An Interdisciplinary Approach to the Electrochemistry at Solid Electrodes

Um enfoque interdisciplinar na eletroquímica em eletrodos sólidos é apresentado levando-se em conta no vas interpretações básicas, tanto para o crescimento de fases quanto para a perturbação de fases sob condições longe do equilíbrio. Para esse propósito, os da dos de imagens derivadas de microscopia de tunelamento por varredura (STM) e de microscopia de força atômica (ATM) em diferentes escalas de comprimento, se tornam extremamente úteis. Esse enfoque proporciona no vas visões no desenvolvimento de rugosidade, decaimento de rugosidade e reatividade em eletrodos sólidos. O estudo de in ter faces móveis por aplicação sequencial de da dos de STM in situ e a interpretação baseada na teoria da escala dinâmica fornece uma descrição qualitativa e quantitativa das irregularidades da superfície, e provê informações sobre o mecanismo de evolução da in ter face sob re gimes estacionários e não-estacionários de rugosidade. Vários exemplos de interesse eletroquímico são discutidos na base deste enfoque interdisciplinar.


Introduction
Elec tro chem i cal re ac tions take place in an in ter fa cial region com prised be tween the elec trode sur face and the bulk of the elec tro lyte so lu tion.The thick ness of the in ter fa cial re gion ex tends from the cross sec tion of the ad sorbed mol e cule on the elec trode up to the liq uid film ad ja cent to the elec trode sur face whose com po si tion, gen er ally, dif fers from that of the so lution [1][2] .Within the in ter fa cial re gion dif fer ent zones re lated to some dom i nant as pects such as the ad sor bate layer, the dou ble layer thick ness, the con cen tra tion gra di ent of re act ing spe cies, and the hy dro dy namic pro file can be dis tin guished (Fig. 1).In this con text, the sim plest ap proach to such a de scrip tion results from a liq uid metal elec trode im mersed in a con duct ing so lu tion.There fore, for many years elec tro chem is try fo cused on the Hg(liq uid)/aque ous dis so lu tion in ter face to neu tral ize the in flu ence of the sur face to pog ra phy on the in ter face proper ties, which is un avoid able when deal ing with solid electrode/so lu tion in ter faces.How ever, less at ten tion has been paid to the ef fect of the elec trode to pog ra phy on those elec trochem i cal re ac tions in volv ing solid elec trodes, de spite the large num ber of pro cesses in which this type of elec trode is em ployed.Solid elec trode sur faces are usu ally of a com pli cated nature.In fact, well-defined solid do mains are gen er ally restricted to very small sur face ar eas, as can be seen by nanoscopy tech niques 3 .Most gen er ally, solid sur faces al ways ex hibit ir reg u lar i ties at dif fer ent scale lengths 4 .
The prob lem of sur face ir reg u lar i ties al ways ap pears in elec tro chem is try when a new solid phase is formed, as in the case of metal electrodeposition or when a phase is de stroyed, as in metal cor ro sion.The same sit u a tion arises in deal ing with het er o ge neous ca tal y sis and electrocatalysis con cern ing real sur face area mea sure ment and sur face re ac tiv ity, two im portant as pects re lated to the spe cific ef fi ciency of the cat a lyst.There fore, knowl edge of the to pog ra phy and sur face struc ture of the solid down to the atomic level is es sen tial in many aspects, such as pre dict ing its be hav ior un der dif fer ent op er at ing con di tions, and cal cu lat ing the energetics of pos si ble ad sorbate struc tures in re la tion to the likely re ac tion path ways [5][6] .All these as pects are of great im por tance to han dling the op timal ki netic con di tions for a par tic u lar elec tro chem i cal re action.To tackle these com plex prob lems an in ter dis ci plin ary ap proach is re quired in which fron tier as pects of phys ics, chem is try, math e mat ics, and mod el ing tech niques con verge.
This pa per sum ma rizes how to deal with ir reg u lar i ties at solid sur faces, such as those en coun tered on solid elec trode sur faces, and how they can be quali-and quan ti ta tively treated.This im plies an in ter dis ci plin ary ap proach to elec trochem i cal prob lems of ba sic im por tance for elec trode de sign and per for mance, and pro vides an ex pla na tion for elec trode be hav ior which has oc ca sion ally been con sid ered "anom alous" from the stand point of clas sic elec tro chem is try.

Interface Motion
Pro cesses such as the for ma tion of a solid phase by ei ther va por phase de po si tion 7 , epitaxial growth in duced by mo lec ular rays 8 , or the growth of metal electrodeposits 9 , and the disso lu tion of a solid phase as it oc curs in cor ro sion in gen eral 10 , in volve an in ter face mo tion.The way in which the in ter face dis place ment oc curs re flects the type of ki net ics obeyed by the sys tem and the dom i nant mech a nism op er at ing in the pro cess.
In fact, the in ter face mo tion can be de scribed by the same tools which are usu ally em ployed for other seem ingly dif fer ent processes such as liq uid flow through po rous me dia, the prop a gation of a flame front, the growth of bac te rial col o nies, or the rup ture of a di elec tric 11 .The com mon fea ture among all of these phe nom ena is the ex is tence of an ir reg u lar mo bile in terface.For such an in ter face the chal lenge is to es tab lish the mech a nism of its for ma tion, as well as its dy namic char ac teris tics 11,12 .The eval u a tion of these char ac ter is tics be comes rele vant to de ter mine the sta bil ity of the mo bile in ter face at ei ther a solid/liq uid or a solid/gas in ter face by us ing pro ce dures which lead to a mi nor per tur ba tion in the sys tem and can be applied ir re spec tive of its na ture.

Growth Modes
The growth of a solid phase im plies a pos i tive shift of the av er age in ter face plane along the z-axis (Fig. 2), whereas the op po site dis place ment ac counts for the phase de struc tion.In both cases, how ever, the over all phe nom e non can be in ves tigated us ing the same for mal ism.
Clas si cal growth mod els for a solid phase are based upon the ex is tence of lo cal equi lib rium on the sur face and fur ther ad mit that the mass trans port par al lel to the mi cro scopic surface is much faster than the flux of par ti cles ar riv ing at the surface.Within this ther mo dy namic frame work sev eral mod els have been pro posed (Fig. 3).Thus, the Volmer-Weber model (1926) 13 1.Scheme of the dif fer ent dis tin guish able thick nesses (x) related to the solid elec trode/so lu tion in ter face.Dis tances are es ti mated for a ro tat ing disc elec trode at ap prox i mately 100 rpm.Fig ure 2. Scheme of the mo bile in ter face.Ar rows in di cate the in ter face mo tion for a de po si tion at a solid sub strate and a solid dis so lu tion process fol lowed by 3d-growth.The Frank-van der Merwe model (1949) 15 con sid ers layer-by-layer growth.
The above men tioned mod els have been ex ten sively used in the in ter pre ta tion of solid phase growth data, de spite the fact that in a num ber of elec tro chem i cal ex am ples pro cesses take place far from con di tions of equi lib rium.Atomic force micros copy (AFM) im ages of Cu films pro duced un der potentiostatic con di tions at a cath odic overvoltage such as ηc = 0.149 V 16 have shown a re mark able change in the to pog raphy dur ing growth, which re sults in a sub stan tial in crease in sur face rough ness.These facts re flect in the shape of the cor respond ing cur rent tran sient it self.In prin ci ple, the value of the ap plied po ten tial, which is far from the equi lib rium po ten tial, and the rel a tively high flux of cop per ions ar riv ing at the surface make it some what un likely that un der these cir cumstances ther mo dy namic mod els could be ap plied.There fore, to an swer the ques tion as to whether the shape of the mo bile inter face could be re lated to the ki net ics of the phys i cal, chem ical or elec tro chem i cal pro cess, an ap proach based upon non-equilibrium con di tions has to be con sid ered 17 .
A sim ple ex per i ment to em pha size the main con cepts involved in the de vel op ment of a mo bile in ter face con sists of pour ing a par tic u late ma te rial onto a solid plane sur face (Fig. 4).Ini tially, the ma te rial forms a small heap which increases both in height, at the ve loc ity v ↑, and in width, at the ve loc ity v →.Ac cord ingly, de pend ing on the v ↑/v → ra tio, two lim it ing sit u a tions in the growth mode of the heap can be distin guished.Thus, for v ↑ > v →, a col umn tends to be formed, whereas for v ↑ < v →, a plane is ap proached.It should be noted that the ve loc ity con cept which is in volved in this ex per i ment can be eas ily gen er al ized to a chem i cal or phys i cal pro cess involv ing at oms or mol e cules.How ever, it should be em phasized that de pend ing on the yard stick used to fol low the in ter face char ac ter is tics, the in ter face can be de scribed às either a dis con tin u ous or a dis crete struc ture.Ac cord ingly, when the yard stick size ex ceeds the size of par ti cles in volved in the in ter face mo tion the struc ture of the in ter face can be an a lyzed in terms of con tin u ous growth modes, whereas atomistic models be come more ad e quate to de scribe dis crete in ter faces.

The Concept of Scaling. Models far from Equilibrium
The con cept of scal ing was in tro duced in the study of mobile in ter faces by Fam ily and Vicsek (1985) 18 for the anal y sis of the to pog ra phy of sur faces un der non-equilibrium con di -tions.In deal ing with con tin u ous mod els the ap pli ca tion of scal ing con cepts can be made di rectly.
The evo lu tion of the ir reg u lar in ter face in con tin u ous mod els is es sen tially rep re sented by two types of com pet i tive con tri bu tions, one re lated to the in crease in rough ness and the other in volv ing sur face smooth ing 11 .The con tri bu tion to the in crease in sur face rough ness is at trib uted to ei ther the stochas tic noise of ar riv ing par ti cles which are in cor po rated into the new phase or to the ran dom na ture of the dis so lu tion process it self.Oth er wise, sur face smooth ing re sults from the contri bu tion of dif fer ent re lax ation pro cesses among which the sur face dif fu sion of im ping ing par ti cles in con tact with the sur face is in cluded.Con tin u ous mod els for growth of new phases have been de vel oped ei ther for homo-or heterodeposits, and among these mod els from Edward-Wilkinson 19 , Kardar, Parisi and Zhang model (1986) 20 , Wolf and Vil lain model (1990) 21 , and Vil lain, Lai and Das Sarma model (1991) 22 are the most ex ten sively used.

A Brief Description of the Dynamic Scaling Theory
Ac cord ing to the dy namic scal ing the ory the dis crete charac ter is tics of the de pos ited ma te rial lead to the ir reg u lar in terface and ex plain its fractal be hav ior.Con sidering an ob ject of length L at time t, the the ory predicts that the in ter face width, W L (t), i.e. the stan dard de vi a tion of the in ter face height (Fig. 5), given by: W obeys the fol low ing equa tion 18 where, h(x i ) is the de posit height mea sured along the x-direction at point x i , <h> is the av er age height, f is a func tion of t/L z , α and z are the rough ness ex po nents, and z rep re sents the scal ing law.
For small val ues of L and con stant t, Eq. 2 re sults in This equa tion rep re sents the steady rough ness re gime which is at tained when the con di tion t >> t x is ful filled.On the other hand, for L ⊗ ◊ the fol low ing re la tion ship is ob tained from Eq. 2: Equa tion 4 rep re sents the non-steady rough ness re gime which is found when t << t x , with t x be ing a tran si tion from the non-steady to the steady rough ness re gime.Ex po nent β is the rough ness growth ex po nent, and α and β are re lated by the ratio z = α/β.The phys i cal mean ing of ex po nents α and β, as well as the non-steady and steady rough ness re gimes, are il lustrated in Fig. 6.In this fig ure the sur face pro file for rough ness de cay is also in cluded.The eval u a tion of these pa ram e ters pro vides a way to de ter mine the pri mary mech a nism in volved in the growth of an ir reg u lar in ter face 18 .

Parameters α and β Predicted for Equilibrium Models
It is in ter est ing to con sider the lim i ta tions of clas si cal equi lib rium growth mod els for those pro cesses in which a mobile in ter face if pro duced far from equi lib rium.Thus, for the growth mech a nism in volved in the Frank-van der Merwe model the dy namic scal ing the ory pre dicts α = β = 0, i.e. , the sur face re mains smooth dur ing the growth pro cess 23 .
Sim i larly, for the mech a nism in volved in the Stranski-Krastanov and the Volmer-Weber mod els, the dynamic scal ing the ory re sults in val ues of α in the range 0.5 ≤ α ≤ 0.6.For these mod els, 3d is lands are formed, the shape of these is lands be ing de ter mined by the en ergy of grow ing facets, in ter face and sub strate.In ad di tion, the co ales cence of islands means that the solid phase has changed from a het er o ge neous to a ho mo ge neous one 23 .All these facts lead to time-dependent val ues of β.

Discrete models
The bal lis tic de po si tion model 24 im plies that ev ery par ticle ar riv ing at the sur faces sticks at the first con tact site (Figs.7a,b).The Monte Carlo sim u la tion of this pro cess in di cates the ex is tence of an L-dependent cor re la tion along the sur face because of the fi nite par ti cle size.There fore, α, β, and z strongly de pend on L, but L → ∞, it re sults in α = 0.33 and β = 0.25 for a 3d pro cess.
The ran dom de po si tion model 11 (Figs.8a,b) as sumes that par ti cles stick at the first deep est site en coun tered in the par ticle dis place ment di rec tion.The at tach ing mech a nism involved is this model leads to a non-steady rough ness re gime in which the value of W L in creases lin early with t 1/2 .Ac cordingly, the value β = 1/2 is ob tained ir re spec tive of the top ographic di men sion.
When a sur face re lax ation term by sur face dif fu sion is included in the ran dom de po si tion model 21 , which im plies the pos si bil ity that the par ti cle will shift a dis tance of l on the surface to in crease its co or di na tion num ber (Figs.9a,b), it re sults in α = 1 and β = 0.25 in 3d.When full sur face dif fu sion is allowed α = 0 and β = 0 in 3d are ob tained 19 .

Continuous models
Con tin u ous mod els for in ter face mo tion are based upon the Langevin equa tion 25 .This equa tion es tab lishes that for an ir reg u lar in ter face the lo cal rate for the height in crease, h(x), is given by ∂ The so lu tion of Eq. 5 only cor re sponds to a non-steady rough ness re gime with β = 0.50.The co ef fi cient ν is the so-called sur face ten sion term.
Based on Eq. 5, the Ed wards and Wilkinson 19 equa tion includes a smooth ing term, η(x,t), which co mes from the stochas tic noise of fi nite ar riv ing par ti cles.Then, By solv ing Eq. 6 for d = 3, the set of ex po nents ob tained is α = 0 and β = 0, i.e. they are equiv a lent to the ran dom de po sition with the full sur face dif fu sion atomistic mod els.
A fur ther im prove ment was made by Kardar, Parisi and Zhang (KPZ) 20 by add ing a non-lineal term to Eq. 6 to ac count for lat eral ef fects in the in ter face growth.Then, the KPZ in terface mo tion equa tion re sults in For d = 3, Eq.7 pre dicts α ≅ 0.38 and β ≅ 0.25 11 .Thus, this equa tion ac counts for the bal lis tic de po si tion and the atomistic growth mod els 26 .
Wolf and Vil lain 21 pro pose a lin ear model in which smooth ing is ex clu sively pro duced by sur face dif fu sion.The cor re spond ing in ter face mo tion equa tion is given by The first term into Eq.8 leads to a non-steady rough ness re gime as it cor re sponds to Fick's law for dif fu sion, whereas the sec ond term op er ates in an op po site way.The so lu tion from Eq. 8 for 3d yields α = 1 and β = 1/4.
Finally, a fur ther im prove ment was made by Lai and Das Sarma whose model in cor po rates a lat eral in ter ac tion term to Eq. 8 22 , At the early stages of growth the so lu tion of Eq. 9 for 3d fur nishes α = 1 and β = 1/4, whereas for t ⊗ ◊, the ex po nents are α = 2/3 and β = 1/5.
In sum, knowl edge of the dy scal ing pa ram e ters offers a sen si tive and con clu sive way to de ter mine the pre vail ing mech a nism in the growth mode of a new phase un der con di tions far from equi lib rium, through the evo lu tion of the in ter face.The ap pli ca tion of the dy namic scal ing the ory to nanoscopy im ag ing data be comes pos si ble be cause of their real space char ac ter is tics and their wide scal ing range.

Examples of the Application of the Dynamic Scaling Theory to Electrochemical Systems
The de vel op ment of the rough ness the ory was ac com panied by ad vances in the de vel op ment of nanoscopy, such as scan ning tun nel ing mi cros copy (STM), atomic force mi croscopy (AFM) and other microscopies de rived from STM and AFM.The avail abil ity of these tech niques al lows us to ob tain the real to pog ra phy of a solid in con tact with any en vi ron ment from the atomic scale up ward.The fol low ing ex am ples describe the ap pli ca tion of the dy namic scal ing the ory to real situ a tions.
This ap proach al lowed us to dis cover new ki netic con tribu tions to elec tro chem i cal re ac tions, to con firm oth ers which had been in di rectly con cluded from elec tro chem i cal ki netic stud ies, and to ex plain "anom a lous" facts.The wide va ri ety of ex per i men tal data gath ered in re cent years con sti tuted good sup port for test ing the va lid ity of the o ret i cal ap proaches and en vis ag ing new ways to im prove the mod el ing of the in ter face mo tion.

Non-stationary and stationary roughness regimes. Silver electrodissolution in aqueous acid solution
Metal electrodissolution in ag gres sive aque ous elec tro lyte me dia im plies the sep a ra tion of par ti cles from the solid phase and the cre ation of va can cies.The over all pro cess is ac com panied by the de vel op ment of an ir reg u lar interphase.A typ i cal ex am ple of this pro cess is the electrodissolution of a Ag sin gle crys tal do main in 0.1 M HClO 4 at 25 °C 10 .This spec i men was sub jected to a galvanostatic electrooxidation at ei ther j < 15 µA cm -2 or j > 15 µA cm -2 , and the amount of electrodissolved Ag in both cases was equiv a lent to ap prox i mately 150 monolayers.
For j < 15 µA cm -2 , the sil ver sur face shows no changes in rough ness (Fig. 10a) as op posed to the re sult ob tained for j > 15 µA cm -2 (Fig. 10b).In the lat ter case, the (W L ) STM vs .t plot shows the ap pear ance of non-steady and steady rough ness growth dur ing electrodissolution (Fig. 10c).This fact makes it pos si ble to eval u ate α and β fol low ing the above-mentioned pro ce dures, re sult ing in α = 0.90 and β = 0.36.The value of α is close to that ex pected for the Vil lain-Wolf model 21 al though the value β = 0.36 ex ceeds the ex pected β = 0.25.How ever, when sur face dif fu sion is re stricted to the ter race sur faces due to the pres ence of en ergy bar ri ers at step edges, the value of β changes from 0.25 to 0.40 as the interlayer mass trans port is pro gres sively hin dered with out any changes in the value of α.There fore, the val ues α = 0.90 and β = 0.36 in di cate that for Ag electrodissolution at j > 15 µA cm -2 , there is a strong con tri bu - tion of sur face atom dif fu sion with re stricted interlayer mass trans port.Such a sit u a tion im plies that a Schwoebel en ergy bar rier is lo cated at the steps 27 .
This re sult con firms pre vi ous con clu sions de rived from elec tro chem i cal ki netic data on sil ver elec trodes es tab lish ing that sur face dif fu sion pro cesses play a key role in the ki net ics of the sil ver elec trode.

A non-stationary roughness regime. The electrodissolution of Cu in acid solutions
Ex per i ments sim i lar to those de scribed above for Ag electrodissolution were made for Cu in the same so lu tions, start ing from a Cu sin gle crys tal do main 28 with char ac ter is tics sim i lar to those re ported for Ag.
Un der open cir cuit con di tions the elec trode sur face becomes con sid er ably mo bile, show ing a trendency to de crease in rough ness due to the dis ap pear ance of the small pits ini tially pres ent on the sur face.
The above de scribed sit u a tion be comes even more complex for j = 6 µA/cm -2 , since in this case a clear non ho mo geneous at tack of the Cu sur face can be ob served yield ing do mains with a fac eted to pog ra phy, do mains with a rel a tively low in crease in rough ness, and do mains where deep etched pits are formed (Fig. 11a).The for ma tion of etched pits can be seen af ter a rel a tively long dis so lu tion time.
The dy namic scal ing anal y sis ap plied to the first and second type at tack at the early stages of the pro cess, leads to a be hav ior which is sim i lar to that al ready de scribed for Ag.In con trast, the over all at tack of Cu leads to a non-steady roughness re gime, as can be seen from the (W L ) STM vs. t plot (Fig. 11b) which ex hib its a con tin u ously in creas ing slope.Therefore, from the dy namic scal ing anal y sis of scan ning force micros copy im ag ing only the value of β can be es ti mated from the log (W L ) STM vs. log L plot for this sys tem.
For those do mains where etched pits are ab sent, β = 0.36.This value in di cates the cru cial role played by Cu atom sur face dif fu sion in the early stages of Cu electrodissolution, but the value β > 0.25 sug gests that sur face dif fu sion also in volves, as in the case of Ag electrodissolution, a Schwoebel type of energy bar rier at the steps.Sur face dif fu sion in this case is prob ably anisotropic, a fact which may ex plain the di rec tional fac et ing pro duced at cer tain do mains of the Cu sur face in the pro cess.
On the other hand, the value of β > 1 re sult ing from the over all electrodissolution in di cates a global pro cess dom inated by the for ma tion of etched pits.This re sult is par tic u larly rel e vant, as it may throw light on the pos si ble or i gin of pit ting cor ro sion of met als.It also em pha sizes the dif fi culty in in terpret ing con ven tional Tafel plots for metal electrodissolution when the real sur face area be comes un cer tain.The con tri bu tion of sur face atom dif fu sion to post-deposition sur face rough ness evo lu tion was stud ied for co lum nar-structured Au films on Au.These sur faces were prepared from the electroreduction of thick hy drous Au ox ides ac cu mu lated on Au by an od iza tion at 2.6 V ( vs. stan dard hydro gen elec trode, SHE) in 0.5 M H 2 SO 4 .This pro ce dure results in the for ma tion of nanometer Au par ti cles (Fig. 12a, t = 0) which co alesce in time yield ing large par ti cles when aged in 0.5 M H 2 SO 4 + 0.1 M NaCl (Fig. 12a) 29 .

The evaluation of z through roughness decay imaging data. Columnar-structured gold electrodes
From the STM im ages the av er age ra dius of grow ing par ticles in the x-direction (r x ) can be ob tained for the coars en ing pro cess.Data fit a r x ~ t 1/4 plot (Fig. 12b), as is ex pected for the case in which the sur face dif fu sion of at oms plays a dom i nant role in the ki net ics of coars en ing.This type of re la tion ship has been pre vi ously ex pressed by Ostwald's rip en ing law of colloids.

The Kinetic Consequences of Surface Fractality. The Enhancement of H-atom Electroadsorption on Pd Electrodes
When a Pd ox ide layer pro duced in an acid so lu tion by the po ten tial re ver sal tech nique us ing E u = 1.7 (SHE), E l = 0 V; f = 1 kHz, τ u /τ l = 9 and t prt = 3 min is electroreduced at a con stant po ten tial or with an ad e quate potentiodynamic sweep, it results in a Pd layer which ex hib its an en hance ment for the H-atom electrosorption pro cess as re vealed by con ven tional  voltammetry and tri an gu larly mod u lated tri an gu lar po ten tial voltammetry 30 .
To ex plain this ap par ently anom a lous ef fect, the pal ladium sur face was in ves ti gated us ing atomic force mi cros copy (Nanoscope III, Dig i tal) un der the con tact mode op er a tion.AFM im ages show an ir reg u lar col umn-like struc ture (Fig. 13a) which was an a lyzed fol low ing the "lakes and is land" proce dure de scribed else where to de ter mine D s , the fractal dimen sion of the sur face 31 .
Ac cord ingly, the value of D s = D l + 1, where D l is the fractal di men sion of the per im e ter of lakes re sult ing from a log(per im e ter) vs. log(area) of lakes plot (Fig. 13b).For the Pd sur face shown in Fig. 13a, D l = 1.57and D s = 2.57.This means that the sur face of Pd is fractal.The phys i cal mean ing of a fractal sur face is that the at oms which con sti tute the ma te rial tend to oc cupy the larg est sur face, and there fore, on this type of sur face it is ex pect that sur face pro cesses will pre vail over bulk dif fu sion pro cesses.This ex plains the par tic u lar char acter is tics of Pd elec trodes pro duced by the above-mentioned pro ce dure, which en tails a net en hance ment of H-electrosorption pro cesses.It should be noted that for Pd, as ob served for Pt co lum nar fractal sur faces 31 , the rough ness decay dis cussed in the pre vi ous sec tion for Au is not rel e vant due to the sur face mo bil ity of Pd and Pt which is slower than that of Au.Sur face mo bil ity must be con sid ered in deal ing with the ef fi ciency of solid cat a lysts in a num ber of chem i cal and electro chem i cal re ac tions.
There fore, the same pro cess may lead to the de vel op ment of dif fer ent sur faces, a fact which un der ad e quate con trol becomes par tic u larly rel e vant for the de sign of solid cat a lysts and electrocatalysts.

Conclusions
• The dy namic scal ing method and fractal ge om e try are pow er ful tools for study ing in ter face dy nam ics and under stand ing the role of sur face ir reg u lar i ties in elec trochem i cal ki netic data.
• Dy namic scal ing anal y sis can be suc cess fully ap plied to scan ning tun nel ing and scan ning atomic nanoscopy im ag ing data, which rep re sent top o graphic data in a real • Rough ness de vel op ment in volves a non-steady and a steady rough ness re gime.
• Un der a non-steady rough ness re gime the value of β can be es ti mated.
• When both the non-steady and the steady rough ness regimes can be de ter mined, then the value of α, β and z can be ob tained.
• From the dy namic scal ing anal y sis of rough ness de cay the value of β can be de ter mined.
• From the val ues of α, β and z, the ki net ics and mech anism of those elec tro chem i cal re ac tions in which an inter face mo tion is in volved can be de duced.
• Sur face atom dif fu sion, a con tri bu tion which is usu ally en vis aged from elec tro chem i cal ki netic data, can be more con vinc ingly dem on strated by the dy namic scaling anal y sis of STM and SFM im ag ing data.
• The ex is tence of a fractal sur face can ex plain some exper i men tal data which have of ten been con sid ered anom a lous in terms of the clas sic mod els.
Fig ure1.Scheme of the dif fer ent dis tin guish able thick nesses (x) related to the solid elec trode/so lu tion in ter face.Dis tances are es ti mated for a ro tat ing disc elec trode at ap prox i mately 100 rpm.

Fig ure 3 .
Fig ure3.Schemes of phase growth mech a nisms based on ther mo dynam ics.

Fig ure 4 .
Fig ure 4. Scheme to il lus trate how a heap of a par tic u late ma te rial is formed by pour ing it on a flat ta ble.Dashed lines in di cate the flux of par ti cles.The shapes of the grow ing heap at times t 1, t 2 and t 3 are shown.Heap growth ve loc ity com po nents par al lel and per pen dic u lar to the ta ble plane are in cluded.At the bot tom, two lim it ing growth modes lead ing to a col umn and to a flat ob ject are de picted.

Fig ure 5 .Fig ure 6 .
Fig ure 5.A def i ni tion of the terms used in the dy namic scal ing anal ysis of a top o graphic pro file at time t.Pro file heights at po si tions x 1 and x2 are in di cated.

Fig ure 7 .Fig ure 8 .
Fig ure 7. Schemes show ing the mo tion of at tach ing par ti cles (a) and re sult ing pro file (b) for a phase growth ac cord ing to the bal lis tic de posi tion model.

Fig ure 9 .
Fig ure 9. Schemes show ing the mo tion of at tach ing par ti cles (a) and re sult ing pro file (b) for a phase growth ac cord ing to the ran dom de posi tion with sur face re lax ation model.

Fig ure 12 .
Fig ure12.( a) Se quen tial in situ STM im ages of a co lumnar-structured gold elec trode sur face im mersed in 0.5 M H 2SO 4 + 0.1 M NaCl; E = 0.6 V ( vs. SHE).(b).Plot cor re spond ing to the change in the par ti cle ra dius mea sured in the x-direction as a function of the im mer sion time.A cor rec tion for the ini tial ra dius of the par ti cle is con sid ered.T = 298 K.

Fig ure 13 .
Fig ure 13.(a) An ex situ AFM im age of a co lum nar struc tured pal ladium elec trode pro duced by the electroreduction of a hy drous pal ladium ox ide layer on pal la dium.(b) Log (per im e ter) vs. log (area) plot to cal cu late the sur face fractal di men sion of the elec trode sur face depicted in (a).The fractal di men sion was cal cu lated fol low ing the proce dure de scribed in the lit er a ture 31 .