JCBPS; Section C; November 2016 – January 2017, Vol. 7, No. 1; 102-109.
E- ISSN: 2249 –1929
Journal of Chemical, Biological and Physical Sciences An International Peer Review E-3 Journal of Sciences Available online atwww.jcbsc.org
Section C: Physical Sciences CODEN (USA): JCBPAT
Research Article
Two-dimensional discrete particle simulation of Townsend gaseous discharge in a micrometer-scale plasma reactor Saurav Gautam1*, Abinish Kumar Dutta2, Deepak Prasad Subedi1 1
Department of Natural Sciences, Kathmandu University, Dhulikhel Kavre, Nepal
2
Department of Mechanical Engineering, Kathmandu University, Dhulikhel Kavre, Nepal
Received: 19 December 2016; Revised: 03 January 2017; Accepted: 05 January 2017
Abstract: Townsend discharge is a multistage gas ionization process that causes a nonconducting gas to become conductive due to an avalanche multiplication of free electrons under acceleration due to an electric field. In this paper, we discuss the methodology and results of a sub-millimeter scale two-dimensional particle simulation of Townsend discharge. Because of the virtually insuperable computational requirements of ordinaryscale particle simulation, almost all the models so far have relied upon particle-in-cell and fluid modeling methods. For computational ease, we have considered 10,000 molecules in a scaled down chamber and simulated the results of electric breakdown under an applied electric field. We have assumed a reasonable 0.001% of these to be already ionized due to background entities like cosmic rays and UV rays from the Sun. Results of this simulation have been found to be closely similar to those predicted by the Townsend theory of gaseous discharge. The concepts used in the simulation and subsequent observations are discussed. Keywords: Particle simulation, Townsend gaseous discharge, Electric breakdown, microplasma
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INTRODUCTION Under normal conditions, gas isolated from external ionizing factors is mostly charge-neutral. But due to the presence of high-energy cosmic rays, solar ultraviolet radiation and radiation from the earth’s interior, there are always a certain number of charged particles/ions (in the order of thousands per cubic centimeter) in the atmosphere1. The atmosphere, however, doesn’t form an ideal insulating medium, since the application of a sufficiently strong electric field causes the charged particles to move in the field’s direction resulting in conduction current. When the voltage difference between the electrodes surrounding the gas-gap reaches a certain critical value, the free electrons in the medium obtain high terminal velocities and thus high kinetic energies which, if greater than the gas ionization potential, result in the formation of even more ions and free electrons which then further ionize other gas molecules resulting in an avalanche current. This phenomenon is known as gaseous breakdown and the corresponding voltage is called the breakdown voltage 2. John Sealy Townsend was the first person to come up with the gaseous breakdown theory through his experiments conducted between 1897 and 1901. He then laid down its theoretical foundation the breakdown criterion, a few years later in 1910 3. This was then widely applied to understanding the phenomena of non-self-sustaining discharge, self-sustaining dark discharge and transition, until Walter Rogowski came along with his modifications incorporating the phenomenon of glow discharge in the theory4. Ordinary Townsend discharge phenomena in any system involve trillions of molecules controlled by a number of independent variables and are, therefore, extremely difficult to simulate down to the scale of every individual particle considered at the same time 5. Even the fastest of the modern supercomputers are unable to perform complete particle simulation of any conductive gaseous system within a justifiable amount of time. Most analyses are therefore reliant on other macroscopic models which rule out certain microscopic variables governing the kinetics of discharge phenomena. The results match the expected trends in general, but don’t fully explain the effects of minor variables. In particle modeling and simulation, the interaction of each particle with other particles and motion is considered and tracked. Since processing the huge amount of information that comes with applying such a method is extremely difficult, the system has to be scaled down to a reasonable size while reducing the particle density at the same time. The essential physics can fairly be captured with a much smaller number of particles than in a real gaseous system and the behavioral trend can still be observed with minor error, irrespective of the numerical deviations 6. METHODOLOGY The number of particles for the simulation was restricted to 10,000 on a 10µm×1µm plane as shown in the Figure 1. Since external ionizing factors like cosmic rays, solar UV rays and radiation from the earth’s interior are constantly present with varying magnitudes; a reasonable assumption has to be made to account for their influence to keep the simulation realistic and free electrons initially available. 0.001% of the gas molecules were assumed to be initially ionized and a corresponding number of free electrons present between the electrodes, which is a reasonable assumption7. According to the original Townsend theory, there are two processes by which the number of charged particles in the gas gap between the electrodes can grow upon the application of an electric field: collision between electrons and neutral gas molecules and the impact of positive ions on the cathode surface, each of them contributing to the creation of more free electrons and ions causing further collisions 8, 9. Thus, three types of particles were considered for the simulation: electrons, positive ions and neutral gas molecules. 103
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The program, written in C++, follows the order as illustrated in the flowchart shown in the Figure 2. It first creates three matrices, each corresponding to a particular type of these particles representing their overall 2N-dimensional state space. Each of these matrices contains four rows; the first and the third representing the x and y components of the positions of the particles and the second and the fourth representing the x and y components of their respective velocities. Each column represents an individual particle of the specified type, as shown in the Figure 3. The program execution begins with the initialization of all these three matrices with arbitrary values of the position coordinates within the interelectrode region. The initial value of velocity is set to zero for all the particles. Gas molecules being electrically neutral show no response to the external electric field. The time evolution of position and velocity coordinates of the electrons and ions is guided by the force due to the field calculated using the Fourth Order Runge-Kutta method. Maxwell-Boltzmann distribution of velocities of particles has not been taken into consideration in this simulation, as was the case with the original Townsend’s proposition. All types of collisions between the particles except those between the electrons and the neutral molecules and those resulting from the impact of cations on the cathode have been neglected. Two different criteria are sequentially tested- collision criterion and ionization criterion. Collision between an electron and a neutral molecule is said to have occurred if the distance between those two particles is less than or equal to the sum of their radii. After the collision criterion is fulfilled, the ionization criterion is tested according to which the collision in which the energy of the colliding electron is greater than or equal to the ionization potential of the neutral molecule causes ionization. The colliding electron failing to satisfy the ionization criterion recoils back with reduced velocity. The average value of the electrical current over a period of time is calculated as the average number of charged particles crossing the electrodes multiplied by the electrical charge and divided by the time interval. Secondary emission yield due to the impact of positive ions on the cathode is also considered in order to make the resulting discharge self-sustaining 10, 11. The algorithm, although simple, covers the most fundamental aspects of Townsend’s theory and is worthy of implementation for a simplified verification of the theory.
Figure 1: Computer aided 3D visualization of particles on the single 10µm×1µm simulation plane. 104
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Figure 2: Flow sequence of the simulation program.
Figure 3: Structure of the matrix containing information about the position and velocity coordinates of individual particles used in the simulation.
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RESULTS AND DISCUSSION The simulation was carried out for different values of the inter-electrode voltage and the resultant values of current were calculated. The graph thus obtained is shown in the Figure 4. At lower values of the interelectrode voltage, the current is markedly low.
Figure 4: The value of current at various inter-electrode voltages across the electrodes in a modeled 10µm×1µm planar plasma reactor. This is because of the relatively low number of free electrons between the electrodes. As the value of the applied voltage is increased, the current increases at a slow rate until the voltage crosses a threshold of 120V, where after the inter-electrode current also rises sharply. At this point, the gas is said to have undergone a breakdown. The curve thus plotted was found to be closely similar to the Townsend curve12. Figures 5(a) and 5(b) represent the distribution of neutral molecules between the electrodes before and after the breakdown respectively. There is no significant difference between the two diagrams, meaning the ionization fraction is so low that it does not affect the number of neutral molecules. Figures 5(c) and 5(d) show the spatial distribution of positive-ions between the electrodes. There are only 10 ions before the breakdown and 141 ions after the breakdown. The distribution of ions appears to be uniform. This behavior may be attributed to their lower mobility in comparison to that of the electrons because of the higher required momentum. Figures 5(e) and 5(f) represent the distribution of electrons in the inter-electrode space. There are 10 electrons in the figure 5(e) and 150 electrons in the figure 5(f). The plot in the figure 6 shows the spatial distribution of electrons between the electrodes after the breakdown condition is met. The curve is exponential as predicted by the Townsend’s equation n = no eαd / [1 − γ (e αd − 1)] , in which n is the number of molecules at a distance d from the cathode, no is the number of molecules leaving the cathode 106
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at any time, α and γ are the Townsend’s first and second ionization coefficients respectively. The value of γ was arbitrarily chosen to be 0.027 at the time of simulation and the value of α was calculated to be 0.0163 from the graph. The close resemblance of the simulation curves in the figures 4 and 6 to each other validates the workability of the simulated model. The apparent minor disagreement may be attributed to the statistical discrepancies resulting from the simplification of the model for simulation 13.
(a). Neutral molecules before breakdown
(c). Ions before breakdown
(e). Electrons before breakdown
(b). Neutral molecules after breakdown
(d). Ions after breakdown
(f). Electrons after breakdown
Figure 5: Spatial distribution of charged particles in the simulation region between the electrodes in a 10µm×1µm plasma reactor.
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Figure 6: Number of electrons at various positions between electrodes after breakdown. CONCLUSION The processing capacity of computers these days is constantly being upgraded. Particle simulation can become an area of great interest and scope for computational physicists in the future. With less simplification required, particle simulations in the future can give a more realistic picture of the microscopic and the resultant macroscopic phenomena in gaseous systems and plasmas. It is hoped that this fairly successful work sets a basis for further developments in this field. ACKNOWLEDGMENTS The authors are grateful to Jeff Pummill and Pawel Wolinski of the High Performance Computing Center at the University of Arkansas for providing access to their cluster computer. Dr Rajendra P. Adhikari, Kathmandu University Department of Natural Sciences, guided in the computational tasks associated with the project. REFERENCES 1. N. V. Ardelyan, V. L. Bychkov, I. V. Kochetov, and K. V. Kosmachevskii, Russian Journal of Physical Chemistry B.,2015, 9, 807. 2. M. M. Pejovic, and R. D. Filipovic, International Journal of Electronics.2007, 67, 251. 3. Von Engel, John Sealy Edward Townsend. 1868-1957. Biographical Memoirs of Fellows of the Royal Society, 1957, 3, 257. 4. X. Lin, F. Wang, J. Xu, Y. Xia, and W. Liu, Plasma Science & Technology,2016, 18, 223. 5. J. M. Dawson, Reviews of modern physics.1983, 55, 403.
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* Corresponding author: Saurav Gautam Department of Natural Sciences, Kathmandu University, Dhulikhel Kavre, Nepal Email :
[email protected] On line publication Date: 05.01.2017
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