Modeling the Influence of the Penetration Channel ’ s Shape on Plasma Parameters When Handling Highly Concentrated Energy Sources

1Department of Welding, Metrology, and Materials Technology, Perm National Research Polytechnic University, Komsomolskiy Prospekt, No. 29, Perm 614099, Russia 2Department of Automation and Telematics, Perm National Research Polytechnic University, Komsomolskiy Prospekt, No. 29, Perm 614099, Russia 3Department of Mechanical Engineering, Rapid Manufacturing Laboratory, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India


Introduction
An important problem in the development of flawless electron beam and laser welding technologies is ensuring that high-quality welds can be consistently reproduced.Work to automate the process using secondary waveform parameters has been pursued for a long time with varied success.Certain successes have been achieved using methods involving registration of the secondary X-ray radiation emitted during welding with highly concentrated energy sources.However, these methods have limitations due to the difficulty placing additional sensors within the vacuum chamber, which creates complications when used in industrial conditions.Moreover, a number of studies indicate that electron beam welding is accompanied by high-frequency processes [1][2][3][4] that often carry the most information about the thermal characteristics of the electron beam's interaction with the metal in the penetration channel.Frequencies up to 20 kHz are inherent to these processes, which makes it possible to study them using X-ray radiation sensors.Methods based on registration of the secondary plasma current originating above the welding zone [5] are promising.Several methods have been developed to study and control electron beam welding [1][2][3][4][5] processes.Similar works in the area of laser welding have been widely conducted recently.However, we must note that the problem of controlling penetration during electron beam welding using the parameters of the secondary plasma current cannot yet be considered fully solved.There are no models that describe the relationship between the parameters of the secondary waveforms in the plasma.To date, there are no universal, reliable, and sufficiently responsive process control systems for making a welded joint with electron beam welding.The accumulated knowledge is frequently tentative for now.
References [1][2][3] describe the results of experimental studies on generating secondary current waveforms in plasma during EBW.Several hypotheses have been put forward to describe the observed phenomena.However, the proposed theories are insufficient to fully interpret the obtained results [4,5].We need a detailed model of the processes occurring in the plasma.There are well-known works that estimate plasma parameters using simplified methods [6][7][8][9][10].The authors of these works propose one-dimensional models that can be used to estimate the concentration and energy of electrons in the plasma above the EBW zone, but they do not make it possible to calculate the parameters of the non-self-sustained discharge.References [11][12][13] describe a two-dimensional model of the formation of the non-selfsustaining discharge in plasma during EBW.However, for practical purposes, [14] is of interest in determining how the waveform's magnitude changes during a change in the position of the beam in the penetration channel and as a function of the channel's geometric dimensions.Direct modeling requires a three-dimensional formulation, which entails unjustifiably large computational costs.This article proposes a method and numeric implementation based on a preliminary solution to the problem in a two-dimensional axisymmetric formulation with subsequent use of the results of this solution for a full description.The results obtained from this solution for measuring secondary electric current in plasma during EBW may be extended to laser welding.

Description of the Model
At the heart of the model are transfer equations for the concentration of electrons   and the average electron energy in the plasma over the EBW zone [14][15][16][17]: where  → D  is the electron stream density,   is the volumetric energy density of the electrons,   ,   are the electrons' diffusion coefficient and mobility coefficient for the energy, R  is the intensity of the electron source,   is the electrons' energy source (describes the energy loss due to inelastic collisions), and  →  is the electric field vector.To describe the mass transfer of heavy plasma particles (ions, neutral unexcited and excited atoms), we use the mass transfer equation for a multicomponent mixture [18][19][20]: The plasma becomes collisionless above the penetration channel.When this happens, the diffusion coefficients in (1) go to zero and the equations degenerate into continuity equations for charge and energy.
A full system of equations and description is given in [14].Calculations are performed in cylindrical coordinates in an axisymmetric formulation (Figure 1).
A non-self-sustaining discharge in plasma is accompanied by the emission of secondary electrons from the surface of the item (cathode).The edge conditions account for the loss of electrons on chamber walls and on the surface of the item due to random motion and thermoemissive effects [19]: and for the electrons' energy flux: where ] ,th is the thermal velocity of electrons,  → D  =  →   / is the electron stream resulting from thermionic emission (only on the walls of the penetration channel), and  →  is the surface normal.  1 presents the plasma-chemical reactions taken into consideration in the calculations.As a result of the plasma's low ionization coefficient, the electrons' interactions with ions and excited atoms are ignored.The energies corresponding to each reaction are the input parameters for the model.To obtain a numeric solution, we used COMSOL 4.4 and the Plasma Module, Molecular Flow, Rarefied Flow modules.The modeling results adequately describe the patterns known from the literature and observed experimentally.
Reference [14] presents the parameters of the plasma, calculated using the presented model, not only above the welding zone but also directly in the vapor channel being formed.The mathematical model makes it possible to calculate the change in plasma parameters and the magnitude of the waveform of the secondary current in the plasma during excitation of a non-self-sustaining discharge by delivering the positive potential to the collector.Modeling the formation of this waveform is of special interest, since it is actively used when building automated control methods for EBW [21].

Influence of the Penetration Channel's Shape on the Magnitude of the Secondary Emission's Waveform
Reference [11] demonstrates that in the upper regions of the penetration channel the electric field reaches a magnitude of 10 6 V/m.The presence of such an electric field significantly lowers the potential barrier and reduces the work to discharge electrons.In this case, the density of the emission is described by the Richardson-Schottky equation where   is the intensity of the electric field.
Figure 2 presents the model's results for the electron emission's current density distributions on the walls of the penetration channel, calculated using (7) with and without the electron emission.The plots demonstrate that, despite the emission's sharp increase due to the Schottky effect in the upper regions of the penetration channel, the emission continues to tend to grow as the depth increases.This result is due to the temperature's prevailing influence on the emission.According to [21], when formulating the problem the temperature is assumed to increase in the direction of the bottom of the channel.
The magnitude of the secondary current in plasma, which is excited when a non-self-sustaining discharge is initiated, is influenced by more than just the electron emission from the walls of the penetration channel.During diffusion of slow electrons in plasma in the penetration channel, some of them escape into the walls of the penetration channel.Consequently, the resulting current density distribution differs from the plots in Figure 2. Figure 3 demonstrates that the influence of electron loss at the walls of the channel reduces the current density when approaching the bottom.At a depth of roughly 9 mm, the current density changes sign for the penetration channel's specified transverse dimensions.The change in sign means that in these regions the electron stream directed at the walls begins to prevail over the stream caused by the emissive phenomenon.The fact that this effect is observed at the bottom of the channel is apparently due to the weaker electric field, which prevents its occurrence.For practical purposes, [22] is of interest in determining how the waveform's magnitude changes during a change in the position of the beam in the penetration channel and as a function of the channel's geometric dimensions.The magnitude of the secondary current in plasma, which is excited when a non-self-sustaining discharge is initiated, is influenced by more than just the electron emission from the walls of the penetration channel.During diffusion of slow electrons in plasma in the penetration channel, some of them escape into the walls of the penetration channel.
During oscillating EBW, the electron beam is moved along the walls of the penetration channel, causing a periodic change in the temperature field.This produces oscillations in the magnitude of the secondary current in the plasma [23].

Results: Transition to a Three-Dimension Formulation
This article proposes a numeric implementation whose first stage involves using the model to determine the statistical probability () of the secondary electrons' passage through the penetration channel from interaction area depth , or, in other words, the ratio of the number of electrons, emitted from a certain surface element at the penetration channel's depth and reaching the collector, to the total number of electrons emitted from this element.Then, we use the discovered function () to numerically calculate the change in the secondary current when a distributed beam is moved about the three-dimensional penetration channel.The value of () may be determined as follows.At depth , a small area Δ = 2Δ (Figure 4) is selected on the wall of the penetration channel.During the calculation, the response Δ  to the waveform's secondary current is increased when the emission's current density is increased in the area Δ by Δ.The probability () of the secondary electrons' passage through the penetration channel from depth  is determined using the following expression: Figure 5 presents the results of the calculation of the relationship between Δ  and the depth  of element Δ.The relationships were calculated for four values of the penetration channel's average radius (0.18 mm, 0.203 mm, 0.345 mm, and 0.428 mm).For each of the calculations, Δ 0 was taken to be 0.1 mm in the upper part of the penetration channel.Based on the requirement that the area of element Δ remains constant as depth  changes, the value of Δ was calculated as Δ = Δ 0 ⋅  0 /, where  0 is the radius of the penetration channel at the top.
Figure 6 presents the probability () of the secondary electrons' passage through the penetration channel from depth .
The plots in Figures 5 and 6 demonstrate that the magnitude of the response Δ  or the probability of electron loss decreases linearly, and at some depth  max they become zero.The value of  max decreases as the penetration channel's transverse dimensions decrease.Given a penetration channel with an average radius of 0.18 mm, only electrons emitted from a depth of no more than 4 mm reach the collector.With wider channels, electron from a depth of 15 mm or more hits the collector.
Knowledge of the probability () of the secondary electrons' passage through the penetration channel from a depth of  makes it possible to calculate the change in the secondary waveform when the beam moves as it oscillates along the walls of the channel.Suppose EBW is being performed with a small oscillation of the beam, according to some trajectory with deflections both along and across the joint.Let OXY be Cartesian coordinates in the horizontal plane.The shape of the penetration channel is described by the function (, ), where   ,   is the electron beam's displacement from its equilibrium position in the penetration channel, and   ( −   ,  −   ) is the electron beam's current density distribution.
Each beam deflection   ,   can be associated with the secondary waveform value   (  ,   ).According to [23], the density of emissive current from the surface of the penetration channel,   (, ), is proportional in the first approximation to the normal of the electric field's electron density   (, )  =   (, ) cos , where  is the beam's angle of incidence on the walls of the penetration channel.If  is an infinitely small element of the surface of the penetration channel, then the current emitted from this element is where   ,   are constants.The share of electron current emitted from the surface of element  and reaching the collector is Finally, the waveform of the secondary electron current as a function of the electron beam's displacement (  ,   ) is described by the integral: ⋅  ( (, ))  , (10) where , , , and  are the boundaries of the region under consideration in the OXY plane.We approximate the beam's current distribution and the shape of the penetration channel using standard Gaussian curves where  is the penetration depth;  0 is a constant that specifies the magnitude of the beam's full current;   ,   are parameters that define the width of the penetration channel;   ,   are parameters that define the transverse dimensions of the electron beam.We will assume   =   and   =   .We approximate the function () in accordance with the modeling results (Figure 6) We approximate the shape of the penetration channel with a Gaussian surface.Figure 7 presents the cross-section (, 0) of the penetration channel used for modeling.
For   (  ,   ), a numeric calculation of formula ( 9) yields the distribution given in Figure 10.This distribution represents the change in the waveform of the secondary current in plasma when the electron beam is shifted by   ,   from its equilibrium position in the penetration channel.The change in the collector's current as the electron beam moves along the -axis is given in Figure 11.The magnitude of the collector's current diminishes as the beam's axis approaches the axis of the penetration channel, where it reaches its minimum.
Each beam deflection corresponds to its own depth, at which the beam's center interacts with the metal on the wall of the penetration channel: (  ,   ). Figure 12 demonstrates the attenuation of the waveform of the collector's current as a logarithmic function of changes in this depth.The waveform's attenuation is calculated as the ratio of the current's maximum current   max , which corresponds to the beam's placement on the surface of the penetration channel, to its magnitude   () given interaction at depth . Figure 12 demonstrates that over most of the interval   max /  () = () may be considered exponential.These results confirm previous experimentally observed patterns (Figure 13).
The theoretical and experimental values are compared in Figure 14.The approximation error was 2.1%.Keep in mind the fact that results calculated using formula (9) depend on the penetration channel's assumed shape.If the penetration channel has a conical shape, the error becomes 7%.

Conclusions
The obtained numeric solution describes the formation of the waveform of the secondary current in plasma during changes to the penetration channel's geometric parameters.We have established that the probability of electron loss decreases linearly as depth  in the channel increases, and at a certain depth  max it becomes zero.We have presented a method for transitioning to a three-dimensional formulation, along with a calculation of the attenuation of the collector's current when the electron beam moves along the depth of the penetration channel.The obtained numeric solutions agree with known experimental data that indicates that the waveform has the greatest amplitude when interacting with the upper areas of the penetration channel and its amplitude decreases nearly exponentially along the depth of the penetration channel.These solutions may be used in researching and automating processes that process materials using concentrated energy sources.

Figure 1 :
Figure 1: Geometry of the computational zone: 1: vapor channel; 2: surface of the sample being welded; 3: electron collector; 4: edges of the calculation zone.

Figure 2 : 2 )Figure 3 :
Figure2: Distribution of the emission's current densities in the penetration channel during excitation of a non-self-sustaining discharge:  0 is the density of the thermionic current;   is the current density, accounting for the Schottky effect.

Figure 4 :
Figure 4: Determination of probability () of the secondary electrons' passage through the penetration channel.

Figure 5 :Figure 6 :
Figure 5: The secondary current waveform's response Δ  to changes in the emission in element Δ as a function of depth  for four penetration channels with varying transverse dimensions.

Figure 7 :
Figure 7: Cross-section of the penetration channel used for modeling.

Figure 13 :Figure 14 :
Figure13: Experimental dependence between the depth of penetration and the secondary waveform parameter ln( max / min ) given welding with oscillation along the joint and different focus settings ( = 3 kW, oscillation frequency  = 678 Hz, and sweep size 2 = 1.3 mm)[24].

Table 1 :
Plasma-chemical reactions taken into consideration.