The Current Distribution*

Transcription

1 Chapter 6: The Current Dstrbuton* 1. Sgnfcance 2. What Controls the Current Dstrbuton? 3. Determnaton of the Current Dstrbuton 4. Controllng Mode3s and Approxmatons 5. Prmary Dstrbuton 6. Secondary Dstrbuton *The followng text s based, n part, on materal that s n press for publcaton n Advances n Electrochemstry and Electrochemcal Engneerng, Vol. 45, Elsever. 6 1

2 1. INTRODUCTION AND OVERVIEW The topc of current dstrbuton modelng s central to the analyss of electrochemcal systems and has been addressed n textbooks (1), revews (e.g., 2 4) and numerous ournal publcatons. Newman s textbook (1) provdes a metculous and comprehensve treatment of the subect. Prentce and Tobas (2) present a revew of the early (up to ~ 1980) publcatons n the area. Dukovc s more recent revew (3) s very comprehensve, provdng crtcal analyss of both the electrochemcal and the numercal aspects of the topc. A recent revew by Schlesnger (4) focuses prmarly on the numercal technques. The present monograph ntroduces the fundamental processes and equatons underlyng the modelng of the current dstrbuton, and crtcally analyzes common assumptons and approxmatons. Focus s placed on dscussng scalng parameters for the characterzaton of the current dstrbuton. Commonly used algorthms for numercal determnaton of the current dstrbuton are compared and a few numercal mplementatons are dscussed. Lastly, the modelng of the current dstrbuton n some specal confguratons and applcatons s ntroduced, emphaszng recent publcatons. Importance of Modelng Identfy crtcal process parameters and ther mpact Relate global measurable parameters to local varables n whch we are nterested Predctve and ratonale desgn & scale-up elmnate tral & error save tme and costs Optmze & control processes Interpretaton of experments 2. SIGNIFICANCE OF MODELING THE CURRENT DISTRIBUTION The current dstrbuton s among the most sgnfcant parameters characterzng the operaton of the electrochemcal cell. The current densty on the electrodes s drectly proportonal to the reacton rate and ts dstrbuton crtcally affects the electrochemcal process. In electroplatng, the depost thckness dstrbuton, and propertes such as the depost surface texture and ts morphology are drectly lnked to the current dstrbuton. When multple smultaneous electrode reactons are present, such as n alloy deposton or n hydrogen co evoluton, the alloy composton n the former case, and the current effcency n the latter, are controlled by 6 2

3 the overpotental dstrbuton, whch, as dscussed below, s drectly related to the current dstrbuton. Electrolytc processes whch do not nvolve deposton are also strongly affected by the current dstrbuton. Examples nclude optmzed utlzaton of catalytc electrodes and the need to prevent the current densty from surgng on electrode sectons, on separators and on membranes. The power requred for operatng an electrochemcal cell, and partcularly the ohmc loss are also dependent on the current dstrbuton. Lastly, the correct nterpretaton of expermental data hnges on understandng the range of current denstes to whch the tested electrode has been subected. The current dstrbuton can be analyzed on dfferent scales. The macroscopc current dstrbuton of the, where the dstrbuton s resolved on length scale of the order of cm, s mportant n characterzng the depost thckness unformty on a plated part, or n selectve platng, where a non unform current dstrbuton s sought. The mcro scale dstrbuton, on the other hand, where the current densty s resolved on sub mm length scales affects prmarly parameters such as the depost texture and roughness, nucleaton and deposton wthn mcron and nano scale features. For many applcatons, numercal smulaton capablty whch provdes the current dstrbuton n a gven confguraton and platng condtons, or for a gven set of such parameters, s suffcent. However, for predctve process desgn and for scale up of cells and processes (and scale down of ndustral processes for laboratory testng) analytcal models that elucdate the dependence of the current dstrbuton on the process parameters are more benefcal. 6 3

4 3. EXPERIMENTAL DETERMINATION OF THE CURRENT DISTRIBUTION Electroplatng processes, where a sold depost s formed and ts thckness can be drectly measured, provde a relatvely convenent means for determnaton of the current dstrbuton. The depost thckness can be measured by a number of commercally avalable devces, based on e.g., x ray fluorescence, beta backscatter, magnetc propertes, or controlled dssoluton. A drect probe based on the nduced feld assocated wth the current flow has been recently ntroduced. Optcal and electron mcroscopy of cross sectoned deposts provde a common means for measurng the depost thckness. Once the depost thckness, d, s measured, t can be related to the current densty,, through Faraday s law: d Mt F Fn [1] Here, t s the platng tme. F s Faraday s constant, M,, and nare the plated metal atomc weght, ts densty, and the number of electrons transferred n the deposton reacton, respectvely. F s the Faradac effcency, accountng for sde ( parastc ) reactons. Metals noble to hydrogen typcally plate from aqueous solutons at F ~ 1 correspondng to close to 100% Faradac effcency (unless they are drven to the lmtng current). When the Faradac 6 4

5 effcency s less than 100%, eq. [1] can be used to determne the current effcency once the current densty s evaluated. For the case of redox or gas evolvng reactons, where no sold depost s formed, the current dstrbuton on the electrodes can be determned usng segmented electrodes, or nsulated probe electrodes. Here, the electrode on whch the current dstrbuton s sought s sectoned nto multple, electrcally solated segments, to whch the current may be ndvdually fed and measured. If multple, narrow, segments are provded, the average segmental current denstes, obtaned by dvdng the segmental currents by the correspondng segmental areas, provde an approxmaton to the current dstrbuton. For the segmented electrode to resemble a contnuous electrode, all segments must be coplanar and essentally equ potental. To assure that the potental of all segments s wthn a few mv, mult channel potentostats must be used. A less costly approach s to connect each segment to the common bus va very low (typcally m) shunt resstor, whch enables the measurement of the current yet ntroduces nsgnfcant voltage varaton. Determnaton of the current dstrbuton Expermental Cross secton back scatter X-ray fluorescence Cuttng and weghng Strpng wth coulometry Isotope deposton Magnetc flux Sound reflecton Drect probe measurement (Hall effect) Sectoned electrode Potental feld mappng Modelng Analytcal Approxmatons Numercal Analoges Electrostatcs Magnetc Electrolytc trough 6 5

6 How can we control the current dstrbuton? Cell Geometry Shelds Theves Shape of the anode Part shape Voltage (current) Non DC waveforms ( pulsng ) Chemstry Knetcs (addtves) Conductvty Complexng agents Flow Agtaton Ultrasound Temperature What controls the current dstrbuton? Why s the current dstrbuton not unform? 1, L 1 1, L 1 2, L 2 2, L 2 6 6

7 V Overpotentals The Drvng Force? O?V? C? a E 0 X Appled Voltage = Standard Potental + Overpotental Actvaton:? a?v = E 0 +? T ()? T =? a () +? C () +? O () a RT F ln 0 Length scale (cm) Tme scale (sec) Lmtng current: L n F D Cb 1 t R Concentraton:? C Ohmc:? O RT C ln 1 n F l κ L What controls the current dstrbuton? Why s the current dstrbuton not unform? I= -? V/R (ohm s law) I V A RA 1 R Assume ohmc control (prmary): V V V R A L A L A In general: V 1 = E 0 + a ( 1 ) + C ( 1 ) + O ( 1 ) V 2 = E 0 + a ( 2 ) + C ( 2 ) + O ( 2 ) 1 L 1 2 L L , L 2 1, L 1 Assumpton: Electrodes are equ-potental,.e. no termnal effect V 1 = V 2 E 0 + a ( 1 ) + C ( 1 ) + O ( 1 ) = E 0 + a ( 2 ) + C ( 2 ) + O ( 2 ) The current densty wll adust to mantan the voltage balance Note: 1 = (x,y,z) 2 system s non-lnear 3 L s a-prory unknown 6 7

8 4. ANALYTICAL DERIVATION OF THE CURRENT DISTRIBUTION Ths topc s covered n sgnfcant detal n Newman s textbooks (1). A summary, relevant to the ensung dscusson s provded here. The Current Densty The current densty s drectly related to the onc flux, N, n the electrochemcal cell. The flux s typcally descrbed n terms of three maor components: dffuson of ons across a concentraton gradent, mgraton of charged ons down the electrc feld, and transport of ons due to bulk electrolyte convecton. Consequently, the flux of an onc speces s gven by: N Dc uzfc cjv [2] The current densty s determned by assgnng the charge Fz to the flux of each speces and summng over all onc speces: F z N [3] Substtutng Eq. [2] nto Eq. [3], 2 2 F zdc F uzcf zcj v Electroneutralty, expressed as: [4] zc 0 [5] s present throughout the cell (except for the vanshngly thn double layer) and renders the last term on the rght of Eq. [4] to zero, provdng for the total current densty: 2 2 F zdc F uzc [6] We also recognze (1) that the electrolyte conductvty,, s gven by: 2 2 F uzc [7] Hence we can rewrte eq. [6] as: 6 8

9 [8] F z D c Eq. [8] ndcates that the current densty s determned by both the potental and concentraton gradents. The explct velocty term s absent from eq. [8] (due to electroneutralty), however, convecton stll affects the current densty by controllng the concentraton feld. It should also be noted that the electrode knetcs whch do not appear explctly n Eq. [8] establsh the boundary condtons requred for ts soluton. As subsequently shown, the electrode knetcs may nfluence qute sgnfcantly the current dstrbuton. Whle representng the current densty as a functon of the potental and concentraton dstrbutons, eq. [8] does not provde the necessary relatonshps requred for solvng the dstrbuton. Ths s derved from the consttutve equatons descrbed below. Ionc Transport Ionc flux s due to: Dffuson + Electrc Mgraton + Convecton +... Dffuson = - D C V Ionc Flux: Mgraton = - U Z FC Z Convecton = C v N = -D C -U Z FC + C v +... [moles/sec cm 2 ] Current densty s due to flux of all charge carryng speces: = Z FN = - FZ D C -F 2 U Z 2 C + F Z C v Electroneutralty: Z C 0 - N + Current densty: = - FZ D C - Boundary layers Compare wth (the dfferental form of) Ohm s law: = - [I = -?V/R] 6 9

10 Materal balance The governng equatons for a cell wth dffuson, mgraton and convecton are derved by performng a materal balance on a volume element, for each of the onc speces: c = N t [9] R R s the rate of speces generaton due to a homogeneous reacton wthn the volume element. Such reactons are uncommon n electrochemcal cells (may be encountered e.g., when a complex dssocates, releasng the onc speces ), and therefore we set dentcally, R =0. When the flux expresson, eq. [2] s substtuted nto eq. [9] the general equaton ( Nernst Planck ) for the concentraton and potental felds s obtaned. c + v c = F (z u c ) + (D c ) t [10] For a mult component electrolyte wth onc speces, equaton [10] represents a system of equatons, one for each onc speces. Snce the potental s present n each equaton, there are a total of +2 unknowns ( speces concentratons, c + the electrostatc potental, the flud velocty, v). The extra equatons requred for solvng the system are the electroneutralty condton [5], and the momentum equaton whch descrbes the flud velocty at all locatons wthn the cell. The momentum equaton s typcally represented n terms of the Naver Stokes approxmaton (5): 2 V 1 p V T V V V ( T) x 2 x x x x x [11] The Naver Stokes equaton s wrtten here for a Cartesan two dmensonal coordnate system where and represent the two axes. Accordngly, v and v are the velocty components n the drectons and. P s the hydrostatc pressure, and and T are the molecular and the turbulent knematc vscosty, respectvely (5). For systems nvolvng forced convecton, the flud flow equatons are typcally decoupled from the electrochemcal process, and can be solved separately. The set of +2 equatons ( eqs. [10] + electroneutralty (eq. [5]) + the momentum equaton [11]) fully descrbe the current, potental and concentraton dstrbutons n the cell as 6 10

11 a functon of tme. boundary condtons. Ths set of equatons must be solved subect to the electrochemcal Materal Balance Rate of accumulaton = net flux n + generaton C t N R C Thn boundary layer: C b N D C U Z F C C V Nernst- Plank Eq. C t D C Z F U C V C Ths s the fundamental equaton to solve Unknowns: Unknown Symbol Number Equatons: Name Number Concentratons C Potental F 1 Velocty V 1 (vector) total +2 Nernst-Plank Electroneutralty 1 Flow 1 (vector) total +2 Boundary condtons Two types of boundares are present n electrochemcal cells: () electrodes and () nsulatng boundares. () Insulator: Here, no current may flow nto the boundary, and accordngly, the current densty normal to the nsulatng boundary s specfed as zero, = 0 n [12] () Electrodes: On an electrode, an expresson for the reacton knetcs that relates the potental to the normal current must be provded: 6 11

12 n e= f(c, ) e [13] Electrode knetcs and overpotentals Typcally, the current densty s related to the overpotental,, whch s the drvng force for the electrochemcal reacton. The overall overpotental at the electrode s gven by: V E [14] V s the electrode potental. E s the thermodynamc equlbrum potental correspondng to the condton of no current flow, and s the electrostatc potental wthn the soluton next the electrode, measured at the outer edge of the mass transport (=concentraton) boundary layer. The total overpotental at the electrode can be further resolved nto two overpotental components, s and c, The frst, s, s the surface (or actvaton ) overpotental, whch relates drectly to the knetcs of the electrode processes. The second overpotental component, c, s the concentraton overpotental, accountng for the voltage dsspaton assocated wth transport lmtatons. The surface overpotental, S, s typcally related to the current densty through the Butler Volmer equaton (6). AF CF n = exp( ) exp( ) 0,e s s RT RT [15] Eq. [15] ncorporates three emprcally measured parameters: the exchange current densty, 0, (gven here n terms of ts value on the electrode, 0,e ) and the anodc and cathodc transfer coeffcents, A and C, respectvely. These parameters are obtaned from polarzaton measurements (7). Often, but not always, A + C = n. It should be noted that whle the Butler Volmer equaton correlates well many electrode reactons, there are numerous others, partcularly when carred n the presence of platng addtves whch do not follow t. The Butler Volmer equaton n the form presented by Eq. [15] relates to pure electrode knetcs and does not consder transport lmtatons, whch cause the concentraton at the electrode, c e, to vary from ts bulk value, c b. Ths concentraton varaton affects mostly two parameters: the exchange current densty, 0, whch s a functon of the concentraton, and the overpotental,, whch now also ncludes the component assocated wth transport lmtatons, C,. We can wrte = S C [16] 6 12

13 Where the concentraton overpotental, C, s gven by: RT ce C = ln [17] nf c b It should be noted that the dvson of the total overpotental nto a pure knetcs component, ( S ) and a mass transport component ( C ) as presented by eq. [16] s somewhat arbtrary and s used manly to characterze the two types of dsspatve processes. Both terms are strongly coupled and t s very dffcult to drectly measure ether component separately. Whle chemcal engneers often dscuss the two overpotental components separately (1), chemsts (e.g., 6, 7) tend to combne both terms together and characterze the electrochemcal system n terms of the total overpotental,. To account for the concentraton varatons whch are always present at electrodes n current carryng cells, a correcton must be ntroduced nto eq. [15]. Ether of two approaches s typcally taken: the frst, more characterstc to engneerng publcatons (1), presents eq. [15] as: o R AF cf C s s e C e RT RT oc, b o R Cb Cb e e [18] Wrtten here for the general frst order reacton: O + ne R [19] and are parameters adustng the value of 0, from ts bulk value to ts value at the electrode and are generally determned emprcally. If the reduced speces does not dssolve wthn the electrolyte (as n most platng systems), the concentraton rato for the reduced speces nvolvng s dentcallyfurthermore (1), for many dvalent ons, Chemsts, tend typcally account for the concentraton varaton at the electrode through a modfed Butler Volmer equaton of the form (6, 7): R AF o C e C RT e oc, e e b R o Cb Cb CF RT [20] It can be shown that the two forms of the modfed Butler Volmer equaton, (eq. [18] and eq. [20]) are dentcal when: 6 13

14 1 C [21] n General Soluton Procedure Once the flud flow equatons [11] are solved and the velocty components are specfed wthn the cell, the system of transport equatons [10], n conuncton wth the electroneutralty condton [5] s solved. The boundary condton at nsulatng boundares s specfed by substtutng equaton [8], representng the current approachng the boundary from the electrolyte, nto eq. [12], statng that the current on the nsulatng boundary s zero: n nsulator = F zdc 0 [22] On electrodes, we equate the current approachng the electrode from the soluton sde (eq. [8]), to the current enterng the electrode, subect to the reacton knetcs equaton: n electrode = F z D c f(c, ) e [23] The functon f(c, e appearng on the rght hand sde of eq. [23] corresponds to the knetcs expressons gven by ether eq. [18] or [20]. We recall that when applyng eq. [18], the overpotental s gven by = V- E - - [24] s C When applyng the knetcs expresson [20], the total overpotental as gven by eq. [14] s used. Clearly, the procedure outlned above s complex. It requres soluton of the flow feld, n conuncton wth the determnaton of the dstrbuton of the electrostatc potental and of all speces concentratons wthn the cell. In addton to the mathematcal complexty, the transport propertes (dffusvtes, moblty) for all speces must be gven. Ths s further complcated by the fact that most practcal electrolytes are concentrated and hence transport nteractons between the speces must be accounted for, requrng the applcaton of the more complex concentrated electrolyte theory (1). Addtonally, the electrode knetcs parameters must be known. However, as dscussed below, smplfcatons are often possble, snce most operatng cells are typcally controlled by ether the electrc potental dstrbuton or by the concentraton dstrbuton, (n conuncton wth the electrode knetcs), and only few systems are nfluenced about equally by both. 6 14

15 Soluton of the Nernst-Plank Equaton Assumpton: Dlute soluton theory Concentraton Profles Example: Bnary Electrolyte (CuSO 4 ) Flow Cell: 0.03 cm V = 0.4 V 40 ms 0.01 cm V = 1 cm/s cathode anode Potental Map: 100 ms M. Menon, PhD Thess, Case Western Reserve Unversty, 1986 M. Menon and U. Landau, J. Electrochem. Soc., 134 (9), (1987) Characterstcs Scales of the Electrochemcal Cell mm nm m mm cm - m V Electrode Concentraton boundary layer Bulk Electrolyte m s mass C e transport: dstrbuton: 2 charge transfer: =f[( m - s ), C e,t, C DCvC t Potental C b Double layer 6 15

16 Thn Boundary Layer Approxmaton A common assumpton n engneerng modelng of electrochemcal cells s the thn boundary layer approxmaton (8). Accordngly, concentraton varatons are assumed to be lmted to a boundary layer along the electrodes, whch s consdered to be much thnner than the wellmxed bulk electrolyte regon. Ths decouplng, depcted schematcally n Fg. 1, elmnates all terms nvolvng concentraton gradents n eq. [10] when the latter s appled to the bulk regon. The concentraton effects are now all lumped wthn the thn boundary layer and ncorporated n the boundary condtons. + well mxed bulk C b 2 = 0 Fg. 1: A schematc of an electrochemcal cell, depctng the thn boundary layer approxmaton. The bulk of the cell s well mxed and all concentraton varatons are assgned to a thn boundary layer next to the electrodes. Typcally, the boundary layer thckness,, s far thnner wth respect to the bulk than llustrated here. In the regon of unform concentraton, Laplace s equaton for the potental holds. Followng ths procedure, and dscardng all terms nvolvng concentraton gradents, the Nernst Planck equaton [10] reduces n the bulk, to Laplace's equaton for the potental. 2 = 0 [25] In dervng eq. [25] we also set the transent concentraton term on the left n eq. [10] to zero, thus consderng only steady state (or pseudo steady state) processes. One may stll apply the approxmaton of the thn boundary layer, as stated by eq. [25], to transent problems, allowng the concentraton wthn the thn boundary layer to vary wth tme. Detaled dscusson of ths class of problems s however outsde the scope of ths revew and can be found n manuscrpts focusng on transents (e.g., 9 12). 6 16

17 Eq. [25] s solved for the bulk regon applyng, n the absence of concentraton gradent n the bulk, smplfed boundary condtons: On nsulators we have a smplfed form of eq. [22]: = 0 0 [26] n nsulator ns ns On electrodes we apply eq. [23], recognzng that now, n the absence of concentraton gradents n the bulk, the current on the soluton sde s drven only by electrc mgraton: n electrode = f(c, ) e [27] As before, the rght hand sde on eq. [27] s gven by ether eq. [18] or [20]. Alternatvely, we can specfy a potental balance at the electrode by applyng (lumped across the thn boundary layer) ether eq. [14] or [24], rearranged n the form of eq. [28]. = V E or = V E S C [28] Snce the soluton on the electrolyte sde nvolves only the potental, as gven by Laplace s equaton [25], matchng the potental n the soluton, at the outer edge of the dffuson layer wth the correspondng potental satsfyng the electrochemcal knetcs on the electrode (eq. [28], s occasonally easer. An mportant addtonal advantage of nvokng the thn boundary layer approxmaton s that the velocty feld needs no longer to be fully computed. The concentraton varaton next to the electrode, requred for the determnaton of the concentraton overpotental and for correctng the knetcs expresson, can often be accounted for wth acceptable accuracy by determnng ust the thckness of the concentraton boundary layer, typcally reported n terms of the equvalent stagnant Nernst dffuson layer, N, along the electrode. The latter s often avalable from correlatons (13), text books (14), expermental measurements (15), or from nterpretaton of computatonal flud dynamcs software output. Obvously, N may vary wth poston along the electrode. 6 17

18 Approxmate Soluton: Thn Boundary Layer Rate of accumulaton = net flux n + generaton Nernst- Plank Eq. C t C t N D C Z F U C V C R C Thn boundary layer: C b 2 = 0 In regon of unform concentraton: C D C t Steady-state: C Z F U C t Z F U C V C 2 0 = 2 = 0 (Laplace s eqn.) Thn boundary layer approxmaton Bulk at unform concentraton Boundary condtons: Insulator: = 0 ( = -? ) = 0 Ths s the fundamental equaton to solve Electrode: = V E 0? a? C = f (V) e.g., Butler Volmer Eqn. = 0 {e (F/RT)? (C E /C B ) e (-F/RT)? }? =? a +? C = V E 0 - C E /C B = 1-/ L O + e R V e E a R O C C b e AF Ce CF 0 exp S exp R O S C b RT C b RT S a C V E C C O e O b 1 L V C 2 = 0 Dstance from electrode 6 18

19 5. COMMON APPROXIMATIONS FOR THE CURRENT DISTRIBUTION Even when applyng the thn boundary layer approxmaton, the equatons requred for solvng the current and potental dstrbutons n the electrochemcal cell yeld a non lnear system requrng teratve soluton. The reason s that the boundary condtons ncorporate the unknown term (the electrostatc potental or the current densty). Whle ths presents no serous hurdle for computer mplemented numercal solutons, analytcal solutons of nonlnear systems are dffcult and generally requre lnearzaton procedure. In order to analytcally characterze features of the current dstrbuton, some smplfyng approxmatons are frequently appled. These are summarzed n Table 1, and dscussed below. Controllng Modes V - = E 0 + a + C Approxmaton Domnant Overpotental Controllng Eqn. Boundary condtons PRIMARY O >> a + C 2 = 0 = V E 0 SECONDARY O ~ a >> C 2 = 0 = V E 0 - a MASS TRANSPORT C v C D 2 C C E << C B or: C >> O + a t C E ~ 0 TERTIARY O ~ a ~ C 2 = 0 = V E 0 - a - C 6 19

20 Table 1: Common approxmatons for modelng the current dstrbuton Approxmaton Prevalng Overpotental Controllng Equaton Boundary condtons Prmary S 2 C 0 V E Secondary ~ S C V E 2 0 S Mass transport C S Tertary * (no approxmaton) vc D C ~ ~ 2 S C C c : 2 e cb or t c ~0 0 V E S C b * The tertary dstrbuton, lsted n the bottom row, represents the formal thn boundary layer soluton wth no further approxmaton. Prmary Dstrbuton: >> S + C Here we assume that the prevalng overpotental s assocated wth the ohmc drop wthn the electrolyte, - [29] A C s C Ths assumpton s tantamount to statng that the electrode reactons are perfectly reversble and knetcs and mass transport lmtatons are both neglgble. Snce the ohmc overpotental s gven by (1), l [30] we may conclude that the prmary dstrbuton s lkely to preval n cells wth large nterelectrode gap, l, and low conductvty. Snce the surface and mass transport overpotentals must be small (n comparson to the ohmc overpotental), t s further expected that a dstrbuton close to prmary wll preval on hghly catalytc electrodes or knetcally reversble reactons, at hgh temperatures (e.g., n molten salts where the knetcs are very fast) and n systems operatng far from the lmtng current, (.e., wth neglgble mass transport lmtatons). 6 20

21 Consequently, Laplace s equaton [25] s solved subect to the boundary condtons: Insulator: 0 [31] Electrode: V E = 0 =V E = constant [32] Eq. [32] s derved from eq. [28] by settng the surface and concentraton overpotentals dentcally to zero ( S + C =0), snce ther magntude s nsgnfcant compared to that of. As a consequence of ths smplfcaton, the potental n soluton next to the electrode, dffers from the electrode potental, V, only by a constant (the equlbrum potental, E). Ths provdes for a smpler and drect soluton of Laplace s equaton. The prmary dstrbuton has some unque characterstcs. Whle the magntudes of the average and the local current denstes depend lnearly on the appled potental across the cell and on the conductvty, the current dstrbuton depends only on the geometry. Perhaps the most notable characterstc of the prmary dstrbuton s that t s typcally non unform, and at certan cell locaton t may exhbt sngulartes. These are ponts wthn the cell where the potental gradent, whch, accordng to eq. [27] s proportonal to the current densty, wll approach nfnty. Hence, whle the potental wthn the cell must be bound between the anode and cathode potentals (after subtracton of the equlbrum potental), the prmary current densty can approach at sngulartes nfntely large values. The sngulartes are located at ntersectons of cell boundares: () an electrode and an nsulator ntersectng at an angle larger than 90 0 () two electrodes ntersectng at an angle larger than The physcal nterpretaton of ths behavor s related to the fact that whle the potental drvng force s fnte, the local resstance becomes vanshngly small at the sngulartes, drvng the current densty to nfntely large values, rrespectve of the counter electrode poston, or the appled voltage. Other specal geometrc confguratons of nterest are: () ntersecton of an electrode and an nsulator formng an angle smaller than 90 0 and (v) two ntersectng electrodes formng an angle smaller than In these two latter cases [() and (v)], the local current densty tends to zero at the ntersecton pont, ndcatng an nfntely large local resstance. In all those cases, the magntude of the ntersecton angle affects only how sharply the current densty approaches ts lmtng values, but not the nature of the lmt tself. Practcal cell boundares do not ntersect at a mathematcally sharp pont. Nonetheless, f the radus of curvature formed by the ntersectng boundares s small, and the ntersecton nvolves the geometrcal features descrbed above, the current densty wll stll become ether very large or vanshngly small at these ponts. Whle clearly beng an approxmaton, the prmary current dstrbuton represents the worst (least unform) current dstrbuton a system may exhbt. The presence of knetcs 6 21

22 lmtatons, as dscussed below wll typcally lead to more unform dstrbutons. It should also be stated that when the system s subect to mass transport control under stagnant dffuson (typcal to mcro scale features), the controllng equaton as dscussed below, s Laplace s equaton for the concentraton: 2 c = 0 [33] Eq. [33], n complete analogy to eq. [25] (Laplace s equaton for the potental), follows also the prmary dstrbuton, and exhbts the same characterstcs as descrbed above. Prmary Dstrbuton Approxmaton Approxmaton Domnant Overpotental Controllng Eqn. Boundary condtons PRIMARY O >> a + C 2 = 0 = V E 0 = V - E 0 - a - C O = = IR = L/ Prevals: Large gaps Low conductvty Hgh current densty Fast knetcs (Hgh T) Unque characterstcs Depends only on cell geometry Potental Feld problem: electrostatcs magnetcs Heat transport by conducton dffuson n solds potental (deal) flow Sngulartes Ponts at whch the current densty ncreases sharply to nfnty (although the voltage s fnte) 6 22

23 Practcal mplcatons assocated wth the characterstcs of the prmary current dstrbuton: Small nodules or roughness elements on electrodes typcally form a sharp tp wth small radus of curvature, causng the (prmary) current densty at the tp to become very hgh. As a consequence, these sharp tps tend to propagate very rapdly (39 42), restrcted only by the knetcs lmtatons at the tp. Consequently, electrodes wth very reversble knetcs, e.g., lthum, slver, lead and znc tend to evolve needles and dendrtc growth qute readly, whle less reversble metals, e.g., nckel or ron do not. Electrodes bounded by co planar nsulators (formng ntersecton), tend to exhbt hgh current denstes close to the edge (lmted only by the knetcs). Therefore, such cell features should be avoded where possble. Slght embedment of the electrode, wll provde a fnte current at the edge whle presentng only mnmal resstance to flow (typcally mnmzed by secondary flow eddes). Non unformtes n the prmary dstrbuton usually orgnate wth geometrc perturbatons and typcally do not propagate very far. In cylndrcal or sphercal felds, generated by e.g., an nsulatng bubble, or sphercal or cylndrcal electrodes, the current dstrbuton perturbaton dmnshes to a small fracton (few %) of ts maxmal value once the dstance from the curved surface exceeds about 3 dameters. As a consequence, n order to effectvely sheld a curved electrode or an edge regon, the sheld must be placed very close to the surface. By the same token, n order to selectvely plate a small feature usng a small dameter wre or a sharp tp electrode, one must place the tp very close to the surface. The prmary dstrbuton s not unque to electrochemcal systems and other physcal systems exhbt the very same dstrbuton. Textbooks avalable n these areas provde nformaton that can be drectly appled to electrochemcal systems operatng under condtons approachng the prmary dstrbuton. Examples nclude heat transfer by conducton (16), dffuson n solds (17), electrostatcs (18), potental (deal) flow (19), and mathematcal texts on the theory of complex varables and conformal mappng. A comprehensve dscusson of the prmary current dstrbuton n electrochemcal systems s provded by Newman (1). 6 23

24 Prmary Dstrbuton - Implcatons Intersectons between cell boundares: Sngulartes: 8 (ndependent of counter electrode) fnte (depends on geometry) = fnte V E A R 0 = = fnte = Prmary Dstrbuton: Nodular ( Dendrtc ) Growth fnte V E A R R R FLAT A C SHARP TIP R, R, R fnte C A R R C 0 RA fnte e 0 C F V E C RT V E A R R R 0 e A C F V E RT Reversble metals (L, Ag, Zn, Pb) tend to evolve dendrtes Platng Addtves Requred to Reduce 0 C 6 24

25 Implcatons: Electrode Insulator Intersectons cathode Cell-Desgn smulaton Implcatons: Short Dstance Propagaton Cell-Desgn smulaton 6 25

26 Implcatons: Short Dstance Propagaton Example: Insulatng Sphere Between Parallel Electrodes Current vectors Equ-potental lnes Implcatons: Part wth a small curvature generates ts own feld Current vectors Equ-potental lnes Cell-Desgn smulaton 6 26

27 Implcatons: Part wth a small curvature generates ts own feld Cell-Desgn smulaton 6 27

28 Unform Prmary Dstrbuton - Only 3 cases: Parallel Planar Bounded Electr. Concentrc Cylnders (Bounded) Concentrc Spheres (Bounded) R0 R ` H R0 R R x = const. 2 1 d L 0 dx L 2 1 I IA R r r I I f ( r ) 2Hr 4 r o Ro o Ro I dr I d dr r 2H d 2 r 4 f ( r ) 2 R IL Ro o ln 2H R R I 1 R ln 2H R o R I 1 1 o 4 R Ro o R I 4 R Ro L x L r R/R 0 r R/R

29 Cylnders 0-1 V 20 lnes k=1 Spheres 0-1 V 10 lnes k=1 Cell-Desgn smulaton 6 29

30 6 30

31 6 31

32 6 32

33 6 33

34 6 34

35 6 35

36 6 36

37 6 37

38 6 38

39 Secondary CD V - = E 0 + a + C Approxmaton Domnant Overpotental Controllng Eqn. Boundary condtons PRIMARY O >> a + C 2 = 0 = V E 0 SECONDARY O ~ a >> C 2 = 0 = V E 0 - a MASS TRANSPORT C >> O + a C E << C B or: C E ~ 0 TERTIARY O ~ a ~ C 2 = 0 = V E 0 - a - C Secondary CD: V - = E 0 + a + C Prevalng: Approxmaton good for almost all applcatons Excepton: L =nfdc b / C b dlute reactant Stagnant soluton Unque characterstcs More unform than prmary - Surface resstance No sngulartes - As I ncreases, a bulds up Dffcult to solve (analytcally): - Non-lnear problem: BC at the electrode - a = f() =- = V-E 0 f() Prmary Secondary 6 39

40 Secondary Dstrbuton: + S >> C Here, both ohmc and knetcs rreversbltes are consdered, however, mass transport lmtatons are assumed neglgble. Snce sgnfcant mass transport effects are present only when operatng close to the lmtng current, e.g., n very dlute solutons, the secondary dstrbuton presents a vald approxmaton for most electrochemcal systems. Here, Laplace s equaton, [25], s solved subect to the boundary condtons: Insulator: 0 [31] Electrode: V E = s [34] where s and are related by: AF CF 0 exp Sexp S RT RT [15] It s sgnfcantly more complcated to analytcally solve the secondary dstrbuton (than the prmary dstrbuton), snce the presence of the knetcs overpotental renders the system non lnear. Avalable analytcal solutons nclude: the dsk electrode (20), Wagner s soluton of the current dstrbuton at a corner (21), current dstrbuton wthn a through hole (22) and a blnd va, and wthn a thn electrolyte layer (23) among many others. Most of the analytcal solutons employ lnearzaton about some average current densty. Computer mplemented numercal solutons of the secondary dstrbuton, (example provded n Fg. 2) are readly obtaned and do not requre lnearzaton. Few examples nclude modelng the current dstrbuton n the wedge (24) and Hull cells (25 27), the rotatng Hull Cell (28), nsde a channel (29), n pattern platng of prnted crcuts (30,31), nsde a va (32), and n the characterzaton of anodzaton applcatons (33), among many others. The secondary dstrbuton ncorporates both the effects of the ohmc resstance, whch renders the prmary dstrbuton ts non unform characterstcs, n combnaton wth the surface resstance assocated wth the lmted reversblty of the electrode knetcs. The latter (wth the excepton of specal cases, nvolvng e.g., the use of unque addtves), leads to a unform dstrbuton. Whle t s dffcult to derve accurate analytcal solutons for the secondary dstrbuton, we can characterze the degree of non unformty by evaluatng the relatve magntude of the resstances assocated wth the surface and ohmc dsspatve processes. 6 40

41 (a) (b) (c) PRIMARY INTERMEDIATE HIGHLY PASSIVATED KINETICS KINETICS Cell Desgn Smulatons Fg. 2: Examples of numercal solutons for the cathodc current dstrbuton on a plate electrode mmersed n a cell wth the counter electrode at the bottom. Three cases are compared: a. (left column): completely reversble knetcs (prmary dstrbuton); b. (center): Intermedate knetcs (Wa ~ 0.2); c. (rght column): Irreversble knetcs (Wa ~ 10). The top row provdes a comparson of the current dstrbuton or the depost profle on the cathode (cross hatched regon). The center row provdes the current dstrbuton along the electrode ( stretched ). The bottom row provdes the correspondng potental dstrbutons. It s evdent that the current dstrbuton unformty ncreases as the electrode knetcs become more passvated. ( Cell Desgn software smulatons [69]). Accordngly, the degree of unformty of the secondary dstrbuton can be characterzed n terms of a dmensonless number, named after Carl Wagner, representng the rato of the surface to the ohmc resstance: 6 41

42 * R S * R Wa [35] The resstances n eq. [35] are specfc resstances (per unt area, cm 2 ), correspondng to the local slope of the polarzaton curve,. The ohmc resstance, whch s a constant, ndependent of the current densty, s gven by: IR R [36] The surface (actvaton) resstance depends on the current densty n a relatvely complex manner, and therefore t s convenent to consder the system n terms of two separate regmes: the Tafel ( hgh feld ) and lnear (low current densty) approxmatons (6): In the Tafel regme, we consder the Butler Volmer equaton n terms of two sub regons, one for hgh anodc polarzaton ( A >> A F/RT), the second for hgh cathodc polarzaton ( A >> C F/RT). It can be shown that these approxmatons are vald when avg >> 0. The mathematcal representatons for the anodc and cathodc Tafel regons are smlar: ST, RT ln a bln F [37] 0 where b = RT/F (2.3*b s the Tafel slope ), a, s a constant, and = A for anodc polarzaton and = C for the cathodc process. We now can derve the surface resstance for the Tafel regme: s RT R ( Tafel) [38] ST, F For the lnear polarzaton regme (=low current densty or mcro-polarzaton, avg << 0 ), recognzng that here, A << ( A + C ) F/RT, the Butler-Volmer equaton can be lnearzed: SL, RT A C F 0 ' b [39] Where b s the lnear polarzaton slope. The lnear regme surface resstance s: 6 42

43 S RT RT R b' ( Lnear) SL, ( ) F nf a c o o [40] Unlke the constant ohmc resstance, the surface (knetcs) resstance decreases wth ncreasng current densty n the Tafel range, but s a constant n the lnear regme. Substtutng the resstances (eq. [36] [40]) nto the Wa number expresson (eq. [35]) we get: Tafel: Lnear: b Wa l and ' b Wa l and RT b [41] F ' b RT [42] nf 0 b (the Tafel polarzaton slope) and b (lnear polarzaton slope) ncorporate the knetcs parameters. A large Wa number desgnates a large surface resstance and a small ohmc resstance, leadng to a unform current dstrbuton. By the same token, a small Wa number s ndcatve of prevalent ohmc overpotental, leadng to a non unform dstrbuton, approachng n the lmt, prmary dstrbuton. Snce the Wa number characterzes the current dstrbuton unformty, a relevant queston s how large must the Wa number be n order to assure a hgh degree of unformty, and conversely, how low should t be to ndcate non unformty. Inspectng numercally derved dstrbutons, we fnd that when the Wa > 5, the dstrbuton s qute unform, and when Wa < 0.2, most dstrbutons exhbt sgnfcant non unformty. Ths becomes also qute evdent by nspectng Newman s classcal analytcal dervaton (20) of the current dstrbuton on a dsk electrode under mxed surface and ohmc control ( secondary dstrbuton ). The current dstrbuton s gven here n terms of a parameter J whch s equvalent to 1/Wa. An mportant ssue s the clear dentfcaton of the characterstc length, l, n the Wa number. In some confguratons the correct selecton of l s ntutvely evdent, but n many others, t s not. For example, the selecton of the characterstc length n the platng of blnd vas has been controversal, wth some authors selectng the va depth, L, as the characterstc length (e.g., 34), whle others (e.g., 22) ndcate that the proper characterstc length for ths confguraton s L 2 /r, where r s the va radus. Akolkar and Landau (35), pont out that the characterstc length can be unambguously dentfed only through an analytcal soluton. 6 43

44 Secondary CD The Wagner Number The degree of current dstrbuton unformty s determned by: R Polarzaton resstace ('chemstry') The Wagner number: Wa = a R Ohmc resstance ('geometry') Wa >> 1 (Wa >5) - Unform (knetcs controlled) Wa << 1 (Wa < 0.2) - Non-unform (ohmc controlled) dstrbuton (~ prmary ) L ( I ) A L R I I A R a a a I ( A ) b Wa L b Wa L ' a ( Tafel) ( Lnear) RT ln F RT 1 b 0 ( Aa) F Aa Aa RT nf RT 1 b' 0 ( Aa) nf 0 Aa Aa Assumng A a b = Tafel slope b = lnear polarzaton slope Secondary CD The Wagner number: Wa >> 1 Wa << 1 Unform Non-unform addtves Plate here V R Wa = R a a Implcatons: For unform dstrbuton: Low Hgh b Wa L b Wa L ' ( Tafel) ( Lnear) Tafel: Hgh Tafel slope small (addtves) Lnear: Low 0 (addtves) Unformty on small scales 6 44

45 Secondary CD The Wagner number: Wa >> 1 Wa << 1 Unform Non-unform R Wa = R a a RT b 0.05 F 0.5 = 0.1 S/cm Wa avg = 0.05 A/cm 2 L macro = 2 cm Wa L macro = 0.02 cm MACRO MICRO b Wa L b Wa L ' ( Tafel) ( Lnear) Secondary CD on a Dsk Electrode 1 J Wa Lnear Polarzaton Tafel Polarzaton af J ( ) 0 A C RT n J avg Fa RT From J. Newman, Electrochemcal Systems, Prentce-Hall, NJ

46 The Throwng Power The Throwng Power ( TP ) s often used by platng practtoners to characterze the ablty of ther bath to coat non unformly accessble parts, unformly. The throwng power relates to the Wagner number ( Wa ) snce both parameters, (the TP and the Wa) are used to a characterze the capablty of the soluton, the platng condtons and the cell to produce a relatvely unform depost thckness, rrespectve of the cell geometry. Whle the Wa number s more precsely defned and used frequently n the scentfc lterature, the TP s not very well defned and s more commonly used n the platng ndustry practce. As we recall, the Wagner number s a measure of the capablty of the bath to produce unform depost on a complex shaped (no unformly accessble) part. It s gven by: b Wa [1] L avg Here, s the conductvty, b s the slope of the polarzaton curve (= /), avg s the average current densty and L s the characterstc length of the system. If the Wa number s hgh, we expect unform coatng rrespectve of the geometry, and f the Wa number s small (compared to 1) we expect non unform platng (and possbly varable alloy composton) when platng a non unformly accessble part. Typcally, based on numercal modelng and experence, Wa > 2.5 corresponds to extremely unform depost Wa = 1~2.5 qute unform depost Wa = varyng thckness (when the part s of rregular shape) Wa = 0.1 ~0.25 qute non unform depost Wa < 0.2 extremely non unform depost The Wa number depends also on a characterstc length L, whch may vary from system to system. The Wa number changes wth the current densty, as expected. For low current denstes Wa s hgh, as the current densty ncreases, the Wa number dmnshes. 6 46

47 If we want to apply the value of the Wa number as reported n one system (ref) to another applcaton, we have Wa Specfc Applcaton L Wa ref Specfc Applcaton Where L s measured n cm. As stated above, the throwng power s used n platng practce. It s not a well defned parameter and usually refers to experments done n the Harng Bloom Cell, where depost weght on two electrodes placed at dfferent dstance from a common anode are measured. The throwng power s typcally reported by percentages between 0 100%, where a low % ndcates poor depost unformty and 100% refers to perfect unformty. Obvously, unless we carry the experment n the specfc cell, the numbers are meanngless, (and often platng soluton supplers use those very ndscrmnately). The throwng power reported for a gven system should therefore be vewed only as a relatve measure for comparson between dfferent electrolytes and the absolute value has lttle meanng. An emprcal correlaton that can be appled to compute the TP, n terms of the Wa number (whch s well defned). TP = 100 * (1 10^( Wa/5)) Ths s a sem emprcal correlaton, whch provdes the correct behavour and lmts: TP [%] Very Unform Very Non Unform Wa 6 47