Numerical Simulation of Hydrogen Plasma in Mpcvd …...LIST OF SYMBOLS 𝐸 electrical field 𝜌 gas density 𝜀0 permittivity of vacuum 𝐵 magnetic field 𝑡 time 𝜇0 permeability

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Open Access Theses Theses and Dissertations

Summer 2014

Numerical Simulation of Hydrogen Plasma inMpcvd ReactorDi HuangPurdue University

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Recommended CitationHuang, Di, "Numerical Simulation of Hydrogen Plasma in Mpcvd Reactor" (2014). Open Access Theses. Paper 441.

01 14

PURDUE UNIVERSITY GRADUATE SCHOOL

Thesis/Dissertation Acceptance

Thesis/Dissertation Agreement.

Publication Delay, and Certification/Disclaimer (Graduate School Form 32)

adheres to the provisions of

Department

Di Huang

NUMERICAL SIMULATION OF HYDROGEN PLASMA IN MPCVD REACTOR

Master of Science in Aeronautics and Astronautics

Alina Alexeenko

Timothy S. Fisher

Li Qiao

Alina Alexeenko

Wayne Chen 07/25/2014

i

NUMERICAL SIMULATION OF HYDROGEN PLASMA IN MPCVD REACTOR

A Thesis

Submitted to the Faculty

of

Purdue University

by

Di Huang

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Aeronautics and Astronautics

August 2014

Purdue University

West Lafayette, Indiana

ii

ACKNOWLEDGEMENTS

I would like to express my deepest appreciation to my advisor, Dr. Alina Alexeenko for

her supportive guidance, constant help, and kindness throughout my pursuit of my degree.

I have the deepest appreciation for being given this great opportunity to complete this

research with her as my advisor.

I would like to thank my committee members, Professor Timothy Fisher and Professor Li

Qiao, for their teaching and serving in my committee. I have great appreciation to Dr.

Abbas Semnani at Birck Nanotechnology Center for his support and assistance in

electromagnetic modeling. I would like to thank Professor Allen Garner at School of

Nuclear Engineering for his support and assistance in utilizing COMSOL Multiphysics.

I have great appreciation to my colleagues, Venkattraman Ayyaswamy, Arnab Ganguly,

Andrew Weaver, Marat Kulakhmetov, Tony Cofer, Cem Pekardan, Devon Parkos, Siva

Sashank, Israel Sebastiao, Andrew Strongrich, Bill O’Neill and Nikhil Varma. They gave

me many opportunities for active discussion and persistent help throughout my research.

iii

TABLE OF CONTENTS

Page

LIST OF TABLES .............................................................................................................. v LIST OF FIGURES ........................................................................................................... vi LIST OF SYMBOLS ....................................................................................................... viii LIST OF ABBREVIATIONS ............................................................................................. x ABSTRACT ....................................................................................................................... xi CHAPTER 1. INTRODUCTION ................................................................................. 1

1.1 Background ............................................................................................... 1

1.2 Motivation ................................................................................................. 2

CHAPTER 2. ELECTROMAGNETIC SIMULATION .............................................. 4 2.1 Experimental System................................................................................. 4

2.2 Computational Domains ............................................................................ 6

2.2.1 3-D Computational Domain ................................................................7

2.2.2 2-D Computational Domain ..............................................................11

2.3 Boundary Conditions............................................................................... 12

2.3.1 Boundary Conditions for 3-D Model ................................................12

2.3.2 Boundary Conditions for 2-D Model ................................................14

2.4 Numerical Simulations and Results ........................................................ 14

2.4.1 3-D Numerical Simulation with ANSYS HFSS ...............................14

2.4.2 3-D Numerical Simulation with COMSOL Multiphysics ................15

2.4.3 2-D Axial Symmetric Simulation with COMSOL Multiphysics ......18

2.5 Assumptions Based on Simulation Results ............................................. 21

CHAPTER 3. PLASMA SIMULATION BASED ON FÜNER’S MODEL ............. 22 3.1 Introduction to the Plasma Model ........................................................... 22

3.1.1 Material Properties Modification Due to Plasma Effects .................23

3.1.2 Füner’s Model of Electron Number Density .....................................24

3.1.3 Drift-diffusion Model of Electron Number Density .........................25

iv

Page

3.1.4 Heat Transfer Model .........................................................................26

3.1.5 Coupling of Models ..........................................................................29

3.2 Computational Model of the Plasma Model ........................................... 32

3.3 Boundary Conditions of the Plasma Model ............................................ 32

3.3.1 Boundary Conditions of Heat Transfer Simulation ..........................32

3.3.2 Boundary Conditions of the Drift-Diffusion Equation .....................33

3.4 Simulation Results of the Plasma Model ................................................ 34

3.4.1 Results of the EM Simulations ..........................................................34

3.4.2 Results of the UDF ............................................................................38

3.4.3 Results of the Heat Transfer Simulation ...........................................45

3.5 Validity Tests for Standing Wave and Sinusoidal Oscillation Field

Assumptions ................................................................................................................. 48

3.5.1 Validity Test for Standing Wave Assumption ..................................48

3.5.2 Validity Test for Sinusoidal Oscillation Field Assumption ..............50

CHAPTER 4. CONCLUSIONS AND FUTURE WORK .......................................... 52 4.1 Conclusions ............................................................................................. 52

4.2 Future Work ............................................................................................ 53

LIST OF REFERENCES .................................................................................................. 55 VITA ................................................................................................................................. 61

v

LIST OF TABLES

Table .............................................................................................................................. Page

Table 2.1 3-D mesh statistic.............................................................................................. 17

Table 2.2 2-D mesh statistic.............................................................................................. 20

Table 2.3 The comparison of solutions from all models .................................................. 20

Table 3.1 Electron including reactions and associated constant parameters [30] ............. 23

Table 3.2 Comparison of the EM simulation results ........................................................ 38

Table 3.3 Comparison of the ne simulation results ........................................................... 42

Table 3.4 Comparison of the Te simulation results .......................................................... 45

Table 3.5 Comparison of the Tg simulation results .......................................................... 48

vi

LIST OF FIGURES

Figure ............................................................................................................................. Page

Figure 1.1 Illustration of plasma environment in AX5200S MPCVD reactor ................... 2

Figure 2.1 The experimental system ................................................................................... 5

Figure 2.2 Schematic diagram of the MPCVD reactor at two stage positions [24]............ 6

Figure 2.3 Models of rectangular waveguide and TE-TM wave convertor ........................ 9

Figure 2.4 Cross-section of the plasma region.................................................................. 10

Figure 2.5 3-D computational domain .............................................................................. 10

Figure 2.6 2-D computational domain .............................................................................. 12

Figure 2.7 3-D model in HFSS including port .................................................................. 13

Figure 2.8 Electrical simulation results (HFSS 3-D) ........................................................ 15

Figure 2.9 3-D mesh built by COMSOL .......................................................................... 16

Figure 2.10 3-D mesh element quality histogram ............................................................. 17

Figure 2.11 Electrical simulation result (COMSL 3-D) ................................................... 18

Figure 2.12 2-D mesh built by COMSOL ........................................................................ 19

Figure 2.13 2-D mesh element quality histogram ............................................................. 19

Figure 2.14 Electrical simulation results (COMSL 2-D) .................................................. 21

Figure 3.1 Comparisons of original RHS and approximation .......................................... 28

Figure 3.2 Loop of solvers ................................................................................................ 29

Figure 3.3 Flow chart of algorithm ................................................................................... 31

vii

Figure ............................................................................................................................. Page

Figure 3.4 BCs of heat transfer simulation ....................................................................... 33

Figure 3.5 BCs of drift-diffusion model ........................................................................... 34

Figure 3.6 EM simulation results for 500 W input power ................................................ 35

Figure 3.7 EM simulation results for 400 W input power ................................................ 36

Figure 3.8 EM simulation results for 300 W input power ................................................ 36

Figure 3.9 Field strength variations on susceptor surface and along reactor axis ............ 37

Figure 3.10 ne simulation results for 500 W input power ................................................ 39

Figure 3.11 ne simulation results for 400 W input power ................................................ 39

Figure 3.12 ne simulation results for 300 W input power ................................................ 40

Figure 3.13 Electron density variations on susceptor surface and along reactor axis ...... 41

Figure 3.14 Te simulation results for 500 W input power ................................................ 42

Figure 3.15 Te simulation results for 400 W input power ................................................ 43

Figure 3.16 Te simulation results for 300 W input power ................................................ 43

Figure 3.17 Electron temperature variations on susceptor surface and along reactor axis 44

Figure 3.18 Tg simulation results for 500 W input power ................................................ 46

Figure 3.19 Tg simulation results for 400 W input power ................................................ 46

Figure 3.20 Heavy species temperature variations on susceptor surface and along reactor

axis .................................................................................................................................... 47

Figure 3.21 Test result for standing wave, with both diffusion and mobility enabled ..... 49

Figure 3.23 Comparison of the diffusion term and mobility term .................................... 50

Figure 3.24 Test result for sinusoidal oscillation field ..................................................... 51

viii

LIST OF SYMBOLS

𝐸�⃗ electrical field

𝜌 gas density

𝜀0 permittivity of vacuum

𝐵�⃗ magnetic field

𝑡 time

𝜇0 permeability of vacuum

𝐽 current density

𝑘𝑠(𝑖,𝑟,𝑑,𝑒) reaction rates

𝐸𝑠(𝑖,𝑟,𝑑,𝑒) threshold energy of reactions

𝜀𝑝 permittivity due to effects of plasma

𝜎𝑝 conductivity due to effects of plasma

𝜔𝑝 angular frequency of plasma

𝑣𝑚 collision rate between electron and heavy particles

𝜔 angular frequency of the microwave

𝑃𝑔 ambient pressure in reactor

𝑇𝑔 temperature of heavy species

𝑛𝑒 number density of electron

ix

𝐸𝑚 threshold field strength to sustain plasma

𝑛𝑒,𝑚𝑖𝑛 minimum number density of electrons estimated

Г�⃗ electron flux

𝑞 source of electrons

𝐷𝑒 diffusion coefficient of electrons

𝜇𝑒 mobility of electrons

𝐼𝑒 ionization rate

𝑅𝑒 recombination rate

𝑛𝑔 number density of heavy species

𝑘 Boltzmann constant

𝐾 thermal conductivity

𝑇𝑒 electron temperature

𝐽𝑒���⃗ electron flux on walls

𝑛�⃗ normal vector of a surface

𝑄𝑒 thermal energy of electrons

𝑢𝑒����⃗ velocity of electrons

𝑄𝑀𝑊 incident power from microwave

𝑄𝑐 energy loss due to electron including reactions

x

LIST OF ABBREVIATIONS

CNT carbon nanotube

CVD chemical vapor deposition

PACVD plasma assisted chemical vapor deposition

EM electromagnetic

MPCVD microwave plasma chemical vapor deposition

UDF user defined function

RHS right hand side

PDE partial differential equation

BC boundary condition

PEC perfect electrical conductor

xi

ABSTRACT

Huang, Di. M.S.A.A., Purdue University, August 2014. Numerical Simulation of Hydrogen Plasma in MPCVD Reactor. Major Professor: Alina Alexeenko. A numerical study was conducted to build a model able to estimate the plasma properties

under different working conditions for pure hydrogen plasma in an AX5200S MPCVD

reactor as part of the synthesis process of diamonds and graphitic nano-petals. A plasma

model based on standing wave assumption and a linear estimation of 𝑛𝑒 and coupled the

electromagnetic simulation, heat transfer simulation and calculations of plasma properties

was built in COMSOL Muitiphysics and tested with six different working conditions.

The reliability of COMSOL EM solver was tested through comparing the simulation

results with a benchmark EM solver, ANSYS HFSS. The validities of two assumptions

made about the electrical field, standing wave assumption and sinusoidal oscillation field

assumption, were tested by a PDE solver in COMSOL for utilizing the drift-diffusion

model of 𝑛𝑒. This numerical model estimated that electrical field ranged from ~9600 V/m

to ~12400 V/m, increased when power input increased and decreased when pressure

increased. The electron density 𝑛𝑒 ranged from 1.33e16 m-3 to 1.73e16 m-3, and electron

temperature 𝑇𝑒 ranged from 1.5 eV to 2.3 eV, both 𝑛𝑒 and 𝑇𝑒 increased when power input

increased and decreased when pressure increased. The gas temperature 𝑇𝑔 ranged from

383 K to 590 K, increased when either power input or pressure increased.

1

CHAPTER 1. INTRODUCTION

1.1 Background

Carbon nanostructures such as carbon nanotubes (CNTs), graphitic nano-petals and few-

layer graphene have desirable mechanical [1, 2], thermal [3, 4] and electrical [5-8]

properties; these properties have made applications such as hydrogen storage devices [9,

10], field emitters [11] and biosensors [12, 13] possible. For example, CNTs with a

coating of graphitic nano-petals are ideal for super capacitor applications because they

have been proven to be efficient nanostructures for maximizing the electrochemical

performance of MnO2 – a substance that is crucial to achieving high specific capacitance

and energy density [14]. Additionally, few-layer graphene has been found to be an

effective ultra-thin oxidation barrier coating in air [15] and under vigorous flow boiling

conditions [16].

Numerous techniques for growing the aforementioned carbon nanostructures have been

invented, such as exfoliation and cleavage [17], arc-discharge [18, 19], laser ablation [20],

thermal chemical vapor deposition (CVD) [21], and plasma-assisted chemical vapor

deposition (PACVD) [22]. This study focuses on the last technique because it provides an

efficient and relatively low-temperature synthesis [23] that is replicable and controllable.

2

During the synthesis process, pure H2 was introduced into the reactor. Microwave

generated by a microwave generator delivers the energy to ignite plasma. After the

environment of the reactor reaches a steady-state, N2 and CH4 gases are introduced into

the reactor. The plasma environment would enhance the dissociation of CH4; the

resultant carbon atoms/ions would deposit on the substrate and complete the synthesis.

Fig. 1.1 shows an illustration of the plasma environment during the synthesis process.

Figure 1.1 Illustration of plasma environment in AX5200S MPCVD reactor

1.2 Motivation

The growth rate and quality of the carbon nanostructures are proven to be highly

dependent on the number density of atomic hydrogen, H, and methyl, CH3, on the surface

of the substrate [25]. These particles are produced by electron including reactions in the

plasma environment, which are closely related to the plasma properties (electron

temperature, 𝑇𝑒 , electron number density, 𝑛𝑒 , and heavy particle temperature, 𝑇𝑔 ). A

numerical model able to estimate these properties is highly desirable for studying the

3

response of plasma properties to different working conditions and for optimizing the

synthesis.

COMSOL Multiphysics was found to be the most suitable numerical tool for this study

because of its ability to simulate all physics phenomena required in a plasma modeling in

a fully coupled manner. A model for the pure hydrogen plasma environment before the

introduction of N2 and CH4 was built for this study to test the accuracy of the governing

equations and boundary conditions (BCs), to understand the validity of the assumptions

and to gain experience of multi-physics modeling with COMSOL.

4

CHAPTER 2. ELECTROMAGNETIC SIMULATION

2.1 Experimental System

In this study, a SEKI AX5200S MPCVD reactor powered by an ASTeX AX2100

microwave generator with up to 1.5 kW (2.45 GHz) output power was used for

synthesizing carbon nanotubes, graphene and graphitic nanopetals over a variety of

substrates under different growing conditions [24]. The experimental system is shown in

Fig. 2.1.

The microwave power was transmitted by a rectangular waveguide, which included three

stabs able to change the internal geometry of the waveguide in order to minimize the

reflection loss of the incident power, from the generator in TE propagating mode and was

converted to a TM mode by a mode convertor structure on top of the reactor. The mode

convertor was bounded by a quartz plate to insulate plasma in a certain volume.

5

Figure 2.1 The experimental system

The plasma region under the quartz plate included gas inlets, a graphite susceptor and a

gas outlet connected to an external mechanical pump. The gas inlets included three inlet

pipes for H2, N2 and CH4 respectively. Each kind of gas could be set to a specific mass

flow rate (in sccm) to optimize the synthesizing. The graphite susceptor could move

along the axis of the reactor away and toward from the quartz window. Fig. 2.2 shows a

schematic diagram of this reactor. A substrate is introduced on the graphite susceptor

stage through a hatch window. As shown in Fig. 2.2, the susceptor is accessible at a stage

height of 0 mm [24], while when plasma was ignited, this height was set to 53 mm [24]

above the original position. The external mechanical pump kept the internal pressure of

this reactor at a specific value required by the synthesizing.

6

Figure 2.2 Schematic diagram of the MPCVD reactor at two stage positions [24]

2.2 Computational Domains

The goal of this study is to build a numerical model that is able to predict the plasma

properties under different working conditions of carbon nano-structure growth inside a

microwave plasma chemical vapor deposition (MPCVD) reactor. Since the plasma was

ignited by microwave power, all plasma properties to be solved were highly dependent on

the electromagnetic (EM) field around plasma region; it is necessary to obtain a solution

of EM field as accurate as possible. However, in order to simplify the starting stage of

this study, all effects on the electromagnetic field due to the appearance of plasma were

ignored therefore a model for pure EM simulation was set up. The governing equations

for this simulation are Maxwell equations:

∇ ∙ 𝐸�⃗ =𝜌𝜀0

∇ ∙ 𝐵�⃗ = 0

∇ × 𝐸�⃗ = −𝜕𝐵�⃗𝜕𝑡

7

∇ × 𝐵�⃗ = 𝜇0(𝐽 + 𝜀0𝜕𝐸�⃗𝜕𝑡

)

where 𝐸�⃗ is the electrical field, 𝐵�⃗ is the magnetic field, 𝜌 is the electrical charge density,

𝜀0 is the permittivity for vacuum, 𝜇0 is the permeability for vacuum, 𝐽 is the local current

density and 𝑡 stands for time. These governing equations were solved in proper

computational domains, resolved the main elements of the experimental system, to

simulate the EM field inside the reactor.

2.2.1 3-D Computational Domain

In order to reduce the complexity of the numerical simulation, the experimental system

was modeled starting from the rectangular waveguide after the three tuning stabs and

some details like the hatch window and the substrate that would not affect the simulation

results as much as others were not included. The computational domain was divided into

four sub-domains: the rectangular waveguide, the TE-TM wave convertor, the quartz

plate and the plasma region.

The rectangular waveguide was modeled as a WR-340 standard calibrated for a work

band of 2.20-3.30 GHz [26]. The cross-section dimension was 86.36 mm X 43.18 mm

[26]; while its length, measured from the schematic diagram from Ref. 27, was set to 297

mm. Vacuum was assigned to the internal volume of the waveguide indicate the free

space inside.

8

The TE-TM wave convertor was able to convert the TE mode microwave propagating

inside the rectangular waveguide to a TM mode microwave necessary for the axial

symmetric structures (rest of the MPCVD reactor). The convertor included a lower

cylindrical perfect electrical conductor (PEC) shell, a PEC hat and a coaxial transmission

line. The lower cylindrical PEC shell had a diameter of 120 mm [27] and height of 142

mm [27]. The PEC hat, placed above the rectangular waveguide, had a diameter of 60

mm and a height of 20.8 mm. The transmission line was constituted by a 30 mm diameter

outer PEC shell and a 10 mm inner diameter inner PEC cylinder; the height of the outer

shell was 5 mm and the height of the inner conductor was 150 mm. In addition, the outer

shell was placed on top of the lower PEC shell while the inner PEC cylinder started from

the same level of the top of the PEC hat. Dimensions of the transmission line and PEC

hat were estimated by measurement of the schematic diagram from Ref. 27. Vacuum was

also assigned to the internal volume of the convertor to indicate free space except for the

inner PEC cylinder of the transmission line, assigned as PEC. Fig. 2.3 shows the

computation models of rectangular waveguide and TE-TM wave convertor.

9

Figure 2.3 Models of rectangular waveguide and TE-TM wave convertor

Below the convertor, a 120 mm diameter [28] quartz plate was introduced to insulate the

plasma region from the convertor. The thickness of the quartz plate was 10 mm,

estimated by measurement of the schematic diagram from Ref. 28. The plasma region

below the quartz plate included a 140 mm diameter [28], 162.2 mm height [28]

cylindrical PEC shell and a 120 mm diameter [28], 12.2 mm thick [28] susceptor stage is

placed 20 mm [28] above the bottom of the PEC cylindrical shell, 2 mm fillets were

added to both edges of the susceptor to reduce the field concentration due to the sharp

edge. The inlet and exit of gases were placed on top and bottom of the plasma region

respectively.

For the pure EM simulation, vacuum was assigned to the internal volume of the plasma

region because plasma effects were temporarily not under consideration while quartz was

10

assigned to the quartz plate. Fig. 2.4 shows the cross-section of the plasma region

including the quartz plate and Fig. 2.5 shows the entire 3-D computational domain.

Figure 2.4 Cross-section of the plasma region

Figure 2.5 3-D computational domain

11

2.2.2 2-D Computational Domain

Consider the plasma simulation required the couple of EM solver, heat transfer solver and

UDFs, it would be a time consuming process to simulate the entire 3-D model with the

three solvers. The fact that the plasma region is cylindrical allows physics only in that

region (heat transfer and plasma) to be simulated in a 2-D axially symmetric

computational domain. However the possibility of simulating EM field in a 2-D model

was not guaranteed and it was necessary to be tested by comparing results with 3-D

models.

This 2-D model excluded the entire rectangular waveguide and some part of the TE-TM

wave convertor. This model started from the top surface of the outer PEC shell of the

coaxial transmission line and was identical to the axial symmetric part of the 3-D model.

Material properties were also identical as used in the 3-D model. Fig. 2.6 shows the 2-D

computational domain.

12

Figure 2.6 2-D computational domain

2.3 Boundary Conditions

The required inputs for the EM solver were material properties and boundary conditions

(BCs). Material properties were assigned to each sub-domain as previously described.

The BCs should also be properly assigned to assure the accuracies of the results.

2.3.1 Boundary Conditions for 3-D Model

ANSYS HFSS automatically assigned PEC to all boundaries as a default setting.

However, in order to indicate the microwave was transferred into the waveguide, a

microwave port BC was assigned to the right end of the rectangular waveguide. HFSS

13

would automatically obtain the proper port mode according to the dimensions of the

waveguide and the frequency of the incident microwave. For the rest boundaries, the

default PEC BCs were kept to indicate EM field could only be normal to the boundary

[29]. Fig. 2.7 indicates the port BC in HFSS.

Figure 2.7 3-D model in HFSS including port

The BCs for 3-D simulations in COMSOL Multiphysics were identical as they were

assigned in HFSS except for the port. The port located on the same surface in COMSOL

as it was in HFSS, but the port mode, incident power and port phase needed to be set

manually. It was set to a TE-10 rectangular port with 500 W incident power and a port

phase equaled to π rad.

14

2.3.2 Boundary Conditions for 2-D Model

Since the 2-D model started from top of the outer PEC shell of the transmission line, the

rectangular port did not exist in this model. The port of the 2-D model was placed on the

top surface of the transmission line included in this model. The port mode was set to

coaxial which always transmitted in a TEM mode and did not require a mode number.

The incident power was 500 W and the port phase was 0 rad. The port for this 2-D model

is labeled in Fig. 2.6.

2.4 Numerical Simulations and Results

The numerical simulations for EM field run on the previously described computational

domains by both HFSS and COMSOL. Since HFSS was commonly used and treated as a

reliable EM solver, the results from HFSS were utilized as a benchmark, while results

from COMSOL were compared with the benchmark to test the reliability of EM solver

built in COMSOL.

2.4.1 3-D Numerical Simulation with ANSYS HFSS

Before simulating, the tested frequency was set to 2.45 GHz which was equal to the

frequency of the microwave generator. The maximum iteration number was set to 25 and

the allowed tolerance was set to 0.03% with the consideration of the balance between

accuracy and computational time. ANSYS HFSS would automatically generate the mesh

with tetrahedral elements and refine it during iterations until a proper mesh size was

obtained to complete the simulation. At last, a scale factor of 500 was needed to be

included in the “edit source” option to resolve the 500 W incident power.

15

Figure 2.8 Electrical simulation results (HFSS 3-D)

The simulation results from HFSS are shown in Fig. 2.8. From the left part of the figure,

a concentration of electrical field could be observed at the center reactor, just above the

susceptor and the maximum value of the field on top surface of the susceptor was ~13000

V/m. In the right part of the figure, the stream line indicated the direction of the field. The

field pointed toward the susceptor would push the positive ions (e.g. CH3+, CH2

+, etc.)

toward the substrate during the synthesis process, and keep the nano-petals continuously

growing. This result agrees with the qualitative understanding of MPCVD mechanism.

2.4.2 3-D Numerical Simulation with COMSOL Multiphysics

In COMSOL, mesh needed to be built manually, and was set to a physics-controlled

mesh with extremely fine mesh size; while the iteration numbers and the tolerance were

managed by COMSOL itself. The mesh method was “Free Tetrahedral” for 3-D domains

that COMSOL would fill the internal volume by tetrahedrons with sizes according to

mesh size. The mesh element quality was defined by:

16

𝑞3−𝐷 =72√3𝑉

(ℎ12 + ℎ2

2 + ℎ32 + ℎ4

2 + ℎ52 + ℎ6

2)1.5

Where V is the volume of the tetrahedron; h1-h6 are the edge lengths. 𝑞3−𝐷 measures the

similarity of a mesh element to a regular tetrahedron, the value is better to be close to 1; a

low mesh element quality may potentially cause convergence issues during simulation.

Fig. 2.9 shows the mesh built by COMSOL; Table 2.1 shows the mesh statistic and Fig.

2.10 shows the mesh element quality histogram.

Figure 2.9 3-D mesh built by COMSOL

17

Figure 2.10 3-D mesh element quality histogram

Table 2.1 3-D mesh statistic

Number of elements 1,380,585

Min element quality 0.1369

Ave. element quality 0.7457

Mesh volume 4,956,000 mm3

The simulation result from COMSOL is shown in Fig. 2.11. By comparing Fig. 2.8 and

Fig. 2.11, the results from both computational tools were qualitatively similar (consider

the position of field concentration and the field direction). However, the maximum value

of the field on top surface of the susceptor was ~16000 V/m. There was a 23.08%

difference between COMSOL and HFSS 3-D simulation results. The surface average of

the field on the susceptor surface, 6643 V/m, was also evaluated by COMSOL.

18

Figure 2.11 Electrical simulation result (COMSL 3-D)

2.4.3 2-D Axial Symmetric Simulation with COMSOL Multiphysics

The 2-D mesh was also built as physics controlled mesh with extremely fine size. The

mesh method was “Free Triangular” that COMSOL would fill the internal area of the 2-D

domains by triangles with sizes according to mesh size. The mesh element quality for a 2-

D mesh was defined by:

𝑞2−𝐷 =4√3𝐴

(ℎ12 + ℎ2

2 + ℎ32)

It measures the similarity of a mesh element to regular trangle, and the criteria is identical

as 3-D mesh. Fig. 2.12 shows the 2-D mesh built by COMSOL; Table 2.2 shows the

mesh statistic and Fig. 2.13 shows the mesh element quality histogram.

19

Figure 2.12 2-D mesh built by COMSOL

Figure 2.13 2-D mesh element quality histogram

20

Table 2.2 2-D mesh statistic

Number of elements 7,138

Min element quality 0.7468

Ave. element quality 0.9786

Mesh area 18,170 mm2

The 2-D simulation result is shown in Fig. 2.14. By comparing Fig. 2.11 and Fig. 2.14

and Fig. 2.8, the 2-D simulation results agreed with the 3-D ones qualitatively in both

field concentration and field direction. The maximum value of field on top surface of the

susceptor was ~12400 V/m and the average over the top surface of the susceptor was

7358 V/m. The difference in maximum value was 5.38% compared to the 3-D simulation

results from HFSS. Table 2.3 includes the comparison among all three computational

models simulated in this chapter. The comparison confirmed the possibility to simulate

the EM field within a 2-D axial symmetric model.

Table 2.3 The comparison of solutions from all models

HFSS 3-D COMSOL 3-D COMSOL 2-D Max field on

susceptor ~13,000 V/m ~16,000 V/m ~12,400 V/m

Averaged field on susceptor 6,643 V/m 7,358 V/m

Diff. in max field (compare with HFSS) 23.08% 5.38%

Diff. in ave. field (compare with COMSOL 3-D)

10.76%

21

Figure 2.14 Electrical simulation results (COMSL 2-D)

2.5 Assumptions Based on Simulation Results

Both HFSS and COMSOL simulated the EM field in frequency domain, therefore the

time dependent nature of the EM field could not be resolved by the results obtained. In

order to utilize the EM solver in the further studies, two assumptions were made based on

the simulation results. The first one, named standing wave assumption, assumed the EM

wave in the reactor would be standing waves everywhere after several reflections;

thereby the direction and magnitude of the EM field would not change in time. The

second assumption, named sinusoidal oscillation field, assumed the field in the reactor

would oscillate sinusoidally with an amplitude equaled the simulation results. The

validitiy of both assumptions were tested in the further stages of this study.

22

CHAPTER 3. PLASMA SIMULATION BASED ON FÜNER’S MODEL

3.1 Introduction to the Plasma Model

In this stage of study, the present of plasma was introduced to the model. Physically,

plasma is a region of ionized gas; and specifically in this study, the ionization level is

around 10-6; and from the experimental observation, the electron density (also the

positive ion density) is at the order of 1016 [ 1𝑚3]. The actual electron density can be

solved by either a drift-diffusion equation or an algebraic simplification called Füner’s

Law [32-34].

Since electrons and ions exist in the plasma region, the gas becomes conductive and

therefore materials properties related to the electrical field (conductivity and permittivity)

will be modified. The electrons under an external electrical field will be accelerated and

collide with other species (ions and neutral molecules); to resolve this phenomenon,

collisional reactions with the energy transfers associated should be under consideration.

In this study, four dominant reactions are modeled, shown in the following Table 3.1.

23

Table 3.1 Electron including reactions and associated constant parameters [30]

Reactions ks(i,r,d,e) [m3/s] Es

(i,d,e) [eV] e + H2 2e + H2

+ 10-14 15.4 e + H2 e + H2

* 6.5 X 10-15 12.0 e + H2 e + 2H 10-14 10.0

H2+ + e H2 10-13

In this table, 𝑘𝑠 is the reaction rate constant and 𝐸𝑠 is the threshold energy associated with

the reaction. The last reaction represents the recombination which will happen once an

electron collides with a H2+ ion, so it does not require threshold energy. Collisions

besides these four kinds are considered as pure energy transfer to neutral species and will

be modeled by a volumetric heat source in a heat transfer solver. However, the

temperature change will affect the collision rate between species in the reactor; therefore

the heat transfer phenomenon should be coupled in the model. Eventually, the plasma

model includes three aspects of simulations, the electromagnetic, the material properties

change due to presence of plasma and the heat transfer.

3.1.1 Material Properties Modification Due to Plasma Effects

The material properties that need to be modified in the electromagnetic simulation due to

plasma effects were conductivity, σp, and permittivity, ԑp. The modifications were given

by the following equations: [31]

𝜎𝑃 =𝜔𝑃

2𝜀0𝑣𝑚(𝜔2 + 𝑣𝑚2)

𝜀𝑃 = 1 −𝜔𝑃

2

(𝜔2 + 𝑣𝑚2)

24

Where 𝜔𝑃 is the angular frequency of plasma, 𝑣𝑚 is the collisional frequency between

electrons and other species and 𝜔 is the angular frequency of the microwave. These

parameters were evaluated by the following equations [30, 31]:

𝜔𝑃 = �𝑒2𝑛𝑒𝜀0𝑚𝑒

𝑣𝑚 = 1.08 × 1010 × �𝑃𝑔𝑇𝑔� 𝑠−1

𝜔 = 2𝜋 × 2.45𝐺𝐻𝑧 = 1.53938 × 1010𝑠−1

where 𝑒 is the electron charge, 𝑛𝑒 is the electron number density and is discussed in Sec.

3.1.2 and Sec. 3.1.3, 𝑚𝑒 is the mass of electron, Pg is ambient pressure in the plasma

region and Tg is the ambient temperature discussed in Sec. 3.1.4.

3.1.2 Füner’s Model of Electron Number Density

The electron number density is related to the electrical field and other parameters in a

complicated method which will be discussed in Sec. 3.1.3; however, this section

introduces an algebraic simplification developed by Füner, et al. [32-34]. It states that the

local 𝑛𝑒 is only related to the local electrical field strength linearly.

𝑛𝑒 = �𝛾 × ��𝐸�⃗ � − 𝐸𝑚� + 𝑛𝑒,𝑚𝑖𝑛 (𝑓𝑜𝑟 �𝐸�⃗ � > 𝐸𝑚)0

where γ = 3 × 1012 [ 1Vm2] [38], Em = 10000 [V

m] and ne,min = 1 × 1016 [ 1

m3] . Em and

𝑛𝑒,𝑚𝑖𝑛 are calibrated with the experimental observation for this study. This simplification

was applied for a steady state simulation of 𝑛𝑒 which required a time-independent

electrical field; therefore, the standing wave assumption was applied.

25

3.1.3 Drift-diffusion Model of Electron Number Density

In reality, 𝑛𝑒 is dependent not only on the electrical field strength. A more accurate

model is proved by the drift-diffusion equation [35]:

𝜕𝑛𝑒𝜕𝑡

+ 𝑑𝑖𝑣Г�⃗ = 𝑞

where Г�⃗ is the local electron flux and 𝑞 is the volumetric local electron source. These

terms are given by the following equations [35]:

Г�⃗ = −𝐷𝑒𝛻𝑛𝑒 − 𝜇𝑒𝑛𝑒𝐸�⃗

𝑞 = 𝐼𝑒 − 𝑅𝑒

The first term of Г�⃗ indicates the electron flux due to the diffusion effect; while the second

term indicates the flux due to electron motion forced by the external electrical field. For

hydrogen, the electron diffusion coefficient, De = 1.3×105

Pg[torr] �cm2

s� [39], the electron mobility,

µe = 0.37×106

Pg[torr] �cm2

Vs� [39]; while Ie and Re are the ionization and recombination terms

given by [30]:

Ie = nengksi exp �−

Esi

Te�

𝑅𝑒 = 𝑛𝑒2𝑘𝑠𝑟

𝑘𝑠𝑖 , 𝑘𝑠

𝑟 and 𝐸𝑠𝑖 are reaction rate coefficients and ionization threshold energy listed in

Table 3.1; 𝑛𝑔 = 𝑃𝑔𝑘𝑇𝑔

is the number density of neutral molecule (H2), where 𝑘 is the

Boltzmann constant and 𝑇𝑒 is the electron temperature discussed in Sec. 3.1.4.

26

This drift-diffusion model is a time-dependent model, but can also simplified to a steady

state one by removing the time derivative term, 𝜕𝑛𝑒𝜕𝑡

. Both versions of this model were

tested and discussed in Sec. 3.5.

3.1.4 Heat Transfer Model

Since plasma contains electrons and heavy species (ions and neutral molecule) and the

velocity of these two components are very different, the temperatures to measure the

thermal motion of particles should be separated into electron temperature, 𝑇𝑒 , and

temperature of heavy species, 𝑇𝑔. 𝑇𝑔, also known as ambient temperature in the plasma

region is simulated by a conductional heat transfer model with a volumetric heat source

given by [36]:

∇�𝐾∇𝑇𝑔� + 𝑄𝑔𝑎𝑠 = 0

𝑄𝑔𝑎𝑠 = 3𝑚𝑒𝑛𝑒𝑣𝑚𝑘�𝑇𝑒 − 𝑇𝑔�

𝑚𝐻2

where 𝐾 is the thermal conductivity and 𝑚𝐻2 is the mass of a hydrogen molecule. The

electron temperature was calculated through the coupling of electron energy and the

microwave power and was given by [30]:

𝜕𝑄𝑒𝜕𝑡

+ ∇(𝑄𝑒𝑢𝑒����⃗ ) = 𝑄𝑀𝑊 − 𝑄𝑐

where 𝑄𝑒 is the thermal energy of electron, 𝑢𝑒����⃗ is the velocity of electron, 𝑄𝑀𝑊 is the

incident power from microwave and 𝑄𝑐 is the power loss to the collisions (e.g. power

consumed by the electron including reactions). Under typical synthesis conditions,

∇(𝑄𝑒𝑢𝑒����⃗ ) is proven to be ten thousand times less than the incident power [30], and under

27

steady-state, the time derivative 𝜕𝑄𝑒𝜕𝑡

is zero. The model to estimate electron energy would

be simplified as:

𝑄𝑀𝑊 = 𝑄𝑐

Substituted with the expression for 𝑄𝑀𝑊 and 𝑄𝑐:

𝑄𝑀𝑊 =𝑒2

2𝑚𝑒

𝑣𝑚𝜔2 + 𝑣𝑚2 �𝐸�⃗ �

2; 𝑄𝑐 = �𝐸𝑠𝑛𝑠𝑘𝑠 𝑒𝑥𝑝 �−

𝐸𝑠𝑇𝑒�

𝑠

The equation to obtain electron temperature was given by:

𝑒2

2𝑚𝑒

𝑣𝑚𝜔2 + 𝑣𝑚2 �𝐸�⃗ �

2= �𝐸𝑠𝑛𝑠𝑘𝑠 𝑒𝑥𝑝 �−

𝐸𝑠𝑇𝑒�

𝑠

The summation was over the first three reactions in Table 3.1. However, the right hand

side (RHS) of the equation has three terms and was hard to solve for 𝑇𝑒 , a one term

approximation was made for the RHS [30]:

𝑝𝐿 𝑒𝑥𝑝 �−𝐸𝐿𝑇𝑒𝛼𝐿

� = �𝐸𝑠𝑛𝑠𝑘𝑠 𝑒𝑥𝑝 �−𝐸𝑠𝑇𝑒�

𝑠

where EL = 18.5327 [eVαL] [30], αL = 0.36757 [30] and pL is evaluated by [30]:

𝑝𝐿 = 2.464 × 1015 × �𝑃𝑔

160 [𝑡𝑜𝑟𝑟]� × �1000 [𝐾]

𝑇𝑔� [𝑒𝑉/𝑠]

28

Figure 3.1 Comparisons of original RHS and approximation

A comparison of the one term approximation and original RHS is shown in Fig. 3.1; it is

shown that the approximation is close enough to the original RHS in the region of this

study (𝑇𝑒 ≈ 2.5 [𝑒𝑉]). With this approximation, the 𝑇𝑒 can be expressed as [30]:

𝑇𝑒 = {𝐸𝐿

ln [𝑝𝐿/𝑝𝑎𝑏𝑠]}1/𝛼𝐿

where 𝑝𝑎𝑏𝑠 is a function of the electrical field strength [30]:

𝑝𝑎𝑏𝑠 =𝑒2

2𝑚𝑒

𝑣𝑚𝜔2 + 𝑣𝑚2 �𝐸�⃗ �

2

29

3.1.5 Coupling of Models

The multi-physics coupling of this plasma model was achieved through the COMSOL

Multiphysics. In this model, the coupling meant solutions from one solver will transferred

to another as an input; and solvers would form a loop, iterated until self-consistent

solutions were obtained. Fig. 3.2 shows the loop of solvers and the solutions transferred

among them. In this stage of modeling the multi-physics coupling, only Füner’s Model

was included in the loop; while the drift-diffusion model was simulated outside of the

loop and was only for testing the validities of the two assumptions made in chapter 2.

Figure 3.2 Loop of solvers

The first step of this model was to obtain an electromagnetic solution from a EM solver

with plasma effects estimated by some initial guesses. This solution was then used to

calculate 𝜎𝑃 , 𝜀𝑃 , 𝑛𝑒 , 𝑄𝑔𝑎𝑠 , 𝑇𝑒 and all the parameters associated with them. The

calculations were completed in UDFs and indicated the beginning of the loop of solvers.

The solutions from the EM solver and the UDFs were then transferred to a heat transfer

30

solver to calculate 𝑇𝑔. After that, the updated 𝑇𝑔 was transferred to the second UDFs and

parameters related to 𝑇𝑔 get renewal. At last, the latest updated 𝑇𝑔 and UDFs solutions

were transferred to the other electromagnetic solver which solves the field with the

effects on 𝜎𝑃 and 𝜀𝑃 . With the end of the loop, solution from the second EM solver,

instead of the pure EM solver, was transferred to the beginning of the loop and the next

iteration started. In order to obtain self-consistent solutions, this solver loop should be run

several times until the current solution was identical as the previous one. Specifically for

this plasma model, number of iterations should be equal or larger than five. Fig. 3.3

shows a flow chart of the algorithm of this model.

31

Figure 3.3 Flow chart of algorithm

Finally, after completed the simulations based on Füner’s Model, the solutions were

transferred to a PDE solver for the drift-diffusion equation. The solutions from the PDE

solver were used to examine the validities of the standing wave assumption and the

sinusoidal oscillation field assumption.

32

3.2 Computational Model of the Plasma Model

The geometrical setup of the computational model were identical as the 2-D simulation of

pure EM field in COMSOL Multiphysics, described in chapter 2, including the sub-

domain setups and mesh setup. However, since the effects on material properties due to

plasma and heat transfer phenomenon were under consideration in the plasma region, the

material used in that sub-domain should be replaced. A user defined gas based on the

hydrogen built in COMSOL material library was introduced to the plasma region. The

conductivity and relative permittivity were set to be 𝜎𝑃 and 𝜀𝑃 calculated by the UDFs

described in Sec. 3.1, and COMSOL automatically completed the evaluation of the

thermal properties of the material.

3.3 Boundary Conditions of the Plasma Model

The plasma model contained three aspects of simulation and each one required a proper

set of boundary conditions. The boundary conditions for EM simulation are identical as

the pure EM simulation described in chapter 2; and the UDFs were a set of algebraic

equations which did not require boundary conditions. BCs of heat transfer simulation and

the drift-diffusion equation were discussed in this section.

3.3.1 Boundary Conditions of Heat Transfer Simulation

The boundaries of the heat transfer simulation contained three parts, the walls of the

reactor, the gas inlet and the gas exit. The walls of the reactor were modeled as thermal

insulations that there was no heat flux through the walls. The gas inlet was modeled as a

constant temperature boundary with 𝑇𝑔 = 293.15 [𝐾], the room temperature; and the gas

33

exit was modeled as an outflow surface that the heat generated by the heat source left the

reactor through the surface to maintain a steady temperature in it. Fig. 3.4 shows the

boundary conditions of the heat transfer simulation.

Figure 3.4 BCs of heat transfer simulation

3.3.2 Boundary Conditions of the Drift-Diffusion Equation

In the drift-diffusion model of electrons, the boundaries of the plasma region were

assumed to be perfect absorption wall that no electron was reflected back to the plasma

region when it hit the boundary. This assumption was equivalent to a free boundary to

electron which meant all electrons would pass through the boundary without any

resistance. The flux on a free boundary [36] was set to be the BC for this model:

𝐽𝑒 =14

(8𝑘𝑇𝑒𝜋𝑚𝑒

)1/2𝑛𝑒𝑛�⃗

34

Where Je is the electron flux, n�⃗ is the normal unit vector of a boundary. Fig. 3.5 shows

the BCs of the drift-diffusion equation.

Figure 3.5 BCs of drift-diffusion model

3.4 Simulation Results of the Plasma Model

Six experiments were run on the plasma system at different power levels (300 W, 400 W

and 500 W) and pressure inside the reactor (10 torr and 30 torr). In order to compare with

the experimental results, six computational cases of the plasma model were tested with

the same operating conditions as the experiments.

3.4.1 Results of EM Simulations

Fig. 3.6 – Fig. 3.8 show the results of EM simulations from the six cases. Each of them

was qualitatively similar with the simulation result shown in chapter 2 in both position of

the field concentration and the field direction. Table 3.2 includes maximum field strength

35

on top surface of the susceptor and average value of field over the top surface of

susceptor for each case. Fig. 3.9 shows the extracts of the field strength on the top surface

of susceptor and along the axis of the reactor. Electrical field strength on top surface of

the susceptor increased when input power increased but decreased when reactor pressure

increased. These trends were to be expected because an increase in input power enhanced

the energy density in the reactor, therefore, the field strength; while an increase in

pressure elevated the collision rate and thereby increased the energy loss of electrons due

to collisions, would result in a reduction of number density of electron. This effect gave

an increment in permittivity as feedback and eventually reduced the field strength.

Figure 3.6 EM simulation results for 500 W input power

36

Figure 3.7 EM simulation results for 400 W input power

Figure 3.8 EM simulation results for 300 W input power

37

Figure 3.9 Field strength variations on susceptor surface and along reactor axis

38

Table 3.2 Comparison of the EM simulation results

10 torr 30 torr

Max Ave. Max Ave.

500 W 12,450 V/m 7,384 V/m 12,400 V/m 7,339 V/m

400 W 11,150 V/m 6,591 V/m 11,110 V/m 6,576 V/m

300 W 9,630 V/m 5,696 V/m 9,633 V/m 5,699 V/m

3.4.2 Results of the UDF

Fig. 3.10 – Fig. 3.12 show the results of ne, Fig. 3.14 – Fig. 3.16 show the results of Te

from the UDFs. In 400 W and 500 W cases, both ne and Te concentrated above the center

of the susceptor, where the plasma was expected to exist. However, in the 300 W cases,

ne and Te were almost zero in the plasma region, indicated that plasma did not ignite in

these cases. Table 3.3 and Table 3.4 include maximum value on top surface of the

susceptor and the average value over the top surface of susceptor of 𝑛𝑒 and 𝑇𝑒. Fig. 3.13

and Fig. 3.17 show the extracts of the electron density and electron temperature on the

top surface of susceptor and along the axis of the reactor. Both of them increased when

the input power increased and decreased when the pressure increased. These trends

behaved similarly as the electrical field which was to be expected because they were

positively correlated to the electrical field strength as described by Füner’s model and the

model to estimate 𝑇𝑒.

39

Figure 3.10 ne simulation results for 500 W input power

Figure 3.11 ne simulation results for 400 W input power

40

Figure 3.12 ne simulation results for 300 W input power

41

Figure 3.13 Electron density variations on susceptor surface and along reactor axis

42

Table 3.3 Comparison of the ne simulation results

10 torr 30 torr

Max Ave. Max Ave.

500 W 1.73e16 1𝑚3 6.60e14 1

𝑚3 1.71e16 1𝑚3 6.47e14 1

𝑚3

400 W 1.34e16 1𝑚3 2.78e14 1

𝑚3 1.33e16 1𝑚3 2.78e14 1

𝑚3

300 W

Figure 3.14 Te simulation results for 500 W input power

43

Figure 3.15 Te simulation results for 400 W input power

Figure 3.16 Te simulation results for 300 W input power

44

Figure 3.17 Electron temperature variations on susceptor surface and along reactor axis

45

Table 3.4 Comparison of the Te simulation results

10 torr 30 torr

Max Ave. Max Ave.

500 W 2.279 eV 0.153 eV 1.684 eV 0.134 eV

400 W 2.029 eV 0.089 eV 1.477 eV 0.081 eV

300 W

3.4.3 Results of the Heat Transfer Simulation

Fig. 3.18 and Fig. 3.19 show the results of the heat transfer simulations exclude the 300

W cases with no plasma ignition. Temperature reached its highest value just above the

center of the susceptor because of the highest Te there generates the largest value of heat

source, and reduced in a radial manner because of the thermal diffusion. Table 3.5

includes maximum value on top surface of the susceptor and the average value of 𝑇𝑔

over the top surface of susceptor. Fig. 3.20 shows the extracts of the heavy species

temperature on the top surface of susceptor and along the axis of the reactor. 𝑇𝑔

increased when the input power increased, which was to be excepted because an increase

in the input power raised 𝑇𝑒 and more energy was available to transfer from electron to

heavy species. 𝑇𝑔 also increased when the pressure increased, which agreed with

exception as well, because an increase in pressure enhanced the collisions between

electrons and heavy particles, therefore, caused more energy transferred from electrons.

46

Figure 3.18 Tg simulation results for 500 W input power

Figure 3.19 Tg simulation results for 400 W input power

47

Figure 3.20 Heavy species temperature variations on susceptor surface and along reactor axis

48

Table 3.5 Comparison of the Tg simulation results

10 torr 30 torr

Max Ave. Max Ave.

500 W 483.4 K 409.3 K 591.1 K 477.8 K

400 W 384.7 K 341.8 K 447.0 K 375.5 K

3.5 Validity Tests for Standing Wave and Sinusoidal Oscillation Field Assumptions

In order to move on to the next stage of this study, the drift-diffusion model of 𝑛𝑒 was

expected to be coupled into the solver loop instead of the Füner’s model. Since the

second term of the local electron flux (the flux due to forced electron motion), described

in Sec. 3.1.3, was highly sensitive to the direction and magnitude of the electrical field in

the reactor, the validity of the assumptions on those two factors became very important.

The validity were tested by a PDE solver set to solve the drift-diffusion equation with the

solutions based on Füner’s model from a 500 W input power, 10 torr reactor pressure

case as initial conditions.

3.5.1 Validity Test for Standing Wave Assumption

The first step of this test was to solve the steady-state version of the drift-diffusion

equation with both terms of the local electron flux. Part a) of Fig. 3.21 shows the solution.

This solution did not have a concentration region of 𝑛𝑒 which did not agree with either

the experimental observation or the qualitative expectation.

49

Figure 3.21 Test result for standing wave, with both diffusion and mobility enabled

A further test was setup by shutting down the forced motion term in the equation, to find

out the causes of the lack of concentration. Part b) of Fig. 3.21 shows the solution. This

solution included a concentration of 𝑛𝑒 in the proper region, which indicated that the

forced motion of electron mainly contributed to the lack of concentration; in other word,

the standing wave assumption was not valid for modeling the forced electron motion.

The next question was on the possibility to ignore the effects due to forced electron

motion in the drift-diffusion model. In order to examine the weight of importance of the

two terms in Г�⃗ , the magnitudes of the terms needed to be compared. Fig. 3.23 plots the

magnitudes of the diffusion term and forced electron motion term in the plasma region.

Through comparison, the forced electron motion term was about two orders of magnitude

larger than the diffusion, which indicated it was impossible to ignore the effects due to

50

forced electron motion. All three steps of the validity test stated that the standing wave

assumption was not valid for the drift-diffusion model.

Figure 3.22 Comparison of the diffusion term and mobility term

3.5.2 Validity Test for Sinusoidal Oscillation Field Assumption

In this test, the electrical field solution from the simulation based on Füner’s model was

multiplied by a factor of sin(𝜔𝑡) to resolve the sinusoidal oscillation of field inside the

reactor. Two steps of tests were setup by different time steps and target times. The first

one simulates 𝑛𝑒 from 0 to 3 × 10−7𝑠 with a time step of 3 × 10−11𝑠, and the second on

was from 0 to 3 × 10−5𝑠 with a time step of 3 × 10−9𝑠; tests with additional time steps

and longer target time are attempted to build but were limited by the disk space on the

work station. Fig. 3.24 shows the results from both of the tests. From comparison of these

results, it was shown that with a longer target time, the maximum value of 𝑛𝑒 decreases

51

from 2.47 × 1019 [ 1𝑚3] to 3.33 × 1017 [ 1

𝑚3] , approaching the experimental observation

value ~1016 [ 1𝑚3]; and the concentration region was shrunken. These trends indicated that

the sinusoidal oscillation field assumption may be valid for the drift-diffusion model, but

a fully coupled model was necessary to compare with the experimental observation and

to confirm this guess.

Figure 3.23 Test result for sinusoidal oscillation field

52

CHAPTER 4. CONCLUSIONS AND FUTURE WORK

4.1 Conclusions

A plasma model was built by coupling the EM simulation, heat transfer simulation and

the estimations of plasma properties based on Füner’s model with assuming standing

waves inside the MPCVD reactor. The multi-physics coupling was implemented through

COMSOL Multiphysics. This study was conducted to better understand the plasma

responses to different working conditions (input power and reactor pressure for the

current study); therefore, cases with different input powers and reactor pressures were

built and tested. The reliability of EM solver built in COMSOL and the validity of

standing wave assumption and sinusoidal oscillation field assumption were also tested

during this study.

The simulation results qualitatively agreed with the theoretical expectations and

experimental observations except for 300 W input power. Electrical field strength,

electron number density, electron temperature and the neutral gas temperature would

increase when the input power increased. Electrical field strength, electron number

density and electron temperature would decrease when the reactor pressure increased

while the neutral gas temperature would increase. The study results from the 300 W input

simulations showed there was no plasma ignited under the working condition which did

53

not agree with the experimental observation. The lack of coincidence was probably

caused by the limit of Füner’s model for low powers.

The reliability of EM solver built in COMSOL was tested by comparing the simulation

results of COMSOL with an identical computational model simulated by ANSYS HFSS.

The differences in the maximum field strength on top surface of the susceptor were

23.08 % for a 3-D model and 5.38 % for a 2-D model, which confirmed the 2-D

simulation in COMSOL was a better choice.

The validity tests for assumptions were necessary in order to utilize the drift-diffusion

model of plasma properties. The two assumptions were tested through a drift-diffusion

PDE solver. The test results showed the standing wave assumption was not valid for the

drift-diffusion model while the sinusoidal oscillation field assumption could be valid but

still need to confirmed in the future works.

4.2 Future Work

The Füner’s model was shown to have limit under low input powers, and the parameters

used in the current study needed further adjustment by comparing the current prediction

with the experimental observations in the plasma region. The future simulation of plasma

able to cover the low input power region needs to include the drift-diffusion model of

plasma properties instead of Füner’s. The model is currently working with sinusoidal

oscillation field assumption. A time-dependent simulation of the multi-physics coupling

of plasma should be built with a long enough target time and small enough time step to

54

obtain a self-consistent solution which should be compared with the experimental

observation to examine the validity of the sinusoidal oscillation field assumption.

LIST OF REFERENCES

55

LIST OF REFERENCES

[1] Shokrieh, M. M., and Rafiee, R., 2010. “A review of the mechanical properties of

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VITA

61

VITA

Di Huang School of Aeronautics and Astronautics, Purdue University

Education B.S., AAE, 2012, Purdue University, West Lafayette, Indiana M.S., AAE, 2014, Purdue University, West Lafayette, Indiana

Research Interests Aerodynamics