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Introduction: Electrocrystallization refers to nucleation and crystal growth occurring on electrodes in electrochemical systems under the influence of an electric field. Nucleation and growth phenomena are involved in many battery systems, where the electron transfer is coupled to various phase transformations occurring during charge and/or discharge in the active electrode materials. For example, in the lead–acid battery the electrochemical reactions involve formation of different electronically conducting and insulating crystal phases (e.g., lead, lead dioxide (PbO2), lead sulphate (PbSO4), which have a decisive influence on the characteristics and operational life of the battery. ...

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  1. Electrocrystalization ¨ ¨ G Staikov, Institute of Solid State Research (IFF-b), Research Center Julich, Julich, Germany & 2009 Elsevier B.V. All rights reserved. Introduction (2) electrodeposition on a substrate of the same metal Me, which corresponds to the growth of the three-di- Electrocrystallization refers to nucleation and crystal mensional (3D) Me crystal phase. The case of electro- growth occurring on electrodes in electrochemical sys- deposition on foreign substrate (1) is more general tems under the influence of an electric field. Nucleation because the advanced deposition stages of the process and growth phenomena are involved in many battery always involve the crystal growth of the 3D Me phase. systems, where the electron transfer is coupled to various The mechanism of initial stages of Me deposition on phase transformations occurring during charge and/or foreign substrates depends strongly on the Me–substrate discharge in the active electrode materials. For example, in interaction and the crystallographic misfit between the the lead–acid battery the electrochemical reactions in- bulk 3D Me crystal phase and the substrate. At a rela- volve formation of different electronically conducting and tively weak Me–substrate interaction, the deposition insulating crystal phases (e.g., lead, lead dioxide (PbO2), process starts at electrode potentials E more negative lead sulphate (PbSO4), which have a decisive influence on than the Nernst equilibrium potential of the 3D Me the characteristics and operational life of the battery. phase (EoE3DMe) with the formation of isolated 3D Electrocrystallization is also a basis for technologically Me crystallites on an unmodified foreign substrate important electrochemical processes such as electro- (Figure 1(a)). This deposition mechanism is not affected deposition, passivation, and electrorefining. The electro- by the Me–substrate misfit and is called ‘Volmer–Weber’ deposition has recently become of particular importance growth mode. In the case of strong Me–substrate inter- for the development of modern micro- and nanosystem action, however, the deposition can start even at elec- technologies. A typical example is the developed elec- trode potentials more positive than the Nernst troplating technology for fabrication of submicron copper equilibrium potential (E > E3DMe) with the formation of on-chip interconnects of microelectronic devices. Elec- Me monolayers (2D Me phases), a phenomenon known trocrystallization processes are not only of technological as underpotential deposition (UPD). The 2D Me phases interest but, in many cases, also offer excellent possibilities formed in the UPD range act as precursors for the for investigating the fundamental aspects of nucleation and crystal growth. This is mainly because in electro- chemical systems the driving force and the rate of crys- 3D Me tallization can be measured and controlled with high accuracy by the electrode potential and the current. Typical electrochemical reactions involving phase for- mation and crystal growth phenomena are: (i) cathodic electrodeposition of metals, semiconductors, and alloys; (a) (ii) anodic electrodeposition of semiconductors; and (iii) anodic formation of insulating layers (passive layers) on 2D Me metallic and semiconducting substrates. The overall electrocrystallization processes are com- plex; in addition, the phase formation phenomena involve ion transport in the electrolyte and charge transfer across (b) the electrode/electrolyte interface. Additional compli- cations can arise due to chemical reaction steps and/or parallel electrochemical reactions. The simplest con- ditions can be realized in the case of electrodeposition of a metal (Me) from an electrolyte containing simple metal ions (Mez þ ): (c) Figure 1 Schematic representation of different mechanisms of Mezþ þ zeÀ -Me ½IŠ initial deposition stages on foreign substrates: (a) Volmer–Weber growth mode, (b) Stranski–Krastanov growth mechanism, and (c) Two situations can be generally considered: (1) formation Frank–van der Merwe growth mode. 3D Me, three-dimensional of the new Me phase on an inert foreign substrate and metal; 2D, two-dimensional metal. 32
  2. Electrochemical Theory | Electrocrystalization 33 subsequent formation of the 3D Me phase at EoE3DMe following expression: and can lead to two different deposition mechanisms depending on the crystallographic Me–substrate misfit. ˜ 4Bvm s3 Fà nc 2 DGðnc Þ ¼ ¼ ze jZj ½2Š In systems with significant Me–substrate misfit, the de- 27ðze jZjÞ2 2 position follows the so-called Stranski–Krastanov growth mode including the formation of unstrained 3D Me where B is a geometrical factor depending on the shape of crystallites on top of a predeposited and strained 2D Me the 3D nucleus (B ¼ 36p for a sphere and B ¼ 63 for a phase (Figure 1(b)). In the rare cases of negligibly small cube), vm the volume of an atom in the nucleus, s the Me–substrate misfit, the UPD monolayers are unstrained average specific surface energy of the nucleus, and F* a and the deposition of Me can take place by a layer- factor, which accounts for the nucleus–substrate inter- by-layer mechanism known as ‘Frank–van der Merwe’ action. The corresponding stationary nucleation rate J growth mode (Figure 1(c)). can be expressed as follows: The above considerations show that electrodeposition of metals includes many 2D and 3D phase formation J ¼ K1 exp½ÀDGðnc Þ=kB T Š ¼ K1 expðÀK2 =Z2 Þ ˜ ½3Š phenomena, which play an important role in determining where K2 ¼ 4Bvm s3 Fà =27ðzeÞ2 kB TT . The preexpo- 2 the deposition kinetics and the evolution of morphology nential factor K1 depends only slightly on Z and, for small and structure of metal deposits. Detailed investigations |Z| values, can be considered as a constant. For small of metal electrodeposition in various systems have nuclei consisting only of several atoms, macroscopic contributed significantly to a deeper understanding quantities such as surface energy and adhesion energy of nucleation and growth and to the development of lose their physical meaning, and eqns [2] and [3] no theoretical models of electrocrystallization at an atomic longer hold good. In this case, the nucleation process can level. Therefore, in the following two sections of this be described by the so-called atomistic nucleation model. article the basic theoretical and experimental aspects of According to this model, the excess energy term F(n) in electrocrystallization will be reviewed and illustrated eqn [1] can be expressed by considering the kinetics and mechanism of nucleation and crystal growth processes in the case of metal elec- X n trodeposition. An approach for studies of electro- FðnÞ ¼ nfk À fi ½4Š i¼1 crystallization processes relevant to the lead–acid battery system will be discussed in a separate section. where fk is the binding energy of an atom in the so- called kink site or ‘half-crystal’ position of the infinitely large bulk crystal and fi represents the binding energy of Formation of New 3D Crystal Phase an atom in position i of the small cluster formed on the foreign substrate. The calculations show that in contrast The driving force of electrochemical formation of a new to the classical nucleation model, nc remains constant in 3D Me crystal phase, the supersaturation Dm, is given by ˜ relatively large overpotential intervals, in which the sta- the relation Dm ¼ ÀzeZ, where e is the elementary ˜ tionary nucleation rate is described by the equation charge and Z ¼ E À E3DMe o0 represents the cathodic overpotential. The formation of the new phase requires 0 J ¼ K1 exp½ÀFðnc Þ=kB T Šexp½ðnc þ 1 À aÞze jZj=kB T Š ½5Š the generation of clusters with a size ensuring their 0 spontaneous growth at the applied overpotential Z. The where the constant K1 includes the number density of ˜ energy of formation DGðnÞ of a cluster consisting of n Me nucleation sites N0 and (1 À a) is the cathodic charge atoms can be generally expressed by the equation: transfer coefficient for the transfer of metal ions from the electrolyte to the nc-size critical nucleus under the action ˜ DGðnÞ ¼ Ànze jZj þ FðnÞ ½1Š of Z. A specific feature of the JÀ|Z| relationship is the where the term F(n) represents the energy excess asso- existence of a critical overpotential |Zcrit|. As illustrated ciated with the creation of new interfaces. At constant Z, schematically in Figure 2, the nucleation rate J is ˜ the function DGðnÞ displays a maximum at a critical practically zero at jZj{jZcrit j and rises exponentially at cluster size nc, at which the cluster can grow spon- jZjcjZcrit j. Based on this specific behavior, a very taneously. This critical cluster is called the nucleus of the powerful double-pulse potentiostatic technique for new phase and determines the energy barrier for nu- experimental investigation of nucleation kinetics was ˜ cleation DGðnc Þ. developed. In this technique, nuclei of the new phase are According to the classical nucleation model, the en- generated applying a short nucleation pulse with suf- ˜ ergy barrier DGðnc Þ for the formation of 3D nuclei on a ficiently high amplitude jZ1 jcjZcrit j. During a following foreign substrate can be related to nc and Z by the second growth pulse with much lower amplitude
  3. 34 Electrochemical Theory | Electrocrystalization jZ2 j{jZcrit j, the nuclei generated by the first pulse grow ln J versus |Z| plot. The experimental ln J À |Z| de- to larger crystallites, which can be microscopically ob- pendence shows two distinct linear regions in agreement served. This allows the determination of the number with eqn [5] and the predictions of the atomistic nucleation density of nuclei N by counting the number of crystallites model. From the slopes of the two straight lines, values on a given substrate area. Experimental N À t depen- of nc ¼ 4 and nc ¼ 1 can be obtained for the number dences at different overpotentials can be easily obtained of silver atoms in critical nucleus in the overpotential by varying the duration of the first nucleation pulse. intervals 25 mV{|Z|{51 mV and 51 mV{|Z|{90 mV, Figure 3 presents the illustrative experimental data respectively. obtained by the double-pulse technique described above for The overall rate of formation of the new 3D Me electrochemical nucleation of silver on a glassy carbon crystal phase on a foreign substrate is determined by both substrate. Experimental plots of the number density of nu- the nucleation rate and the growth rate of newly formed clei N versus time t (nucleation pulse duration) at different crystallites. Two limiting cases of nucleation are usually overpotentials Z (nucleation pulse amplitudes) are shown in considered. The so-called ‘instantaneous nucleation’ re- Figures 3(a) and 3(b). The data for the stationary nucle- fers to the case when all nucleation sites N0 are converted ation rate (J ¼ dN/dt) determined from the slopes of the to nuclei virtually instantaneously at t ¼ 0, while in the linear N À t dependences are presented in Figure 3(c) in a case of so-called progressive nucleation, the nucleation sites are converted to nuclei progressively during the whole period of observation. In the very initial stage of phase formation, the indi- vidual small crystallites grow independently and the fraction of the substrate surface covered by them or their diffusion zones is negligibly small. Under these con- Nucleation rate J ditions, the current–time relationships can be expressed by a general time law i(t)Bt p, where the power p depends on the rate-determining step of metal deposition reaction (charge transfer, ion transport) and on the nucleation crit type (instantaneous or progressive). For example, in the case of crystallite growth controlled by hemispherical diffusion, the initial current density i(t) is expressed by Overpotential Figure 2 A schematic representation of the dependence of nucleation rate J on the overpotential |Z|. jiðt Þj ¼ N0 Kd ðZÞt 1=2 ½6Š 50 mV 45 mV 80 40 mV ) 2 60 35 mV 13 N (cm 40 30 mV 25 mV 20 11 0 0 40 80 120 160 200 240 In J 9 (a) t (ms) 90 mV 80 80 mV 70 mV 60 mV 7 55 mV ) 2 60 N (cm 40 5 20 0 3 0 2 4 6 8 10 40 60 80 100 (b) t (ms) (c) (mV) Figure 3 Kinetics of electrochemical nucleation of silver on a glassy carbon electrode: (a) and (b) dependences of number of nuclei N on time t for various overpotentials |Z|, and (c) dependence of the stationary nucleation rate J on the overpotential |Z|. Adapted from Milchev A and Vassileva E (1980) Electrolytic nucleation of silver on a glassy carbon electrode. Part II. Steady state nucleation rate. Journal of Electroanalytical Chemistry 107: 337–352.
  4. for the case of instantaneous nucleation and by 2J ðZÞKd ðZÞ 3=2 jiðt Þj ¼ t ½7Š 3 for the case of progressive nucleation. In these equations the growth constant Kd ðZÞ is given by Kd ðZÞ ¼ pzevm ð2DcÞ3=2 ½1 À expðzeZ=kB T ފ3=2 , where D and c are 1=2 the diffusion coefficient and the concentration c of metal ions in the electrolyte, respectively. As can be seen from eqn [7], in the case of progressive nucleation the nucleation rate J can be estimated from the slopes of the |i| versus t3/2 plots of initial parts of experimental current transients. This technique has been successfully used for investigation of nucleation kinetics in many electro- chemical systems with fast charge-transfer reactions and low concentrations c. As mentioned earlier, eqns [6] and [7] are applicable only in the very initial stages of the phase formation process. In the advanced deposition stages, however, it is also necessary to take into account the reduction in the substrate surface area available for nucleation due to the spreading and overlap of growing crystallites and their diffusion zones. In the existing theoretical models, the actual fractional surface coverage y(t) is usually described by the Kolmogorov–Avrami relation: yðt Þ ¼ 1 À exp½Àyext ðt ފ ½8Š where yext(t) is the so-called extended fractional coverage representing the total surface area that would be covered by all growing crystallites disregarding their interference and overlap. Different theoretical expressions for the overall current transients have been derived on the basis of eqn [8] using various approximations for the shape of growing crystallites and their diffusion zones. A good fitting has been obtained in many cases between experimental and theore- tical transients. Nevertheless, it is often recommended that the used theoretical models and the fitting parameters be checked, whenever possible, using independent measure- ments including direct microscopic observations. In the advanced growth stages the surface of the crystal consists usually of low-index faces called singular faces, which determine the crystal growth rate. The most
  5. 36 Electrochemical Theory | Electrocrystalization 20 80 2 ) i (mA cm 60 I (nA) 40 10 20 0 0 0 10 20 30 40 0 10 20 30 40 50 t (ms) t (s) Figure 5 Current density transient for the multinuclear Figure 4 Current transients during the mononuclear layer-by- multilayer growth of a quasi-perfect (screw dislocation–free) layer growth of a quasi-perfect (screw dislocation–free) Ag(100) Ag(100) single crystal face at an overpotential. Adapted from single crystal face at an overpotential Z ¼ À 6 mV. Electrode area Budevski E, Obretenov W, Bostanov V, et al. (1989) Noise A ¼ 3  10À4 cm2. The insets indicate the locations of growing 2D analysis in metal deposition – Expectations and limits. islands. Adapted from Budevski E, Obretenov W, Bostanov V, Electrochimica Acta 34: 1023–1029. et al. (1989) Noise analysis in metal deposition – Expectations and limits. Electrochimica Acta 34: 1023–1029. Carlo simulations using various approximations. Ac- conditions. Each current transient is characterized by a cording to the theoretical calculations, the overall tran- charge density equivalent to that for the formation of one sient is characterized by an initial current increase monolayer (qmon), which demonstrates clearly that the followed by several oscillations before reaching the mononuclear layer-by-layer growth mechanism operates steady-state current density given by in this case. The shape and the duration of each current transient depend on the location of growing 2D islands jiss j ¼ bqmon ðbJVp Þ1=3 2 ½14Š on the crystal face as illustrated in Figure 4 (insets). Before the interference of a growing 2D island with the where b is a constant, which varies from theory to theory face boundaries, the growth current I1(t) increases lin- and has a value close to unity. early with time according to the equation: Figure 5 presents a typical experimental current transient obtained on a screw dislocation–free Ag(100) jI1 ðt Þj ¼ 2qmon bVp2 t ½13Š face at an overpotential Z ¼ À 14 mV. As seen, the tran- sient shows the characteristic current oscillations pre- where b is a factor that depends on the geometric shape dicted by the theoretical calculations for the case of 2 of the growing 2D island (b ¼ p for a disk). The propa- multinuclear multilayer growth. The product ðbJVp Þ can gation rate Vp can be evaluated from the slope of initial be evaluated from the very initial rising part of the cur- linear parts of current transients using eqn [13]. A de- rent transient, which is described by jiðt Þj ¼ jqmon jbJVp t 2. 2 tailed statistical analysis of current–time records with The analysis of experimental data for |iss| and ðbJVp2 Þ 2 sufficiently large number of nucleation events showed obtained at different Z in an ln |iss| versus ln ðbJVp Þ plot that under these conditions the process of 2D nucleation shows a linear dependence with a slope of 1/3 in good occurs randomly in space and time. The mean nucleation agreement with eqn [14]. time tn and the corresponding 2D nucleation rate J can ¯ The growth mechanisms with 2D nucleation dis- be obtained from the statistical analysis of experimental cussed above operate rarely in real systems because real results. crystals are usually not perfect and contain screw dis- With increasing |Z|, the nucleation rate J increases locations. It is well known that screw dislocations play an much faster than the propagation rate Vp and tn becomes ¯ important role in determining the mechanism of growth much shorter than tp ð¯ n {tp Þ. In this case the deposition t of real crystals. A screw dislocation intersecting a singular of each monolayer involves the formation of large crystal face gives rise to the appearance of a step, which number of nuclei. During the growth process the growth originates from the intersection point (Figure 6(a)). After morphology of the crystal face has a multilevel structure, application of a cathodic overpotential, this step winds up which is formed due to the nucleation occurring on top into a spiral and the growth takes place by the so-called of growing 2D islands of the preceding monolayer. This spiral growth mechanism (Figure 6(b)). A specific feature multinuclear multilayer growth has been investigated of this mechanism is that during the growth process, the extensively by different analytical methods and Monte spiral step does not disappear and the crystal face can
  6. Electrochemical Theory | Electrocrystalization 37 grow continuously even at very low |Z| without the need where i(0) represents the current density at t ¼ 0. This of 2D nucleation. The height of spiral steps is related to current–time relationship is derived on the basis of the Burgers vector of screw dislocation and for metal Kolmogorov–Avrami relation (eqn [8]) considering a crystals is usually equal to the height of the monatomic large number of dislocations randomly distributed on the steps. The growth spirals consisting of monatomic steps crystal face. appear as growth pyramids with a slope determined by Figure 7(a) shows a current transient obtained on a the step distance (the distance between spiral turns) ds(Z). ‘quasi-perfect’ Ag(111) crystal face intersected by only At low cathodic overpotentials |Z|{kBT/ze the steady- one screw dislocation. At short times, before the inter- state current density for the spiral growth is given by ference of the new growth pyramid with the face 2 boundaries, the initial growth current I(t) is described by Vp ðZÞ qmon kV 2 jiss j ¼ qmon ¼ Z ½15Š ds ðZÞ 19e I ðt Þ ¼ I ð0Þ þ ½iss À ið0ފbVp t 2 2 ½17Š where e is the specific edge energy and kV the propa- gation rate constant of monatomic steps defined by kV ¼ Vp(Z)/|Z|. The parabolic |iss| À Z2 dependence where I(0) and i(0) ¼ I(0)/A are the current and the predicted by eqn [15] has been found experimentally by current density at t ¼ 0. Figure 7(b) represents the initial electrochemical growth of ‘quasi-perfect’ Ag(100) and part of experimental current transient in an I versus t 2 Ag(111) crystal faces with limited number of screw dis- plot. From the analysis of observed linear I À t 2 de- locations. As seen from eqn [15], the steady-state current pendence and from the steady-state current obtained at density iss does not depend on the number density Nd of long times, the parameters (kV and e) characterizing the screw dislocations intersecting the crystal face. On the spiral growth process can be determined. contrary, the current density transient i(t) for the spiral growth depends on Nd and can be expressed by the equation Electrocrystallization Processes Relevant to the Lead–Acid Battery System iðt Þ ¼ ið0ÞexpðÀNd bVp t 2 Þ 2 þ iss ½1 À expðÀNd bVp t 2 ފ 2 ½16Š As mentioned in the ‘Introduction’ section, the electro- chemical reactions occurring during the operation of the lead–acid battery involve the formation of nonmetallic crystal phases such as the semiconducting lead dioxide and the insulating lead sulfate. The investigation of kinetics and mechanism of these electrocrystallization processes under real operation conditions of the lead– acid battery is a very difficult task. An effective approach (a) (b) to overcome some of the arising problems is described in Figure 6 Schematic representation of the spiral growth this section. This approach is based on the application of mechanism: (a) crystal face with a step originating from a screw microelectrodes and allows the study of nucleation and dislocation and (b) spiral growth of the crystal face. growth kinetics of single growth centers of the new 3D 1.0 0.5 0.8 0.45 I ( A) I ( A) 0.6 0.4 0.4 0.2 0.35 0 5 10 15 20 0 4 8 12 (a) t (s) (b) t 2 (s2) Figure 7 Current transient for the spiral growth of a Ag(111) crystal face intersected by only one screw dislocation (initial overpotential |Zi| ¼ 0.25 mV; final overpotential |Zf| ¼ 1.15 mV ) (a) overall current transient and (b) I vs t 2 plot of the initial part of the transient. Adapted from Staikov G, Obretenov W, Bostanov V, Budevski E, and Bort H (1980) Current transients for the electrolytic growth of crystal faces intersected by screw dislocations. Electrochimica Acta 25: 1619–1623.
  7. 38 Electrochemical Theory | Electrocrystalization 12 • 3 • 10 • • • I 1/2 (nA1/2) 8 2 • I (nA) 6 • • • 4 1 • • 2 t0 • • 0 0 40 80 120 20 40 60 80 (a) t (s) (b) t − t 0 (s) Figure 8 Current transient for the electrocrystallization of a-PbO2 on a carbon disk microelectrode (radius 4 mm) at an overpotential Z ¼ 400 mV: (a) I vs t transient and (b) I1/2 versus (t À t0) plot of the transient. Adapted from Li LJ, Fleischmann M, and Peter LM (1989) Microelectrode studies of lead–acid battery electrochemistry. Electrochimica Acta 34: 459–474. crystal phase by electrochemical and microscopic the birth of the first nucleus t0 by analyzing sufficient measurements. number of nucleation events at different overpotentials. Figure 8 shows a typical current transient obtained Carbon microelectrodes modified with electrodeposited during anodic electrocrystallization of a-PbO2 on a single lead dioxide centers were also applied successfully carbon disk microelectrode with a 4 mm radius. The for investigation of the reductive transformation of corresponding electrochemical reaction is given by lead dioxide to lead sulfate in sulfuric acid electrolyte solutions. Pb2þ þ 2H2 O-PbO2 þ 2eÀ þ 4Hþ ½IIŠ The driving force for this electrocrystallization process is Conclusions given by the anodic overpotential Z ¼ E À E3DPbO2 > 0 where E3DPbO2 represents the Nernst equilibrium po- This article reviews the present knowledge of nucleation tential of the 3D lead dioxide phase. As seen in and crystal growth phenomena involved in electro- Figure 8(a), after application of the overpotential at t ¼ 0 crystallization. The theory and experiment of electro- a sharp current increase is observed at a relatively long chemical nucleation have been well developed. A time t0 corresponding to the birth of the first stable significant contribution has been made by the intro- growth center of lead dioxide. Figure 8(b) shows that the duction of the atomistic nucleation theory. The mech- initial rising part of the transient in Figure 8(a) is in anisms and kinetics of crystal growth are well understood agreement with the current–time relation for kinetically and have been experimentally verified in the case of (charge transfer) controlled growth of a single hemi- electrocrystallization of silver on isolated silver single- spherical lead dioxide center given by crystal faces. Although the considerations in this article are restricted to the simple case of electrocrystallization nF 3 of metals, the presented basic theoretical and experi- I ðt Þ ¼ 2p V ðt À t0 Þ2 ½18Š mental concepts are general and can be applied to other vM g electrocrystallization systems. In real systems, however, In this equation, n ¼ 2 is the number of transferred and in particular in the battery systems electro- electrons, F the Faraday number, vM the molar volume of crystallization processes are complex and are often the lead dioxide deposit, and Vg ¼ Vg(Z) the overpotential coupled to other electrochemical and chemical rea- dependent growth rate. The deviation from the linear ctions. Therefore, developments of specific sophisticated I1/2 versus (t À t0) plot at (t À t0) > 40 s observed in experiments and theoretical models are necessary to Figure 8(b) is due to the birth of a second lead dioxide elucidate the kinetics and mechanism of electrocry- growth center. Subsequent examination of lead dioxide stallization. deposit by scanning electron microscopy (SEM) con- firmed the formation of a second growth center. SEM studies also confirmed the hemispherical shape of the Nomenclature growing centers. From the slopes of experimental linear I1/2 versus (t À t0) plots and eqn [18], the overpotential Symbols and Units dependence of the growth rate Vg(Z) was determined. A area of a crystal face The nucleation rate can be extracted from the times for B,b geometrical factors
  8. Electrochemical Theory | Electrocrystalization 39 c concentration of metal ions in the Abbreviations and Acronyms electrolyte 2D two-dimensional ds step distance SEM scanning electron microscopy D diffusion coefficient of metal ions in the UPD underpotential deposition electrolyte e elementary charge E electrode potential E3DMe equilibrium potential of a 3D metal See Also: Chemistry, Electrochemistry, and phase Electrochemical Applications: Lead; Silver; fmon frequency of generation of monolayers Electrochemical Theory: Electrokinetics; Kinetics; F Faraday number History: Electrochemistry; Recycling: Lead–Acid i current density Batteries: Overview; Secondary Batteries – Lead–Acid i¯ mean current density Systems: Overview. iss steady-state current density I current J stationary nucleation rate Further Reading K1, K10 preexponential factors in nucleation rate equations Armstrong RD and Harrison JA (1969) Two-dimensional nucleation in K2 nucleation constant electrocrystallization. Journal of the Electrochemical Society 116: kB Boltzmann constant 328--331. Bockris JO’M and Despic AR (1970) The mechanism of deposition and Kd growth constant dissolution of metals. In: Eyring H, Henderson D, and Jost W (eds.) n number of atoms in a metal cluster Physical Chemistry – An Advanced Treatise, vol. IXB, pp. 611--730. nc number of atoms in a nucleus New York: Academic Press. Bockris JO’M and Razumney GA (1967) Fundamental Aspects of N number density of nuclei Electrocrystallization. New York: Plenum Press. N0 number density of nucleation sites Bostanov V, Staikov G, and Roe DK (1975) Rate of propagation of Nd number density of screw dislocations growth layers on cubic crystal faces in electrocrystallization of silver. Journal of the Electrochemical Society 122: 1301--1305. t time (nucleation pulse duration) Budevski E (1983) Deposition and dissolution of metals and alloys. Part T temperature A: Electrocrystallization. In: Conway EB, Bockris JO’M, Yeager E, vm volume of an atom in the crystal phase Khan SUM, and White RE (eds.) Comprehensive Treatise of Electrochemistry, vol. 7, pp. 399--450. New York and London: Vg overpotential-dependent growth rate Plenum Press. Vp rate of propagation of monatomic steps Budevski E, Bostanov V, and Staikov G (1980) Electrocrystallization. qmon charge density needed for deposition of Annual Review of Material Science 10: 85--112. Budevski E, Staikov G, and Lorenz WJ (1996) Electrochemical one monolayer Nucleation and Growth: An Introduction to the Initial Stages of Metal z charge number Deposition. Weinheim: VCH. D G ˜ energy of formation of a metal cluster Budevski E, Staikov G, and Lorenz WJ (2000) Electrocrystallization: Nucleation and growth phenomena. Electrochimica Acta 45: D l˜ supersaturation 2559--2574. e specific edge energy of monatomic Fischer H (1954) Elektrolytische Abscheidung und Elektrokristallisation steps von Metallen. Berlin: Springer. Fischer H (1969) Electrocrystallization of metals under ideal and real g overpotential conditions. Angewandte Chemie 8: 108--119. gcrit critical overpotential Fleischmann M and Thirsk HR (1960) Anodic electrocrystallization. h actual fractional surface coverage Electrochimica Acta 2: 22--49. Fleischmann M and Thirsk HR (1963) Metal deposition and hext extended fractional surface coverage electrocrystallization. In: Delahay P (ed.) Advances of j v propagation rate constant of Electrochemistry and Electrochemical Engineering, vol. 3, monatomic steps pp. 123--210. New York: John Wiley & Sons. Fleischmann M, Pons S, Sousa J, and Ghoroghchian J (1994) r average surface energy of a nucleus Electrodeposition and electrocatalysis – The deposition and s¯n mean nucleation time dissolution of single catalyst centers. Journal of Electroanalytical sp mean propagation time Chemistry 366: 171--190. Harrison JA, Rangarajan SK, and Thirsk HR (1966) Some problems of /i binding energy of an atom in the i th electrodeposition. Journal of the Electrochemical Society 113: position of a cluster 1120--1129. / binding energy of an atom in a kink-site Harrison JA and Thirsk HR (1971) The fundamentals of metal k deposition. In: Bard AJ (ed.) Electroanalytical Chemistry, vol. 5, position pp. 67--148. New York: Marcel Dekker. U energy excess connected with the Kaischew R and Budevski E (1967) Surface processes in creation of new interfaces electrocrystallization. Contemporary Physics 8: 489--516. * Li LJ, Fleischmann M, and Peter LM (1989) Microelectrode studies of U factor accounting for the nucleus– lead–acid battery electrochemistry. Electrochimica Acta 34: substrate interaction 459--474.
  9. 40 Electrochemical Theory | Electrocrystalization Milchev A (1991) Electrochemical phase formation on a foreign Staikov G, Lorenz WJ, and Budevski E (1999) Low-dimensional metal substrate – basic theoretical concepts and some experimental phases and nanostructuring of solid surfaces. In: Lipkowski J and results. Contemporary Physics 32: 321--332. Ross PN (eds.) Imaging of Surfaces and Interfaces, pp. 1--56. New Milchev A (2002) Electrocrystallization: Fundamentals of Nucleation and York: Wiley-VCH. Growth. Boston/Dordrecht/London: Kluwer Academic Publishers. Staikov G and Milchev A (2007) The impact of electrocrystallization on Paunovic M and Schlesinger M (1998) Fundamentals of nanotechnology. In: Staikov G (ed.) Electrocrystallization in Electrochemical Deposition. New York: Wiley-Interscience. Nanotechnology, pp. 3--29. Weinheim: Wiley-VCH. Scharifker B and Hills G (1982) Theoretical and experimental studies of Thirsk HR and Harrison JA (1972) A Guide to the Study of Electrode multiple nucleation. Electrochimica Acta 28: 879--889. Kinetics. New York: Academic Press. Staikov G (1995) Fundamentals of electrodeposition of metals. Vetter K (1967) Electrochemical Kinetics. New York: Academic Press. In: Gewirth AA and Siegenthaler H (eds.) Nanoscale Probes of Solid/ Volmer M (1939) Kinetik der Phasenbildung. Leipzig/Dresden: Teodor Liquid Interface, pp. 193--213. Dordrecht/Boston/London: Kluwer Steinkopf. Academic Publishers. Walsh FC and Herron ME (1991) Electrocrystallization and electro- Staikov G (2002) Nucleation and growth in microsystem echnology. chemical control of crystal growth: Fundamental considerations In: Schultze JW, Osaka T, and Datta M (eds.) Electrochemical and electrodeposition of metals. Journal of Physics D: Applied Microsystem Technologies, pp. 156--184. London and New York: Physics 24: 217--225. Taylor & Francis.
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