass, in large covective currents, the whole cell. Naturally to say, such convective motions not only modify the transport of the depositing cations, but may markedly in#uence the physicochemical properties of the solution close to the metallic aggregate.
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Now, let's turn to the second family of mechanisms, i.e. those related to the electrochemical reactions taking place at the growing cathode. Under the usual standard conditions to which our experiments refer, the interfacial kinetics is assumed to be fast as compared to the previously mentioned transport mechanisms. However, this does not mean that we can completely forget the electrochemical processes that occur at the cathode. The most important role is played by possible concurrent reactions [7,17}21] accompanying the reduction process itself of the depositing cations. Several reactions could be considered depending on the composition of the solution or even the presence of impurities, the most important, however, being hydrogen evolution from water reduction. The implications of such a reaction will be emphasized later on in Sections 7 and 8, when referring to morphological transitions and singularly to the "ngering growth mode. Let's convert the picture just given of a typical electrodeposition process into a sort of simpli"ed and comprehensive model. We will build it for the simplest situation corresponding to the electrodeposition of a binary electrolyte, i.e., composed simply of the cations to be deposited and the corresponding anions. Furthermore and for the sake of simplicity we will neglect the presence of any concurrent reaction di!erent from the pure reduction of the depositing cations. We start by writing down the basic transport equation for the depositing cations in terms of the three previously mentioned ingredients, i.e. di!usion, migration and convection (Nernst}Planck equation [22]). Thus, the #ux of cations, J , is expressed in terms of the concentration gradient, C , the electric "eld, E, and the advection velocity, V, zFD C E#C V , J "!DC # R¹ being such a relation readily transformed into a bulk equation zF ) (DC )# ) C V . * C " ) (DC )# R R¹
(1)
Now, we have to supplement such a bulk relation with boundary conditions for the #ux of cations at the interface. Actually this is formulated as a conservation equation ov ) n"!MJ ) n , (2) relating the growth velocity, v, to the #ux of cations. In turn such a #ux is related to the current density, j j"nFJ , (3) where n is the number of electrons involved in the cation reduction and F is the Faraday constant. Finally, experimental constraints "x this last quantity directly if we work under what is called galvanostatic control (prescribed current intensity) or indirectly in terms of the applied potential di!erence, *<, j"iE(*<), if we take this last one, *<, as a control parameter as we mostly do in our experiments. In this last equation, i is the conductivity of the solution. Just by looking at the simpli"ed model we have just proposed, we can advance a sort of general reasoning that will enable us to interpret, at least in an approximate way, the principle of morphological selection we need for in view of the morphological diversity observed in our experimental system.
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Let's forget at this point about convective e!ects (actually one could always imagine himself working with thin enough cells to supress them, while still one would observe the basic morphologies in Fig. 1), and concentrate on two particularly di!erent growth textures: the so-called open fractal and the homogeneous, or dense parallel, one. They, respectively, correspond to very di!erent growth modes and we assume that they both result from a di!erent balance between the otherwise unique set of ingredients participating in the whole process. As already mentioned, the open fractal morphology develops with very small growth velocities, typically a few microns per second. This in turn result in large di!usion lengths of the order of hundreds of microns. It seems then natural to consider such a growth mode as the natural candidate to be examined when looking for morphological features typical of a Laplacian growth mode. Actually, this conclusion is supported by the evident signatures of self-similarity appearing for those highly branched structures. Contrarily, the dense parallel morphology appears with large growth velocities, at least one order of magnitude larger than in the previous case, resulting in very small di!usion lengths, an order of magnitude smaller, i.e. several tenths of microns. Consequenly, di!usive-like features are going to be restricted to such small length scale in this case and thus will be rather unobservable, but contrarily such a growth mode seems particularly suited to be examined based on pure migration e!ects. The arguments just stated are going to be exploited to resolve separately in the next two sections these two signi"cative ECD morphologies. 5. Fractal characteristics of the open ECD morphology Right after the "rst experiments were done on quasi-two-dimensional electrocrystallization, fractal concepts were applied to quantitatively characterize the highly rami"ed open morphology obtained in ECD [3,23]. More precisely, de"nitive signatures of DLA [24] scaling were evidenced with a fractal dimension close to . Our aim was to extend such an analysis in two ways: (i) "rst we wanted to check the true self-similar nature of the geometrical mass measure of the grown branched aggregates by applying to them a multifractal analysis, and (ii) we aimed at analysing such deposits, not only from a static point of view but also from a dynamic perspective, by resolving their distribution of local growth velocities in terms of multifractal concepts. Let's go "rst with the static analysis. It was performed using a box-counting algorithm [8], restricted purely to the contour of typical tree-like aggregates grown from zinc sulphate solutions. A digitized image of the examined pattern, as that presented in Fig. 2, is covered with a mobile grid composed of square boxes of "xed size l (alternatively "xed mass), taken as the scaling variable. By de"ning e"l/¸ (¸ linear size of the grid), the set of scaling exponents q(q) is obtained from the scaling relation ,C Z(q)" pOJ eOO , (4) G C G where p is the relative measure of the cluster's contour contained in the ith cell. In the case of the G box-counting "xed-mass algorithm, one has 1!q(q)"lim N
log[1/N(p) ,N e\O(p)] G * , log p
(5)
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Fig. 2. Quasi-two-dimensional open fractal deposit (512 pixels"6.5 mm) obtained in zinc electrodeposition from a non-deaerated ZnSO solution. Cell thickness: 70 lm. Electrode separation: 3 cm. [ZnSO ]"0.4 M. *<"5 V.
where N(p) is the number of boxes with measure p. Due to the unavoidable existence of large and small experimental cut-o!s, one is compelled in practice to check any fractal or multifractal scaling limited to a range of length scales. We optimized such a choice by the computation of local slopes and gain the necessary statistics by moving the grid over 50 positions. Di!erent patterns corresponding to slightly di!erent experimental conditions, together with di!erent magni"cations of the same pattern, were examined in the same way [6,25]. Typical results for the pattern in Fig. 2 are shown in Fig. 3. There we plot q(q), a convenient rede"nition of the set of fractal dimensions D , O i.e. q(q)"(q!1)D . On what follows we use indistinctely q or q(q). The linear behavior there O O evidenced is a clear indication of a true self-similar measure, the di!erent values of D practically O collapsing around a value of 1.61$0.02 in agreement with previous results [23,26] and DLA simulations [23,24]. More interesting is the dynamic analysis [27}29]. Here a multifractal analysis was applied to the growth probability distribution (GPD) of an electrodeposited pattern (this measure is called the harmonic measure in a DLA-like context [30,31]). This means, in short, that the measure p now refers rather than to the cluster's contour mass to the local growth probability distribution. Actually such analysis was done in two di!erent and complementary ways. The "rst procedure is completely experimental. It consists in taking successive pictures of a growing branched-like pattern. The length of segments drawn between two successive perimeters, taken on perpendicular directions to any point in the surface contour, give us an estimation of the distribution of local growth velocities. In order to obtain a large number of growth sites, this procedure is repeated by comparing a set of successive images with a "xed original one, "nally averaging the computed growth velocities over time. The second related procedure of evaluating the GPD is in fact partially
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Fig. 3. Indices q versus q for the open-fractal morphology when the mass-distribution measure is analyzed. Two algorithms are used: "xed size (䢇) and "xed mass (*). The regression range for the "xed-size algorithm is 68 lm4l41104 lm. The regression range for the "xed-mass algorithm is the standard one.
experimental since it relies mostly on a computer resolution. It amounts to solve numerically the Laplace equation in the bulk region limited by two Dirichlet conditions taken, respectively, at the contour of the experimental aggregate (to which we applied the previous analysis) and at an outer contour far away from it. The GPD is then computed in terms of the normal gradient of the Laplacian "eld. The results of such a double-way analysis, applied to a similar pattern to that in Fig. 2, are summarized in Fig. 4 and Table 1. In spite of the experimental limitations, mainly related to the crucial acquisition and digitization techniques necessary to resolve the smaller GPD values, i.e. those characterizing the negative q indices, evidences of a Laplacian multifractality exist for the GPD of open-fractal ECD deposits, at least taking the GPD measure as indicator and given the spatial resolution and corresponding length scale chosen for such an analysis.
6. A simpli5ed model for homogeneous (dense parallel) growth As anticipated, the main features of the growth dynamics of dense parallel or homogeneous deposits can be analysed considering that migration is the main contribution to the transport of electroactive metal cations to the electrode surface [17]. This leads us to propose a drastic simpli"cation of the general set of equations formulated in Section 4. Later on this will be checked
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Fig. 4. Indices q versus q for the open fractal morphology when the GPD measure is assumed Laplacian. The discrete points stand for the values computed with "xed size (䢇) and "xed mass (*). The regression ranges are the same as in Fig. 3.
Table 1 Some representative multifractal indices in the q-representation for the open fractal morphology when the GPD measure is either assumed Laplacian or calculated from experimentally determined growth velocities. The regressions in the box-counting "xed-size algorithm are achieved in the range 60 lm4l41100 lm. (a ,d/dq(q ); f ,qa !q ). O O O O O q inf
0 1 3 sup
Singularity exponent
Open fractal (ECD) Laplacian GPD
Open fractal (ECD) Growth velocities distribution
q q a f (a ) D f (a )"D D a D "f (a )"a q q q a f (a ) D
!3 !7.5$ 0.8 6.7$0.2 0.10$0.01 1.9$0.2 1.53$0.01 3.41$0.09 0.94$0.01 1.51$0.02 15 8.8$0.2 0.61$0.01 0.07$0.01 0.63$0.01
!1 !3.9$0.2 3.02$0.07 0.40$0.07 1.96$0.08 1.43$0.02 1.82$0.02 1.05$0.02 1.53$0.06 15 8.3$0.4 0.53$0.03 !0.05$0.03 0.59$0.03
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Fig. 5. Diagram of morphologies for copper electrodeposition from a 0.05 M CuSO aqueous solution containing Na SO in a parallel cell. Cell thickness: 100 lm. Electrode separation: 4 cm.
for the experimental system CuSO /Na SO (Fig. 5). A sort of ideal ohmic model can thus be built taking into account the following assumptions: (i) metal electroreduction is the only reaction on the electrode surface and only takes place on the growing front of the deposit; (ii) concentration of metal cation on the electrode surface and in the solution embedding the deposit vanishes; (iii) ionic migration towards the electrode surface is the limiting step of the whole process; (iv) contributions due to the concentration layer are neglected and thus, the conductivity and the electric "eld are considered uniform throughout the whole cell; (v) convection e!ects are neglected; (vi) electroneutrality is ful"lled in any point of the solution; and (vii) the zone between the cathode and the advancing front is considered to be an equipotential region. This model has been developed for a solution containing an inert electrolyte with the same anion as the binary electroactive salt [17]. Our aim in choosing this system is to be able to gain an extra parameter to check the e!ect of conductivity on the growth velocity. Simple manipulations with the transport and balance equations which follow from the assumptions stated above lead to a simple expression for the growth velocity, v (u #u )E > C , v"u E# \ GL \ C
(6)
where u are the ionic mobilities of the ions of the inert electrolyte, E is the electric "eld, and C and C are the concentrations of electroactive and inert electrolyte, respectively. This equation shows GL
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Fig. 6. Growth velocities of homogeneous electrodeposits versus Na SO concentration. Cell thickness: 100 lm. Electrode separation: 4 cm. [CuSO ]"0.05 M. triangles: 7 V. circles: 5 V. crosses: 4 V.
that the growth velocity is proportional to the electric "eld, and that, for a binary electrolyte solution, it is purely determined by the anion mobility. Actually, this last result had been previously derived by di!erent groups [32}34], but applied to binary electrolyte solutions. This approach has been experimentally checked with the CuSO /Na SO system. The addition of sodium sulphate to a 0.05 M copper sulphate solution leads to a change in the morphology of the deposit as shown in the diagram of Fig. 5. At low sodium sulphate concentrations, open fractal patterns are formed resulting in smaller growth velocities than those expected for homogeneous deposits. At high sodium sulphate concentrations, hydrogen evolution coexists with a new ECD "nger-like morphology (see Section 8), and only at intermediate values of this parameter, homogeneous structures are formed on the cathode surface. For this last morphology, we test the model proposed above by plotting the growth velocity in front of sodium sulphate concentration (Fig. 6). In spite of the limited range of experimental conditions where our model can be checked, growth velocity values reasonably agree with the model predictions. Actually, slopes (0.22 and 0.14 lm s\ mM\ at 7 and 5 V, respectively) fall quite close to the theoretical values (0.20 and 0.15) [17]. Finally, it is worth remarking at this point that this model reasonably predicts the kinetics of this growth mode but not its "lamentary structure. One could conceive that the spacing between "laments and their own existence is related to electroconvection e!ects that take place in front of and between the tips of the growing "laments [12,13].
7. Dynamical morphological transitions A common feature in the thin layer electrodeposition experiments is the occurrence of abrupt changes in the morphology of the deposit reproducing the anode symmetry and appearing at
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Fig. 7. Morphological transitions in a copper electrodeposition experiments from a 0.05 M CuSO aqueous solution. Electrode separation: 4 cm. Cell thickness: 300 lm. Picture width: 16 mm. (A) *<"30 V. (B) *<"60 V.
a reproducible stage of its development (Fig. 7) [6,35}37]. These changes can a!ect the rami"cation degree of the deposit and its colour and texture, and can appear for any initial morphology of the deposit. In spite that they have been mainly evidenced and examined for the homogeneous growth mode, they have been also reported for open fractal and "nger-like deposits. The main characteristics of these transitions suggest that they are due to cationic fronts coming from the anode. When these fronts meet the growing deposit, the chemical environment of the metal cation reduction process is modi"ed, inducing a change in the composition and therefore, in the observable features of the deposit. With the primary aim of investigating such morphological transitions, we conducted a systematic series of experiments for copper deposition from an aqueous copper sulphate solution in the concentration range 0.05}0.1 M [38]. In these experiments, anode dissolution increases the copper ion concentration in the anodic region together with the proton concentration due to copper ion hydrolysis. This prompts the formation of an acid front travelling from the anode towards the cathode (i.e. contrary to the advancing deposit) which is at the origin of the transition. The previous considerations clearly lead us to consider in more detail the chemical features of ECD. For the experiments just mentioned we investigated the composition of the initial deposit. The result was that instead of a pure copper deposit it is actually a mixture of copper and cuprous oxide. This "nding can be interpreted as originated from a competition between the two electrode reactions 2Cu>#2OH\#2e\PCu O#H O , Cu>#2e\PCu .
(7) (8)
At the pH values of the initial 0.05 M CuSO solution (pH&4.5), the contribution of the "rst reaction is not negligible at all, but when the acid front meets the growing deposit, this concurrent
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process is hindered and the direct reduction of copper ions to metallic copper is enhanced. This has been also veri"ed by phase analysis of the deposit before and after the transition using X-ray powder di!raction. These measurements allow us to conclude that the content of cuprous oxide in the deposit obtained from a 0.05 M copper sulphate solution is higher before (&40%) than after (&20%) the transition has occurred. This is in complete accordance with: (i) the in situ observation that the initial deposit is reddish and dull and turns to a red and bright structure with a metallic appearance when the transition has been completed; and (ii) the formation of this metallic deposit at high CuSO concentrations (i.e. lower pH values) where the transition is not observed ([CuSO ]'0.2 M). The second point to discuss is the position of the transition. It is clear that this will depend on the growth velocity of the deposit and that of migration of the acid front. If we accept the previously proposed ohmic model, the position of the transition would depend on the mobility of the ions responsible for the anodic front and the growth velocity of the deposit, proton and sulphate respectively. If we de"ne the position of the transition, j, as the distance from the transition to the cathode relative to the electrode separation, an straightforward calculation leads to u \ . (9) j" u #u \ & The experimental values of j are close to 0.25 at low potential values (<430 V), but increase with potential up to 0.43 at 100 V. However, when gravity-induced convection e!ects are completely suppressed placing the cell in a vertical con"guration with the cathode facing downwards, the position of the transition is not potential dependent with values very close to those obtained at low potential values (0.25) (Fig. 8). Thus, the potential dependence of j must be attributed to gravity
Fig. 8. Morphological transitions in a copper electrodeposition experiment from a 0.1 M CuSO aqueous solution in a vertical cell with the cathode facing downwards. Electrode separation: 4 cm. Cell thickness: 100 lm. Picture width: 16 mm. (A) *"60 V. (B) *"105 V.
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driven convection, and the value of 0.25 has to be considered as the genuinous value for pure ionic migration, i.e. in absence of any convection e!ect. Taking as sulphate mobility the value given by the growth velocity of the deposit, proton mobilities derived from this value of j are in reasonable agreement with those obtained with di!erent experimental techniques for similar systems. Let's "nish the section by stressing the role of the buoyancy-driven convection. As mentioned in Section 4, concentration gradients developing at both electrodes create rotating rolls that propagate through the solution. Looking in particular at the anodic region, where such a propagation is not a!ected by the growing deposit, it is found that the width of the propagating roll, l, depends on time as [14,15] lJt? with a(1. The rate of this motion will increase with the density gradient in the anodic region and therefore with the applied potential. Thus, it is reasonable to believe that at high potential values the protons originated in the anodic region will be at least transiently somehow trapped, decreasing their transport rate, and then delaying the position of the transition.
8. Finger-like morphology When small amounts of an alkaline sulphate are added to a 0.05 M copper sulphate solution, a deposit with new and totally unusual morphology in the "eld of ECD [17] is formed on the cathode (see Fig. 5). These aggregates consists of an array of rounded-like patterns that enclose a soft and very highly rami"ed microstructure (Fig. 9). The main morphological characteristics of this growth mode are: (i) a grey and dull appearance with a non-metallic aspect; (ii) the coexistence with hydrogen bubbles, and (iii) their regular pro"le, with rounded tips at the centimeter scale,
Fig. 9. Finger-like deposits obtained in a ECD experiment from a 0.05 M CuSO #3 mM Na SO aqueous solution. Cell thickness: 100 lm. Electrode separation: 4 cm. *<"25 V. Picture width: 16 mm.
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recalling those of viscous "ngering commonly ascribed to hydrodynamic phenomena [39,40]. For this reason, this texture is hereafter referred as the "nger morphology. Finger-like electrodeposits are an excellent example to stress the interplay between some of the ECD mechanisms mentioned before. In particular, migration, convection and chemical reactions play a crucial role. On what follows, we propose a sort of qualitative intrepretation of the origin of such a morphology. The addition of the inert electrolyte increases the conductivity of the solution and in turn the electric current. This prompts water reduction and hydrogen evolution as a concurrent electrochemical reaction. This reaction originates a basic medium around the growing interface that, in turn, explains the formation of copper hydroxylated species accumulating in a sort of hydrogel boundary layer surrounding the deposit, as is clearly observed in Fig. 10. These species are reduced to cuprous oxide, which is the predominant component of "nger-like deposits (see Eq. (7)). In contrast, common copper aggregates (i.e. open or homogeneous) are richer in copper than in cuprous oxide. This fact explains the di!erent color and texture of both type of microstructures. The special physico-chemical properties of the boundary layer are responsible for the enhancement of the e!ective surface tension associated with the interfacial region separating the depleted solution embedding the growing deposit, from the bulk. Such a surface tension acts as a restoring force opposite to the driving "eld, which is nothing but the electric force genuinous to any ECD experiment. Note that such a picture leads us directly to look for similarities between the phenomenon at hand and the usual "ngering instabilities in #uid displacement. Actually, we can even take for granted from such similar scenario the standard linear dispersion relation, to analyze the initial growth dynamics in our "ngering experiments [18]. One important remark is worth
Fig. 10. An optical image showing the hydrogel boundary layer surrounding a "nger-like deposit obtained from a 0.05 M CuSO #7 mM Na SO aqueous solution. Cell thickness: 250 lm. Electrode separation: 4 cm. *<"12 V. Picture width: 4 mm.
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making at this point. Since the value of the e!ective surface tension can not be calculated by any reasonable means, we assume that it is going to be inversely proportional to the mixing layer width, l [41]. According to parallel experimental observations, this last quantity is in turn related to the @J growth velocity according to a power law as v?. In such a way one can formally predict a value for the parameter b, de"ned in the adapted dispersion relation, NJk J(vl ) Jv@, as
@J b"(a!1)/2a, where k represents the most unstable wave number. Taking a"1.5 as determined
previously, bK0.16. This was experimentally checked by conducting a series of experiments varying the applied potential di!erence, and recording the number of incipient "ngers, N, at di!erent growth velocities, v. These results "tted a power law, N&v@ with bK0.18. Another experimental observation reinforcing the particular role of the mixing layer in the origin of this morphology in ECD, concerns the singular e!ect of convection. It has been mentioned before (see Section 7) that gravity-induced convection can be largely suppressed by turning the cell into a vertical con"guration with the cathode facing downwards. Under this arrangement, the deposits display their typical "ne texture but enclosed this time by a planar front which is free of any long wavelength modulation (Fig. 11). This behaviour can be understood by considering that in absence of convection e!ects the boundary layer would be largely compressed and its endowed e!ective surface tension increased up to a value such that the "nger structure cannot be developed, in spite of the fact that the microstructural texture is largely preserved. The "nal aspect we want to refer to in relation with the "nger growth mode concerns the morphological transitions they experience [7]. Complementary to what has been said in the last section one observes a richer scenario when examining such transitions in ECD "ngers. Actually, there are two transitions appearing at di!erent potential ranges. Both di!er in their position and in the changes they cause to the initial deposit (Fig. 12). At low potential values, the transition simply
Fig. 11. Deposit obtained in an ECD experiment from a 0.05 M CuSO #2 mM Na SO aqueous solution in a vertical cell with growth directed downwards. Cell thickness: 100 lm. Electrode separation: 4 cm. *<"25 V. Picture width: 16 mm.
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Fig. 12. Morphological transitions experimented by "nger-like deposits obtained from a 0.05 M CuSO # 2 mM K SO aqueous solution in a parallel cell. Cell thickness: 100 lm. Electrode separation: 4 cm. (A) *"20 V. (B) 10 V.
re#ects the depletion of alkaline ions due to their migration to the cathodic region and a typical copper deposit is recovered after the morphological change. This is in good agreement with its position relative to the distance between the electrodes. Furthermore this would explain the observed #at boundary separating the two morphologies. At high potential values, the proton concentration in the anodic region increases prompting the formation of an acid front which neutralizes the basic layer around the deposit and, in many cases, even leading the penetration inside the "nger microstructure (Fig. 12). In fact, this transition is the same as that discussed in Section 7.
9. Summary A review of di!erent aspects of the phenomenon of quasi-two-dimensional electrocrystallization has been presented. Emphasis has been devoted to the understanding of the delicate coupling between the di!erent transport mechanisms and chemical processes, which "nally results in the diverse growth modes observed in the experiments. In particular fractal and multifractal features associated to di!usive-like growth processes have been examined for the open and highly rami"ed patterns, obtained at very low applied potentials and high concentrations from a binary electrolyte. By invoking pure ionic migration concepts, we were able to predict the growth velocity, and its dependence on the conductivity properties of the solution, for the so-called dense parallel or homogeneous morphology, obtained contrarily at low concentrations when high enough potentials are applied to these systems. Buoyancy-driven convection e!ects and the role of chemical reactions are stressed in relation to both, morphological transitions and the appearance of a particularly interesting and unexpected
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"nger-like morphology obtained by electrolizing electrolytes containing a small amount of an added inert ionic substance.
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