Chromate conversion coatings formation on zinc studied by electrochemical and electrohydrodynamical impedances

Article abstract of DOI:10.1149/1.1528196

The formation of chromate conversion coatings on zinc was studied by chronopotentiometric, electrochemical, and electrohydrodynamic impedances, and interfacial pH measurements. The electrochemical experiments were performed with a rotating disk electrode of pure zinc, and the pH measurements were obtained with a zinc deposit on a gold grid electrode in a submerged impinging jet cell. The electrolyte was an industrial chromate bath. The experimental results were achieved for different immersion times, temperatures, and rotation speeds. Kinetic reactions and physical model for the chromate layer formation on zinc were proposed, and the electrochemical and electrodynamic impedances were well simulated.

Full text of DOI:10.1149/1.1528196

B16
Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
0
013-4651/2002/150͑1͒/B16/10/$7.00 © The Electrochemical Society, Inc.
Chromate Conversion Coatings Formation on Zinc Studied
by Electrochemical and Electrohydrodynamical Impedances
a,b
b,
a, ,z
a,c
A. A. O. Magalh
and O. E. Barcia
˜
aes, B. Tribollet, * O. R. Mattos, * I. C. P. Margarit,
d
aPrograma de Engenharia Metalurgica e de Materials/Dipartamento de Engenharia Metal u´ rgica e de
Materials/Instituto Alberto Luiz Coimbra de P o` s Gradua c¸ ao e Pequ `ı sa de Engenharia, Escola de Engenharia
˜
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
b
Unit e´ Propre de Recherche 15 du Centre National de la Recherche Scientifique/Physique des Liquides et
Electrochimie, Universit e´ Paris VI, Paris, France
c
Escola de Qu ´ı mica Departamento de Processos Inorg aˆ nicos, dInstituto de Qu ´ı mica, Departamento de
Fis ´ı co-Qu ´ı mica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
The formation of chromate conversion coatings on zinc was studied by chronopotentiometric, electrochemical, and electrohydro-
dynamic impedances, and interfacial pH measurements. The electrochemical experiments were performed with a rotating disk
electrode of pure zinc, and the pH measurements were obtained with a zinc deposit on a gold grid electrode in a submerged
impinging jet cell. The electrolyte was an industrial chromate bath. The experimental results were achieved for different immer-
sion times, temperatures, and rotation speeds. Kinetic reactions and physical model for the chromate layer formation on zinc were
proposed, and the electrochemical and electrohydrodynamic impedances were well simulated.
©
2002 The Electrochemical Society. ͓DOI: 10.1149/1.1528196͔ All rights reserved.
Manuscript submitted January 18, 2002; revised manuscript received July 22, 2002. Available electronically December 13, 2002.
Several sectors of automobile, aerospace, and electrical appli-
the two main stages for the formation of a layers of chromatation are
the dissolution of zinc in an acidic electrolyte and the formation of a
precipitate of trivalent chromium.
ances industries improve the anticorrosive properties of their galva-
nized products by application of chromate conversion coatings on
zinc deposits. The objective is to decrease the formation of corrosion
products, and, thereby, to avoid modification of their properties dur-
ing transportation and storage. The conversion treatments can also
act as a support for the application of painting if direct adherence of
the organic coatings on the substrate is not satisfactory. The prin-
ciple of a conversion treatment consists basically in transforming the
surface of the metallic substrate from the active state to the passive
state. The formed layer is composed of a combination of reduced
species products from ions present in the conversion bath and, in
smaller quantities, corrosion products from the substrate.
Trivalent chromium can form different types of oxides and/or
hydroxide compounds. During reduction of hexavalent chromium
ϩ
there is a consumption of H , which increases the local pH and
allows precipitation of a gel-like film.
The aim of this work is to employ infrequently used modern
electrochemical techniques in the chromatation process studies. We
hope that this approach can be useful for further research concerning
nontoxic alternatives for conversion baths.
Experimental
The chromate coating is usually applied on galvanized steels
through simple immersion in an acidic solution that contains chro-
mium hexavalent species. The quality and the efficiency of these
layers are strongly dependent on the bath composition, the surface
of substrate, and the parameters of the process. Many studies in the
literature concern the mechanisms of deterioration and the develop-
ment of control and assessment methods for the quality of these
All the electrochemical measurements were performed with a
three-electrode cell. The working electrode was manufactured from
pure zinc ͑99.9%͒ rod. This electrode was covered by a cataphoretic
2
paint and coated with a resin epoxy to expose a 0.27 cm cross-
sectional area to the electrolyte. The electrode surface treatment pro-
tocol consisted of polishing with emery paper 600 and 1200, rinsing
with double-distilled water, rinsing with ethanol, and finally drying
with air. The reference electrode was a saturated calomel electrode
1
-8
conversion layers. Even though chromatation is used in industry
its deposition mechanism is not yet completely understood.
͑SCE͒ and the counter electrode was a large platinum grid.
The chemical composition of these layers is not yet well defined;
the difficulties of a composition survey of this type are due mainly
to the thinness of the layers. The main parameters that control the
composition of the chromatation layers on zinc are the quality of the
zinc surface, the composition of the bath, the pH of the solution, the
immersion time in the bath, the bath temperature, and the thickness
of the layer. According to the majority of works, the main constitu-
ent of the formed layer is trivalent chromium in the form of
The electrolytic solution was a 5% v/v solution of a concentrated
commercial product, which consisted primarily of sodium chromate
and sulfuric acid. This solution is used in the chromatation step of an
electrogalvanized steel industrial process in a Brazilian metallurgy
society. The chromate conversion coatings formed by this solution
with temperature between 47 and 48°C and pH between 2.96 and
.00, showed good corrosion resistance as presented previously.1
3
The electrochemical tests were conducted at pH 3.0, at tempera-
tures of 25 and 47°C, controlled by a thermostatic bath.
9
-12
4,7,13
3,14,15
oxides,
hydroxides,
and complex.
The hexavalent chro-
mium species is found in least quantity in the form of complex
compounds with trivalent chromium or simply absorbed in the layer.
Only trace amounts of zinc and such activator anions as phosphates,
nitrates, sulfates, chlorides, and fluorides are present.
Although there are a great variety and complexity of reactions
proposed for the zinc/chromate solution interface,16 this formation
obeys a zero net current as a result of the compensation of a reduc-
tion component from the electrolyte and an oxidization component
of the metal. A consensus exists in the literature that maintains that
The zinc rotating disk electrode ͑RDE͒ open-circuit potential
͑OCP͒ evolution in the chromate solution was achieved using a po-
tentiostat Ominimetra model PG-19 for 30 min of immersion for
four different rotation speeds ͑0, 250, 500, and 1000 rpm͒. A dc
motor with a servo system controlled the RDE and the rotation rate
was measured with a tachometer made by Asservissement Eletron-
ique under a CNRS/France license. The ac impedance measurements
were conducted at the OCP of the system under potentiostatic regu-
lation at three different immersion times: 2, 15, and 30 min, and
three rotation speeds: 250, 500, and 1000 rpm. The electrohydrody-
namic impedance spectroscopy ͑EHD͒ measurements were con-
ducted under potentiostatic regulation. The rotation rate was initially
fixed at 1000 rpm, and after 15 and 30 min of immersion, the OCP
*
Electrochemical Society Active Member.
E-mail: Omattos@metalmat.ufrj.br
z

Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
B17
corresponding to this rotation was applied and the measures were
done reducing the rotation rates. The rotation rates measured were
40, 480, and 720 rpm. The ac impedance measurements were ob-
2
tained for frequencies varying from 65 kHz to 100 mHz using a
sinusoidal amplitude of 10 mV applied by the generator of a
frequency-response analyzer ͑FRA, Solartron 1250͒ controlled with
an IBM PC using software developed in the UFRJ Corrosion Labo-
ratory in Brazil. For EHD measurements, a sinusoidal speed modu-
lation of 10% around a mean value was applied by the system de-
scribed above.
The interfacial pH measurements were obtained using a working
electrode with an 82 mesh gold grid ͑Goldfellow, wire diam: 0.06
mm, open area 64%͒. A 0.5 mm diam gold wire was welded onto the
periphery of the gold grid to provide an electrical connection. A zinc
deposit was formed on the gold grid using a solution of 350 g/L of
ZnSO •7H O and 30 g/L (NH ) SO pH 4.0 at 25°C and a current
4
2
4
2
4
2
density of 180 mA/cm for 7 min. After deposition, the grid was
dried with an argon jet and contacted with the flat end of a combined
pH electrode and fixed to the lateral wall of the pH electrode with
Teflon tape. The pH electrode/zinc deposit grid electrode assembly
was fixed in a submerged impinging jet cell. The jet cell and all the
17
pH setup are the same as used by Deslouis et al. The OCP of the
grid electrode in the chromate solution was monitored against a
saturated sulfate electrode ͑SSE͒ with an electronic potentiostat
͑
͑
Sotelem͒ and the pH was measured simultaneously with a pH meter
LPH 230T Radiometer͒.
Results and Discussion
OCP.—Figure 1 presents the evolution of the OCP as a function of
immersion time in the chromate solution for the stationary electrode
͑
͑
0 rpm͒ and with rotation speeds of 250, 500, and 1000 rpm at 25
Fig. 1a͒ and 47°C ͑Fig. 1b͒. During the first seconds of immersion
at 47°C, the potential decreased toward more cathodic values. After
this reduction, the potential stabilized for all rotation speeds to val-
ues close to Ϫ1.32 mV, and the potential remained steady during
the entire immersion time. It is supposed then that the film covers
the substrate surface completely, and the dissolution of zinc contin-
ues to take place through the formed layer. The potential stabiliza-
tion speed is greater for the smallest rotation speeds. The potential
of a stationary electrode stabilizes more quickly than the potential of
an electrode rotating at a speed of 1000 rpm. This behavior is more
evident when the temperature of the solution is reduced. The poten-
tial curves at 25°C are presented for the same rotation speeds as
shown for 47°C. The influence of the speed variation of the elec-
trode in the process appears more clearly, it changes not only the
time of stabilization, but also the values of the potential. With these
results it is possible to identify some effects of temperature and mass
transport in the process. First, the rotation speed influences film
formation, maybe acting on zinc dissolution rate at the interface.
Second, the temperature, when it increases from 25 to 47°C, pro-
vides an increase of the zinc dissolution velocity, which is almost
the same for all rotations tested.
Figure 1. OCP of zinc RDE in function of immersion time in the chromate
solution at ͑a͒ 25 and ͑b͒ 47°C.
quency limit of the impedance associated to diffusion is directly
proportional to the thickness of the layer. The reduction of the size
of the capacitive diffusive loop when the rotation speed of the elec-
trode increases ͑see Fig. 2c and d͒ can be assigned to diffusion
within the solution and/or to diffusion through a film whose thick-
ness decreases with the rotation speed. This last hypothesis probably
can occur if the layer is a gel-like film.
Due to the mixed potential of the electrode, the inductive loop
can be explained by either considering the anodic component sepa-
rately from the cathodic ͑adsorption of an activating species or de-
sorption of a blocking species with increasing overpotential of each
component͒ or considering the adsorption of only one and same
species interacting with the two components. Two cases can be con-
sidered: (i) the anodic adsorption ͑therefore cathodic desorption͒ of
a blocking species for the two reactions ͑passive layer͒. The induc-
tive loop appears then if the dominant effect is in the cathodic com-
ponent or (ii) or the anodic desorption ͑therefore cathodic adsorp-
tion͒ of a blocking species for the reactions ͑chromate layer͒. The
inductive loop appears then if the dominant effect is in the anodic
component.
AC and EHD impedance measurements.—The electrical impedence
spectroscopy ͑EIS͒ diagrams obtained during the chromatation pro-
cess are presented in Fig. 2. The diagrams present a relatively com-
plex shape that suggest the presence of two capacitive loops at high
frequencies ͑HF͒ followed by an inductive loop at low frequencies
͑
LF͒. The first capacitive loop at higher frequencies is assigned to
the charge-transfer resistance of the electrochemical reactions in par-
allel with the double layer ͑dl͒ capacitance. The second capacitive
loop at intermediate frequencies presents a slight inclination that
could correspond to a diffusion phenomena. This diffusion could
take place in the electrolyte and/or in the chromate film.
The increase of the size of the diffusive loop with immersion
time of the electrode in the solution ͑see Fig. 2a and b͒ can be
explained by diffusion in a layer of increasing thickness, by diffu-
sion in a porous film in which the pores contract or by diffusion in
a gel-like film that becomes increasingly cross-linked. The low fre-
Only a weak effect of temperature is seen in these impedance
diagrams by comparing the diagrams obtained at 25 ͑Fig. 2b and d͒
and 47°C ͑Fig. 2a and c͒. Therefore, it can be considered that the
effect of the mass transport in this system has a higher influence
than the temperature effect in the kinetics of chromate film forma-
tion. Thus, the EHD technique seems well adapted to this problem.
Figure 3 presents the EHD of zinc RDE immersed in the chro-
mate solution at 25 and 47°C. The shape of the experimental dia-

B18
Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
Figure 3. Experimental EHD diagrams after 15 min immersion in the solu-
tion at ͑a͒ 25 and ͑b͒ 47°C. ͑-᭹-͒ 240, ͑-छ-͒ 480, and ͑-᭿-͒ 720 rpm.
shift of characteristic frequencies toward lower values as rotation
speed increased is qualitatively consistent with diffusion through a
1
8
porous layer. In conclusion, this qualitative analysis in the shape of
the EHD seems to show that the layer of chromatation possesses a
mixed behavior in relation to diffusion. The diffusion is affected by
both a partially blocked electrode and a porous layer. However, at
4
2
7°C ͑Fig. 3b͒, the behavior is quite different from that seen at
5°C; an increase of the impedance modulus at intermediate fre-
quencies is observed. Such behavior was never modeled for any
system. We proposed a model that takes into account the oxidization
6
ϩ
of zinc and the reduction of Cr simultaneously, where the process
of film formation occurs at the OCP in the electrode interface.
Local pH measurements.—Figure 4 shows the behavior of the pH
interfacial and the rest potential of the gold grid with zinc deposited
in the chromate solution. At immersion of the electrode in the solu-
tion, a reduction of pH occurs, and, after some seconds, a plateau is
seen around pH 4.0. At the same time, there is a reduction of the
potential followed by a plateau around Ϫ1.32 Vsse
.
The pH of the bulk solution is 3.0, so the elevated pH plateau
ϩ
shows that there is consumption of H at the zinc/electrolyte inter-
face. Such behavior is in agreement with the literature, which shows
ϩ
consumption of H at the interface providing the precipitation of
4
,6,19
trivalent chrome products on the zinc surface.
The potential
plateau is the same as obtained for the pure zinc electrode ͑Fig. 1͒
and corresponds to the formation of chromate film on a zinc surface
that dissolves at the same time. After the first pH plateau there is an
abrupt reduction for values around pH 3.2 that corresponds to direct
contact of the solution with the gold substrate. This is due to the
dissolution of the zinc in the electrode, masking the value of actual
pH on the surface of the remaining zinc. After 15 min of immersion,
the potential begins to increase indicating, at the end of the test, that
the zinc is totally dissolved from the surface electrode. The pH stays
at the value of bulk solution.
Kinetic and Physical model
Figure 2. Experimental electrochemical impedances diagrams at 1000 rpm
after ͑-᭺-͒ 2, ͑-छ-͒ 15, and ͑-ᮀ-͒ 30 min of immersion in the solution at ͑a͒
The chemistry of chromate baths is complex and involves many
2
0
equilibrium reactions. However, the chromatation itself occurs as a
heterogeneous process at the metal/solution interface. In this case,
anodic reactions play an important role. Due to the complexity of
the process, it is difficult to generalize results obtained with
4
7°C ͑a͒ and ͑b͒ 25°C and diagrams after 30 min at ͑-᭺-͒ 250, ͑-छ-͒ 500,
and ͑-ᮀ-͒ 1000 rpm in the solution at at 47°C ͑c͒ and ͑d͒ 25°C ͑frequencies
in hertz͒.
2
1
aluminum or others metals to this study concerning zinc. Actually,
the interfacial pH of electrochemical processes depends significantly
2
2
gram at 25°C ͑Fig. 3a͒ shows two time constants, one at low fre-
quencies and another at high frequencies, that do not vary with the
increase of the immersion time. In a preliminary analysis, such ob-
servations indicate that the chromate layer is partially blocking. The
on the metal and/or bath.
The model proposed below is very simple and did not take into
account the complexity of the chromatation process. Moreover, a
kinetic model is never unique and the main purpose of this model is

Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
B19
Figure 5. Physical model scheme for the formation of chromate conversion
layer on Zn in an acid chromate solution.
ϩ
The H concentration gradient ͑Fig. 5͒ is distributed between the
chromate layer, where mass transport is controlled by molecular
diffusion, and the electrolyte diffusion layer, where mass transport is
controlled by convective diffusion. For this model, the transport
becomes one-dimensional ͑1-D, along the axial coordinate of RDE͒
where ␭ is the chromate layer thickness, ␦ is the diffusion layer
ϩ
thickness, C is the concentration of the H in the bulk solution, and
ϱ
ϩ
C the concentration of H at the zinc/chromate layer interface.
0
Steady-state equations.—From the model of four kinetic reac-
tions proposed above the steady-state faradaic current may be ob-
tained
i
2
0
ϭ 2k ͑1 Ϫ ␪͒ ϩ 2k ͑1 Ϫ ␪͒ Ϫ 2k ␤␪C Ϫ 3k ͑1 Ϫ ␪͒
1
2
22
3
Figure 4. OCP and pH evolution of a zinc electrodeposited grid in the
chromate solution during ͑a͒ entire time of measurements and ͑b͒ the first 10
min.
F
2
4
Ϫ
5
0
ϫ ͓CrO ͔C ϩ 2k ␤␪
͓5͔
4
where ZnO is an insoluble adsorbed species whose surface coverage
is ␪, ␤ being the total number of sites available for adsorption. F is
to verify whether the qualitative ideas presented during the analysis
of the new results obtained by ac and EHD can be supported by an
electrochemical model.
The proposed kinetic reaction model considers the following re-
actions
the Faraday number, and the rate constants k obey the Tafel law,
i
which are written in the standard way: k ϭ k exp(b E), with bi
i
i0
i
ϭ nF/RT. The mass balance for the ZnO species is
d␪
k
1
2
0
␤
ϭ k ͑1 Ϫ ␪͒ Ϫ k ␤␪C
͓6͔
2
22
2
ϩ
Ϫ
dt
Zn ——→ Zn ϩ 2e
͓1͔
͓2͔
k
2
Under the assumption of a steady state (d␪/dt ϭ 0) and that Re-
action 2 is at equilibrium
ϩ
Ϫ
Zn ϩ H O ꢀ ZnO ϩ 2H ϩ 2e
2
(␪)
k
2
2
k2
k
3
␪
ϭ
͓7͔
2
Ϫ
ϩ
Ϫ
2
0
CrO4 ϩ 5H ϩ 3e ——→ Cr͑OH͒ ϩ H O
͓3͔
͓4͔
k ϩ k ␤C
2 22
3
2
k
4
2
ϩ
Ϫ
The overall steady-state current is equal to zero (i ϭ 0). k30 can be
Zn ϩ ZnO(␪) ——→ Zn ϩ ZnO(␪) ϩ 2e
derived from Eq. 5 as
2
The first reaction corresponds to zinc dissolution. The second
2k k C ϩ 2k k
1
22
0
4
2
4,14
reversible reaction corresponds to zinc oxide formation
as an
k30 ϭ
ͩ
2Ϫ
7
0
ͪ
exp͑b3E͒
͓8͔
3
k ͓CrO ͔C
22
4
insoluble adsorbed species whose surface coverage is ␪. The third
reaction corresponds to a formation of a precipitated trivalent chro-
ϩ
Under these conditions, the H consumption is due only to Reaction
3. By taking into account Eq. 7, the H flux is given by
mium ͓assumably Cr(OH) ] product from the reduction of hexava-
3
ϩ
ϩ
lent chromium species and consumption of H . This reaction can be
replaced by any other source of Cr͑VI͒, for example Cr O . For
2
7
Ϫ
2Ϫ
7
0
2
¯
dCHϩ
5k3k22␤͓CrO4 ͔C
the present purpose we need only a Cr͑VI͒ source and, as we are
Df
ϭ
͓9͔
ͯ
2
0
dy
k ϩ k ␤C
2
working with Na CrO , Reaction 3 was taken. The last reaction
0
22
2
4
is an autocatalytic reaction of zinc dissolution. A reaction as
xϪ
ϩ
Cr(VI) ϩ 4H ϩ 3e → Cr(OH) ϩ yH O could well be used
where Df is the diffusion coefficient relative to the chromate film.
The H flux at the surface can be expressed as
3
2
ϩ
replacing Reaction 3.

B20
Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
¯
where the function WH corresponds to the EHD given in Ref. 23, 24
dC
Hϩ
C Ϫ C0
Cϱ Ϫ C␭
␭
Df
ͯ
ϭ Df
ϭ Ds
͓10͔
as W (␰) and Ϫ1/␪Ј(0) is the usual dimensionless convective dif-
0
dy
␭
␦
0
fusion impedance which is a function of the dimensionless fre-
quency p ͑␻/⍀͒ and the Schmidt number of the diffusing
2
3,24
ϩ
where D is the diffusion coefficient relative to the electrolyte dif-
fusion layer, and C␭ is the H concentration at the interface chro-
mate layer/electrolyte. Thus
s
species.
electrolyte.
Position ␭ indicates the concentration gradient in the
ϩ
Chromate layer/electrolyte interface.—At the chromate layer/
ϩ
Df
electrolyte interface, the balance of the flux of H is given by
2
4
Ϫ
7
0
¯
dC
Hϩ
␭
Cϱ Ϫ C0
5k3k22␤͓CrO ͔C
˜
˜
Hϩ
dC
Hϩ
dC
Df
ͯ
ϭ
ͩ
Ds
ͪ
ϭ
2
0
dy
Df
Ds
␦
k ϩ k ␤C
2
Ds
ͯ
ϭ Df
ͯ
͓18͔
22
0
dy
dy
ϩ
␭ϩ
␭Ϫ
␭
␦
͓11͔
Ϫ
where ␭ indicates the concentration gradient in the chromate layer.
2
Diffusion in the chromate layer.—In the chromate layer, the diffu-
sion of H ion is given by
Under assumptions that Cϱ ӷ C and ␪ Ӷ 1 (k Ӷ k ␤C )
0
2
22
0
ϩ
Df
ץ2CHϩ
ץt2
ץCH
ϩ
␭
Cϱ
ϭ Df
͓19͔
͓20͔
2
Ϫ
5
ͩ
Ds
ͪ
ϭ 5k3͓CrO4 ͔C0
͓12͔
ץt
Df
Ds
␦
ϩ
␭
␦
By using an axial dimensionless distance defined by
y
and an expression can be obtained for C that is a function of rota-
␹ ϭ
0
␭
tion speed and potential of the electrode ͓C ϭ f(⍀,E)͔
0
ץC
Hϩ
ץC
Hϩ
1/5
Df
ϭ
ͩ
ͪ
ͩ
ͪ
ץt
ץt
y
␹
␭
Cϱ
1
C ϭ
0
ͩ
Ds
ͪ
͓13͔
2Ϫ
ͫ
Df
Ds
␦
5k ͓CrO4
͔
ͬ
3
The diffusion equation becomes
ϩ
␭
␦
2
˜
ץ C
Hϩ
˜
ϭ j␮C
Hϩ
ץ␹2
where
1/3
3
Ds
v
ͱ
⍀
where
␦
ϭ
ͩ
ͪ
͓14͔
0
.51023v
␭2
␻
␮
ϭ
͓21͔
͓22͔
Nonsteady-state equations.—The concentration distribution in the
diffusion layer is governed by the 1-D nonsteady equation of con-
vective diffusion, such that the usual mass-balance equation can be
integrated in the electrolyte on a normal distance to the chromate
layer
Df
The general solution is
˜
ϭ G exp ͱj␮␹ ϩ H exp Ϫ ͱj␮␹
C
Hϩ
ץ2CH
ץy2
ϩ
where G and H may be obtained by considering the boundary con-
ditions.
ץCH
ϩ
ץCH
ϩ
ϩ Vy
ϭ Ds
͓15͔
ץt
ץy
At the zinc interface (␹ ϭ 0)
The development of V is provided in Ref. 23, 24. Upon introducing
˜
C ϭ G ϩ H
0
y
¯
CHϩ
ϭ CHϩ ϩ Re ˜c exp j␻t and in terms of the dimensionless
͕ ͖
and
variables and parameters: (i) a ϭ 0.51023, (ii) ␩ ϭ y/␦, (iii),
2
1/3
and (iv) ␰ ϭ ␻␦ /D ϭ 3.258 pSc . Where Sc ϭ ␷/D is the
s
˜
Hϩ
ץC
Schmidt number, and the equivalent Nernst diffusion layer thickness
ϭ
ͱ
ͱ
j␮ G Ϫ H
͒
͓23͔
ͯ
ͯ
͑
ץ␹
is ⌫͑4/3͒ times ␦, Eq. 15 becomes
0
2
Thus
˜
˜
˜
˜
Hϩ
d C
Hϩ
dC
Hϩ
⌬⍀ 3fЈ͑p͒
dC
ϩ 3␩2
Ϫ j␰CHϩ ϭ Ϫ
␩ Ds
2
˜
d␩2
d␩
¯
a
d␩
˜
ץCHϩ
j␮
⍀
ϭ
͑G Ϫ H͒
͓24͔
͓25͔
͓26͔
ץy
␭
0
where
At the chromate layer/electrolyte interface (␹ ϭ 1)
¯
dC
Hϩ
Cϱ Ϫ C␭
ϭ
exp Ϫ ␩3
͓16͔
C ϭ ͑G exp
ͱj␮ ϩ H exp Ϫ ͱj␮͒
˜
␭
d␩
⌫͑4/3͒
thus
By using the same procedure as described in Ref. 23, 24, the con-
vective diffusion equation for H is obtained as
ϩ
˜
ͱ
j␮
␭
dC
Hϩ
ͯ
ϭ
͑G exp j␮ Ϫ H exp Ϫ j␮͒
ͱ ͱ
dy
␭Ϫ
˜
˜
¯
ϩ
dC
Hϩ
C ␪Ј͑0͒
⌬⍀ dCH
␭
ͯ
ϭ
ϩ
ͯ
WH
͓17͔
Chromate layer/zinc interface.—Due to the electrochemical reac-
dy
␦
¯
⍀
dy
␭ϩ
␭ϩ
tions, the nonsteady kinetic equations are

Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
B21
1
2
2Ϫ
5
Ϫ␭M
␭N
1
2
˜
͓Ϫ2͑k1 ϩ k ͒ Ϫ 2k22␤C ϩ 3k ͓CrO ͔C ϩ 2k ␤͔
˜
4
␪ ϩ
˜
Eϩ
i
2
0
3
4
0
˜
˜
C
0
RtF
H ϭ
E Ϫ
Ϫ
͓36͔
ϭ
ͩ
ͪ ͩ
ͪ
F
ͭ
ͮ
2Df
ͱ
j␮
2Df j␮
ͱ
2
4
Ϫ
4
0
˜
ϩ͓Ϫ4k22␤␪C0 Ϫ 15k3͑1 Ϫ ␪͓͒CrO ͔C ͔C0
Upon including Eq. 25, 26, 35, and 36 in Eq. 19
͓
27͔
˜
dC
Hϩ
ϭ ͑M cosh
ͱ
˜
j␮͒E
where
Ds
ͯ
dy
␭ϩ
2
0
R ϭ ͑F
͕
2b k ͑1 Ϫ ␪͒ ϩ 2b k ͑1 Ϫ ␪͒ ϩ 2b k ␤␪C
t
1
1
2
2
22 22
Df
ͱ
␭
j␮
2
4
Ϫ
5
0
͒Ϫ1
ͱj␮ ϩ
sinh ͱj␮ͪ
˜
C
0
ϩ
ͩ
N cosh
ϩ 3b k ͑1 Ϫ ␪͓͒CrO ͔C ϩ 2b k ␤␪
͖
͓28͔
3
3
4 4
͓
37͔
is the charge-transfer resistance. From Eq. 2
2
0
2
0
Upon introducing the term dC
˜
Hϩ
/dy ͉␭ϩ ͑Eq. 17͒ in Eq. 37
˜
˜
˜
j␻␤␪ ϭ ͓b k ͑1 Ϫ ␪͒ ϩ b k ␤␪C ͔E Ϫ ͑k ϩ k ␤C ͒␪
2
2
22 22
2
22
Df
ͱ
␭
j␮
˜
Ϫ ͑2k ␤␪C ͒C
͓29͔
ͱ
˜
ͱj␮ ϩ
ͱ
˜
0
2
2
0
0
͑M cosh j␮͒E ϩ N cosh
sinh j␮ C
ͩ
ͪ
or
˜
˜ ¯
Hϩ
1
C ␪Ј͑0͒
␭
⍀ dC
˜
␪
˜
˜
0
ϭ AE ϩ BC
͓30͔
ϭ
ϩ
WH
͓38͔
ͯ
ͫ
ͬ
Ds
␦
¯ dy
⍀
␭ϩ
where
˜
Finally, introduction of the term C of Eq. 25 into Eq. 38 yields
2
␭
b k ͑1 Ϫ ␪͒ ϩ b k ␤␪C
0
2
2
22 22
A ϭ
2
j␻␤ ϩ k ϩ k ␤C
1
⍀
˜
¯
dC
Hϩ
1
␭
tanh
ͱ
j␮ Ϫ␪ 0͒
1
2
22
0
Ј͑
˜
E
WH
Ϫ M 1 ϩ
ͯ
ͭ
ͫ
ͬ
ͮ
cosh
ͱ
j␮
¯
⍀
dy
j␮
Ds
Df ͱj␮
␦
Ds
␭ϩ
and
˜
C0 ϭ
Df
ͱ
␭N tanh
ͱ
j␮
Ϫ␪Ј͑0͒
1
Ϫ2k ␤␪C
22
0
N ϩ
tanh
ͱ
j␮ ϩ
ϩ 1
ͩ
ͪ
ͫ
ͬ
B ϭ
2
0
␭
Df ͱj␮
␦
Ds
j␻␤ ϩ k ϩ k ␤C
2
22
͓
39͔
˜
dCHϩ
Df
ͯ
AC impedance model.—To obtain an expression for electrochemical
dy
0
impedance, Eq. 30 is incorporated into Eq. 27 to provide
2
Ϫ
5
2
˜
͓
Ϫ5b3k3͑1 Ϫ ␪͓͒CrO ͔C Ϫ 2b2k2͑1 Ϫ ␪͒ ϩ 2b22k22␤␪C ͔Eϩ
˜
4
0
0
i
2
2Ϫ
5
ϩ͓Ϫ5k3͓CrO2 ͔C Ϫ 2k2 ϩ 2k22␤C ͔␪ϩ
Ϫ
5
0
2
ϭ ͓Ϫ2͑k1 ϩ k2͒ Ϫ 2k22␤C0 ϩ 3k3͓CrO4 ͔C0 ϩ 2k4␤͔
˜
0
ϭ
F
4
ͭ
ͮ
2
4
Ϫ
4
0
˜
ϩ͓25k3͑1 Ϫ ␪͓͒CrO ͔C ϩ 4k22␤␪C0͔C0
1
˜
˜
˜
E ϩ ͓Ϫ4k ␤␪C Ϫ 15k ͑1 Ϫ ␪͒
ϫ͑AE ϩ BC ͒ ϩ
0
22
0
3
ͱ
␭
j␮
RtF
ϭ Df
͑G Ϫ H͒
͓
31͔
2
4
Ϫ
4
0
˜
ϫ͓CrO ͔C ͔C
0
͓40͔
Eq. 17, 18, 23, 25-27, 30, and 31 provide a set of eight equations
˜ ˜
and the faradaic impedance (1/ZF ϭ i/E) becomes
˜
˜
with eight unknowns (i, ␪, C , C , dC /dy ͉␭
G, and H), which can be solved.
The set of eight equations can be simplified through the following
steps. Introduction of Eq. 30 into Eq. 31 yields
Hϩ
Ϫ
, dC /dy ͉␭ ,
Hϩ
ϩ
0
␭
Ϫ2͑k1 ϩ k2͒ Ϫ 2k22␤C ϩ 3k3͓CrO 24 Ϫ͔C ϩ 2k4␤͔
2
5
͓
0
0
˜
C0
1
˜
ϫ
A ϩ B
ϩ
ϩ
i
ͩ
ͪ
ͱ
␭
j␮
ϭ F
˜
E
RtF
˜
˜
˜
E
ME ϩ NC ϭ D
͑G Ϫ H͒
͓32͔
͓33͔
0
f
˜
Ά
2Ϫ
4
4
0
C0
·
ϩ͓Ϫ4k22␤␪C0 Ϫ 15k3͑1 Ϫ ␪͓͒CrO ͔C ͔
˜
E
where
͓
41͔
2
4
Ϫ
5
0
M ϭ ͓Ϫ5b k ͑1 Ϫ ␪͓͒CrO ͔C Ϫ 2b k ͑1 Ϫ ␪͒
3
3
2 2
For the electrochemical impedance the rotation speed modulation is
2
0
ϩ 2b k ␤␪C ͔
22
22
˜
zero (⍀ ϭ 0), thus Eq. 39 becomes
2
Ϫ
5
0
2
0
ϩ A͓Ϫ5k ͓CrO4 ͔C Ϫ 2k ϩ 2k ␤C ͔
3
2
22
␭
tanh
ͱ
j␮ Ϫ␪Ј͑0͒
1
ϪM 1 ϩ
ͭ
ͫ
ͬ
ͫ
ͮ
and
˜
Df ͱj␮
␦
Ds
C0
ϭ
N ϭ ͓25k ͑1 Ϫ ␪͓͒CrO 24 ͔C ϩ 4k ␤␪C ͔
Ϫ
4
0
3
22
0
˜
Df
ͱj␮
␭N tanh
ͱj␮
Ϫ␪Ј͑0͒
1
N ϩ
tanh
ͱ
j␮ ϩ
ϩ 1
ͩ
ͪ
ͬ
2
Ϫ
5
2
0
␭
Df
ͱj␮
␦
Ds
ϩ B͓Ϫ5k ͓CrO4 ͔C Ϫ 2k ϩ 2k ␤C ͔
͓34͔
͓35͔
3
0
2
22
͓
42͔
Incorporation of Eq. 23 into Eq. 32 yields
The total impedance (Z ) expression of the system is
T
␭
M
␭N
1
2
˜
E ϩ
˜
0
G ϭ
ϩ
C
1
ͩ
ͪ ͩ
ͪ
2
Df
ͱ
j␮
2Df
ͱ
j␮
Z ϭ
T
͓43͔
1
ϩ j␻Cd
and
ZF

B22
Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
˜
The nonsteady kinetic equations under potentiostatic control (E
ϭ 0) is, for the anodic part
˜
i
a
˜
ϭ ͓Ϫ2͑k ϩ k ͒ ϩ 2k ␤͔␪
͓47͔
1
2
5
F
and, for the cathodic part
˜
i
c
2
0
2Ϫ
4
5
0
˜
ϭ ͓Ϫ2k ␤C ϩ 3k ͓CrO ͔C ͔␪ ϩ ͓Ϫ4k ␤␪C
2
2
3
22
0
F
2
Ϫ
4
˜
Ϫ 15k ͑1 Ϫ ␪͓͒CrO ͔C ͔C
0
͓48͔
͓49͔
3
4
0
˜
Eq. 30 for E ϭ 0 becomes
˜
␪
˜
0
ϭ BC
where
Ϫ2k ␤␪C
0
2
2
Figure 6. Comparison between simulated electrochemical impedance dia-
ϭ
͓50͔
2
0
grams obtained with ͑
perimental impedance obtained after 15 min immersion in the solution at
7°C at 1000 rpm ͑frequencies in hertz͒.
͒ ZdeLevie , ͑
͒ without Zdelevie, and ͑᭺͒ ex-
j␻␤ ϩ k ϩ k ␤C
2
22
4
and Eq. 39 becomes
˜
¯
1
⍀
dCHϩ
1
WH
ͯ
where C is the dl capacitance.
d
¯
j␮
⍀
dy
Ds
cosh
ͱ
␭
ϩ
In the experimental impedance diagrams, the high frequency part
of the second capacitive loop, which has the appearance of a diffu-
sion impedance, does not exhibit an angle of 45° with respect to
real-axis but, rather, an angle close to 22.5°. To provide a better
match between simulations and experiments, it was necessary to
take into account this behavior which may be linked to the 3-D
geometry of a zinc electrode ͑roughness or porosity͒. For the porous
˜
C0 ϭ
Dfͱj␮
␭
␭N tanh j␮
ͱ
Ϫ␪Ј͑0͒
1
N ϩ
tanh
ͱ
j␮ ϩ
ϩ 1
ͩ
ͪ
ͩ
ͪ
Df ͱj␮
␦
Ds
͓
51͔
52͔
2
4
Ϫ
4
0
where N ϭ ͓25k ͑1 Ϫ ␪͓͒CrO ͔C ϩ 4k ␤␪C ͔
3
22
0
25
model of de Levie, when the pore is considered as semi-infinite the
2
Ϫ
5
0
2
0
overall impedance varies as Z1/2 where Z is the corresponding im-
ϩ B͓Ϫ5k ͓CrO ͔C Ϫ 2k ϩ 2k ␤C ͔
3
4
2
22
pedance on a plane electrode. It was shown that this behavior is the
same for all pore geometry.26 In a potential and/or concentration
distribution along the pore the model is much more complex, Los
and Hitz et al. proposed the use of two constant phase elements
͓
˜
Including the term ␪ ͑Eq. 51͒ in Eq. 47 and 48, expressions are
obtained for the anodic EHD influence
27,28
͑
CPEs͒ to take this distribution into account.
In the present case,
to limit the number of unknown parameters we prefer to consider
that the overall impedance is proportional to the square root of the
total impedance given by expression 43
˜
˜
0
i
C
a
ZEHDa
ϭ
ϭ F ͓Ϫ2͑k ϩ k ͒ ϩ 2k ␤͔B
͓53͔
ͭ
1
2
4
ͮ
˜
⍀
˜
⍀
1/2
Zac ϭ KZ
͓44͔
and the cathodic influence
T
˜
i
c
2
0
2Ϫ
5
where K is a constant. The dissolution of zinc does not occur on
plan-to-plan but on the walls of a semi-infinite porous medium,
ZEHDc
ϭ
ϭ F ͓͑Ϫ2k ␤C ϩ 3k ͓CrO ͔C ͔B
2
9,30
ͭ
22
3
4
0
˜
⍀
and in the present case clearly the diffusive impedance responses do
not have an angle of 45° with respect to real axis as is shown in Fig.
˜
4
C
0
1
/2
2Ϫ
6
. It can be seen that it is necessary to consider Z to simulate the
ϩ ͓Ϫ4k22␤␪C0 Ϫ 15k3͑1 Ϫ ␪͓͒CrO4 ͔C0͔͒
T
ͮ
˜
⍀
capacitive diffusion loops detected in the experimental results.
͓
54͔
EHD impedance model.—The simulation of electrohydrodynamical
impedance takes into account simultaneously, but separately, the an-
odic and cathodic reactions in the process of formation of the film.
So it is necessary to separate the anodic and cathodic influence from
the total steady-state faradaic current. The anodic steady-state fara-
daic current is
The total EHD (ZEHDT) is obtained by the sum of anodic and ca-
thodic EHD impedances divided by the same impedance in the low
dimensionless frequency limit.
˜
˜
i
c
i
a
͑
͑
p͒ ϩ
0͒ ϩ
͑p͒
͑0͒
ͩ
ͩ
ͪ
ͪ
ia
ZEHD
T͑
p
͒
˜
⍀
˜
⍀
ϭ 2k ͑1 Ϫ ␪͒ ϩ 2k ͑1 Ϫ ␪͒ ϩ 2k ␤␪
͓45͔
1
2
4
F
ϭ
͓55͔
ZEHDT͑0͒
˜
i
˜
i
c
a
and the cathodic is
˜
⍀
˜
⍀
ic
2
0
2Ϫ
5
The simulations of Zac obtained for the set of parameters show in
Tables I-IV are given in Fig. 7. The parameters k and C0 were
ϭ Ϫ2k ␤␪C Ϫ 3k ͑1 Ϫ ␪͓͒CrO ͔C0
͓46͔
2
2
3
4
F
3
0

Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
calculated by Eq. 8 and 11, respectively. The parameter k varies
B23
3
15
15
between 2.5 ϫ 10 and 5.0 ϫ 10 , and C varies between 4.8
0
Ϫ5
Ϫ5
ϫ 10 and 6.38 ϫ 10
M. These variations depend on the po-
tential and the rotation speed of the electrode. The values of C0
yield a variation of pH between 4.32 and 4.20, which are consistent
with the experimental results obtained by local pH measurements
͑
Fig. 4͒ that represent the pH measured at the chromate layer/
electrolyte interface. The convective diffusion coefficient D is
s
Ϫ4
2
Ϫ1
10
cm
s . The chromate layer diffusion coefficient that corre-
sponds to the experimental conditions at 25°C is 3.3
Ϫ8
cm2
s , at 47°C the diffusion coefficient is assumed to
Ϫ1
ϫ 10
be 6.1 ϫ 10
Ϫ8
cm2
Ϫ1
s , considering that increases of temperature
increase the diffusion in a gel-like film.
It is shown in Fig. 7a and b that the impedance increases for the
same rotation speed with the increase of chromate layer thickness,
and it is directly proportional to immersion time in the chromate
bath as shown in experimental results ͑Fig. 2a and b͒. The Zac simu-
lation in Fig. 7c and d, analogous to the experimental results with
rotation speed variation for the same immersion time ͑Fig. 2c and
d͒, were obtained by the variation of chromate and convective elec-
trolyte layer thickness. The decrease of chromate layer thickness
Table I. Reaction rates parameters kio „mol cm sÀ1… used for
À2
ac and EHD impedance simulations. k30 is claculated by Eq. 4.
k10
k20
k220
k40
8
.0 ϫ 10Ϫ1
2.6 ϫ 1010
2.1 ϫ 10Ϫ16
1.4 ϫ 1011
Table II. Kinetic parameters b used for ac and EHD impedance
i
simulations in function of solution temperatures from experimen-
tal results.
Temperature
°C͒
b
V
b
(V
b
(V
b
(V
b
4
(V )
1
2
22
Ϫ1
3
͑
Ϫ1
Ϫ1
Ϫ1
Ϫ1
(
)
)
)
)
2
4
5
7
18.0
16.8
37.8
35.3
40.0
37.3
3.0
2.8
30.0
28.0
Table III. Chromate layer thickness and parameters used for the
simulations of experimental impedances at 1000 rpm in function
of immersion times, solution temperatures, and electrode poten-
tials.
Temperature
°C͒
Immersion time
E
͑V͒
␭
͑cm͒
͑
͑min͒
Ϫ5
Ϫ5
Ϫ5
2
5
2
Ϫ1.309
Ϫ1.317
Ϫ1.319
2.0 ϫ 10
5.0 ϫ 10
7.0 ϫ 10
1
3
5
0
Ϫ5
Ϫ5
Ϫ5
4
7
2
5
0
Ϫ1.320
Ϫ1.322
Ϫ1.324
2.2 ϫ 10
5.7 ϫ 10
7.8 ϫ 10
Figure 7. Simulated electrochemical impedance diagrams at 1000 rpm for
different chromate layers thickness ͑legends͒ corresponding to the experi-
mental temperature condition of ͑a͒ 47°C and ͑b͒ 25°C and simulated dia-
grams obtained for different chromate layers thickness ͑legends͒ at 250 rpm
1
3
͑
͒, at ͑gray dashes͒ 500 rpm, and ͑
͒ 1000 rpm corresponding to
Table IV. Chromate layer thickness and parameters used for the
simulations of experimental impedances at 30 min in function of
rotation speed, solution temperatures, and electrode potentials.
experimental temperature condition of ͑c͒ 47°C and ͑d͒ 25°C after 30 min
͑frequencies in hertz͒.
Temperature
°C͒
Rotation speed
E
͑V͒
␭
͑cm͒
͑
͑rpm͒
with increase of rotation rate was assumed because the layer is pre-
sumed to be a gel-like film, and the convective electrolyte layer
thickness calculated by Eq. 14 is inversely proportional to rotation
speed.
The simulations of ZEHD were obtained with the same set of
parameters used for the Zac simulations, but in this case the anodic
and cathodic effects were analyzed independently. The cathodic and
anodic currents in function of electrode rotation rates calculated by
Ϫ5
Ϫ5
Ϫ5
2
5
250
Ϫ1.325
Ϫ1.322
Ϫ1.319
9.4 ϫ 10
8.7 ϫ 10
7.0 ϫ 10
500
1000
Ϫ5
4
7
250
Ϫ1.325
Ϫ1.324
Ϫ1.324
10.5 ϫ 10
8.2 ϫ 10
7.8 ϫ 10
Ϫ5
500
Ϫ5
1000

B24
Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
Figure 8. ͑square symbols͒ Anodic and ͑triangle symbols͒ cathodic currents
in function of rotation speed simulated for temperatures conditions of ͑open
symbols͒ 25 and ͑solid symbols͒ 47°C.
Figure 10. Simulated total EHD diagrams corresponding to the experimen-
tal conditions of ͑a͒ 47 and ͑b͒ 25°C.
Eq. 45 and 46 are presented in Fig. 8. These simulations were ob-
tained by the variation of rotation rates with fixed potentials corre-
spondent to a rotation rate of 1000 rpm at 25 and 47°C. The current
shifts are larger when rotation speeds decrease and for the condi-
tions of 47°C are greater than 25°C. The nonnormalized modulus of
EHD of cathodic and anodic parts calculated by Eq. 53 and 54 are
ϩ
presented in Fig. 9, the Schmidt number corresponding to H diffu-
sion used to simulate the experimental conditions was 100 at 25°C
and 50 at 47°C. The calculations allow identification of the different
effects of cathodic and anodic reactions in the electrohydrodynami-
cal behavior in the zinc interface with a chromate film formed in a
way that cannot be observed by experimental results. The behavior
of the cathodic curves in terms of modulus values is completely
different between the temperature conditions simulated. The
ϩ
changes are due to the temperature dependence of H diffusion
coefficient and to the current shifts show in Fig. 8. When the ca-
thodic and anodic electrohydrodynamical effects are combined into
the total EHD of the system ͑Eq. 55͒, the responses are obtained
͑Fig. 10͒ that are equivalent to experimental results ͑Fig. 3͒. The
diagrams shown in Fig. 10b are analogous to those obtained in ex-
perimental conditions at 25°C, showing the same two constant
times, the diagrams being shifted toward lower values as rotation
rate is increased, and phase curves that increase continuously. The
simulated diagrams of Fig. 10a are in good agreement with their
analogous experimental diagrams at 47°C. They present the same
‘
‘hump’’ in intermediary frequencies that is the beginning of the
6
ϩ
total response of the oxidization of zinc and the reduction of Cr
on the surface of zinc.
Conclusions
Based on the above results and discussion, the following points
can be outlined: (i) the Zac simulation based with only four reac-
tions and the mass-transport model proposed for the chromatation
reactions on zinc are in good agreement with the different experi-
mental results; (ii) the ZEHD simulation of the separation of anodic
and cathodic reactions effects was favorable for explain the electro-
hydrodynamical behavior for different conditions of chromate for-
mation. This aspect seems to have fundamental importance in the
kinetic role of the chromatation process and could be key as a future
candidate of nontoxic conversion baths.
Acknowledgments
The Brazilian authors thank CNPq, FAPERJ, CAPES, and FUJB.
The authors express their thanks to Mark Orazem for his critical
reading of the manuscript.
Figure 9. Anodic and cathodic EHD in function of rotation speed simulated
corresponding to the experimental temperature conditions of ͑a͒ 25 and ͑b͒
47°C.

Journal of The Electrochemical Society, 150 ͑1͒ B16-B25 ͑2003͒
B25
The Federal University of Rio de Janiero assisted in meeting the publi-
cation costs of this article.
s
␭
ϱ
relative to convective diffusion layer
at the interface chromate layer/electrolyte
in the bulk solution
List of Symbols
Superscripts
a
constant ͑0.51023͒
Ϫ1
Ϫ
mean value
complex quantity of small amplitude
bi kinetic parameter of reaction i, V
C
Cd double layer capacitance, F cm
D
E
F
ϩ
Ϫ3
ϳ
concentration of H , mol cm
Ϫ2
2
Ϫ1
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␮
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ϩ
ϩ
H
o