Effect of Voltage transfer Curve on Pattern
Formation in Dielectric Barrier Glow Discharge
Xiaoxi Duan, Feng He, and Jiting Ouyang*
School of science, Beijing Institute of Technology, Beijing 100081, China
(*Author to whom correspondence should be addressed; [email protected])
Abstract: The voltage transfer curve reveals the important relationship between
the voltage transfer in a pulsed discharge and the initial voltage across the gas
gap in dielectric barrier glow discharge. Based on its bifurcation characteristic
and considering the coupling term as a short-range activator and long-range
inhibitor, a discrete one-component coupled-map model is developed to study the
pattern formation in barrier discharge system. The stability of spatially structures
and the bifurcation condition are investigated. The initial uniform discharge
evolves to the eventual patterned discharge in one- and two- dimensional
simulations. The evolution processes show that, the wavelike patterns firstly form
near the boundary of the electrode and then occupy the whole space of the
electrode. The results of this model are well consistent with the pervious
experimental and fluid modeling results, and provide a new method to investigate
the pattern formation in discharge systems.
Keywords: Dielectric barrier discharge, coupled-map model, voltage transfer
curve, lateral inhibition, pattern formation
1. Introduction
Dielectric barrier discharge (DBD) is an excellent
pattern forming system which shows interesting
phenomena such as self-generated oscillation, stable
pattern formation or chaotic behavior under different
conditions [1, 2]. Especially in glow mode, the
discharge develops generally synchronously in the
whole space, with a single current pulse in each half
period [2]. This allows one to achieve a favorable
observation for the pattern formation in experiments.
The voltage transfer curve is an important
characteristic in glow DBD systems [3]. It is well
used to define the stable operating conditions in
plasma display panel (PDP). It has been verified that
the distribution of the wall charge density is in good
accordance with the discharge structure both in
experiments and simulations [4]. Therefore a
dynamic model taking the wall voltage as the sole
parameter is helpful to simplify the discharge model.
In this work, we employ a simple coupled-map
model [5] based on the voltage transfer theory. The
processes of the pattern formation in glow DBD
systems are achieved from this model.
2. Description of the coupled-map model
In this model, the spatio-temporal evolution of DBD
is governed by the total voltage across the gas gap
(i.e., the sum of the applied voltage and the wall
voltage) at position x before each pulsed discharge,
written as the general form of
V i +1 (x) = f (V i (x);VS (x)) + c(x, V i ) .
(1)
Where Vi(x) denotes the voltage across the gas gap
at position x before the ith breakdown. Reaction
term f represents the ideal voltage transfer caused by
voltage V(x) at an applied voltage VS(x). The
function c is a coupling term that specifies the
interaction of the voltage at different position x. The
bold V refers to the gas gap voltage at the positions
around x. To simplify, we consider a square-wave
driven DBD system in which the voltage is a
constant during any discharge pulse. Then, the
reaction term f can be written as follows:
f (V i (x);VS (x)) = 2VS (x) − V i (x) + g (V i (x)) .
(2)
Where g(Vi(x)) represents the function of the voltage
transfer curve for DBD systems. It is defined as the
change of the wall voltage during a discharge pulse
at a given voltage. The typical voltage transfer curve
for an infinity plate DBD system can be obtained by
the fluid model. Figure 1 shows a voltage transfer
curve for DBD in 3 KPa Ne (parallel electrodes
structure, two 1 mm-wide dielectric layers are
spaced by a 3 mm-wide gas gap, the relative
permittivity of the dielectric layer is 10). This curve
can be well fitted by the following polynomial
expression
V ≤ 150
⎧0
⎪
−4 3
2
⎪-1.66 × 10 V +0.104V
g (V ) = ⎨
150 < V ≤ 280 . (3)
−
18.9
V
+
1050.5
⎪
⎪⎩V
V > 280
voltage transfer g(V) (V)
300
250
voltage transfer
Polynomial fitting curve
200
The coupling term c is derived from the fluid
modeling of discharge in a non-uniform electric field,
as shown in figure 2. This figure shows that at the
positions where there are obvious voltage gradients
along the dielectric surface, the voltage transfer
deviates from the ideal value calculated from the
voltage transfer curve. The coupling term c can be
considered as a short-range activator and long-range
inhibitor from the deviation curve. The edge area of
the higher voltage gradient is the source of this
effect. The modeling also indicates that the strength
of the coupling term c(x) relates to the voltage
difference closed to position x. Hence, the coupling
term c can be described as follows:
c ( x, V i ) =
Δx
⎧⎪
⎫⎪ (4.1)
−
i
2σ 2
γ 1 ∑ ⎨Vm (x + Δx)e
[ a − V i ( x + Δx)]⎬,
0 ≤ Δx ≤ r ⎪
⎩
⎭⎪
2
c ( x, V i ) =
Δx
⎧⎪
⎫⎪
−
2
−γ 2 ∑ ⎨Vm i ( x + Δx)e 2σ [V i ( x ) − b]⎬.
Δx > r ⎪
⎩
⎭⎪
2
100
50
150
200
V (V)
250
300
(4.2)
Where Δx is the deviation vector from x, Δx is the
distance between (x+Δx) and x. γ1 and γ2 are
constant coefficients. A symmetrical exponential
distribution is assumed and the parameter σ
determines standard deviation. The amplitude Vm(x)
is defined as the difference between the value of the
voltage at position x and the average voltage at x
Vm (x) = V (x) − V (x + Δx) .
150
0
100
the wall voltage before each pulsed discharge would
be just equal to the applied voltage.
(5)
In equation (5) only the positive value is considered
and we set Vm(x) = 0 when Vm(x) < 0. The last term
( a − V(x +Δx )) in function (4.1) is included
Figure. 1 The voltage transfer curve of an infinite plate DBD
system (obtained from 2-D fluid model in 3 KPa Ne), and its
fitting curve.
because the effect of the activate effect decrease
when the voltage increase and is close to the
completed voltage transfer value. While the factor
(V (x) − b) in (4.2) indicates that the effect of the
This curve indicates that the breakdown voltage of
the DBD system is 150 V. When the total voltage V
is above 280 V, the voltage transfer is complete and
lateral inhibition is more obvious when the voltage
is higher which could result from the transverse field
becoming stronger due to the localized discharge.
200
Voltage (V)
160
120
80
40
0
0
8
16
24
Position (mm)
32
40
V: initial voltage across the gas gap
g1(V): caculated voltage transfer from voltage transfer curve
g2(V): simulated voltage transfer by fluid model
g2(V)-g1(V)
Figure 2. Voltage transfer from a non-uniform initial voltage
distribution.
300
150
0
80
V S=152 V
V S=160 V
i=2
20
0
A
B
150
-20
D
C
200
250
300
i=4
Voltage (V)
i+1
δV=V
i=0
40
i
-V (V)
60
initial voltage is constructed and out of the electrode
boundary the voltage decreases exponentially. The
disturbance to the uniformity status firstly forms at
the edge region because the obvious voltage grads at
these places provide a bigger value Vm(x). Due to the
effect of lateral inhibition the voltage at the sites
near the boundary decrease and a new boundary
forms in the inner of the electrode. In the same
mechanism the maximum of the voltage forms one
by one towards the central of the electrode and
finally the wavelike patterns form in the whole
electrode.
350
i
V (V)
i=6
i=8
i = 10
-40
Figure 3. Typical phrase diagrams of equation (1) at VS = 152
V / VS = 160 V. The arrows indicate the direction of the
discharge evolution at given voltages.
The stability of this model is plotted at given applied
voltages VS for a uniform system (i.e., the coupling
term c is not considered in the variation term of δV),
as shown in figure 3. One sees that for an applied
voltage slightly higher or lower than the breakdown
voltage, (e.g. VS = 152 V), the discharge shows a bistability property and the points “A” and “D” are the
two stable states for the DBD. The characteristic is
well discussed in Ref [3]. For a higher voltage (e.g.
VS = 160 V), there is only one stable point (i.e., the
point “D”) and the variation of the voltage δV is
always above zero. Under this condition, the
patterned structures, once formed, must satisfy the
bifurcation condition at some positions
δ V + c(x,V i ) ≤ 0 .
(6)
3. Results and discussion
Based on the model described above, the evolution
of the patterned DBD can be observed. The typical
evolution of a 1D pattern is shown in figure 4. In
these results, a finite-width electrode with a uniform
i = 12
-r
r
i = 14
Figure 4. Time evolution of the patterned discharge in a onedimensional model. Parameters: γ1=2.07×10-4, γ2=1.33×10-3,
a=480, r=3, σ=5, b=150.
When we decrease the effect of the lateral inhibition
(i.e., reduce the value of σ or γ2), different stable
states of the DBD can be observed，as shown in
figure 5. One can see that strong lateral inhibition is
important for the pattern formation because the
coupling term has a low absolute value at the
inhibited positions and the bifurcation condition is
easy to be satisfied. Furthermore, the places near the
boundary are more unstable than the inner places. It
results from that the system has a lower value of δV
from the phrase diagram during the first few
breakdowns. It is helpful to satisfy the bifurcation
condition at the positions near the boundary of the
electrode where the initial voltage gradient forms.
The evolution of the pattern in the 2D model under
the same condition as in figure 4 (only the radius of
the electrode is set to a half value) is achieved, as
shown in figure 6. One sees that the initial
uniformity transits to the concentric ring structure
after about 10 breakdowns and then the rings
become unstable and some spots appear. Eventually,
the stable hexagonal patterns are achieved.
Comparing with the previous results obtained from
experiments and fluid modeling, the discrete onecomponent model shows well-matched spatiotemporal evolution of the patterned discharge.
Although in a real DBD system, the discharge
development are much more complicated and many
input parameters such as the gases, the pressure, and
the characteristics of the power source must be
considered, the process of the discharge and effect of
all the parameters can be involved in our model in a
phenomenological way. This simplified model will
be helpful to discuss the formation and behavior of
the patterns.
4. Conclusions
Figure 5. Stable structures of the DBD in a one-dimensional
model for (a) different γ2, σ = 5.0 and (b) different σ, γ2 =1.33×
10-3. The other parameters are the same as in figure 4.
A discrete one-component model based on the
voltage transfer curve and the effect of lateral
inhibition in a glow DBD system is developed. The
bifurcation condition for the pattern formation is
discussed from the phase diagram. The one- and
two- dimensional pattern evolution from the initial
uniform voltage distribution are observed. The
wavelike patterns evolve from the edge of the
electrode to the center. During this process, the
strong effect of the lateral inhibition plays a major
role in pattern formation.
Acknowledgement
This work was supported in part by the National
Science Foundation of China under Grant No.
10875010.
References
Figure 6. Time evolution of the patterned DBD in a twodimensional model. The time t at which the figures are taken are
as follows: (a) i = 10; (b) i = 20; (c) i = 40; (d) i = 60.
The evolution of the pattern structures in our model
has the similar processes as the reaction-diffusion
systems [6]. But differently, the voltage across the
gas gap can be considered playing the both roles of
activator and inhibitor.
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