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jPergamon

Elrcr rochmca Acm Vol 40 No 4 393 401p 1995

Copyright < 1995 Elsewermnce Ltd.

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reserved

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MODELING OF IMPEDANCE MECHANISMS IN

ELECTROPOLISHING

MICHAEL MATLOSZ

Laboratoire des sciences du genie chimique CNRS, Ecole nationale suparieure des industries chimiques,

1, rue Grandville B. P. 451, F-54001 Nancy, France

(Received 21 April 1994)

Abstract-Recent theoretical results concerning mechanisms for mass-transport-limited electrochemical

polishing are summarized and discussed. The underlying physical bases for both salt-film and acceptor

models are presented with particular emphasis on the differences

in their alternating-current impedance

behavior. Direct comparison of the dependence on operating conditions of characteristic features of the

impedance diagrams in the high-frequency range provides a clear experimental basis to distinguish

among the various mechanisms that can lead to the polishing phenomenon in specific cases.

Key words: mathematical modeling, alternating current impedance, electropolishing, anodic dissolution,

kinetic mechanism.

NOMENCL TURE

Greek symbols

Tafel constant

bulk acceptor concentration/mol cmS3

characteristic

capacity of the high-

frequency loop/F cm- ’

double-layer

capacity of the metal/

electrolyte interface/F cm- ’

dif n coefficient of the acceptor species/

Faraday constant/96 487 C molt ’

current density/A cm- *

pre-exponential

factor (solid-state

conduction), or exchange-current density

(Tafel kinetics)/A cm _ 2

steady-state current density/A cm - ’

rate constant for cation solvation by the

acceptor/s

’

number of acceptor molecules needed to

solvate a single cationic species

high-frequency impedance limit/ohm-cm2

diameter of the high-frequency impedance

loop/ohm-cm*

resistance of the porous film (per unit

a

Y

6

parameter in the solid-state conduction

expression/cm V- ’

dimensionless parameter in the model

developed in reference[6]

diffusion-layer thickness/cm

porous-film porosity

electrical permittivity of the compact layer/

Fcm-’

surface coverage of adsorbed cations

thickness of the compact film/cm

thickness of the porous film/cm

potential at the inner limit of the porous

layer or potential at the metal/electrolyte

interface/V

potential at the outer limit of the porous

layer/V

potential at the reference

electrode

position/V

angular frequency/rad s-

’

INTRODUCTION

length)/ohm-cm

Electropolishing is a surface finishing process

ohmic resistance of the electrolyte/ohm-

cm2

based on anodic dissolution of a metal or alloy in an

appropriately chosen electrolyte. Applications of the

ohmic resistance for a primary current

distribution/ohm-cm2

technique are numerous and range from the pol-

ohmic resistance for a uniform current

ishing of stainless steel cutlery to the preparation of

distribution/ohm-cm2

samples for transmission electron microscopy. The

polishing phenomenon is characterized by the elimi-

time/s

nation of micro-roughness (leveling) and the absence

c/c,, dimensionless surface concentration

of crystallographic and grain-boundary attack

of the acceptor

(brightening) and results in the production of

electrode potential/V

smooth, bright surfaces. A complete summary and

metal cation valence

discussion of the scientific literature concerning elec-

393

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394

M.

MATLOSZ

tropolishing from its patented publication by

Jacquet in 1930[1-33 up to and including research

results from the mid-1980s can be found in the

review by Landolt[4].

Mass-transport limitations for anodic dissolution

are generally believed to be responsible for electro-

polishing, and this view is supported by the observ-

ation in numerous experimental systems of polishing

for anodic dissolution along a limiting-current

plateau. Leveling behavior for anodic dissolution at

the limiting current can be interpreted as the prefer-

ential dissolution of protrusions on the order of the

diffusion layer due to their greater accessibility for

diffusive transport. Brightening can also be inter-

preted as a result of mass-transport control, but on a

smaller scale where diffusion is essentially isotropic

and independent of the crystallographic orientation

and grain structure of the metallic surface.

Whether or not anodic dissolution is mass-

transport controlled depends on the experimental

system. Unlike cathodic limiting-current plateaux in

electrodeposition, which are the inevitable result of

the depletion of metal cations in the diffusion layer

near the electrode surface, anodic limiting-current

plateaus do not necessarily appear with increasing

overpotential in all cases since the surface concentra-

tion of dissolving metal ions will generally rise with

increasing anodic current. Mass-transport-limited

anodic dissolution requires therefore the presence of

an additional mechanistic step, such as the precipi-

tation of a salt film (which limits the surface concen-

tration to the saturation value of the metal cations)

or a diffusion limitation for transport of an acceptor

molecule necessary for solvation.

It is of consideral scientific and technological

interest to be able to determine clearly which of the

possible mechanisms is at work in a given polishing

system in order to understand the chemistry

involved and the role of the various operating

parameters. For this purpose, several studies of salt-

film and acceptor systems have been undertaken in

the past decade, including theoretical work on the

shapes and sizes of the characteristic loops of the

impedance diagrams measured along the hmiting-

current plateau. The results of two of these studies,

summarized here, yield considerable insight into the

transport mechanisms involved and provide a solid

basis for determining the likely mechanism in a given

experimental system.

The two models chosen for discussion have been

studied theoretically in some detail over the past

several years and represent special limiting cases of

the salt-film and acceptor approaches. The first, the

duplex salt-film model proposed by Grimm et aI.[S],

attempts to characterize the role of compact and

porous layers in the frequency response of complex

precipitate films. The second, the adsorbate-acceptor

mechanism proposed by Matlosz et a/.[6], examines

the role of adsorbed intermediates and acceptor-

molecule transport in the behavior of polishing

systems in the absence of films. Both types of mecha-

nism have been observed and studied in experimen-

tal polishing systems.

The objective of this article is to present the

underlying physical bases of the models and the

principal results of the theoretical analyses in order

to obtain a clear pictue of the differences and simi-

larities in the approaches. For this reason, quaht-

ative arguments will be favored and mathematical

developments limited to the greatest extent possible.

In the interest of clarity, some notation has been

modified slightly with respect to the original refer-

ences, and model behavior has been simplified some-

what in certain cases. In particular, the intermediate

surface reaction step discussed in the acceptor model

in [6], and which is not strictly necessary for pol-

ishing, has been eliminated in the present discussion

(an approach equivalent to the case of y = 0 in[6]).

More detailed mathematical treatments and addi-

tional theoretical support for the conclusions report-

ed here can be found in[5] and[6].

THE DUPLEX SALT-FILM MODEL

A physical picture of the duplex salt-film model is

represented schematically in Fig. 1. The origin of the

anodic limiting current is the salt-film precipitate

DUPLEX SALT FILM MODEL

oxidation

Fe + Fe+* +

2 e

precipitation

dissolution

‘r

r’

w

(diffusion) Cl -

sat

L

Reference

k:

- Cbulk

compact

dielectric

film

diffusion

layer

Fig. 1. Schematic illustration of the duplex salt-film model.

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Modeling of impedance mechanisms

395

which fixes the metal cation concentration at its

saturation value at the salt-film/electrolye interface.

The rate of transport of the cations across the diffu-

sion layer into the bulk electrolyte limits the anodic

dissolution rate. The precipitate itself is composed of

two regions. In the porous-film region, the pores of

the precipitate are filled with electrolyte solution (at

the saturation concentration) and the mobile charge

carriers (anions and cations) transport the current by

migration in the electric field in the pores. In order

to explain the high electric resistance of the precipi-

tate films, the porous layers are generally attributed

a rather low porosity and taken to be several

microns thick. In fact, however, it is not possible to

distinguish clearly whether an increase in film resist-

ance is due to an increase or decrease in porous film

resistance is due soley to a change in film thickness.

In the compact-film region, the precipitate forms a

solid dielectric barrier through which the cations are

transported by solid-state ionic conduction in the

presence of a much higher electric field. Due to the

lower mobility of the ions for solid-state transport,

compact films are generally considered to have

thicknesses on the order of 10 nm, much thinner than

corresponding porous layers. Limiting cases of inter-

est for the duplex model arise if one or the other of

the two regions can be neglected. In such cases, one

may speak of a porous-film model or a compact-film

model, and most of the results presented here will be

restricted to one or the other of these limiting cases.

Although the duplex model is viewed here as two

distinct layers, it is possible to consider an alterna-

tive (and perhaps more realistic) interpretation as a

single film of variable properties with a gradual

change from a region of high porosity to a region of

low porosity with simultaneous solid-state and

liquid-state transport. Both interpretations result in

essentially the same behavior, however, with the

presentation as two separate regions offering the

advantage of clearly separating solid-state and

liquid-state effects. For this reason, the description

below will be restricted to the two-film approach.

Discussion of the physical basis of the model is

somewhat simplified if one considers the dissolution

of a single metal into a binary electrolyte. Following

Grimm et 0/.[5], the presentation here will focus on

the dissolution of Fe into an electrolyte of FeCI,.

Transport processes are taking place in three distinct

layers: the electrolyte diffusion layer, the porous-film

layer, and the compact-film layer. In the electrolyte

diffusion layer, metal cations Fe’* are transported

outward toward the bulk electrolyte by both diffu-

sion in the concentration field and migration in the

electric held, whereas the counterions Cl --, whose net

steady-state fux must be zero, are drawn simulta-

neously outward by diffusion and inward by migra-

tion. The migration of both ions contributes to the

electrolyte conductivity and to net current flow. The

resulting steady-state concentration profile is consis-

tent with electroneutrality and shows a rise in FeCI,

concentration from its bulk value up to a saturation

concentration at the salt-flim/electrolyte interface.

Transport processes in the binary electrolyte in the

pores of the outer salt film are somewhat different

from those in the liquid diffusion layer. Contrary to

the diffusion layer, the concentration in the pores of

the outer film cannot vary with position, since rapid

equtlibratton with the solid prectpttate and the

requirement of the electroneutrality force the con-

centration of the electrolyte in the pores to remain at

the same saturation value throughout. As a result,

diffusional transport for both ions is eliminated, only

migration of the ions in the electric field remains,

and the porous layer acts as a simple ohmic resistor.

Since Fe”

cations and Cll anions have similar

mobilities, it is reasonable to assume that migration

will cause cations to be transported outward

through the film by migration, whereas anions will

tend to be drawn inward. There is clearly no net

transport of chloride ions into the dissolving iron

electrode, and the chloride transported into the

porous film must therefore precipitate out at the

bottom of the pores. Since (for constant film

porosity) the film thickness must remain constant at

steady state, one sees clearly that the precipitation of

freshly formed FeCl, beneath the film must be com-

pensated by equivalent dissolution of the precipitate

at the outer-film/electrolyte interface. Physically, the

porous-film model represents a dynamic process

with a salt film of constant thickness continually

being renewed by precipitation in the inner region

beneath the pores and dissolution at the outer edge

along the film/electrolyte interface. The fact that the

layer thickness is conserved by precipitation of

FeCI, solid beneath the film is an interesting result,

and one might speculate as to whether the mechani-

cal pressure built up by the precipitating layer

beneath the film may not be responsible in some way

for cracks and fissures leading to film porosity.

Under the transient conditions resulting from

potential modulation, the dynamic steady state of

film renewal is perturbed, and the resulting imped-

ance response depends on the nonzero rates of chlo-

ride transport and the corresponding variations in

film thickness with time. During a growth phase, for

example, a net flux of Cl- is drawn toward the elec-

trode surface by an increase in migration that is not

totally compensated by diffusion since the concentra-

tion profiles are fixed by the saturation and bulk

values. During a thinning phase, the net flux of Cl

is reversed and the thickness of the film decreases.

For the inner compact layer, it is possible to

imagine that only the Fe+2 cations are capable of

significant mobility in the solid phase (transference

number of Cll zero in the solid). In that case, the

migration flux of the iron cations corresponds to the

net anodic current and the film is not renewed as in

the case of the porous outer region. If chloride ions

do have significant mobility in the soild, their trans-

port through the compact film will result in renewal

of the film at constant steady-state thickness similar

to that described for the porous layer. In the most

general case of a complete duplex film with Cl

mobility in both layers, the renewal rates of the

layers will depend on the solid-state and liquid-state

transference numbers for chloride ions and material

balances at the interface between the two regions.

Figure 2 shows the relative contributions of the

porous layer, compact layer and electrolyte solution

to the overall potential drop observed in an experi-

mental polishing system. In the example shown, the

overpotential for Fe dissolution at the inner metal/

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396

M. MATLOSZ

anode

$0

POTENTIAL DROP : SALT FILM MODEL

Reference

Fig. 2. Sources of potential drop in salt-film systems.

compact-film interface has been neglected, and the

role of the potential drop across the compact layer

has been emphasized. For typical high-field conduc-

tion of metallic cations, the steady-state current

density transported across the compact layer can be

represented as follows,

i,, = i,

exp(“(‘, ‘O))

where

(V - +o)/

denotes the electric field strength

across the compact layer of thickness A,, x a con-

stant characterstic of the jump spacing for solid-state

ionic transport in the precipitate crystal, and i, a

kinetic constant related to the number of charge car-

riers present in the compact film.

For operation along the limiting-current plateau,

increasing the applied potential does not change the

current density (which is limited by the transport of

Fe+’ across the electrolyte diffusion layer), and con-

sequently the electric field strength in equation (1)

must be conserved. This is accomplished by an

increase in the thickness 1, which compensates for

any increase in

V - qSo

in the absence of a porous

layer.

For transient operation, the charging of the dielec-

tric layer must be taken into account along with the

high-field conduction. The thicker the dielectric film,

the smaller the charging current necessary to estab-

lish the steady-state field. With this interpretation,

the current density under transient or modulated

conditions can be expressed by addition of a second

term to equation (I) as follows:

where Em_,,,

denotes the electrical permittivity of the

compact film.

When a porous layer is present, the current

density through the porous region can be expressed

by the equivalent of Ohm’s law:

where R, denotes the porous-film resistance (per unit

length), and 1, the porous-film thickness. The resist-

ance R, is a function of the electrolyte conductivity

and film porosity and tortuosity. The total potential

drop

V - q5p

across a duplex film is divided between

the compact and porous layers, with an increase in

applied potential generally resulting in an increase in

the thickness of both regions.

In the special case of a porous film only, the over-

potential for charge transfer at the metal electrolyte

solution within the pores is generally not negligible.

In this case, one can reinterpret the potential 4. as

that of an appropriate reference electrode located in

the electrolyte solution in the pores just outside the

electric double layer at the metal/electrolyte inter-

face. For Tafel kinetics, the following steady-stae

expression can be used

t =

i, exp(b( V - ))

(4)

where b denotes a Tafel constant,

i,

an appropriate

exchange-current density, and t: the film porosity

(included to represent the fact that charge transfer

occurs only on the fraction of the metal surface

exposed to electrolyte solution in the pores). Use of

the same symbols

i,

and b. is intended to emphasize

the considerable similarity between the expressions

for high-field conduction (compact film) and Tafel

kinetics (porous film). Despite these similarities, there

are important differences between the two limiting

cases. In the case of a porous layer alone with Tafel

kinetics, for example, it is not possible to alter the

value of the Tafel costant h by changing the film

thickness, whereas in a compact film model the

equivalent constant r/i, in the solid-state conduc-

tion expression is inversely proportional to thick-

ness. This difference is essential in the determination

of the size of the high-frequency loop in the imped-

ance diagrams because of its influence on the effec-

tive charge-transfer resistance.

In the absence of a compact film, the value of

V - 4. cannot vary along the limiting-current

plateau, and all of the necessary potential drop must

be obtained by thickening of the porous layer alone.

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Modeling of impedance mechanisms

397

In addition, the double-layer capacity in the absence

of a compact film no longer represents the charging

of a solid dielectric material, but rather the charging

of a metal-electrolyte interface on the exposed area

beneath the pores. This charging contribution can be

added to the Tafel expression in the porous film

model to yield

:

i

- = i0 ew(W - I) +

c,,

d(V - 44

E

dr

(9

where C,, denotes the capacity of the electric double

layer at the metal/electrolyte interface.

A final source of potential drop is the electrolyte

solution between the electrode surface and the posi-

tion of the reference electrode used to measure the

applied potential. The current density can be

expressed in terms of the electrolyte resistance as

follows:

(6)

where R, represents the effective ohmic resistance of

the electrolyte from the dissolving metal electrode to

the position of the reference electrode in the bulk

electrolyte. This ohmic resistance depends on the

geometry of the experimental system and on the con-

ductivity of the electrolyte, and can be affected by

current-distribution effects which differ considerably

between the porous-film and compact-film models.

In particular, in high-frequency impedance measure-

ments,

R,

represents essentially the ohmic resistance

for the passage of the double-layer charging current

through the electrolyte. In the case of a totally

compact dielectric film, it is the salt film itself which

is charged to create the electric field necessary for

solid-state ionic conduction. As a result, the surface to

be charged is in direct contact with the ohmic resist-

ance of the bulk electrolyte solution. The resistance

R, should correspond in that case to the equivalent

resistance measured in the absence of a film and

should not be affected by the thickness of the

compact layer.

The case of a porous or duplex film is different. If

the porous layer is thick enough to present a signifi-

cant ohmic resistance, that resistance will be placed in

series with the bulk electrolyte since double-layer

charging takes place beneath the porous layer. This

additional resistance has two effects. The first effect

is a linear increase in the high-frequency ohmic

resistance with increasing film thickness, an increase

not observed for compact films. The second effect is

more subtle and is related to the influence of the film

resistance on the distribution of the current lines in

the bulk electrolyte. In the case of a disk electrode,

for example, the presence of a significant surface

resistance in series with double-layer charging will

tend to create a more uniform distribution of current

lines by drawing more current toward the center of

the disk. In the limit where the porous film resistance

is significant enough to create a completely uniform

distribution along the outer film surface, the effective

ohmic resistance for the electrolyte alone (for a refer-

ence placed far from the surface) will increase by

approximately 27 per cent compared to the primary

resistance in the absence of a film. This effect has

been examined experimentally by Grimm et aI.[5] by

extrapolation of the measured ohmic resistance in

the presence of the film to the limit of zero film

thickness and comparison of the result with the mea-

sured electrolyte resistance in the absence of a film.

THE ADSORBATE ACCEPTOR MODEL

Figures 3 and 4 present schematically the

adsorbate-acceptor model in a similar manner to the

duplex film model in Figs 1 and 2. The system pre-

sented shows uniform anodic dissolution of a metal-

lic alloy M (for example Fe-15Cr) in an electrolyte

containing a small quantity of acceptor species A.

One can imagine (as in[6]) that the electrolyte is a

concentrated solution of phosphoric and sulfuric

ADSORBATE ACCEPTOR MODEL

solvation :

Reference

I-

' dl

d

adsorbate

double

diffusion

layer

layer

‘MA bulk

M = Fe-i 50

A = H,O

Fig. 3. Schematic illustration of the adsorbate-acceptor model

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398

M. MATLOSZ

POTENTIAL DROP : ACCEPTOR MODEL

Fig. 4. Sources of potential drop in acceptor systems.

acid, and that the acceptor is water or a water-related

species necessary for solvation of the dissolving

cations. Contrary to the salt-film models, where the

surface concentration is fixed by a saturation value,

in this case the situation corresponds much more

closely to the traditional limiting current observed in

cathodic metal deposition. The limiting current is

reached when the concentration of the acceptor

species A drops to near zero at the electrode/solution

interface. Due to the absence of the salt films, the

transport processes are limited to diffusion of the

acceptor species through the electrolyte diffusion

layer.

At the electrode surface, the dissolution mecha-

nism consists of oxidation of the metal to adsorbed

cations followed by solvation of the adsorbed ions

by the acceptor species. Whereas in the salt-film

model it is the thickening of the salt layer which

leads to an increase in potential drop along the

limiting-current plateau, in the adsorbate-acceptor

model it is the accumulation of adsorbed ions on the

surface which leads to blocking of the surface and a

subsequent increase in overpotential for metal disso-

lution.

For transient measurements, a clear consequence

of the absence of solid films on the electrode surface

can be observed in the double-layer charging of the

interface. The situation is similar to the porous-film

model in the absence of a compact layer in that the

charging phenomenon is related to the double-layer

capacity of the metal/electrolyte interface. In the

adsorbate acceptor model, however, the entire elec-

trode surface is available and the double-layer con-

tribution can be expressed simply as follows:

4, = C,,

d(I’ - 40)

dt

(7)

where C,, denotes the double-layer capacity of the

metal/electrolyte interface in the presence of the

adsorbed cations. The presence of the adsorbed

cations is not expected to alter dramatically the

value of C,, in comparison with a typical metal/

electrolvte interface. however. and the exnerimental

measurements in[7] tend to support this view for the

Fe-15Cr system. In salt-film systems, on the other

hand, the measured capacities depend on film

properties (thickness, porosity, dielectric constant)

and can be much lower. In the experimental studies

reported in[5], for example, the values of effective

double-layer capacity (and the corresponding RC

time constants) were so low that complete measure-

ment of the high-frequency loops in the impedance

diagrams could not be achieved in the frequency

range of the experimental apparatus.

The adsorbate-acceptor model exhibits specific

characteristic behavior for the charge-transfer resist-

ance as well. For metal dissolution with Tafel

kinetics, for example, but with the additional

requirement that free surface sites must be available

for the adsorbed oxidized species, the steady-state

rate of anodic oxidation will drop with increasing

coverage as follows:

i,, = i, exp(b(V - &J) . (1 - 0)

(8)

where 0 denotes the surface coverage by the

adsorbed intermediates. The current density at a

given potential depends not only on the kinetic

expression, but also on the mass balance for adsorb-

ate and acceptor species. At steady state, the rate of

production of adsorbed species by equation (5) must

equal their rate of consumption by solvation. In par-

ticular, with kinetics for solvation first order in both

acceptor molecules and adsorbed cations:

s= c uO

where z denotes the cation valence, F the Faraday

constant,

k

a rate constant for solvation (assumed

results from the equivalence of acceptor transport

independent of potential), cb the bulk concentration

of acceptor molecules, and u the ratio of surface to

bulk acceptor concentration. Equation (6) illustrates

the coupling that occurs in the acceptor mechanism

between the rate of mass transport of acceptor

species and the rate of cation solvation. Determi-

nation of the surface concentration of acceptor (u)

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Modeling of impedance mechanisms

399

and the steady-state current:

43s

DC ~ - u)

-=

i F

rnd

where D denotes the acceptor diffusion coefficient, 6

a diffusion-layer thickness and M the number of

acceptor molecules needed to solvate a single

cationic species.

Along the limiting-current plateau (u < l), the

current density is essentially constant and limited by

the rate of arrival of acceptor molecules at the elec-

trode surface. The surface coverage rises quickly, and

the group 1 - U (the fraction of free surface sites)

drops exponentially with increasing potential:

1 - 0 = 5 exp(-b(V - &)).

As a result, and contrary to what one would expect

for normal Tafel kinetics in the absence of surface

blocking, the effective charge-transfer resistance does

not drop with increasing potential, but instead

remains perfectly constant along the limiting-current

plateau. This feature is characteristic of a blocking

mechanism and can be observed in porous-salt-film

models as well. In the porous-film models, however,

the effect is not due to a rise in surface overpotential.

Rather, a change in applied external potential simply

does not alter the true surface overpotential beneath

the film, since all of the potential difference is elimi-

nated by additional ohmic drop due to growth of the

porous layer.

The situation for a compact salt film is qualit-

atively different. Increasing applied potential in a

compact film system results in an increase in thick-

ness 1,) and a corresponding increase in the charge-

transfer resistance through the group &/u. As a

result, a change in applied potential results in a

strong variation in charge-transfer resistance which

is observed neither in porous-film models nor in

acceptor models.

NYQUIST DIAGRAM

SALT FILM MODEL

DISTINGUISHING FEATURES OF THE

IMPEDANCE DIAGRAMS

The features pointed out in the discussion above

concerning ohmic resistances, charge-transfer resist-

ances, and double-layer capacities can be examined

directly with impedance measurements, and can be

used to establish experimental criteria to distinguish

between the different models. Figures 5 and 6 present

the general shapes of the impedance diagrams

obtained with the salt-film and adsorbate-acceptor

systems, respectively. The most important overall

feature of the diagrams concerns the difference in

low-frequency behavior between the adsorbate-

acceptor model and the salt-film models. Despite the

complexity of the transport processes involved, the

impedance behavior of the salt-film models is rela-

tively simple. At high frequencies, charge-transfer

processes (Tafel kinetics for the porous film, solid-

state ionic conduction for the compact film) domi-

nate the response since the film does not have time

to grow or shrink and remains essentially unchanged

during modulation. As the lower frequencies are

approached, film thickening and thinning processes

begin to play an ever greater role, and at the very

lowest frequencies the system behaves essentially as a

pure capactor with the film growing and shrinking in

response to the oscillating signal.

The adsorbate-acceptor system is quite different.

At the highest frequencies charge-transfer and

double-layer processes dominate, but as the fre-

quency is lowered additional features related to time

constants for the rate of change of surface coverage

begin to appear. At the very lowest frequencies, a

large third loop appears containing contributions

from an underlying Warburg-Nernst impedance.

The Warburg-Nernst loop is due to variations in

surface acceptor concentration as a result of the

oscillations in surface flux. It is interesting to note

that such effects are impossible in the salt-film

models, since the surface cation concentration in

those models is fixed at its saturation value and

Fig. 5. General shape of the Nyquist diagrams in salt-film systems.

EA 40 4 O

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400

M. MATLOSZ

NYQUIST DIAGRAM : ACCEPTOR MODEL

Fig. 6. General shape of the Nyquist diagrams in acceptor systems.

cannot be affected by the oscillating flux. This differ-

ence between the salt-film and acceptor models is

very striking, but it can only be observed with low-

frequency data that may be difficult to obtain in a

polishing system undergoing high-rate dissolution. It

is therefore of interest to compare more easily

obtainable and reproducible data that can be mea-

sured at high frequencies.

At high frequencies, all of the models yield

Nyquist diagrams of essentially the same shape, a

semicircle displaced to the right of the origin. The

position of the high-frequency limit of the semicircle

R,,

the diameter of the circle R, and the effective

capacity C,, determined from the angular frequency

at the top of the semicircle w = l/R,C,, provide

three characteristic values that can be measured

easily and compared in light of the theory for the

different models. Table 1 summarizes the results

from the theoretical analyses in [S] and [6].

Since all three models yield similar diagrams at

high frequency, it is not possible to distinguish

between them based on the shape of a single imped-

ance spectrum. The results in Table 1 can be used,

however, to determine expected trends for targeted

experiments aimed at comparing impedance spectra

obtained at different steady-state operating condi-

tions. Consider, for example, a series of impedance

diagrams obtained at different total applied poten-

Table 1. Characteristic features of the high-frequency loops

Salt-Film Models

Porous Film Compact Film

Adsorbate-

Acceptor

Model

tials along the limiting-current plateau. Under condi-

tions of constant

convective diffusion, the

limiting-current value is essentially unchanged. For

compact or porous-film models, increasing total

applied potential at constant limiting current density

must result in an increase in the corresponding film

thicknesses. Similarly, for the adsorbate-acceptor

model, an increase in applied potential should result

in an increase in the adsorbate surface coverage. The

results in Table 1 for an increase in 1, and 1, indi-

cate an increase in

R,

but no change in

R,

and C,

for a porous-film model, whereas for a compact film

the same experiment should yield an increase in

R,,

a decrease in C, but no change in

R,.

For an

adsorbate-acceptor model, none of the parameters

should vary.

An alternative series of impedance diagrams can

be obtained at a fixed applied potential (corrected

for the ohmic drop i,,

R,),

but at varying convective

diffusion conditions. For a rotating disk electrode,

for example, one can increase the disk rotation

speed, which will result in an increase in the steady-

state current density i,, (proportional to the square

root of disk rotation speed). For constant applied

potential, the incease in steady-state current will lead

to a decrease in salt-film thickness and both factors

will play a role in the behavior of the diagrams. For

a porous-film system, Table 1 reveals that

R,

and

R,

should decrease while C, should remain unchanged.

For a compact film,

R,

should remain unchanged,

C, increase and

R,

decrease. Finally, for the

adsorbate-acceptor model,

R,

and C, should remain

unchanged, while

R,

should decrease (propor-

tionately to the square root of the disk rotation

speed). Each of these effects has a clear explanation

in the underlying physical models as described above

and can be related directly to the parameters of the

experimental system under study. Experimental

investigations of this type have been undertaken on

both salt-film and acceptor systems and have pro-

vided considerable support for the physical pictures

of the polishing mechanisms described above.

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Modeling of impedance mechanisms 401

CONCLUSIONS

tion will be of general use in the study of a wide

range of electropolishing systems of practical and

Theoretical analyses of the underlying physical scientific importance.

bases of simple salt-film and acceptor models have

resulted in identification of important qualitative dif-

ferences between the models that are independent of

the specific property parameters for a given electro-

polishing system. Targeted experiments aimed at

studying the variation in alternating current imped-

ance spectra as a function of steady-state operating

conditions such as applied potential and hydrody-

namic conditions can be used to elucidate the

mechanism. The results of such experimental studies

can provide fundamental information concerning the

mechanisms of electropolishing on a qualitative basis

without recourse to extensive fitting of the model

parameters. It is hoped that the physical picture of

the polishing process presented here and the corre-

sponding methodology for experimental investiga-

REFERENCES

1. H. Figour and P. A. Jacquet, French Patent No. 707526

(1930).

2. P. A. Jacquet, Nature 135, 1076 (1935).

3. P. A. Jacquet, Trans. Electrochemical Society 69, 639

(1936).

4. D. Landolt, Electrochim. Acta 32, 1 (1987).

5. R.-D. Grimm, A. West and D. Landolt, J. electrochem.

Sot., 139, 1622 (1992).

6. M. Matlosz, S. Magaino and D. Landolt, J. electrochem.

sot., 141,410 (1994).

7. S. Magaino, M. Matlosz and D. Landolt, J. electrochem.

Sot. 140, 1365 (1993).