Download FREQUENCY RESPONSE METHOD FOR MODELLING

FREQUENCY RESPONSE METHOD FOR MODELLING
OF PLASMA ELECTROLYTIC OXIDATION PROCESSES
A. Yerokhin1, E.V. Parfenov2 and A. Matthews1
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.
12
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
(UAC) is comparable with DC magnitude (UDC). 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+
1
<
2
<
U+
3 ….
1
<
2
<
3 ….
UAC
UAC
UDC
UDC
t
U-
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
UDC 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 ( jω)  Z (ω)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 (ω)
I (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 :
13
u(t)=UAsin(t+U),
i(t)=IAsin(t+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.
14
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)
Ro
Fig.
R1
R2
R3
Element
Freedom
Value
Error
Error %
Ro
Free(+) C1
12
N/A C3
N/A
C2
R1
Free(+)
60
N/A
N/A
(b)
R2
Free(+)
746
N/A
N/A
R3
Fixed(X)
0
N/A
N/A
Ro
R1
R2
C3
Fixed(X)
0 Value
N/A
N/AError %
Element
Freedom
Error
C2
Free(+)
7.38E-05
N/A
N/AN/A
C1
C2
Ro
Free(+)
12
N/A
C1
Free(+)
9.7E-08
N/A
N/A
R1
Free(+)
60
N/A
N/A
R3
C1
Free(+)
9.7E-08
N/A
N/A
Data File:
R2
Free(+)
746
N/A
N/A
C3
Circuit Model File:
C:\Users\owner\Documents\Alex\MDO\Parfenov\2007\for_eis_spline\model2_Exp2-12.md
C2
Free(+)
7.38E-05
N/A
N/A
Mode:
Run Fitting / All Data Points (1 - 1)
R3
Fixed(X)
0
N/A
N/A
(c)
Maximum Iterations:
1000
C3
Fixed(X)
0
N/A
N/A
Optimization Iterations:
100
of Fitting:
Complex used for modelling PEO processes: (a) –
4. Type
Basic
types
of
equivalent
circuits
Element
Freedom
Value
Error
Error %
Data
Type File:
of Weighting:
Data-Proportional
Ladder;
(b) – Voigt;
(c) – mixed parallel-serial.
Ro
Free(+)
12
N/A
N/A
Circuit Model File:
C:\Users\owner\Documents\Alex\MDO\Parfenov\2007\for_eis_spline\model2_Exp2-1
R1
Free(+)
60Fitting / All Data N/A
N/A
Mode:
Run
Points (1 - 1)
C1
Free(+)
9.7E-08
N/A
N/A
Maximum
Iterations:
1000
R2
Free(+)
N/A
N/A
Optimization
Iterations:
100746
Type
of
Fitting:
Complex
C2
Free(+)
7.38E-05
N/A
N/A
15
Type
Data-Proportional
R3 of Weighting: Fixed(X)
0
N/A
N/A
C3
Fixed(X)
0
N/A
N/A
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 UDC 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.
104
-175
103
2
102
-150
1
2
101
100
-125

1
10
102
103
104
105
106
Frequency (Hz)
-100
3
-75
0
Phase Angle
Z''
3
|Z|
-200
-50
1
-25
0
50
100
150
200
2
-50
3
-75
100
0
1
-25
1
10
2
10
3
10
4
10
105
106
Frequency (Hz)
Z'
(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. UDC=500 V; Uac=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.
16
104
-2000
-75
103
1
-1500
1
|Z|
-1750
2
2
102
Z''
3
3
-1250
Z''
101
100
101
102
103
104
105
106
Frequency (Hz)
-1000
0
-750
0
75
0
Phase Angle
Z'
-500

-250
0
0
500
1000
1500
2000
3
-25
2
1
-50
-75
100
101
102
Z'
103
104
105
Frequency (Hz)
(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; Uac=50 V. (1) – UDC = 450 V; (2) –
UDC = 500 V; (3) – UDC = 550 V.
Finally, a severely deformed arc shape resembling number ‘8’ can sometimes be
observed, in particular under high UDC 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.
17
106
REFERENCES
1.
2.
3.
4.
Yerokhin AL, Nie X, Leyland A, Matthews A and Dowey SJ. Surf. Coat.
Technol., 1999 (122) 73-93
Parfenov EV, Yerokhin AL and Matthews A: Surf. Coat. Technol. 2007 (201)
8661-8670
Raju GG: Dielectrics in Electric Field. New York-Basel, Marceld Ekkeirn,Inc.,
2003
Barsoukov E and Macdonald JR: Impedance Spectroscopy: Theory, Experiment
and Applications. New York, Wiley-Interscience, 2005
18