A Novel Approach to Microwelding Using Electron Beam
Satya S. Gajapathi
Ulterra Drilling Technologies
Leduc, Alberta, Canada
Abstract
Integrating components at micro and nano levels is one of the current barriers to high-tech applications such as MEMS and NEMS, especially in the micro-electronics and medical industries. The
current state of the art in microwelding technology has an emphasis on laser and electron beams, but
these technologies are limited in their application because of excessive surface ablation when the beam
power density is very high. Preliminary work performed as part of this project indicated that excessive
ablation could be avoided in the especial combination of process parameters that result in the electron
beam acting as a volume heat source. At the time this analysis was started, no existing electron beam
welding (EBW) machine or electron microscope was capable of testing the desired range of parameters
experimentally.
To explore the possibility of EBW in the micro-scale in the absence of experimental equipment,
this project developed a realistic numerical model to perform a detailed feasibility study. For the
first time an electron beam is modeled as a volumetric heat source for welding purposes; the physical
representation of the beam is based on the Kanaya-Okayama theory to bring generality across diverse
families of base materials. This work also proposes for the first time a set of parameters for microwelding
in which the time constant for thermal diffusion is comparable or smaller than the beam dwell time.
The numerical model is used to optimize two key welding parameters: beam voltage and beam travel
speed. The approximations made in the model are tested through detailed analysis of particular cases,
confirming that evaporation is negligible, and phase change is of secondary relevance. Based on the
model, a set of critical experiments are proposed in the range that should be attainable in EBW
machines available only within the last two years.
Introduction
The growing demand for miniaturization of products has led to the development of techniques that can
integrate them at their sizes. Microwelding is one such technique that is of great significance in the
electronics and micro fabrication industry where it has been used for packaging, electrical interconnect,
device fabrication, etc. [1]. Laser and electron beams are commonly used for microwelding because of their
ability to achieve extremely small spot size, high power density, and high speed beam deflection [2].
For the electronics industry to continue obeying Moore’s law (which states that the number of transistors that fit into an integrated circuit will double every year), it is required that the components be
successfully integrated at increasingly smaller scales. This necessitates the laser and electron beams to
focus at micro and nano level spot sizes. As the power density is inversely proportional to the beam focus
area, having smaller spot sizes result in critically high power densities. Conventionally, power densities
above 10−13 W m−2 are considered unsuitable for welding due to excessive vaporization, as described by
Lancaster [3] and illustrated in Fig. 1. However, the future of microwelding technology depends on using
such high power densities in a controlled manner to overcome vaporization issues and successfully weld at
micron and sub-micron levels.
The current work proposes a novel concept of using the volume heating capabilities of electron beam
to successfully employ high power densities for welding without excessive vaporization. The fast moving
This is a summarized paper of the Masters thesis titled “Heat Transfer Analysis of Microwelding Using Tuned Electron
Beam" completed under the supervision of Dr. Patricio F. Mendez and Dr. Sushanta K. Mitra at the University of Alberta,
Canada in August 2011.
1
No welding
possible
Vaporisation
dominated
1013
1012
Electron
beam
Power density (W m-2)
1011
Vaporisation
conduction
and melting
(keyholing)
1010
CO2 laser
109
Radial
conduction
dominated
with melting
Plasma
108
107
Arc processes
106
105
No welding
possible
Negligible
melting
104
Figure 1: Chart depicting welding power density along with associated heat transfer phenomena [3]
electrons in an electron beam are capable of penetrating a few microns when impinged onto a solid.
Electron beams can uniquely qualify as a volume heating source when the desired weld depth is of the same
order of the electron penetration depth. The volume heating phenomenon prevents excessive vaporization
at high power density levels as the incoming energy is absorbed in few microns of material depth instead
of irradiating just the top surface. Such a volume heating phenomena of electron beam is not valid in
macro welding scenario where a few microns of electron penetration is negligible compared to the required
weld depth of many millimeters.
The electron beam can be used to heat a solid volumetrically by optimally balancing the beam voltage
that controls the depth of electron penetration and the beam spot size and travel speed that governs the
amount of heat diffusion in the solid. The previous studies on micro electron beam welding (µEBW) have
not considered balancing the welding parameters to avail the volume heating capabilities of the electron
beam [4–9]. In the present work, a numerical model of µEBW has been developed accounting for the
intrinsic volume heating characteristics of an electron beam. A novel volumetric heat source model has
been developed, based on Kanaya-Okayama electron penetration theory [10]. Heat diffusion in the solid
during microwelding has been numerically solved to simulate the process. The optimum beam voltage and
beam travel speed are obtained by numerical analysis of µEBW based on the minimum heat input and
appropriate process stability.
The model used for optimization incorporates important approximations that are confirmed as valid
through the analysis of specific cases and upper-bound estimates. Among the key approximations are the
consideration of heat conduction only, neglecting evaporation heat losses and enthalpy of phase transformation.
Based on the optimization results and incorporating phase-transformation effects and variable thermophysical properties, proof-of-concept experiments are proposed to validate this study experimentally.
2
Mathematical Model of µEBW
In µEBW, as the electron beam strikes the solid surface, the electrons penetrate into the body and generate
heat in the interaction volume by colliding with the lattice [2]. The heat generated due to the thermal
interactions causes to melt the material that finally solidifies to form fusion weld, as the beam traverses
along the line of joining. The three dimensional thermal interaction of the electron beam with the solid
and the modeling of heat transfer process during weld formation is discussed in this section.
Modeling of Electron Beam Heat Source
Electron beam interaction with a solid is a complex process resulting in many physical phenomena occurring simultaneously. A fraction of the incoming electrons bounce back as backscattered electrons. A part
of the incoming electron energy is also lost in the process of forming secondary electrons, plasmon excitation, thermionic emission, x-rays etc., that are explained in great detail in [11] and [12]. The remaining
electrons collide with the solid lattice and are absorbed as thermal energy, which makes the electron beam
act as a heat source. Figure 2 shows a schematic of the electron beam interaction with a solid.
Figure 2: Electron beam interaction with a solid
Modeling electron beam interaction with a solid as a volumetric heat source requires describing the
beam power distribution on the surface and energy decay along the penetration. A circular Gaussian
function is widely accepted to represent the variation of beam energy in the cross section of the beam.
The thermal interactions of the electrons along the depth of the material, however, is more complicated.
One can represent the electron beam heating with the volumetric heat generation term, q as:
( 2
)
x + y2
q = qmax exp −
F (z)
(1)
2σ 2
where, qmax is the coefficient which represents maximum beam power absorbed per unit volume of the
substrate, x, y, and z are coordinates along the beam motion, normal to the beam motion, and along
3
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 half maximum (FWHM)
of the Gaussian function as follows: FWHM=2 2 ln 2σ=2.35σ. In this work, the beam diameter (d) is
represented by the FWHM.
The function F (z) captures the distribution of absorbed energy along the depth of the material in
normalized form. Kanaya and Okayama derived a theoretical expression for the fraction of absorbed
energy as a function of depth, which can be written as [10]:
(
)
EA (z ∗ )
γz ∗
= 1 − (1 − z ∗ )3/5 exp −
E0
1 − z∗
[ ∫ ∗
)
))]
(
(
(
∗
EB 6 z
1.9γ
6
1.9γz ∗
1.9γz
−
dz ∗ +
exp −
1 − exp −
(2)
E0 5 0 (1 − z ∗ )7/6
1 − z∗
1 − z∗
5 × 25/6
where, EA (eV) is the total absorbed electron energy within the normalized distance z ∗ from the target
material surface, EB (eV) is the mean backscattered energy of the electrons, E0 is the incident energy, and
γ is a constant which accounts for the effects of diffusion loss due to the multiple collisions for returning
electrons and energy retardation due to the electronic collisions. The normalized depth variable, z ∗ , is
defined as z ∗ =z/R where, R is the maximum electron penetration depth in cm and is expressed as [10]:
R = 2.76 × 10−11
AV 1.67
ρZ 0.889
(3)
where, A, the atomic weight, in g/mole; V , the beam voltage, in V; Z is the atomic number; and ρ is the
density in g cm−3 .
The derivative of the absorbed energy in equation (2) with respect to the depth provides the rate at
which the electron transfers its energy to the material and can be written as [10]:
[
)
(
)(
dEA
E0
1
γz ∗
γ
3
=
exp −
+
dz
R (1 − z ∗ )2/5
1 − z∗
1 − z∗ 5
)
(
)]
(
EB 6 × 1.9
γ
1
1.9γz ∗
∗ 5/6
+
− (1 − z )
(4)
exp −
E0
5 (1 − z ∗ )2
1 − z∗
25/6
The function F (z) can now be described by normalizing the variation of absorbed energy along the depth
by its maximum value, as:
F (z) = (
(
where, the peak value of the absorbed energy,
dEA
dz
)
dEA
dz)
dEA
dz
max
max
(5)
, can be determined by substituting the normalized
depth variable in equation (4) by the normalized depth value at which the maximum energy dissipation
occurs; the normalized depth value (zE /R) can be determined using the following expression derived by
Kanaya and Okayama [10]:
(
)
1 + 2γ − 0.21γ 2
zE
=
(6)
R
2 (1 + γ)2
The coefficient qmax in equation (1) can now be described as the product of the maximum current
flux (Jmax ) and the maximum absorbed electron energy per unit depth, and can be represented using the
following expression.
)
(
dEA
(7)
qmax = Jmax ×
dz max
4
Also, the variation of the current flux (J) of the electron beam on the surface of the target is represented
by a Gaussian distribution as:
x2 + y 2
J = Jmax exp(−
)
(8)
2σ 2
The total beam current, I, can be obtained by the area integration of the current flux.
∫ ∞∫ ∞
x2 + y 2
I=
Jmax exp(−
)dxdy
(9)
2σ 2
−∞ −∞
The maximum beam current flux for a Gaussian distribution is obtained by solving equation (9) as:
I
2πσ 2
Using equations (1), (5), and (7), the volumetric heat source term can now be written as:
Jmax =
q = Jmax exp(−
dEA
x2 + y 2
)×
2
2σ
dz
(10)
(11)
Modeling of Heat Transfer During Microwelding
In this study, the electron beam is used to form a butt type fusion weld between two brick shaped solids.
A simplified representation of the process is shown in Fig. 3, where the heat source travels at speed U
along the centerline. The shape of the heat distribution on the surface of the solid and along the beam
penetration is pointed out in the same figure.
y
q
x
q(0,0,z)
(C
on Mat
sta eri
nt al I
Te nle
mp t
e ra
x
tur
e)
Shape of the electron beam
(fixed heat source)
z
te
ula
Ins
ps
d to
c
urfa
e
y
z
(T Ma
he te
rm ri a
al l O
In ut
s u le
la t t
io
n)
Ins
te
ula
d
e
s id
s
e
ie c
rk p
Wo
tion
mo
Figure 3: Schematic of the model showing the brick shaped solid, heat distribution behavior of electron
beam, and the pertinent boundary conditions valid in a moving reference frame
Fourier heat conduction equation is used to describe the heat transport in the solid, as the electron
beam traverses along the line of joining. A steady temperature field can be obtained by attaching the
coordinate system to the moving heat source. The governing equation can be written as:
( 2
)
∂ T
∂2T
∂2T
∂T
k
+
+
+ q + ρcp U
=0
(12)
2
2
2
∂x
∂y
∂z
∂x
where, q represents the volumetric heat generation by electron beam as described in equation (11), T is
the temperature of the body, and k, ρ, and cp are the thermal conductivity, density, and heat capacity of
the material.
5
Numerical Analysis
The governing equation (12) is solved numerically using a finite elements based software COMSOL
MultiphysicsTM . The GMRES (Generalized Minimal Residual) method is employed to solve the system of linear equations where “algebraic multigrid" method is used as a preconditioner. Such a technique
is ideally suited for solving elliptic partial differential equations, as in the given case. The relative tolerance
is specified as 10−6 for this study.
The computational domain here is only one-half of the welding arrangement shown in Fig. 3, owing to
the symmetry of geometry along the centerline. A non-uniform mesh with very high grid density at the
center of the beam is used to resolve the high heat concentration in the region; Fig. 4 depicts a typical grid
used in the current analysis. The size of the domain is chosen large enough to consider it as a semi-infinite
solid; the top, rear, bottom and side faces can be considered thermally insulated as they are far from the
heat source. The front face of the solid is imposed with a constant temperature boundary condition, in
accordance with the current case of coordinate system attached to the moving heat source. The face along
the centerline can be applied symmetry condition. Figure 3 points out the boundary conditions imposed
on different surfaces of the computational domain. The numerical solution is obtained considering the
constant room temperature properties of silicon, listed in Table 1, as an example.
Figure 4: Computational domain meshed with non-uniform tetrahedral elements
Table 1: Properties for Silicon at 27◦ C [13]
Properties
Thermal conductivity
Density
Specific heat
Thermal diffusivity
Melting temperature
Initial temperature
Atomic mass
Atomic number
Symbol
k
ρ
cp
α
Tm
T0
A
Z
6
Values
148
2330
712
8.92×10−5
1412
27
28
14
Units
[W m−1 K−1 ]
[kg m−3 ]
[J kg−1 K−1 ]
[m2 s−1 ]
[◦ C]
[◦ C]
[g mol−1 ]
Determination of Optimum µEBW Parameters
The two critical parameters of µEBW are the beam voltage which governs the beam penetration (R)
relative to the maximum melting depth (zm ) and the beam travel speed that regulates the rate of heat
diffusion into the solid during the process. The Peclet number (Pe=U d/α), which can be defined as the
ratio of rate of heat flow by convection to the rate of heat flow by diffusion, is used to represent the beam
travel speed in the non-dimensional form. The critical parameters are represented in the non-dimensional
form such that the results hold generality and can be valid for any type of material/process settings. The
numerical model of µEBW is now used to obtain the optimum process parameters.
In µEBW, both the relative beam penetration (R/zm ) and Pe parameters govern heat diffusion into
the solid and have to be optimized together as a system; the criteria of optimization is obtaining a given
weld depth with least heat input. The heat input in welding can be defined as the relative amount of
energy supplied per unit length of the weld [14]: HI=V I/U . A constant beam diameter of 10 µm is chosen
in this study. The beam voltage of the process is varied to obtain different electron penetrations (R) for
a parametric study, using the relationship in equation (3). The beam current is adjusted suitably so that
the maximum melting depth (zm ) obtained of all the microwelding processes remains constant at 2.5 µm.
The heat input for each process with variable beam penetration is also compared against three different
beam traveling speeds corresponding to Pe=10, 100, and 300; the results are shown in Fig. 5.
The higher heat input values for the processes having Pe=10 in Fig. 5 is because of the large amount
of heat being quickly diffused outside the weld region. The electron interactions and hence the increase in
electron range are secondary in such conduction dominated processes due to which the heat input curve
is close to being straight. There is a significant drop in the heat input curve as the Pe of the process is
increased to 100. Also, it is observed that the heat input is minimum corresponding to the relative beam
penetration of R/zm =2. The high heat input requirements for smaller beam penetrations (R/zm <2) can
be attributed to the large dependence on heat conduction to propagate the electron energy till the melting
depth, where as, at larger beam penetrations (R/zm >2), high energy electrons travel beyond the melting
depth, causing unwanted heating of the additional material volume.
The processes with Pe beyond 100, for example Pe=300, does not result in any significant gain in the
heat input especially at larger beam penetrations (R/zm >1.2), as seen in Fig. 5. The weld cross sections
for the two lowest heat input processes in the same figure are compared which correspond to Pe=100 and
300 and have R/zm =2. It is observed that the Pe=300 process with a slightly lower heat input also has a
smaller weld cross section as compared to the process with Pe=100. For the same electron penetration, the
processes with higher Pe results in less heat diffusion and hence smaller weld cross sections are obtained.
Thus, an optimum relative beam penetration of 2 and Pe of 100 is determined through this study.
Extension of the Model Beyond Original Approximations
The µEBW model used in the previous section is incomplete as important welding phenomena such as
melting and evaporation are not incorporated into it. However, the numerical analysis has been carried
out to optimize the welding parameters. This is based on the hypothesis that the two critical parameters,
namely, the beam penetration and Pe, govern the heat diffusion into the solid. The energy required for
melting the solid during welding and heat loses due to evaporation are compensated by the beam current.
In this section, a case study has been carried out to incorporate melting and evaporation phenomena in the
µEBW model. The optimum relative beam penetration and Pe obtained in the previous section are used.
The beam current is increased such that the desired weld depth is obtained. The amount of vaporization
losses occurring from the weld surface is measured to ensure the choice of optimal welding conditions lead
to microwelding successfully.
The melting phenomenon can be modeled by accounting for the latent heat of melting in the Fourier
7
0.5
Pe=10
Pe=100
Pe=300
0.4
HI (J m
-1
)
0.3
0.2
A
B
0.1
0.0
0.5
m)
2
3
melting line
4
3
2
weld width (in
2.0
2.5
1
1
2
3
4
0
3.0
B
0
1
4
R / zm
weld depth (in
m)
1.5
A
0
weld depth (in
1.0
4
3
2
weld width (in
m)
1
0
m)
Figure 5: Variation of heat input with the relative electron penetration range for three different Pe
processes of 10, 100, and 300. The weld cross sections having the same weld depth of 2.5 µm are shown
for the specific processes marked as A and B.
heat conduction equation in moving coordinate system as:
(
)
(
)
(
)
∂
∂T
∂
∂T
∂
∂T
∂T
km
+
km
+
km
+ q + ρm cp m U
=0
∂x
∂x
∂y
∂y
∂z
∂z
∂x
(13)
where, the thermal conductivity (km ) and density (ρm ) of the material are defined such that they assume
a value based on the solid/liquid fraction. The knowledge of the solid fraction (fs ) and liquid fraction (fl )
at any spatial location in the material can be used to define the properties as:
km = fs ks + fl kl ;
ρm = fs ρs + fl ρl
(14)
where, ks , ρs are thermal conductivity and density of the solid state and kl , ρl are the thermal conductivity
and density of the liquid state. The solid and liquid fractions can be modeled using a Heaviside step
function (H) imposed at the average melting temperature (Tm ), as follows:
{
0 if T < Tm
H(T ) =
(15)
1 if T > Tm
8
The liquid fraction in the material is represented directly by a Heaviside step function. The remaining
volume of solid fraction can be represented by subtracting the liquid fraction from the whole, as:
fs = 1 − fl = 1 − H
fl = H;
(16)
The specific heat capacity of the material (cpm ) is defined similarly to depend on solid/liquid fraction. Also,
the latent heat of melting (Lm ) is incorporated into the specific heat capacity term using a Dirac-delta
function (D) as:
cp m = fs cp s + fl cp l + DLm
(17)
The Dirac delta function is represented as the derivative of the Heaviside step function (D =
integral over the temperature range is equal to 1, as follows:
∫ −∞
DdT = 1
dH
dT )
and its
(18)
∞
The evaporation phase transition condition is incorporated into the numerical model at the surface of
the molten weld pool. The amount of material evaporated per unit area and unit time can be calculated
using the Langmuir’s rate of evaporation equation, as [15]:
√(
)
Mw
′′
(19)
ṁ = (Pv − P0 )
2πRTlv
where, ṁ′′ is the rate of mass flux in kg m−2 s−1 ; Pv is vapor pressure of the material in N m−2 ; P0 is
the surrounding pressure; Mw is the molecular weight of the material in kg mol−1 ; R is the universal gas
constant in J mol−1 K−1 ; Tlv is the temperature of the molten material at the liquid-vapor interface. The
surrounding pressure is considered negligible especially in electron beam welding operations owing to the
high vacuum environment in the beam chamber. The vapor pressure of a given material is a function of
temperature and can be calculated using the following empirical relation [16]:
log Pv = 5.006 + A +
E
B
+ C log T + 3
T
T
(20)
The heat flux due to evaporation can be estimated as the product of evaporative rate of mass flux and
the latent heat of vaporization (Lv ), as:
q ′′ = ṁ′′ Lv
(21)
Equation (21) is used to implement evaporation as a heat flux boundary condition at the surface of the
weld pool in the numerical heat transfer model of µEBW.
Numerical simulation of µEBW is performed accounting for the phase transition conditions. The
optimum relative beam penetration of 2 and Pe of 100 is used in the numerical study; a constant beam
diameter of 10 µm is used and beam current is adjusted to obtain a weld depth of 2.5 µm. Commercially
pure titanium (grade - 3) is considered as the substrate in this case study whose material properties are
listed in Table 2.
The effect of evaporation is evaluated in the form of maximum depth of mass loss (λ) in the weld pool.
The rate of evaporative mass flux can be obtained on the top surface of the substrate using equations (19)
and (20), after the temperature field is solved numerically. A 3-D color map showing the rate of mass flux
on the top surface of the substrate (x − y plane) is plotted in Fig. 6; the negative sign of the mass flux
indicates that the material is removed from the substrate. The maximum rate of evaporative mass flux is
observed to be 0.02 kg m2 s−1 , which is typical even in macro scale evaporation of titanium alloys [15].
Figure 6 depicts the distribution of rate of evaporative mass flux in a quasi steady-state system, where
the heat source appears to be stationary with respect to the coordinate frame. Hence, the rate of mass
flux varies from higher values near the heat source to negligible values in the far away region. In practical
9
Table 2: Properties of CP titanium (grade-3) used in the study [17]
Properties
Thermal conductivity
Density
Specific heat
Solidus temperature
Liquids temperature
Average melting temperature
Latent heat of melting
Latent heat of vaporization [18]
Initial temperature
Atomic mass
Atomic number
Symbol
ks
kl
ρs
ρl
cp s
cp l
Ts
Tl
Tm
Lm
Lv
T0
A
Z
Values
15.5
28.5
4518
4151
526.6
967.5
1668
1686
1677
3.48×105
9.19×106
27
47.867
22
Units
[W m−1 K−1 ]
[kg m−3 ]
[J kg−1 K−1 ]
[0 C]
[0 C]
[0 C]
[J kg−1 ]
[J kg−1 ]
[0 C]
[g mol−1 ]
Figure 6: Contour of the rate of evaporative mass flux on the welding surface is shown
situation, the depth of the pit on the weld surface caused by evaporated mass loss continually increases
within the time frame it is in molten state. To obtain depth of mass loss profile linearly along any line
parallel to beam travel, the rate of mass flux distribution is to be integrated over the length of the molten
pool and divided by the density of the material. Only the maximum depth of material loss, which occurs
along the centerline of the substrate, is of interest in the current study. Therefore, the rate of mass flux
10
along the line of beam motion (y = 0) is integrated as:
∫
1
ṁ′′ (x, 0)dx = λ
ρU x
(22)
Equation (22) is numerically integrated to obtain the maximum depth of material loss as λ=6.96×10−13
m. It is to be noted here that the maximum depth of material loss is even smaller than a diameter of an
atom (about 10−10 m), which indicates that the evaporation loss on the surface is negligible. The effect of
evaporation in the proposed µEBW is negligible because of the relatively low temperatures obtained on
the surface.
Proposed Proof of Concept Experiments
The µEBW model, that considers three dimensional electron beam heat source and takes into account the
melting and evaporation phenomena occurring during welding, closely represents the practical situation.
However, this approach of volume heating in microwelding leads to an uncommon combination of small
beam diameters and high beam currents in electron beam; the conventional macro-scale EBW machine
has high current (∼ 25 mA) and large beam diameter (∼ 1 mm), where as, the typical scanning electron
microscope (SEM) used for imaging has small current (∼ few nA) and very small beam diameter (∼ 1
nm). Thus, the current research will pave way towards the development of next generation electron beams
that will revolutionize welding at micron scales and smaller.
Latest-generation EBW machines commercially available in the last two years should be capable of
testing the concepts described in this work; this can be done not quite in the micron scale but in a relative
larger scale. The electron beam set up can achieve beam voltage of 140 kV maximum, beam current 40 mA
maximum, travel speed of 25.4 m s−1 , and beam diameter of 0.2 mm. The optimum welding conditions
are proposed using the current approach that falls within the available window of this set up. Choosing
the material as commercially pure titanium (properties listed in Table 2), the optimum microwelding
parameters are found to be: V =140 kV, I= 2.5 mA, d=0.2 mm, and U =3.4 m s−1 . The shape of the
weld pool as predicted by the numerical model is shown in Fig. 7, which is desired to be matched with
experimental data when available to validate the proposed concept.
Discussion
The proposed technology could have a revolutionary impact on assembly at the micro, and perhaps nano,
scales. There is currently no reliable approach to fusion joining such that an Ohmic joint is assured and
Schottky joints are avoided.
The combination of small beam size, and relatively high currents and travel speeds that result for
the analysis presented above is outside the range of all but the newest EBW machines. A numerical
analysis was undertaken to justify experiments in such an unexplored regime, and the detailed analysis
has confirmed the feasibility anticipated by previous order of magnitude work. The sophistication of
numerical analysis has enabled an optimization analysis, aiming at suggesting parameters most likely to
result in successful welds. The optimization analysis was performed in a simplified system of constant
thermophysical properties, no melting, and no evaporation. Further analysis indicated that evaporation
is negligible, while phase change is secondary, and not expected to alter the optimization conclusions. A
detailed study presented in the thesis (not included here for brevity) concluded that fluid flow in the weld
pool is expected to change the ratio of depth to width of the bead, and that beam focusing could be
challenging at such high currents and small beam spot size. Beam focusing could possibly be helped by
slightly reducing the vacuum level with inert gas. This approach would also be useful to evacuate electrons
in less conductive materials such as silicon; this is the same approach used in environmental SEM.
The effect for surface active elements in causing undercutting or humping still needs to get explored.
Similarly, the short time scales proposed might play a role in the kinetics of phase transformations. It is
11
0.00
Tl
weld depth (mm)
0.02
0.04
0.06
Ts
CP Ti-Grade3
V=140 kV
I=2.5 mA
d=0.2 mm
0.08
U=3.4 m/s
0.10
0.10
0.08
0.06
0.04
0.02
0.00
weld width (mm)
Figure 7: Solidus (Ts ) and liquidus (Tl ) isotherms representing the weld shape in the titanium substrate
possible that ferritic steels might not transform to austenite in such fast heating, and thus avoid undesirable
transformation products during the fast cooling. It is also possible that the melting temperature is raised
at such fast heating rates. While all these considerations can be included in the numerical model, at this
point the most productive path seems to be to use the model results to perform feasibility experiments.
Conclusions
For the first time it is established that welding is possible in the high heat intensity conditions without
excessive vaporization losses, that has been considered impossible before. The key to employ high intensity
heat sources for welding is volumetric heating. Such an approach has important implications for welding
at micro and, possibly, nano scales.
A comprehensive numerical model of µEBW is developed for the first time in this research, which accurately represents the three dimensional heat distribution of electron beam and accounts for predominant
heat transfer phenomena during the process. The numerical model is used to optimize the welding parameters that can be used to obtain a micro-weld of given size in a solid. The optimal welding parameters are
reported in terms of non-dimensional Pe, that indicates the beam scan speed based on the beam size and
material properties, and relative beam penetration, which guides the choice of the beam voltage based on
the material properties and weld depth required. The following are the important conclusions:
1. The optimum Peclet number of µEBW is found to be 100. For Pe larger than 100, the process
requirements (e.g. beam velocity) increases without meaningful gains in heat input or weld shape.
At Pe below 100, there is excessive heat lost into the substrate surrounding the weld.
2. The optimum value of electron penetration depth in the solid is found to be twice that of the
required weld depth. At electron penetrations below the optimum, the process starts to resemble
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a traditional surface heating process with high heat input such as laser welding consistent with
potentially excessive surface ablation. At electron penetrations above the optimum, the process
results in excessive direct heating beyond the depth of the weld, possibly damaging the surroundings
of the device being welded.
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Related Publications
i. S. S. Gajapathi, S. K. Mitra, and P. F. Mendez. Modeling of micro electron beam welding with
melting and evaporation. In ASME International Mechanical Engineering Congress and Exposition,
Paper no. IMECE2012-88427, Houston, Texas, US, 9-15 Nov., 2012 (accepted).
ii. S. S. Gajapathi, S. K. Mitra, and P. F. Mendez. Part - 1: Development of a new heat source model
applicable to micro electron beam welding. Science and Technology of Welding and Joining, 17(6),
pp. 429–434, 2012.
iii. S. S. Gajapathi, S. K. Mitra, and P. F. Mendez. Part - 2: Application of kanaya-okayama heat source
in modeling micro electron beam welding. Science and Technology of Welding and Joining, 17(6),
pp. 435–440, 2012.
iv. P. F. Mendez, S. S. Gajapathi, and S. K. Mitra. Thermal profile of high voltage EBW in the
submillimeter scale. In 65th Annual Assembly of the International Institute of Welding - SC MICRO,
Denver, Colorado, US, 10-11 July, 2012.
v. P. F. Mendez, K. E. Tello, and S. S. Gajapathi. Generalization and communication of welding
simulations and experiments using scaling analysis. Trends in Welding Research, Chicago, Illinois,
US, 4-8 June, 2012.
vi. S. S. Gajapathi, S. K. Mitra, and P. F. Mendez. Controlling heat transfer in micro electron beam
welding using volumetric heating. International Journal of Heat and Mass Transfer, 54(25-26),
pp. 5545–5553, 2011.
vii. S. S. Gajapathi. Heat transfer analysis of microwelding using tuned electron beam. Msc. Dissertation,
University of Alberta, 2011 (Advisors: Prof. P. F. Mendez and Prof. S. K. Mitra).
viii. S. S. Gajapathi, S. K. Mitra, and P. F. Mendez. Modeling of micro-welding process using electron
beam under high peclet number. In ASME International Mechanical Engineering Congress and
Exposition, Paper no. IMECE2010-39248, Vancouver, Canada, 12-18 Nov., 2010.
ix. S. S. Gajapathi, S. K. Mitra, and P. F. Mendez. Heat transfer aspects of micro electron beam welding.
In FABTECH/AWS Annual Meeting, Atlanta, Georgia, US, 2-4 Nov., 2010.
x. S. S. Gajapathi, P. F. Mendez, and S. K. Mitra Analytical method to study the temperature distribution of a moving heat source electron beam micro-welding. In 7th International Conference on Heat
Transfer, Fluid Mechanics and Thermodynamics, Antalya, Turkey, 19-21 July, 2010.
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