Pattern formation during electropolishing

PHYSICAL REVIEW B 15 NOVEMBER 1997-IVOLUME 56, NUMBER 19

Pattern formation during electropolishing

Vadim V. Yuzhakov, Hsueh-Chia Chang,* and Albert E. MillerDepartment of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556

~Received 7 April 1997!

Using atomic force microscopy, we find that the surface morphology of a dissolving aluminum anode in acommercial electropolishing electrolyte can exhibit both highly regular and randomly packed stripe and hex-agonal patterns with amplitudes of about 5 nm and wavelengths of 100 nm. The driving instability of thispattern formation phenomenon is proposed to be the preferential adsorption of polar or polarizable organicmolecules on surface ridges where the contorted double layer produces a higher electric potential gradient. Theenhanced relative coverage shields the anode and induces a smaller dissolution rate at the ridges. The insta-bility is balanced by surface diffusion of the adsorbate to yield a length scale of 4p(Ds /kd)1/2, whereDs is thesurface diffusivity andkd is the desorption coefficient of the adsorbate, which correlates well with the mea-sured wavelength. A long-wavelength expansion of the double-layer field yields an interface evolution equa-tion that reproduces all of the observed patterns. In particular, bifurcation analysis and numerical simulationyield a single voltage-dependent dimensionless parameterj that measures a balance between smoothing ofadsorbate concentration by electric-field-dependent surface diffusion and fluctuation due to interfacial curva-ture and stretching. Randomly oriented stripes are favored at largej ~low voltage!, while random hills domi-nate at smallj ~high voltage! with perfectly periodic stripes and hexagonal hill patterns within a small windownear j51. These predictions are in qualitative and quantitative agreement with our measurements.@S0163-1829~97!04744-9#

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INTRODUCTION

With the introduction of atomic force microscopy~AFM!,the surface morphology of metal surfaces can now be exined at nanoscales. Not surprisingly, a myriad of pattehave been uncovered with very rich pattern formationnamics. Surface structures on single-crystal electrodes hbeen shown1 to change with applied voltage and adsorbainduced mobility as surface atoms rearrange to form stingly regular stripes and dots with amplitudes of 0.3 Å awavelengths of 25 Å. Such stripe/dot patterns are alsoserved when surfactants self-assemble at solid-liqinterfaces.2 These aggregates form micelle cylinders, hemcelles, and other patterns a few angstroms deep and abonm long.

Most scrutiny of such patterns exploits the double-larepulsion between the tip of the atomic force microscopethe sample. Such double layers appear on the electrodeface when one or two layers of polar water molecules3 orsolvated ions4 are electrostatically packed against the eltrode. This well-packed~Helmholtz! layer of oriented polarmolecules or ions greatly reduces the transport of ions toelectrode interface such that the current through the layerbe small. As such, the layer functions as a molecular-selectrochemical capacitor between the anode surface prand the closest packed anions. Due to the small separathe capacitance can reach as high as 100mF cm22 by charg-ing the electrode at a small current (;1 A/cm2). At smallapplied voltage (,1 V!, the potential dropDcH across theHelmholtz layer is significant compared to the overall droMoreover, this dropDcH can vary as the dissolution reactioproceeds. Any charge carried by the ion can either chadischarge the electrochemical capacitor or be consumethe dissolution reaction, which is usually assumed to be

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activated process with an activation energy that dependDcH .5 When the capacitance dynamics is driven by thetwo parallel mechanisms, fronts and patterns can appe6

However, at the large applied voltage (.10 V! encounteredin electropolishing, the packing is saturated and the tocharge packed against a flat electrode is at its maximvalue and it remains constant. As a result,DcH remains con-stant and is much smaller than the total applied voltaMoreover, under this diffusion-controlled condition of eletropolishing, the average dissolution rate is equal exactlythe diffusion flux of water molecules or solvated ions, whiparticipate in the dissolution reaction, outside the Helmholayer. In this diffusive layer outside the Helmholtz layer, telectric potential decays to the zero reference value ofelectrolyte over a small distance of about 10–100 nm. Sithe current is small, the flux of ions driven by both the eletric potential and the concentration gradient must alsosmall. As such, the anion concentration decreases awaythe anode while the cation concentration increases to couthe potential gradient. This quasisteady charge separatiousually described by the Poisson-Boltzmann distributio7

such that charge at any position scales as sinh(c/dE2) or c/dE

2

for dilute electrolytes, wheredE; 1–20 nm, the Debyethickness of the double layer measuring the characterlength of decay for the potential in the diffusive layer. Cosequently, a large electric field is induced at the interfabetween the Helmholtz and diffusive layers.

From AFM studies, it is now known that this large eletric field introduced by the double layer can fundamentaaffect the adsorption affinity of polar or polarizable organadsorbates and ionic surfactants. Since the former can mlize surface atoms and trigger surface reconstruction wsurfactant aggregation and phase change are obviouslysitive to surfactant-electrode interaction, the observed p

12 608 © 1997 The American Physical Society

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56 12 609PATTERN FORMATION DURING ELECTROPOLISHING

terns are extremely sensitive to the applied voltagespecifies the strength of the electric field at the double la

Electropolishing of metals to produce optically reflectisurfaces is designed carefully such that only rough bumpthe electrode are dissolved away. This requires a slow dilution process that reaches a maximum of about 1.5 A/c2

at 70 V. Much of the resistance is traditionally attributedthe existence of a hydrodynamic diffusion boundary layer8 ofthicknessdD . Since the ions must transport through thboundary layer before reacting, a slow transport would thyield a slow dissolution process. However, double laymust also exist during most electropolishing processes wsuch small currents.~Electropolishing conditions are quitsimilar to surface chemistry experiments with AFM deteable double layers.! This then offers additional resistanceion transport, in series with the hydrodynamic diffusioboundary layer. In fact, it is quite possible that this doulayer offers a resistance comparable to or larger thanhydrodynamic diffusive boundary layer. A slow dissolutiorate is only achieved when the ion concentration andpH ofthe electrolyte are carefully controlled. These factorsknown to affectdE as well as the dissolution chemistry anhence the ion transport rate to the anode. Because ofresistance of the double layers (dE;10 nm! on the anodeand of the hydrodynamic diffusion boundary layers (dD;1mm!, the dissolution rate is controlled by the transportions9 and not by the dissolution reaction, whose detailedcroscopic mechanism remains unknown.10,11 It is the trans-port through the hydrodynamic diffusion layer that preferetially dissolves macroscopic roughness with an amplitudeexcess of 100 nm, approaching that ofdD . Smaller rough-ness on the order ofdE , however, would not be affected bthe transport enhancement due to change indD in light of thelength scale separationdE /dD!1. It is hence quite conceivable that optically reflective metals from electropolishistill possess irregularity of the amplitudedE or smaller. Forroughness of this amplitude to ‘‘self-assemble’’ into regupatterns, however, there must exist an instability mechanat the length scale ofdE to preferentially select/amplify certain length scales.

In solidification processes, transport in the liquid phasea maximum at the phase interface is enhanced either duthe reduction indD for high Peclet transport or in diffusionlength in globally diffusive transport.12 As such, the unstablesituation of a maximum growing faster than a minimumthe surface occurs and solidification processes have aspectrum of patterns driven by this instability. In dissolutior melting, however, the enhanced transport from a intecial maximum serves to diminish its relative growth rate wrespect to the minimum. Dissolution processes are tknown to produce very smooth and stable interfaces. Aresult, any pattern that appears at thedE level must be ofchemical origin with a double layer playing a paramourole. They are also quite unexpected for a dissolution pcess, especially in electropolishing, which is supposedyield a smooth surface.

As early as 1956, Welsh13 reported observation of smalscale patterns on electropolished Al, but dismissed it as relar incoherence in the bulk grain boundary of the metal. Wshall offer more direct evidence of stripe and dot~hexagon!patterns on polished Al. More importantly, we shall demo

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strate strong correlation between the length scales (;100nm! and patterns of these structures and the applied poteand hence disprove the theory that they are uncovered iherence of the bulk grain boundaries. Instead, the strvoltage dependence suggests a double-layer driven instity mechanism is responsible for the surprising patterns. Otheoretical analysis will further support the proposed mecnism by providing an accurate prediction of the small voltawindows where regular stripes and hexagons can beserved.

I. EXPERIMENTS

Aluminum foil ~Johnson Mattley! of 99.997% purity wasused as the anode in a LECO EP-50 electropolisher apptus ~LECO is a commercial electropolisher vendor!. The an-ode is a circular disk whose exposed diameter is 6 mm anseparated from the grade 304 stainless-steel cathodemm. The electrolyte flows between the anode and cathsuch that the average liquid velocity is 0.1 m/sec. The etropolisher is maintained at a constant temperature of 15by a coolant.

The Al coupons were first degreased with trichlorethylene and then electropolished in the LECOL1electrolyte. The electrolyte consists of 70.0 volof ethanol CH3CH2OH!, 13.8 vol % of distilled water,10.0 vol % of Butyl Cellusolve @or 2-butoxyethanolCH3~CH2!3OCH2CH2OH], and 6.2 vol % of perchloric acid(HClO4). The ethanol acts as a high-kinematic-viscosity awetting solvent for the molecules/ions involved in the disslution. The high viscosity ensures that the momentum bouary layer is thicker than that of the diffusion boundary layon the disk such that the ion flux to the electrode is naffected by bulk turbulence and eddies, which can introdlarge-scale nonuniformity. According to the bulk kineticstudy of Vidal and West,11 water molecules are the bulacceptor molecules that provide the necessary ions to rwith Al 31, the aluminum ion released by the dissolvinganode. The perchloric acid is to achieve the proper lowpH toensure that Al ionizes into Al31 and does not form otheions or oxides in the anodization reaction. The perchloacid ions act essentially as catalysts on the metal that relthe Al31 ions.~A Pourbaix diagram determining when Al31

ions are formed can be found in Ref. 14!. Since perchloricacid is a strong oxidizer, it can often oxidize violentlyelevated temperatures. In addition to careful temperacontrol, Butyl Cellusolve is added as a stabilizer.

Both ethanol and Butyl Cellusolve are polar molecuwith permanent dipoles. In fact, the stronger dipole onButyl Cellusolve lies on the branch that is identical to ethnol and hence has the same dipole moment. Moreover,dipole moment of ethanol is numerically close to thatwater @5.64310230 J/~V/m! compared to 6.17310230 J/~V/m!, respectively#. For this reason, it is quite conceivable ththe well-packed species near the electrode that producedouble-layer effect are not only the proposed wamolecules3 or solvated atoms,4 but also the more abundanethanol molecules. Solvated ions from dissociated waperchloric acid, and Al31 can also be present in the doublayer. We shall not speculate on the chemical compositionthe double layer but merely stipulate its presence in

llusolve,

12 610 56YUZHAKOV, CHANG, AND MILLER

FIG. 1. Plethora of surface features that are produced on an aluminum surface by electropolishing in perchloric acid, Butyl Ceethanol, and distilled water under various voltages and for various durations.

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model. Nor can we detect any significant hydrophobichydrophilic parts in any of our species that will allow theto aggregate and form surface micellar structures suchsurfactants or lipids.

We find our patterns to be insensitive to the electrolflow velocity, suggesting thatdD does not play a role. Theyare, however, strong functions of the applied voltage, sgesting the paramount role of the double layer. We haveto alter the composition of the electrolyte. Since this mchange the reaction chemistry as well as double-layer thness, the effect of electrolyte composition requires moretailed knowledge about the dissolution kinetics than is crently available. We hence use the same electrolyte inexperiments and vary only the voltage and the polishtime. The dissolution chemistry then presents itself asunknown but constant factor in our model, which is a phycal model describing how interfacial curvature affectslocal electric field and in turn the local dissolution rate.

AFM examination of the samples prior to electropolishireveals a rough surface with random nonuniformities gre

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than 50 nm peak to valley over a 6-mm square scan areaElectropolishing this surface dramatically reduces this heirange for all applied voltages studied to less than 3 nm apolishing. Persistent regular patterns of this latter height svive, however. As shown in the AFM images in Fig. 1, thstripe structure appears first for any applied voltage but aa short transition time it may evolve into another structuThese asymptotic structures are usually established aftesec from the beginning of electropolishing. Representatop-view AFM images of the final surface structures oserved for each of the electropolishing voltages examinedpresented in Fig. 2. It can be seen from this figure that thexist four distinctly different types of final surface structurerandom stripes for low voltage, regular stripes and reguhexagons for intermediate voltage, and random hills~whichwe shall refer to as random hexagons! for high voltage.

A two-dimensional spatial Fourier transform of the heagonal pattern at 60 V in Fig. 3 demonstrates the extreperiodicity of this pattern. Amplitudes of the surface strutures were measured with precision and they were found

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be in the range of 1–5 nm with a nonmonotonic dependeon voltage. However, the wavelengths are two ordersmagnitude larger and can also be measured with great psion. They are observed to increase monotonically withspect to applied voltage as seen in Fig. 4.

II. MODEL

The characteristic amplitude of the observed pattern isthe order ofdE . To obtain an instability mechanism, anvariation of the anode from the planar geometry mustamplified by the double layer such that a maximum~ridge!

FIG. 2. Four typical patterns on the electropolished aluminobtained for the applied voltage of~a! 40 V, random stripe pattern~b! 50 V, regular stripe pattern;~c! 60 V, regular hexagon patternand ~d! 70 V, random hills.

FIG. 3. Image of the first 47347 modes of the two-dimensionaFourier spectrum for the AFM image of the 333 mm2 aluminumsurface electropolished during 30 sec at 60 V.

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on the electrode dissolves slower and a minimum~valley!dissolves faster. Such variations can distort the electric fiwithin the diffusive layer and this must be how interfacidistortion transmits the initial disturbance to subsequentsolution dynamics. Due to the high voltage with saturapacking, the potential dropDcH within the Helmholtz layerremains constant and is negligible compared to the totalplied voltage. Also, since the Helmholtz layer is thin compared todE and the characteristic wavelength of the patterDcH remains constant over the entire patterned surfaceing dissolution. Pattern formation is then driven only by sptial and temporal variations of the field in the outside diffsive layer. For the dilute electrolyte used, we can useDebye-Huckel approximation in the Poisson equation thdescribes the potential field in the diffusive layer of tdouble layer,

¹ III2 c5

c

dE2

,

cuz5h5cs , ~1!

cuz→`50,

where, neglectingDcH , cs is the potential at the anode relative to the bulk electrolyte, which has a constant potentialto zero~the potential gradient in the bulk electrolyte is neligible relative to the large potential gradient in the diffusivlayer!, and h is the interfacial position measured from thmean flat position. The term on the right-hand side of Eq.~1!represents the charge distribution and is proportional toc.For a flat interfaceh50, the solution to Eq.~1! is simply

c05csexpS 2z

dED , ~2!

which yields an electric field on the metal of

E052dc

dz Uz50

5cs

dE

. ~3!

~Since the Helmholtz layer is only a few angstroms and imuch smaller thandE , we have omitted it in the analysis!Since the amplitude of the patterns is much smaller thanwavelength, we carry out a long-wavelength expansionEq. ~1! in the Appendix to obtain a relationship between telectric field at the anode interface and the interfacial shh,

E52dc

dn Uz5h

'E01E1 , ~4!

whered/dn is the normal gradient and the deviation electfield E1!E0 relative to the planar case is

E152E0

2@~¹h!21dE¹2h#, ~5!

where ¹, unlike ¹ III in Eq. ~1!, is the two-dimensionalLaplace operator. It is hence clear thatE1 /E0 is of the order(h/l )2 or (dE /l )2, where l is the characteristic wavelength. As is observed, this long-wavelength parametesmall and of order 1022– 1021; this justifies a long-


FIG. 4. Measured wavelength of the stripe and hexagonal patterns as a function of the applied voltageU. The solid curve and lines arethe theoretical predictions; typical AFM images at 30 sec are also shown.

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wavelength expansion to simplifyE in the double layer andthe subsequent derivation of a long-wavelength evolutequation for the anode interface. We shall use subscripsuperscript 0 to denote the planar zero order state anddenote the next-order correction that introduces interfacurvatures.

The first term in Eq.~5! represents the ‘‘stretching’’ o‘‘tension’’ term, which describes how the interface lengtheat the nodes. This stretched interface hence decreaselocal potential gradient~field! due to the larger interfaciaarea. The second term is the curvature term which descrhow a point on the anode ‘‘sees’’ the surface around itthis point is at a maximum (¹2h,0), the receding surfacearound it enhances the field gradient. The opposite is trua minimum where the gradient is reduced by the protrudsurface around it.

There are several possible ways this slightly distorelectric field can affect the local dissolution rate. Itknown15 that the dissolution rateKr depends on the anodelectric field in the usual threshold Arrhenius form, knownthe Tafel dependence,

Kr5Kr* expS b rE

kBTD ,

whereb r is a proportionality coefficient that is typically othe order of 10231 J/~V/m! for most metal anodes.18 Thepreexponential factorKr* denotes the fieldless dissolutiorate, which contains the weak dependence on bulk mtransport through the diffusion boundary layerdD and thestronger dependence on electrolyte composition, whichchange the dissolution chemistry. It is a nominal fieldlelimit of the dissolution rate since the dissolution chemiscan disappear entirely when no voltage is applied. We hinstead assumed that the pertinent chemistry remains in pfor the entire range of voltages. We are unable to detect

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pattern, for example, at zero voltage in Fig. 4. Neverthelethe extrapolation of the data to zero voltage representsnominal limit that we shall use an asterisk to denote.

The dependence on voltage, however, is entirely inactivation energy. The rationale is that the dissolution retion evolves through a bottleneck of ionic or polar transitistate whose energy, relative to the undissolved anode, isportional to the electric field. We can then carry out a lonwavelength expansion of the dissolution rateKr'Kr

01Kr1 to

see how the deviation fieldE1 affects it, how local tensionand curvature affects dissolution. Due to the exponentialpendence on field, we find as in all subsequent expansionfield-dependent rates thatKr

1 is proportional toKr0 ,

Kr1

Kr0'

Kr2Kr1

Kr0

'b r

kBTE11

1

2S b r

kBTD 2

E12 , ~6!

where we have carried out the expansion to second ordeE1 . It is then clear that the maximum with a positiveE1would have a higher dissolution rate, a stable negative feback scenario.

Double layers are also known to induce phase changeself-aggregating surfactants such that cylindrical and hespherical surface micelles can form. If the increased fi(E1.0) at a maximum can precipitate such a transition,micelles can shield the maximum from further dissolutioHowever, such micelles require surfactants that can inteamong themselves attractively to self-assemble. We findevidence of such surfactant features in the chemical spepresent. In particular, Butyl Cellusolve, the most likely cadidate, does not have a significant hydrophobic tail thatfavor a micelle aggregation to shield the tails from watThe fact that stripes precede all other patterns~Fig. 1! alsosuggests that if a surface phase transition is driving thetern formation process, the micelles need to break up inmittently along the ridge after they have aggregated to p

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duce the ridges. This is a very unlikely scenario andlength scales2 reported for such micelle structures are alsoorder of magnitude smaller than our patterns. We hencemiss surfactant aggregation as the cause of our patterns

However, adsorbates do not have to form surface miceto affect dissolution adversely. Adsorbed organic molecucan change the energy of the surface metal atoms onelectrode and cause small-scale~25 Å wavelength and 1atom in amplitude! atomic rearrangement.1 The sameadsorbate-metal interaction in the form of either electrostadsorption or chemisorption can also stabilize the surfatom and reduce its dissolution rate. The adsorbed orgmolecule can further physically or chemically shield the sface to reduce the transport rate of the ions involved indissolution reaction. If such shielding molecules adsorbthe maxima with higher electric field (E1.0) more readilythan the minima with negativeE1, the necessary positivfeedback is then available to provide a instability mecnism. This is more reasonable than aggregation of surfactsince adsorption and desorption occur more readily. Thshielding molecules can even be the same ones that pacsurface to produce the double layer. Enhanced packing, ifpacking species are not involved in the dissolution procecould reduce the dissolution rate by suppressing the ion flThey must, however, be polar or polarizable molecules sthat their adsorption rate is enhanced by the electric field

In our electrolyte, water, ethanol, and Butyl Cellusolare all possible candidates for the shielding molecule. Enol is, however, in abundance and they can saturate theface such that small variations inE would not produce asignificant variation in ethanol coverage. Water by itselfafter ionization is supplying the reactant for dissolution; ithence unlikely that enhanced water coverage can reducaction. It is then possible that it is the Butyl Cellusolve thaplaying the role of the shielding molecule. A second possiscenario is that water can appear within complexes ofvated ions that have been speculated to be the packingcies for the double layer.1 Such symmetric polar complexehave fewer degrees of freedom than water, ethanol, or BCellusolve and hence can reach the anode more readily.ter molecules in such complexes can also be chemicstable and hence do not participate in the dissolution retion. In this scenario, it can be the favorable packing of svated ion complexes at the maxima that reduces the disstion. Yet another possibility with a solvated-ion Helmhollayer is that ethanol, Butyl Cellusolve, or even water mecules can compete with the solvated ions for the metalface. If they succeed, the packing will have defects at lotions where these molecules are either electrostaticadsorbed or site specifically chemisorbed. Chemisorptionthese shielding molecules can stabilize surface Al atoagainst dissolution. The short hydrophobic tail of ethanol aButyl Cellusolve can also repel any solvated ion, suchOH2, and hence reduce its flux to the surface to participin the dissolution. Our suspicion is that Ethanol and/or BuCellusolve are the most likely shielding molecules that creduce ion flux and/or stabilize surface Al atoms. Howevlike identifying the packing species for the double layer, texact identity of the shielding molecule~s! and how it re-duces the local dissolution rate are a formidable kineproblem that we shall undertake in the future.

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It is quite apparent and reasonable, however, that sucmolecule~or molecules! does exist and we shall model iteffect without specifying its identity. To produce a lengscale selection process and explain the appearance ofregular structures from an irregular surface, we must aintroduce a short-wavelength cutoff mechanism to couwith the long-wavelength instability. While the enhancdissolution rate is a stabilizing mechanism, it occurs at lawavelengths and at the same order as the long-waveleshielding instability mechanism. The only short-wavelengcutoff possible is surface diffusion, as in other interfacpattern evolution.16 Hence the fractional surface coveragethe shielding molecule can be described by the quasisteequation

Ds~E!¹2u1ka~E!2kd~E!u50. ~7!

The adsorption rateka is a function of the electrolyte composition, but since the composition does not vary in time,do not consider it explicitly. It can also be a function of thdiffusion boundary layerdD , but we find such a dependencweak. We have also assumed the coverage is small suchthe adsorption rate is independent ofu. The desorption ratehowever, must be a function ofu. Expanding about the planar case again and imposing the natural orderka2ka

05ka1;u1 such that the variation in the coverageu is

proportional to the variation inka , we obtain to orderu1

Ds0¹2u11ka

12kd0u150. ~8!

Note that correction to the diffusivity and desorption raconstants due to deviation from the planar geometry isimportant. The adsorption rate of an organic moleculecompetition with another one has a field-dependent acttion energy that is proportional to the difference in their pmanent and induced dipole moments.17 Since all three polarmolecules, water, ethanol, and Butyl Cellusolve, haroughly the same permanent dipoles, we assume that thesorption rate of the shielding molecule is dependent onlythe difference in its polarizability relative to the others, trelative polarizability,

ka5ka* expS aE2

2kBTD , ~9!

wherea is the difference in effective polarizability.An effective a can be obtained by using the Deby

Langevin equationa5a01P2/3kBT, which contains boththe difference in permanent dipole momentsP and the dif-ference in polarizabilitiesa0. SinceP is about 3310231 J/~V/m! @the dipole moments of water and ethanol are 6310230 J/~V/m! and 5.64310230 J/~V/m!, respectively#, thedifference in the permanent dipoles does not play a role cpared to the difference in the polarizabilities, whicha0'4.7310240 J/~V/m! 2 for water and ethanol.

Expandingka aboutE0 to obtainka1 to O(E1

2), we obtain

ka15

aE0

kBTka

0E111

2

a

kBTS aE0

2

kBT11D ka

0E12 .

Substituting this into Eqs.~5! and ~8!, one obtains

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S Ds0

kd0 D ¹2u2u5B1F ~¹h!21dE¹2h

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by omitting terms higher order thanO(h2). This restricts ourtheory to small-amplitude, nearly sinusoidal shapes wweak nonlinear effects. Since we can obtain an accuratefile of our observed patterns, this seems like a good apprmation considering the measured small amplitude relativthe wavelength. This weakly nonlinear approximatigreatly simplifies our analysis and the same simplificationEq. ~10! describes how the coverage of shielding molecchanges due to variation of the interfacial position fromplanar geometry. The first term on the right-hand side incates that the coverage will decrease at the nodes due tdecreased field from interface stretching. The second tindicates that the coverage increases~decreases! at a maxi-mum ~minimum! due to curvature-enhanced~reduced! field.The last term is a higher-order curvature term that increathe coverage at both the maxima and the minima. Themensionless parameterB1 represents the sensitivity of thadsorption rate to field changes, a normalized activationergy,

B15Ka0~E0!

aE02

kBT, ~11!

where Ka05ka

0/kd0 is the equilibrium constant for the

adsorption-desorption process.It is now a matter of relating the coverage of the shield

molecule to the dissolution rate. The Tafel expression~5! fordissolution rate assumes a planar equilibrium coverageu0 .Hence the deviation dissolution rate from the planar cas

]h

]t5Kr

0u12Kr1 . ~12!

The first term is obviously due to molecular shielding, whthe second term is the change in the dissolution rate duthe unshielded field variations on a nonplanar anode in~6!. Combining Eq.~6! with the field-interface correlation~5!, one gets

Kr1

Kr0

52B2F ~¹h!21dE¹2h2B2

2~dE¹2h!2G , ~13!

where again we have neglected terms ofO(h3) and higherand the parameterB2 measures the sensitivity of unshieldedissolution to field changes: It is simply the normalized ativation energy

B25b rE0

2kBT. ~14!

For most dissolution reactions,B2 is a very small parameterThe estimated values for Al in most electrolytes ran0.05–0.15.18 Hence we shall safely omit the third term on th

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right-hand side of Eq.~13!. The high-order nonlinear mechanism arises solely from the shielding mechanism descriin Eq. ~10!.

Inserting the coverage-interface correlation~10! and thedissolution-interface correlation of Eq.~13! into the evolu-tion equation~14!, we obtain the final weakly nonlinear evolution equation

]h

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kd0 D ¹2

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0B1F2~12n!~¹h!22~12n!dE¹2h

2nS Ds0

kd0 D dE¹4h2nS Ds

0

kd0 D ¹2~¹h!2

1dE

2

4S aE0

2

kBT11D ~¹2h!2G , ~15!

wheren5B2 /B1 measures the ratio of the activation enegies for the two key processes: dissolution and adsorptUsing the scalings

t5n~Ds

0/kd0!

Kr0B1~12n!2dE

t,

~x,y!5S n

12n D 1/2S Ds0

kd0 D 1/2

~X,Y!,

h5dEH,

we obtain a dimensionless, weakly nonlinear evolution eqtion with only two parameters

Ht2~12n!

n¹2Ht52~¹H !22¹2H2¹4H2¹2~¹H !2

12j~¹2H !2, ~16!

where

j5dE

2

8nS kd

0

Ds0D S aE0

2

kBT11D ~17!

contains two ratios. The parameter (Ds0/kd

0)1/2 represents thediffusion length on a planar anode for the shielding molecover its lifetime. HencedE

2(kd0/Ds

0) represents the squareratio of the double-layer thickness to the diffusion lengMore accurately,j measures the coupling between diffusioand interface curvature and stretching as we shall demstrate subsequently. The parametern5B2 /B1, as mentionedearlier, represents the ratio of the activation energies ofstabilizing dissolution rate to that of the destabilizing adsotion. The evolution equation~16! contains terms similar tothose in the Kardar-Parisi-Zhang equation16 ~and its exten-sion to include surface diffusion19! for interfacial evolutionof epitaxial films. However, field variation~due to interfacialcurvature and stretching! affects the activation energies oboth the adsorption of shielding molecules and the dissotion rate, with the former having the higher sensitivityensure an instability mechanism. As a result, we obtaiunique ¹2Ht due to the coupling between diffusion an

a-roertioiththt-

thatth

icaam

tethanth

inain

imndle

teles

ataiat

ttess

e.oest

ulWn

tith

ee

rmics,in-ialigidndthe

erctsn-

onif

hann

-

drierinear

nsn ine isionrep-andtest

aceanyar

paceove


shielding. We also obtain higher-order terms such¹2(¹H)2 and (¹2H)2 from the higher-order field dependence of adsorption and dissolution processes and fhigher-order diffusion-adsorption coupling. Such highorder terms are necessary to allow a quantitative predicof the various patterns appearing at different voltage. Wout higher-order terms, we would only be able to predictcritical voltage for pattern formation, but not how the paterns depend on the voltage.

One observation can already be made. Since Eq.~16! isinvariant to (X,Y)°(2X,2Y), we expect all one-dimensional stripe patterns to be symmetric aboutmaxima and each hill in the two-dimensional hexagonal pterns to be nearly radially symmetric. This arises becausestretching at the nodes is independent of the direction ofslope. All slopes of the same amplitude would have identfield deviation from the mean and hence dissolve at the srate. If the dissolution mechanism dominates (n.1), thenodes will grow and if the shielding mechanism domina(n,1), the nodes will dissolve. We hence expect bothstripe and hill patterns that appear for the latter case to hsharper crests due to this enhanced symmetric dissolutiothe nodes. Another significant observation concernshigher-order ¹2(¹H)2 and (¹2H)2 terms that dictate asecond-order transition. In particular, they determwhether stripes will break into hexagonal hills or remainstripes. Since the evolution of well-developed stripesvolves dissolution of their ridges whereHX vanishes and(HY)2 is of the same magnitude at the disturbance maxand minima along the ridge~assuming the ridges line up ithe Y direction!, the (¹H)2 term does not participate anonly the two higher-order nonlinear terms are involved. Alinear terms are also ineffective for these developed stripIn the evolution equation,2¹2(¹H)2 produces a pertinen22(¹2H)2 term at any maximum to compete with th2j(¹2H)2 term. The first promotes ridge dissolution, whithe second favors ridge growth. One hence expects stripedominate at largej and expects them to break up into hillssmallerj. A simple physical view of this is to notice thatridge would only dissolve at certain positions if preferentdissolution at the nodes breaks the symmetry betweenmaxima and the minima along the ridge such that the lahas a lower curvature, smaller adsorption, and higher dilution. This is favored by the2¹2(¹H)2 term when adsorp-tion is more field sensitive than dissolution: The reducfield at the stretched nodes actually promotes dissolution

The large-time dynamics governing reorientationstripes and self-assembly of hills is also governed by thtwo terms. From the previous arguments, we expect hillsconnect into irregular stripes at largej and stripes to breakup into random hills at smallj. It is only whenj is close tounity, when these two opposing forces balance, that regstripe and hexagonal hill patterns can be established.hence expect the windows of perfectly regular stripe ahexagonal patterns to appear nearj51. The (¹2H)2 termcomes from the second-order field dependence in dissoluand adsorption and is explicitly obtained by squaringcurvature term¹2h in the perturbation fieldE1 in Eq. ~5! forboth dissolution~6! and adsorption~10!. The¹2(¹H)2 term,on the other hand, comes from a coupled effect betwdiffusion and stretching@(¹h)2 in Eq. ~5!# on the induced

s

m-n-e

et-heele

seveofe

es-

a

ls.

to

lher

o-

d

feo

are

d

one

n

coverage variation of the shielding molecule. The first teappears in any second-order curvature-driven dynamwhile the second term is unique to interfacial dynamicsvolving a coupling between surface diffusion and interfacstretching. Both terms render the developed patterns rsuch that the pattern dynamics do not evolve indefinitely aseem to reach steady patterns after some transients. Incurrent context, this coupling is offered by the double-layelectric field that is affected by stretching and in turn affethe diffusing shielding molecule. It is a unique physical iterfacial dynamics offered by electropolishing.

III. ANALYSIS AND SIMULATION

It is more apparent from the unnormalized evolutiequation~15! that the planar geometry is linearly unstablethe adsorption rate is more sensitive to the electric field tthe dissolution raten,1. For the linearized scaled versio~16!, a normal modeH;exp@lt1i(kX1 lY)# expansionyields the growth rate

l5q22q4

11~12n!

nq2

, ~18!

whereq25k21 l 2 is the generalized wave number. All Fourier modes with wave numbers (k,l ) within the circle

k21 l 251 ~19!

are hence unstable with the maximum growth rate at

qm2 5km

2 1 l m2 5

An

11An. ~20!

Our bifurcation analysis will first focus on the stripe anhexagonal patterns, which, as shown from the spatial Foutransform in Fig. 3, consist of discrete Fourier modes withthe unstable circle. Since stripes and hexagonal hills appwithin a small window of the applied voltage while patteroutside resemble irregular stripes and hexagons, as seeFig. 4, the selected pattern over the entire range of voltagexpected to be determined by weakly nonlinear interactbetween these two basic patterns. Both patterns can beresented by the discrete modes within the unstable bshown in Fig. 5, where we have chosen six discrete fasgrowing modes along the circle described by Eq.~20!. Thesymmetry of these discrete points in the wave-number spreflects the hexagonal symmetry. The stripes consist oftwo of the six, while the hexagonal hills result from a linecombination of all six modes~or three complex modes!. Thissignature pattern of the hexagon is also seen in Fig. 3.

We hence expandH(X,Y,t) of Eq. ~16! by these threeactive ~master! complex modes, their passive~stable! over-tone, and any other passive modes in the wave-number sof Fig. 5 that satisfies the resonance condition with the abmodes

kp6kq6kr50, ~21!

-e

he

muicdeThn

thfo

u

e

tam

an

lm-

n-

foren-

e

es

tive

chsses

nt

viahi; th


wherek5(k,l ) is the wave-number vector. A total of 9 complex modes, three active and six passive, are used as seFig. 5. The expansion then becomes

H~r ,t !5(j 51

3

$Wjexp~ ik j•r !1Vjexp~2ik j•r !

1Vjexp@ i ~k j2k j 11!•r #%, ~22!

wherer5(X,Y), uk j u5qm of ~20!, Wj , Vj , and Vj are thecomplex amplitudes of the unstable master modes, tovertones and the other resonant modes, respectively.

By substituting Eq.~22! into the interfacial evolutionequation~16! and collecting terms corresponding to the saFourier mode, we then obtain nine coupled amplitude eqtions for all the modes. However, at large time, the dynamis dominated by the active modes with the passive moslaving to the dominant dynamics of these active modes.‘‘adiabatic invariance’’ describing this slaving principle cabe derived explicitly using center manifold projection.20–22

We omit the tedious algebra here and simply presentresulting set of three couple complex amplitude equationsthe three active modes. One of them is

]W1

]t5

16

81W11g1W2* W3* 2g2uW1u2W1

2g3~ uW2u21uW3u2!W1 . ~23!

Due to the hexagonal symmetry, the other amplitude eqtions for W2 andW3 can be obtained from Eq.~23! by ob-vious permutation of indices.

The coupling coefficientsg j are functions ofn andj. Asseen earlier, the parametern, representing the ratio of thsensitivity of the dissolution and adsorption processeselectric-field variation, must be less than unity for patternsappear. We shall show in Sec. IV that, for reasonable pareter values for the electropolishing of Al, the values ofnremain relatively constant at around 0.5 for the importvoltage range ofUP ~20 V, 80 V!. This observation allows

FIG. 5. Modes used in the representation of the relative detion H for nonlinear analysis of pattern selection. The area witthe neutral stability circle represents the region of active modesother area is the region of passive modes.

n in

ir

ea-sse

er

a-

too

-

t

us to fix n and vary onlyj in our analysis and numericasimulations. In fact, since the maximum-growing wave nuber ~20! and its growth rate are functions only ofn1/2, weshall choosen50.64 for convenience. The actual depedence ofn on E0 is

n~E0!5b r

2aE0Ka0~E0!

and hence this simplification may not be permissibleother system with a more sensitive adsorption rate depdence on the field.

With n50.64, the nonlinear terms (¹H)2, ¹2(¹H)2, and(¹2H)2 of the evolution equation yield the following simplj dependence for the coupling coefficientsg j :

g154

81~16j217!,

which represents quadratic coupling between active mod

g258

891~16j25!~728j!,

g357

621~16j29!~15216j!,

which represent the degree of cubic coupling between acmodes.

After transformation to the polar coordinate system suthat Wj[Rjexp(iwj), Eq. ~19! yields a system of equationfor the real part of the amplitudes and the sum of the phaF5w11w21w3 for the active modes:

]R1

]t5

16

81R11g1R2R3cosF2g2R1

32g3~R221R3

2!R1 ,

]R2

]t5

16


32g3~R321R1

2!R2 ,

~24!

]R3

]t5

16


32g3~R121R2

2!R3 ,

]F

]t52g1

R12R2

21R22R3

21R32R1

2

R1R2R3

sinF.

Thus the phaseF relaxes monotonically to 0 wheng1.0and relaxes monotonically top wheng1,0. In both cases,g1cosF5ug1u and only the amplitude dynamics is importaat long times,

]R1

]t5

16

81R11ug1uR2R32g2R1

32g3~R221R3

2!R1 ,

]R2

]t5

16

81R21ug1uR3R12g2R2

32g3~R321R1

2!R2 ,

]R3

]t5

16

81R31ug1uR1R22g2R3

32g3~R121R2

2!R3 .

~25!

-ne


This system of equations has the following four types of steady-state solutions:~i! stripes of amplitude

R15A 22

~16j25!~728j!, R250, R350, ~26!

or any circular permutation thereof;~ii ! structures with hexagonal symmetryR15R25R35R, where

R5253u16j217u1A251 137 1191173 019 616j2127 139 584j2

234 4051102 688j270 912j2; ~27!

o

sul

ar

a.f

wroF

elitthunneonorsroblapal

at

eso

oiillalinrin

p of

-s.

ght

noed toble,ri--ndct.rage

eddlepear

~iii ! mixed states in which two of three amplitudes are nequal to zero and one is equal to zero; and~iv! flat stateR15R25R350.

Solutions~ii ! and~iv! are unstable for any value ofj. Therelative stability of solutions~i! and ~ii ! can be determinedby linearization of the system of equations~25! about boththe stripes and hexagonal hill solutions. The eigenvaluethe resulting Jacobian were defined analytically. The resshow that the stripe solution can exist forjP~0.32,0.88! andthe hexagon solution can exist forjP~0.53,0.92!. However,the stripe solution is stable only forjP~0.68,0.88! and thehexagonal solution is stable forjP~0.57,0.88!. That regularpatterns do not exist forj,0.57 andj.0.88 is consistentwith our experimental data.~The parameterj will be shownto decrease with voltage.! That the regular patterns appeonly within a small window nearj51 is also consistent withour earlier speculation. However, that stable hexagonsstripes coexist for 0.68,j,0.88 requires further analysisWe computed the eigenvectors of all the steady states0.57,j,0.68 and 0.68,j,0.88. These eigenvectors allous to decipher the direction of the trajectory along heteclinic connections between the steady states. As seen in6, we find all initial conditions near the origin~flat state! toevolve first towards a stripe pattern along the unstable eigvector of the origin in both regions, regardless of the stabiof the stripe. In the first region, the stripe is unstable andtrajectory comes close to it but then departs along itsstable eigenvector towards the stable hexagon. Thismiss corresponds to how the stripes break up into hexaghill patterns. In the second region, however, the trajectstays with the stable stripe pattern and never approachehexagon despite the latter’s stability. Although the heteclinic connections are actually stable and unstamanifolds,20 the eigenvectors yield simple and accurateproximations of them. They indicate that if the anode ismost uniformly flat initially, stripes will be selected forjP(0.68,0.88), while hexagons will be the final selected stfor jP(0.57,0.68).

Actually, the above bifurcation analysis only indicatthat regular stripe patterns are expected within the windjP(0.68,0.88) and the hexagonal patterns withinjP(0.57,0.68). Since we have subjected the patterns to nfrom only certain discrete modes, it is not clear if they wremain stable in the presence of more wideband noise. Itdoes not indicate which patterns will exist outside the wdow jP(0.57,0.88), although they certainly cannot be peodic states. We answer both questions by carrying out americal simulation of Eq.~16!. We use a finite-difference

t

ofts

nd

or

-ig.

n-ye-aralythe-e--

e

w

se

so--u-

numerical scheme with a grid size of 0.589 and a time steDt51023. Computations are done forj50.8842,j50.7664,j50.6235, andj50.4733.~These strange numbers will be shown to correspond to convenient voltage!The computation domain is large enough to contain eiwavelengths of the fastest modeqm . Small random devia-tions were adopted as initial conditions.

It is quite clear from the snapshots shown in Fig. 7 thatregular structure appears forj50.8842. Instead, the averagseparation tends to increase, while contorted stripes tenpersist. Although regularly spaced stripes are not possiisolated hills still tend to connect and form randomly oented stripes. Forj50.7664 in Fig. 8, the hills also reconnect to form stripes, but the latter now strive to stretch aform regular patterns by expelling any remaining defeThere is also no coarsening trend that increases the ave

FIG. 6. Phase-space trajectories for~a! 0.57,j,0.68 and~b!0.68,j,0.88. The origin is the unstable planar interface, fixpoints on the axes are stripe patterns, the fixed point in the midis the hexagonal patterns, and unstable mixed patterns also apin ~b!.


FIG. 7. Numerical simulations of the electrode surface evolution from random initial conditions atj50.8842~40 V!.




arec

eemow

is

a

thdyo-1ro

nn-ythomro

ghennm

the

thatthe

a-at-

theo-ump-thefini-

n-

p-

at

length scale. Such late-stage stretching dynamics towregular stripe patterns is extremely slow and some defremain att52000. Forj50.6235~Fig. 9! when we expectregular hexagons, the initial dots do not connect and insttend to form randomly symmetric hills. These hills also seto strive to form regular hexagonal patterns with a very sldynamics. Finally, forj50.4733~Fig. 10!, the disconnecteddots are much more irregular, while stripe patterns are ming entirely.

While we clearly see tendencies to form isolated hillssmallj and longer stripes at largej, the formation of regularstripes and hexagons at the intermediate windows inj, assuggested by the bifurcation analysis, is not clear fromabove simulations with random initial conditions. To remethis, we accelerate the long-time dynamics by using randinitial conditions only within a narrow strip while the remaining domain is initially smooth. As is evident in Figs. 1and 12, regular straight stripe patterns propagate away fthis strip. These stripes survive forj50.7664 and break upinto perfect hexagonal hills forj50.6235. Such a selectioof regular patterns from a propagating front has been seeother systems.23 Although we have exploited this phenomenon here to expedite the pattern formation process, it mafact happen in the actual electropolishing process ifdominant noise source for the microscopic dynamics is frthe intersection of crystal defects with the surface that pduce straight boundaries~slip lines, twin boundaries, etc.!.Such slip steps distort the double-layer thickness and trignonuniform dissolution along the straight domain wall. Tresulting patterns will then resemble those in Figs. 11 a12. Finally, cross-section profiles of the regular stripes apatterns from our simulations in Figs. 11 and 12 are co

dsts

ad

s-

t

e

m

m

in

ine

-

er

dd-

pared in Fig. 13 to the measured ones obtained fromAFM images in Figs. 2~b! and 2~c!. Careful analysis of theobtained profiles shows the presence of the sharp crestswe have attributed earlier to the enhanced dissolution atnodes forn,1.

IV. COMPARISON WITH EXPERIMENTAL DATA

To actually obtain quantitative prediction for the mesured wavelength and voltage windows for the various pterns in Fig. 4, we need to relatej and n to the appliedvoltage with some reasonable approximations. Givencomplexity of the adsorption-desorption and diffusion prcesses and the lack of knowledge about them, some asstions are necessary, but we shall be careful to invoke onlymost reasonable and least crucial ones. From these detions

n~E0!5b r

2aE0Ka0~E0!

,

j5dE

2

8nS kd

0

Ds0D S aE0

2

kBT11D , ~28!

whereKa0(E0) is the adsorption-desorption equilibrium co

stant atE0 defined in Eq.~11! and the reference diffusionlength scale of the shielding moleculel D5(Ds

0/kd0)1/2 is also

a function ofE0 .We first relate the planar electric field to the voltage a

plied to the electropolishing cellU by assuming thatDcH isnegligible and there is a double layer of equal thicknessthe cathode such that



FIG. 11. Numerical simulations of the electrode surface evolution from a domain boundary atj50.7664~50 V!.


FIG. 12. Numerical simulations of the electrode surface evolution from a domain boundary atj50.6235~60 V!.

r incrfatioocaEathth-

leiff

se

l-

ofnt

lyn-

oee

w-

E05U

2dE

. ~29!

This should be a good approximation with only some errothe factor 2, which does not affect the voltage dependeWe then assume that the adsorption, desorption, and sudiffusion are activated processes with a common transistate. This is not unreasonable if the diffusing shielding mecule has to hop through the bulk to reach a different lotion on the surface. We shall also assume, as before in~9!, that this transition state of the shielding molecule hnearly the same dipole moment as other polar moleculescompete for the surface such that only the energy ofinduced dipole momentaE0

2 appears in the activation energy. However, unlikeka(E0) in Eq. ~9!, we expectkd

0(E0)and Ds

0(E0) to have activation energies that scale as2E02

instead of1E02 . From Fig. 4, however, the length sca

(Ds0/kd

0)1/2 must increase with voltage. This will occur onlythe activation energy ofkd

0 is larger in magnitude than that oDs

0 . This then yields

DS0

kd0

5DS*

kd*expS b

aE02

kBTD 5

DS*

kd*expS b

aU2

4dE2kBT

D , ~30!

whereb is a positive coefficient of unit order. We also uEq. ~9! to rewriten as

n5b rdE

aUKa*expS 2

aU2

8dE2kBT

D . ~31!

ne.cenl--q.sate

The difference in polarizability between the shielding moecule and watera was already estimated to be 4.7310240

J/~V/m! 2 in Eq. ~9!. The coefficientb is a unit order quantityand will be set directly to one without significant lossinformation. FordE , we use a reasonable and convenievalue of 10 nm for dilute electrolytes. The coefficientb r formost field-assisted metal dissolution is estimated18 to be3.7310231 J/~V/m!. These assignments would completespecify n in Eq. ~31! if the fieldless adsorption-desorptioequilibrium constantKa* is known. Specification of the fieldless diffusion length (DS* /kd* )1/2 in addition to Ka* wouldalso specifyj. Hencen and j are now dependent on twparametersKa* andDS* /kd* . We note that the dissolution ratKr

0 is not required if we are not concerned with the timscale. In any case,Kr

0 is measurable from the current.In dimensional variables, the wavelength of fastest gro

ing mode with dimensionless wave number

qm5A An

11An

of Eq. ~20! is

l ~U !52pADS0

kd0A An

12An

52pA An

12AnADS*

kd*expS aU2

4dE2kBT

D . ~32!

the

a

foth

r

hgesi

er

vsa

ns

reple,

-

wsrom

heyis athatraterns

o

nlyin-

ran-l orrn

rn

l


Since the wavelength in the limit of zero voltagel (0)555nm can be estimated by extrapolating the data in Fig. 4,provides an additional condition and reduces the degrefreedom to a single oneKa* .

We hence adjustKa* until Eq. ~32! can satisfactorily fitthe wavelength data. Due to the scatter in Eq.~32!, a widerange ofKa* seems to yield a satisfactory fit. We chooseconvenient value ofKa* 51/3. While we can find no literaturevalue for the adsorption-desorption equilibrium constantour molecules on Al, this value is quite comparable toreported value of ethanol on silica gel.24

With the given value ofKa* , n(U) in Eq. ~31! is com-pletely specified and is plotted in Fig. 14. Note that theexists a threshold voltage atUc1522.7 V for pattern forma-tion whenn51. We are unable to polish our anode at suclow voltage and hence could not verify this onset voltaHowever, the experimental data for high voltages is content with the value of the upper critical voltageUc2567.5 V(n51/4), where the resonant modesVj lose their stabilityand hence the patterns become nonregular. We obshowever, thatn remains in the range~0.3, 0.8! for the perti-nent range from 30 V to 60 V. We hence choose the connient value ofn50.64 indicated in Sec. III. The final resultfor wavelength and voltage windows are off by no more th10% by changing this value by 0.1.

FIG. 13. Comparison of the profiles for~a! and ~b! the stripepattern with ~c! and ~d! profiles for the regular hexagon patteobtained from~a! and ~c! AFM data and~b! and ~d! numericalsimulations.

isof

re

e

a.

s-

ve,

e-

n

This convenientn value then offers compact expressiofor all the pertinent quantities

ADS* /kd* 54.33 nm, ~33!

l ~U !54pADS* /kd* expS aU2

4dE2kBT

D , ~34!

j5~1.0413.0731024U2!exp~23.4331024U2!,~35!

whereU is the applied voltage~in volts!, a is estimated inEq. ~9!, and dE is taken to be 10 nm. These estimates aquite consistent with typical measured values. For examrecent measurements forn alkanes on Ru~001! in vacuumyield an extrapolated value ofADS* /kd* 54.84 nm formethane.25 The fieldless diffusion length of 4.33 nm, corresponding to the measured wavelengthl (0) of 55 nm in Fig.4, is hence reasonable.

The curve in Fig. 4 is drawn from Eq.~34!, which wasobtained by adjusting only a single model parameterKa* . Amore stringent test is the prediction of the voltage windowhere regular stripes and hexagonal hills are predicted. FEq. ~35!, sincej decreases with increasingU, the windowfor stripes (0.68,j,0.88) now corresponds to 40.4 V,U,56.2 V, while the window for hexagonal hills(0.57,j,0.68) now lies in the range 56.2 V,U,63.4 V.These predicted windows are also shown in Fig. 4 and tare in excellent agreement with the observed patterns. Ittestimony to the robustness and accuracy of the modelone single adjustable parameter can provide such accupredictions of not only the length scale but also the patteas functions of the applied voltage. Thej values~0.8842,0.7664, 0.6235, and 0.4733! in the simulations correspond twhole-figure voltages~40, 50, 60, and 70! in the variouspattern windows. As seen before, the simulation not oproduces periodic stripes and hexagons in the proper wdows but also shows random stripes at low voltage anddom hills at high voltage. We have not attempted to modemeasureKr

0(E0) to also specify the actual times each patte

FIG. 14. Dependence of the parametern on the applied voltageU; the lower critical voltageUc1522.7 V and the upper criticavoltageUc2567.5 V.

e

eliotehea

.

nactueeit

andate-mt

owr,fononorltere

tdkor-o

hel-it

lf

r

hat


appears in a particular experiment. Nevertheless, our msured current value at 60 V indicatesKr

0 ~60 V!;0.4mm/sec.This translates the dimensionless timet5500 for the middlesnapshot in the bottom row in Figs. 9–12 to approximatt560 sec, which is consistent with the observed transittimes in Fig. 1. A more detailed correlation is not attempdue to the ambiguity about the actual initial condition. Twavelength and voltage-dependent pattern selectionhowever, dependent only on the activation energy ofKr

0 andnot the actual dissolution rate. Knowledge aboutj and ncompletely specifies the evolution, but not the time scale

V. DISCUSSION

The most attractive application of the periodic stripe ahexagonal hill patterns, like self-assembly of regular surfmicelle patterns by surfactants, is to use them to manufacnanoelectronic functional devices. However, Al itself donot serve this purpose and one can only hope to use thpatterns as templates for other materials. For this reason,better to develop patterns on Zr since anodization ofmetal surface with the hexagonal hills has been observeproduce a porous oxide film with a highly regimented arrof deep pores26 and zirconia film is a much better substrathan porous alumina.27 Our preliminary electropolishing experiments of Zr coupons shown in Fig. 15 show the sapattern dependence on voltage as Al, but we have yepinpoint the voltage window for regular hexagons. Ifn isclose to zero, it is quite conceivable that such a windcannot be reached with the same electrolyte. Howevedoes look like the same coupling between the diffusion oshielding molecule with interfacial curvature and tensi~stretching! is a general mechanism for pattern formatiduring electropolishing. We intend to use our current theto design the most desirable patterns for any metal by aing the electrolyte composition, which changes the adsotion, desorption, and diffusion rates and the Debye thickndE and by adjusting the temperature and voltage.

ACKNOWLEDGMENTS

Several colleagues in our laboratory have contributedthis effort. D.-F. Yue28 first observed the patterns reportehere and we benefited from discussions with O. OrlichenA. Indeikina, and G. Banerjee. K. Krischer from Fritz-HabeInstitut der Max-Planck-Gessellschaft pointed out the rolethe Helmholtz layer. H.-C.C. was partially supported by tNSF, Grant No. CTS95-22277. V.V.Y. and A.E.M. acknowedge financial support from the Midwest SuperconductivConsortium, MISCON ~DOE Contract No. DE-FG02-90ER45427!. We are also grateful to Professor E. E. Wofor the use of an AFM purchased with NSF funds.

APPENDIX: LONG-WAVELENGTH EXPANSIONOF THE POISSON EQUATION

Since the characteristic interfacial amplitude isdE whilethe characteristic wavelengthl is much longer thandE , weuse the scaling

~ x,y!5~x,y!/l ,

a-

ynd

re,

deresAlisyto

y

eto

ita

yr-

p-ss

o

,

f

y

z5z/dE ,

h5h/dE

to normalize Eq.~1! to

]2c

] z21«2S ]2c

] x 21

]2c

] y 2D 5c,

cu z5 h5cs ,

cu z→`50.

Expanding c in the long-wavelength paramete«5dE /l !1, c'c01«2c1, we obtain forO(«0)

]2c0

] z25c0,

c0u z5 h5cs ,

c0u z→`50.

FIG. 15. AFM images of the surface patterns on zirconium twas electropolished at~a! 60 V and~b! 70 V.

io

hie

,

led

md


Note that thisc0 is distinct from c0 in the text, which isstrictly for the planar geometry. This leading-order equatcan be solved to yield

c05csexp~ h2 z !,

where h5 h ( x , y ).Continue to the next nontrivialO(«2); we obtain

]2c1

] z252¹2c01c1,

c1u z5 h50,

c1u z→`50,

where

¹2c05S ]2

]x21

]2

] y 2D c0

5csexp~ h2 z !@~¹h!21¹2 h # .

The first term in the square brackets denotes a line stretcterm that arises when the Poisson equation is perturbwhen Laplacian is operated on the leading-order fieldc0 thathas an exponential dependence on the interfacial positionh .

s

n

ngd,

The next-order field termc1 can also be readily obtained

c15cs

2~ z2 h !exp~ h22 z !@~¹h!21¹2 h #.

Hence the field is given by

E'2]c

] zU

z5 h

'2]c0

] zU

z5 h

2«2]c1

] zU

z5 h

5 E01 E1

and hence

E05cs ,

E152«2cs

2@~¹h!21¹2 h #.

Since l is unspecified, we need to return to the unscavariables to yield

E152E0

2@~¹h!21dE¹2h#,

which is the deviation field due to interfacial variation frothe flat geometry in Eq.~5! with the signature stretching ancurvature terms.

e

J.

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c.

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16M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett.56, 889~1986!.

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Pattern formation during electropolishing