Calculation of the electrostatic field distribution formed by the generator of the off-electrode plasma

Abstract

A calculation of the electrostatic field distribution in the electrode system of a high-voltage gas-discharge device is made. The application of the conformal mapping method in order to obtain an analytical description of the of equipotentials and field lines distribution is described. The figures of the electrostatic field distribution are calculated, which made it possible to determine their relationship with the cathode-anode distance, the voltage at the electrodes and the hole diameter in the anode of the gas discharge device. The electrostatic field distribution of the device forming the off-electrode plasma is analyzed.

Keywords

• plasma
• high-voltage gas discharge
• equipotentials
• field lines
• conformal mapping
• the Schwarz-Christoffel integral

1. Introduction

to the region of the Z field with the electrodes A1 A2 (cathode), A3 A4 (anode) (fig. 2) with internal angles  k at the vertices. with internal angles  k at the vertices ( 2 )

At the first stage, the vertices A A A A

1 2 3 4 of the Z-plane are associated with certain points of the real axis of the plane .

According to the theorem of uniqueness of a conformal mapping for a present correspondence of three arbitrarily chosen boundary points, for example, 0, 1, ∞, we can obtain the correspondence [11]:

A1 A2 A3 A4 0

0 1 a2  .

According to the technique developed in [10-12], the angles μk are determined, which complement the internal angles αk at the vertices of the quadrangle A A A A

1 2 3 4 to π. Considering the inner region of a quadrilateral and moving in the positive direction of traversing its boundary, i.e. counterclockwise, we find the angles: 1  1/ 2 ( 1  1  1  1/ 2) ;  2  1 ( 2  1  2  0);  3  1 ( 3  1   3  2) ;  4  3 / 2 ( 4  1   4  1/ 2) .

To find the mapping function of a domain bounded by a polygon A A A A 1 2 3 4 the Schwarz-Christoffel integral [11] is used: 1 Z  C  ( a1 )11 (  a2 ) 2 1...(  an ) n 1 d  C1, ( 1 )

In the expression ( 1 ) instead of а1 – an we substitute the corresponding points 0, 1, а2, ∞. According to [10], the factor related to the vertex а4 in the Schwarz-Christoffel integral is omitted, since а4 = ∞.

In this case, the expression ( 1 ) has the form:

  1    a2 

Z  С 0 2   11  a2 d  C1  С 0   1  d  C1 Let   x 2 , then:

 ( x 2  a 2 ) Z  C  0 ( x 2  1) x dx 2  C1 2С

1    Ca 2  1ln

 1    C .

1

The value of the constant coefficient C1 is determined from the correspondence of the points A1 ↔ 0, which allows us to write the equation:

Z  2C  0  C(a 2  1) ln Mathematical Modeling / M.A. Markushin, V.A. Kolpakov, S.V. Krichevskiy 1  0 1   ] r  rei  C(1  a 2 )(i ).

This allows us to write the expression:

Equating (3) and (4), we obtain:

Thus, the change of the coefficient value a2 can be described by the equation: h The correspondence of the points a2 and А3 makes it possible to transform expression ( 2 ) to the form: ih  С a 2  1i . ( 7 ) Z  2С   h 

 1  ln   1      .

Thus, expressions ( 5 ), ( 6 ) allow us to find a constant C whose value depends on the design parameters D and h.

At the second stage, an additional mapping of the half-plane Im > 0 is applied to the strip 0  Im  V with cuts along the corresponding rays (fig. 3). In this case we have a capacitor with infinite plates in the plane  .

Considering only the right triangle with vertices В1В2В3 because of the electrode design symmetry, we put the points 0, 1, ∞ lying on the real axis ω in correspondence to these vertices [11]:

B1 B2 B3 0 1  .

The inner angles  k at the vertices of the triangle В1В2В3 and the angles  k' , that complement the angles  k to π, are defined similarly to αk, µk,:  2'  1 ( 2  1   2'  0) ;  3'  1/ 2 ( 3  1  3'  1/ 2) ; 1'  1/ 2 ( 1  1  1'  1/ 2) .

Mathematical Modeling / M.A. Markushin, V.A. Kolpakov, S.V. Krichevskiy   ] r  rei  C (i )  C i , 2 2 The final function conformally mapping the half-plane  to the strip 0  Im  V has the form:

V  

 Using ( 7 ) and ( 9 ), we obtain a system of equations: ln 1 

 1    2V 

arcth  .

  2V    arcth  h  1 

Ln   1       

From (10) we find

    Z  2С  th    h ln  1  th 2V   2С  th    h  . ( 11 ) 2V   1  th    2V V

 2V 

By separating the real and imaginary parts of equation ( 11 ), the parametric equations of lines of equal potential and field lines of force are found. After separation, we obtain a system of equations describing the coordinates of the electric field distribution in the electrode system of a gas-discharge device: which allows us to obtain equality: Solving this equality, we define C 2 :

  iV . iV  C i .

2 C2 

V  ( 9 ) ( 10 ) ( 12 ) u v ch  cos

V V

Substituting the parameters h, V, D into the expressions ( 5 ), ( 6 ) and system ( 12 ) and changing the values of the variables v and u with the necessary step, we can determine the number of the field lines and equipotential distribution (fig.4 and fig.5).

Changing the voltage at the electrodes does not lead to a change in the field configuration, but it affects the energy of charged particles. Thus, the equations system ( 12 ) allows to obtain the electrode system configuration to form the required electric field by varying the parameters h, V, D.

3. Analysis of the field lines and equipotentials distribution

The initial coordinate (x = x0, y = y0 = 0) of the rectilinear segment of the field line can be determined with the aid of the system ( 12 ), giving the values u = u0 and v0 = 0. Then, searching further all the values of v = v1-vn for which the coordinate x = x0 is constant, and the y varies in the limits y1-yn. Further on, comparing the obtained maximum value of yn with the mean free path of the electron kλе (k = 1,2,3) and the potential at the given point with the ionization energy of the working gas atom (molecule) Ei, we verify the fulfillment of the condition for the emergence of an high-voltage discharge γQ ≥ 1 [ 8] is, where γ is the number of electrons knocked by one ion from the cathode (γ-process), Q is the number of positive ions formed by the electron on the trajectory of its motion due to inelastic collisions with atoms and molecules of the working gas (α-process). The energy 3rd International conference “Information Technology and Nanotechnology 2017” 97

Mathematical Modeling / M.A. Markushin, V.A. Kolpakov, S.V. Krichevskiy accumulated by the electron on the mean free path must be higher than the ionization energy of the working gas atom, and the energy of the positive ion bombarding the cathode must be sufficient for the emission of electrons necessary for sustaining the self-dependent discharge. Analogously, changing the values u = u1-un for v0 = 0, the corresponding x = x1-xn are determined. Further on, searching further values of v = v1-vn for each x, we find y = y1-yn = 0-kλе. In other words, by repeating the comparison process, we can find all field lines with initial coordinates x0 ,..., x , on the rectilinear segments of which the e ionization process takes place (α-process), and, accordingly, the length of the cathode region x  2x where the electron e emission from the cathode (γ-process) takes place [13]. Fig.5. Field lines and equipotentials distribution in the electrode system of the gas-discharge device obtained by the equations system ( 12 ): a – h = 1 mm,

D = 3 mm, V = 1200 V; b – h = 2.7 mm, D = 3 mm, V = 1200 V; c – h = 4 mm, D = 3 mm, V = 1200 V.

Mathematical Modeling / M.A. Markushin, V.A. Kolpakov, S.V. Krichevskiy

In order to compare the maximum values of yn with kλe, it is necessary to find the mean free path of an electron. Using the expression λe = 1/(Ni) [14], we obtain the value 0.203 cm, which makes it possible to determine x  318 μm. The calculated value of x correlated well with the experimental data of [9], namely, the size of the region on the cathode surface with intense sputtering by positive ions is 300 μm (fig. 6).

Fig 6. The profile of the etching pit on the surface of the cathode formed by positive ions.

This value is comparable with the size of the region x on which the rectilinear segments of field lines correspond to the value kλе and the condition for the emergence of an high-voltage discharge is observed.

4. Conclusions

The parametric equations system presented in this paper makes it possible to simulate the of the field lines and equipotentials distribution in the electrode system of the off-electrode plasma generator and to monitor the dependence of this distribution on the design parameters of the system: the anode-cathode distance, the hole diameter in the anode, and also on the applied voltage at the electrodes. In addition, estimates are made in this paper: the length of the rectilinear segments of the field lines on which the condition is satisfied, the size of the cathode spot x within which the γ process is realized. The discrepancy between the calculated value and the experimental value does not exceed 6%, which indicates that the model corresponds to the actual physical processes occurring in the electrode system of a high-voltage gas discharge. Therefore, it becomes possible to optimize the devices design forming the off-electrode plasma without costly experimental investigations.

Acknowledgments References

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