The Mathematical Model of Arc Discharge in Metal Vapors at Active Gases over Crucible for Technological Process of Electron Beam Deposition of Ceramic Coatings

The mathematical model of arc discharge in the metal vapors, propagated in the soft vacuum in the medium of active gases, is presented in this chapter. Such type of discharge is widely used in advanced electron beam technologies for obtaining the coating of new types from nanostructurized materials, especially ceramics coating. As electron beam sources for evaporation of refractory metals in this technological process the high voltage glow discharge electron guns are widely and effectively used. But the aim of applying of additional low-voltage arc discharge under the crucible is stimulating and maintaining the chemical reactions between metal vapors and residual gas in the vacuum technological chamber. In formed model for calculation of electric field distribution the analytical solving of Poisson equation is used, and the spatial distribution of discharge current density is defined on the base of the equation of uncontineously of electrons and ions flows. All analytical relations for auxiliary geometry of electrodes system with of electron beam technologies for obtaining different types of nanostructurized ceramics coatings.


INTRODUCTION
Today in the technologies of obtaining the nanostructurized ceramic thin films is widely used the reactive deposition.
The main physical processes of reactive films deposition are  [1 -4].
Since the traditional electron guns with heated cathodes can't long time stably operated at the medium of active gases, often for providing the reactive deposition of ceramic thin films applied the high voltage glow discharge electron guns (HVGDEG). Generally, such guns are always characterized by the simplicity of construction, cheapness of the gun and evacuation equipment, as well as by the stable operation in the soft vacuum, range of 1 -10 Pa, at the medium of active gases [5,6]. In additional, effective control of beam power both by changing the operation pressure at the gun chamber [7] and by lighting the additional low-voltage discharge [8] is also possible.
Advanced technology of obtaining nanostructurized ceramic coatings with using HVGDEG can be successfully applied for many industrial applications in mechanical engineering, automotive industry, aerospace industry, as well as instrumentmaking and in electronics industry. The ceramic films, obtained with applying reactive deposition by using HVGDEG as the source of electron beam, are usually characterized by the high mechanical strength, heat resistance, as well as by the good dielectric properties. Therefore, using of such ceramics films and coatings is very perspective at cutting instruments, as thermoinsulation coating in engines, as well as thin dielectric films in the modern electronic devices, including microwave devices for communication systems [9][10][11].
Unfortunately, further development of advanced technology of reactive deposition of thin films with using HVGDEG is hampered by the lack of precision and adequate mathematical models of this process. The absence of such kind mathematical model does not allow to elaborate effectively the electron beam technological equipment for reactive deposition of nanostructurized thin films. Since the basic principles of simulation of high voltage glow discharge (HVGD) electron sources generally have been considered at the paper [6], for forming the complex model of process of reactive deposition of thin films considering the basic principles of simulation the arc discharge in the metal vapors over the crucible is necessary.
Therefore, the aim of this chapter is considering these basic physical principles for such kind of model, obtaining necessary analytical relations and analyzing obtained simulation results.

BASIC STATEMENT OF CONSIDERED PROBLEM
The generalized scheme of electrodes system for lighting and maintaining arc discharge in the metal vapors, which included surface of evaporated metal, located in crucible, and additional ring-like electrode with the positive potential relatively to crucible, is presented in Fig. 1 [5]. Basic geometry parameters of depicted at Fig. 1     Basic equations, which are corresponded to described above physical presumptions of considered simulation problem, will be presented at the next sections of this chapter.

Equations for calculation the distribution of electric field
Generally, the equation for defining of electric field at the electrodes system, which structure and geometry parameters are presented at . (1) where U(r,z) -the electric potential, ρ -space charge density [12,13].
Analytical solving of equation (1) for electrodes system with the plane and thin-ring electrodes, like plotted at Fig. 2, is written as follows [12,13]: where ε -dielectric constant of residual gas,  Defining the pressure of saturated vapor, concentration of

metals' atoms and necessary power of electron beam
The pressure of saturated vapor at the system of reactive deposition of cheremic films, which general constructive scheme is presented at Fig.1 and the geometry parameters are defined at Fig. 2, in the general case can be defined with using Mendeleev -Clapeyron equation as follows [14 -16]: where p s -pressure of saturated vapors, ρ v -density of vapors, Therefore, taking into account relations (7) and (8), the analytical relation for concentration of the atoms of metal vapors over the crucible is written as follows: 1 .
Concentration of metal's atoms n m , have been defined with using the equation (9), is used later by substitution to the differential equation (1) for defining the space charge in simulated electrodes system of reactive deposition of thin films. Dependence n m (T ev ), have been obtained with using relation (9) is presented at Fig. 3.
It is also important, that concentration of vapor atoms n m is strongly depended on power of electron beam and defined by simple relation [14 -16]: where q ev -specific heat of vaporization, m m -mass of metal atoms.
Taking into account relation (9), the dependence P ev (T ev ) for obtaining the regime of saturation vapors, can be written as follows: Dependence Pev(Tev), obtained with using relation (11) for titanium evaporation which have thermodynamic parameter   atoms (9) and other relations of gas dynamic [14 -16]. Firstly, the current density of arc discharge has to be estimated on the base of fundamental relations of gas-discharge theory, presented in monographs [18 -20]. Generally, the current density of arc discharge can be estimated as follows [18 -20]:  For model of single-charge ionization, taking into account the law of charges equilibrium, such relation is always holds [18 -20]: . e im ig n n n = + Taking into account, that concentration of gas atoms is defined as [14 -16]: 0 g g p n kT = (14) relation (13), with taking into account (9), (14), can be rewritten as: where β m -level of ionization of metal vapor atoms, β g -level of ionization of gas atoms.
Therefore, equation (12), with taking into account (9,14,15), can be rewriting as follows: For simplifying obtained equation (16) and further solving the differential equation (1), the following coefficient have been introduced: With using substitutions (17), equation (16) has been rewritten in the simplified form as follows: With known the spatial distribution of electrical potential φ(r), the current-voltage characteristic of considered arc discharge at the metal's vapors can be obtained with using equation (18). Now, for further analytical solving of differential equation (1), the components of obtained equation (18) It is also well-known fact, that the space charge, formed by the moving charged particles in the electrodes' system, is defined form the current density of this particles by the following relations [12, 13, 18 - where s -sort of the particles, v s -velocity of the particles, q scharge of the particles.
Therefore, for obtaining the values of space charges, formed by the metals' ions ρ im , by the residual gas ions ρ ig and by the electrons ρ e , corresponding to equations (19,20), such analytical relations can be written: Substituting coefficients C 1 , C 2 and C 3 from relations (17) to relations (21) and making corresponded analytical transformers, the following relations to the values of space charges of ions and electrons can be written as follows: Taking into account the law of equilibrium of charged particles, defined by the relations (13,17), from obtained relations (22), after corresponded mathematical transformers, the formula for summarized space charge ρ Σ can be wrote as follows: For further theoretical analysis, it is very important, that the coefficients of the model C 1 , C 2 , and C 3  On the base of obtained relations (22, 23) the potential distribution in the considered electrodes system, which basic construction and main geometry parameters are presented at Fig. 1 and Fig. 2, can be defined by analytical solving the differential equation (1).

Defining the distribution of electric field
Taking into account the simple analytical relation (22), differential equation (1) can be written for the considered system of reactive deposition of thin films as follows:  where E 0 and U 0 -the constants, which are defined by initial conditions (25).
From the equations (25 -27), such analytical relation to the value E 0 can be written: Taking into account obtained relation (28) and substitution the value of E 0 into equation (27), following analytical relation can be obtained: From obtained equation (29) the direct analytical relation for defining unknown value U d from the basic discharge electric and geometry parameters is written as follows: Taking into account obtaining relations (28, 30) for the coefficients E 0 and U 0 , finally analytical relation (27) for the potential distribution φ(r) can be rewritten as follows: Let assume in the equation (31) the following substitute: After that substitution, formula (31) can be considered as standard cubic equation with the following coefficients: Analytical solving of the set of equations (33) with using wellknown Cordano formulas [22] give such result: ( )  The simulation results for distribution of potential at the plane of ring electrode location, obtained with using relation (34), as well as results of calculation of current density distribution, will be presend at the next part of this chapter.

Simulation results and its analyze
The results of simulation of distribution the electris field U(r) at the plane of simmetry of ring electrode of simulated electrodes system, designed for reactive deposition of thin films at the soft vacuum, is presented at Fig. 6. These results have been obtained with using the set of equations (23, 34) for different values of arc discharge voltage Ud. These results was obtained for such dischartge parameters: 1.
The inner radius of ring electrode R r = 0.05 m.

2.
Power of electron beam P b = 20 kW.

3.
Pressure of residual gas at the technological chamber p g = 5 Pa.

4.
Level of ionization of metal vapor atoms β m = 0.8.

5.
Level of ionization of residual gas atoms β g = 0.75. This physical effect is also described mathematically with using equations (2, 3) [12,13]. For free space, without charged particles, and for the values of voltage at the ring electrode U d range of 50 -100 V, the sagging of electric field potential is generally grater, up to 15 -20 V [12,13]. Therefore, in the considered conditions of maintaining the arc discharge the smallest value of potential sagging near the axis of electrodes' system can be explained by the influence of space charge of the positive ions of metal vapors and residual gas [18 -20].
As for space charge of electrons, its influence in the discharge systems is always smaller by the reason of small mass of electrons and its high velocity and negligible [18 -20]. Also, it is important on the theoretical point of view, that derivation of electric potential on the axis of electrodes system is also 0 ( ) 0. r dU r dr = = This important condition is always satisfied for axial electrodes systems [12,13].
With known dependence of potential distribution at the plane of symmetry of ring electrode the value of current density of arc discharge can be defined with using equation (18).
Corresponded graphic dependences, obtained as results of using described simulation technique, are presented at Fig. 7, a, b. The values of the coefficients C 1 and C 2 for considered parameters of arc discharge, pointed out above, was Clear, that generally dependences of arc discharge current density, presented at Fig. 7, similar to dependences of electric potential distribution, given at Fig. 6. This fact can be simply explained by analyzing and rewritten the equation (18). Really, dependence j d (φ(r)), defined with using equation (18), with knowing coefficients C 1 , C 2 and C 3 , can be rewritten in the simplified form as follows:  jd(r), for different voltage of arc discharge maintaining U d , are given at Fig. 6 and Fig. 7.
The main advantage of proposed model of arc discharge for reactive deposition of nanostructurized ceramic films is its simplicity, because it based only on analytic relations and don't include iterative numerical calculations. But in any case, it generally corresponded to the main physical principles of electric fields theory [12,13], thermodynamic and gas dynamic theory [14,15], as well as to the theory of gas discharges [18 -20]. Therefore, generally obtained simulation results for distribution of electric field and arc discharge current density at the ring plane are looks plausible. For improving the accuracy of proposed model more precision values of the used semiempirical coefficients C 1 , C 2 , C 3 , K, S 1 and S 2 can be defined empirically for the specific technology and for metals and gases, applied for obtaining the highquality nanostructure ceramic films [1 -4]. Therefore handbooks on basic physics [17], on physics of gas discharges [18 -20], as well as on gas dynamics [14 -16] can be used.
It also should be pointed out, that for realizing the technology of electron-beam deposition of nanostructurized films the HVGDEG can be successfully applied, because the electron guns of such type are reliably and stably operated at the medium of active gases [5 -8].