Model for Predicting Thickness of
Electron Beam Physical Vapor Deposited Thermal Barrier Coatings
by
C. Colette Opsahl
an Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Venkat Seetharaman, Project Adviser
_________________________________________
Ernesto Gutierrez-Miravete, Co-Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2011
© Copyright 2011
by
C. Colette Opsahl
All Rights Reserved
i
CONTENTS
LIST OF FIGURES .......................................................................................................... iii
LIST OF TABLES ............................................................................................................. v
ACKNOWLEDGMENT .................................................................................................. vi
NOMENCLATURE ........................................................................................................ vii
KEYWORDS .................................................................................................................... ix
ABSTRACT ...................................................................................................................... x
1. INTRODUCTION ....................................................................................................... 1
1.1
BACKGROUND ............................................................................................... 3
1.2
COATINGS ....................................................................................................... 5
1.3
TBC PROCESSING TECHNOLOGIES ........................................................... 8
1.4
COATING CHARACERISTICS ..................................................................... 10
2. METHODOLOGY .................................................................................................... 12
2.1
THICKNESS DISTRIBUTION ....................................................................... 12
2.2
INCLINED PLATE DEPOSITION ................................................................. 15
3. RESULTS AND DISCUSSION ................................................................................ 17
4. CONCLUSION.......................................................................................................... 22
4.1
FUTURE WORK AND MODEL IMPROVEMENT ...................................... 22
REFERENCES ................................................................................................................ 23
APPENDIX A: RESULTS .............................................................................................. 24
ii
LIST OF FIGURES
Figure 1: Thermal Barrier Coating System (drawing not to scale) ................................... 1
Figure 2: Thornton zone model of physically vapor deposited structure .......................... 2
Figure 3: Typical Gas turbine jet engine ........................................................................... 3
Figure 4: Gas turbine blade................................................................................................ 4
Figure 5: Temperature capabilities of several classes of alloys ........................................ 5
Figure 6: Common coating processes ................................................................................ 6
Figure 7: Thermal Barrier Coating system ........................................................................ 7
Figure 8: EB-PVD coating chamber .................................................................................. 9
Figure 9: EB-PVD TBC on a metallic substrate .............................................................. 10
Figure 10: Knudsen cosine law schematic....................................................................... 11
Figure 11: horizontal plate schematic .............................................................................. 12
Figure 12: model mesh schematic ................................................................................... 13
Figure 13: thin plate model example ............................................................................... 14
Figure 14: inclined flat plate schematic ........................................................................... 16
Figure 15: 1 ingot source centered................................................................................... 17
Figure 16: ingot source centered, n=9 ............................................................................. 17
Figure 17: inclined thin rectangular plate schematic ....................................................... 18
Figure 18: inclined plate, n=1 .......................................................................................... 18
Figure 19: dual ingot source ............................................................................................ 19
Figure 20: dual ingot source, n=1 .................................................................................... 20
Figure 21: dual ingot source, n=9 .................................................................................... 20
Figure 22: dual ingot, closely spaced .............................................................................. 21
Figure 23: dual ingot, closely spaced, n=1 ...................................................................... 21
Figure 24: Right side ingot .............................................................................................. 24
Figure 25: Right side ingot, n=1 plot ............................................................................... 24
Figure 26: Right side ingot, n=4 plot ............................................................................... 25
Figure 27: Right side ingot, n=9 plot ............................................................................... 26
Figure 28: Center ingot, n=1 ............................................................................................ 27
Figure 29: Center ingot, n=1 plot .................................................................................... 27
iii
Figure 30: Center ingot, n=4 plot .................................................................................... 28
Figure 31: Center ingot, n=9 plot .................................................................................... 29
Figure 32: Left side ingot ................................................................................................ 30
Figure 33: Left side ingot, n=1 plot ................................................................................. 30
Figure 34: Left side ingot, n=4 plot ................................................................................. 31
Figure 35: Left side ingot, n=9 plot ................................................................................. 32
Figure 36: 3 ingots ........................................................................................................... 33
Figure 37: 3 ingots, n=1 plot ........................................................................................... 33
Figure 38: 3 ingots, n=4 plot ........................................................................................... 34
Figure 39: 3 ingots, n=9 plot ........................................................................................... 35
Figure 40: 2 ingots ........................................................................................................... 36
Figure 41: 2 ingots, n=1 plot ........................................................................................... 36
Figure 42: 2 ingots, n=4 plot ........................................................................................... 37
Figure 43: 2 ingots, n=9 plot ........................................................................................... 38
Figure 44: 2 ingots centered ............................................................................................ 39
Figure 45: 2 ingots centered, n=1 plot ............................................................................. 39
Figure 46: 2 ingots centered, n=4 plot ............................................................................. 40
Figure 47: 2 ingots centered, n=9 plot ............................................................................. 41
iv
LIST OF TABLES
Table 1: Example coating thickness for thin plate........................................................... 15
Table 2: Right side ingot, n=1 ......................................................................................... 24
Table 3: Right side ingot, n=4 ......................................................................................... 25
Table 4: Right side ingot, n=9 ......................................................................................... 26
Table 5: Center ingot, n=1 ............................................................................................... 27
Table 6: Center ingot, n=4 ............................................................................................... 28
Table 7: Center ingot, n=9 ............................................................................................... 29
Table 8: Left side ingot, n=1............................................................................................ 30
Table 9: Left side ingot, n=4............................................................................................ 31
Table 10: Left side ingot, n=9.......................................................................................... 32
Table 11: 3 ingots, n=1 .................................................................................................... 33
Table 12: 3 ingots, n=4 .................................................................................................... 34
Table 13: 3 ingots, n=9 .................................................................................................... 35
Table 14: 2 ingots, n=1 .................................................................................................... 36
Table 15: 2 ingots, n=4 .................................................................................................... 37
Table 16: 2 ingots, n=9 .................................................................................................... 38
Table 17: 2 ingots centered, n=1...................................................................................... 39
Table 18: 2 ingots centered, n=4...................................................................................... 40
Table 19: 2 ingots centered, n=9...................................................................................... 41
v
ACKNOWLEDGMENT
I would like to thank my advisor, Dr. Venkat Seetharaman, for his support and
guidance during this project. I would also like to thank Russell Beers, David Litton, and
Benjamin Zimmerman from Pratt &Whitney for all of their advice and encouragement
they have given me in order to complete this project.
vi
NOMENCLATURE
APS
°C
Al
Co
Cr
d
Air Plasma Spraying
degrees Celsius (temperature)
Aluminum
Cobalt
Chromium
coating thickness
do
EB-PVD
fcc
h
coating thickness (original)
Electron Beam Physical Vapor
Deposition
face-centered cubic
height
ho
in
height (original)
inches
kB
kW
LLPS
M
m
m
Mpa
n
n'
Ni
P
r
SZM
T
TBC
TGO
torr
vol
VPS
W
YSZ
α
θ
μm
ρ
Boltzmann constant
kilowatt (power)
Low-Pressure Plasma Spary
mass
meter
molecular weight
Mega Pascal (pressure)
vapor characteristic
evaporation coefficient
Nickel
pressure
radial distance
Structure Zone Model
temperature
Thermal Barrier Coating
thermally grown oxide
Torr (pressure)
volume
Vapor Plasma Spray
coating deposition rate
Yttria-stabalized zirconia
alpha angle (radians)
theta angle (radians)
micrometer (distance)
density
vii
Φ
phi angle (radians)
viii
KEYWORDS
Aerospace
Coating Thickness Prediction
Electron Beam Physical Vapor Deposition
Gas Turbine Engine
Thermal Barrier Coatings
ix
ABSTRACT
Physics-based models can be used to predict the deposition coating thickness
profiles of thermal barrier coatings (TBCs) using electron beam physical vapor
deposition (EB-PVD). Coating thickness is generally a strong function of the orientation
and distance of the substrate relative to the evaporation source. A uniform coating
thickness is desirable due to its increased temperature reduction capability, which results
in a reduced local variation in temperature, reduced distortion and porosity, and
improved thermal fatigue life. TBCs allow the turbine blades to operate at a higher
temperature, thus increasing the efficiency of the engine. A simple mathematical model
was developed to relate the coating thickness with the relative position and orientation of
the substrate and the vapor sources. Several geometries were investigated for their
influence on coating thickness uniformity. Maximum level of uniformity was achieved
for the case of two coating sources (ingots) placed directly below the plate at equal
distance
of
0.101m
(4in)
offset
x
from
the
plate
center
point.
1. INTRODUCTION
Gas turbine engines operate in very aggressive environments, exposed to high
temperatures, extreme temperature gradients, high pressures, resulting in oxidation and
corrosion damage. Because gas path temperature in these engines can surpass 1650ºC, it
is necessary to have components with exceptionally high-temperature capability.
Nickel-based single-crystal superalloys are the material of choice based on their
optimized mechanical strength in creep and fatigue. It is common to have a gas stream
temperature exceed the melting temperature of the superalloys used; therefore, a thin
ceramic/metallic coating is required to reduce the metal temperature and provide
environmental protection.
In industry today, there are many different types of coatings applied in a variety
of ways. The main type of coating is thermal barrier coating. Thermal Barrier Coating
(TBC) is a type of ceramic/metallic coating system used to protect the substrate against
oxidation and corrosion by reducing the component temperature which results in an
increase in component life. As shown in Figure 1, TBC consist of a combination of
multilayer coatings, with each layer having a specific role and purpose.
Ceramic Top coat
Thermally grown oxide
Metallic Bond coat
Superalloy substrate
Figure 1: Thermal Barrier Coating System (drawing not to scale)
The progression of TBCs has mainly focused on zirconia-yttria ceramics, mostly
because there are only a few materials that satisfy the requirements of TBCs.
Yttria-
stabalized zirconia (YSZ) has been determined to be the most appropriate top coat
material for the gas turbine engine application based on its thermodynamic properties,
phase stability, and low thermal conductivity.
Processing of zirconia is extremely difficult because of the amount of energy
required to melt the raw material. There are several types of processing applications
1
available: electron beam physical vapor deposition (EB-PVD), air plasma spray (APS),
vapor plasma spray (VPS), low-pressure plasma spray (LLPS). Electron Beam Physical
Vapor Deposition is the chosen type of processing used to deposit the TBC system due
to its columnar and polycrystalline microstructure and high thermal efficiency. During
this type of processing, the superalloy component is rotated and tilted in the vapor of an
evaporated partially stabilized zirconia molten pool heated by high-energy electron beam
guns. The vapor condenses onto the component to gradually grow a coating.
The microstructure of the coating is heavily influenced by the deposition
processing. The characteristics and strength of the coating can vary depending on how
the coating deposits on the component.
For example, Figure 2 describes the
microstructure zones of the coating as a function of pressure and temperature, (Bose,
2007).
Figure 2: Thornton zone model of physically vapor deposited structure [2]
Studies have been conducted to determine the correlation between film structure and
deposition parameters. Thornton’s model describes that the thermal barrier coating can
be systematically represented by a single structure zone model (SZM). Figure 2 is
important in understanding coating thickness, grain structure, and porosity. With these
2
characteristics plotted, one can determine that EB-PVD is the most suitable processing
method.
The physical vapor deposition process is controlled by the electron beam induced
evaporation, vapor phase interactions, and target absorption characteristics. Depending
on the operating temperature and pressures in the coater, the governing physics may vary
from simple ideal gas behavior to highly complex solid-liquid-vapor interactions. By
varying the input parameters of the EB-PVD, such as temperature, pressure, and
component orientation, the resulting coating microstructure will be modified.
This project will focus on comparing the normalized coating thickness as a
function of component orientation and ingot location, based on an analytical approach.
1.1 BACKGROUND
The gas turbine engine is an internal combustion engine that extracts chemical
energy from fuel and converts it to mechanical energy using air as the working fluid to
drive the engine and propeller, which then propels the aircraft. As depicted in Figure 3,
air moves from left to right, (Bose, 2007).
Figure 3: Typical Gas turbine jet engine [3]
Cold air is pulled in through the Intake and then delivered to the Compressor. The
Compressor rotor blades convert mechanical energy into gaseous energy, which greatly
increases pressure. Next, the air goes through the Combustion Chambers where it is
mixed with fuel and ignited. The air/fuel mixture expands and releases heat. This
mixture gets accelerated through the Turbine section. The Turbine takes the air/burned
fuel mixture and converts it into mechanical energy. By expanding the hot, high3
pressure gasses to a lower temperature and pressure, energy is created. The last section
of the engine is the Exhaust.
The gas that passed through the Turbine is discharged
through the Exhaust, creating thrust and propelling the aircraft forward.
The turbine section is made up of a disk or hub that holds many turbine blades, as
seen in Figure 4. The turbine section is the location where temperature and pressure
become reduced, which is converted into energy.
Figure 4: Gas turbine blade [3]
The gas temperature in current turbine engines can exceed 1650 ºC. The higher the gas
temperature, the more power is produced, thus increasing engine efficiency. However,
the materials used for the turbine blade limit the maximum temperature at which the gas
turbine can operate.
As more high-performing materials are developed for manufacturing turbine
blades, the blade’s performance and efficiency will improve. Gas turbine blades must be
made from an alloy with high-temperature capabilities. The most relevant alloys for this
application include nickel and cobalt based alloys. These metals are chosen based on
their high melting point, strength, ductility, toughness, and low density. Figure 5 shows
different classes of alloys as a function of temperature.
4
Figure 5: Temperature capabilities of several classes of alloys [2]
These high performing alloys are also known as “superalloys”. Nickel (Ni) and cobalt
(Co) superalloys exhibit high strength over a wide range of elevated temperatures. Bose
(2007) states specifically, “Ni exhibits fcc crystal structure and Co is hexagonal close
packed at room temperature,” (p. 23). Although superalloys were developed to achieve
high strength and ductility, they have limited resistance to oxidation and high
temperature corrosion. Oxidation is the event of a metal or alloy being exposed to
oxygen or oxygen-containing gasses at elevated temperatures converting the metallic
elements into their oxides. Bose (2007) describes oxidation as “the loss of load-bearing
capability of the original metal or alloy component, eventually resulting in component
failure,” (p. 29).
In addition to exposing the metal to an oxidizing atmosphere, other
environmental factors in the form of gases create solid particles and result in corrosion,
which ultimately reduces component strength. A substance that contains both aluminum
(Al) and chromium (Cr) is needed to prevent oxidation and corrosion.
However,
superalloys with high levels of Al or Cr tend to exhibit poor ductility and crack growth
resistance. Therefore, in order to create a superalloy with load capability and resistance
to environmental degradation, application of a coating is required.
1.2 COATINGS
The substrate mechanical properties such as tensile, creep, and fatigue strength
are provided by the superalloy. Protection from the environment is provided by a thin
ceramic coating. There are several different types of coatings available to protect the
5
component surface. Important coating requirements include oxidation and corrosion
resistance, phase stability, adhesion, and structural properties. The optimum coating
process depends on the application and component design.
Figure 6: Common coating processes [2]
Diffusion coating, as shown in Figure 6, is a metallic coating used for oxidation
resistance by applying a layer of oxide scale formers on the surface of the substrate. For
diffusion coatings, the substrate participates in the coating formation. Another type of
metallic coating available for high temperature applications is called “overlay” coatings.
Compared to diffusion coatings, overlay coatings are not dependant on the substrate
alloys and can be used in a larger range of applications. Although diffusion and overlay
coatings offer oxidation and corrosion resistance, there is still a need for reducing
substrate surface temperatures. Figure 6 describes a second type of coating process
called ceramic coatings. This category includes thermal barrier coatings, whose primary
purpose is to reduce component temperatures and thereby increasing life.
TBCs are used to reduce surface temperature of the substrate to which they are
applied. The coating consists of multiple layers, each one having its own function.
6
Figure 7: Thermal Barrier Coating system [2]
Figure 7 describes the 3 layers to the coating system: substrate, bond coat, and top coat.
The substrate is chosen based on the required structural strength of the application. Most
typical materials used are nickel and cobalt based superalloys. The bond coat is used to
provide oxidation resistance to the substrate. The bond coat is generally made of a
NiCoCrAlY (Nickel, Cobalt, Chromium, Aluminum, Yttrium) alloy. The thickness of
this coating will be 50 to 125 μm.
During engine operation, the bond coat can
experience temperatures past 700 ºC, creating bond coat oxidation, which is the third
layer of the coating system. The thermally grown oxide (TGO) forms between the
ceramic top coat and bond coat due to high temperature oxidation of the bond coat. The
TGO is important, as Bose (2007) states “binding the ceramic layer to the metallic bond
coat deposited on the substrate,” (p.175). This layer is critical to the spallation life of the
TBC. The thickness can be expected to be 0.05 to 10 μm. The TGO layer consisting of
mostly α alumina prolongs the delamination failure of the TBC system. The top coat is
the layer that provides the thermal insulation. The ideal material for this layer is zirconia
based on its melting point and phase transformations. Due to its phase transformations,
zirconia needs to be combined with a stabilizer, usually yttria. As a result of this
zirconia and oxide combination, yttria-stabalized zirconia has become the most desirable
high-temperature application.
In order for the TBC to be effective in reducing extreme surface temperatures, the
7
coating needs to be applied correctly to the substrate. Without proper application, the
TBC is unable to provide improvement to engine efficiency. TBCs are typically based
on zirconia and can only be deposited by processing that is capable of adding enough
energy to the raw material.
1.3 TBC PROCESSING TECHNOLOGIES
In operation today, there are two processes used to apply TBCs: air plasma spraying
(APS) and electron beam physical vapor deposition. The plasma spraying process is the
spraying of molten coating material onto a surface. The coating raw material is melted
into the form of powder and is injected into high temperature plasma jets where it is then
heated and accelerated.
The hot material impacts the substrate, rapidly cools, and
solidifies to form the coating.
The microstructure of the plasma sprayed YSZ is
characterized as having good adhesive strength (20-70 MPa) and porosity (10-15 vol %),
(Bose, 2007). The microstructure of the plasma sprayed TBC can be compared to the
microstructure produced by EB-PVD processing.
Another processing method used for delivering localized high energy to zirconia
by using high power of focused electrons is called Electron Beam Physical Vapor
Deposition. This technique is used for preparing thin-film materials with structural
control at the atomic or nanometer scale, (Martin-Palma and Lakhtakia, 2010). In this
specific application, the method of EB-PVD processing requires a melting pool of the
raw material coating that is contained in an evacuated chamber, as described in Figure 8.
This melt pool is produced by the localized heating imparted by high-energy electron
beams. This pool generates a vapor and as the substrate is being held over the pool, the
coating is formed on the surface by deposition of vapor molecules.
8
Figure 8: EB-PVD coating chamber [2]
The EB-PVD coater consists of several key components and functions:
1.) Coating Chamber: the coating chamber is evacuated and is set at a pressure of
about 10-4 torr. Because the vapor molecules follow the ideal gas laws, the
molecules are able to travel a large distance from the molten pool to the
substrate.
2.) EB guns: there are as many as six electron guns which provide an electron beam
of high energy, with an average power rating of 50 kW per gun.
3.) Raw Material: the coating raw material is in the form of ingots, which are
enclosed in a water cooled crucible.
4.) Part manipulator: part that is being coated requires rotation movement as well as
tilting movement. In this case, there are two shafts that move, called stings.
5.) Multiple ingots: by having multiple ingots in the chamber, the coating
composition is able to be varied.
Advantages and disadvantages of EB-PVD process depend on the material to be
evaporated and the desired microstructure. The EB-PVD process offers a flexible
deposition rates, dense coatings, strong metallurgical bonding, columnar and
polycrystalline microstructure, and high thermal efficiency. In addition, good adhesion
can be obtained at higher substrate temperatures due to diffusion bonding. Coating
density is also an important characteristic of EB-PVD process. The coating can be
altered by changing the deposition rate, method of rotation, and angle of coating
9
incidence. The main disadvantage of EB-PVD process is the high capital equipment
cost and the limitation of being only a line-of-sight process. A different coating process
would need to be used for more complex shapes.
1.4 COATING CHARACERISTICS
The main requirement of EB-PVD processing is to produce TBC microstructures
that are designed to be tolerant of thermo-mechanical strains imposed on the TBC during
service. A typical micrograph of an EB-PVD TBC is shown in Figure 9.
Figure 9: EB-PVD TBC on a metallic substrate [7]
A typical EB-PVD coating thickness is approximately 100 μm.
TBCs of various
application thicknesses have different thermal properties; therefore, in order to produce a
coating with the appropriate mechanical strain properties, an approximate thickness of
100 μm is required. The YSZ grows in a columnar microstructure perpendicular to the
substrate, “consisting of independent columns that are narrow in width at the base and
wider near the surface,” (Johnson, Ruud, Bruce, and Wortman (1998). Because of the
columnar microstructure, the coating provides strain tolerance. This columnar
microstructure is important in aligning the intercolumnar pores perpendicular to the
plane of the coating as its thickness increases.
The elongated intercolumnar pores
increase the conformity of the coating in the plane of the substrate, leading to improved
spallation lifetimes of the TBC system. The coating deposition rate, W can be explained
by the Hertz-Knudsen equation, as described in Equation (1), where n’ is the
10
dimensionless evaporation coefficient, kB is the Boltzmann constant, m is the molecular
weight of the deposition species, P is the absolute pressure in the coater chamber, (Bose,
2007).
𝑊 = 𝑛′ (𝑃∗ − 𝑃)(𝑚⁄2𝜋𝑘𝐵 𝑇)1/2
(1)
It can then be determined that a faster deposition rate can be achieved at a higher
vacuum level (very low pressure).
Another model that describes the deposition profile of the TBC is the Knudsen
cosine law, as described in Equation (2) and Figure 10, where d is the coating thickness,
M is the mass of the depositing species, ρ is the density of the coating, r is the radial
distance from the emission source to received surface, and θ and φ are the angles
between the radial distance and receiving surface normal.
Figure 10: Knudsen cosine law schematic [1]
𝑑 = (𝑀⁄𝜌){(𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜑)/𝜋𝑟 2 }
(2)
The EB-PVD processing is ideal due to the creation of columnar microstructures. In
addition to the deposition rate, there are other factors that can impact the TBC
microstructure: EB gun power, substrate preheating, chamber temperature control,
chamber pressure, etc. This project only focuses on the impacts of geometry and
orientation of the substrate relative to the ingot. In this application, the vapor source
comes from raw material containing alloy powder in the form of ingots.
It is
determined that the performance of the thermal barrier coating is dependent on the
ceramic coating thickness.
The ability to predict coating thickness is critical in
understanding and optimizing the EB-PVD process variables.
11
2. METHODOLOGY
2.1 THICKNESS DISTRIBUTION
The deposition thickness of the TBC on a substrate can be described by the model
created by Hertz, Knudsen, and Langmuir. According to Nicholls (1998), the thickness
distribution is seen as “being deposited from an ideal point source evaporator” (p. 16-5)
which is defined by the inverse square law, as seen in Equation (3),
2
𝑑 ℎ
𝑑̅ = 𝑑 ∙ ℎ𝑜2 𝑐𝑜𝑠 𝑛 𝜃 ∙ cos 𝛼
0
(3)
Where 𝑑̅ is defined as a dimensionless coating thickness, d is the coating thickness at a
distance h from the vapor source, do is the thickness at a distance ho from the vapor
source. θ and α are defined by the source to substrate geometry. As seen in Figure 11, α
is the angle between the vapor trajectory and the normal to the thin plate and n is a nondimensional constant. The goal of this project is to model the most uniform coating on
the substrate. The substrate is initially modeled as a thin rectangular plate, 0.33m length
by 0.18m width (13in length by 7in width), perpendicular to the ingot in the EB-PVD
coater.
Figure 11: horizontal plate schematic
12
As seen in Equation (3), n is a function of the vapor stream density distribution, where n
is greater than or equal to unity, (Heisig, 1982). As n increases, the source vapor plume
becomes more focused. By varying the location of the ingot relative to the substrate and
changing the evaporation characteristics of the vapor source, the coating thickness can
be altered significantly. Microsoft Office Excel is used to create this model, first by
creating a coarse mesh on the thin plate, as seen in Figure 12.
Figure 12: model mesh schematic
Next, Equation (3) is used to define 𝑑̅ , the final coating thickness.
The coating
thickness, 𝑑̅, is calculated at each node in the mesh and transposed into matrix form.
The final result is a 0.33m by 0.18m (13in by 7in) matrix, defining the coating thickness
profile. Next, a 3D surface plot is generated showing the variation of the dimensionless
coating thickness with distance along the length. This methodology is repeated for
several different ingot orientations and plate inclinations.
13
For example, the following procedure was followed to determine the coating
thickness at node (9,6).
Figure 13:
thin plate model example
Referencing Equation (3), d0 is shown as the coating thickness directly over the vapor
source. For this project, do is assumed to be one (1). The next step in the model is
defining ho, which is the distance of the flat plate above the coating source. For this
project, ho is chosen to be 0.3048m (12 in). In addition to the distance of the plate
relative to the vapor source, the distance of the node (9, 6) relative to the vapor source is
defined as h. For this project,
ℎ = √(𝑟𝑥2 + 𝑟𝑦2 ) + ℎ𝑜2 ,
(4)
rx is the distance from the center node ( 7, 4) to the calculated node (9, 6), in the x
direction, and ry is the distance from the center node ( 7, 4) to the calculated node (9,6) in
the y direction. As seen in Equation (4), ℎ = √(22 + 22 ) + 122 = √152 = 12.33.
Referencing Equation (3), cos (θ) is defined using h and ho.
θ = cos−1
ho
h
12
= cos −1 12.33 = .23 radians
(5)
Because the flat plate is parallel to the vapor source, α is equal to θ. Lastly, n is defined
as the vapor source plume projection. As n increases, the vapor source becomes more
focused. For this example, n is equal to 9. Combining all of these variables together, as
defined in Equation (3), the coating thickness at node (9, 6) is calculated in Equation (6).
14
122
𝑑 = 1 ∙ 12.332 𝑐𝑜𝑠 9 (0.23) ∙ cos(0.23) = 0.72
(6)
This example is repeated 90 more times to get the coating thickness at each node in the
13 by 7 mesh of the flat plate, as calculated in Table 1.
Table 1: Example coating thickness for thin plate
0.20
0.23
0.25
0.26
0.25
0.23
0.20
0.28
0.33
0.37
0.38
0.37
0.33
0.28
0.38
0.46
0.51
0.53
0.51
0.46
0.38
0.49
0.60
0.67
0.70
0.67
0.60
0.49
0.60
0.72
0.81
0.85
0.81
0.72
0.60
0.67
0.81
0.92
0.96
0.92
0.81
0.67
0.70
0.85
0.96
1.00
0.96
0.85
0.70
0.67
0.81
0.92
0.96
0.92
0.81
0.67
0.60
0.72
0.81
0.85
0.81
0.72
0.60
0.49
0.60
0.67
0.70
0.67
0.60
0.49
0.38
0.46
0.51
0.53
0.51
0.46
0.38
0.28
0.33
0.37
0.38
0.37
0.33
0.28
0.20
0.23
0.25
0.26
0.25
0.23
0.20
After the matrix of coating thickness is created, a 3D surface plot of the thickness across
the entire plate is generated.
2.2 INCLINED PLATE DEPOSITION
During the coating process, the part to be coated is fixtured in a part manipulator.
The substrate can be not only rotated but also tilted, potentially creating a more uniform
coating. In addition to modeling the substrate perpendicular to the vapor source, the
coating thickness is also predicted by varying the inclination of the substrate.
15
Figure 14: inclined flat plate schematic
Figure 14 describes the inclined plate model. In this model, referencing Equation (3),
the substrate is inclined at a 45 degree angle from the horizontal. As in the horizontal
plate schematic, Figure 11, the plate mesh is also 0.33m by 0.18m (13in by 7in). Using
basic trigonometry functions to calculate θ and α, the coating thickness profile can be
predicted and will follow the same steps as described in Equation (6).
16
3. RESULTS AND DISCUSSION
By solving the preceding equations, the coating deposition profile can be predicted.
The most classic example is seen in Figure 15 where the vapor source is directly
centered to the thin plate.
Figure 15: 1 ingot source centered
Figure 16 shows the deposition profile exhibiting the classic bell shape curve with the
peak rate of deposition directly above the ingot source.
Figure 16: ingot source centered, n=9
It can be seen that having one vapor source and evaporation characteristic of n =9, a
uniform coating thickness is not created. The next thickness profile is described by the
17
thin plate rotated at a 45º angle from the horizontal, as shown schematically in Figure
17.
Figure 17: inclined thin rectangular plate schematic
Nicholls (1998) states “when an inclinded substrate is located over the vapor source, the
peak deposition rate measured on the substrate is displaced from the centerline of the
evaporant source.” Figure 18 shows the “maximum coating thickness measured on the
substrate is reduced as the vapor flux arrives at an oblique incident vaport flux
[resulting] in an assymetric coating thickness distribution on the substrate,” (p. 16-7).
Figure 18: inclined plate, n=1
18
As seen in the bell shaped evaporation profile, Figure 16, and the incline plate profile,
Figure 18, a uniform coating profile is difficult to achieve. There are two options for
creating a more uniform coating: changing the location of the ingots relative to the
substrate and changing the vapor source characteristics.
Figure 19 details for the thin plate located above two ingot sources that are equally
spaced from the center of the plate.
Figure 19: dual ingot source
The coating thickness profiles for n =1 and n =9 are shown in Figure 20 and Figure 21,
respectively. It is clear that the distribution for n =9 (Figure 21) is worse than that for n
=1 (Figure 20). This result clearly does not produce a uniform coating, with an even
worse profile defined in Figure 21.
19
Figure 20: dual ingot source, n=1
Figure 21: dual ingot source, n=9
The final experimental approach for seeking the most uniform coating is to place the two
20
ingots more closely together, as seen in Figure 22.
Figure 22: dual ingot, closely spaced
Here, the deposition coating profile is found to be the most uniform, having a relatively
constant thickness along the entire plate, as seen in Figure 23.
Figure 23: dual ingot, closely spaced, n=1
Although there are many options in creating a uniform coating, only a few were
documented as part of this project. Review Appendix A for all research results.
21
4. CONCLUSION
Mathematical prediction of the deposition thickness of a thin TBC in an EB-PVD
coater is critical for obtaining process control. It is also necessary for understanding the
microstructure evolution in the ceramic coating.
It can be seen that by altering the
substrate orientation inside the evacuated EB-PVD chamber, the coating thickness can
be varied significantly. In order to predict the most uniform coating thickness, the
substrate can either be positioned on an incline above the vapor source, or the coating
chamber can be equipped with multiple ingots, or the vapor source deposition (n) can be
changed. Using multiple vapor sources allows the area of a uniform deposition to
increase by ensuring the vapor plumes from each ingot source overlap each other.
From the analyses presented in this report, it is clear that coating microstructure is
dependent on vapor source position in the EB-PVD coating chamber. Even though the
current model for predicting TBC thickness deposited did not take into account other
processing parameter variables, the coating profile uniformity was still demonstrated.
The benefits of EB-PVD TBCs thickness uniformity can be correlated to improved
temperature capability and ultimately improved engine efficiency.
4.1 FUTURE WORK AND MODEL IMPROVEMENT
Although there was much success in being able to model the coating thickness
profile for an EB-PVD TBC, testing of a real substrate and coating is needed to validate
the thickness coating profiles.
Actual coating thickness measurements along the
substrate would be key in understanding the process control. I would also recommend
continuing using the model generated and begin to include other factors that impact
coating uniformity, such as temperature, pressure, deposition rate, and evaporation rate.
22
REFERENCES
[1]
[2]
Bernier, J.S., Weir, W.C.S., Fontecchio, M., Sisson, Jr., R.D. Deposition rates of EB-PVD
TBC's on Cylindrical Surfaces. Worcester: Worcester Polytechnic Institute.
Bose, S. (2007). High Temperature Coatings . Boston: Elsevier Inc.
[3]
Gas Turbine. (n.d.). Retrieved November 5, 2011 from Wikipedia:
http://en.wikipedia.org/wiki/Gas_turbine
[4]
Heisig, U., Panzer, S. (1982).Electron Beam Technology . Berlin: John Wiley & Sons, Inc.
[5]
Johnson, C.A., Ruud J.A., Bruce R., Wortman, D. (1998).Relationship between residual stress,
microstructure and mechanical properties of electron-beam physical vapor deposition thermal
barrier coatings. Surface and Coating Technology,108-9 , 80-85.
[6]
Martin-Palma, R. and Lakhtakia, A. (2010). Nanotechnology: A Crash Course.
Bellingham: Society of Photo-Optical Instrumentation Engineers.
[7]
Nicholls, J.R., Pereira, V., Lawson, K.J., Rickerby, D.S. (Eds.). (1998). Proceedings from RTO
AVT Workshop '98: Process Control of Deposition Profiles in the Manufacture of EB-PVD
Thermal Barrier Coatings. Brussels, Belgium: RTO MP-9.
23
APPENDIX A: RESULTS
1.) All the data in this section refer to a thin plate with the vapor source on the far
right side, as seen in Figure 24.
Dimensionless Coating Thickness
Figure 24: Right side ingot
1.10
0.90
0.70
0.50
0.30
0.10
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 25: Right side ingot, n=1 plot
Table 2: Right side ingot, n=1
𝑑̅ =
0.24
0.24
0.25
0.25
0.25
0.24
0.24
0.28
0.29
0.29
0.30
0.29
0.29
0.28
0.32
0.34
0.35
0.35
0.35
0.34
0.32
0.38
0.40
0.41
0.41
0.41
0.40
0.38
0.44
0.46
0.47
0.48
0.47
0.46
0.44
0.51
0.53
0.55
0.56
0.55
0.53
0.51
24
0.58
0.61
0.63
0.64
0.63
0.61
0.58
0.65
0.69
0.72
0.73
0.72
0.69
0.65
0.73
0.77
0.80
0.81
0.80
0.77
0.73
0.79
0.84
0.87
0.89
0.87
0.84
0.79
0.84
0.90
0.93
0.95
0.93
0.90
0.84
0.87
0.93
0.97
0.99
0.97
0.93
0.87
0.89
0.95
0.99
1.00
0.99
0.95
0.89
Dimensionless Coating Thickness
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 26: Right side ingot, n=4 plot
Table 3: Right side ingot, n=4
𝑑̅ =
0.08
0.08
0.09
0.09
0.09
0.08
0.08
0.11
0.11
0.12
0.12
0.12
0.11
0.11
0.14
0.15
0.16
0.16
0.16
0.15
0.14
0.18
0.20
0.21
0.21
0.21
0.20
0.18
0.24
0.26
0.27
0.28
0.27
0.26
0.24
0.31
0.33
0.35
0.36
0.35
0.33
0.31
25
0.39
0.42
0.45
0.46
0.45
0.42
0.39
0.48
0.53
0.56
0.57
0.56
0.53
0.48
0.57
0.63
0.68
0.69
0.68
0.63
0.57
0.66
0.74
0.79
0.81
0.79
0.74
0.66
0.74
0.83
0.89
0.91
0.89
0.83
0.74
0.79
0.89
0.95
0.98
0.95
0.89
0.79
0.81
0.91
0.98
1.00
0.98
0.91
0.81
Dimensionless Coating Thickness
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 27: Right side ingot, n=9 plot
Table 4: Right side ingot, n=9
𝑑̅ =
0.01
0.01
0.02
0.02
0.02
0.01
0.01
0.02
0.02
0.03
0.03
0.03
0.02
0.02
0.03
0.04
0.04
0.04
0.04
0.04
0.03
0.05
0.06
0.07
0.07
0.07
0.06
0.05
0.09
0.10
0.11
0.11
0.11
0.10
0.09
0.13
0.15
0.17
0.17
0.17
0.15
0.13
26
0.20
0.23
0.25
0.26
0.25
0.23
0.20
0.28
0.33
0.37
0.38
0.37
0.33
0.28
0.38
0.46
0.51
0.53
0.51
0.46
0.38
0.49
0.60
0.67
0.70
0.67
0.60
0.49
0.60
0.72
0.81
0.85
0.81
0.72
0.60
0.67
0.81
0.92
0.96
0.92
0.81
0.67
0.70
0.85
0.96
1.00
0.96
0.85
0.70
2.) All the data in this section refer to a thin plate with the vapor source centered, as
seen in Figure 28.
Dimensionless Coating Thickness
Figure 28: Center ingot, n=1
1.00
0.90
0.80
0.70
0.60
0.50
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 29: Center ingot, n=1 plot
Table 5: Center ingot, n=1
𝑑̅ =
0.58
0.61
0.63
0.64
0.63
0.61
0.58
0.65
0.69
0.72
0.73
0.72
0.69
0.65
0.73
0.77
0.80
0.81
0.80
0.77
0.73
0.79
0.84
0.87
0.89
0.87
0.84
0.79
0.84
0.90
0.93
0.95
0.93
0.90
0.84
0.87
0.93
0.97
0.99
0.97
0.93
0.87
27
0.89
0.95
0.99
1.00
0.99
0.95
0.89
0.87
0.93
0.97
0.99
0.97
0.93
0.87
0.84
0.90
0.93
0.95
0.93
0.90
0.84
0.79
0.84
0.87
0.89
0.87
0.84
0.79
0.73
0.77
0.80
0.81
0.80
0.77
0.73
0.65
0.69
0.72
0.73
0.72
0.69
0.65
0.58
0.61
0.63
0.64
0.63
0.61
0.58
Dimensionless Coating Thickness
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 30: Center ingot, n=4 plot
Table 6: Center ingot, n=4
𝑑̅ =
0.39
0.42
0.45
0.46
0.45
0.42
0.39
0.48
0.53
0.56
0.57
0.56
0.53
0.48
0.57
0.63
0.68
0.69
0.68
0.63
0.57
0.66
0.74
0.79
0.81
0.79
0.74
0.66
0.74
0.83
0.89
0.91
0.89
0.83
0.74
0.79
0.89
0.95
0.98
0.95
0.89
0.79
28
0.81
0.91
0.98
1.00
0.98
0.91
0.81
0.79
0.89
0.95
0.98
0.95
0.89
0.79
0.74
0.83
0.89
0.91
0.89
0.83
0.74
0.66
0.74
0.79
0.81
0.79
0.74
0.66
0.57
0.63
0.68
0.69
0.68
0.63
0.57
0.48
0.53
0.56
0.57
0.56
0.53
0.48
0.39
0.42
0.45
0.46
0.45
0.42
0.39
Dimensionless Coating Thickness
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 31: Center ingot, n=9 plot
Table 7: Center ingot, n=9
𝑑̅ =
0.20
0.23
0.25
0.26
0.25
0.23
0.20
0.28
0.33
0.37
0.38
0.37
0.33
0.28
0.38
0.46
0.51
0.53
0.51
0.46
0.38
0.49
0.60
0.67
0.70
0.67
0.60
0.49
0.60
0.72
0.81
0.85
0.81
0.72
0.60
0.67
0.81
0.92
0.96
0.92
0.81
0.67
29
0.70
0.85
0.96
1.00
0.96
0.85
0.70
0.67
0.81
0.92
0.96
0.92
0.81
0.67
0.60
0.72
0.81
0.85
0.81
0.72
0.60
0.49
0.60
0.67
0.70
0.67
0.60
0.49
0.38
0.46
0.51
0.53
0.51
0.46
0.38
0.28
0.33
0.37
0.38
0.37
0.33
0.28
0.20
0.23
0.25
0.26
0.25
0.23
0.20
3.) All the data in this section refer to a thin plate with the vapor source on the far
left side, as seen in Figure 32 .
Dimensionless Coating Thickness
Figure 32: Left side ingot
1.00
0.80
0.60
0.40
0.20
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 33: Left side ingot, n=1 plot
Table 8: Left side ingot, n=1
𝑑̅ =
0.89
0.95
0.99
1.00
0.99
0.95
0.89
0.87
0.93
0.97
0.99
0.97
0.93
0.87
0.84
0.90
0.93
0.95
0.93
0.90
0.84
0.79
0.84
0.87
0.89
0.87
0.84
0.79
0.73
0.77
0.80
0.81
0.80
0.77
0.73
0.65
0.69
0.72
0.73
0.72
0.69
0.65
30
0.58
0.61
0.63
0.64
0.63
0.61
0.58
0.51
0.53
0.55
0.56
0.55
0.53
0.51
0.44
0.46
0.47
0.48
0.47
0.46
0.44
0.38
0.40
0.41
0.41
0.41
0.40
0.38
0.32
0.34
0.35
0.35
0.35
0.34
0.32
0.28
0.29
0.29
0.30
0.29
0.29
0.28
0.24
0.24
0.25
0.25
0.25
0.24
0.24
Dimensionless Coating Thickness
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 34: Left side ingot, n=4 plot
Table 9: Left side ingot, n=4
𝑑̅ =
0.81
0.91
0.98
1.00
0.98
0.91
0.81
0.79
0.89
0.95
0.98
0.95
0.89
0.79
0.74
0.83
0.89
0.91
0.89
0.83
0.74
0.66
0.74
0.79
0.81
0.79
0.74
0.66
0.57
0.63
0.68
0.69
0.68
0.63
0.57
0.48
0.53
0.56
0.57
0.56
0.53
0.48
31
0.39
0.42
0.45
0.46
0.45
0.42
0.39
0.31
0.33
0.35
0.36
0.35
0.33
0.31
0.24
0.26
0.27
0.28
0.27
0.26
0.24
0.18
0.20
0.21
0.21
0.21
0.20
0.18
0.14
0.15
0.16
0.16
0.16
0.15
0.14
0.11
0.11
0.12
0.12
0.12
0.11
0.11
0.08
0.08
0.09
0.09
0.09
0.08
0.08
Dimensionless Coating Thickness
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 35: Left side ingot, n=9 plot
Table 10: Left side ingot, n=9
𝑑̅ =
0.70
0.85
0.96
1.00
0.96
0.85
0.70
0.67
0.81
0.92
0.96
0.92
0.81
0.67
0.60
0.72
0.81
0.85
0.81
0.72
0.60
0.49
0.60
0.67
0.70
0.67
0.60
0.49
0.38
0.46
0.51
0.53
0.51
0.46
0.38
0.28
0.33
0.37
0.38
0.37
0.33
0.28
32
0.20
0.23
0.25
0.26
0.25
0.23
0.20
0.13
0.15
0.17
0.17
0.17
0.15
0.13
0.09
0.10
0.11
0.11
0.11
0.10
0.09
0.05
0.06
0.07
0.07
0.07
0.06
0.05
0.03
0.04
0.04
0.04
0.04
0.04
0.03
0.02
0.02
0.03
0.03
0.03
0.02
0.02
0.01
0.01
0.02
0.02
0.02
0.01
0.01
4.) All the data in this section refer to a thin plate with 3 vapor sources, as seen in
Figure 28.
Dimensionless Coating Thickness
Figure 36: 3 ingots
2.00
1.50
1.00
0.50
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 37: 3 ingots, n=1 plot
Table 11: 3 ingots, n=1
𝑑̅ =
1.70
1.80
1.87
1.89
1.87
1.80
1.70
1.81
1.91
1.98
2.01
1.98
1.91
1.81
1.89
2.01
2.08
2.10
2.08
2.01
1.89
1.96
2.08
2.15
2.18
2.15
2.08
1.96
2.01
2.13
2.21
2.24
2.21
2.13
2.01
33
2.04
2.16
2.24
2.27
2.24
2.16
2.04
2.05
2.17
2.25
2.28
2.25
2.17
2.05
2.04
2.16
2.24
2.27
2.24
2.16
2.04
2.01
2.13
2.21
2.24
2.21
2.13
2.01
1.96
2.08
2.15
2.18
2.15
2.08
1.96
1.89
2.01
2.08
2.10
2.08
2.01
1.89
1.81
1.91
1.98
2.01
1.98
1.91
1.81
1.70
1.80
1.87
1.89
1.87
1.80
1.70
Dimensionless Coating Thickness
2.00
1.50
1.00
0.50
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 38: 3 ingots, n=4 plot
Table 12: 3 ingots, n=4
𝑑̅ =
1.27
1.42
1.51
1.55
1.51
1.42
1.27
1.37
1.53
1.63
1.67
1.63
1.53
1.37
1.45
1.61
1.72
1.76
1.72
1.61
1.45
1.51
1.68
1.79
1.83
1.79
1.68
1.51
1.55
1.72
1.84
1.88
1.84
1.72
1.55
34
1.57
1.75
1.86
1.91
1.86
1.75
1.57
1.58
1.76
1.87
1.92
1.87
1.76
1.58
1.57
1.75
1.86
1.91
1.86
1.75
1.57
1.55
1.72
1.84
1.88
1.84
1.72
1.55
1.51
1.68
1.79
1.83
1.79
1.68
1.51
1.45
1.61
1.72
1.76
1.72
1.61
1.45
1.37
1.53
1.63
1.67
1.63
1.53
1.37
1.27
1.42
1.51
1.55
1.51
1.42
1.27
Dimensionless Coating Thickness
2.00
1.50
1.00
0.50
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 39: 3 ingots, n=9 plot
Table 13: 3 ingots, n=9
𝑑̅ =
0.90
1.09
1.23
1.28
1.23
1.09
0.90
0.97
1.17
1.32
1.37
1.32
1.17
0.97
1.01
1.22
1.37
1.42
1.37
1.22
1.01
1.04
1.25
1.40
1.46
1.40
1.25
1.04
1.06
1.28
1.43
1.49
1.43
1.28
1.06
35
1.08
1.30
1.46
1.51
1.46
1.30
1.08
1.09
1.31
1.47
1.52
1.47
1.31
1.09
1.08
1.30
1.46
1.51
1.46
1.30
1.08
1.06
1.28
1.43
1.49
1.43
1.28
1.06
1.04
1.25
1.40
1.46
1.40
1.25
1.04
1.01
1.22
1.37
1.42
1.37
1.22
1.01
0.97
1.17
1.32
1.37
1.32
1.17
0.97
0.90
1.09
1.23
1.28
1.23
1.09
0.90
5.) All the data in this section refer to a thin plate with 2 vapor sources, as seen in
Figure 40.
Dimensionless Coating Thickness
Figure 40: 2 ingots
1.30
1.25
1.20
1.15
1.10
1.05
1.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 41: 2 ingots, n=1 plot
Table 14: 2 ingots, n=1
𝑑̅ =
1.12
1.19
1.23
1.25
1.23
1.19
1.12
1.15
1.22
1.27
1.28
1.27
1.22
1.15
1.17
1.23
1.28
1.29
1.28
1.23
1.17
1.17
1.24
1.28
1.30
1.28
1.24
1.17
1.17
1.23
1.27
1.29
1.27
1.23
1.17
36
1.16
1.23
1.27
1.28
1.27
1.23
1.16
1.16
1.22
1.27
1.28
1.27
1.22
1.16
1.16
1.23
1.27
1.28
1.27
1.23
1.16
1.17
1.23
1.27
1.29
1.27
1.23
1.17
1.17
1.24
1.28
1.30
1.28
1.24
1.17
1.17
1.23
1.28
1.29
1.28
1.23
1.17
1.15
1.22
1.27
1.28
1.27
1.22
1.15
1.12
1.19
1.23
1.25
1.23
1.19
1.12
Dimensionless Coating Thickness
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 42: 2 ingots, n=4 plot
Table 15: 2 ingots, n=4
𝑑̅ =
0.89
0.99
1.06
1.09
1.06
0.99
0.89
0.90
1.00
1.07
1.09
1.07
1.00
0.90
0.88
0.98
1.04
1.07
1.04
0.98
0.88
0.84
0.94
1.00
1.02
1.00
0.94
0.84
0.81
0.89
0.95
0.97
0.95
0.89
0.81
37
0.78
0.86
0.91
0.93
0.91
0.86
0.78
0.77
0.85
0.90
0.92
0.90
0.85
0.77
0.78
0.86
0.91
0.93
0.91
0.86
0.78
0.81
0.89
0.95
0.97
0.95
0.89
0.81
0.84
0.94
1.00
1.02
1.00
0.94
0.84
0.88
0.98
1.04
1.07
1.04
0.98
0.88
0.90
1.00
1.07
1.09
1.07
1.00
0.90
0.89
0.99
1.06
1.09
1.06
0.99
0.89
Dimensionless Coating Thickness
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 43: 2 ingots, n=9 plot
Table 16: 2 ingots, n=9
𝑑̅ =
0.71
0.86
0.97
1.02
0.97
0.86
0.71
0.69
0.84
0.95
0.99
0.95
0.84
0.69
0.63
0.76
0.86
0.89
0.86
0.76
0.63
0.55
0.66
0.74
0.76
0.74
0.66
0.55
0.47
0.56
0.62
0.64
0.62
0.56
0.47
38
0.41
0.49
0.54
0.56
0.54
0.49
0.41
0.39
0.46
0.51
0.52
0.51
0.46
0.39
0.41
0.49
0.54
0.56
0.54
0.49
0.41
0.47
0.56
0.62
0.64
0.62
0.56
0.47
0.55
0.66
0.74
0.76
0.74
0.66
0.55
0.63
0.76
0.86
0.89
0.86
0.76
0.63
0.69
0.84
0.95
0.99
0.95
0.84
0.69
0.71
0.86
0.97
1.02
0.97
0.86
0.71
6.) All the data in this section refer to a thin plate with 2 vapor sources centered, as
seen in Figure 44.
Dimensionless Coating Thickness
Figure 44: 2 ingots centered
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 45: 2 ingots centered, n=1 plot
Table 17: 2 ingots centered, n=1
𝑑̅ =
1.15
1.22
1.27
1.28
1.27
1.22
1.15
1.21
1.28
1.33
1.35
1.33
1.28
1.21
1.25
1.33
1.38
1.40
1.38
1.33
1.25
1.28
1.36
1.41
1.43
1.41
1.36
1.28
1.30
1.38
1.43
1.44
1.43
1.38
1.30
39
1.31
1.38
1.43
1.45
1.43
1.38
1.31
1.31
1.39
1.44
1.45
1.44
1.39
1.31
1.31
1.38
1.43
1.45
1.43
1.38
1.31
1.30
1.38
1.43
1.44
1.43
1.38
1.30
1.28
1.36
1.41
1.43
1.41
1.36
1.28
1.25
1.33
1.38
1.40
1.38
1.33
1.25
1.21
1.28
1.33
1.35
1.33
1.28
1.21
1.15
1.22
1.27
1.28
1.27
1.22
1.15
Dimensionless Coating Thickness
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 46: 2 ingots centered, n=4 plot
Table 18: 2 ingots centered, n=4
𝑑̅ =
0.91
1.02
1.09
1.12
1.09
1.02
0.91
0.93
1.04
1.11
1.13
1.11
1.04
0.93
0.92
1.02
1.09
1.12
1.09
1.02
0.92
0.90
1.00
1.06
1.08
1.06
1.00
0.90
0.88
0.97
1.03
1.05
1.03
0.97
0.88
40
0.86
0.95
1.01
1.03
1.01
0.95
0.86
0.86
0.95
1.01
1.03
1.01
0.95
0.86
0.88
0.97
1.03
1.05
1.03
0.97
0.88
0.90
1.00
1.06
1.08
1.06
1.00
0.90
0.92
1.02
1.09
1.12
1.09
1.02
0.92
0.93
1.04
1.11
1.13
1.11
1.04
0.93
0.91
1.02
1.09
1.12
1.09
1.02
0.91
0.87
0.97
1.04
1.06
1.04
0.97
0.87
Dimensionless Coating Thickness
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Thin Plate length
Figure 47: 2 ingots centered, n=9 plot
Table 19: 2 ingots centered, n=9
𝑑̅ =
0.72
0.87
0.98
1.03
0.98
0.87
0.72
0.70
0.85
0.96
1.00
0.96
0.85
0.70
0.65
0.78
0.88
0.92
0.88
0.78
0.65
0.58
0.69
0.78
0.81
0.78
0.69
0.58
0.51
0.61
0.68
0.70
0.68
0.61
0.51
41
0.48
0.56
0.62
0.64
0.62
0.56
0.48
0.48
0.56
0.62
0.64
0.62
0.56
0.48
0.51
0.61
0.68
0.70
0.68
0.61
0.51
0.58
0.69
0.78
0.81
0.78
0.69
0.58
0.65
0.78
0.88
0.92
0.88
0.78
0.65
0.70
0.85
0.96
1.00
0.96
0.85
0.70
0.72
0.87
0.98
1.03
0.98
0.87
0.72
0.68
0.83
0.94
0.97
0.94
0.83
0.68