Porous Anodic Aluminum Oxide: Anodization and Templated Synthesis of Functional Nanostructures

70 Pages·2014·58.54 MB·English

Review
pubs.acs.org/CR
Porous Anodic Aluminum Oxide: Anodization and Templated
Synthesis of Functional Nanostructures
Woo Lee*,†,‡ and Sang-Joon Park†
†
Korea Research Institute of Standards and Science (KRISS), Yuseong, 305-340 Daejeon, Korea
‡
Department of Nano Science, University of Science and Technology (UST), Yuseong, 305-333 Daejeon, Korea
6.2.3. Morphological Instability U
6.3. Steady-State Pore Formation W
6.3.1. Joule’s Heat-Induced Chemical Dissolu-
tion W
6.3.2. Field-Assisted Oxide Dissolution W
6.3.3. Average Field Model for Steady-State
Pore Structure X
6.3.4. Direct Cation Ejection Mechanism Y
6.3.5. Flow Model for Steady-State Pore
Formation Z
7. Self-Ordered Porous Anodic Aluminum Oxide
(AAO) AA
7.1. Mild Anodization (MA) AB
7.2. Hard Anodization (HA) AD
CONTENTS
7.3. Pulse Anodization (PA) AF
1.Introduction A 7.4. Cyclic Anodization (CA) AI
2.Types of Anodic Aluminum Oxide (AAO) B 7.5. Anodization of Thin Aluminum Films De-
3.Ionic Conduction in Anodic Oxide Films D posited on Substrates AI
3.1. High-Field Conduction Theory D 8. Long-Range Ordered Porous AAO AL
3.2. Elementary Interfacial Reactions D 9. AAO Template-Based Synthesis of Functional
3.3. Transport Numbers E Nanostructures AP
3.4. Stress-Driven Ionic Transport F 9.1. Electrochemical Deposition (ECD) AP
4.Electrolytic Breakdown G 9.2. Electroless Deposition (ELD) AS
4.1. Factors Influencing Breakdown G 9.3. Sol−Gel Deposition AS
4.1.1. The Nature of Anodized Metal H 9.4. Surface Modification AU
4.1.2. Electrolyte Conditions H 9.5. Template Wetting AZ
4.1.3. Current Density (j) H 9.6. Mask Techniques BA
4.1.4. Other Factors Influencing Breakdown H 9.7. Chemical Vapor Deposition (CVD) BB
4.2. Models for Breakdown I 9.8. Atomic Layer Deposition (ALD) BC
4.2.1. Electron Avalanche Multiplication I 10. Closing Remarks and Outlook BE
4.2.2. Stress-Driven Breakdown J Author Information BF
5.Structure of Porous Anodic Aluminum Oxide Corresponding Author BF
(AAO) K Notes BF
5.1. General Structure K Biographies BG
5.1.1. Pore Diameter (D ) K Acknowledgments BG
p
5.1.2. Interpore Distance (D ) M Abbreviations BG
int
5.1.3. Barrier Layer Thickness (t ) M References BI
b
5.2. StructureofPoreWall(AnionIncorporation) M
5.3. Effect of Heat Treatments P
6.GrowthofPorousAnodicAluminumOxide(AAO) P 1. INTRODUCTION
6.1. Stress Generation in Anodic Oxide Films P
In ambient atmospheres, aluminum becomes rapidly coated
6.1.1. Volume Expansion P with a compact 2−3 nm thick oxide layer. This native oxide
6.1.2. Stress Measurements Q
layerpreventsthemetalsurfacefromfurtheroxidation.Because
6.1.3. Effects of External Stresses on Pore
of the surface native oxide, aluminum generally has good
Growth S
corrosion resistance. However, local corrosion of metal can
6.2. Initial-Stage Pore Formation T
occur in rather aggressive outdoor environments, containing
6.2.1. Qualitative Description on Pore Forma-
tion T
Received: January2, 2014
6.2.2. Kinetics of Porosity Initiation T
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corrosive chemicals (e.g., chlorides or sulfates). In 1857, Buff discussed (section 5). Anodization of aluminum is a volume
first found that aluminum can be electrochemically oxidized in expansion process, and thus is accompanied by mechanical
an aqueous solution to form an oxide layer that is thicker than stresses. Recent studies have indicated that the stresses have
the native one.1 This phenomenon has been called “anodiza- profoundimplicationsnotonlyontheionictransport,butalso
tion” because the aluminum part to be processed constitutes on the self-ordering behavior of oxide nanopores. We will
the anode in an electrolytic cell. In the early 1920s, the discuss in detail the effect of stress on pore growth (section
phenomenonobservedbyBuffwasexploitedforindustrialscale 6.1), the kinetics of pore initiation, and morphological
applications, for example, protection of seaplane parts from instability associated with the early stage of anodization
corrosive seawater.2 In general, the anodic aluminum oxide (section 6.2), and recent models describing steady-state pore
(AAO) films form with two different morphologies (i.e., formation (section 6.3). After that, recent progress on
nonporous barrier-type oxide films and porous-type oxide anodization of aluminum used in fabricating self-ordered
films) depending mainly on the nature of the anodizing porous AAO and also for engineering internal pore structures
electrolyte.3 Because the process was first implemented for willbediscussed(section7).Inaddition,variousapproachesto
protection purposes, the anodization of aluminum and its long-range order porous AAO will be reviewed (section 8). In
alloys, particularly porous-type anodization, has received the last part of this Review (section 9), various chemical
considerable attention in the industry because of its extensive approaches for the syntheses of low-dimensional functional
practical applications. Many desirable engineering properties nanostructures and the fabrications of advanced nanodevices
such as excellent hardness, corrosion, and abrasion resistance will be discussed. These approaches include electrochemical
can be obtained by anodizing aluminum metals in acid deposition (ECD), electroless deposition (ELD), sol−gel
electrolytes.4 In addition, due to its high porosity, the porous deposition, surface modification, template wetting, shadow
oxidefilmsformedonthemetalsserveasagoodadhesionbase mask techniques, chemical vapor deposition (CVD), and
for electroplating, painting, and semi-permanent decorative atomic layer deposition (ALD). Chemistry issues encountered
coloration. The anodized products can be easily found in in the template-based synthesis of functional nanostructures
electronic gadgets, electrolytic capacitors, cookware, outdoor will be discussed in detail. Finally, we will present the
products, plasma equipment, vehicles, architectural materials, challenges and future prospects of the field (section 10).
machine parts, etc. Recently, this nearly century-old industrial
processhasbeendrawingincreasingattentionfromscientistsin 2. TYPES OF ANODIC ALUMINUM OXIDE (AAO)
the field of nanotechnology. This trend originated with the Anodization ofaluminuminaqueous electrolytesforms anodic
seminal works of Masuda and co-workers, who reported on oxide films with two different morphologies, that is, the
self-ordered porous AAO in 19955 and the subsequent nonporous barrier-type oxide films and the porous-type oxide
development of the two-step anodization process in 1996.6 films. The chemical nature of the electrolytes mainly
PorousAAOfilmgrownonaluminumiscomposedofathin determines the morphology of AAOs.3,7,8 A compact non-
barrieroxidelayerinconformalcontactwithaluminum,andan porous barrier-type AAO films can be formed in neutral
overlying, relatively thick, porous oxide film containing electrolytes (pH 5−7), such as borate, oxalate, citrate,
mutually parallel nanopores extending from the barrier oxide phosphate,adipate,tungstatesolution,etc.,inwhichtheanodic
layer to the film surface.7 Each cylindrical nanopore and its oxideispracticallyinsoluble.9,10Meanwhile,porous-typeAAOs
surrounding oxide region constitute a hexagonal cell aligned are formed in acidic electrolytes, such as selenic,11 sulfuric,12
normal to the metal surface. Under specific electrochemical oxalic,12 phosphoric,7,12,13 chromic,12,14 malonic,12,15−17 tarta-
conditions, the oxide cells self-organize into hexagonal close- ric,12,18citric,12,17−20malicacid,12,18etc.,inwhichanodicoxide
packed arrangement, forming a honeycomb-like structure.5−7 isslightlysoluble.Earlymodelsdescribinganodicoxidegrowth
Pore diameter and density of self-ordered porous AAOs are were developed on the basis of the barrier-type oxide.21−24
tunable in wide ranges by properly choosing anodization Moreover, in the early stage of porous-type oxide growth, the
conditions: pore diameter = 10−400 nm and pore density = formation of the initial barrier oxide is followed by the
108−1010porescm−2.Thenovelandtunablestructuralfeatures emergenceofincipientpores.Therefore,inthisReview,wewill
of porous AAOs have been intensively exploited for synthesiz- mentionthebarrier-typeoxidegrowthtotheextentneededfor
ing a diverse range of nanostructured materials in the forms of understanding porous-type oxide formation. Some excellent
nanodots, nanowires, and nanotubes, and also for developing review articles covering the barrier-type anodic oxide films are
functional nanodevices. given in refs 3 and 25.
TheobjectiveofthisReviewistoprovideasolidinformation The two types of anodic oxides (i.e., barrier- vs porous-type
source for researchers entering this field and to establish a AAO) differ in their oxide growth kinetics. In the case of
broad and deep knowledge base. This Review introduces the barrier-type oxide formation under potentiostatic conditions
fundamental electrochemical processes associated with anodic (i.e.,U=constant),currentdensity(j)decreasesexponentially
oxidation of aluminum, and discusses the recent progress on with time (t). Correspondingly, the film growth rate decreases
anodization of aluminum for the development of ordered almost exponentially with time (t), which places a limit on the
porous AAOs, and nanotechnology applications of porous maximum film thickness obtainable for barrier-type AAO films
AAOs.WeorganizethisReviewasfollows:afterdiscussingthe (Figure 1). It has been experimentally verified that the
growth characteristics of two different types of AAOs (section thickness of barrier-type film is directly proportional to the
2), we will describe the theory of ionic conductions and appliedpotential(U).Ontheotherhand,currentdensity(j)in
elementary interfacial reactions (section 3), followed by porous-type anodization under potentiostatic conditions
electrolytic breakdown (section 4) to understand the remainsalmostconstantwithinacertainrangeofvaluesduring
fundamental electrochemistry associated with anodic oxidation the anodization process, due to the constant thickness of the
of aluminum. Next, the electrochemical factors defining the barrierlayerattheporebottom.Thethickness oftheresulting
geometric and chemical structures of porous AAOs will be porousoxidefilmislinearlyproportionaltothetotalamountof
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aluminum, new oxide forms above and below the marker
layer(Figure2b).Themarkerlayerislocatedatadepthof40%
of the film thickness in a plane corresponding to that of the
original metal surface. On the other hand, when a barrier-type
film is formed at 60% current efficiency (η), the plane of the
j
marker layer is immobile and 40% of the Al3+ cations are shed
into the electrolyte via direct cation ejection mechanism
withoutcontributingtotheoxideformation(Figure2c).Inthis
case, anodic oxide grows at the metal/oxide interface via the
inward migrationofO2−/OH−ions. Whenporous-type anodic
oxide forms at 60% current efficiency, the marker plane is
located abovethat oftheoriginalmetal surface (Figure2d).In
this case, the metal/oxide interface is also the oxide growth
front, and 40% of Al3+ ions are ejected into the solution.
Because cations are being shed into the electrolyte, the
current efficiency (η) of porous-type oxide growth is typically
j
much lower than that of the barrier-type. Accordingly, the
Pilling−Bedworth ratio (PBR = the ratio of molar volume of
the grown oxide to molar volume of the consumed metal; see
Figure 1. Two different types of anodic aluminum oxide (AAO) section 6.1.1) for the initial barrier oxide formation in porous-
formed by (a) barrier-type and (b) porous-type anodizations, along typeoxidegrowthattheearlystageofanodizationislowerthan
withtherespectivecurrent(j)−time(t)transientsunderpotentiostatic thatfor barrier-typeoxide growth:PBR=0.90forporous-type
conditions. oxide growth at η = 53.5% in phosphoric acid solution and
j
PBR=1.7forbarrier-typeoxidegrowthatη =100%inneutral
charge(i.e.,anodizationtime,t)involvedintheelectrochemical j
adipate solution.9,33 Shimizu et al.33 suggested that the initial
reaction.
barrier oxide grows under increasing tensile stress (PBR < 1),
Radiotracer studies, employing an immobile marker (125Xe),
which causes local oxide cracking most probably at the
haveindicatedthat,inthecaseofbarrier-typeoxideformation,
randomly present metal protrusions. The generated surface
anodic alumina grows simultaneously at the oxide/electrolyte
interface and at the metal/oxide interface, through Al3+ egress cracks were considered to be local paths for electrolyte
and O2−/OH− ingress, respectively, under a high electric field penetration, causing non-uniform local thickening of the initial
(E).26−28Inthecaseofporous-typeanodicaluminaformation, barrieroxide.Non-uniformthickeningoftheinitialoxidecauses
concentration and redistribution of the current lines into the
ontheotherhand,oxidegrowsatthemetal/oxideinterfacevia
theinwardmigrationofO2−/OH−ions.18Otracerstudieshave relatively thin oxide regions between the protrusions (i.e., a
shownthat outwardly migrating Al3+cations do not contribute local increase in electric field, E). Consequently, localized
totheoxidegrowthattheoxide/electrolyteinterface,butareall scalloping of the metal/oxide interface takes place. Shimizu et
shed into the anodizing electrolyte via direct ejection al.33 pointed out that the non-uniform thickening of anodic
mechanism (see section 6.3.4).10,29−31 Otherwise, egressing oxide(i.e.,morphologicalinstability)intheinitialbarrieroxide
Al3+ ions would form anodic alumina at the oxide/electrolyte is“oneofthemostdistinctlydifferentgrowthfeaturesbetween
interface to heal any developing or embryonic pores there. porous-andbarrier-typeAAOfilms”.Unlikeporous-typeoxide
Schematic diagrams illustrating the dimensional changes of growthinacidelectrolytes,anodicoxidesinneutralelectrolytes
aluminumduringthebarrier-typeandporous-typeanodicoxide grow highly uniformly on surface finished aluminum,
formationareshowninFigure2.32Animmobilemarkerlayeris maintaining flat metal/oxide and oxide/electrolyte interfaces.
implanted into the starting aluminum with a native oxide layer Eventhesmoothingofinitiallyroughaluminumsurfacesduring
(Figure2a).Whenabarrier-typefilmisformedat100%current the growth of barrier oxide films has been experimentally
efficiency (η) by anodization of the marker-implanted observed.34
j
Figure2.Schematicdiagramsillustratingdimensionalchangesofanaluminumspecimenfollowinganodizing.(a)Initialaluminumwithathinair-
formedoxidefilm.Thereddashedlinerepresentsanimmobilemarkerlayerimplantedintotheinitialaluminumwithathinair-formedoxidefilm.
(b)Anodizedat100%efficiencywithformationofabarrier-typeanodicfilm.(c)Anodizedatjustabove60%efficiencywithformationofabarrier-
typeanodicfilm.(d)Anodizedat60%efficiencywithformationofaporousanodicfilm.Reproducedwithpermissionfromref32.Copyright2006
TheElectrochemical Society.
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3. IONIC CONDUCTION IN ANODIC OXIDE FILMS in the oxide is high enough (e.g., 106−107 V cm−1), the ionic
current density (j) can be expressed as25
3.1. High-Field Conduction Theory
⎛ W ⎞⎛αazFE⎞
When a valve-metal is anodized under either potentiostatic or j = vρa exp⎜⎝− ⎟⎠⎜⎝ ⎟⎠
galvanostatic condition, anodic oxide film forms on the metal. RT RT (2)
For anodizing aluminum (Al) and tantalum (Ta), an empirical where v is the hopping attempt frequency of the ion, ρ is the
exponentialdependence oftheionic current density(j) onthe density of concentration of mobile charge in C cm−3, a is the
electric field (E) is established. Ionic current density (j) under hopping inter-distance, W is the hopping activation energy at
high-fieldconditions,whichisthecaseforanodicoxidegrowth, zero field, α is a parameter describing the asymmetry of the
can be associated with the movement of charged ions in the activation barrier at non-zero field, z is the valence of the
barrier oxide, and can be related to the potential drop (ΔU) mobileions,andFisFaraday’sconstant.Fromeqs1and2,the
across the barrier oxide through the exponential law of following relations can be obtained:
Güntherschulze and Betz, as follows:21,35 ⎛ ⎞
W
j = υρa exp⎜⎝− ⎟⎠
j = j exp(βE) = j exp(βΔU/t ) 0 RT (3)
0 0 b (1)
αazF
where j and β are material-dependent constants at a given β =
tempera0ture, and ΔU/t is the effective electric field (E, RT (4)
b
typically 106−107 V cm−1) impressed on the barrier layer with Because the parameter a can be related to the inter-atomic
thicknesst .Foranodicalumina,alargerangeofj andβvalues distance in the oxide, one can expect that the electric field
b 0
hasbeenreported:j =3×104to1×10−18Acm−2andβ=0.1 strength (E) increases when the oxygen ion density increases
0
× 10−6 to 5.1 × 10−6 cm V−1.25 (i.e., a decrease in parameter a) provided that the other
Foranodicoxidationofmetalinanelectrolyte,threetheories parameters are constant. Equation 1 can be modified to obtain
basedonthefollowingpossiblerate-determiningstepsforoxide a Tafel equation:
formation have been developed:3 ion transfers (i) across the
lnj = lnj + βE
metal/oxideinterface(Mott−Cabreratheory),23,24(ii)through 0 (5)
the oxide bulk (Verwey theory),22 and (iii) across the oxide/ For a constant oxide thickness t , a constant Tafel slope β is
b
electrolyte interface (Dewald theory).36,37 In the point defect obtained.
model of Macdonald et al.,38 the oxide film is assumed to The electric field (E) in the oxide can be related to the
contain a high concentration of non-interacting positive and applied(ormeasured,inthepotentiostaticcondition)electrode
negative point defects, and the rate-determining step for the potential (U). The measurable potential drop between the
oxidegrowthisassumedtobethetransportofmetalandoxide metal and the electrolyte is equal to
vacanciesacrosstheoxidefilm.Allofthesetheoriescanexplain
the empirical exponential relationship proposed by Günter- U = ΔU + Φm/o + Φo/e (6)
shultz and Betz. On the other hand, transient experiments whereΔUisthepotentialdropintheoxide,andΦ andΦ
m/o o/e
favorably indicate that the rate-determining step is the arethepotentialdropsatthemetal/oxideandoxide/electrolyte
movement of charged ions within the oxide.25 interfaces, respectively.39 In a typical anodization, the potential
On the basis of the rate-determining movement of ions drops at the metal/oxide and oxide/electrolyte interfaces are
within the oxide, the high-field model relates the parameters j quitesmall,ascomparedtotheseveraltensofvoltsofpotential
0
and β in eq 1 to the nature of oxide materials. The high-field drop in the oxide (i.e., ΔU ≫ Φ + Φ ). Therefore, the
m/o o/e
conduction model is based on a hopping mechanism, in which followingapproximationfortheelectricfield(E)ispossiblefor
theactivationenergyforhoppingionsisdependentonelectric the high-field ionic transport:
fieldE(Figure3).25Ionsatregularsitesorinterstitialpositions
jump to vacancies or other interstitial positions in their E = ΔU/tb ≈ U/tb (7)
neighborhood.Themodelassumesthattheoxideisdefect-free where t is the thickness of oxide.
b
and of homogeneous composition. When the electric field (E) 3.2. Elementary Interfacial Reactions
Aswasalreadydiscussedinsection2,itisnowwidelyaccepted
that for the anodic growth of alumina both Al3+ cations and
oxygen-containinganions(e.g.,O2−orOH−)aremobilewithin
the anodic oxide under high electric field (E).10,26−28,40 Al3+
ions migrate outwardly toward the oxide/electrolyte interface,
while O2− or OH− anions move inwardly toward the metal/
oxideinterface.Therefore,onecanconsiderboth(i)themetal/
oxide and (ii) the oxide/electrolyte interfaces as the growth
front of anodic oxide during anodization of a valve-metal. For
anodizing aluminum, the following elementary reactions are
consideredtobepossiblyoccurringattheinterfaces(Figure4).
(i) At the metal/oxide interface:
Al → Al3+ + 3e−
Figure3.Influenceoftheelectricfieldstrength(E)ontheactivation (ox) (8)
energy of hopping ions. Reproduced with permission from ref 25. 2Al3+ + 3O2− → Al O
Copyright 1993Elsevier. (ox) (ox) 2 3 (9)
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Figure 4. Schematic diagrams showing elementary interfacial reactions for (a)barrier-type and (b) porous-type anodic oxide.
(ii) At the oxide/electrolyte interface: and t− = 1 − t+ for anion. Transport numbers can be
determined by employing a “marker layer”, whose position in
2Al(3o+x) + 3O(2o−x) → Al2O3 (10) the anodic oxide film indicates the extent of oxide that was
formed at each interface. If the metal ions are the only mobile
Al2O3 + 6H(+aq)→ 2Al(3a+q) + 3H2O(1) (11) species, new oxide should be formed at the oxide/electrolyte
interface on top of the marker layer. On the other hand, if
Al(3o+x) → Al(3a+q) (12) oxygen anions are the only mobile species, the new oxide
should be formed at the metal/oxide interface below the
2O2− + O + 4e− marker layer. Davies et al.26 stated that the ideal marker atoms
(ox) 2(g) (13)
for determination of transport numbers should fulfill the
2H O + O2− + OH− + 3H+ followingrequirements:“Themarkersshouldbe(i)uncharged,
2 (1) (ox) (ox) (aq) (14) sothattheydonotmigrateintheoxideundertheinfluenceof
Reactions 9 and 10 correspond to the formation of anodic the applied field; (ii) large in size, so that they do not diffuse
oxide at the metal/oxide and oxide/electrolyte interfaces, significantly within the oxide lattice; (iii) present in trace
respectively. Reaction 11 describes dissolution of anodic amount,sothatthemacroscopicpropertiesofthetaggedoxide
alumina by Joule’s heat-induced oxide dissolution and/or remain unaltered; and (iv) detectable, in order to assess their
field-induced oxide dissolution, which will be discussed in depth in the oxide.”
section6.3.1andsection6.3.2,respectively.Ontheotherhand, Theseconditionscanbesatisfiedbyimplantingradioisotopes
reaction 12 occurs through field-assisted direct ejection of Al3+ 125Xe inert gas atoms or 222Rn, which are heavier than typical
ions from the metal/oxide interface through oxide into the valve-metals and oxygen, in a preformed thin oxide film and
electrolyte, which will be discussed in detail in section 6.3.4. subsequently anodized.26,27,42 Radioactive tracers allow the
Reactions 11−13 decrease the net current efficiency (η)
j positionoftheburiedmarkertobeassessedbymonitoringthe
associated with the anodic oxide formation. Reaction 14 energy of emitted α- or β-particles.26,27,42,43 Other techniques
describes the heterolytic dissociation of water molecules at the
to measure the buried marker position in oxide include
oxide/electrolyteinterface,whichsuppliesoxygenanionstothe Rutherford backscattering spectrometry (RBS)40 or direct
metal/oxide interface to form anodic oxide. By assuming that
observation of voids formed by implanted Xe by employing
all oxide anions from the dissolution of Al2O3 at the oxide/ cross-sectionaltransmissionelectronmicroscopy(TEM).28,44A
electrolyte interface migrate to the metal/oxide interface to
representativecross-sectionalTEMimageshowinganimmobile
reformAl2O3,andthatalloxideanionsfromthedissociationof Xe marker layer is given in Figure 5. The sample in the figure
watercontributetotheoxideformation,Suetal.proposedthe
wasformedinnear-neutralpotassiumphosphateelectrolyteata
following overall reaction at the oxide/electrolyte interface:41
high current efficiency (η ≈ 100%).44 The approximately 10-
j
Al O + nH O → 2Al3+ + (3 − n − x)O2− nm-thick straight Xe marker layer is located at about the
2 3 2 (1) (aq) (ox) midpoint of the film. The anodic oxide above the marker layer
+ xOH(−ox)+ (2n − x)3H(+aq) (15) formed by the field-driven outward migration of Al3+ ions and
that beneath the marker layer by the field-driven inward
where n denotes the amount of water dissociated per mole of migrationofoxygencarryinganions,O2−/OH−.Assumingthat
Al2O3 that is dissolved at the same time. Su et al. claimed the all egressing Al3+ ions contribute to the oxide formation, the
field-dependent nature of the heterolytic dissociation of water cation transport number was directly estimated to be t+ =
in reaction 14 and related the dissociation rate of water to the 0.49.44Foranodizingconditionsunderwhichoxidegrowswith
porosity (P) of AAO, which will be touched upon in section
appreciable metal dissolution, however, TEM-based direct
7.1.
measurement may underestimate the cation transport number.
3.3. Transport Numbers In such cases, Al3+ ions dissolved in anodizing electrolyte
Asmentionedinprevioussections,anodicoxideformationcan shouldbequantifiedtoestimatetheequivalentoxidethickness.
occur at both the metal/oxide and the oxide/metal interfaces. Davies et al.26 pointed out that the location of an immobile
The relative amount of mobile ions transported to the oxide marker in anodic oxide markedly depends on the anodization
forminginterfacesiscalledthe“transportnumber”:t+forcation conditions, such as current density (j) and the nature of
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Figure 5. Cross-sectional transmission electron microscopy (TEM)
imageshowingimmobile125Xemarkerlayer.Thesample(i.e.,barrier-
typeanodicoxidefilm)wasformedataconstantcurrentdensityof1
mA cm−2 to 100 V in near-neutral potassium phosphate electrolyte. Figure6.Schematicsshowing(a)themovementofAl3+andO2−ions
Adapted from ref 44 with permission. Copyright 1987 Taylor & during the re-anodizing process and (b) the corresponding cell
Francis (www.tandfonline.com). potential (U)−time (t) curve. Reproduced with permission from ref
46. Copyright 1978Elsevier.
electrolyte. The authors performed transport number experi- ⎡ dh+ dh−⎤ jM
ments using radioisotope 125Xe marker atoms and also ρ⎢P + ⎥ =
⎣ ⎦
quantitativeanalysesondissolvingAl3+ionsduringanodization dt dt nFk (16)
in two different near-neutral electrolytes (i.e., sodium whereρ(=2.95gcm−3)isthedensityofoxide,Pistheporosity
tetraborate and ammonium citrate). According to their ofporousAAO,dh+/dtanddh‑/dtare,respectively,theratesof
experiments, both metal and oxygen ions are mobile during
the increase of the barrier oxide thickness at the oxide/
oxidegrowth.Intheboratesolution,anodicaluminagrewwith
electrolyte and metal/oxide interfaces, j is the current density,
high current efficiencies (i.e., negligible cation loss), and thus
MistheatomicweightofAl,n(=3)isthenumberofelectrons
the immobile 125Xe markers were completely buried in the involved in oxidation reaction, F is Faraday’s constant, and k
oxide.Themeancationtransportnumber(t+)wasestimatedto
(=0.505) is the weight fraction of aluminum in the oxide. The
be t+ = 0.58 for anodic alumina formed at the current density
range of 0.1−10 mA cm−2. On the other hand, in citrate cation transport number is given by the ratio of the weight of
new oxide formed within the pores per unit time to the total
solution,theamountofaluminumpassedintothesolutionwas
weight of new oxide formed per unit time:46
as high as 40% of the total oxidized metal at low current
density, but decreased as the current density increased. The ⎛ dh+⎞ ⎛ dh+ dh−⎞
125Xe markers in anodic alumina remained very close to the t+= ⎝⎜P dt ⎠⎟/⎝⎜P dt + dt ⎠⎟ (17)
outersurface.Thecationtransportnumbervariedwithcurrent
density,fromaboutt+=0.37at0.1mAcm−2tot+=0.72at10 The slopes m and m of the U−t transient in Figure 6b are
1 2
mA cm−2. given by46
ionIns cthanecbaeseeostfimpoartoeudsbAyAtOhe,thsoe-ctraallnesdpo“rptonreu-mfilblienrgsomfemthoobdil”e, m = 1 ⎛⎜dh+ + dh−⎞⎟
which was originally used to determine the porosity (P) of 1 AR⎝ dt dt ⎠ (18)
porous AAO by Dekker and Diddelhoek.45,46 In this method,
aluminum is first anodized in an acid electrolyte to form 1 ⎛ dh+ dh−⎞
m = ⎜P + ⎟
porous-type anodic oxide and subsequently re-anodized in a 2 AR⎝ dt dt ⎠ (19)
neutralelectrolyteelectrolytetoformbarrier-typeoxideundera
galvanostatic condition. During anodizations, potential (U)− where ARis the anodizing ratio (=the ratio ofthe barrier layer
time (t) transients are monitored. During the barrier-type thicknesstothecellpotential,innmV−1),andassumedtobea
anodizing (i.e., re-anodizing process), new oxide gradually constant. From eqs 16−19, the porosity (P) ofporous AAO is
forms simultaneously within the pores and underneath the given by
barrier layer of the pre-formed porous anodic oxide, because
bothAl3+andO2−ionscontributetotheoxideformationatthe P = t+(m2/m1)
metal/oxide and oxide/electrolyte interfaces, respectively. As a 1 − (1 − t+)(m /m) (20)
2 1
result,thecellpotentialgraduallyincreaseswithtimeduringthe
For porous AAO formed in 1.125 M oxalic acid at 30 V,
re-anodizing process. Figure 6 schematically shows (a) the
(mt)ovpermoefinletodfuAriln3+ganthdeOr2e−-aannoddi(zbin)gthpercoeclelspso.4t6enTtihale(Uno)n−-tziemroe Tofakmahoabsihlei aAndl3+NaangadyaOm2a−reiopnosrteadrethta+t t=he0t.r4anasnpdortt−nu=mb0e.6rs,
respectively.46
value of U at t = 0 is due to the original barrier layer of pre-
formed porous AAO. The complete filling of pores is 3.4. Stress-Driven Ionic Transport
accompanied by the change of the slope in the U−t curve at The high-field conduction model describes the relation
time t due to the sudden increase of the oxide/electrolyte between ionic current density (j) and the electric field (E)
p
interfacialarea.Forthetimet<t ,thefollowingrelationcanbe well. However, stress gradients in the oxide may possibly
p
obtained:46 contribute to the ionic transport. Hebert and Houser47,48 have
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developed a model for ionic transport in growing amorphous totensiletransitionat0.5−1.0mAcm−2),whileaveragetensile
anodic alumina films, in which ion migration in the oxide is stressoftheorderof50MPawaspredictedabove1mAcm−2,
driven by gradients of mechanical stress as well as electric whichisingoodagreementwiththeexperimentalstressdataof
potential.Italsoconsiderstheviscoelasticcreepoftheoxide.In Bradhurst and Leach.50 In addition, by taking into consid-
other words, both stress gradient-driven ionic migration and eration the viscous flow of oxide material, the model predicted
stress gradient-driven creep are considered in the model. It is the increases of the cation transport number (t+) as a function
assumed that stress originates at the metal/oxide interface due of current density (j). On the basis of experimental evidence
to the volume change upon oxidation. For stress gradient- that cation transport number (t+) is largely dependent on the
driven ionic migration, the empirical high-field conduction electrolyte condition, Hebert and Houser pointed out that the
relation is generalized by considering the dependence of the oxide viscosity and conduction parameters may depend on the
ionic current density on the gradient of the ionic chemical solution composition as a result of electrolyte anion
potential ∇μ:47−49 incorporation into the anodic oxide film. They suggested that
i
bulky electrolyte anions disrupt the local packing of oxygen
Ji = −2|∇∇μμi|Ciui0 sinh⎜⎛⎝RaT|∇μi|⎟⎞⎠ (21) aiodndsitiaonndalinfrfleueenvocelutmraensinptoortthperoapmerotriepshobuystohxeidine.t4r8oduction of
i
whereJ,C,andu0are,respectively,theflux,theconcentration,
and thei prie-exponiential velocity of ion “i” (i = M and O for 4. ELECTROLYTIC BREAKDOWN
metal and oxygen, respectively), and a is the migration jump When valve-metals (e.g., Al, Ta, Nb, Zr, etc.) are anodized
distanceintheoxide.Thechemicalpotentialμisrelatedtothe under galvanostatic conditions, the thickness ofthe oxide films
mean stress (σ) and electrical potential (ϕ) as follows:48 increases linearly with time. Correspondingly, the applied
potential (U) increases linearly with time to keep the electric
μi = ui0 + ziFϕ − Vi̅σ (22) field(E)constantduringtheprocess.Underthiscondition,the
where u0, z, and V̅ are the standard chemical potential, the anodizing potential (U) finally reaches a value at which visible
i i i
chargenumber,andthemolarvolumeofioni,respectively.For sparking on the anode starts appearing, and local thickening,
barrier-type anodic alumina film, the mean normal stress is cracking, blistering, or even burning of oxide film commences.
definedaccordingtoσ=1/3(σ +σ )=2/3σ ,wherex-and This local event is called “electrolytic breakdown”, which not
y-directions are parallel to thxxe intyyerface.47 xFxor the stress only prevents the uniform growth of anodic films over the
gradient-driven oxide creep, the model enforces the con- macroscopic metal surface, but also permanently degrades the
servation of electrical charge and volume and the momentum dielectricpropertiesoftheoxide.Theanodizingpotentialatthe
balance in a Newtonian fluid. For galvanostatic anodization of onset of this local event is called breakdown potential (UB).
aluminum at the applied current density j, the constraint of Because the oxide thickness increases linearly with the
charge conservation can be written as follows: anodizing potential (U) in galvanostatic conditions, the
breakdown is dependent on the oxide thickness and occurs at
j = −2FJO + 3FJM (23) acriticaloxidethickness.Breakdownduringanodizationcanbe
associated with a number of phenomena. These include the
On the other hand, the volume balance is
appearance of visible sparking/luminescence,51−59 the local
jΩM = −V̅ J − v crystallization of oxide,60−66 oxygen evolution at the
3F OO (24) anode,63,67,68 retardation of potential rise,69,70 occurrence of
whereΩ isthemolarvolumeoftheAlatominthemetaland audible cracking,71 and rapid voltage fluctuations.69,70,72 In
M porous AAO growth, breakdown can occur under high current
v is the creep velocity in the oxide. By employing the Maxwell
density anodizing conditions.4 If the reaction heat cannot be
viscoelasticmodelandalsobyassumingalargeelasticmodulus,
the momentum balance in a Newtonian fluid is expressed as47 acdauesqeualoteclayldiniscsriepaasteedinfrocomndthuectiavnitoydean, deleactcruorlyretenth“erautningawmaayy”
0 = 1∇σ + ∇2v + 1∇(∇·v) process. This results in local thickening or burning of anodic
η 3 (25) oxide,terminatinguniformgrowthofporousAAO.Theanodic
oxide in the burnt area exhibits typically a different color from
where η is the viscosity.
the burnt-free areas. For a given anodizing electrolyte, on the
For porous AAO film growing under steady-state,47 the
other hand, porous AAO formed at a potential just below
model predicted that a large compressive interfacial stress
breakdownvalue(i.e.,U<U )exhibitsthebestself-orderingof
causes the lateral flow of oxide materials from the center of pores (section 7.1).18 ImprovBing the breakdown characteristics
pore base toward the cell boundaries and the upward flow in of anodic oxide films through proper control of the electrolyte
the pore wall oxide, as in the flow pattern experimentally
composition, surface state of the starting aluminum, and
observed from W tracer studies (see section 6.3.5). Simulation
reaction heat can allow one not onlyto explore new anodizing
results indicated that the stress field driving the flow results
conditions for self-ordered pore growth, but also to engineer
from the following three origins: “the volume expansion internal pore structures (see sections 7.2−7.4). In this section,
occurring at the metal/oxide interface, nonlinearity of the we discuss some of the electrochemical factors influencing
equations governing conduction of mobile ions (i.e., Al3+ and
breakdown, and models that explain the breakdown phenom-
O2−/OH−), and incorporation of electrolyte-derived anionic
ena.
species within the anodic oxide near the oxide/electrolyte
interface”.47 4.1. Factors Influencing Breakdown
For barrier-type anodic alumina film,48 the model predicted In general, the breakdown potential (U ) is dependent on the
B
the average stress in the oxide to be compressive when the natureofthemetalbeinganodized,thecurrentdensity(j),and
current density is smaller than 0.5 mA cm−2 (i.e., compressive the composition (or resistivity) of the electrolyte. Meanwhile,
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the electrolyte temperature, stirring rate, and history of anodic film. For anodic oxide of aluminum, some authors have
oxide have no influence on the breakdown potential. reported that anion concentration (C −) influences U .76,77
A B
4.1.1. The Nature of Anodized Metal. Wood and Kato et al. showed that at a fixed solution resistivity U
B
Pearson investigated metals whose anodization in 3% decreases linearly with an increase in the logarithm of the
ammonium tartrate ended in sparking, and associated the anionconcentration,ormorespecificallytheanionchargewith
breakdown potential (U ) with the ionic bonding character- the following relation:77
B
istics of the anodic oxides by employing the criteria of Pauling
andWells.TheyestablishedadescendingorderofU according UB = A − BlogCA− (27)
B
tothemelting pointofthecorresponding oxide:Zr(300V)> On the basis of anodization experiments with tantalum in
Al(245V)>Ta(200V)>Nb(190V).72However,Alwittand sulfuric, phosphoric, and hydrochloric acids, Arifuku et al.78
Vijh reported a different descending order of UB for reported that UB is dependent upon the detailed distribution
anodizations of the same metals in the same conditions: Al profiles of incorporated anions in the anodic oxide. Later, the
(350 V) > Zr (315 V) > Ta (275) > Nb (190 V).73 They role of incorporated electrolyte species in the electrical
correlated the increase in UB with the increasing heat of breakdown was emphasized by Albella et al., who have put
formation per equivalent (−ΔHf/equiv) of oxide, which is forward a theory of avalanche breakdown during anodic
approximatelyequaltoone-halfthevalueoftheforbiddenband oxidation.79−81
gapoftheoxide.Further,theynotedthatthedependenceofUB 4.1.3. Current Density (j). For tantalum anodization in
on the band gap would reflect the electronic nature of the ammonium sulfate, Yahalom and co-workers76,82 reported that
breakdownphenomena.Assuch,ratherconflictingreportshave the breakdown potential (U ) is almost independent of the
B
beenpublishedforthedependenceofUBontheintrinsicsolid- current density (j). For anodic films on aluminum, Ikonopisov
statepropertiesofanodicoxides.Iknopisovetal.69pointedout etal.69alsoreportedthata500-foldincreaseofcurrentdensity
that the dependence of UB on the nature of the metal is (j) only lowers the breakdown potential (UB) by 15%. On the
considerably smaller than the dependence on the electrolyte other hand, Di Quarto et al.74,75,83 pointed out the occurrence
resistivity (ρe). of two different kinds of breakdown, that is, “mechanical” and
4.1.2.ElectrolyteConditions. Earlystudies havereported “electrical” breakdown. For anodic oxides of tungsten,74
that the breakdown potential (UB) increases linearly with the zirconium,75 and titanium84,85 under limited conditions, they
logarithm of the electrolyte resistivity (ρe) with the following noted that anodic oxides grew with an increasing number of
equation: defects at a retarded rate (i.e., reduced slope in U−t curve)
during galvanostatic anodizations, until electrical breakdown
UB = A + Blogρe (26) (EB) eventually occurs with visible sparks. They termed this
characteristic growth as mechanical breakdown (MB). For
where A and B are the constants depending on the electrolyte
electrical breakdown (EB), they reported that current density
composition and the anodized metal.69,71,74,75 Figure 7 shows (j) has little effect on the breakdown potential (U ), which is
the dependence ofU onlog ρ during anodization ofNb, Ta, EB
B e in line with the reports of Yahalom et al. and Ikonopisov et
Al, and Zr.69 It appears from the figure that the different al.69,76,82Inthecaseofmechanicalbreakdown(MB),however,
influencesofρeonUBdefeatattemptstosetthemetalsinseries theyobservedthatcurrentdensity(j)hasasignificanteffecton
with respect to the breakdown characteristics of their anodic
the breakdown potential (U ) according to the following
MB
equation:
UMB = AMB + BMB logj (28)
whereA andB areconstants,whichdependmainlyonthe
MB MB
kind of anion in the electrolyte and slightly upon pH and
concentration of electrolyte: B > 0 for anodic oxides of
MB
zirconium and titanium75,85 and B < 0 for anodic oxide of
MB
tungsten.74
4.1.4. Other Factors Influencing Breakdown. The
surface state of the starting metal (i.e., the surface defects
(flaws),purity,processinghistory,etc.)alsostronglyinfluences
the breakdown potential (U ).61,86 In general, the surface
B
defects unavoidably cause a decrease of the breakdown
potential (U ) with the commencement of sparks.87 On the
B
other hand, post-breakdown anodization experiments have
shown that breakdown characteristics are independent of the
historyoftheanodicoxidefilm.72,88,89Whenavalve-metalwas
anodizedinelectrolyteAuntilbreakdownoccurredatU ,and
B,A
thentheresultingsamplewasre-anodizedinelectrolyteBwith
ahigherbreakdownpotential(U ),thefilmformationduring
B,B
the post-breakdown anodization continued at normal kinetics
until breakdown occurred at U .88,89 Temperature (T) is one
B,B
of the easily controllable parameters of the electrolyte.
Figure 7. Dependence of the breakdown potential (U ) on the
B
logarithmofelectrolyteresistivity(ρ)foranodizationsofTa,Nb,Al, Ikonopisov formulated the temperature dependence of the
and Zr in solutions of ammoniumesalicyalte in dimethylformamide. breakdownpotentialUB(section4.2.1).90However,achangein
Reproduced with permission fromref 69. Copyright 1979 Elsevier. the electrolyte temperature can alter both the electrolyte
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Wreshisetnivittyhe(ρed)eapnedndtehnecperoopferteyleocftrtohleytgerorweisnisgtiavnitoydic(ρox)ideis. je,x=0 = α1exp[α2E1/2] (31)
e
considered, no clearly pronounced dependence of the break- lnj = β/T + β
down potential (U ) on the temperature (T) was ob- e,x=0 1 2 (32)
B
t4a.i2n.eMd.7o1d,7e7,l9s1for Breakdown je,x=0 = γ1ρe−γ2 (33)
4.2.1. Electron Avalanche Multiplication. The first From eqs 30 and 31, the dependence of breakdown potential
attempt to develop a quantitative model of breakdown was (U ) on the electric field (E) is given by
B
made by Iknopisov.90 He considered experimentally observed
U = (ψ/rq)(lnj − lnα) − (ψα /rq) E
breakdown characteristics, and noting that the breakdown B i e,B 1 i 2 (34)
potential (U ) mainly depends on the nature of the anodized
B This equation explains a slight decrease of U with increasing
metal and the electrolyte resistivity (ρ), he inferred that B
e current density (j). Regarding the relation between the
breakdown is dependent upon the solid-state properties of the
breakdown potential (U ) and temperature (T), the following
anodic oxide and is controlled by electrochemical reactions at B
expression is obtained by combining eqs 30 and 32:
the oxide/electrolyte interface. In his model, the initial
electrons are injected from the electrolyte into the oxide U = (ψ/rq)(lnj − β/T − β )
conduction band (CB) by either a Fowler−Nordheim or a B i e,B 1 2 (35)
Schottky mechanism (Figure 8). The injected electrons Equation 35 predicts that UB is dependent on the temperature
(T), which conflicts with experimental observations.71,77,91 For
thisdiscrepancy,Ikonopisovpointedouttheinterplaybetween
temperature (T) and the solution resistivity (ρ). For the
e
dependence of breakdown potential (U ) on the electrolyte
B
resistivity (ρ), from eqs 30 and 33, one may obtain
e
U = (ψ/rq)(lnj − lnγ + γ lnρ)
B i e,B 1 2 e
= (ψ/rq)(lnj − lnγ) + (2.3ψγ/rq)logρ
i e,B 1 i2 e (36)
Equation 36 has exactly the same form as eq 26, which
describes an empirical relation between U and ρ.
B e
Although Ikonopisov’s model explains some of the
experimental results, it has been criticized by many authors.
Shimizupointedouttheunrealisticvalueofthemeanfreepath
λ(E) of ionized electrons, which from eqs 26, 29, and 36 is
given by
λ(E) = 1/α(E) = ψ/rqE = (1/2.3E)(B/γ) (37)
i 2
By using the experimental values from Ikonopisov et al. for E
(=8.7×106Vcm−1)andB/γ (=1000V),Shimizuobtainedλ
2
Figure 8. Schematic representation of the band structure and the =500nm,whichroughlycorrespondstothethicknessofoxide
avalanche breakdown in Ikonopisov’s model. Reproduced with films formed up to the potential 400 V and indicates the
permissionfromref79.Copyright1984TheElectrochemicalSociety. absence of the electron avalanche capable of causing the
breakdown.94Albellaetal.questionedtheoriginofelectronsin
Ikonopisov’s model.87 They pointed out Ikonopisov’s model
accelerate and multiply in avalanche during their travel in the
lacked a reasonable explanation of the role of the electrolyte
oxideofthickness(tox)totheanode,untiltheavalanchecurrent and the absence of specific electrochemical reactions required
reaches a critical value for breakdown. In this multiplication
for the injection of the initial electrons.
process, the electronic current (je) depends on the travel Albella et al. explicitly considered the effect of the anodizing
distance (x) with x = 0 being the oxide/electrolyte interface,
electrolyte by posulating that the initial electrons for the
which can be expressed by
avalanche come from the electrolyte species incorporated into
j = j exp[α(E)t ] = j exp[rqEt /ψ] anodic oxide.79−81 The incorporated electrolyte species act as
e,x=t e,x=0 ox e,x=0 ox i (29)
ox impurity centers close to the oxide conduction band (CB),
where α(E) is the impact ionization coefficient at the electric releasingelectronstotheconductionbandviathefield-assisted
field E, r is a recombination constant (r < 1), q is the electron Poole−Frenkelmechanism(Figure9).80,87,95,96Inthemodelof
charge, and ψi is the threshold energy for impact ionization. Albella et al., the total current density (jt) consists of three
Breakdownoccursiftheelectroniccurrent(je)exceedsacritical components:
valuej atacriticaloxidethickness(t ).BecauseU =Et ,
e,B ox,B B ox,B j = j + j + j
the breakdown potential (U ) is given by t 1 2 e (38)
B
UB = (ψi/rq)(lnje,B − lnje,x=0) (30) dwehnesrietyjc1onissutmheedobxyidtahtieoinnccourrproernatteddeenlseicttyr,oljy2teissptehceiescuarnrdenist
The dependences of j on the electric field (E),69,92 assumed tobea constantfaction γofj (i.e., j =γj ), andj is
e,x=0 1 2 1 e
temperature (T), and electrolyte resistivity (ρ )93 were the electronic current density. The electronic current density
e
empirically determined to be, respectively: (j) at the anode can be expressed as
e
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1 + φγ
V(t) = Kjt
1 + γ t (45)
γη E
ΔV(t) = [exp(αU/E) − 1]
1 + γ α (46)
The factor (1 + φγ)/(1 + γ) in eq 45 describes the correction
of the anodizing rate due to the incorporation of electrolyte
species. On the other hand, eq 46 enforces deviation of
potential from the linearity due to the avalanche effect.
Accordingly, eq 44 predicts a gradual decrease of slope (dU/
dt) of the potential−time curve during galvanostatic anodiza-
tion.Albellaetal.confirmedthevalidityofeq44byfittingiton
theexperimentalresultsoftantalumanodization(Figure10).80
Figure 9. Band diagram showing the avalanche multiplication of
electrons in the model of Albella et al. The impurity level in
conduction band (CB) is denoted by “A”. Reproduced with
permission fromref 81. Copyright 1987 Elsevier.
j = j exp[αt ]=j exp[αβU]
e e,x=0 ox e,x=0 (39)
whereαistheimpactionizationcoefficient,andβistheratioof
the oxide thickness to the anodization potential (U) and is
equal to the inverse of the electric field E (i.e., β = 1/E = AR,
the anodizing ratio). Because the initial electronic current
originatesfromincorporatedspecies,j shouldbeaconstant
e,x=0
fraction η of oxyanion current (j ) and is j = ηj = ηγj .
2 e,x=0 2 1
Under the assumption that the critical current density is a
fraction z of the oxidation current j , the breakdown potential
1
(U ) should satisfy the following relation:
B
j exp[αβU ] = zj
e,x=0 B 1 (40)
Figure10.Experimentalresultfortheevolutionofthepotential (U)
Accordingly, the breakdown potential (U ) is given by asafunctionoftime(t)duringtantalumanodizationin1.2MHPO
B 3 4
at 1.78 mA cm−2. The theoretical curve (solid line) has been fitted
UB = (1/αβ)ln(zj1/j0) = (E/α)ln(z/ηγ) (41) according to eqs 44−46. Reproduced with permission from ref 80.
Copyright 1985 Elsevier.
The time derivative of the potential is given by
dU = ⎛⎜⎜ 1 ⎞⎟⎟⎜⎛⎝E⎟⎞⎠⎛⎜⎜ M1j + M2j ⎞⎟⎟ Fevuorltuhteior,nsbyinfidttiiffnegrenetq e4le4ctroonlytetheconecxepnetrrimatieonntsal(Cpo),tenthtieayl
dt ⎝ρox⎠ F ⎝x1y11 x2y2 2⎠ (42) obtained a relation between γ and C:95
where ρox is the oxide density, F is the Faraday constant, and γ ≈ aCb (47)
M and M are the molecular weights of the oxide and the
1 2
with a and b being electrolyte-dependent constants. From eqs
incorporated species, respectively, whose corresponding anion
41 and 47, the concentration dependence of the breakdown
andcationvalencesarex ,y andx ,y ,respectively.Combining
1 1 2 2
eqs 38 and 42 yields the following differential equation: potential (UB) is given by
dU/dt = Kj(1 + φγ)[1 + γ + γηexp(αβU)]−1 UB ≈ (E/α)[ln(z/ηa) − blnC] (48)
t (43)
which is in good agreement with the experiments.
where K is the unitary rate of anodization for oxide without 4.2.2.Stress-DrivenBreakdown.Sato97distinguishedfive
electrolyte incorporation and given by M1E/x1y1ρoxF, and φ is different possible contributions to the mechanical stresses in
theratiooftheequivalentweightoftheincorporatedspeciesto anodic oxide: (a) electrostriction pressure, (b) interfacial
that of oxide, that is, φ = (M2/x2y2)/(M1/x1y1). Assuming a tension of the film, (c) internal stress caused by the volume
constant field in the oxide, the integration of eq 43 yields the expansion, (d) internal stress due to partial hydration/
relation between the anodizing potential (U) and time (t): dehydration of the anodic oxide, and (e) local stress caused
by impurities. By considering the first two contributions as the
U(t) = V(t) − ΔV(t) (44)
most general factors for breakdown, hemathematically derived
with V(t) and ΔV(t) given by a thermodynamic model of stress-driven breakdown, and
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