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Three-dimensional nucleation with diffusion-controlled growth: Simulation of hierarchical diffusion zones overlap
Journal of Electroanalytical Chemistry 631 (2009) 22–28
Contents lists available at ScienceDirect
Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem
Three-dimensional nucleation with diffusion-controlled growth: Simulation of hierarchical diffusion zones overlap D. Mazaira, C. Borrás, J. Mostany *, B.R. Scharifker Departamento de Química, Universidad Simón Bolívar, Apartado 89000, Caracas 1080A, Venezuela
a r t i c l e
i n f o
Article history: Received 9 December 2008 Received in revised form 5 March 2009 Accepted 11 March 2009 Available online 19 March 2009 Keywords: Electrocrystallization Nucleation Growth Monte Carlo KJMA theory
a b s t r a c t We show that a major difficulty persists for the comprehensive description of multiple nucleation on active sites followed by three-dimensional diffusion-controlled growth, namely the unfeasible consideration of a uniform height for all diffusion zones, which is required for proper consideration of the process in analogy to a two-dimensional transformation within the standard model. Through numerical simulations of the Monte Carlo type we illustrate that a hierarchical model of overlap, with earlier diffusion zones prevailing over later ones, may lead to overcoming this major limitation. The results obtained have been compared with those given by current formulations of the standard model, showing that hierarchical overlap of diffusion zones offers a robust and improved description of multiple nucleation with threedimensional diffusion-controlled growth. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Electrochemical phase formation processes are matters of intense current interest, as detailed knowledge of the kinetics of such processes and the mechanisms involved are needed for current and envisaged developments in energy conversion and storage, electrocatalysis, and in general for the development of nanostructures. Phase formation processes generally occur by way of nucleation and growth mechanisms. Under thermodynamic conditions ensuring the viability of a phase transformation, nuclei are stable structures arising from the clustering of atoms or molecules constituting centers for the growth of the transformed new phase. In general phase transformations may arise from changes in pressure, temperature, composition or, in electrochemical systems, by changes of the electric potential, driven by charge transfer reactions. Hence nuclei for the formation of a new phase may arise at the interphase between ionically and electronically conducting phases during the course of electron transfer reactions [1]. Thus depending on the electrochemical conditions allowing the formation of a new phase, nuclei will appear on the electrode surface at a certain rate. Furthermore, clustering processes occur heterogeneously on the electrode surface, hence they will most likely appear on low energy locations such as, e.g., surface defects, or active sites on the surface. The new phase propagates from these centers, growing through incorporation of ions from solution via charge transfer reactions. * Corresponding author. Tel.: +58 2129063993; fax: +58 2129063969. E-mail addresses: [email protected], [email protected] (J. Mostany). 0022-0728/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2009.03.004
In broad terms we may classify the growth of nuclei into two distinctive categories, either limited by the rate at which ions may incorporate to the new phase, i.e., charge transfer controlled, or limited by the rate of mass transport of depositing species from the bulk of the solution to the electrode surface, i.e., diffusion controlled [2]. For nuclei developed to sufficiently large sizes during deposition of most metals from their ions in aqueous solutions or molten salts, the rates of charge transfer and incorporation of discharged species into the lattice of the deposited phase are fast in relation to mass transport from solution to the surface of growing centers, hence diffusion control prevails. As Compton and Hyde [2] have shown in their 2003 review, the description of the potentiostatic transient has been a controversial theme for the past 50 yr, and is still generating questions and models. The main difficulty is that even though nucleation occurs on the surface of the electrode, the growth of nuclei and the mass transport supporting it extends towards the bulk of the solution generating concentration gradients with locally spherical symmetry. In order to consider their mutual interference, Scharifker and Hills [3] conceived planar diffusion zones on the electrode surface towards which the same amount of depositing material would flow through planar diffusion as that flowing by spherical diffusion to growing nuclei. In this fashion the locally spherical diffusion around individual nuclei has been regarded as planar to the segments of the electrode surface area covered by diffusion zones, with mass transport in the direction normal to the electrode surface. The time-dependent radius rd(t) of diffusion zones has been obtained from the expression of mass transport under semi-infinite
D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
23
Nomenclature nucleation frequency per active site (s1) concentration of electroactive species in the bulk of solution (mol cm3) diffusion coefficient (cm2 s1) Faraday constant (C mol1) current associated to a diffusion zone k (A) diffusion flux to a hemispherical diffusion zone (mol cm2 s2) diffusion flux to an equivalent planar diffusion zone (mol cm2 s2)
A c D F ik Jr Jp
k M N0 r0 rd u d
q hk
= (8pcM/q)1/2 (dimensionless) molar mass of electroactive species (g mol1) number density of active sites (cm2) nucleus radius (cm) diffusion zone radius (cm) nucleus birth time (s) width of diffusion layer (cm) density of electrodeposit (g cm3) fractional coverage of diffusion zone k (dimensionless)
diffusion conditions and zero surface concentration of electroactive species to small hemispheres of radius r0, growing under steady state mass transport conditions [4,5]:
r 0 ðtÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DcMt
q
ð1Þ
and considering around them circular areas such as to equate the amount of material transported to them by planar flux to that flowing by way of spherical diffusion to the growing nuclei, yielding [6]:
rd ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8pcMDt
q
ð2Þ
in order to describe the growth of an isolated nucleus, without consideration of the mutual interference between neighbouring centers. 1.1. Overlap of planar diffusion zones Scharifker and Mostany [6] (henceforth referred to as SM) took on the concept of planar diffusion zones developed in [3] but with explicit consideration of the decaying number density of sites available for nucleation. Following arguments developed by Evans [7] leading to results equivalent to those of Kolmogorov [8], Johnson and Mehl [9] and Avrami [10–12] (commonly known as KJMA theory), they obtained a general expression for the current transient embracing instantaneous and progressive nucleation as particular cases at opposite extremes of the N0/A ratio, with very high nucleation frequencies on a small number density of active sites characterizing instantaneous nucleation, and low nucleation frequencies on very large number densities of sites on the surface corresponding to progressive nucleation. Sluyters-Rechbach et al. (henceforth SRWBS) [13] appropriately considered that the KJMA theory could not be applied to diffusion zones with different heights, representing non-uniform gradients across the surface, cf. Fig. 1, as the SM model would imply, and this has prompted the development of alternative formulations of the standard model, striving to overcome these difficulties [14]. Heerman and Tarallo [15] (HT) took on the inconsistencies pointed out by SRWBS about the standard model and considering that proper application of the KJMA theory to planar diffusion zones requires that their height should be uniform disregarding their distinct times of birth during progressive nucleation, arrived to an expression identical to that obtained earlier by Mirkin and Nilov [16] who did not support their integral equation analysis on the a priori assumption that the current should conform to that given by the Cottrell equation in the diffusion zones. This approach gives a overestimated current density that Heerman et al. [17] deemed resulted from inaccurate detailed mass balance at individual diffusion zones when these are born at different times; thus
Fig. 1. Schematic representation of mass transport to diffusion zones on the electrode surface, with diffusion widths related to their ages, see text.
they defined a single planar diffusion zone for the overall ensemble of growing nuclei instead of manifold zones associated to each of the growing centers. This introduces a uniform diffusion layer, related to the mean concentration field brought in by Bobbert et al. [18]. In 2004 Matthijs et al. (MLMH) [19] considered the ‘‘phantom nuclei” (cf. [10–12]) needed to recover the random distribution required for proper application of the KJMA theory to progressive nucleation. Further below we show that the MLMH conjecture fits well experimental and simulated data approaching progressive nucleation, however it only applies to this limiting case and hence it is not possible to obtain A and N0 separately using this approach. However this model can describe well the behaviour at progressive nucleation limit, where all the previous approaches become inappropriate, as discussed below. 1.2. A hierarchical model for the overlap of diffusion zones with different heights All the previously described treatments use the KJMA theory [7–12] to take account of overlap of planar diffusion zones. Since proper consideration within this framework only holds within the dimensional domain of the phase transformation, in this case for the overlap of circular regions within the plane, it is hence required that all diffusion zones have the same width. This is readily
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D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
realized when all nuclei are born at the same time, i.e., when nucleation is instantaneous. Yet uniform width of diffusion zones will hardly be the case if nuclei started growing at different times, and it stems clear from the succeeding formulations of the standard model, that the requirement of uniform width for all diffusion zones poses a major difficulty for the treatment of the current transient within the framework of the KJMA theory when nucleation is not instantaneous. Circumventing this difficulty requires either compelling uniform width for all diffusion zones to retain use of the KJMA theory, or discarding the use of the KJMA theory to account for the overlap of diffusion zones of variable width. From the discussion above it is apparent that this latter path is now necessary; thus we will now turn to describe a hierarchical model accounting for overlap of diffusion zones born at various times, and consequently with differing widths of their respective diffusion layers. We considered a set of diffusion zones expanding on the surface of the electrode, with radial rates at time t for nuclei born at times u, given by rd(t,u) = [kD(t u)]1/2, and width
dðt; uÞ ¼ ½pDðt uÞ1=2
ð3Þ
We also considered that once a diffusion zone has been established, the growth of a nucleus and a diffusion zone within it has no bearing on mass transport, and hence current flow is unaffected by nuclei appearing, or newer diffusion zones growing, within the already established one. Therefore whereas in the usual unranked situation described by the KJMA theory growth ceases as circles impinge each other during radial growth, within the hierarchical model older diffusion zones with larger radii and taller widths prevail over younger and consequently shorter ones. Hierarchy upon impingement then implies that at short times during progressive nucleation there will be a spread of diffusion zones widths, though at longer times the older diffusion zones will run over younger ones and eventually, as t ? 1, only the oldest zone will survive and the width will attain uniformity all over the surface approaching that corresponding to planar diffusion to the entire surface given by the Cottrell equation. Analytical solutions for the hierarchical overlap of diffusion zones postulated here are as yet not available. Thus we have explored this model by way of Monte Carlo type simulations on a small square segment of surface (104 cm2) with periodical boundary conditions to reduce border effects [20]. Simulations were run with various ratios N0/A of number densities of active sites to nucleation frequencies in order to cover the range from instantaneous to progressive nucleation.
ik ðtÞ ¼ zFcD1=2 ðpðt uÞÞ1=2 hk
ð5Þ
and the complete transient was built adding all contributions. Several nucleation frequencies A were used to cover the range from instantaneous to progressive nucleation. Nucleation sites were established randomly on the surface and new nuclei were introduced according to the nucleation frequency without taking account if the active site had been covered previously by an already established diffusion zone, since consideration of hierarchy will anyway take the nuclei as phantom in these cases. N0 was kept constant throughout at 1 106 cm2. Since theories taking into account overlap have been developed for infinite planes, proper definition of boundary conditions is of importance and more so given the small size of the simulation cell considered for the calculations reported in this work. In order to minimize effects due to the borders of the simulation cell, we have used periodic boundary conditions [21,22]. Thus we created eight copies of the simulation cell, with diffusion zones spilling over the borders of the central cell being reintroduced from the opposite side. These toroidal boundary conditions simulate a surface with finite size but no boundaries. The simulation parameters were chosen to correspond to the electrodeposition of mercury at room temperature from 5 mM Hg22+ aqueous solution in the presence of excess supporting electrolyte. 3. Results and discussion Fig. 2 shows the radial growth rate drd/dt of a set of diffusion zones introduced at regular intervals of 0.4 s in the simulated area of 1 104 cm2 during nucleation at frequency A = 0.025 s1 onto N0 = 1 106 sites/cm2. Because of the parabolic growth rate rd(t,u) = [kD(t u)]1/2, younger diffusion zones grow at a given time faster than older ones. Therefore younger diffusion zones born inside already established ones may conceivably overflow these. It has been established that such phantom nuclei may account for up to 4% error in the estimation of the coverage using the KJMA theory [23], an effect not arising in the present digital simulation using the hierarchical concept of overlap. Fig. 3 shows the coverage of individual diffusion zones during hierarchical overlap, resulting from simulation of the process with A = 0.025 s1, N0 = 1 106 cm2. The coverage corresponding to the oldest nucleus in the set increases constantly, whereas the coverage of younger zones increase linearly with time at short times,
2. Simulation method Monte Carlo simulations were carried on a square surface of 1 104 cm2 with periodic boundary conditions. One hundred pairs of (x, y) coordinates were generated in each simulation, on which nuclei were introduced at regular intervals determined by the nucleation frequency A. Each nucleus developed a diffusion zone around it, with radius given by the standard model taking into consideration its age:
pðrd;SM Þ2 ¼ ð2pÞ3=2 DðcM=qÞ1=2 ðt uÞ
ð4Þ
Hierarchy was established for the calculation of the surface areas covered by each diffusion zone that determined the currents corresponding to each. When diffusion zones impinged with each other, then the overlapped area was assigned to the older one. The contribution of each diffusion zone k to the current density of the ensemble was then determined as that corresponding to planar diffusion to the fraction of area covered by k,
Fig. 2. Rate of radial growth of individual diffusion zones as a function of time, during simulation in 1 104 cm2 with A = 0.025 s1 and N0 = 1 106 cm2.
D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
Fig. 3. Coverage of the simulated area of 1 104 cm2 by individual diffusion zones, during hierarchical overlap with A = 0.025 s1 and N0 = 1 106 cm2.
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Fig. 5. Current contributions by individual diffusion zones according to (14), during hierarchical overlap as shown in Fig. 4, with A = 0.025 s1 and N0 = 1 106 cm2.
but then fall at longer times as they are ingested by older zones, vanishing in the limit of t ? 1, when only the oldest of the diffusion zones survives. The ‘phantom’ nuclei also generate diffusion zones but they do not contribute to the coverage; only the diffusion zone with highest hierarchy at a given point contributes to the coverage. Fig. 4 shows the coverage of the simulation area with diffusion zones as a function of time for different values of the A/N0 ratio. As the nucleation frequency increases, total coverage of the surface with diffusion zones is reached faster and eventually, at very long times, the oldest diffusion zone covers the entire surface. The currents attending the growth of diffusion zones, given by (14), are shown in Fig. 5. The resulting overall current transients obtained summing the individual current contributions at every time step are shown in Fig. 6 for several values of the ratio A/N0, with N0 kept constant at N0 = 1 106 cm2 throughout, while A was varied from 0.025 to 100 s1.
Fig. 4. Total fractional coverage of the surface with diffusion zones for different values of A, as indicated in s1. N0 = 1 106 cm2.
Fig. 6. Potentiostatic current transients from simulations with hierarchical overlap of diffusion zones, with N0 = 1 106 cm2 and A as indicated, between 0.025 and 0.75 s1 (a), and 1 and 100 s1 (b).
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D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
3.1. Comparison of transients arising from hierarchical and nonhierarchical models The various formulations of the standard model describing nucleation with diffusion-controlled three-dimensional growth consider non-hierarchical overlap of diffusion zones in terms of the KJMA theory, and the chief differences among them rest in the way the widths of diffusion zones born at different times are homogenized. The hierarchical overlap introduced in the present work avoids mean field approximations [17,18] or reducing all disks to a uniform width, notwithstanding their different ages due to progressive nucleation. We followed two approaches to compare transients obtained from simulations of hierarchical overlap with theoretical models considering the KJMA non-hierarchical overlap. First the transients obtained from simulations of hierarchical overlap of diffusion zones with given A and N0 values were graphically compared with those predicted by the various formulations of the standard model in terms of non-hierarchical KJMA overlap. Then in a second approach and for a more quantitative assessment of differences among the various models, transients generated from simulations with hierarchical overlap of diffusion zones were analysed using the theoretical expressions of the models corresponding to KJMA non-hierarchical overlap, in order to obtain values of the kinetic parameters A and N0 that were contrasted to the values of A and N0 actually used to generate the simulated transients. The first approach is illustrated in Fig. 7, where transients obtained from simulations with different nucleation frequencies are shown. Nucleation shifts from progressive to instantaneous as A increased. For N0/A 1 nucleation is progressive and as shown in
Fig. 7a and b, the currents obtained from simulations of hierarchical overlap approach those predicted by the MLMH conjecture [19], however attaining higher maximum currents and approaching faster the Cottrell limit corresponding to planar diffusion to the entire electrode surface. The SM formulation of the standard model does not consider the spread in birth times of diffusion zones occurring during progressive nucleation, thus the corresponding currents remain invariably below the Cottrell limit at all times. The HT expression on the other hand does take into account the successive appearance of nuclei and homogenizing the width of diffusion zones, yields higher diffusion currents at long times, an effect not shown by the transients generated considering hierarchical overlap of diffusion zones with non-uniform widths. Fig. 7c and d show that the various expressions based upon the KJMA theory, with the exception of the MLMH conjecture, coincide with each other and also with the simulated results considering hierarchical overlap, as the instantaneous limit at N0/A ? 0 is reached, where ranking between diffusion zones vanishes when all nuclei are born at the same time. Having compared graphically the current transients arising from simulations of hierarchical overlap of diffusion zones at given A and N0 values, with those obtained from the various expressions for unranked KJMA overlap, we will now assume no prior knowledge of the kinetic parameters, and obtain their values from analysis of the result of simulations using the formulations devised for unranked KJMA overlap. This will provide further insights on the relation between the hierarchical overlap postulated here and that of the KJMA theory. When analytical methods were available, as in the SM case, A and N0 where obtained from {im, tm} values at the current maximum; other cases were examined fitting the theoret-
Fig. 7. Comparison of transients from simulations with hierarchical overlap of diffusion zones (.), with different formulations of the unranked, standard model with overlap accounted for by the KJMA theory: SM (j), HML (N), MLMH (I), and HTMN (d). N0 = 1 106 cm2 and A = 0.025 s1 (a), 0.25 s1 (b), 2.0 s1 (c), and 20 s1 (d). The diffusion current to a uniformly accessible planar electrode is also shown (broken line).
D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
Fig. 8. Nucleation frequencies Amod obtained from analysis of simulations with hierarchical overlap of diffusion zones using the SM (s), HT (4) and HML (5) formalisms, as a function of the nucleation frequency Asim in simulations with N0 = 1 106 cm2. The continuous line corresponds to Amod = Asim
Fig. 9. Number densities of active sites for nucleation N0 obtained from analysis of simulations with hierarchical overlap of diffusion zones using the SM (s), HT (4) and HML (5) formalisms, as a function of the nucleation frequency Asim in simulations with N0 = 1 106 cm2.
ical expressions non-linearly to the simulated data, leaving A and N0 as the sole fitting parameters. The result of such analysis is shown in Figs. 8 and 9. Fig. 8 shows the nucleation frequencies Amod found from the different formulations of the standard model using unranked KJMA overlap, correlated with the corresponding values Asim used in simulations with hierarchical overlap, whereas Fig. 9 shows the N0 values found from the simulated transients using the various expressions as a function of Asim. Figs. 8 and 9 show the confluence of the SM, HT and HML formulations of the KJMA unranked overlap together with hierarchical overlap at the instantaneous limit, where differences between widths of diffusion zones vanish as the distribution of nuclei ages narrows when these are born at the onset of the process. In contrast, fresh information becomes apparent from the present analysis at the opposite extreme of progressive nucleation. The SM model yields lower nucleation frequencies A than those actually used to generate the
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Fig. 10. Stationary nucleation rates AN0 obtained from analysis of simulations with hierarchical overlap of diffusion zones using the SM (s), HT (4) and HML (5) models, as a function of the nucleation frequency Asim in simulations with N0 = 1 106 cm2. The continuous line corresponds to the AN0 product used in the simulations.
transients with hierarchical overlap, while significantly overestimating the number densities of active sites N0. Conversely, the HT and HML models overestimate A and yield lower values for N0. Since none of these models contemplates hierarchical overlap, then it is not expected that they should predict appropriately the A and N0 values used to generate the simulated data; however, when the product of A and N0 is considered, then the differences between the SM, HT, HML models vanish, as shown in Fig. 10, and moreover the AN0 product obtained is indeed close to that corresponding to the simulations with hierarchical overlap. This result is significant, suggesting that the value of the product AN0 determined from the various models accounting differently for overlap of diffusion zones is robust and model-independent, while conversely indicating also that establishing precisely and unequivocally the values of A and N0 requires determining which of the models affords the most fitting description of reality. As a final remark, we need to point out that some of the differences observed may have arisen from the different mathematical methods employed to obtain A and N0 from the respective models. A and N0 were found from the SM model solving a system of nonlinear equations using the coordinates of the current maximum [6], while in the case of the HML [17] and HT [15] models the kinetic parameters were obtained from non-linear fitting of the whole data sets to the theoretical equations, using the Levenberg–Marquardt algorithm. Values obtained from the MLMH model have not been included in this particular analysis because the cubic polynomial developed as approximate solution [19] holds only before the current maximum, a time span in general too short for comparison with other models. 4. Conclusions We presented a novel description of Multiple nucleation with three-dimensional diffusion-controlled growth, based upon hierarchical overlap of planar diffusion zones associated to growing nuclei, wherein older zones prevail over younger ones when they impinge and overlap. Through digital simulations we have shown the main features of such model, and the results arising from hierarchical overlap were compared both qualitatively and quantitatively with models based upon application of the KJMA theory
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D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
for unranked overlap. Hierarchical overlap of diffusion zones avoids the need of homogenization of diffusion zones through mean field approximations or other measures, considering also realistically the effects of dispersion in the age of nuclei on their growth rates, of up most importance in the limit of progressive nucleation. In the course of this study we have shown that current models provide for the robust evaluation of the product AN0 of the nucleation frequencies and the number density of active sites for nucleation, and that while hierarchical overlap of diffusion zones may provide a germane model for the unequivocal evaluation of A and N0 from single-step, potentiostatic experiments, a closed form expression for the current transient is not yet available. This will be a next step ahead, and work is currently in progress in our laboratory following this direction.
Acknowledgements The authors are very grateful to the members of the Electrochemistry Group at Simon Bolivar University for stimulating and productive discussions, specially to Michele Milo for his contribution to this work. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jelechem.2009.03.004.
References [1] B.R. Scharifker, J. Mostany, in: A.J. Bard, M. Stratmann, E.J. Calvo (Eds.), Encyclopedia of Electrochemistry, vol. 2, John Wiley & Sons, New York, 2003, p. 512. [2] M.E. Hyde, R.G. Compton, J. Electroanal. Chem. 549 (2003) 1. [3] B. Scharifker, G. Hills, Electrochim. Acta 28 (1983) 879. [4] G.J. Hills, D.J. Schiffrin, J. Thompson, Electrochim. Acta 19 (1974) 657. [5] S. Fletcher, J. Chem. Soc., Faraday Trans. 79 (1983) 467. [6] B.R. Scharifker, J. Mostany, J. Electroanal. Chem. 177 (1984) 13. [7] U.R. Evans, Trans. Faraday Soc. 41 (1945) 365. [8] A.N. Kolmogorov, Bull. Acad. Sci. USSR, Math. Nat. 3 (1937) 355. [9] W.A. Johnson, R.F. Mehl, Trans. Am. Ins. Min. Met. Eng. 135 (1939) 416. [10] M. Avrami, J. Chem. Phys. 7 (1939) 1103. [11] M. Avrami, J. Chem. Phys. 8 (1940) 212. [12] M. Avrami, J. Chem. Phys. 9 (1941) 177. [13] M. Sluyters-Rehbach, J.H.O.J. Wijenberg, E. Bosco, J.H. Sluyters, J. Electroanal. Chem. 236 (1987) 1. [14] B.R. Scharifker, J. Mostany, M. Palomar-Pardave, I. Gonzalez, J. Electrochem. Soc. 146 (1999) 1005. [15] L. Heerman, A. Tarallo, J. Electroanal. Chem. 470 (1999) 70. [16] M.V. Mirkin, A.P. Nilov, J. Electroanal. Chem. 283 (1990) 35. [17] L. Heerman, E. Matthijs, S. Langerock, Electrochim. Acta 47 (2001) 905. [18] P.A. Bobbert, M.M. Wind, J. Vlieger, Physica A 146 (1987) 69. [19] E. Matthijs, S. Langerock, E. Michailova, L. Heerman, J. Electroanal. Chem. 570 (2004) 123. [20] D. Mazaira, Dissertation, Universidad Simón Bolívar, Caracas, 2006. [21] B.R. Scharifker, Acta Científica Venezolana 35 (1984) 211. [22] B.D. Ripley, Spatial Statistics, John Wiley & Sons Inc., New York, 1981. [23] W.H. Mulder, J.H.O.J. Wijenberg, M. Sluyters-Rehbach, J.H. Sluyters, J. Electroanal. Chem. 270 (1989) 7.
Contents lists available at ScienceDirect
Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem
Three-dimensional nucleation with diffusion-controlled growth: Simulation of hierarchical diffusion zones overlap D. Mazaira, C. Borrás, J. Mostany *, B.R. Scharifker Departamento de Química, Universidad Simón Bolívar, Apartado 89000, Caracas 1080A, Venezuela
a r t i c l e
i n f o
Article history: Received 9 December 2008 Received in revised form 5 March 2009 Accepted 11 March 2009 Available online 19 March 2009 Keywords: Electrocrystallization Nucleation Growth Monte Carlo KJMA theory
a b s t r a c t We show that a major difficulty persists for the comprehensive description of multiple nucleation on active sites followed by three-dimensional diffusion-controlled growth, namely the unfeasible consideration of a uniform height for all diffusion zones, which is required for proper consideration of the process in analogy to a two-dimensional transformation within the standard model. Through numerical simulations of the Monte Carlo type we illustrate that a hierarchical model of overlap, with earlier diffusion zones prevailing over later ones, may lead to overcoming this major limitation. The results obtained have been compared with those given by current formulations of the standard model, showing that hierarchical overlap of diffusion zones offers a robust and improved description of multiple nucleation with threedimensional diffusion-controlled growth. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Electrochemical phase formation processes are matters of intense current interest, as detailed knowledge of the kinetics of such processes and the mechanisms involved are needed for current and envisaged developments in energy conversion and storage, electrocatalysis, and in general for the development of nanostructures. Phase formation processes generally occur by way of nucleation and growth mechanisms. Under thermodynamic conditions ensuring the viability of a phase transformation, nuclei are stable structures arising from the clustering of atoms or molecules constituting centers for the growth of the transformed new phase. In general phase transformations may arise from changes in pressure, temperature, composition or, in electrochemical systems, by changes of the electric potential, driven by charge transfer reactions. Hence nuclei for the formation of a new phase may arise at the interphase between ionically and electronically conducting phases during the course of electron transfer reactions [1]. Thus depending on the electrochemical conditions allowing the formation of a new phase, nuclei will appear on the electrode surface at a certain rate. Furthermore, clustering processes occur heterogeneously on the electrode surface, hence they will most likely appear on low energy locations such as, e.g., surface defects, or active sites on the surface. The new phase propagates from these centers, growing through incorporation of ions from solution via charge transfer reactions. * Corresponding author. Tel.: +58 2129063993; fax: +58 2129063969. E-mail addresses: [email protected], [email protected] (J. Mostany). 0022-0728/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2009.03.004
In broad terms we may classify the growth of nuclei into two distinctive categories, either limited by the rate at which ions may incorporate to the new phase, i.e., charge transfer controlled, or limited by the rate of mass transport of depositing species from the bulk of the solution to the electrode surface, i.e., diffusion controlled [2]. For nuclei developed to sufficiently large sizes during deposition of most metals from their ions in aqueous solutions or molten salts, the rates of charge transfer and incorporation of discharged species into the lattice of the deposited phase are fast in relation to mass transport from solution to the surface of growing centers, hence diffusion control prevails. As Compton and Hyde [2] have shown in their 2003 review, the description of the potentiostatic transient has been a controversial theme for the past 50 yr, and is still generating questions and models. The main difficulty is that even though nucleation occurs on the surface of the electrode, the growth of nuclei and the mass transport supporting it extends towards the bulk of the solution generating concentration gradients with locally spherical symmetry. In order to consider their mutual interference, Scharifker and Hills [3] conceived planar diffusion zones on the electrode surface towards which the same amount of depositing material would flow through planar diffusion as that flowing by spherical diffusion to growing nuclei. In this fashion the locally spherical diffusion around individual nuclei has been regarded as planar to the segments of the electrode surface area covered by diffusion zones, with mass transport in the direction normal to the electrode surface. The time-dependent radius rd(t) of diffusion zones has been obtained from the expression of mass transport under semi-infinite
D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
23
Nomenclature nucleation frequency per active site (s1) concentration of electroactive species in the bulk of solution (mol cm3) diffusion coefficient (cm2 s1) Faraday constant (C mol1) current associated to a diffusion zone k (A) diffusion flux to a hemispherical diffusion zone (mol cm2 s2) diffusion flux to an equivalent planar diffusion zone (mol cm2 s2)
A c D F ik Jr Jp
k M N0 r0 rd u d
q hk
= (8pcM/q)1/2 (dimensionless) molar mass of electroactive species (g mol1) number density of active sites (cm2) nucleus radius (cm) diffusion zone radius (cm) nucleus birth time (s) width of diffusion layer (cm) density of electrodeposit (g cm3) fractional coverage of diffusion zone k (dimensionless)
diffusion conditions and zero surface concentration of electroactive species to small hemispheres of radius r0, growing under steady state mass transport conditions [4,5]:
r 0 ðtÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DcMt
q
ð1Þ
and considering around them circular areas such as to equate the amount of material transported to them by planar flux to that flowing by way of spherical diffusion to the growing nuclei, yielding [6]:
rd ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8pcMDt
q
ð2Þ
in order to describe the growth of an isolated nucleus, without consideration of the mutual interference between neighbouring centers. 1.1. Overlap of planar diffusion zones Scharifker and Mostany [6] (henceforth referred to as SM) took on the concept of planar diffusion zones developed in [3] but with explicit consideration of the decaying number density of sites available for nucleation. Following arguments developed by Evans [7] leading to results equivalent to those of Kolmogorov [8], Johnson and Mehl [9] and Avrami [10–12] (commonly known as KJMA theory), they obtained a general expression for the current transient embracing instantaneous and progressive nucleation as particular cases at opposite extremes of the N0/A ratio, with very high nucleation frequencies on a small number density of active sites characterizing instantaneous nucleation, and low nucleation frequencies on very large number densities of sites on the surface corresponding to progressive nucleation. Sluyters-Rechbach et al. (henceforth SRWBS) [13] appropriately considered that the KJMA theory could not be applied to diffusion zones with different heights, representing non-uniform gradients across the surface, cf. Fig. 1, as the SM model would imply, and this has prompted the development of alternative formulations of the standard model, striving to overcome these difficulties [14]. Heerman and Tarallo [15] (HT) took on the inconsistencies pointed out by SRWBS about the standard model and considering that proper application of the KJMA theory to planar diffusion zones requires that their height should be uniform disregarding their distinct times of birth during progressive nucleation, arrived to an expression identical to that obtained earlier by Mirkin and Nilov [16] who did not support their integral equation analysis on the a priori assumption that the current should conform to that given by the Cottrell equation in the diffusion zones. This approach gives a overestimated current density that Heerman et al. [17] deemed resulted from inaccurate detailed mass balance at individual diffusion zones when these are born at different times; thus
Fig. 1. Schematic representation of mass transport to diffusion zones on the electrode surface, with diffusion widths related to their ages, see text.
they defined a single planar diffusion zone for the overall ensemble of growing nuclei instead of manifold zones associated to each of the growing centers. This introduces a uniform diffusion layer, related to the mean concentration field brought in by Bobbert et al. [18]. In 2004 Matthijs et al. (MLMH) [19] considered the ‘‘phantom nuclei” (cf. [10–12]) needed to recover the random distribution required for proper application of the KJMA theory to progressive nucleation. Further below we show that the MLMH conjecture fits well experimental and simulated data approaching progressive nucleation, however it only applies to this limiting case and hence it is not possible to obtain A and N0 separately using this approach. However this model can describe well the behaviour at progressive nucleation limit, where all the previous approaches become inappropriate, as discussed below. 1.2. A hierarchical model for the overlap of diffusion zones with different heights All the previously described treatments use the KJMA theory [7–12] to take account of overlap of planar diffusion zones. Since proper consideration within this framework only holds within the dimensional domain of the phase transformation, in this case for the overlap of circular regions within the plane, it is hence required that all diffusion zones have the same width. This is readily
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realized when all nuclei are born at the same time, i.e., when nucleation is instantaneous. Yet uniform width of diffusion zones will hardly be the case if nuclei started growing at different times, and it stems clear from the succeeding formulations of the standard model, that the requirement of uniform width for all diffusion zones poses a major difficulty for the treatment of the current transient within the framework of the KJMA theory when nucleation is not instantaneous. Circumventing this difficulty requires either compelling uniform width for all diffusion zones to retain use of the KJMA theory, or discarding the use of the KJMA theory to account for the overlap of diffusion zones of variable width. From the discussion above it is apparent that this latter path is now necessary; thus we will now turn to describe a hierarchical model accounting for overlap of diffusion zones born at various times, and consequently with differing widths of their respective diffusion layers. We considered a set of diffusion zones expanding on the surface of the electrode, with radial rates at time t for nuclei born at times u, given by rd(t,u) = [kD(t u)]1/2, and width
dðt; uÞ ¼ ½pDðt uÞ1=2
ð3Þ
We also considered that once a diffusion zone has been established, the growth of a nucleus and a diffusion zone within it has no bearing on mass transport, and hence current flow is unaffected by nuclei appearing, or newer diffusion zones growing, within the already established one. Therefore whereas in the usual unranked situation described by the KJMA theory growth ceases as circles impinge each other during radial growth, within the hierarchical model older diffusion zones with larger radii and taller widths prevail over younger and consequently shorter ones. Hierarchy upon impingement then implies that at short times during progressive nucleation there will be a spread of diffusion zones widths, though at longer times the older diffusion zones will run over younger ones and eventually, as t ? 1, only the oldest zone will survive and the width will attain uniformity all over the surface approaching that corresponding to planar diffusion to the entire surface given by the Cottrell equation. Analytical solutions for the hierarchical overlap of diffusion zones postulated here are as yet not available. Thus we have explored this model by way of Monte Carlo type simulations on a small square segment of surface (104 cm2) with periodical boundary conditions to reduce border effects [20]. Simulations were run with various ratios N0/A of number densities of active sites to nucleation frequencies in order to cover the range from instantaneous to progressive nucleation.
ik ðtÞ ¼ zFcD1=2 ðpðt uÞÞ1=2 hk
ð5Þ
and the complete transient was built adding all contributions. Several nucleation frequencies A were used to cover the range from instantaneous to progressive nucleation. Nucleation sites were established randomly on the surface and new nuclei were introduced according to the nucleation frequency without taking account if the active site had been covered previously by an already established diffusion zone, since consideration of hierarchy will anyway take the nuclei as phantom in these cases. N0 was kept constant throughout at 1 106 cm2. Since theories taking into account overlap have been developed for infinite planes, proper definition of boundary conditions is of importance and more so given the small size of the simulation cell considered for the calculations reported in this work. In order to minimize effects due to the borders of the simulation cell, we have used periodic boundary conditions [21,22]. Thus we created eight copies of the simulation cell, with diffusion zones spilling over the borders of the central cell being reintroduced from the opposite side. These toroidal boundary conditions simulate a surface with finite size but no boundaries. The simulation parameters were chosen to correspond to the electrodeposition of mercury at room temperature from 5 mM Hg22+ aqueous solution in the presence of excess supporting electrolyte. 3. Results and discussion Fig. 2 shows the radial growth rate drd/dt of a set of diffusion zones introduced at regular intervals of 0.4 s in the simulated area of 1 104 cm2 during nucleation at frequency A = 0.025 s1 onto N0 = 1 106 sites/cm2. Because of the parabolic growth rate rd(t,u) = [kD(t u)]1/2, younger diffusion zones grow at a given time faster than older ones. Therefore younger diffusion zones born inside already established ones may conceivably overflow these. It has been established that such phantom nuclei may account for up to 4% error in the estimation of the coverage using the KJMA theory [23], an effect not arising in the present digital simulation using the hierarchical concept of overlap. Fig. 3 shows the coverage of individual diffusion zones during hierarchical overlap, resulting from simulation of the process with A = 0.025 s1, N0 = 1 106 cm2. The coverage corresponding to the oldest nucleus in the set increases constantly, whereas the coverage of younger zones increase linearly with time at short times,
2. Simulation method Monte Carlo simulations were carried on a square surface of 1 104 cm2 with periodic boundary conditions. One hundred pairs of (x, y) coordinates were generated in each simulation, on which nuclei were introduced at regular intervals determined by the nucleation frequency A. Each nucleus developed a diffusion zone around it, with radius given by the standard model taking into consideration its age:
pðrd;SM Þ2 ¼ ð2pÞ3=2 DðcM=qÞ1=2 ðt uÞ
ð4Þ
Hierarchy was established for the calculation of the surface areas covered by each diffusion zone that determined the currents corresponding to each. When diffusion zones impinged with each other, then the overlapped area was assigned to the older one. The contribution of each diffusion zone k to the current density of the ensemble was then determined as that corresponding to planar diffusion to the fraction of area covered by k,
Fig. 2. Rate of radial growth of individual diffusion zones as a function of time, during simulation in 1 104 cm2 with A = 0.025 s1 and N0 = 1 106 cm2.
D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
Fig. 3. Coverage of the simulated area of 1 104 cm2 by individual diffusion zones, during hierarchical overlap with A = 0.025 s1 and N0 = 1 106 cm2.
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Fig. 5. Current contributions by individual diffusion zones according to (14), during hierarchical overlap as shown in Fig. 4, with A = 0.025 s1 and N0 = 1 106 cm2.
but then fall at longer times as they are ingested by older zones, vanishing in the limit of t ? 1, when only the oldest of the diffusion zones survives. The ‘phantom’ nuclei also generate diffusion zones but they do not contribute to the coverage; only the diffusion zone with highest hierarchy at a given point contributes to the coverage. Fig. 4 shows the coverage of the simulation area with diffusion zones as a function of time for different values of the A/N0 ratio. As the nucleation frequency increases, total coverage of the surface with diffusion zones is reached faster and eventually, at very long times, the oldest diffusion zone covers the entire surface. The currents attending the growth of diffusion zones, given by (14), are shown in Fig. 5. The resulting overall current transients obtained summing the individual current contributions at every time step are shown in Fig. 6 for several values of the ratio A/N0, with N0 kept constant at N0 = 1 106 cm2 throughout, while A was varied from 0.025 to 100 s1.
Fig. 4. Total fractional coverage of the surface with diffusion zones for different values of A, as indicated in s1. N0 = 1 106 cm2.
Fig. 6. Potentiostatic current transients from simulations with hierarchical overlap of diffusion zones, with N0 = 1 106 cm2 and A as indicated, between 0.025 and 0.75 s1 (a), and 1 and 100 s1 (b).
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3.1. Comparison of transients arising from hierarchical and nonhierarchical models The various formulations of the standard model describing nucleation with diffusion-controlled three-dimensional growth consider non-hierarchical overlap of diffusion zones in terms of the KJMA theory, and the chief differences among them rest in the way the widths of diffusion zones born at different times are homogenized. The hierarchical overlap introduced in the present work avoids mean field approximations [17,18] or reducing all disks to a uniform width, notwithstanding their different ages due to progressive nucleation. We followed two approaches to compare transients obtained from simulations of hierarchical overlap with theoretical models considering the KJMA non-hierarchical overlap. First the transients obtained from simulations of hierarchical overlap of diffusion zones with given A and N0 values were graphically compared with those predicted by the various formulations of the standard model in terms of non-hierarchical KJMA overlap. Then in a second approach and for a more quantitative assessment of differences among the various models, transients generated from simulations with hierarchical overlap of diffusion zones were analysed using the theoretical expressions of the models corresponding to KJMA non-hierarchical overlap, in order to obtain values of the kinetic parameters A and N0 that were contrasted to the values of A and N0 actually used to generate the simulated transients. The first approach is illustrated in Fig. 7, where transients obtained from simulations with different nucleation frequencies are shown. Nucleation shifts from progressive to instantaneous as A increased. For N0/A 1 nucleation is progressive and as shown in
Fig. 7a and b, the currents obtained from simulations of hierarchical overlap approach those predicted by the MLMH conjecture [19], however attaining higher maximum currents and approaching faster the Cottrell limit corresponding to planar diffusion to the entire electrode surface. The SM formulation of the standard model does not consider the spread in birth times of diffusion zones occurring during progressive nucleation, thus the corresponding currents remain invariably below the Cottrell limit at all times. The HT expression on the other hand does take into account the successive appearance of nuclei and homogenizing the width of diffusion zones, yields higher diffusion currents at long times, an effect not shown by the transients generated considering hierarchical overlap of diffusion zones with non-uniform widths. Fig. 7c and d show that the various expressions based upon the KJMA theory, with the exception of the MLMH conjecture, coincide with each other and also with the simulated results considering hierarchical overlap, as the instantaneous limit at N0/A ? 0 is reached, where ranking between diffusion zones vanishes when all nuclei are born at the same time. Having compared graphically the current transients arising from simulations of hierarchical overlap of diffusion zones at given A and N0 values, with those obtained from the various expressions for unranked KJMA overlap, we will now assume no prior knowledge of the kinetic parameters, and obtain their values from analysis of the result of simulations using the formulations devised for unranked KJMA overlap. This will provide further insights on the relation between the hierarchical overlap postulated here and that of the KJMA theory. When analytical methods were available, as in the SM case, A and N0 where obtained from {im, tm} values at the current maximum; other cases were examined fitting the theoret-
Fig. 7. Comparison of transients from simulations with hierarchical overlap of diffusion zones (.), with different formulations of the unranked, standard model with overlap accounted for by the KJMA theory: SM (j), HML (N), MLMH (I), and HTMN (d). N0 = 1 106 cm2 and A = 0.025 s1 (a), 0.25 s1 (b), 2.0 s1 (c), and 20 s1 (d). The diffusion current to a uniformly accessible planar electrode is also shown (broken line).
D. Mazaira et al. / Journal of Electroanalytical Chemistry 631 (2009) 22–28
Fig. 8. Nucleation frequencies Amod obtained from analysis of simulations with hierarchical overlap of diffusion zones using the SM (s), HT (4) and HML (5) formalisms, as a function of the nucleation frequency Asim in simulations with N0 = 1 106 cm2. The continuous line corresponds to Amod = Asim
Fig. 9. Number densities of active sites for nucleation N0 obtained from analysis of simulations with hierarchical overlap of diffusion zones using the SM (s), HT (4) and HML (5) formalisms, as a function of the nucleation frequency Asim in simulations with N0 = 1 106 cm2.
ical expressions non-linearly to the simulated data, leaving A and N0 as the sole fitting parameters. The result of such analysis is shown in Figs. 8 and 9. Fig. 8 shows the nucleation frequencies Amod found from the different formulations of the standard model using unranked KJMA overlap, correlated with the corresponding values Asim used in simulations with hierarchical overlap, whereas Fig. 9 shows the N0 values found from the simulated transients using the various expressions as a function of Asim. Figs. 8 and 9 show the confluence of the SM, HT and HML formulations of the KJMA unranked overlap together with hierarchical overlap at the instantaneous limit, where differences between widths of diffusion zones vanish as the distribution of nuclei ages narrows when these are born at the onset of the process. In contrast, fresh information becomes apparent from the present analysis at the opposite extreme of progressive nucleation. The SM model yields lower nucleation frequencies A than those actually used to generate the
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Fig. 10. Stationary nucleation rates AN0 obtained from analysis of simulations with hierarchical overlap of diffusion zones using the SM (s), HT (4) and HML (5) models, as a function of the nucleation frequency Asim in simulations with N0 = 1 106 cm2. The continuous line corresponds to the AN0 product used in the simulations.
transients with hierarchical overlap, while significantly overestimating the number densities of active sites N0. Conversely, the HT and HML models overestimate A and yield lower values for N0. Since none of these models contemplates hierarchical overlap, then it is not expected that they should predict appropriately the A and N0 values used to generate the simulated data; however, when the product of A and N0 is considered, then the differences between the SM, HT, HML models vanish, as shown in Fig. 10, and moreover the AN0 product obtained is indeed close to that corresponding to the simulations with hierarchical overlap. This result is significant, suggesting that the value of the product AN0 determined from the various models accounting differently for overlap of diffusion zones is robust and model-independent, while conversely indicating also that establishing precisely and unequivocally the values of A and N0 requires determining which of the models affords the most fitting description of reality. As a final remark, we need to point out that some of the differences observed may have arisen from the different mathematical methods employed to obtain A and N0 from the respective models. A and N0 were found from the SM model solving a system of nonlinear equations using the coordinates of the current maximum [6], while in the case of the HML [17] and HT [15] models the kinetic parameters were obtained from non-linear fitting of the whole data sets to the theoretical equations, using the Levenberg–Marquardt algorithm. Values obtained from the MLMH model have not been included in this particular analysis because the cubic polynomial developed as approximate solution [19] holds only before the current maximum, a time span in general too short for comparison with other models. 4. Conclusions We presented a novel description of Multiple nucleation with three-dimensional diffusion-controlled growth, based upon hierarchical overlap of planar diffusion zones associated to growing nuclei, wherein older zones prevail over younger ones when they impinge and overlap. Through digital simulations we have shown the main features of such model, and the results arising from hierarchical overlap were compared both qualitatively and quantitatively with models based upon application of the KJMA theory
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for unranked overlap. Hierarchical overlap of diffusion zones avoids the need of homogenization of diffusion zones through mean field approximations or other measures, considering also realistically the effects of dispersion in the age of nuclei on their growth rates, of up most importance in the limit of progressive nucleation. In the course of this study we have shown that current models provide for the robust evaluation of the product AN0 of the nucleation frequencies and the number density of active sites for nucleation, and that while hierarchical overlap of diffusion zones may provide a germane model for the unequivocal evaluation of A and N0 from single-step, potentiostatic experiments, a closed form expression for the current transient is not yet available. This will be a next step ahead, and work is currently in progress in our laboratory following this direction.
Acknowledgements The authors are very grateful to the members of the Electrochemistry Group at Simon Bolivar University for stimulating and productive discussions, specially to Michele Milo for his contribution to this work. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jelechem.2009.03.004.
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