A. Yerokhin 1 , E.V. Parfenov 2 and A. Matthews 1
1
Department of Engineering Materials, University of Sheffield, Sheffield, U.K.
2
Ufa State Aviation Technical University, Ufa, Russian Federation
Abstract. The paper discusses feasibilities of the frequency response method for modelling of plasma electrolytic oxidation of valve metals. Issues related to specifics of frequency modulation in PEO, identification of equivalent circuits and problems arising during modelling are discussed. It is shown that frequency response method represents a powerful tool for investigation of PEO process that opens up new prospects in achieving new insights into discharge characteristics and mechanisms.
The method can be utilised in development of advanced PEO coatings for protective and functional purposes.
INTRODUCTION
Plasma electrolytic oxidation (PEO) technology offers unique possibilities in surface engineering of a wide range of metals and alloys, including Al, Ti, Mg, Zr, Nb and some other materials. PEO films show great promise not only as superior protective
(i.e. wear- and corrosion-resistant) coatings but also as functional materials with interesting biological, catalytic and photoactive properties [1]. Importantly, complex chemical compounds can be formed in situ , with high rate and at low cost.
Development of advanced PEO coatings is associated with achievement of desirable surface chemistry, phase composition and morphology. This requires careful selection of electrolyte composition and current regimes. The former is relatively straight forward and the latter represents major challenge, as no scientifically justified methodology for selection of the current mode and electrical characteristics of PEO is currently available. The trial and error approach prevailing in this aspect of the process development substantially retards application of sophisticated current regimes, in particular those based on pulsed and pulsed-reversed current waveforms.
Additional challenges exist in process controllability. They are associated with the lack of understanding of discharge mechanisms in systems with fluid electrodes and with the absence of reliable diagnostic tools.
These challenges can be addressed by frequency response (FR) evaluation of PEO process. The FR method is based on the application to the studied system of the input signal, containing frequency modulated AC component, followed by analysis of an output characteristic. The input characteristic is represented by appropriate excitation signal (e.g. electrical, the case of PEO) and the output one – by another electrical
(most often) or different characteristics (e.g. by light emission). Original methodology for evaluation of electrical frequency response has been recently developed by the authors and applied to the investigation of PEO process on Al alloys [2]. In this paper, a detailed discussion is given to the analysis of FR data, their use for simulation of
PEO process and problems arising.
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SPECIFICS OF FREQUENCY MODULATION IN PEO TECHNOLOGY
For frequency response studies in PEO, the modulation is usually realised by controllable frequency sweep of the main voltage waveform from the lower to the higher limit. The modulation is possible in large and small and signal modes (Fig 1).
For the large signal mode, the magnitude of the frequency swept AC component
(U
AC
) is comparable with DC magnitude (U
DC
). This mode is used for identification of PEO systems that operate in AC, pulsed and pulsed-reversed regimes, providing conditions of dynamic breakdown [3].
U +
<

<
 U +
<

<
 t t
U t t
(a) (b)
Fig. 1.
Two types of frequency swept voltage waveforms used to obtain frequency response of PEO process: (a) large signal and (b) small signal modulation .
In the small signal mode, the magnitude of AC perturbation is significantly less than
DC component. This mode provides a steady state operation conditions defined by
U
DC
value, with discharge characteristics corresponding to static breakdown. A similar approach is utilised in a well-known impedance spectroscopy [4]. In the following sections it will be shown that static and dynamic conditions are described by substantially different models.
Ideally sine waveform of the AC component should be used; alternative shapes, such as square, trapezoidal or triangle, must be carefully tailored to eliminate secondary harmonics. However the application of alternative waveforms in PEO research can be often justified by wider availability of appropriate equipment (Fig. 2) [2].
Experimental procedure for determination of frequency response in PEO is discussed elsewhere [2]. FR is represented by the following complex impedance function:
Z

j

Z
e j

(
ω
)
, (1) where ω is the radian frequency, Z(ω) and φ(ω) are the impedance magnitude and the phase angle as functions of frequency, j
 
1 . The impedance components can be obtained directly from the voltage and current signals. The magnitude Z is defined as the absolute value of impedance at a frequency of interest

and calculated as a ratio of effective voltage to current values:
Z (
ω
)

U
I (
(
ω
)
ω
)
(2)
The phase angle

is defined as the phase shift between the voltage and the current sine waves u (
 t ) and i (
 t ) at the frequency of interest

:
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u (
 t )= U
A i (
 t )= I
A sin( sin(

 t t

=

U
– 
I
+
+


U
),
I
),
(3)
(a)
(b)
Fig. 2.
Typical examples of (a) frequency swept square pulse voltage and (b) corresponding current waveforms in PEO Al [2].
Figure 3 illustrates typical behaviour of impedance magnitude and phase angle depending on PEO processing time. It shows that between 1 and 5 kHz the impedance magnitude develops a maximum which increases with processing time. The phase angle is negative, indicating an active-capacitive type of frequency response. The phase angle is close to zero at low frequencies and shifts towards more negative values with frequency increased above 1 kHz, indicating increasing role of capacitive component.
Fig. 3.
Typical example of frequency response evolution during PEO of Al in the large signal mode: (a) – impedance magnitude; (b) – phase angle.
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In general, this type of behaviour is common for electrochemical processes. This gives grounds for application of well-developed methods of electrochemical impedance spectroscopy for modelling of PEO process. Crucial aspect of such modelling is identification of physically sound equivalent circuit, adequately reflecting the electrochemical system under investigation.
IDENTIFICATION OF EQUIVALENT CIRCUITS
The active-capacitive behaviour of the frequency response indicates that the model of
PEO process can be represented by a set of resistors and capacitors, forming characteristic active-capacitive arcs on the complex (Z’;Z’’) plane (so called ARC sub-circuits). The following 3 types of equivalent circuits can be composed of ARC elements for modelling of PEO (Fig. 4): (a) Ladder; (b) Voigt and (c) mixed parallelserial circuit, often used for simulation of corrosion breakdown in passive oxide films on valve metals. With properly selected values of resistances and capacitances, all 3 basic models can, in principle, fit the obtained frequency response data. This causes major ambiguities in modelling that can only be resolved with development of comprehensive physical model of the process.
Current understanding of the mechanisms of charge transfer in PEO allows suggesting the Ladder model to describe the most adequately physical processes taking place at the surface of the active electrode. In this case, Ro, R1, R2 and R3 can be attributed to the processes of charge transfer in the bulk of electrolyte, at the electrolyte-discharge, discharge-oxide and oxide-substrate interfaces respectively and C1, C2 and C3 – to corresponding partial capacitances of the multilayer system. However as show numerous exercises, this model is by no means universal, describing all varieties of situations arising in PEO. The most common deviations from the basic model (Fig.
4a) are discussed in the following section.
Ro R1 R2 R3
C3
C2
C1
(a)
Element
Ro
Ro
R1
R2
Freedom
R1
Free(+)
Free(+)
Free(+)
C1
Value
12
60
746
R2
C2
Error
N/A
N/A
N/A
R3
C3
Error %
N/A
N/A
N/A
(b)
R3
Ro
Fixed(X)
R1
0
0
Value
C1 12
60
R2
N/A
Error
C2 N/A
N/A
N/A
N/A
Error %
N/A
N/A
N/A
N/A
C1
Data File:
R2
Circuit Model File:
C2
Mode:
Free(+)
Free(+)
Free(+)
R3 Fixed(X)
Maximum Iterations:
C3 Fixed(X)
Optimization Iterations:
R3
C3
9.7E-08 N/A N/A
746 N/A N/A
C:\Users\owner\Documents\Alex\MDO\Parfenov\2007\for_eis_spline\model2_Exp2-12.mdl
7.38E-05 N/A
Run Fitting / All Data Points (1 - 1)
0
1000
0
100
N/A
N/A
N/A
N/A
N/A
(c)
Fig. 4 . Basic types of equivalent circuits used for modelling PEO processes: (a) –
Ro
Data-Proportional
12
Error
N/A
Error %
N/A
Free(+) N/A
Free(+)
Type of Weighting: Fixed(X)
C3 Fixed(X) 0
15
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
Data File:
Circuit Model File:
Mode:
Maximum Iterations:
Optimization Iterations:
Type of Fitting:
Type of Weighting:
C:\Users\owner\Documents\Alex\MDO\Parfenov\2007\for_eis_spline\model2_Exp2-12.mdl
Run Fitting / All Data Points (1 - 1)
1000
100
Complex
Data-Proportional

PROBLEMS IN MODELLING PEO PROCESSES
The first type of problems is caused by ambiguities at the boundary conditions, Z(0) and Z(

), lying outside the bandwidth of frequency sweep (Fig. 5). These can be overcome by additional experimentation under DC polarisation and by analytical evaluations. Analysis of the circuit structure shows that both at very low and very high frequencies the phase angle approaches to zero. Therefore the value of the system conductivity under DC polarisation can be used for evaluation of Z(0) and the value of electrolyte resistance – for Z(

).
Other problems are associated with degeneration of certain ARC elements under specific conditions of PEO. Typical example is given by curve (1) in Fig. 6(a) and corresponds to the moment of time when relatively low U
DC
value becomes insufficient to maintain discharge conditions on the oxidising surface. A similar situation is observed at the treatment in the large signal mode that is associated with dynamic breakdown conditions. In these cases, appropriate adjustments to the equivalent circuit are necessary to account for substantially different physical mechanisms of oxidation process.
Yet another type of problems relates to the deformation of RC arcs on the complex plots (Figs. 5 and 6) that is manifested in depressed and skewed arcs (curve (1) in Fig.
5 and curve (2) in the inset to Fig. 6 respectively). This is caused by deviation of characteristic relaxation times of relevant processes, which is due to either nonuniform conditions (e.g. macroscopically uneven coating thickness or electric field) or stochastic phenomena (e.g. randomly distributed microdischarge events) at the sample surface. The former can be eliminated by appropriate design of electrolytic cell and careful selection of electrolyte composition. The latter represents genuine interest and can be described by distributed elements, e.g. constant phase, Davidson-Cole or
Havriliak-Negami elements [3], introduced into the equivalent circuit. A justification of such modification should however be provided by additional experimentation.
(a) (b)
Fig. 5 . Complex (a) and Bode (b) plots of PEO process of Al obtained by frequency swept method in the small signal mode. U
DC
=500 V; U ac
=50 V. (1) – 1 min; (2) – 8 min; (3) – 16 min. Blank data points are directly evaluated from the experiments; filled points are obtained from the additional DC experiment and the evaluation of electrolyte resistance. Lines represent interpolation outside the frequency bandwidth.
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(a) (b)
Fig. 6 . Complex (a) and Bode (b) plots of PEO process of Al obtained by frequency swept method in the small signal mode. t = 8 min; U ac
=50 V. (1) – U
DC
= 450 V; (2) –
U
DC
= 500 V; (3) – U
DC
= 550 V.
Finally, a severely deformed arc shape resembling number ‘8’ can sometimes be observed, in particular under high U
DC
polarisation conditions (curve (3) in the inset to Fig. 6). Along with the above reasons, this can indicate an appearance in the system of negative differential resistance (NDR). The NDR could be caused by several reasons, including surface conductivity, volume charge and (most importantly) plasma discharge in both diffused and filamented form [3].
CONCLUSIONS
Frequency response represents a powerful tool for investigation of PEO process. Both conditions of static and dynamic breakdown can be reproduced and modelled. The modelling performed with the aid of methodological apparatus developed for impedance spectroscopy opens up new prospects in achieving new insights into discharge characteristics and mechanisms of PEO. This can be crucial for the development of advanced PEO processes based on application of pulsed and pulsedreversed current modes.
ACKNOWLEDGEMENTS
The work was supported by the UK Engineering and Physical Sciences Research
Council through the Platform Grant No RA105224. The authors are grateful to the
Russian Federal Agency for Science and Innovation and British Royal Society for funding the fellowships of Dr. E.V. Parfenov in the University of Sheffield where the research was carried out.
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REFERENCES
1.
Yerokhin AL, Nie X, Leyland A, Matthews A and Dowey SJ. Surf. Coat.
Technol ., 1999 (122) 73-93
2.
Parfenov EV, Yerokhin AL and Matthews A: Surf. Coat. Technol . 2007 (201)
8661-8670
3.
Raju GG: Dielectrics in Electric Field. New York-Basel, Marceld Ekkeirn,Inc.,
2003
4.
Barsoukov E and Macdonald JR: Impedance Spectroscopy: Theory, Experiment and Applications. New York, Wiley-Interscience, 2005
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