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

Authors: Dmitriy N. Trushnikov; Ekaterina S. Salomatova; Igor I. Bezukladnikov; Igor L. Sinani; K. P. Karunakaran
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

In our work to formulate a scientific justification for process control methods when processing materials using concentrated energy sources, we develop a model that can calculate plasma parameters and the magnitude of the secondary waveform of a current from a non-self-sustained discharge in plasma as a function of the geometry of the penetration channel, thermal fields, and the beam’s position within the penetration channel. We present the method and a numeric implementation whose first stage involves the use of a two-dimensional model to calculate the statistical probability of the secondary electrons’ passage through the penetration channel as a function of the interaction zone’s depth. Then, the discovered relationship is used to numerically calculate how the secondary current changes as a distributed beam moves along a three-dimensional penetration channel. We demonstrate that during oscillating electron beam welding the waveform has the greatest magnitude during interaction with the upper areas of the penetration channel and diminishes with increasing penetration channel depth in a way that depends on the penetration channel’s shape. When the surface of the penetration channel is approximated with a Gaussian function, the waveform decreases nearly exponentially.


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## Body

## 1. 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 [1–4] 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 [1–5] 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 [1–3] 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 [6–10]. 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 [11–13] describe a two-dimensional model of the formation of the non-self-sustaining 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.


## 2. Description of the Model

At the heart of the model are transfer equations for the concentration of electronsne and the average electron energy in the plasma over the EBW zone [14–17]: (1)∂ne∂t+∇·Γ→e=Re,∂nε∂t+∇·-nεμεE→-Dε∇nε+eE→·Γ→e=Rε,where Г→e is the electron stream density, nε is the volumetric energy density of the electrons, Dε, με are the electrons’ diffusion coefficient and mobility coefficient for the energy,Re is the intensity of the electron source, Rε is the electrons’ energy source (describes the energy loss due to inelastic collisions), and E→ 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 [18–20]:(2)ρ∂ωk∂t+ρu→·∇ωk=∇·j→k+Rk,where j→k expresses the density of the mass stream of the kth component; ωk is the mass concentration of the kth component; Rk is the intensity of the source of the kth component; u→ is the average velocity vector; ρ is the mixture density.The electric field is determined from the Poisson equation(3)∇2U=-ρ0ε0.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 (Figure1).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.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]:(4)-n→·Γ→e=12νe,thne+neμeE→·n→-n→·Γ→tand for the electrons’ energy flux:(5)-n→·Γ→ε=56νe,thnε+nεμεE→·n→,where νe,th is the thermal velocity of electrons, Г→t=j→t/e is the electron stream resulting from thermionic emission (only on the walls of the penetration channel), and n→ is the surface normal.Table1 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.Table 1
Plasma-chemical reactions taken into consideration.
Reaction
Formula
Type of interaction
1
e
+
A
⟶
e
+
A
Elastic
2
e
+
A
⟶
e
+
A
∗
Excitation
3
e
+
A
⟶
2
e
+
A
+
IonizationReference [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].


## 3. 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 106 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(6)jt=AT2exp⁡-eφ-eEkkbT,where Ek is the intensity of the electric field.Figure2 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.Figure 2
Distribution of the emission’s current densities in the penetration channel during excitation of a non-self-sustaining discharge:jt0 is the density of the thermionic current; jt is the current density, accounting for the Schottky effect.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 Figure2. 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.Figure 3
Resulting distribution of the current density on the walls of the penetration channel.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].


## 4. Results: Transition to a Three-Dimension Formulation

This article proposes a numeric implementation whose first stage involves using the model to determine the statistical probabilityp(z) of the secondary electrons’ passage through the penetration channel from interaction area depth z, 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 p(z) to numerically calculate the change in the secondary current when a distributed beam is moved about the three-dimensional penetration channel.The value ofp(z) may be determined as follows. At depth z, a small area ΔS=2πrΔl (Figure 4) is selected on the wall of the penetration channel. During the calculation, the response ΔIk to the waveform’s secondary current is increased when the emission’s current density is increased in the area ΔS by Δj. The probability p(z) of the secondary electrons’ passage through the penetration channel from depth z is determined using the following expression:(7)pz=ΔIkΔjΔS.Figure 4
Determination of probabilityp(z) of the secondary electrons’ passage through the penetration channel.Figure5 presents the results of the calculation of the relationship between ΔIk and the depth z of element ΔS. 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, Δl0 was taken to be 0.1 mm in the upper part of the penetration channel. Based on the requirement that the area of element ΔS remains constant as depth z changes, the value of Δl was calculated as Δl=Δl0·r0/r, where r0 is the radius of the penetration channel at the top.Figure 5
The secondary current waveform’s responseΔIk to changes in the emission in element ΔS as a function of depth z for four penetration channels with varying transverse dimensions.Figure6 presents the probability p(z) of the secondary electrons’ passage through the penetration channel from depth z.Figure 6
Probabilityp(z) of the secondary electrons’ passage through the penetration channel from depth z for channels with differing radii.The plots in Figures5 and 6 demonstrate that the magnitude of the response ΔIk or the probability of electron loss decreases linearly, and at some depth zmax they become zero. The value of zmax 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 probabilityp(z) of the secondary electrons’ passage through the penetration channel from a depth of z 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. LetOXY be Cartesian coordinates in the horizontal plane. The shape of the penetration channel is described by the function z(x,y), where τx,τy is the electron beam’s displacement from its equilibrium position in the penetration channel, and jb(x-τx,y-τy) is the electron beam’s current density distribution.Each beam deflectionτx,τy can be associated with the secondary waveform value Ik(τx,τy). According to [23], the density of emissive current from the surface of the penetration channel, jt(x,y), is proportional in the first approximation to the normal of the electric field’s electron density jbx,yn=jbx,ycos⁡ψ, where ψ is the beam’s angle of incidence on the walls of the penetration channel. If dS is an infinitely small element of the surface of the penetration channel, then the current emitted from this element is (8)dIt=Ktjtx,ydS=Ktjtx,ydxdycos⁡ψ=Ktejbx-τx,y-τydxdy,where Kt, Kte are constants.The share of electron current emitted from the surface of elementdS and reaching the collector is (9)dIk=Ktejbx-τx,y-τypzdxdy.Finally, the waveform of the secondary electron current as a function of the electron beam’s displacement(τx,τy) is described by the integral:(10)Ikτx,τy=∫ab∫cdKtejbx-τx,y-τy·pzx,ydxdy,where a, b, c,andd are the boundaries of the region under consideration in theOXY plane.We approximate the beam’s current distribution and the shape of the penetration channel using standard Gaussian curves(11)jbx,y=jb0·exp⁡-x22σbx2-y22σbx2,zx,y=H·exp⁡-x22σx2-y22σy2,where H is the penetration depth; jb0 is a constant that specifies the magnitude of the beam’s full current; σx, σy are parameters that define the width of the penetration channel; σbx, σby are parameters that define the transverse dimensions of the electron beam. We will assume σx=σy and σbx=σby.We approximate the functionp(z) in accordance with the modeling results (Figure 6)(12)pz=1-zzκp,z≤zκp0,z>zκp.We approximate the shape of the penetration channel with a Gaussian surface. Figure7 presents the cross-section z(x,0) of the penetration channel used for modeling.Figure 7
Cross-section of the penetration channel used for modeling.Functionp(z(x,y))=p(x,y) has a zone of insensitivity (Figures 8 and 9).Figure 8
Two-dimensional distribution of the functionp(z(x,y)).Figure 9
Cross-section of the functionp(z(x,y)) when y = 0 mm.ForIk(τx,τy), 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 τx,τy from its equilibrium position in the penetration channel. The change in the collector’s current as the electron beam moves along the x-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.Figure 10
DistributionIk(τx,τy), obtained through numeric methods.Figure 11
Graph of the change in the collector’s current when the electron beam intersects the penetration channel as it moves along thex.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:z(τx,τy). 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 Ikmax, which corresponds to the beam’s placement on the surface of the penetration channel, to its magnitude Ik(z) given interaction at depth z. Figure 12 demonstrates that over most of the interval Ikmax/Ik(z)=f(z) may be considered exponential.Figure 12
Graph of the attenuation of the collector’s current as the depth of the electron beam’s area of interaction with the walls of the penetration channel increases.These results confirm previous experimentally observed patterns (Figure13).Figure 13
Experimental dependence between the depth of penetration and the secondary waveform parameterln⁡(Smax/Smin) given welding with oscillation along the joint and different focus settings (P = 3 kW, oscillation frequency f = 678 Hz, and sweep size 2A = 1.3 mm) [24].The theoretical and experimental values are compared in Figure14. 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%.Figure 14
Dependence of the collector’s current on the depth of the electron beam’s interaction with the metal in the penetration channel. 1: theoretical curve and 2: curve based on experimental data.


## 5. 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 depthz in the channel increases, and at a certain depth zmax 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.



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