Part I: Development of new heat source model applicable to micro electron beam welding

Introduction

Micro electron beam welding (μEBW) is a relatively new welding technology that uses heating from an electron beam to weld micrometre scale sections. Accurate representation of the beam profile is central to the understanding and improvement of the process. This work, for the first time, formulates a theoretical model to represent the volumetric heat distribution of an electron beam based on the electron penetration theory of Kanaya and Okayama.1

Because the electrons penetrate the free surface of the material for a depth of a few micrometres, there is a radical difference between micro- and macroelectron beam welding. In μEBW, electron penetration and weld penetration are comparable,2, 3 so the incident beam is best approximated as a volumetric heat source. On the other hand, the effect of electron penetration is insignificant in conduction dominated macrowelding due to the large size of the weld compared to the beam penetration. Three-dimensional volumetric heat source models are often used to represent keyhole mode macrowelding, where the heat penetration occurs as a result of high intensity heat source. There are significant conceptual differences between the volumetric sources used for modelling the keyhole and the focus of this paper. The three-dimensional electron beam heat source models used for macroscale in the past are either empirical or semiempirical. Goldak et al. 4 first developed a three-dimensional double ellipsoidal heat source model that was widely used in the analysis of conduction mode electron beam welding.5^– 7 Later, the conical heat source model developed by Wu et al. 8 became popular in modelling deep penetration electron beam welds. However, such models are not explicitly defined and depend on the feedback data from ad hoc experiments in the form of size and position of the ellipsoids and/or cones.

The heat source models used in studying μEBW until now are still in their early stages. Hwang and Na9 modelled the energy input from the beam as heat generation in a specific volume of square cross-section inside the solid. Similarly, Knorovsky et al. 10 considered heat generation in a cylindrical region of given radius and height inside a material within which the beam power is equally distributed. Both studies did not account for the variation in electron energy across the cross-section of the beam or along its penetration. Recently, Gajapathi et al. 3 used an electron beam heat source model that considers the exponential decay of electron energy along the depth of the material. The previous models do not account for a very important factor of electron beams: the maximum rate of energy deposition occurs below the free surface.

The behaviours of electrons–solid interactions are best predicted using Monte Carlo simulations.11 The variation in absorbed electron energy along the depth, as obtained from the simulations, can be used to model the trend of electron energy decay using a polynomial best fit. Such an approach has been applied in several studies;12, 13 however, using the technique requires to run new simulations each time the material properties and/or the configuration of the beam are changed. An analytical formula describing the electron decay trend, proposed by Schiller et al., 14 is reported to compare reasonably well with the Monte Carlo results for specific cases.12 The formula has been widely applied in the heat source modelling of electron beam.15, 16 However, this analytical model is very simplistic in nature and does not account for the influence of material atomic number Z or atomic mass A on the electron–solid interactions. None of the approaches discussed provides a general analytical expression indicating the influence of beam voltage V, density ρ, atomic mass and atomic number of the material on the trend of electron energy decay.

In the present work, the Kanaya and Okayama electron penetration theory1 is used to represent the electron energy deposition along the depth of the solid. The Kanaya–Okayama model is preferred over the previously discussed models because it accounts for the predominant physics of electron–solid interactions. Moreover, it provides a closed form expression that is reported to have good agreement with the experimental results.1 The heat source model is further used to relate the heating efficiency of the electron beam to the material atomic number.

Mechanism of electron beam heating

In an electron beam column, an electrostatic field accelerates the electrons to collide with the target material and transfer their kinetic energy. Not all of the incoming energy is converted into heat in the material during the interactions. The loss of incident beam energy is due to backscattered electrons, secondary electrons, X-ray generation and electromagnetic radiation, as shown in Fig. 1. All the aforesaid processes can be explained by considering the two major classifications of electronic collisions with the substrate:14, 17

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Figure 1

Electron beam interactions with solid

The amount of energy attributed to each of the physical processes depicted in Fig. 1 depends on both the material properties and the electron beam acceleration voltage. Typically, the energy loss occurring due to secondary electrons, X-ray generation and electromagnetic radiation is <0·5% of the incoming energy; in contrast, backscattered electrons can have a high share of the incident energy (up to 40%).14 The remaining electron energy is transferred into the substrate, gradually being absorbed along the depth. The fraction of electrons that do not shed all their energy within a given thickness of material and emerge out are called ‘transmitted electrons’. Because electron transmission happens in samples much thinner than those of interest for μEBW, they are not an issue in this study. The maximum depth until which the electrons penetrate can be defined as the electron penetration range R, which depends on both the beam voltage and the material properties. In the voltage range of 10–1000 keV, it can be expressed as1 (1) where R is given in cm, A is the atomic weight in g mole⁻¹, V is the beam voltage in V, Z is the atomic number and ρ is the density in g cm⁻³.

Heat source modelling

The modelling of the electron beam as a volumetric heat source requires describing the power distribution on the surface and energy decay along the penetration. While a circular Gaussian function is widely accepted to represent the variation in energy over the cross-section of the beam, the thermal interactions of the electrons along the depth of the material are more complicated. One can represent the electron beam heating with the volumetric heat generation term q as (2) where q_max is the coefficient that represents the maximum beam power absorbed per unit volume of the substrate, x, y and z are the coordinates along the beam motion, normal to the beam motion, and along the depth pointing from the target surface into the matter respectively and σ is the standard deviation of the Gaussian function. The standard deviation is related to the full width at half maximum (FWHM) of the Gaussian function as follows: FWHM = 2(2 ln 2)^1/2 σ = 2·35σ. In this work, the beam diameter is represented by the FWHM.

The beam diameter is considered constant for a given beam configuration, and the widening of the beam due to electron scattering is not taken into account. Such an approximation is valid when the beam diameter is many times larger than the depth of electron penetration. It is established by the Kanaya–Okayama modified diffusion model1 that the radius of electron scattering is less than or equal in magnitude to the electron penetration depth. As the overall diameter of the beam is chosen increasingly larger than the electron penetration depth, the radius of electron scattering at the beam periphery reduces to a small fraction of the beam diameter and hence can be ignored. The beam diameter is chosen at least two times the electron penetration depth in this study so that the effects of beam scattering are not significant.

The function F(z) captures the distribution of energy along the depth of the material, and it is the key concept of this paper. This function can be derived by taking into account the physics of electron–solid interactions described in the previous section. It represents the trend of only the absorbed energy along the depth of the solid. An energy balance of the system helps to determine the total amount of energy absorbed within a given material depth by subtracting the transmitted fraction (electrons travelling beyond the depth considered but captured by material at greater depths) and backscattered fraction of the electrons from the incident energy. The effects of secondary electron, X-rays and electromagnetic radiations are ignored due to their negligible influence in the energy balance. Kanaya and Okayama obtained a theoretical expression for the fraction of absorbed energy as a function of depth, which can be written as1 (3) where E_A (eV) is the absorbed electron energy within the distance z* = z/R from the target material surface, E_B (eV) is the mean backscattered energy of the electrons, E₀ is the incident energy and γ is a constant that accounts for the effects of diffusion loss due to the multiple collisions for returning electrons and energy retardation due to electronic collisions.

The differentiation of the absorbed energy in equation (3) with respect to depth provides the trend of its variation along the depth of the solid. The absorbed energy per unit depth, also known as the stopping power, signifies the rate at which the electron transfers its energy to the material and can be written as1 (4) The normalised energy distribution curve along the depth of the material, represented by the function F(z), is described by dividing the stopping power function in equation (4) by its maximum value as (5) where the peak value of the stopping power curve (dE_A/dz)_max can be found out with the knowledge of the depth z_E at which the maximum energy dissipation occurs. The following expression reported by Kanaya and Okayama can be used to determine z_E 1 (6) The coefficient q_max in equation (2) can be described as the product of the maximum current flux J_max and the maximum absorbed electron energy per unit depth and can be represented using the following expression (7) Moreover, the variation in the current flux J of the electron beam on the surface of the target is represented by a Gaussian distribution as (8) The total beam current I can be obtained by area integration of the current flux (9) The maximum beam current flux for a Gaussian distribution is (10) Using equations (2), (6) and (7), the volumetric heat source term can now be written as (11) The heat source model described in equation (11) is valid for welding in conduction mode, which is the desired case for μEBW. The model breaks down for cases when the electron penetration is significantly smaller than the weld penetration. This results in steep thermal gradients and high surface temperature values, with the resulting evaporation and possibility of a keyhole. The operating parameters under which this present model may fail are discussed in detail in part II of the work.

Heating efficiency of electron beam

The incoming energy of the electrons is not entirely absorbed in the solid. The heating capability of the electron beam is characterised by a heating efficiency factor η, which can be defined as the relative amount of power absorbed in the material. For a flat free surface, the absorbed beam power can be obtained by the volume integration of the heat source term in equation (11), which can be used to describe η as follows (12) This procedure is not valid for a keyhole type of EBW because not all backscattered electrons are lost as assumed here. On integrating equation (12), η can be written as (13) where E_A/E₀ represents the fraction of incident energy absorbed in the material. It is important to note here that the thickness of the materials in consideration is greater than the electron penetration range such that there is no loss of energy due to electron transmission. The total amount of absorbed energy can be obtained by integrating the stopping power in equation (4) through the depth until the electron penetration range. Numerical integration has been performed to solve for the definite integral as the antiderivative of the same is not obvious; a code based on Simpson's method for numerical integration has been developed using the Fortran 77 programming language.

The two unknowns that are required to determine the heating efficiency values, as shown in equation (4), are γ and E_B/E₀. The constant γ is a function of atomic number of the material only and can be evaluated using the following expression1 (14) The mean backscattered energy fraction E_B/E₀ depends on both the target material and the incident energy; it can be evaluated using dedicated experiments for a given combination of material and incident energy.18^– 22 Published experiments indicate that within the ranges of atomic number and incident energy considered, the backscattered energy depends almost exclusively on atomic number, and any dependence on incident energy cannot be well discriminated from experimental error. Sternglass20 has observed that the mean backscattered energy obeys an approximately linear trend with the increase in material atomic number irrespective of the incident energies between 0·2 and 32 keV. Kanaya and Okayama1 compared different experimental findings of mean backscattered energy for materials having an atomic number of up to 100 and incident energies in the range of 2–70 keV and obtained a more generalised relationship describing the dependence of relative mean backscattered energy on the atomic number of the material only; the relationship they obtained can be approximated by the following second order polynomial (15) Figure 2 shows the heating efficiency calculated as a function of the material atomic number using the relative mean backscattered energies from equation (15).

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Figure 2

Heating efficiency of normally incident electron beam as function of atomic number of irradiated material. Continuous line represents current calculations showing heating efficiency independent of input beam voltage. Curve shows good agreement with experimental data points at different beam voltages and is valid in range of 10–1000 keV, where Kanaya–Okayama electron penetration theory is valid. Data points in plot corresponding to experiments by *Sternglass20 do not fall into 10–1000 keV beam voltage range but included only for comparison purpose

There are very few published values of electron beam heating efficiency tabulated as a function of incident energy. However, the available data on backscattering phenomenon can be used to estimate the heating efficiency of the process by rewriting the expression in equation (2) as η = 1−(I_B E_B/IE₀), where I_B/I is the relative backscattering current. Such a consideration is valid when the power loss due to processes other than backscattering is negligible.

For a normally incident beam, the relative backscattering current, also known as the backscattering coefficient, does not depend on the incident energy of the beam but varies with the properties of the target material.23, 24 Archard23 proposed the backscattering coefficient as a function of material atomic number only based on the composite theory of diffusion (valid for high atomic numbers) and elastic collisions (valid for low atomic numbers). A fourth order polynomial is used to best fit the relationship curve between the backscattering coefficient and the material atomic number,23 which can be written as (16) It is noteworthy that the backscattering coefficient can be calculated based on the atomic number of target material using equation (16) irrespective of the beam incident energy.

To calculate the relative backscattering power loss and hence the heating efficiency, the relative backscattered current needs to be multiplied by the mean relative backscattered energy. The mean relative backscattered energy values have been obtained from the literature18^– 22 for different combinations of target material and incident energy. For a given material, the product of the backscattered coefficient and the mean values of relative backscattered energy provides the relative backscattering power loss. Direct experimental measurements of the backscattering power loss have also been reported by Reichelt25 for nickel and tantalum at 10 keV incident energy. The heating efficiency is then calculated by subtracting the relative backscattering power loss from the whole. The experimental results of heating efficiency for specific materials are compared against the current findings in Fig. 2. The present calculation of the electron beam heating efficiency matches the experimental findings within the margin of ±10% (see Fig. 2) for a wide range of incident energies and materials.

Effect of voltage on heating efficiency

In this study, the heating efficiency is expressed as a function of material atomic number only. It would be reasonable to expect the acceleration voltage to also play a role. The validity of the approximations made in the current study is discussed in this section by comparing the present calculation with the experimental values.

Figure 3 illustrates the experimental results of mean backscattered energy normalised by the approximate values obtained using equation (15). The horizontal axis represents the incident energy of the beam. It is observed that there is no clear dependence of the mean backscattered energy on the incident energy of the beam in the range of 10–1000 keV; however, the mean backscattered energy seems to increase with the increase in incident energy at low accelerating voltages (<10 keV). Moreover, the differences in experimental conditions contribute largely to the mean backscattered energy measurements.26 For example, there is a considerable amount of difference in the relative mean backscattered energy values reported separately by Kulenkampff and Spyra19 and Niedrig22 for copper (Z = 29) in the incident energy range of 20–40 keV, as seen in Fig. 3.

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Figure 3

Ratio of experimental relative mean backscattered energy to values given by equation (15) is plotted against beam incident energy for several material atomic numbers: there is no clear pattern observed of backscattered energy variation with beam incident energy

The dependence of backscattered energy on incident energy is rather complex. The experimental conditions have seemingly larger impact on the backscattered energy measurements as compared to the incident energy by itself. In such a situation, the heating efficiency values proposed in this study as a function of material atomic number only provide an approximate estimation for many practical applications.

Discussion

The mathematical model of electron–condensed matter interactions in the range of 10–1000 keV incident energy used in this study accounts only for the predominant physics of backscattering, absorption and transmission of electrons.1 The other processes occurring simultaneously such as emission of secondary electron and X-rays were considered secondary.

The description of electron heating efficiency as a function of atomic number only is a consequence of describing the incident energy dependence of heating efficiency as a function of atomic number. The determination of dependance of efficiency on the incoming energy would require to assemble a large amount of experimental data of mean backscattered energy for different incident energy beams and target materials; the collected data can then be arranged in a plot of relative mean backscattered energy against the incident energy of the beam similar to Fig. 3. Advanced statistical methods should be used to best fit the closest relative mean backscattered energy based on the incident energy of the beam and atomic number of the material.

The current model considers an electron beam perpendicular to the surface and would not be as accurate in keyhole penetration mode, where efficiency is expected to be higher. The model proposed here, as it is based on the Kanaya–Okayama approach, does not take into account the effect of alloying and microstructure of the substrate. It is reasonable to expect, however, that the use of average values for the volume of interaction of the electrons will still produce closer estimations.

Conclusions

In this work, a volumetric heat source model has been developed theoretically to represent electron beam heating. The energy decay trend during electron beam penetration and the fact that the maximum of the incoming electron energy is deposited underneath the surface have been accounted using the electron penetration theory proposed by Kanaya and Okayama. Such a heat source model is valid for studying electron beam welding in conduction mode when the scale of the target material is comparable to the beam penetration.

Using the model developed, a theoretical approach has been proposed, for the first time, to calculate the heating efficiency of an electron beam normally incident on a solid or liquid metal. The heating efficiency values obtained depend only on material atomic number and provide an average estimate for a wide range of beam incident energies.