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NUMERICAL MODELING OF THE PLASMA-PARTICLE INTERACTIONS OFAEROSOL VAPORIZATION IN A LASER-INDUCED PLASMA
By
PHILIP B. JACKSON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2011
c⃝ 2011 Philip B. Jackson
2
This work is dedicated to Knicole, whose love and support made its completion possible.
3
ACKNOWLEDGMENTS
I would first like to thank all of my past and present labmates for their friendship,
encouragement, and most of all, for their help. I thank Bret Windom and Prasoon
Diwakar, who are not only great researchers, but who would also provide a laugh and
kind words when I needed it most. I thank Kibum Kim for being a kind and helpful
colleague, roommate, and golf partner. I thank Soupy Dalyander and Patrick Garrity for
the study sessions in preparation for the qualifying exam. I also thank Michael Asgill,
Michael Bobek, and Richard Stehle for their support during the last year of my research.
I would especially like to thank Leia Shanyfelt for being a wonderful friend and
colleague, and for introducing me to two of my now favorite past-times, Lost and World
of Warcraft.
I also would like to thank my parents for their constant encouragement and support
during my time at the University of Florida. I thank my mother for her unconditional love
and pride, and for always reminding me to use my common sense. I thank my father for
his seemingly endless wisdom. No matter how much I learn, he always seems to come
up with new insights I never would have considered.
I owe a special debt of gratitude to Knicole Colon. So much of what I’ve accomplished
over the last two years is due to her influence in my life. Her work ethic is to me a
standard to which I will always seek to achieve.
I thank Dr. Jill Peterson for her guidance and support during my master’s research.
If she had not believed in me, I would not be where I am today.
Lastly, I would like to thank Dr. David Hahn for providing as much guidance and
direction as only the most dedicated of mentors. I thank him for his endless willingness
to inspire and to help and mostly for his patience over the last several years.
4
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1 Laser Induced Breakdown Spectroscopy of Aerosol Systems . . . . . . . 121.2 The Philosophy and Design of a Numerical Model . . . . . . . . . . . . . 131.3 Scope of the Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 REVIEW OF LITERATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Laser-Induced Breakdown Spectroscopy . . . . . . . . . . . . . . . . . . 182.1.1 Laser-Induced Plasma Diagnostics . . . . . . . . . . . . . . . . . . 182.1.2 Local Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . 20
2.2 The Current State of Aerosol LIBS . . . . . . . . . . . . . . . . . . . . . . 222.3 Laser-Induced Plasma Modeling . . . . . . . . . . . . . . . . . . . . . . . 242.4 Inductively-Coupled Plasma Modeling . . . . . . . . . . . . . . . . . . . . 282.5 Early Laser-Induced Plasma Behavior . . . . . . . . . . . . . . . . . . . . 31
3 COMPUTATIONAL FUNDAMENTALS . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Numerical Considerations in Atomic Emission Spectroscopy . . . . . . . . 343.1.1 The Boltzmann Distribution and Partition Functions . . . . . . . . . 343.1.2 The Saha Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.3 Determining Electron Density and Ionization State Distributions . . 363.1.4 Spectral Line Broadening and the Calculation of Voigt Functions . 44
3.2 Numerical Techniques for the Solution of Partial Differential Equations . . 483.2.1 Finite Difference Methods versus Finite Element Methods . . . . . 483.2.2 The Explicit Finite Difference Method . . . . . . . . . . . . . . . . . 493.2.3 Deriving the Discretization Equations for One-Dimensional Conduction
through a Spherically Symmetric Medium . . . . . . . . . . . . . . 503.2.4 The Implicit Finite Difference Method . . . . . . . . . . . . . . . . . 553.2.5 The Tridiagonal Matrix Algorithm . . . . . . . . . . . . . . . . . . . 563.2.6 The SIMPLE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 583.2.7 The SIMPLER Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 593.2.8 Solving for Roots of Non-Linear Equations . . . . . . . . . . . . . . 61
3.2.8.1 The bisection method . . . . . . . . . . . . . . . . . . . . 613.2.8.2 Fixed-point iteration . . . . . . . . . . . . . . . . . . . . . 62
5
3.2.9 Calculation of Higher-Order Legendre Polynomials . . . . . . . . . 623.3 Automated Peak Detection Algorithms . . . . . . . . . . . . . . . . . . . . 65
3.3.1 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.2 Baseline Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.3 Peak Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3.4 Peak Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 THE STATIC, CONDUCTIVE PLASMA MODEL . . . . . . . . . . . . . . . . . . 76
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 The Problem Statement and Simplifying Assumptions . . . . . . . . . . . 764.3 Numerical Formulation and Implementation . . . . . . . . . . . . . . . . . 78
4.3.1 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1.1 The explicit finite difference formulation . . . . . . . . . . 794.3.1.2 The implicit finite difference formulation . . . . . . . . . . 80
4.3.2 Mass Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.2.1 The explicit finite difference formulation . . . . . . . . . . 824.3.2.2 The implicit finite difference formulation . . . . . . . . . . 83
4.3.3 Temperature Dependent Material Properties . . . . . . . . . . . . . 844.3.3.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.3.2 Specific heat capacity . . . . . . . . . . . . . . . . . . . . 864.3.3.3 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . 864.3.3.4 Mass diffusion coefficient . . . . . . . . . . . . . . . . . . 86
4.3.4 Determining Ionization State Distributions . . . . . . . . . . . . . . 884.3.5 Simulation of Plasma Radiative Emission . . . . . . . . . . . . . . 90
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.4.1 The Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . 904.4.2 The Concentration Field . . . . . . . . . . . . . . . . . . . . . . . . 914.4.3 Electron Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 MODELING AEROSOL VAPORIZATION WITHIN THE LASER-INDUCEDPLASMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1 Overview of the Aerosol Vaporization Process . . . . . . . . . . . . . . . . 1075.2 Instantaneous Aerosol Vaporization . . . . . . . . . . . . . . . . . . . . . 1085.3 Linear Aerosol Vaporization . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 Heat- and Mass-Transfer Modeling of Aerosol Vaporization . . . . . . . . 110
5.4.1 Temperature Increase to the Melting Point . . . . . . . . . . . . . . 1115.4.2 The Melting Process . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.3 Temperature Increase to the Boiling Point . . . . . . . . . . . . . . 1135.4.4 The Vaporization Process . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.4.1 Heat transfer limited vaporization . . . . . . . . . . . . . . 1145.4.4.2 Mass transfer limited vaporization . . . . . . . . . . . . . 116
5.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6
6 INVESTIGATION OF PLASMA INCEPTION . . . . . . . . . . . . . . . . . . . . 128
6.1 Introduction and Motivation for Early Plasma Studies . . . . . . . . . . . . 1286.2 Experimental Apparatus and Methods . . . . . . . . . . . . . . . . . . . . 1296.3 Data Processing and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.3.1 Automated Peak Detection . . . . . . . . . . . . . . . . . . . . . . . 1326.3.2 Plasma Inception Characteristics . . . . . . . . . . . . . . . . . . . 135
6.4 Experimental Results and Discussion . . . . . . . . . . . . . . . . . . . . 1366.5 Theoretical Considerations and Conclusions . . . . . . . . . . . . . . . . 1386.6 A Note on Spherical Aberration . . . . . . . . . . . . . . . . . . . . . . . . 141
7 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.2 Suggestions for Future Research . . . . . . . . . . . . . . . . . . . . . . . 158
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7
LIST OF TABLES
Table page
4-1 Summary of parameters used in the evaluation of diffusion coefficientby Chapman-Enskog theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8
LIST OF FIGURES
Figure page
2-1 Schematic of a typical LIBS experimental setup. . . . . . . . . . . . . . . . . . 33
3-1 Comparison of Doppler, Lorentzian, and Voigt profile functions. . . . . . . . . . 72
3-2 The Voigt profile function for various values of the damping parameter, a. . . . 73
3-3 Control volume for a general interior node. . . . . . . . . . . . . . . . . . . . . . 74
3-4 The first six Legengre polynomials of the first kind. . . . . . . . . . . . . . . . . 75
4-1 Argon gas density, ρ, as a function of temperature. See Fujisaki (2002). . . . . 94
4-2 Specific heat capacity, Cp, of argon as a function of temperature. . . . . . . . . 95
4-3 Thermal conductivity, k , of argon as a function of temperature. . . . . . . . . . 96
4-4 Mass diffusion coefficient as a function of temperature. . . . . . . . . . . . . . 97
4-5 Plasma temperature distribution evolution with time for a flat initial profile. . . . 98
4-6 Plasma temperature distribution evolution with time for aparabolic initial profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4-7 Change in temperature with time at three locations in the plasma. . . . . . . . 100
4-8 Concentration distribution of cadmium at early times. . . . . . . . . . . . . . . . 101
4-9 Concentration distribution of cadmium at later times. . . . . . . . . . . . . . . . 102
4-10 Temporal evolution of cadmium concentration at three locationswithin the plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4-11 Evolution of electron density with time on a logarithmic scale. . . . . . . . . . . 104
4-12 Evolution of electron density with time on a uniform scale. . . . . . . . . . . . . 105
4-13 Temporal evolution of electron number density at threelocations in the plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5-1 Total aerosol mass in the plasma volume. . . . . . . . . . . . . . . . . . . . . . 121
5-2 Simulated cadmium concentration throughout the plasma after 1µs. . . . . . . 122
5-3 Simulated cadmium concentration throughout the plasma after 5µs. . . . . . . 123
5-4 Simulated cadmium concentration throughout the plasma after 10µs . . . . . . . 124
5-5 Simulated cadmium concentration throughout the plasma after 15µs . . . . . . . 125
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5-6 Simulated cadmium concentration throughout the plasma after 20µs . . . . . . . 126
5-7 Simulated cadmium concentration throughout the plasma after 30µs . . . . . . . 127
6-1 Schematic of experimental LIBS apparatus for plasma inception study. . . . . . 142
6-2 Evolution of laser-induced plasma in nitrogen over its lifetime. . . . . . . . . . . 143
6-3 Laser-induced plasma formation in nitrogen at early times . . . . . . . . . . . . 144
6-4 Line profile across the CCD showing early plasma inceptionfeatures in nitrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6-5 Line profile across the CCD showing early plasma inceptionfeatures in argon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6-6 Line profile across the CCD showing early plasma inceptionfeatures in helium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6-7 Collection of 30 plasma inception images in nitrogen in relation to the laserbeam profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6-8 Collection of 30 plasma inception images in argon in relation to the laser beamprofile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6-9 Collection of 30 plasma inception images in helium in relation to the laser beamprofile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6-10 In relation to the beam profile, plasma inception events occur past the focalpoint, where the plasma forms at the focal point. . . . . . . . . . . . . . . . . . 151
6-11 Summary of plasma inception statistics for nitrogen, argon, and helium in relationto the laser beam profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6-12 Simulated image of the distribution of photon density across several pixels ofthe CCD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6-13 Simulated distribution of the probability of a multi-photon ionization event innitrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6-14 Simulated distribution of the probability of a multi-photon ionization event inargon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6-15 Simulated distribution of the probability of a multi-photon ionization event inhelium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
NUMERICAL MODELING OF THE PLASMA-PARTICLE INTERACTIONS OFAEROSOL VAPORIZATION IN A LASER-INDUCED PLASMA
By
Philip B. Jackson
December 2011
Chair: David W. HahnMajor: Mechanical Engineering
Laser-Induced Breakdown Spectroscopy (LIBS) is a powerful and well-established
atomic emission diagnostic for the identification and analysis of unknown samples.
Recent research efforts have shown that LIBS is useful for both qualitative identification
and for the quantitative measurement of relative as well as absolute analyte concentration
regardless of analyte state. More recently, much interest has been directed toward the
use of LIBS in the analysis of aerosol systems, including those generated by laser
ablation (LA-LIBS). While LIBS offers many advantages as a diagnostic tool, there
are several difficulties that limit its capability and robustness. Chief among these are
matrix effects and incomplete or inhomogeneous sample vaporization. In an effort to
fully understand, and eventually mitigate, these difficulties, the current work seeks to
design and implement a numerical model that describes the complex plasma-particle
interactions that govern the LIBS of aerosol systems. The model incorporates
the processes of heat transfer, hydrodynamics, mass diffusion, vaporization, and
electromagnetism. The model considers the fundamental physics of three distinct
regimes: the global plasma environment, the local particle behavior, and the initial
nature of plasma inception.
11
CHAPTER 1INTRODUCTION
1.1 Laser Induced Breakdown Spectroscopy of Aerosol Systems
Laser-Induced Breakdown Spectroscopy (LIBS) is a diagnostic tool used for the
identification and analysis of unknown samples. Since its discovery as an analytical
method in the early 1960s, LIBS has found ever-increasing exposure in the laboratory
and in the field. Among its many advantages, LIBS is a real-time technique that can
be applied in situ with little or no sample preparation. As such, it has the capability
of analyzing samples in any state, be it solid, liquid, or gas. Recently, LIBS has been
applied to the analysis of aerosol systems as well, including aerosols generated by laser
ablation, in a technique called LA-LIBS.
The primary challenges to the accuracy and robustness of the LIBS technique
are difficulties such as matrix effects and fractionation. Matrix effects describe a broad
class of phenomena whereby the signal behavior of the analyte is affected by the
presence of additional matrix constituents. Fractionation is essentially the incomplete
or inhomogeneous vaporization of a sample within the plasma and results in an analyte
response that is not reflective of the true sample stoichiometry. The analyte signal then
provides a misleading view of sample makeup. Unfortunately, both of these effects
establish a limit to the effectiveness of the LIBS technique in analyzing general systems.
Traditionally, researchers have relied on certain simplifying assumptions in LIBS
that form a fundamental basis on which the diagnostic is built. With the consideration of
several of the aforementioned difficulties on the LIBS of aerosol systems, it is becoming
increasingly apparent that these assumptions may warrant reevaluation as to their
validity. It may be found that not only do these assumptions yield an inexact picture
of the physics, but it is possible that the relaxation of these assumptions, or even the
adoption of new ones, may lead to the improvement of the diagnostic.
12
Chief in those assumptions are that the processes of heat transfer from the
plasma into the discrete analyte particle, and mass transfer from the particle into the
laser-induced plasma, occur instantaneously. In fact, however, the heat transfer from
the laser plasma to the aerosol particle occurs over a finite time (Hohreiter, 2006).
Even though that time may be small when compared to the plasma lifetime, it may
not be small enough to be considered instantaneous. Also, as mass is liberated from
the surface of the particle it diffuses throughout the plasma volume over a finite time.
Although the diffusion of particle mass is rapid, it may not be so rapid when compared
to the speed of plasma expansion as to be assumed instantaneous. In light of the
problems of matrix effects and inhomogeneous vaporization, the true time scales of
heat and mass transfer may not only need to be a consideration, but may also lead to an
explanation of their existence.
Reevaluation of the key assumptions in LIBS may provide researchers with a
more complete picture of the rapid and complex processes that govern the method.
In addition, such insight, while providing fundamental knowledge, may also be used
to combat some of the difficulties of LIBS, and especially those of aerosol LIBS. An
improved understanding of the fundamental physics may lead to methods to lower
detection limits, methods to reduce uncertainty in quantitative measurements, and
techniques to build more robust field-deployable systems.
The objective of the current research is to develop a rigorous, fundamental model
to describe the plasma-particle interactions of particle vaporization in LIBS in order
to provide the community with more complete knowledge and ultimately improve the
effectiveness of the diagnostic.
1.2 The Philosophy and Design of a Numerical Model
Toward this end, the current study seeks to develop and implement a complete
mathematical model for the synthesis of the variety of processes that take place during
the LIBS of aerosol systems. The processes of heat transfer, hydrodynamics, mass
13
diffusion, and even electromagnetics, each describe the many different physical
phenomena observed in aerosol LIBS. First, the various modes of heat transfer
must be examined. A laser-induced plasma is a short-lived, high-temperature gas in
which conduction, convection, and radiation modes may all play appreciable roles.
Furthermore, heat transfer from the plasma to the aerosol particle is one of the chief
mechanisms by which vaporization occurs. It is noted here that based on the large
mismatch in the plasma volume (the larger) and the laser focal volume (the smaller) that
direct laser-particle interactions are much less likely than plasma-particle interactions.
Highly coupled to the temperature problem are the hydrodynamics of the system.
Laser-induced breakdown induces a rapid plasma expansion, so much so that shock
waves are produced. The large velocity gradient, therefore, will have significant effects
on the temperature field and the distribution of mass within the plasma. Also important
to the transport of material throughout the plasma volume is mass diffusion which
greatly influences vaporization in the immediate vicinity of an aerosol particle. Lastly,
electromagnetic forces may greatly affect the plasma’s behavior, especially with regard
to the early dynamics. The large electromagnetic field generated from the incident
laser pulse itself influences the breakdown event and therefore the initial plasma
characteristics.
With this in mind, the current modeling efforts categorize the problem into three
sub-models that are implemented independently: a global model, a local model, and
an initial model. The global model describes the physical environment throughout the
laser-induced plasma as distributions of temperature, electron density, and mass that
has been liberated from an aerosol particle. Once the global environment is established,
the local model describes the vaporization kinetics of a single aerosol particle subjected
to the local conditions of current plasma location. While the local model depends upon
the global model, the converse is not true. Lastly, in order to determine the temporal
progression of both global and local variable distributions, the initial conditions must
14
first be prescribed. Due to the complexity of modeling considerations during the early
times of plasma life, which are characterized by non-equilibrium dynamics, the initial
conditions are prescribed based on empirical observations. An experimental study into
the growth and behavior of the laser-induced plasma in its early lifetimes is performed to
provide insight into how a complete model of aerosol LIBS may incorporate a description
of plasma inception.
Like any numerical model there are two challenges that must be addressed when
one discusses the correctness of the model: physical correctness and numerical
correctness. First, the model must by physically correct. That is, the governing
equations and fundamental processes considered must indeed represent the correct
physical principles at work. Much care has been taken to justify the use of each
fundamental principle and equation employed in the current modeling treatment,
and each is discussed as they arise. Secondly, the model must exhibit numerical
correctness. That is, the solution procedure must provide numerical values that
accurately satisfy the equations upon which they are based within acceptable numerical
uncertainty. Each numerical technique that is used here is widely accepted as a correct
technique and is independently verified through the use of benchmarking examples.
Lastly, it is important that any good numerical model achieve two objectives: (1) it
must agree with and support (or in certain cases, challenge) current accepted research,
and (2) it must be able to make testable predictions. Much research is currently being
undertaken to more fully understand the physics of aerosol LIBS. As such, much data
exists by which the current model may be verified. Many model output quantities may
be compared with various current studies to validate the model, such as: temperature
measurements, diffusion characteristics, and even spectral signatures. Finally, once the
model has been validated, it may be used to investigate new situations that inspire new
experiments in a further attempt to provide insight into the complicated physics of the
15
phenomena. This, above all, is the most important goal both of numerical modeling in
general and of the current efforts.
1.3 Scope of the Current Work
The present study seeks to provide the reader with the description, design, and
implementation of a rigorous numerical model for the analysis of aerosol LIBS. This
study is organized in a ”bottom-up” fashion with each new chapter building upon the
work of each previous chapter.
Chapter 2 begins with a review of several important, fundamental topics included
both for completeness and for reference. First, the basics of LIBS are covered along
with a discussion of a few important concepts in the quantification of atomic emission
spectroscopy in general. Second, several basic numerical techniques are examined
that are implemented throughout the present study. These techniques are provided
here in their general forms so their implementation in specific facets of the model may
be better understood. Lastly, the chapter is concluded with a discussion of automated
peak detection algorithms that find use in the analysis of data taken in the current
experimental study of plasma inception.
After the review of several fundamental concepts, Chapter 3 describes the current
state of research into which the present study is placed. First, current trends in LIBS
research are discussed, as is the present role of aerosol LIBS. Next, several recent
modeling efforts in LIBS and related techniques, such as Inductively-Coupled Plasma,
Atomic Emission Spectroscopy (ICP-AES), are discussed. Lastly, the chapter is
concluded with a discussion of the present understanding of early plasma behavior
and non-equilibrium considerations.
With the preliminary basis and motivation for the current study established, Chapter
4 begins the description of modeling efforts by detailing the method for simulating the
global plasma environment. The physics of the global plasma model are described in
detail. Included are discussions of the relative importance of conduction, convection,
16
and radiation heat transfer modes, the effects of temperature dependent properties,
and the necessary considerations for the effective implementation of these models. The
implications of various physical phenomena and modeling methodologies are discussed
including the roles of compressibility effects, the roles of electromagnetic forces, and
single-fluid representations versus ion-neutral representations.
With the global environment established, Chapter 5 examines the local environment
in the immediate vicinity of a single aerosol particle. The kinetics of aerosol vaporization
are investigated along with their effect, if any, on the global environment with reference
to accepted models of aerosol vaporization. Individual processes of melting, evaporation,
and diffusion are discussed. The competing roles of heat transfer-limited vaporization
and mass transfer-limited vaporization are also discussed.
Chapter 6 turns attention to the investigation of the behavior of early plasma
lifetimes and the study of the plasma inception event itself. An experimental study is
presented to investigate the earliest breakdown events and the subsequent growth of
the plasma in several different gases. In this study numerous images of initial plasma
breakdown are automatically processed to compile statistics on the variations of early
behavior in the various gases. The implication this behavior may have on the current
understanding of plasma inception is introduced.
Lastly, Chapter 7 summarizes the most important points and conclusions of the
present work, suggests refinements that may improve the sophistication of the present
model and discusses various avenues of interest that may inspire future work.
17
CHAPTER 2REVIEW OF LITERATURE
2.1 Laser-Induced Breakdown Spectroscopy
2.1.1 Laser-Induced Plasma Diagnostics
In laser-induced breakdown spectroscopy (LIBS), a high-energy laser pulse is
focused to a point. At that point the power density becomes sufficiently high to induce
the breakdown of whatever medium is present, and a high-temperature plasma results.
The atomic emission from the plasma is collected and used for various qualitative and
quantitative diagnostics. Figure 2-1 shows a typical LIBS laboratory configuration where
plasma emission is collected in back scatter through the use of a pierced mirror.
The first laser-induced plasma to be used in the laboratory was produced in the
early 1960s (Miziolek, 2006). Since then, the LIBS technique has found widespread
use in the analytical laboratory as an attractive method for analyzing materials. Like
many other methods of atomic emission spectroscopy, the primary goal of LIBS is the
identification and analysis of an unknown sample.
Over the past several decades the LIBS method has proven to be useful as a
robust qualitative diagnostic for the detection of the presence of unknown sample
constituents. Spectra of collected emission can be observed for the presence of
peaks at the characteristic wavelength of a given element. LIBS uses libraries of
elemental signatures, and combinations of such signatures, to identify samples ranging
in complexity from single-species samples to complex biological samples. In more
recent years, LIBS has been shown to provide valuable quantitative analysis as well.
Based on relative peak intensities and spectral line broadening, researchers have
been able to use LIBS to determine relative and even absolute concentrations of the
constituents in a sample (Miziolek, 2006).
As an analytical tool, LIBS has many advantages over other methods of elemental
analysis. First of all, no sample preparation is required for LIBS as it is a technique that
18
can be performed on virtually any sample, in any state. LIBS has been demonstrated on
solid surfaces, in liquids, in gases, and, most recently, on aerosol systems (Hohreiter,
2004). Moreover, LIBS proponents state that it is capable of in situ analysis, in that the
laser plasma, as the excitation source, is focused onto the sample, rather than bringing
the sample to the excitation source as in many other atomic emission spectroscopy
methods. The only preparation that is required is optical access to the sample. This is
especially beneficial in situations that may be hazardous to human life. Lastly, a LIBS
analysis is fast. Due to the aforementioned lack of sample preparation and delivery time,
and the fact that the laser-plasma itself is short-lived, a single LIBS measurement can
be made virtually instantly. Many LIBS analyses require an ensemble of shots and then
batch processing of the resulting data. Most automated identification and chemometric
routines are fast enough that LIBS is described as a real-time technique.
LIBS is not a perfect diagnostic tool, however. Many challenges still exist to
improve the robustness of the technique, especially in quantitative analysis. The first
challenge to LIBS analysis is the issue of sample non-homogeneity. The LIBS plasma
is small, and, as such, probes a small point in space that may not contain elemental
constituents that are perfectly representative of the overall sample. Related to this
is the concept of fractionation. Fractionation is essentially the non-uniform analyte
response of constituents in the plasma. For example, varying vaporization rates of
plasma constituents alters the elemental excitation of the constituents, and therefore
non-uniform vaporization and diffusion can yield misleading results for the relative
concentrations that are calculated.
Matrix effects also limit the LIBS diagnostic as is the case in many other analytical
methods. Matrix effects occur when the presence of the various sample constituents
affects the signal of the specific element of interest. Two samples that contain the same
concentration of a given element may easily yield different absolute signal strengths
in the same LIBS setup depending on the state of the sample. Matrix effects are not
19
completely limiting, however. Often matrix-dependant calibration is performed to help
mitigate these effects using matrix-matched standards.
2.1.2 Local Thermodynamic Equilibrium
Global thermodynamic equilibrium exists in a medium that is in thermal equilibrium
(constant temperature), mechanical equilibrium (constant pressure), and chemical
equilibrium (constant concentration). Such a homogeneous and constant system
allows for several equilibrium relations to be employed to describe the system. On a
molecular level, thermodynamic equilibrium implies that all collisional and radiative
processes balance one another out. In equilibrium, ionization events are equally
frequent as recombination events, and radiation emitted is equal to radiation absorbed
(Lochte-Holtgreven, 1995).
Global thermodynamic equilibrium therefore implies a system is static and
unchanging. While this state may seem uninteresting, facets of such a concept may
be employed in truly dynamic systems, allowing one to accurately describe all the
complexities of a varying system while still taking advantage of the simple equilibrium
relations. Such is the case in the concept of local thermodynamic equilibrium. In local
thermodynamic equilibrium (LTE), a single point in the system is assumed to be in
thermodynamic equilibrium with some small region about that point in time and space.
In this sense, thermodynamic equilibrium holds at each single point, while still allowing
for the thermodynamic state to vary from one point to the next.
From a molecular viewpoint in a plasma, local thermodynamic equilibrium no longer
requires collisional and radiative processes to balance one another. Rather, collisional
processes are assumed to dominate the plasma kinetics (Lochte-Holtgreven, 1995).
The question remains, when is the local thermodynamic equilibrium assumption a
valid one, and when is it not? If local thermodynamic equilibrium results when collisional
processes dominate radiative processes in the plasma kinetics, then it is reasonable
to assume that one may require the electron number density to be sufficiently high to
20
ensure a high collision rate. This line of thinking leads to the popular McWhirter criterion
(Miziolek, 2006) for establishing local thermodynamic equilibrium:
ne ≥ 1.6× 1012T 1/2(∆E)3, (2–1)
where ∆E is the energy transition of a line in eV, and T is the temperature in K. It is
important to note that the McWhirter criterion is a necessary, but insufficient, criterion
for assuming local thermodynamic equilibrium (Tognoni, 2006). There has been much
recent discussion on developing sufficient conditions for which local thermodynamic
equilibrium can confidently be assumed to hold. Despite the difficulty in establishing
precise metrics for the LTE assumption, researchers are currently confident that local
thermodynamic equilibrium holds for all but the earliest of plasma lifetimes.
Assuming local thermodynamic equilibrium holds ultimately allows the statistics
of microscopic states to follow certain standard relations. Once local thermodynamic
equilibrium is established, the population distribution of excited states of a species may
be described by the Boltzmann formula, and the population distribution of the different
ionization states of a species may be described by the Saha equation. Both of these
relations are discussed in detail in subsequent sections. Indeed it is only when these
and other equilibrium conditions hold that temperature may be defined as a single,
unique quantity at a point (Lochte-Holtgreven, 1995).
Variations from local thermodynamic equilibrium assume that population and
velocity distributions are not given by the relations mentioned above. When local
thermodynamic equilibrium does not hold, the very concept of temperature is called
into question. Common non-equilibrium models simplify this difficulty by allowing for two
distinct temperatures to exist at each point, an electron temperature, Te , and a heavy
particle temperature, Tp, which are determined from unique distribution relations for
each species (Povarnitsyn, 2007).
21
2.2 The Current State of Aerosol LIBS
The study of the response, characteristics, and latest improvement of the laser-induced
breakdown of aerosol based samples is just one small corner of the overall LIBS
community. It is, however, a field with vast exposure in the literature. The first reported
case of the use of a laser-induced plasma diagnostic for the study of an aerosol sample
can be traced back to Radziemski et al. (1983)
In 1983 Radziemski et al. developed time-resolved measurements of the presence
of several elements in aerosols. Local thermodynamic equilibrium was assumed
throughout their experiment, with increased confidence in this assumption after the
first 1µs . The collected spectra were used to calculate the plasma temperature and
electron density. A simple hydrodynamic model was also implemented to predict
plasma temperature and size. The study also represents the first use of LIBS for in situ
measurements of aerosols.
Since then, the use of LIBS on aerosol systems has continued to grow. In
1998, Hahn studied the use of LIBS for the sizing of single aerosol particles. Of
particular interest was the use of LIBS, not just to qualitatively determine the elemental
composition of a single aerosol particle, but to provide a quantitative analysis of the
mass concentration of the particle. Calibration was performed as a two-step process
where LIBS spectra were compared, first, to that of known mass concentration, and
second, to that of known particle size and composition.
Later, in 2001 Carranza, et al. used aerosol LIBS to study the detection of trace
concentrations of the constituent elements, such as magnesium and aluminum,
characteristic of fireworks, in ambient air for the Fourth of July holiday period. Increases
in signal response for these elements were observed over three orders of magnitude.
The measurements also employed a real-time conditional data analysis scheme to
increase the effective analyte signal’s response based on whether or not an individual
LIBS measurement (i.e. that from a single laser-induced plasma) could be classified as
22
a particle ”hit”. This greatly reduced the number of total spectra in the ensemble average
and limited the ensemble to spectra that could yield useful information. The real-time
nature of the experiment and it’s use of conditional analysis has shown that LIBS of
aerosols has become a more competitive diagnostic over the years.
In 2002 Carranza and Hahn investigated an upper-particle size limit for complete
aerosol vaporization. The size limit was determined by deviation from linear mass
response in the atomic emission of silicon. In addition, the fundamental mechanism by
which vaporization occurs is assumed to be controlled by plasma-particle interaction
rather than by laser-particle interactions based on the comparison of aerosol sampling
measurements with Poisson statistics. As such, the spatial and temporal evolution of
the plasma becomes more important to the overall process and is dicussed in detail.
Thermophoretic forces and vapor expulsion dynamics are mentioned to have important
implications to LIBS.
The fundamental processes that govern the LIBS of aerosols was investigated
further by Hohreiter and Hahn in 2004 with the ultimate goal to understand and thus
improve the factors affecting the quantitative precision of the diagnostic. Spectral
and temporal effects of particle presence or absence were studied. Laser cavity
seeding produced no significant improvement over the possible analyte precision,
however marked improvement was noticed when concomitant aerosols from the sample
stream were removed. The plasma-particle interactions in similar experiments were
further investigated by the authors in 2006. The interaction between the plasma
and individual particle mass controls the rate of particle vaporization and diffusion
throughout the plasma volume thereby influencing the spectroscopic signal measured.
Finite time scales of these processes are discussed along with the issue of spatial
non-homogeneity and the influence of localized effects.
In 2009 Hahn summarizes the community’s efforts over the past decade understand
and improve the use of LIBS as a diagnostic for aerosol systems. The importance of
23
understanding the many fundamental processes that govern the complex plasma-particle
interactions are emphasized. Also, Hahn challenges several of the key assumptions
employed during the earlier days of the diagnostic and suggests that critical evaluation
of the assumptions are typical at this stage of a scientific method’s lifetime. Growth and
improvement of the diagnostic into the future, then, is assured as much work must still
be done to understand how the fundamental physics of aerosol LIBS ultimately leads to
analyte response.
2.3 Laser-Induced Plasma Modeling
Several models have been developed in recent years in an attempt to better
understand and predict various aspects of laser-induced breakdown spectroscopy.
While many of these investigations all inherently share consideration of the same
physics, the specifics of each model and their assumptions have varied significantly.
This is expected since the fundamental processes that govern the entire LIBS evolution
are numerous and computationally costly. A full model that seeks to contain each
fundamental process for a variety of species over the entire plasma lifetime with
dependence on space and wavelength is ambitious almost to the extent of being
unwieldy. Despite these modeling difficulties, many successful LIBS models can be
found in the literature.
In 1996, Ho et al., published a study on the numerical modeling of the energy-matter
interactions of a laser-induced plasma with a solid surface. While LIBS analysis of solid
surfaces has been covered in the literature in great detail, few LIBS models that couple
mass, momentum and energy conservation in multiple phases are found. In the Ho
model, heat is transferred to the solid surface and phase transitions are allowed as the
solid converts to liquid and ultimately to the vapor phase. Several layers are considered
and therefore the transport equations are solved as piecewise functions through these
layers. Radiation and absorption mechanisms are considered throughout the plasma,
while maintaining the assumption of local thermodynamic equilibrium. Compressibility
24
effects are also considered and as such the model produces an effective approximation
to the behavior of the spherical shock wave propagating about the plasma.
The specific problem of the expanding plasma and shockwave interacting with
the surrounding gas was studied by Itina, et al. in 2003. The gas dynamics of the
laser plume expansion into both vacuum and dense background gas are considered.
In addition, two different numerical methods are used to develop a hybrid model that
describes both continuum and molecular regimes. First, the authors solve the gas
dynamic equations of mass, momentum, and energy conservation. This gives a view of
the problem from a continuum or macroscopic view point. Second, the authors use the
Direct Simulation Monte Carlo approach to obtain a microscopic view of the physics. Of
particular interest to the authors was the mixing of laser plume and ambient species to
describe experimentally observed phenomena.
The gas dynamics of plasma expansion is again considered by Mazhukin et
al. (2003). In this model the plasma is assumed to be non-stationary, radiative, and
represented with a two-dimensional axially symmetric grid. The plasma is modeled to
impinge upon a solid sample surface comprised primarily of aluminum. The authors find
that the radiative characteristics of the plasma dominate over convective mechanisms
and thus drive the evolution of the plasma expansion. Non-equilibrium effects are
considered on the spectral dependence of the radiation both emitted and absorbed by
the plasma. The plasma is assumed optically thick.
A rigorous plasma model was developed by Gornushkin, et al. first in 2001 that
forms much of the inspiration of the current work. Also assumed as optically thick,
the first plasma model envisioned by Gornushkin, et al. considers both convective
and radiative modes of heat transport. As in the previous models considered so far
in this review, local thermodynamic equilibrium is considered through much of the
work. Here, however, plasma expansion is not found from the solution of governing gas
dynamic equation but rather prescribed through set functions with empirically chosen
25
parameters. The plasma expansion radius, as well as the temperature profile throughout
the plasma volume at any time is prescribed based on empirical measurements. Based
on this largely empirical model, the distributions of constituent species of silicon and
nitrogen, and their neutral and ionized states are calculated. Of primary interest is the
calculation of the spectral dependence of the emitted radiation. As a result, the atomic
line profiles are calculated with the inclusion of line broadening mechanisms such as
Stark broadening and Doppler broadening. The result is a series of synthetic spectra
based on the model’s input quantities.
Since its first inception, the model by Gornushkin, et al, has undergone several
revisions in recent years. Of particular interest is a study published in 2004 where much
of the semi-empirical nature of the model was removed in favor of a strict solution of the
gas dynamic equations. Again, radiative transfer and convective heat transfer modes
are considered to dominate. The gas dynamic equations are solved as a laser-induced
plasma is created on the surface, and completely vaporizes a spherical particle. The
plasma is assumed to be in local thermodynamic equilibrium throughout. Plasma
radiation is calculated as a function of spectral dependence to generate synthetic
spectra. While much of the empirical nature of the model has been removed, some
is still retained by way of the prescription of plasma initial conditions. The model is
defined to begin at some small time after breakdown has occurred. As such the initial
plasma temperature profile is prescribed along with the initial plasma radius and velocity.
Experimental verification of the model was exhaustively performed in 2005.
More cases of particle sensitive plasma models have been found in the literature
in more recent years. Bleiner et al. developed a mathematical model of laser-assisted
particle sampling in 2004. Particles of various size distributions are modeled in an
expanding laser plume to examine their influence on micro-particle formation and the
ablation of solid material. It was found that local plasma conditions drive the kinetics of
the micro-processes rather than bulk laser-plume characteristics. The author specifically
26
addresses the use of laser based techniques for the sampling of discrete points and the
benefits of mathematically modeling the behavior.
A further refinement to the work of Gornushkin et al. was published by Kazakov, et
al. in 2006. Again the dynamics of a convective, radiative plasma gas are considered,
but in this case, the plasma environment expands not into vacuum, but into ambient
gas. As a result, the model includes compressibility effects and is able to predict the
formation of the spherical shockwave that propagates along with plasma expansion. The
initial plasma dynamics are still defined based on semi-empirical observation and the
model is only applicable after the laser pulse has vanished. The evolutions of atomic
and ionic line profiles are also computed.
In 2007, a study was performed by Povarnitsym, et al. demonstrated several
non-equilibrium characteristics of laser plasmas, though the study was specific to
those created from pulsed lasers in the femto-second range. The model assumed
the existence of two separate temperatures, the electron temperature and the heavy
particle temperature. The model describes the hydrodynamic motion of the plasma and
accounts for laser energy absorption and conduction through a solid sample target.
Phase transitions throughout the sample are considered and tracked using a high-order
multi-material Gudunov method. The model is used primary to describe the ablation and
fragmentation of the target with respect to measured stresses and observed ablation
depth.
More recently, a study was performed by Dalyander, et al. that also served as
significant inspiration to the current work. The authors develop a finite difference solution
to the conduction equation to describe the temperature difference in a stationary
laser-induced plasma that does not expand with time. The model was developed for the
specific purpose to understand the role that finite vaporization and diffusion rates play
in the nature of aerosol-based LIBS measurements. A particle consisting of cadmium
and magnesium is introduced into the center of the plasma mesh and is allowed to
27
vaporize linearly with time. The resulting mass diffusion throughout the plasma volume
is calculated. Based on equilibrium considerations the distribution of neutral atoms
and ions is calculated. From this atomic emission is estimated and used to assess the
distinction between global temperature evolution and local temperature characteristics.
2.4 Inductively-Coupled Plasma Modeling
While LIBS is the atomic emission diagnostic that is primarily under consideration
in the current work, there are several other techniques within the wide field of atomic
emission spectroscopy whose studies are also relevant. In addition, many other
plasma-based techniques exist in the analytical community. While operating temperatures,
lifetimes, and other characteristics of plasmas created from the various sources may
differ, the fundamental processes governing plasmas all share certain common physics.
As such the present research in the fields of other plasma techniques and various model
features may yield useful insight into current efforts in LIBS.
A plasma-based technique that sees large exposure in the literature is Inductively-Coupled
Plasma Atomic Emission Spectroscopy (ICP-AES). The creation of an inductively
coupled plasma is drastically different than the formation of the laser plasma. An ICP
is a sustained plasma created from a strong electromagnetic field that induces and
maintains a relatively large (in comparison to a laser-induced plasma) plasma core.
Current efforts in aerosol analysis and also modeling in the field of ICP-AES lend much
to the current study.
Perhaps the largest contribution to the present study from the ICP community
comes in the form of theoretical models for the vaporization kinetics of solute particles.
In 1987, Hieftje, et al. developed two contrasting models for the vaporization of single
particles entrained in analytical flames or plasmas. The formulation considered
that while heat transfer and mass transfer were both important mechanisms in the
vaporization and liberation of mass from a single particle, only one mechanism would
be rate limiting and therefore solely govern the rate of particle radius decrease. Their
28
arguments also considered the role that the particle size plays in the determination
of these rate constants. In fact, whether heat transfer-limited or mass transfer-limited,
both large particle and small particle regimes and expression were defined for each
mechanism. The model found difficulties in determining exactly in what regime a
given particle may fall, but it created a foundation for a series of follow-up theoretical
formulations that solved the problem more succinctly.
In 1998, Horner and Hieftje developed a numerical simulation of the ICP environment
and its interaction with their previously derived aerosol vaporization kinetics. Two types
of simulations were performed, one where single aerosols were entrained in the ICP, and
one where many-particle distributions were entrained. They determined that changes
to plasma operating conditions, and thus plasma properties affected the vaporization
characteristics appreciably. It was also found that from the previous studies of various
particle regimes and mechanisms that small-particle heat transfer limited vaporization
seemed to drive the observed behavior. The many-particle simulations were used
for comparison directly with experimental results. The chief goal of the investigation
is similar to the present study, namely to determine the mechanism by which matrix
interference affects spectroscopic measurements.
In 2008, an additional refinement to the aerosol vaporization model was made by
the introduction of a more rigorous description of the vaporization kinetics of earlier
phase transitions than the evaporation phase. Particle-vaporization kinetics are modeled
as a series of sequential steps that describe each transition from solid to liquid and
from liquid to vapor in detail. Model input values consist of plasma operating conditions
and location within the plasma, as well as characteristics of the particles themselves,
such as diameter and composition. In addition, their earlier assessment of what particle
regime and what mechanism dominates in an ICP analysis is revised showing that
either may be important and controlling. Since either process might limit the rate
29
of vaporization, both are considered based on automated criteria during simulation
execution.
The effect of aerosol droplets and vaporization mechanics on an ICP were
also investigated by Hobbs and Olesik in 1992. Large signal fluctuations in analyte
response were observed during ICP mass spectrometry. These signal fluctuations
were investigated exhaustively and it was found that the presence of incompletely
dissolved droplets and partially vaporized solid particles affected the analyte response
a great deal. The authors also found that these effects were dependent on composition
and that in some cases no adverse or enhancing effects were found. In some cases,
opposite effects of signal enhancement were observed. The effects were attributed to
a general class of behaviors known as matrix effects. Matrix effects are discussed in
detail previously. Studies such as these provided useful observations for the theoretical
investigation of matrix effects in atomic emission spectroscopy in later years.
In 1997, Olesik discussed the motivations behind theoretical investigation of
individual particle histories. Olesik stated that the analytical signals observed during
ICP-AES were products of a series of kinetic processes that controlled the vaporization
of droplets and particles from which the analytes come. Particle surface temperature
is first raised to the melting point, when phase transition to liquid occurs. The liquid
particle then increases in temperature until the boiling point is reached whereby particle
evaporation kinetics take over. Particle vaporization, he reasoned, was limited either
by heat transfer to the surface of the particle or by mass transfer. These vaporization
kinetics depend on local plasma conditions rather than bulk properties. Olesik also
discussed the effects that non-ideal vaporization kinetics have on analyte signal.
In the next few years several imaging studies were performed to obtain a better
picture of these kinetics described by Olesik and Heiftje. Houk et al, in 1997, performed
a series of high speed photographic studies of the history of solid particles and liquid
droplets in an ICP. They found that not only were individual particle histories important
30
to the continued study of ICP-AES, but that individual particle calibration was desired for
the isolation of ideal behavior.
Several other ICP studies have produced results that have inspired investigations
in LIBS analysis. In 2006 Hergenroder proposed that hydrodynamic sputtering is
responsible for fractionation in various plasma studies. His model is based on the
solution of a three dimensional heat conduction equation with moving interface
boundaries. Particle vaporization kinetics are considered with specific interest on forced
inhomogeneous vaporization where a fraction of analyte material is evaporated while
a fraction remains solid. The model was used to identify optimal operating conditions
to avoid such behavior. In a similar investigation, Bleiner et al. studied, by numerical
simulation, the effect of surface melting and vaporization during laser ablation, also
with the purpose of examining fractionation. It was found that at high irradiance, phase
explosion and droplet expulsion greatly enhance ablation rate and affect ideal sampling
conditions.
2.5 Early Laser-Induced Plasma Behavior
The study of early laser plasma behavior is another area with little exposure in
the literature in recent years. Non-equilibrium considerations, coupled with the rapid
transient nature of these regimes of plasma life, make studies of early plasma dynamics
difficult.
A related study that has ceased to be common in the literature in recent years
concern the characterization of plasma shape. One such study performed by Beduneau
and Ikeda in 2003, while not specifically focused on early plasma lifetimes, lends
information toward the understanding the plasma formation. In the study images
and emission spectra were collected for a variety of laser energies and optical
configurations. It was found that not only was good reproducibility found for early
stages of breakdown but that the characteristics of size and location depend greatly
on the operating conditions. High ionization levels in the early plasma was found to be
31
confirmation of the electron cascade mechanism for plasma formation. Ionization was
also used to explain the asymmetry of plasma shape.
In 1988, a study by Carls and Brock was performed that used a computer model to
investigate laser-induced plasma formation and the explosion of aerosol droplets within
it. The model described the formation and evolution of the plasma and the fluid flow that
results. Still, the one component lacking is the initial breakdown event, which is instead
represented by an empirical initial condition.
Lastly, in 2008, a study was performed by Diwakar and Hahn in which early
laser-induced plasma dynamics were considered. The motivation for the study was
that only by understanding the mechanisms of plasma creation and evolution can the
fundamental processes of laser-induced breakdown spectroscopy be understood. The
first 100 ns of plasma lifetime were considered to describe the early plasma. During
the first 50 ns, significant Thomson scattering was observed and the electron number
density was calculated. The highly transient nature of electron density was used to
suggest that plasma dynamics at early times were in fact non-equilibrium dominated.
Additional measurements by Stark broadening were made and seemed to corroborate
this conclusion. Deviation from local thermodynamic equilibrium within the first 10 ns
of plasma lifetime was then discussed as it pertains to the plasma-particle interactions
present in LIBS measurements.
32
Figure 2-1. Schematic of a typical LIBS experimental setup where collection is taken inback-scatter.
33
CHAPTER 3COMPUTATIONAL FUNDAMENTALS
3.1 Numerical Considerations in Atomic Emission Spectroscopy
3.1.1 The Boltzmann Distribution and Partition Functions
One of the most useful relations that may be employed once local thermodynamic
equilibrium has been established is the Boltzmann equation. For a species in LTE, the
Boltzmann equation represents the distribution of the population at each excited state for
each energy level. The Boltzmann equation is commonly written as:
nin=giU(T )
exp
(− EikT
), (3–1)
where n is the total number density for the entire species and ni is the number density of
the species that is excited to the i -th energy level. The term gi is the degeneracy of the
i -th level, U(T ) is the species internal partition function, Ei is the energy of the i -th level,
k is Boltzmann’s constant, and T is the temperature (Lochte-Holtgreven, 1995).
The internal partition function itself is of interest as it is the most difficult term of the
Boltzmann distribution to calculate. The partition function is the sum over all possible
microstates and is given by:
U(T ) =∑i
gi exp
(− EikT
). (3–2)
To calculate the partition function for a species requires, in theory, the sum over an
infinite number of energy levels. Attempts to perform such a calculation often produce
exorbitantly high values for the partition function and the sum diverges. This and other
common difficulties in calculating partition functions are alleviated with the use of
polynomial approximations. Irwin represents the internal partition functions of several
species with polynomial fits (Irwin, 1980) of the form:
34
lnU =
5∑i=0
ai(lnT )i , (3–3)
where the coefficients ai are tabulated in the article (Irwin,1980). The polynomial
approximations are considered accurate and provide a computationally inexpensive
method for calculating partition functions.
3.1.2 The Saha Equation
A useful relation for the relative magnitude of consecutive ionization stages of any
element in a plasma is given by the Saha equation. Derived in 1920 by the astronomer
Megh Nad Saha, the Saha equation was first used in the study of stellar atmospheres.
The Saha equation is derived from equilibrium considerations, and so for it to hold
true, the plasma under consideration must be assumed to be in local thermodynamic
equilibrium. Here, the plasma’s kinetics are assumed to be dominated by collisional
interactions rather than by radiative processes (Lochte-Holtgreven, 1995).
A common representation of the Saha equation is:
nenznz−1
= 2Uz(T )
Uz−1(T )
(2πmekT
h2
)3/2exp
(−χz−1 − ∆χz−1
kT
), (3–4)
where ne is the electron number density of the plasma, and nz and nz−1 are the
number densities of the z-th and z − 1-th ionization stage, respectively. Here, Uz is
the partition function for the z-th ionization stage, me is the rest mass of the electron,
k is Boltzmann’s constant, and h is Planck’s constant. The term χz−1 is the ionization
energy of the z − 1-th stage and ∆χz−1 is the reduction of the ionization energy due to
the presence of the plasma microfield. Note that as written above, the right side of the
Saha equation is entirely (except for the ∆χz−1 term) a function of temperature, and can
be written in a more succinct form:
nenznz−1
= Sz−1 (T ) . (3–5)
35
Also note that in the above equation z = 1 corresponds to the neutral atom, z = 2 to the
first ionization state, and so on. Hence, the expression z − 1 represents the charge on
the species.
For the current purposes, the Saha equation will be used to solve for the ionization
state distributions of a multi-component plasma where multiple ionization states are
allowed to exist in equilibrium. One Saha equation may then be written for each
elemental plasma constituent for each pair of consecutive ionization states. For
example, in a two-component plasma where the first two ionization states (z = 2, 3)
are considered, four distinct Saha equations can be written that must be solved
simultaneously, along with other conservation equations, to uniquely determine the
ionization state distributions. This topic will be discussed more thoroughly in the next
section.
Lastly, note that, for the current purposes, the reduction in ionization energy, ∆χz−1,
will be neglected. For most practical applications of laser plasmas the reduction in
ionization energy is only on the order of about 0.1 eV (Miziolek, 2006). Neglecting this
term amounts to a change in the true ionization energy of only about 1% in the worst
case. This simplification is justified when one considers that ∆χz−1 is a function of the
electron number density, ne . While many relations exist to describe this dependence
(Lochte-Holtgreven, 1995), the computational cost of performing this calculation while
solving for the ionization states is unwarranted when one considers its negligible
numerical effect.
3.1.3 Determining Electron Density and Ionization State Distributions
If both the temperature field and the concentration field of species are known, the
distribution of neutral atoms and ions can be found. Assuming the plasma dynamics
are collision-dominated (local thermodynamic equilibrium), the relationship between
the number densities of two consecutive ionization states is given by the Saha equation
(Radziemski, 1989):
36
nenznz−1
= 2Qz(T )
Qz−1(T )
(2πmekT
h2
)3/2exp
(−∆Ez−1kT
)= Sz−1 (T ) . (3–6)
Here, z = 1 corresponds to the neutral atom, z = 2 corresponds to the first ionization
state and so on, such that the expression z − 1 represents the charge on the species.
Also, ne is the electron number density, nz is the number density of species z , Qz(T )
is the internal partition function of species z , me is the rest mass of the electron, k is
Boltzmann’s constant, h is Plank’s constant, and ∆Ez is the ionization energy of species
z . Since the right hand side of the equation is completely defined by temperature one
may represent the Saha equation by:
nenznz−1
= Sz−1 (T ) ,
where
Sz−1 (T ) = 2Qz(T )
Qz−1(T )
(2πmekT
h2
)3/2exp
(−∆Ez−1kT
).
As an example, consider a plasma environment that consists of two elements,
argon and magnesium, that may exist as either neutral or singly ionized atoms. In this
case one may write two Saha equations, one for each element:
neArII
ArI= SAr,I (T ) and ne
MgII
MgI= SMg,I (T ) . (3–7)
While SAr,I and SMg,I are completely determined by temperature, the number densities
ne , ArI, ArII, MgI, and MgII are all unknown. Since the total number densities of each
species, irrespective of ionization state, are known from the concentration distribution,
one may close the system and solve for all the unknowns by also considering the
conservation of species and the conservation of charge. Conservation of species for
argon and magnesium are given by the following two relations:
37
ArT = ArI + ArII and MgT = MgI +MgII. (3–8)
Conservation of charge is then simply:
ne = ArII +MgII. (3–9)
With the two Saha equations, two equations for the conservation of species, and a
single equation for the conservation of charge, all unknowns can be determined. First
solve equations 3–7 for the neutral species to get:
ArI = neArII
SAr,Iand MgI = ne
MgII
SMg,I. (3–10)
Substituting equations 3–10 into 3–8 gives:
ArT = ArII(1 +
neSAr,I
)and MgT = MgII
(1 +
neSMg,I
). (3–11)
Solving each of these for the first ionization states and substituting into 3–9 gives:
ne =ArT(1 + ne
SAr,I
) + MgT(1 + ne
SMg,I
) . (3–12)
The only unknown in the relation above is the electron number density ne . This equation
can be solved numerically by a numerical root-finding method. Moreover, a unique
solution is guaranteed to be found from the set of positive real numbers as will be
discussed later in this section. Once ne has been determined, all the other unknown
number densities can be found sequentially from equations 3–10 and 3–11.
In a similar fashion, this system may be solved for an arbitrary number of participant
species with an arbitrary number of ionization states. In general the Saha equation is
given by:
38
nenj ,znj ,z−1
= Sj ,z−1, (3–13)
where nj ,z is the number density of species j in state z . Note that j = 1, 2, 3, ..., J, where
J is the total number of species present, and z = 1, 2, 3, ...,Z + 1, where Z is the highest
ionization state considered. With J species and Z ionization states, there are then JZ
Saha equations in our system (Gornushkin, 2004).
Conservation of species is given by:
Z+1∑z=1
nj ,z = Nj , (3–14)
where Nj is the total number density of species j . Since there are J species in the
system, there are J species conservation equations.
Conservation of charge is then given by:
J∑j=1
Z+1∑z=1
(z − 1)nj ,z = ne. (3–15)
The system is now closed with JZ + J + 1 equations and JZ + J + 1 unknowns. The
solution of the system begins by multiplying equation 3–15 by ne ,
J∑j=1
Z+1∑z=2
(z − 1)nenj ,z = n2e . (3–16)
Next, multiply equation 3–13 by z − 1(nj ,z−1) and sum over all z ’s and all j ’s, to yield:
J∑j=1
Z+1∑z=2
(z − 1)nenj ,z =J∑j=1
Z+1∑z=2
(z − 1)Sj ,z−1nj ,z−1. (3–17)
Substituting equation 3–16 into equation 3–17 gives:
n2e =
J∑j=1
Z+1∑z=2
(z − 1)Sj ,z−1nj ,z−1. (3–18)
Multiplying equation 3–14 by ne gives:
39
Z+1∑z=1
nenj ,z = nenj ,1 +
Z+1∑z=2
nenj ,z = neNj . (3–19)
Substituting equation 3–13 into 3–19 gives:
nenj ,1 +
Z+1∑z=2
Sj ,z−1nj ,z−1 = neNj . (3–20)
Continuing to expand this sum, yields:
nenj ,1 + Sj ,1nj ,1 +
Z+1∑z=3
Sj ,z−1nj ,z−1 = neNj , (3–21)
nenj ,1 + Sj ,1nj ,1 + Sj ,2nj ,2 +
Z+1∑z=4
Sj ,z−1nj ,z−1 = neNj . (3–22)
Which, by 3–13, becomes:
nenj ,1 + Sj ,1nj ,1 + Sj ,2Sj ,1nj ,1ne
+
Z+1∑z=4
Sj ,z−1nj ,z−1 = neNj . (3–23)
Continuing, the system becomes:
nenj ,1 + Sj ,1nj ,1 + Sj ,2Sj ,1nj ,1ne
+ Sj ,3nj ,3 +
Z+1∑z=5
Sj ,z−1nj ,z−1 = neNj , (3–24)
nenj ,1 + Sj ,1nj ,1 + Sj ,2Sj ,1nj ,1ne
+ Sj ,3Sj ,2nj ,2ne
+
Z+1∑z=5
Sj ,z−1nj ,z−1 = neNj , (3–25)
nenj ,1 + Sj ,1nj ,1 + Sj ,2Sj ,1nj ,1ne
+ Sj ,3Sj ,2ne
Sj ,1nj ,1ne
+
Z+1∑z=5
Sj ,z−1nj ,z−1 = neNj . (3–26)
Which more concisely becomes:
40
nj ,1
ne +Z+1∑z=2
z−1∏i=1
Sj ,i
nz−2e
= neNj . (3–27)
Rearranging gives:
nj ,1 =Nj1 +
Z+1∑z=2
z−1∏i=1
Sj ,i
nz−1e
. (3–28)
Now, consider again equation 3–15. Expanding the sum with respect to z yields:
ne =
J∑j=1
nj ,2 +
J∑j=1
Z+1∑z=3
(z − 1)nj ,z . (3–29)
Substituting 3–13 yields:
ne =
J∑j=1
Sj ,1nj ,1ne
+
J∑j=1
Z+1∑z=3
(z − 1)nj ,z . (3–30)
Continuing to expand the sum yields:
ne =
J∑j=1
Sj ,1nj ,1ne
+
J∑j=1
2nj ,3 +
J∑j=1
Z+1∑z=4
(z − 1)nj ,z , (3–31)
ne =
J∑j=1
Sj ,1nj ,1ne
+
J∑j=1
2Sj ,2nj ,2ne
+
J∑j=1
Z+1∑z=4
(z − 1)nj ,z , (3–32)
ne =
J∑j=1
Sj ,1nj ,1ne
+
J∑j=1
2Sj ,2ne
Sj ,1nj ,1ne
+
J∑j=1
Z+1∑z=4
(z − 1)nj ,z , (3–33)
ne = nj ,1
Z+1∑z=2
J∑j=1
(z − 1)
z−1∏i=1
Sj ,i
nz−1e. (3–34)
41
Substituting 3–28 into 3–34 finally yields:
ne =
Z+1∑z=2
J∑j=1
Nj(z − 1)z−1∏i=1
Sj ,i
nz−1e
1 +Z+1∑w=2
w−1∏k=1
Sj ,k
nw−1e
. (3–35)
This is a nonlinear algebraic equation for ne whose coefficents grow in complexity as
one increases the number of the participating species and whose order grows as new
ionization states are added. The form of the equation, however, suggests that under
certain conditions one will always find a viable solution via fixed-point iteration (Atkinson,
1978). Since the desire is to determine the distribution of ionization states based on
calculated temperature and concentration fields, the equation above will be executed for
a variety of different conditions. Those conditions may or may not result in an equation
that a fixed-point iteration method is guaranteed to find a solution for for a given choice
of the initial guess.
Recall that fixed-point iteration is a procedure for solving a nonlinear algebraic
equation in the form:
xn+1 = g(xn),
of which Newton’s method is a common example. Atkinson (1978) describes conditions
for g(x) that guarantees fixed-point iteration will converge upon a unique solution.
First, assume that g(x) is continuously differentiable on [a,b], that g ([a, b]) ⊂ [a, b],
and that
Maxa<x<b|g′(x)| < 1.
Then (i) x = g(x) has a unique solution α in [a, b] and (ii) for any choice x0 in [a, b], with
xn+1 = g(xn), n ≤ 0,
limn→∞xn = α.
42
If one takes (0,∞) as the domain, then it follows that g([0,∞]) ⊂ [0,∞]. Therefore, to
show that the relation above, ne = g(ne), has a unique solution that is guaranteed to be
found by fixed-point iteration (since g(ne) is continuously differentiable in (0,∞)), it must
be shown that
Max0<x<∞|g′(ne)| < 1.
Let g(ne) be written as:
g(ne) =
Z+1∑z=2
J∑j=1
Nj(z − 1)z−1∏i=1
Sj ,i(nz−1e +
Z+1∑w=2
nz−we
w−1∏k=1
Sj ,k
) . (3–36)
Differentiating gives:
g′(ne) =
Z+1∑z=2
J∑j=1
−Nj(z − 1)
z−1∏i=1
Sj ,i(nz−1e +
Z+1∑w=2
nz−we
w−1∏k=1
Sj ,k
)2((z − 1)nz−2e +
Z+1∑w=2
(z − w)nz−w−1e
w−1∏k=1
Sj ,k
).
(3–37)
Rearranging:
g′(ne) =
Z+1∑z=2
J∑j=1
−Nj(z − 1)
z−1∏i=1
Sj ,i(nz−1e +
Z+1∑w=2
nz−we
w−1∏k=1
Sj ,k
)2((z − 1)nz−2e +
Z+1∑w=2
(z − w)nz−w−1e
w−1∏k=1
Sj ,k
).
(3–38)
Finally, multiplying the top and bottom by nZe yields the result:
43
g′(ne) =
Z+1∑z=2
J∑j=1
−Nj(z − 1)z−1∏i=1
Sj ,i
((z − 1)nz+2Z−2e +
Z+1∑w=2
(z − w)nz+2Z−w−1e
w−1∏k=1
Sj ,k
)(nz+Z−1e +
Z+1∑w=2
nz+Z−we
w−1∏k=1
Sj ,k
)2 ,
(3–39)
which is a rational fraction whose polynomial order in the denominator exceeds the
polynomial order of the numerator. The fraction then tends to 0 as ne tends to ∞.
Therefore it appears that, while not rigorously proven, Maxa<x<b|g′(x)| < 1 is satisfied for
sufficiently large ne .
3.1.4 Spectral Line Broadening and the Calculation of Voigt Functions
In spectroscopy, when atomic emission, absorption, or fluorescence are observed,
thin spectral lines are obtained. In the most ideal of atomic or microscopic processes,
these spectral lines would be just that, infinitely thin lines of a finite magnitude positioned
on a single frequency. In reality, however, the signals obtained, while they still may be
considered thin, are slightly broadened producing a peak with a distinct shape or profile.
Spectral lines are then representations, not of a single frequency, but of a distribution of
frequencies about the peak. A variety of mechanisms are responsible for spectral line
broadening and each produce their own characteristic profile shapes.
One of the fundamental reasons for the existence of spectral line broadening is
due to the inherent variability in the population of an atom’s excited energy levels as
described by the Heisenberg uncertainty principle (Ingle, 1988). The uncertainty in
the population of energy states of active transitions leads to a frequency distribution of
emitted photons and therefore to spectral lines that cover a distribution of frequencies
or wavelengths. Since the population of excited states are determined by several
processes, both collisional and radiative excitation and deactivation, then spectral line
broadening can be separately attributed to these processes as well. The most dominant
44
of the lifetime effects come from the deactivation of the excited state due to collisions
and is termed collisional broadening.
Collisional broadening or pressure broadening was first described in 1905 by H.
A. Lorentz who showed that the width of spectral profiles is related to the frequency of
atomic collisions (Lochte-Holtgreven, 1995). The term collisional broadening is used
to describe effects from collisions that occur both between different atoms as well as
between like atoms. Mathematically, the spectral profile that results from collisional
broadening takes the form of a Lorentzian function that can be written in the following
general form,
SL(ν) =2/(π∆νL)
1 + [2(νm − ν)/∆νL]2,(3–40)
where νm is the central frequency and ∆νL is the half-width. The other lifetime effects,
such as from spontaneous or stimulated emission can also be represented by
Lorentzian profiles and often the most dominant of these effects can be assumed to
be independent. The result is that a single Lorentzian function, with an appropriate
composite half-width, can be used to model all of the effects together. For example,
natural broadening, which results from the natural decay of the excited-state population
due to spontaneous emission, is often a negligible effect in comparison to collisional
broadening.
Another dominant source of spectral line broadening comes from the Doppler effect.
The atoms and ions that are present in spectroscopic observations are always in motion
with some distribution of velocities. Because of the Doppler effect, the distribution
of velocities results in the statistical variation of observed frequencies. According to
Maxwell’s law the distribution of velocities is Gaussian in nature. If it can be assumed
that the velocity of a single atom does not change while it radiates then the resulting
distribution of frequencies is also Gaussian. A general form for a spectral line under
Doppler broadening is
45
SD(ν) =2√ln 2
∆νD√πexp−4(ln 2)(ν−νm)2/(∆νD)
2, (3–41)
where ∆νD is the half-width. While there are many other phenomena that lead to
spectral line broadening, such as Stark broadening that results from systems with
permanent dipole moments, the current study will chiefly consider only collision and
Doppler broadening.
In reality, true spectral profiles are usually neither purely Gaussian in shape nor
Lorentzian. Rather a combination of the two profiles is needed to produce a better fitting
shape and model the effects of collisional and Doppler broadening simultaneously. Such
a combination is described by the Voigt function which is named after Woldemar Voigt’s
work of the late 19th century.
The Voigt profile is therefore a convolution of Lorentzian and Gaussian profiles,
assuming the two effects are independent, and is given by the following formulation
SV (ν) =2√ln 2
∆νD√πK(a, νr). (3–42)
The quantity K(a, νr) is known as the Voigt integral and is defined as
K(x , y) =y
π
∫ ∞
−∞
exp(−t2)(x − t)2 + y 2
dt, (3–43)
where t is a dummy variable of integration over all frequencies and a is the damping
constant. The damping constant, a, is related to the Lorentzian and Doppler half-widths
by
a =√ln 2∆νL∆νD. (3–44)
The Voigt function represents a combination of the Lorentzian profile and the
Doppler profile. The three profiles are shown together and normalized in 3-1. Qualitatively,
the normalized Lorentzian profile shape tends to favor its tails for a decrease in
46
amplitude about the mean when compared with the Gaussian profile shape. The
Voigt function, as a combination of the Lorentzian and Gaussian profile shapes, allows
for an extra degree of variablity by altering the dominance of each component. The
damping parameter, a, determines the relative effects of each profile as shown in 3-2.
From a practical perspective, the Voigt profile formulation above poses an additional
challenge in that the numerical calculation of the Voigt integral can be costly. An efficient
method for calculating the Voigt integral is desirable and necessary in any practical
situation where multiple high-resolution Voigt functions must be calculated.
There have been many studies that describe efficient calculations of the Voigt profile
function, the one adopted here is an implementation of the Humlicek algorithm that has
been accelerated by Kuntz to fit the needs of optical spectroscopy (Kuntz, 1997). The
method developed by Kuntz divides the x-y plane into four regions and approximates the
Voigt integral in each region by a rational polynomial expression. For example, Region 1
is defined by the expression |x |+ y > 15 and the following parameters within this region
are developed:
a1 = 0.2820948y + 0.5641896y3 (3–45)
b1 = 0.5641896y (3–46)
a2 = 0.25 + y2 + y 4 (3–47)
b2 = −1 + 2y 2 (3–48)
Using these parameters the Voigt function is then approximated by the following rational
expresssion:
47
K(x , y) =a1 + b1x
2
a2 + b2x2 + x4. (3–49)
This implementation of Humlicek’s algorithm is considerably more efficient
numerically than evaluation by elementary integration techniques. This is especially
beneficial in the current study, where routines are developed to numerically fit experimentally
observed spectral windows to sets of theoretical profiles described by Voigt functions.
Algorithms are implemented to find the best fit of a Voigt profile to a peak of interest.
These methods are implemented on a host of individual peaks over many sets of
spectral windows such that the savings in computation time are considerable.
Overall the investigation of spectral line broadening is important to the field of
quantitative spectroscopy. Theoretical investigations of a spectral peak’s width provides
additional information the location and amplitude of the peak alone cannot provide.
Spectral profile widths can be used to provide estimates of quantitative data such as
particle density, relative concentration, and temperature.
3.2 Numerical Techniques for the Solution of Partial Differential Equations
3.2.1 Finite Difference Methods versus Finite Element Methods
In computational fluid dynamics and heat transfer, the two main choices for the
method of formulation are finite difference methods (FDM) and finite element methods
(FEM). Both methods discretize the pertinent partial differential equations into a system
of algebraic equations, but the underlying principle by which this occurs is quite different.
In finite difference methods, derivative approximations are used at nodal grid points
to reduce the partial differential equation to an algebraic one. Finite element methods
model the function itself between grid points using some type of profile assumption.
While finite difference methods tend to have more popularity in the study of fluid
flow and heat transfer, both methods have merit. It is interesting to note that most
researchers in these computational arenas rarely cross-implement methods (White,
1974), and indeed the practical differences between the two do not warrant advantages
48
of one over the other in considering a specific problem. Thus, to endeavor to solve
a series of partial differential equations means to make a choice as to the method of
formulation one will employ.
While the comparative advantage of one method over the other does not prohibit
one method from finding use in any given field, each method does have its specific
benefits. For fluid flow and heat transfer the finite difference method formulations
tends to follow a more logical derivation. On the other hand, finite element method
formulations, which use some variational method in their derivation, do not lend as easily
to a physical interpretation (Patankar, 1980).
For these reasons, only the finite difference method has been used in the present
study to formulate the solution of the partial differential equations governing heat
transfer, mass transfer, and fluid dynamics.
3.2.2 The Explicit Finite Difference Method
Finite difference methods for time dependent partial differential equations fall
into a spectrum of explicit-ness in their formulation. A finite difference approximation
may be fully explicit, fully implicit, or may fall somewhere between the two extremes,
a formulation known as a Crank-Nichols method. Moreover, in the solution of more
complex systems of partial differential equations, solution methods may see the use
of a combination of explicit, implicit or Crank-Nichols methods for each equation or
for different terms in any given equation. As a consequence, both explicit and implicit
methods are described here.
Explicit finite difference methods calculate the values of the nodal unknowns
for a given time step based purely on their values at the previous time. Implicit finite
difference methods, on the other hand, calculate the values of the nodal unknowns
simultaneously for any single time step, and are only minimally dependent on the values
of the previous time step.
49
The discretization equation for an explicit finite difference scheme will have the
following form (Patankar, 1980):
aiTp+1i = ai−1T
pi−1 + ai+1T
pi+1 + bT
pi + c .
As a consequence the solution of such a set of discretization equations is quite simple
and straightforward. Beginning with some initial condition, T 0i , the values of the
nodal unknowns at the next time step, T 1i , are simply calculated by evaluating each
discretization equation explicitly. There is no need for an iterative procedure. Even in
the case where the coefficients are a function of the dependent variable themselves, the
coefficients are evaluated as functions of the values at the previous time step. Solving
the equations in this manner requires computational time of complexity O(N).
It should be noted that one major disadvantage plagues explicit finite difference
methods. That is, in general, explicit methods are only conditionally stable. The time
steps must be sufficiently small to guarantee a physically meaningful solution is
achieved from solving the equations explicitly. For the discretization equation above,
this can be achieved by requiring that each coefficient ai and b be positive. This makes
sense, as one expects an increase in any given nodal temperature to produce a definite
increase in the new nodal value. In the case of one-dimensional conduction in cartesian
coordinates, for example (Incropera, 2002),
Fo ≤ 12
is a sufficient criterion for physically meaningful stability, with Fo being the non-dimensional
time defined by Fo = αt/L2c .
3.2.3 Deriving the Discretization Equations for One-Dimensional Conductionthrough a Spherically Symmetric Medium
As a first approach, the finite difference equations for a one-dimensional simplification
are derived for the current problem. The plasma will be assumed to be spherically
50
symmetric, that is, temperature will be dependent purely on the radial coordinate.
Conduction will be the only mode of heat transfer considered, except for the outermost
boundary which will lose heat radiatively to the environment. Furthermore, the medium
is assumed homogeneous, isotropic, stationary, free of heat generation, and at local
thermodynamic equilibrium. Nodes are numbered 0, ...,m for a total of m + 1 nodes,
where node 0 is at the symmetry boundary, or the center of the plasma volume, and
node m represents the outer edge of the plasma.
Three discretization equations will be solved: one for the symmetry boundary, one
for the outer, radiative boundary, and one that is valid for all remaining, internal nodes.
The scheme is started by solving the discretization equation for the internal nodes
1, ..., n − 1, n, n + 1, ...,m − 1. The control volume representative of each of the internal
nodes is given in 3-3.
Writing an energy balance for a control volume around node n, yields:
q|n− 12+ q|n+ 1
2= ρVCp
T p+1n − T pndt
, (3–50)
where q|n− 12
is the total energy entering the control surface at n − 12. Finite difference
simplifications of Fourier’s law then take the following form:
q|n− 12= kn− 1
2
T pn−1 − T pndr
4π(rn− 12)2, (3–51)
q|n+ 12= kn+ 1
2
T pn+1 − T pndr
4π(rn+ 12)2, (3–52)
Substituting 3–51 and 3–52 into 3–50 , and introducing an expression for the volume of
the finite difference element gives:
kn− 12
T pn−1 − T pndr
4πr 2n− 1
2+kn+ 1
2
T pn+1 − T pndr
4πr 2n+ 1
2= ρ4
3π(r 3
n+ 12− r 3n− 1
2)CpT p+1n − T pndt
. (3–53)
51
Note that:
r 2n+ 1
2=
(rn +
dr
2
)2, (3–54)
r 2n− 1
2=
(rn −
dr
2
)2, (3–55)
r 3n+ 1
2− r 3
n− 12=
(rn +
dr
2
)3−(rn −
dr
2
)3. (3–56)
It follows since rn = ndr :
r 2n+ 1
2= dr 2
(n +1
2
)2, (3–57)
r 2n− 1
2= dr 2
(n − 12
)2, (3–58)
r 3n+ 1
2− r 3
n− 12= dr 3
[(n +1
2
)3−(n − 12
)3]. (3–59)
Substituting 3–59 into 3–53 and dividing by 4πdr 2 yields:
kn− 12
T pn−1 − T pndr
(n − 12
)2+ kn+ 1
2
T pn+1 − T pndr
(n +1
2
)2=
1
3ρCpdr
[(n +1
2
)3−(n − 12
)3]T p+1n − T pndt
. (3–60)
Rearranging gives:
kn− 12dt
ρCpdr 2(T pn−1 − T pn )
(n − 12
)2+kn+ 1
2dt
ρCpdr 2(T pn+1 − T pn )
(n +1
2
)2=
1
3
[(n +1
2
)3−(n − 12
)3](T p+1n − T pn ). (3–61)
Recall the non-dimensional Fourier number is written as:
Fo =kdt
ρCpdr 2. (3–62)
52
Therefore
Fon− 12(T pn−1 − T pn )
(n − 12
)2+ Fon+ 1
2(T pn+1 − T pn )
(n +1
2
)2=
1
3
[(n +1
2
)3−(n − 12
)3](T p+1n − T pn ). (3–63)
Also note that one can reduce the cubic term as follows:[(n +1
2
)3−(n − 12
)3]= n3 +
3
2n2 +
3
4n +1
8−(n3 − 3
2n2 +
3
4n − 18
)(3–64)
[(n +1
2
)3−(n − 12
)3]= 3n2 +
1
4. (3–65)
Therefore, as dr becomes small:
[(n +1
2
)3−(n − 12
)3]= 3n2. (3–66)
One can also reduce the squared terms in a similar fashion:
(n +1
2
)2= n2 + n +
1
4= n(n + 1), (3–67)
(n − 12
)2= n2 − n + 1
4= n(n − 1). (3–68)
Substituting these results into equation 3–64 gives:
Fon− 12(T pn−1 − T pn )n(n − 1) + Fon+ 1
2(T pn+1 − T pn )n(n + 1) = n2(T p+1n − T pn ). (3–69)
Rearranging one arrives at the final result for the discretization equation for each of the
internal nodes:
T p+1n = T pn +1
n
[Fon− 1
2(T pn−1 − T pn )(n − 1) + Fon+ 1
2(T pn+1 − T pn )(n + 1)
]. (3–70)
53
A similar procedure will be followed for deriving the discretization equation for the
symmetrical boundary node. The energy balance for the symmetry node is:
q| 12= ρVCp
T p+10 − T p0dt
. (3–71)
By Fourier’s law:
k 12
T p1 − Tp0
dr4π
(dr
2
)2= ρCp
4
3π
(dr
2
)3T p+10 − T p0dt
. (3–72)
Rearranging and writing in terms of the Fourier number, yields:
T p+10 = T p0 + 6Fo (Tp1 − T
p0 ) . (3–73)
Lastly the discretization equation for the outer boundary node will be derived. The
plasma exchanges heat by radiation to the environment. Writing an energy balance for
this element gives:
q|m− 12+ q”Rrπr
2m = ρVCp
T p+1m − T pmdt
. (3–74)
Here the heat flux due to radiation is given by:
q”R = ϵσ(T 4∞ − T 4m),
where ϵ is the emissivity (assumed here to be 1 for a perfect black body emitter), and
σ is the Stephan-Boltzmann constant. Substituting this expression into the above
equation, along with Fourier’s law, results in:
km− 12
T pm−1 − T pmdr
4πr 2m− 1
2+ ϵσ(T 4∞ − T 4m)4πr 2m = ρ
4
3π(r 3m − r 3
m− 12)CpT p+1m − T pmdt
. (3–75)
Rearranging and simplifying gives:
54
km− 12
ρCp
T pm−1 − T pmdr
r 2m− 1
2+
ϵσ
ρCp(T 4∞ − T 4m)r 2m =
1
3(r 3m − r 3
m− 12)T p+1m − T pmdt
, (3–76)
km− 12
ρCp
T pm−1 − T pmdr
dr 2(m − 1
2
)2+
ϵσ
ρCp(T 4∞−T 4m)dr 2m2 =
1
3dr 3
[m3 −
(m − 1
2
)3]T p+1m − T pmdt
,
(3–77)
Fom− 12(T pm−1−T pm)
(m − 1
2
)2+
ϵσdt
ρCpdr(T 4∞−T 4m)m2 =
1
3
[m3 −
(m − 1
2
)3](T p+1m −T pm).
(3–78)
Rearranging one last time one arrives at the final result for the discretization equation for
the radiation boundary node:
T p+1m = T pm +3[
m3 −(m − 1
2
)3][Fom− 1
2(T pm−1 − T pm)
(m − 1
2
)2+
ϵσdt
ρCpdr(T 4∞ − T 4m)m2
].
(3–79)
3.2.4 The Implicit Finite Difference Method
It should be noted that the discretization equations derived in the previous section
were an example of an explicit finite difference formulation. In the implicit finite difference
formulation, the new nodal unknowns are written in terms of each other and must be
calculated simultaneously. The discretization equation for the implicit formulation will
have the following form (Patankar, 1980):
aiTp+1i = ai−1T
p+1i−1 + ai+1T
p+1i+1 + bT
pi + c .
Hence, the nodal unknowns must be solved simultaneously for each new time step.
There are many strong differences between the implicit and explicit finite difference
formulations. First, since the implicit method requires a simultaneous solution of the
discretization equations for each time step, the computation expense will, in general,
55
be greater than the explicit method. However, the benefit is that implicit schemes are
unconditionally stable (Incropera, 2002). That is, no matter how great the time step,
physically realistic solutions are guaranteed to be found.
One may write the results of the previous section’s derivation, in the manner of the
implicit method as follows. The discretization equation for the internal nodes becomes:
T p+1n = T pn +1
n
[Fon− 1
2(T p+1n−1 − T p+1n )(n − 1) + Fon+ 1
2(T p+1n+1 − T p+1n )(n + 1)
]. (3–80)
The discretization equation for the symmetry boundary node is:
T p+10 = T p0 + 6Fo(T p+11 − T p+10
). (3–81)
And the discretization equation for the radiation boundary node is:
T p+1m = T pm+3[
m3 −(m − 1
2
)3][Fom− 1
2(T p+1m−1 − T p+1m )
(m − 1
2
)2+
ϵσdt
ρCpdr(T 4∞ − T p+1m
4)m2
].
(3–82)
While, in general, schemes to solve matrix equations resulting from implicit finite
difference methods have computational complexity O(N2) or O(N3), certain simple
algorithms can be found for limiting cases. Such an algorithm is described for pure
conduction in the next section.
3.2.5 The Tridiagonal Matrix Algorithm
Once a partial differential equation is approximated by a series of finite difference
equations, whether in an explicit, implicit, or Crank-Nichols method, that system of
equations must be solved simultaneously for the unknown values of the dependent
variable at each node. There are numerous general methods that may be used to
solve such systems, several of which will be discussed in the present work. Methods
for solving systems of algebraic equations simultaneously can be grouped into two
56
categories: direct methods and iterative methods. Direct methods are simply those
employing a finite, deterministic, procedural solution to the system that requires no
iterative convergence. Iterative methods, on the other hand, require a procedural
calculation to be performed iteratively until an accepted convergence has been
achieved.
The first method to be discussed is a direct method that may be used to solve a
system of equations that, when written in matrix form, produce a tridiagonal matrix.
Such systems are commonly encountered in solving heat conduction equations.
First, the discretization equations are written in the following form (Patankar, 1980):
aiTi = biTi+1 + ciTi−1 + di .
Each discretization equation is, in general, only dependent on three consecutive nodal
unknowns, thereby producing a tridiagonal matrix when written as a matrix equation.
Here, the nodes i vary as 1, 2, 3, ... ,N. Note that for the special case of the boundary
equations, the coefficients c1 and bN are set as:
c1 = 0 and bN = 0.
Consider the discretization equation for the boundary at node 1. That equation
has as its unknowns T1 and T2. That relation may be substituted into the discretization
equation for node 2, resulting in an equation of two unknowns, T2 and T3. In general
each nodal equation can then be rewritten in the form:
Ti = PiTi+1 +Qi
where the coefficients Pi and Qi are given by the following relations:
Pi =bi
ai − ciPi−1
57
Qi =di + ciQi−1ai − ciPi−1
The solution of the system of equations is then found as follows (Patankar, 1980).
1. Calculate all Pi ’s and Qi ’s from i = 1 to i = N from the equations above.
2. Note that since bN = 0, then PN = 0 and therefore TN = QN .
3. Solve backwards for Ti from i = N − 1 to i = 1.
This simple algorithm is straightforward and relatively inexpensive computationally.
3.2.6 The SIMPLE Algorithm
The Tridiagonal Matrix Algorithm is used to solve a system of discretization
equations for the special case in which they produce a tridiagonal matrix equation.
Such a system is often encountered when solving heat conduction or mass diffusion
problems. In general, the solution of a partial differential equation that contains
convective terms and non-linear source terms produces a set of discretization equations
that are not so easily solved. In addition, many practical problems require the solution
of multiple partial differential equations simultaneously. One procedure for solving such
problems is the Semi-Implicit Method for solving Pressure-Linked Equations, or SIMPLE.
The SIMPLE algorithm was specifically designed to solve the Navier-Stokes
equations for the unknown velocity distribution when the pressure field is also unknown
(Patankar, 1980). In a two-dimensional flow situation, for example, the system of
equations consists of the continuity equation and two momentum equations (one for
each coordinate direction). These three equations are necessary to solve for the two
unknown velocity components and the pressure field. The problem’s chief difficulty
appears when one attempts to solve the discretization equations without regard to the
physics of the situation. Care must be taken if one is to obtain physically meaningful,
converged solutions.
In the SIMPLE algorithm, a first guess to the pressure field is used to solve the
momentum equations. The continuity equation is then solved producing a correction
58
to the pressure field. The corrected pressure is then used to calculate a corrected
velocity which becomes the final value for the velocity at the end of a given iteration. The
procedure is repeated iteratively until convergence is found. The steps in the SIMPLE
algorithm for a one-dimensional case are outlined below (Patankar, 1980).
1. Guess the pressure field p∗.
2. Solve the momentum equation to obtain the velocity field u∗.
aiu∗i = ai+1u
∗i+1 + ai−1u
∗i−1 + b + (pi − pi+1)Ai
3. Solve the continuity equation for the pressure correction, p′.
aip′i = ai+1p
′i+1 + ai−1p
′i−1 + b
4. Calculate the corrected pressure pi = p′i + p∗i .
5. Calculate the corrected velocity ui = u∗i +Aiai(p′i − p′i+1).
6. Once the velocity field is known, solve for other unknowns, such as T in the energy
equation, etc.
7. Repeat from step 2, until a converged solution is achieved.
The SIMPLE algorithm is a powerful tool for solving numerous partial differential
equations simultaneously. In the present work, the SIMPLE algorithm, or more precisely
the modified SIMPLER algorithm, is used to find the unknown velocity, pressure, and
temperature by solving the momentum, continuity, and energy equations simultaneously.
3.2.7 The SIMPLER Algorithm
The SIMPLER algorithm is a useful revision to the SIMPLE algorithm and stands
for Semi-Implicit Method for the solution of Pressure-Linked Equations, Revised. The
chief advantage of SIMPLER over SIMPLE are its improved convergence. Although the
computational effort required for one iteration of SIMPLER is larger than that of SIMPLE,
the faster rate of convergence of SIMPLER results in faster total computational times
over SIMPLE.
59
The major difference in the SIMPLER procedure is the manner in which the velocity
field is corrected. In SIMPLER, one starts with a guess for the velocity field and uses
this velocity to approximate the pressure field. With the pressure field at hand, the
momentum equations are solved for velocity. The velocity field is then corrected, but it is
no longer necessary to correct the pressure. The procedure is then repeatedly iteratively
until convergence. The steps in the SIMPLER algorithm are outlined in more detail
below (Patankar, 1980).
1. Guess the velocity field.
2. Calculate the pseudovelocity field,
ui =ai+1ui+1 + ai−1ui−1 + b
ai
3. Solve the continuity equation to obtain the pressure field, p∗, where the coefficients
are calculated from the pseudovelocities,
aip∗i = ai+1p
∗i+1 + ai−1p
∗i−1 + b
4. With p∗ known, solve the momentum equation for u∗,
aiu∗i = ai+1u
∗i+1 + ai−1u
∗i−1 + b + (pi − pi+1)Ai
5. Solve the continuity equation to obtain the pressure correction, p′,
aip′i = ai+1p
′i+1 + ai−1p
′i−1 + b
6. Correct the velocity field, but do not correct the pressure field. p = p∗,
ui = u∗i +Aiai(p′i − p′i+1)
7. Once the velocity field is known, solve for other unknowns, such as T in the energy
equation, etc.
8. Repeat from step 2, until a converged solution is achieved.
60
Note that SIMPLER does not rely on a guessed pressure field, but rather a guessed
velocity field as its first step as in the SIMPLE algorithm.
3.2.8 Solving for Roots of Non-Linear Equations
Many numerical endeavors require the routine solution for the roots of non-linear
algebraic equations. As such, it is necessary to include a short description of two
root-finding methods used commonly in the present work. The bisection method and
fixed-point iteration will be discussed.
3.2.8.1 The bisection method
The bisection method is a procedure that guarantees one to find a root given that a
function, f (x), is continuous on an interval [a, b], such that
f (a)f (b) < 0.
If the interval [a, b] can be chosen such that only one root is present, then the bisection
method can be guaranteed to find it. The steps in the bisection method are outlined
below (Atkinson, 1978).
1. Let c = (a + b)/2.
2. If (b − c)/c ≤ tolerance, then root = c and exit.
3. If f (b)f (c) ≤ 0, then a = c , otherwise b = c .
4. Return to step 1.
Essentially the bisection method halves the interval of interest for every iteration
through the algorithm. The interval is halved continuously until the desired tolerance
is achieved, calculated as the percent change from one guess c to the next. When the
tolerance is reached, the guessed root is the midpoint of the interval of interest in which
the root is known to lie.
The bisection method is not the fastest method of convergence, but it is the most
dependable in that it will always find a root in the given interval [a, b] if one exists.
61
3.2.8.2 Fixed-point iteration
The second root-finding method to be discussed is the general fixed-point iteration,
of which Newton’s method is an example. In fixed-point iteration one solves an equation
x = g(x) by performing the following iteration:
xn+1 = g(xn),
where x0 is an initial guess. Iteration of the equation above is performed until the error,
|xn+1 − xn|/xn+1, is sufficiently small. The benefit of fixed-point iteration is that for certain
functions its convergence is quite rapid. Unfortunately, in certain situations, the method
may fail to find a root and so a discussion of the uniqueness of solution is warranted.
It can be shown (Gerald, 1997) that if g(x) and g′(x) are continuous on an
interval [a, b] and if |g′(x)| < 1 for all x in [a, b], then the method of fixed-point
iteration will converge to a root in that interval. This condition, while sufficient, is not
always necessary in that a root may still be found even if |g′(x)| > 1. For practical
implementations where this condition may not apply, it is useful to examine if consecutive
xn values converge, that is: |x3 − x2| < |x2 − x1|.
The method of fixed-point iteration is used often in the current work to solve for the
electron number density as described earlier in the chapter.
3.2.9 Calculation of Higher-Order Legendre Polynomials
Legendre polynomials of the first kind, Pn(x) are solutions to Legendre’s equation:
d
dx
[(1− x2) d
dxPn(x)
]+ n(n + 1)Pn(x) = 0. (3–83)
Ordinary differential equations that can be reduced to this form occur frequently in
fluid dynamics and heat transfer, especially when formulating the conservation equations
in spherical coordinates. General solutions to the conservation equations often take
the form of a series of orthogonal basis functions. When those solutions involve the
62
Legendre polynomials as basis functions, it is necessary to evaluate a series of the
polynomials, Pn(x), to a sufficiently large n to guarantee convergence.
A general expression for the definition of each Legendre polynomial for any n is
given by Rodrigues’ formula
Pn(x) =1
2nn!
dn
dxn[(x2 − 1)n
]. (3–84)
where the domain is usually |x | < 1. The Legendre polynomials are then simple
to determine out to any necessary n by Rodrigues’ formula. The first six Legendre
polynomials are shown below and plotted in 3-4
P0(x) = 1 (3–85)
P1(x) = x (3–86)
P2(x) =1
2(3x2 − 1) (3–87)
P3(x) =1
2(5x3 − 3x) (3–88)
P4(x) =1
8(35x4 − 30x2 + 3) (3–89)
P5(x) =1
8(63x5 − 70x3 + 15x) (3–90)
The evaluation of these polynomials is trivial and the implementation of these calculations
within a numerical scheme is straightforward. A difficulty arises, however, when
calculating the Legendre polynomials for increasingly high values of n. In many practical
engineering applications the calculation of only the first ten Legendre polynomials in
63
the series may be needed to achieve good convergence. However, there are also many
practical cases, such as near boundary conditions or discontinuities, where the series
must be carried out to an excess of 100 terms or more in order to converge.
Consider, for example, the Legendre polynomial for n = 17 whose highest order
term is
P17(x) =1
229, 3764, 083, 810, 885x17 + · · · (3–91)
The most straightforward procedure for the numerical calculation of the series of
Legendre polynomials would be to store the appropriate coefficients for each term and
evaluate the standard form of each polynomial directly at each x required. But already,
at n = 17, the integer polynomial coefficients require at least ten digits of precision.
Moreover, the evaluation of the 17th power of an x within |x | < 1 may easily fall close
to machine precision. Ultimately, calculating the Legendre polynomials in this way,
arguably the most straightforward evaluation procedure, is prohibitive past n = 17 or
n = 18 due to the limitations of machine precision, which on many computers is around
17 to 18 digits.
Instead, one can take advantage of an alternate expression for determining the
Legendre polynomials. The Legendre polynomials are also obtainable from a recurrence
relationship given by:
Pn+1(x) =2n + 1
n + 1xPn(x)−
n
n + 1Pn−1(x). (3–92)
Using the recurrence relation we can calculate the Legendre polynomials at any x for
any sufficiently large n without the need to explicitly define each polynomial. It is easy
to see that with P0(x) = 1 and P1(x) = x , the repeated application of 3–92 n − 1
times results in the direct calculation of Pn(x) for any single x . In addition, since each
|Pn(x)| < 1, there is no danger of reaching machine precision.
64
It may, at first, appear that calculating the Legendre polynomials through the
recurrence relation is less efficient, since in the direct calculation the polynomial
coefficients are already defined, stored in memory, and may be accessed through
look-up. But this is not the case. The most efficient algorithms for the direct calculation
of a polynomial, such as Horner’s method, have a time complexity of Θ(n) (Horowitz,
1998). It is easy to see upon inspection that the evaluation of 3–92 for any n is also
done in Θ(n) time. In addition the direct calculation of a series of polynomials whose
coefficients are stored in memory requires Θ(n log n) space at best, whereas with the
recurrence relation only Θ(1), or constant space is needed.
The use of the recurrence relationship, 3–92 is therefore a more efficient method,
both in space and time, for calculating Legendre polynomials for any arbitrary n that
does not encounter the machine’s limits of precision.
3.3 Automated Peak Detection Algorithms
It is often desirable to automate the process of detecting peaks and recording
their characteristics in any spectroscopic application. This is especially desirable when
precise peak information must be taken from ensembles that contain numerous spectra
to the extent that peak detection by hand is not practical purely for time purposes.
However, automated peak detection methods are not without their difficulties.
Automating peak detection routines has many advantages and disadvantages. The
advantages are that peak detection is automated and can be completed in a fraction
of the time it takes the same process to be done by hand, the human bias is removed
(by a great deal, but not perfectly so) from the peak detection process, each peak is
known with the same certainty, and that certainty can be quoted confidently to within
fractions of pixels. The disadvantage of peak detection algorithms is the complexity of
said algorithms that is necessary to achieve confidence that one had detected every
important peak and not detected any false peaks. Removing the human bias from peak
detection is a two-way street. In order to ensure that all important peaks have been
65
correctly identified, it is often necessary to fine-tune certain parameters of the algorithm,
whose optimum values might differ from one case to another. These optimum values
are certainly not known beforehand. In addition, these parameters must not be defined
too conservatively. If not strict enough, the algorithm’s parameters may detect too many
peaks from what it should recognize as lower frequency noise.
All classes of automated peak detection algorithms must consist essentially of
three main components: smoothing, baseline correction, and peak finding (Yang, 2009).
All raw spectra that one might process with an automated peak detection routine are
presumed to come from real sources, such as atomic emission spectra, mass spectra,
or others. As such, all real raw spectra are known to contain some level of noise at
varying frequencies. The smoothing process is essentially designed to remove all noise
above a certain frequency. By applying some type of low-pass smoothing filter to the
data, much of the small, peak-like noise can be removed.
Baseline correction is essentially a means to normalize the spectra. One would
expect, or desire, that noisy sections of data, or sections that contain no peak information,
should be close to zero. All real data contain some baseline offset or continuum spectra
that must be removed to make the process of actually identifying peaks easier. Often
baseline correction can be achieved with a simple subtraction if baseline data is close to
uniform. There are cases, however, when baselines exhibit monotonically increasing or
decreasing behavior and the algorithm for baseline reduction grows in complexity.
Finally, once the baseline has been removed and the data smoothed to eliminate
obvious noisy fluctuations, only then can the actual peak finding routine be employed
with relative ease. In all but the most ideal or well behaved of cases, smoothing and
baseline correction will still leave some local maxima in the data that are not true
peaks one would want to detect. The peak finding process usually then consists of two
steps: identifying all local maxima and then determining which of the local maxima are
important and which are not important. The step of determining which local maxima are
66
worthy of being found as a peak is accomplished by the use of a threshold value defined
in any number of ways. All local maxima above this threshold value are counted as a
peak, while all the remaining local maxima below this value are not.
3.3.1 Smoothing
One of the simplest filters used for smoothing data is the Moving Average filter.
Each data point is recalculated to be a moving average of its surrounding k data points
given by the following formula,
x ′[n] =1
2k + 1
k∑i=−k
x [n − i ],
where x [n] represents the data before smoothing and x ′[n] represents the data after
smoothing. The parameter k determines the size of the filter width and therefore the
intensity of the smoothing effect. The filter width is given by the expression 2k + 1, which
is the number of points included in the moving average. The greater the value of k , the
greater the filter width, and the more intense the smoothing effect. The choice of the
parameter k is then paramount to the effectiveness of the smoothing operation. Too
high a value of k may reduce features that should be detected as peaks, while too low
a value increases the strain on the peak finding algorithm performed later and could
potentially result in the detection of a false peak.
Smoothing filters are also written as a convolution of the original data vector to the
filter window as seen below,
x ′[n] = x [n] ∗ w [n],
where w [n] is the filter window, which for the moving average filter is given by:
w [n] =1
2k + 1for − k ≤ n ≤ k .
67
Other smoothing filters offer much in the way of more robust smoothing, where values
of filter parameters can be chosen that produce good behavior for a given class of raw
data. Additional smoothing filters will be discussed in the future.
3.3.2 Baseline Correction
Once raw data is smoothed, the next step is to correct for the baseline. Baseline
correction essentially consists of two steps: determining the baseline of the data
and then the actual removal of the baseline. The second step is usually just a simple
subtraction. The first method we will discuss for the detection and removal of the
baseline is the Monotone Minimum method (Yang, 2009). The Monotone Minimum
method is most useful for a baseline whose behavior is monotonically decreasing from
the start to the end of the data. For optimal effectiveness of the Monotone Minimum
method one may wish to reorder the raw data depending on the baseline’s apparent
behavior. As a starting point we’ll suggest that the data points be reversed if the final
data point is greater than the initial data value (this correction, of course, assumes that
peak information is not contained in the first or last data point in the spectra). In other
words, if x [N] > x [0], then let
x ′[n] = x [N − n].
This reversal guarantees that if the baseline shows either a monotonically increasing or
decreasing behavior, that the baseline-corrected data will be ordered appropriately.
To determine the baseline, the difference between each consecutive data point is
first calculated to determine the slope s[n] at each point n given by:
s[n] = x [n + 1]− x [n].
Next the slope vector is scanned. If the slope of a point is negative, the value of those
points will be taken as baseline. If the slope of a point is not negative, the value of that
point will be the baseline for all subsequent points until a data point is found such that
68
its value is lower than the baseline. Once the baseline is determined, the baseline
corrected data is then given by:
x ′[n] = x [n]− b[n],
where b[n] is the baseline vector. If the data vector was reversed to ensure a monotonically
decreasing baseline above, then the baseline corrected vector must be re-reversed, as a
final step, to preserve the original pixel space.
3.3.3 Peak Finding
Once the raw data vector has been smoothed and the baseline corrected, it is then
possible to identify the available peaks. Smoothing and baseline correction may be
applied successively to produce a satisfactorily conditioned data vector. Here one will
assume that all smoothing and baseline corrections have been completed. At this stage,
the peak finding algorithm consists of two main parts: determining all local maxima and
determining which local maxima are peaks and which are not.
The determination of all local maxima is at this point a relatively trivial step. A local
maximum point is the point where the slope changes from positive to negative. Once all
local maxima have been identified one must then use some criteria to choose if a local
maximum is indeed worthy of being designated as a peak or if the local maximum is a
remnant of some lower frequency noise. Usually this decision process is based on a
simple threshold value that is a characteristic of the relative strength of a peak. Above
this threshold value the peak is ”strong” enough to be detected. Below this threshold
value, the peak is not ”strong” enough to be detected.
The decision criteria discussed here is the shape ratio. The area under the curve for
each local maximum will be determined. The criteria will be determined by the ratio of
each area to the maximum area found. Put another way, if:
AnAmax
> kT ,
69
then An represents a peak. Here kT is some threshold value that must be chosen.
Typically kT will be the most important factor that determines the sensitivity of the overall
peak detection algorithm and should be chosen with caution.
There are several other criteria by which one may choose if a given local maximum
qualifies as a peak. Employing several different criteria simultaneously may help to instill
confidence that no true peaks are neglected and no false peaks are detected. Absolute
peak intensity is one additional criteria that may used. Similar to the shape ratio, if the
absolute intensity ratio of any peak to the maximum peak is sufficiently large, that peak
may be a true peak. One may choose peak width, or the left-hand and right-hand peak
slopes as the criteria. In this case, a peak must be sufficiently wide in comparison to the
widest peak to be identified as a true peak.
3.3.4 Peak Idealization
Once a peak has been identified it is often necessary to do additional processing
on that feature depending on the application. One may wish to work with the original or
conditioned data and it is necessary to keep track of several characteristics of the peak
in addition to simply its location, such as its FWHM, its peak intensity, the location of
its endpoints and others. A useful alternative is to fit a characteristic profile to the peak
once it is identified . It is advantageous in many applications to have a simple closed
expression for each peak rather than a data vector depending on the analysis to be
done. The choice of the mathematical form a peak should take is subjective and should
be determined based on the underlying physical basis for the feature. The analysis of
physical images in the current study produce peaks that tend to fit well with Gaussian
profiles.
Each peak detected by the current algorithm that corresponds to a physical feature
is fit to a Gaussian function of the form:
g(x) = Ae−b(x−x0)2
70
where A is the maximum value of the peak, b is a value related to the width of the peak,
and x0 is the peak’s center location.
The analysis of spectra tends to produce peaks that may deviate from pure
Gaussian behavior. Instead Voigt profiles, which are combinations of both Gaussian
and Lorentzian profiles, are used to model peaks detected from spectroscopic data and
are discussed in 3.1.4.
71
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Frequency (νr = ν − ν
m)
Nor
mal
ized
Pro
file
Fun
ctio
n (S
)
Doppler ProfileLorentzian profileVoigt Profile
Figure 3-1. Comparison of Doppler, Lorentzian, and Voigt profile functions.
72
−8 −6 −4 −2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a = 0
a = 0.5
a = 1.0
a = 2.0
Frequency (νr = ν − ν
m)
Nor
mal
ized
Voi
gt F
unct
ion
(SV)
Figure 3-2. The Voigt profile function for various values of the damping parameter, a.
73
Figure 3-3. Control volume for a general interior node.
74
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
µ
Pn(µ
)
P0(µ)
P1(µ)
P2(µ)
P3(µ)
P4(µ)
P5(µ)
Figure 3-4. The first six Legengre polynomials of the first kind.
75
CHAPTER 4THE STATIC, CONDUCTIVE PLASMA MODEL
4.1 Overview
The first step toward the development of a rigorous model of the plasma-particle
interaction in aerosol LIBS is the design of a model to describe the plasma environment.
The plasma model describes the global environment in which the vaporization model,
described in the next chapter, will be contained. The complete model will be a synthesis
of these two regimes: the global model and the local model.
The global plasma model begins as a simple case to which additional complexities
and sophistications will be applied gradually. Building a simple, and therefore simply
testable, model and increasing sophistication gradually is necessary to ensure the
model behaves appropriately.
Here a simple plasma model is implemented, where the plasma is modeled as
a static, conductive gas. The temperature distribution in space and time is found by
solving the equation of heat transfer. The distribution of species concentration is found
by solving the equations of mass diffusion. The ionization state distributions and excited
energy level distributions are found from the Saha and Boltzmann relations. Finally, the
emitted intensity is calculated and used to simulate the experimental measurement of
temperature.
The numerical formulation that follows is implemented in the C/C++ programming
language and executed on a machine using a 2.6 GHz Intel Core 2 Quad processor. All
post-processing is done in Matlab.
4.2 The Problem Statement and Simplifying Assumptions
The plasma environment is modeled as a one-dimensional, time-dependent,
spherically symmetric system. As such, model input parameters and output quantities
will, in general, vary with both radius from the plasma center and time. The system is
assumed to be static, that is, the velocity field is zero everywhere and no convective
76
terms appear in either the energy transport equation or the mass transfer equation.
Local thermodynamic equilibrium is assumed to hold at all nodes for all times throughout
the modeling process.
The temperature field is found by solving the equation of energy transport, written
for a one-dimensional, time-dependant spherically, symmetric system as shown below:
1
r 2∂
∂r
(kr 2
∂T
∂r
)+ q = ρCp
∂T
∂t, (4–1)
Here it is assumed that conduction is the only mode of energy transport. The convective
terms, while playing a potential role in the physics, will not be modeled here for the
sake of simplicity and computational cost. Many laser-induced plasmas are modeled
as optically thin, and as such, the radiative terms in the energy equation can be shown
to be negligible for all but the earliest of plasma lifetimes (Gornushkin, 2001). The
radiative terms in the energy equation will be addressed again during the study of
plasma inception.
The species concentration distribution is found by solving the equation of mass
transfer, written for a one-dimensional, time-dependant, spherically symmetric system as
shown below:
1
r 2∂
∂r
(DABr
2∂CA∂r
)+ NA =
∂CA∂t. (4–2)
Since no bulk velocity field is assumed in this case, the only mode of mass transport is
through mass diffusion.
The material comprising the plasma is assumed to be pure argon gas. The
solution of the energy equation is then a statement of argon temperature at each
point in the plasma. Analyte species in the plasma may be either of two components:
cadmium or magnesium. These elements are used in the present study primarily
because experimental data exists for partial validation (Diwakar, 2007). The species
77
concentration of each throughout the plasma volume is found independently as a first
approximation.
Lastly, it is important to note that all material properties, and hence the coefficients
of the partial differential equations, are allowed to be functions of temperature. This
amounts to an energy equation with non-constant coefficients and a mass diffusion
equation with non-constant coefficients that is coupled to the energy equation.
Based on these considerations, the equations of heat transfer and mass diffusion
are then solved for the temperature and species concentration distributions using both
explicit and implicit finite difference formulations.
4.3 Numerical Formulation and Implementation
4.3.1 Heat Transfer
The energy equation to be solved is given by 4–1 in one-dimensional, spherical
coordinates. The partial differential equation is solved using finite difference approximations.
The problem domain is defined to be a sphere with a radius of 1.5mm, with temperature
evaluated at 101 nodes. Starting at this size neglects approximately the first 100ns of
rapid plasma expansion. The grid spacing is therefore:
∆r =1.5mm
101− 1= 15µm (4–3)
The simulated time is allowed to encompass a total of 30µs, evaluated at 30,001
temporal nodes. The time resolution is therefore:
∆t =30µs
30001− 1= 1ns (4–4)
Since the energy transport equation is second order in space and first-order in
time, two boundary equations and a single initial condition are required to uniquely
solve for the temperature distribution. The boundary node at r = 0 is taken as a
spherical symmetry condition (i.e., ∂T∂r
∣∣r=0= 0), which is mathematically implemented
78
as an insulated boundary. The boundary node at r = R loses heat by radiation to the
environment at temperature, T∞,
− k ∂T
∂r
∣∣∣∣r=R
= ϵσ(T (R)− T∞)4 (4–5)
4.3.1.1 The explicit finite difference formulation
The problem is first solved using an explicit finite difference formulation for simplicity.
The finite difference equations are derived using a control volume method described in
detail in Section 2.2.3. The finite difference equation for the internal temperature nodes
are given by the following:
T p+1n = T pn +1
n
[Fon− 1
2(T pn−1 − T pn )(n − 1) + Fon+ 1
2(T pn+1 − T pn )(n + 1)
], (4–6)
where the Fourier number is:
Fo =k∆t
ρCp∆r 2. (4–7)
The discretization equation for the symmetry boundary node is:
T p+10 = T p0 + 6Fo (Tp1 − T
p0 ) , (4–8)
The discretization equation for the radiation boundary node is:
T p+1m = T pm +3[
m3 −(m − 1
2
)3][Fom− 1
2(T pm−1 − T pm)
(m − 1
2
)2+
ϵσdt
ρCpdr(T 4∞ − T 4m)m2
].
(4–9)
Explicit finite difference formulations are attractive as their solution procedure is
simple. For each new time step, the discretization equations can be solved sequentially
for each node without the need for iteration. Convergence issues are therefore
79
avoided. Stability, on the other hand, is not. Explicit finite difference schemes are
”conditionally stable”, meaning that the numerical parameters must be chosen with
specific considerations to avoid physically unrealistic solutions.
In the present case, it is required that each coefficient of the discretization equations
above be positive. This is satisfied by applying the condition that Fo ≤ 1/2. Since
material properties cannot be prescribed arbitrarily, this is essentially a limitation on the
temporal and spatial grid spacing. For a spatial grid spacing of ∆r = 15µm, stability may
be guaranteed for time steps less than 26ns.
4.3.1.2 The implicit finite difference formulation
The finite difference approximation was also formulated implicitly and compared
to the explicit formulation. In general, implicit finite difference formulations are more
numerically accurate to true solutions and have the benefit of being ”unconditionally
stable”. Unconditional stability implies that any choice of grid spacing in space or time
will yield a physically realistic solution. The drawback of implicit methods are that, in
general, they must be solved using iterative methods and therefore may require more
computational time than explicit formulations.
Fortunately, many implicit finite difference formulations that involve conduction
or diffusion terms only produce systems that may be solved by the Tridiagonal Marix
Algorithm (TDMA). Since the current case falls into this category of problems, little
increase in execution time was found from the explicit to the implicit formulations for any
given time step. In addition, since the implicit formulation may be computed over fewer
time steps in the same domain, the total execution time may be reduced.
The implicit finite difference formulation is described in Section 2.2.4. The resulting
finite difference equation for each of the internal nodal temperatures is given by:
T p+1n = T pn +1
n
[Fon− 1
2(T p+1n−1 − T p+1n )(n − 1) + Fon+ 1
2(T p+1n+1 − T p+1n )(n + 1)
]. (4–10)
80
The discretization equation for the symmetry boundary node is:
T p+10 = T p0 + 6Fo(T p+11 − T p+10
), (4–11)
and the discretization equation for the radiation boundary node is:
T p+1m = T pm+3[
m3 −(m − 1
2
)3][Fom− 1
2(T p+1m−1 − T p+1m )
(m − 1
2
)2+
ϵσdt
ρCpdr(T 4∞ − T p+1m
4)m2
].
(4–12)
4.3.2 Mass Diffusion
The problem of mass diffusion is directly comparable to heat transfer as their
governing equations take the same form. As such, the same methods used for
the solution of heat transfer problems may be used for the solution of mass transfer
problems. Here, the mass diffusion equation is solved for the concentration of several
species within the plasma domain. The mass diffusion equation to be solved is given by
4–2. The problem domain is defined in the same manner as the discretization used for
the solution of the energy equation, namely:
∆r = 15µm and ∆t = 1ns (4–13)
The boundary condition at the center, r = 0, is, again, defined to by a symmetry
boundary condition to preserver the spherical symmetry of the system. The boundary
condition at the outer node, r = R, is defined to be diffusion out into an environment of 0
concentration.
The current problem considers three constituent species to be present. The plasma
carrier gas is pure argon. A particle of varying composition is placed at the center of
the plasma environment and consists of some mixture of cadmium and magnesium as
noted before. Once matter, be it cadmium or magnesium, is vaporized and liberated
81
from the particle (a process to be discussed in detail in Chapter 5) it diffuses throughout
the plasma environment.
The current model is then that of the diffusion of two species into a third: (1) the
diffusion of cadmium into argon and (2) the diffusion of magnesium into argon. Each
process will be solved independently and it is assumed that the presence of either
species does not effect the diffusion behavior of the other.
4.3.2.1 The explicit finite difference formulation
The mass diffusion equation is first solved by way of an explicit finite difference
formulation in much the same way as the energy equation. The finite difference equation
for the internal nodes of the cadmium concentration can be shown to be:
C p+1Cd ,n = CpCd ,n+
∆t
n∆r 2
[DCd→Ar ,n− 1
2(C pCd ,n−1 − C
pCd ,n)(n − 1) +DCd→Ar ,n+ 12 (C
pCd ,n+1 − C
pCd ,n)(n + 1)
].
(4–14)
The discretization equation for the symmetry boundary node is:
C p+1Cd ,0 = CpCd ,0 +
6DCd→Ar∆t
∆r 2(C pCd ,1 − C
pCd ,0
). (4–15)
The discretization equation for the outer boundary node, for the diffusion of mass into
zero cadmium concentration, is:
C p+1Cd ,m = CpCd ,m +
∆t
m∆r 2
[DCd→Ar ,m− 1
2(C pCd ,m−1 − C
pCd ,m)(m − 1)−DCd→Ar ,mC pCd ,m(m + 1)
].
(4–16)
The finite difference equations for the diffusion of magnesium into argon can be written
similarly.
The issue of stability must again be considered. The choice of discretization steps
to ensure the stability of the energy equation to not necessarily guarantee the stability
of the mass diffusion equation. Using a similar argument as before, one finds that to
82
ensure the stability of the explicit scheme for mass diffusion each coefficient in the
equations above must be positive. For typical values of the diffusion coefficient, it may
be shown that stability is guaranteed for time steps less than 9ns. While this is a much
more strict requirement on the time step than was found for the stability analysis for the
energy equation, it is still satisfied by the time resolution of 1ns chosen above.
4.3.2.2 The implicit finite difference formulation
The implicit finite difference formulation was again applied to the mass diffusion
equation to remove the requirement of stability and reduce the number of time steps
necessary to arrive at an accurate solution. Since the diffusion equations of each
species result in matrix systems that are tridiagonal, the TDMA method may be used for
their solution just as was done for the solution of the implicit discretization of the energy
equation.
The implicit finite difference equation for each of the internal nodes for the mass
diffusion of cadmium into argon is given by:
C p+1Cd ,n = CpCd ,n+
∆t
n∆r 2
[DCd→Ar ,n− 1
2(C p+1Cd ,n−1 − C
p+1Cd ,n)(n − 1) +DCd→Ar ,n+ 12 (C
p+1Cd ,n+1 − C
p+1Cd ,n)(n + 1)
].
(4–17)
The discretization equation for the symmetry boundary node is:
C p+1Cd ,0 = CpCd ,0 +
6DCd→Ar∆t
∆r 2(C p+1Cd ,1 − C
p+1Cd ,0
). (4–18)
The discretization equation for the outer boundary node, for the diffusion of mass into
zero cadmium concentration, is:
C p+1Cd ,m = CpCd ,m +
∆t
m∆r 2
[DCd→Ar ,m− 1
2(C p+1Cd ,m−1 − C
p+1Cd ,m)(m − 1)−DCd→Ar ,mC p+1Cd ,m(m + 1)
].
(4–19)
83
The discretization equations for the diffusion of magnesium into argon may be written
similarly.
4.3.3 Temperature Dependent Material Properties
Finite difference formulations for the solution of partial differential equations reach
an extra level of complexity when the coefficients of the equations are themselves
functions of the unknown nodal quantities. Each of the discretization equations written in
this chapter may be written in the following form:
aiTp+1i = ai−1T
pi−1 + ai+1T
pi+1 + bT
pi + c , (4–20)
in the case of an explicit formulation, and as,
aiTp+1i = ai−1T
p+1i−1 + ai+1T
p+1i+1 + bT
pi + c , (4–21)
in the case of an implicit formulation.
If the coefficients, ai , in these equations are constant, then the solution procedures
that have been described may be implemented to provide physically realistic solutions.
If the coefficients are not constant, but functions of the temperature, ai = ai(Ti), then
additional considerations must be made.
Typically, the procedure for the solution of finite difference equations with non-constant
coefficients follows that for constant coefficients, except for one additional iterative
procedure. At each new time step, the coefficients are evaluated based on the
temperature at the previous time as an initial guess. That is, api = api (T
p−1i ). The
coefficients are calculated in this manner and the nodal temperatures are solved. The
new nodal temperature will, in general, not be the same temperature as in the previous
time step. This new temperature is used to re-evaluate the coefficients and the process
is solved iteratively in this manner until the temperature no longer changes. Allowing
for the coefficients to be dependent upon temperature is an introduction of an iterative
procedure at each time step regardless of solution procedure.
84
This iterative solution may be avoided, however, if the time step is taken as
sufficiently small. If the time steps are small enough that the coefficients do not change
appreciably from one step to the next, then the coefficients may be approximated
from the temperature values of the previous time step. In this case, it is said that the
coefficients ”lag” behind the temperature solution by one time step.
Lastly, it is important to note that since the stability of explicit finite difference
formulations depend on the value of the coefficients of the discretization equation, it
is desirable to employ implicit formulations when the coefficients are strong functions
of temperature. If temperature varies over a broad range of values, as is the case in
a laser plasma, the stability criterion may be difficult to achieve, requiring prohibitively
small time steps. Using implicit finite difference formulations avoids to problem of
temperature-dependent coefficients from ”breaking” the solution procedure.
For the current purposes, temperature dependant properties yield non-constant
coefficients in the discretization equations.
4.3.3.1 Density
The temperature values in a laser-induced plasma vary greatly in a short distance
and over a short period of time. The pertinent problem domain will see temperatures
ranging from room temperature to tens of thousands of degrees. Because of this, the
material properties cannot be taken as constant. Instead they will be allowed to be
functions of temperature.
The first property examined is the argon gas density. Fujisake (2002) implements
a simulation of an argon plasma used in welding that uses a temperature dependent
model for argon density given by:
ρ = 1.783(273/T − 2.06× 10−7T + 6.72× 10−11T 2 − 5.21× 10−15T 3) (4–22)
85
This density model for argon is only valid below 15000 K. It is desired to develop
property models for the current purposes allowing for a maximum temperature of about
30000 K. Since density decreases monotonically with increasing temperature, density
is modeled to decrease down to a critical value, below which the density cannot fall.
This value is taken as the density at 15000 K as given in the model above. Density is
constant for increasing temperature beyond this point as shown in Figure 4-1.
4.3.3.2 Specific heat capacity
The specific heat capacity for an argon plasma is given by Maouhoub (1999) based
on measurements taken in plasma arcs at atmospheric pressure. Local thermodynamic
equilibrium is assumed. The specific heat values used for calculations in the current
study are taken as piecewise linear fits to Maouhoub (1999) and are shown in Figure
4-2.
4.3.3.3 Thermal conductivity
The thermal conductivity values for argon used in the present study are given
by Atsuchi, et al. (2005). There, the authors model an induction thermal plasma in
an investigation of non-equilibrium behavior for temperatures ranging up to 15000 K.
Thermal conductivity as a function of temperature for pure argon plasmas is shown in
Figure 4-3
The thermal conductivity is calculated as an approximation to the Chapman-Enskog
method. The values for thermal conductivity are modeled as constant above 15000 K.
4.3.3.4 Mass diffusion coefficient
The mass diffusion coefficient is in general a function of temperature and a function
of the two constituents in the diffusion process. Often the mass diffusion coefficient
may be modeled as a simple power law function based on a single reference value as
described in Incropera and Dewitt (2002). This relationship is written as:
D(T ) = Dref
(T
Tref
)3/2. (4–23)
86
As a first approximation the diffusion coefficient is calculated based on this relation
and is allowed to hold for both the diffusion of cadmium into argon and the diffusion
of magnesium into argon. Reference values are taken as Tref = 15000K and Dref =
0.04m2/s based on order of magnitude estimates.
While this temperature dependance suffices for temperatures below Tref , the value
of the diffusion coefficient quickly grows for temperature values much larger than this.
These values grow rapidly enough to induce unstable behavior in the explicit finite
difference solution and lead to impractically low choices for time resolution. In addition,
no useful physics are modeled by this relation.
Chapman-Enskog theory is used to model theoretical values for the diffusion
coefficients. The diffusion coefficient is calculated by:
DAB =3
16
(4πkBT/MWAB)1/2
(p/RuT )πσABΩDfD , (4–24)
where MWAB is the harmonic mean of the molecular weights of species A and B, σAB is
the arithmetic mean of the hard sphere collision diameters of species A and B, and ΩD
is a dimensionless empirical fit to temperature. The parameter ΩD is given by:
ΩD =1.06036
(T ∗)0.15610+
0.19300
exp(0.47635T ∗)+
1.03587
exp(1.52996T ∗)+
1.76474
exp(3.89411T ∗). (4–25)
The non-dimensional temperature, T ∗ is calculated from the Lennard-Jones energy for
each species by:
T ∗ =kBT
(ϵAϵB)1/2. (4–26)
The pertinent properties for the evaluation of Chapman-Enskog derived diffusion
coefficients are presented in Table 4-1
The diffusion coefficients calculated for each of these methods are shown in Figure
4-4 over the estimated temperature range expected in a laser-induced plasma.
87
It is also important to note that the mass diffusion equation is coupled to the energy
equation due to the temperature dependence of the mass diffusion coefficient. Since the
converse is not true, the energy equation may simply be solved first, at each time step,
and the resulting temperature may be used to evaluate the diffusion coefficient for the
solution of the mass transfer equation.
4.3.4 Determining Ionization State Distributions
Once the temperature and species concentration distributions are known, one
may then calculate the ionization state distribution of each species. Section 2.1.5
describes this process in detail. Here, a specific case is considered using the results
of section 2.1.5 as the solution. Consider a three-component plasma, where first- and
second-ionization states are allowed (z = 1, 2, 3). In this case, one may write three
species conservation equations:
ArT = ArI + ArII + ArIII, (4–27)
MgT = MgI +MgII +MgIII, (4–28)
CdT = CdI + CdII + CdIII. (4–29)
One may write six versions of the Saha equation, two for each species:
neArII
ArI= SAr,I (T ) , (4–30)
neArIII
ArII= SAr,II (T ) , (4–31)
neMgII
MgI= SMg,I (T ) , (4–32)
88
neMgIII
MgII= SMg,II (T ) , (4–33)
neCdII
CdI= SCd,I (T ) , (4–34)
neCdIII
CdII= SCd,II (T ) . (4–35)
Lastly, the system of equations is closed by considering the conservation of charge,
which is simply:
ne = ArII +MgII + CdII + 2ArIII + 2MgIII + 2CdIII
Recall the general solution to the system of equations is given by:
ne =
Z+1∑z=2
J∑j=1
Nj(z − 1)z−1∏i=1
Sj ,i
nz−1e
1 +Z+1∑w=2
w−1∏k=1
Sj ,k
nw−1e
. (4–36)
The for present case under consideration, this equation becomes:
ne =ArT(1 + ne
SAr,I
) + MgT(1 + ne
SMg,I
) CdT(1 + ne
SCd,I
).
This equation is now a function of ne alone and may be solved by a numerical procedure
such as the bisection method or fixed-point iteration. Once the electron number density,
ne is found, the ionization state distributions can be readily calculated.
89
4.3.5 Simulation of Plasma Radiative Emission
Once the temperature distribution and concentration distributions of neutral atoms
and ions are all known, the plasma composition is then fully determined. The scheme
is now in a position to simulate the act of spectroscopy by calculating the radiative
emission one would measure with a spectrometer. The simulated emission can be used
with common laboratory metrics to calculate temperature and electron density as one
would do in an experiment. With quantities such as temperature and electron density
known from theory, one may then assess the validity of such metrics.
The emitted intensity of a species from some excited state, i , to the ground state
may be calculated from:
Iij = Aijni(T , ne), (4–37)
where Aij is the transition probability and ni is the number density of excited state i .
With the total number density of each neutral atom and ion known, the number density
of each species in each excited state may be given by the Boltzmann relation in the
assumption of local thermodynamic equilibrium:
nin=giU(T )
exp
(− EikT
). (4–38)
Once the intensity distribution is known, the total intensity for each transition of each
species may be calculated as a volume-weighted average of the intensity distribution
(Dalyander 2008).
4.4 Results and Discussion
4.4.1 The Temperature Field
The temperature distribution as it changes with time is shown in Figure 4-5, where
the initial temperature profile is assumed to be a constant 15000 K throughout the
plasma volume. As the plasma loses heat by radiation to the environment at the outer
90
boundary a steep temperature gradient is observed. The innermost boundary exhibits a
flat gradient consistent with the symmetry condition imposed at that point.
The temporal evolution of the temperature distribution for the case where the initial
condition is prescribed as a parabolic profile is shown in Figure 4-6.
Figure 4-7 shows the temporal evolution of the plasma temperature at three points
within the plasma: the plasma center, halfway between the center and the edge, and
the plasma edge. The bulk temperature, estimated as an average value weighted
by the volume of each discretized control volume, is also shown. The temperatures
monotonically decay with time with the volume-weighted temperature more closely
following the temperature of the plasma core.
4.4.2 The Concentration Field
The distribution of cadmium atoms as it changes with time is shown in Figure 4-8
for early plasma lifetimes corresponding to the vaporization phase of the particle. Mass
enters the plasma volume from the center node, diffuses throughout the plasma volume
until finally diffusing out of the plasma from the outer boundary. A thorough discussion of
particle vaporization is included in the next chapter.
At longer times, after the particle has been fully vaporized, the concentration
field begins to settle. With no more cadmium atoms being added to the system, the
concentration gradually decreases through diffusion from the outer boundary and is
shown in Figure 4-9
Figure 4-10 shows the change in cadmium concentration with time at three
locations in the plasma volume: at the plasma center, halfway between the center
and the edge, and at the plasma edge. The cadmium concentration at the plasma
center gradually increases due to the net increase in cadmium atoms generated at that
location from the vaporization process and diffusion of those atoms to the surrounding
plasma. A marked change in behavior for the plasma center concentration is seen due
91
to the conclusion of the vaporization process. At that point the cadmium concentration
decreases monotonically.
The concentration of cadmium atoms at the plasma center and at the outer edge
both increase rapidly at early times due to the influx of mass from the center node due
to vaporization. Some time after the vaporization process completes, the concentration
at these locations begins to gradually decrease. The rate of diffusion of mass out of
the total plasma volume is observed to be significantly less than the rate of mass influx
through vaporization.
The concentration of magnesium atoms, while at different absolute vales, follows
the same behavior.
4.4.3 Electron Density
The distribution of electron number density is shown in Figure 4-11. Note that the
electron number density is highly dependant on temperature.
Figure 4-12 conveys the same information as Figure 4-11 except that the y-axis is
given on a uniform scale rather than logarithmic. The electron number density decays
rapidly with time in a similar fashion as the temperature profile. Electron number density
drops close to zero at the outer boundary of the plasma and retains a zero gradient at
the center corresponding to the symmetry condition.
The electron number density at three locations in the plasma are shown in Figure
4-13. The figure shows the temporal evolution of electron number density at the plasma
center, at halfway between the center and plasma edge, and at the plasma edge.
Electron density decays rapidly with time, with the centerline values greatly exceeding
that of the outer edge.
92
Table 4-1. Summary of parameters used in the evaluation of diffusion coefficientby Chapman-Enskog theory
i MWi σi ϵi[g/mol] [ang] [K]
Ar 39.948 3.408 119.9Cd 112.411 2.606 1227Mg 24.3050 2.926 1614
93
0.5 1 1.5 2 2.5 3
x 104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T [K]
ρ [k
g/m
3 ]
Figure 4-1. Argon gas density, ρ, as a function of temperature. See Fujisaki (2002).
94
2000 4000 6000 8000 10000 12000 140000
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
11000
T [K]
Cp [J
/kg−
K]
Figure 4-2. Specific heat capacity, Cp, of argon as a function of temperature. SeeMaouhoub (1999).
95
0 5000 10000 150000
0.5
1
1.5
2
2.5
3
T [K]
k [W
/m−
K]
Figure 4-3. Thermal conductivity, k , of argon as a function of temperature. See Atsuchi(2005).
96
0 0.5 1 1.5 2 2.5 3
x 104
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
T [K]
D [m
2 /s]
DD
Ar,MgD
Ar,Cd
Figure 4-4. Mass diffusion coefficient as a function of temperature.
97
0 0.5 1 1.5
x 10−3
0
5000
10000
15000
r [m]
T [K
]
0.1 µs1 µs5 µs10 µs15 µs20 µs30 µs
Figure 4-5. Plasma temperature distribution evolution with time for a flat initial profile.
98
0 0.5 1 1.5
x 10−3
0
0.5
1
1.5
2
2.5
x 104
r [m]
T [K
]
0 µs1 µs5 µs10 µs15 µs20 µs30 µs
Figure 4-6. Plasma temperature distribution evolution with time for a parabolic initialprofile.
99
0 0.5 1 1.5 2 2.5 3
x 10−5
0
0.5
1
1.5
2
2.5x 10
4
time [s]
tem
pera
ture
[K]
Model Temperatures
Volume WeightedPlasma centerHalfway to edgePlasma edge
Figure 4-7. Change in temperature with time at three locations in the plasma. Alsoshown is the volume weighted temperature’s evolution with time.
100
0 0.5 1 1.5
x 10−3
106
108
1010
1012
1014
1016
1018
C [#
/m3 ]
x [m]
1 µs5 µs10 µs15 µs
Figure 4-8. Concentration distribution of cadmium at early times.
101
0 0.5 1 1.5
x 10−3
1013
1014
1015
1016
1017
1018
C [#
/m3 ]
x [m]
15 µs20 µs30 µs
Figure 4-9. Concentration distribution of cadmium at later times.
102
0 0.5 1 1.5 2 2.5 3
x 10−5
100
102
104
106
108
1010
1012
1014
1016
1018
1020
time [s]
C [#
/m3 ]
Plasma centerHalfway to edgePlasma edge
Figure 4-10. Temporal evolution of cadmium concentration at three locationswithin the plasma.
103
0 0.5 1 1.5
x 10−3
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
n e [#/m
3 ]
x [m]
1 µs5 µs10 µs15 µs20 µs30 µs
Figure 4-11. Evolution of electron density with time on a logarithmic scale.
104
0 0.5 1 1.5
x 10−3
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
21
n e [#/m
3 ]
x [m]
1 µs5 µs10 µs15 µs20 µs30 µs
Figure 4-12. Evolution of electron density with time on a uniform scale.
105
0 0.5 1 1.5 2 2.5 3 3.5
x 10−5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
21
time [s]
n e [#/m
3 ]
Plasma centerHalfway to edgePlasma edge
Figure 4-13. Temporal evolution of electron number density at three locations in theplasma.
106
CHAPTER 5MODELING AEROSOL VAPORIZATION WITHIN THE LASER-INDUCED PLASMA
5.1 Overview of the Aerosol Vaporization Process
The plasma model described in Chapter 4 represents a simulation of the plasma
properties as they vary throughout its volume and as time passes. Energy transport and
mass transport by diffusion are allowed to govern the behavior and state of the species
within. The current model has considered a plasma gas comprised of argon in which
cadmium and magnesium atoms are diffused. The plasma model describes the global
distribution of temperature and concentration without regard to how the analyte species
of cadmium and magnesium come to be present.
This chapter discusses a model of aerosol vaporization within the laser-induced
plasma that considers not the global plasma environment, but only those conditions at
a local point that will govern the liberation of particle mass. The aerosol vaporization
model considers a single, stationary particle to be present at the plasma center. The
particle is comprised of equal amounts of cadmium and magnesium by mass. The
particle will vaporize gradually allowing more and more mass to be liberated from the
surface of the particle. Once that mass is liberated it becomes part of the global plasma
model and is allowed to diffuse throughout the plasma volume based on the theory
discussed in Chapter 4.
The interface between the local model of aerosol vaporization and the global
model of the plasma environment exists in the generation terms of the central node
discretization equations. Recall the central node discretization equations of Chapter 4
for the distribution of energy derived based on the symmetry boundary condition:
C p+1Cd ,0 = CpCd ,0 +
6DCd→Ar∆t
∆r 2(C pCd ,1 − C
pCd ,0
)(5–1)
If mass generation is allowed in the central node only, the discretization equation
becomes:
107
C p+1Cd ,0 = CpCd ,0 +
6DCd→Ar∆t
∆r 2(C pCd ,1 − C
pCd ,0
)+ NCd (5–2)
In this case, the generation term provides an increase (or decrease) in the concentration
of the central node. The results of this chapter will provide a description of the mass
generated as that which is liberated from the aerosol particle during the vaporization
process.
The vaporization process will be discussed in three contexts: Instantaneous
vaporization, linear vaporization, and a kinetic model of vaporization. Instantaneous
vaporization implies that mass is liberated from the particle, not truly instantaneously,
but rather instantaneously in comparison to the analytical time scales of Laser-Induced
Breakdown Spectroscopy. Next, linear vaporization of the aerosol particle will be
discussed where mass is liberated from the particle at a prescribed linear rate with time
for a simple comparison with the instantaneous rate. Finally, a rigorous kinetic model
of aerosol vaporization will be considered where each transition of aerosol phase is
considered.
5.2 Instantaneous Aerosol Vaporization
When one considers instantaneous vaporization, or any instantaneous process, it is
understood that the process does not truly take place instantly, but rather very quickly in
comparison to the period of time under consideration. No physical process that involves
the transport of mass can truly occur instantaneously since general relativity dictates
that mass and information can travel no faster than the speed of light.
In the case of aerosol vaporization in a laser-induced plasma, an instantaneous
vaporization rate implies that the process takes place over a time scale that is much
smaller than the analytical time scale of spectroscopy. Ideally, this is how the analytical
community views the vaporization of mass from aerosol particles in LIBS, as a process
that completes rapidly and fully before the analytical signal is collected.
108
This is assumed to be the ideal behavior for the sake of diagnostic feasibility.
Particle mass is liberated from the aerosol rapidly and that mass is distributed
throughout the plasma environment so quickly that the analyte signal read from
spectroscopic measurements is assumed to describe a uniform condition within
the plasma. The analyte signal, then, may be represented by a linear function of its
concentration within the plasma, and therefore directly related to the particle mass and
size.
The question of whether aerosol vaporization occurs so rapidly is the major issue
dealt with in this chapter. It is assumed, that while indeed rapid, the aerosol vaporization
process occurs at finite rates and that this deviation from ideal behavior does affect the
analyte signal.
5.3 Linear Aerosol Vaporization
The first step in modeling a description of aerosol vaporization more detailed
than the ideal assumption that it occurs instantaneously is logically to consider that
vaporization occurs linearly with time. In fact, many real kinetic vaporization processes
will show strong linear behavior in certain conditions. Here, as an initial attempt at
complexity, it will be considered that a single aerosol particle in a laser-induced plasma
will lose mass linearly with time at a rate that will be prescribed based on empirical
observations.
The aerosol particle will be assumed to vaporize completely over a period of time,
tv , and therefore the change in particle mass as a function of time, t, is given by the
following expression:
dm
dt=4
3πr 3p ρp
t
tv(5–3)
where m = m(t) is the particles mass as a function of time, rp is the particle radius, and
ρp is the particle density.
109
The particle under consideration here is composed of both cadmium and magnesium
defined to be in equal amounts by mass. The particle radius is taken to be rp = 100nm
and the time for total vaporization is set to be tv = 15µs . With this assumed linear
vaporization model, the amount of total mass in the plasma volume is shown in Figure
5-1.
Recall that instantaneous vaporization, considered as the ideal behavior, assumes
that vaporization and the diffusion of mass throughout the plasma volume is very rapid.
As such the entire aerosol particle’s mass would be distributed evenly throughout the
plasma volume. This is represented by the flat line in Figure 5-1. The case of linear
vaporization with a finite diffusion coefficient, as described in Chapter 4, is given by
the dotted line. Note that the total mass in the plasma volume increases during the
vaporization time, tv , and then decreases afterward as no more matter is added to
the plasma, yet matter is allowed to diffuse out to the environment. The non-linear
nature of the curve during the vaporization period shows the balance between the mass
influx from vaporization and the diffusion of mass out of the plasma. To show that the
vaporization process is indeed linear, another case is considered where mass is allowed
to diffuse throughout the plasma volume, but not out of the plasma volume. This is
represented in Figure 5-1 by the dashed line. The total mass increases linearly with
time, until the vaporization time is exceeded at which time the total mass remains at a
constant value consistent with the value from the ideal case.
5.4 Heat- and Mass-Transfer Modeling of Aerosol Vaporization
While many processes of aerosol vaporization may indeed yield linear behavior
with time, the modeling of linear vaporization at a prescribed rate lacks the rigor of the
kinetic theories of heat and mass transfer. Considered next is the complete aerosol
vaporization process modeled as a series of 4 steps. Each transition is considered
sequentially and is assumed to be independent of the next.
110
At the start of the simulation, t = 0 the particle is assumed to be at a uniform
temperature equal to room temperature. It’s sudden introduction into the plasma
environment, whose temperature greatly exceeds the boiling point of the particle, will
induce phase change in the following 4 steps:
1. Particle temperature increases to the melting point, Tm
2. Phase change from solid particle to liquid particle
3. Particle temperature increases to the boiling point, Tb
4. Phase change from liquid particle to vapor
Particle diameters under consideration here, rp = 100nm, are much smaller than the
discretized spacial steps assumed in the global plasma model, dr = 15µm. As such the
gaseous particle mass that is liberated from the particle surface in the last step of the
vaporization process yields the value of mass generation included in the global model.
5.4.1 Temperature Increase to the Melting Point
At time, t = 0, the aerosol particle, assumed to be perfectly spherical, is at
a uniform temperature equal to that of the ambient environment, T∞. Upon its
exposure to the laser-induced plasma, the first step that it will undergo is to increase
its temperature to the melting point. Here, it is assumed that the particle remains at
a uniform temperature throughout its volume as that temperature increases, since
Bi << 1, where Bi = (hD/3k) is the Biot number. For a given time step, the total change
in particle temperature during this process is given by the following expression:
∆TP =(Tg − TP)
e
[3hcM∆Z
ρs rCp,l Vcc
] (5–4)
where Tg is the local plasma temperature, TP is the particle surface temperature, M
is the molecular weight of the particle species, ∆Z is the grid spacing, ρs is the density
of the solid particle, and Vcc is the velocity of the particle during the time step under
consideration. The quantity, hc is a heat transfer coefficient based on the motion of the
particle through the plasma environment and is given by the following expression:
111
hc =Kg2r(2 + 0.515
√Re) (5–5)
where Kg is the thermal conductivity of the plasma evaluated at Tg (Horner 2007).
This kinetic process assumes that heat transfer is the limiting, and therefore,
governing mechanism for temperature increase. The process is assumed to be driven
by the difference in temperatures, Tg − TP , even though in reality a small layer of vapor
at some temperature between the two surrounds the particle.
Lastly, it is important to note that this phase may or may not be significant to the
overall vaporization process. Based on plasma temperature under consideration in
Chapter 4, this step in the process may take as little as a few nanoseconds or up to as
much as several hundred nanoseconds to complete.
5.4.2 The Melting Process
Once the aerosol particle has reached the melting point, the phase transition of
solid to liquid occurs. This process is modeled as a simple change of phase with all
other thermodynamic and mechanical traits remaining constant. The particle’s shape
remains spherical and the particle does not lose mass, it merely changes from solid to
liquid. This transition is therefore significantly simpler to calculate than the transition
from liquid to gas, where mass liberation does indeed occur.
It is assumed here, and throughout the rest of the chapter, that sublimation, that
is the transition from solid directly to gaseous species, does not occur. Sublimation
typically occurs at pressures much higher than that experienced by the laser-induced
plasma at atmospheric pressure.
The total time required for the particle to melt from solid to liquid is given by the
following relation:
∆tmelt =2ρsr∆Hfus
(Tg − Tm)Mhc(5–6)
112
where Tm is the melting point of the particle, ∆Hfus is the latent heat of fusion for the
particle, and hc is the same heat transfer coefficient described previously. The time
required for this transition to occur is typically greater than the individual time steps
described for the global plasma model. One may therefore calculate the total amount of
mass that has melted in a given time step by the following equation:
∆Mmelt =4πρs3
(∆Z(Tg − Tm)Mhc2Vccρs∆Hfus
)3(5–7)
Again, this model assumes that the particle temperature is uniform throughout and equal
to Tm.
5.4.3 Temperature Increase to the Boiling Point
Only after the particle has completely changed phase to liquid, is the temperature
increase to the boiling point considered. This process can be calculated in much the
same way as the increase in temperature to the melting point. The relation describing
the temperature increase in this phase is given by the following relation:
∆TP =(Tg − TP)
e
[3hcM∆Z
ρl rCp,gVcc
] (5–8)
This relation is almost identical to Equation 5–4 except that the particle density is now
that of a liquid, and the specific heat is that of a gas.
This process, much like the transition to the melting point is usually rapid as it
occurs before the plasma temperature has decreased significantly either through the
loss of energy to the previous vaporization steps or to the expansion and cooling of the
plasma to the environment.
5.4.4 The Vaporization Process
The last step in the overall vaporization process is the actual phase change of liquid
particle mass to gaseous particle mass and its subsequent liberation from the sphere
of influence of the particle. The evaporation phase is by far the most complex, and
therefore most computationally taxing portion of the overall vaporization model.
113
The vaporization process occurs when heat is transferred to molecules at the
surface of the liquid particle. Those liquid particles reach the boiling point and
move away from the surface at a velocity determined by the boiling point and at a
rate determined by the local vapor pressure of the material. The heating of surface
molecules and the subsequent liberation of those molecules is a process that combines
mass transfer and heat transfer. Since these processes are coupled together, the total
process is limited by the slower of the two. The remainder of this chapter will be devoted
to the determination of which mechanism limits the vaporization process and therefore
determines the rate of mass evaporated.
5.4.4.1 Heat transfer limited vaporization
First, one considers the case where vaporization is limited by the effects of heat
transfer. Heat conducts from the bulk plasma gas to the surface of the spherical particle
which causes its radius to change with time, often written as a quadratic expression
similar to the following:
r 2 = r 20 − kHT ,lt (5–9)
where kHT ,l is the heat-transfer limited rate of vaporization for large-particles. The
”large-particles” qualifier will be discussed below. The determination of this rate constant
is found from kinetic arguments that model the transfer of heat from the plasma gas
through a vapor layer and into the molten particle mass. This relation is given below:
kHT ,l =2MΛKg(Tg − TP)
∆Hvapρl(5–10)
where Kg is the conductivity of either the plasma or particle, whichever is lowest. The
quantity, Λ is the mass counterflow coefficient and is given by the following expression:
Λ =ln(1 + ∆Hov
∆Hvap
)∆Hov∆Hvap
(5–11)
114
where ∆Hov is the overall heat of vaporization given by the following expression:
∆Hov = Cp,g(Tg − TP) + β∆Hat + ϵ∆Hion (5–12)
where β is the fraction of species atomized, defined here to be unity and ϵ is the fraction
of singly ionized particles calculated from the Saha-Boltzmann equation. With the heat
transfer-limited vaporization rate constant now known based on these equations, the
mass lost by the particle per unit time is given by the following relationship:
dm
dt= −2πρlkHT ,l r =
(−4πMΛKg(Tg − TP)
∆Hvap
)r (5–13)
An important distinction needs to be made in regard to the validity of these relations.
This argument for the heat transfer limited rate constant for vaporization requires that the
particle be large in comparison to the mean free path of the plasma environment, such
that the situation falls within a continuum description. If a particle is small in comparison
to the mean free path, then heat transfer behaves slightly differently, in fact it is slowed in
comparison to the continuum heat transfer. This phenomenon is known as the Knudsen
effect.
The Knudsen number is a non-dimensional ratio representing the relationship
between the particle mean free path of the plasma and the length scale characteristic of
particle diameter, and is written as:
Kn =λ
2r(5–14)
Typically if the Knudsen number is smaller than about 0.001 it is stated that the particle
falls within the large-particle regime. If the Knudsen number is larger than this quantity,
then small-particle, or Knudsen effect, considerations need to be made. The Knudsen
effect is quantified as a correction to the heat transfer calculated in a continuum regime.
Since the heat transfer-limited rate of vaporization is directly related to the amount of
115
heat transferred, one may write the Knudsen effect in terms of the vaporization rate
constants as:
kHT ,lkHT ,s
=1
1 + Z∗
r
(5–15)
where Z ∗ is a quantity known as the temperature jump distance that describes the
distance over which the temperature changes from that at the particle’s surface to the
plasma gas. The temperature jump distance is given by:
Z ∗ =
(2− aa
)(γ
1 + γ
)(4Kg
ρgvgCp,g
)(5–16)
where a is the thermal accommodation coefficient, taken to be 0.8, and gamma is the
specific heat ratio, which for argon gas is 5/3.
Together, the temperature jump distance and Knudsen effect comprise a correction
to the previously calculated vaporization rate constant.
5.4.4.2 Mass transfer limited vaporization
The evaporation process is most likely heat transfer limited if there is a steep
temperature gradient around the particle, which usually occurs if the boiling point is
well below the plasma gas temperature. If the boiling point is close to, or exceeds
the local gas temperature, then evaporation transition is likely mass transfer limited.
Like the heat transfer-limited vaporization mechanism, the mass transfer process
occurs by different kinetics in the large-particle in Knudsen regimes. Therefore, both a
large-particle vaporization rate constant and a small-particle vaporization rate constant
will be developed.
In general, the change in radius with time of a particle under mass transfer-controlled
vaporization is given by the following expression:
dr
dt=
−MαPs
(2πMRTg)1/2ρl(1− α/2)(1 + αvgr
(1−α/2)D12
) (5–17)
116
where α is the evaporation coefficient, Ps is the saturated vapor pressure, and vg is the
aerosol particle velocity. This expression may be evaluated as given, or can be simplified
into large-particle and small-particle expressions.
In the large-particle regime, for Knudsen numbers smaller than about 0.001, it can
be shown that:
αvgr >> D12(1− α/2) (5–18)
Therefore, the particle radius as a function of time can be written similarly as the
large-particle case for heat transfer-limited vaporization as:
r 2 = r 20 − kMT ,lt (5–19)
where the vaporization rate constant, kMT ,l is given by:
kMT ,l =2MρsD12ρlRTg
(5–20)
In the case of small particles, the opposite condition, of 5–18 is true, namely:
αvgr << D12(1− α/2) (5–21)
In this case, the change of particle radius with time is known to follow a linear change
with time of the form:
r = r0 − kMT ,st (5–22)
where the vaporization rate constant, kMT ,s is given by:
kMT ,s =Mρs
ρl(2πMRTg)1/2α
1− α/2(5–23)
117
Transition regimes between Knudsen numbers about 0.001 and 0.1 are difficult to place
in either the large- particle or small-particle approximations. Therefore, for questionable
particle Knudsen numbers, the general equation for the change of particle radius for
time, given by 5–17, must be solved explicitly.
5.5 Results and Discussion
This chapter has outlined in detail a method for defining the individual transitions
that take place during the aerosol vaporization process. A particle, once introduced into
the plasma environment, increases in temperature to its melting point, undergoes phase
change from solid to liquid, then increases in temperature to its boiling point, and finally
undergoes phase change from liquid to vapor.
Emphasis has been placed on the fact that the particle vaporization kinetics are
a local process and are therefore governed not by the bulk plasma conditions, but
only the local conditions in the vicinity of a particle. Care must be taken, then, when
implementing the current vaporization model within the context of the global plasma
model introduced in Chapter 4. The global plasma temperature profile is solved for
first as described in Chapter 4. Once the temperature is known near the location of the
aerosol particle, the change in state of the particle, based on the transitions described in
this chapter, is then calculated.
As each time step passes, the particle’s history progresses sequentially along the
four transition steps of the present kinetic model. First, at the initial model time step,
the particle is assumed to have just been instantaneously introduced into the plasma
environment. Its temperature corresponds to that of the ambient environment, T∞.
During the first time step, its change in temperature is calculated based on the equations
for that transition. Each new time step increases the particle temperature until the
boiling point has been reached and at that point the program flow directs the local model
into the next transition. Each time step melts more and more of the particle, until it is
118
completely melted. Once the particle has become fully liquid, program flow directs the
next time step to the final step of vaporization.
Once the evaporation step is reached, only then is particle mass added into
the bulk plasma discretization equations by way of their generation terms. First, the
Knudsen number is calculated for the particle based on its current radius at that time
step. Based on this value, the proper regime, whether large-particle or small-particle is
assumed. Then, the heat transfer-limited vaporization rate constant is determined, kHT ,l
or kHT ,s . Next, the mass transfer-limited vaporization rate constant is determined, kMT ,l
or kMT ,s . Since the vaporization process is governed by whichever mechanism, heat
transfer or mass transfer, is slowest, the lower of the two rate constants is chosen as the
appropriate rate constant.
Based on this choice, the amount of particle mass that is liberated is calculated at
each time step and used as the value for the generation terms in the global model of
mass diffusion as discussed previously. The particle’s new radius is calculated and used
for the next time step until the particle is completely vaporized.
Since the diffusion of particle mass throughout the plasma volume is dependent
on the available mass that is close to, but liberated from, the vaporizing particle, the
diffusion is also dependent on the means by which particle vaporization occurs. The
atomic emission and LIBS response of an aerosol system is therefore dependent upon
the vaporization process as well. And indeed the different methods for numerically
modeling vaporization, whether instantaneously, linearly, or from a rigorous heat- and
mass-transfer scheme, affect the LIBS response.
Figure 5-2 shows the resulting mass diffusion throughout the simulated laser-induced
plasma volume in the first microsecond for the heat- and mass-transfer vaporization
model. The figure follows the radial symmetry of the model and shows the concentration
of cadmium liberated from a single aerosol particle located in the center of the plasma
volume on a logarithmic scale.
119
As time passes, particle vaporization continues simultaneously with the diffusion of
mass into the plasma environment as shown in Figure 5-3 after 5µs, in Figure 5-4 after
10µs, and in Figure 5-5 after 15µs.
By about 20µs after the initiation of the laser-induced plasma, the particle has fully
vaporized releasing no new cadmium atoms into the plasma volume as shown in Figure
5-6. The diffusion process continues, however, as the cadmium concentration seeks
equilibrium with the surroundings. After 30µs, as shown in Figure 5-7 the cadmium
concentration is approaching uniformity.
120
0 0.5 1 1.5 2 2.5 3
x 10−5
0
0.5
1
1.5
2
2.5
3
3.5x 10
−15
time [steps]
mas
s [g
]
InstantNo lossDiffusion out
Figure 5-1. Total aerosol mass in the plasma volume.
121
Figure 5-2. Simulated cadmium concentration throughout the plasma after 1µs .
122
Figure 5-3. Simulated cadmium concentration throughout the plasma after 5µs .
123
Figure 5-4. Simulated cadmium concentration throughout the plasma after 10µs.
124
Figure 5-5. Simulated cadmium concentration throughout the plasma after 15µs.
125
Figure 5-6. Simulated cadmium concentration throughout the plasma after 20µs.
126
Figure 5-7. Simulated cadmium concentration throughout the plasma after 30µs.
127
CHAPTER 6INVESTIGATION OF PLASMA INCEPTION
6.1 Introduction and Motivation for Early Plasma Studies
The present study has, as many before it, sought to model and understand
the various plasma-particle interactions present in LIBS and other plasma based
techniques. The various modeling efforts discussed in Chapter 2 have all considered
several of the most important mechanisms in the processes of laser plasma expansion,
plasma cooling, aerosol vaporization and radiative emission. Each of these studies has,
understandably, required the use of several simplifying assumptions to validate the use
of many fundamental theories of spectroscopic applications, such as the assumption of
local thermodynamic equilibrium.
One aspect of the laser-induced plasma behavior and its affect on plasma-particle
interactions that has largely been left unconsidered, with respect to the development of
a rigorous analytical model, is the area of plasma inception and of early plasma lifetime.
There are several reasons why this is so. First, the inception and early lifetimes of the
laser-induced plasma occur on a time scale on the order of less than 100 ns. This time
scale is almost always significantly shorter than the analytical time scales involved in
plasma diagnostics. The physical considerations that dominate during this time period
are thusly thought to be generally less important than those in later times.
Secondly, when one does consider the physics of plasma inception and early
plasma behavior, one notices that non-ideal, or at least, non-equilibrium effects are
likely to dominate in this regime. As such the modeling of the physics are much more
complicated and based on less direct or deterministic methods than the modeling of
efforts after this time period. It may be argued, then, that modeling efforts in this regime
offer little to the accuracy of existing plasma models and would incur relatively significant
computationally expense.
128
The contrary argument is made here, however. The plasma models that ignore
the physics of plasma inception and the dynamics of early plasma expansion must be
dependent upon some empirical information on which to base the initial conditions of the
model. Such a procedure is certainly valid, however, when the goal is to understand the
plasma-matter interactions on a fundamental level, any dependence on experimental
data to the model input is a limitation to its scope. This is especially true when one
considers the extent to which the long term behavior of many such numerical systems
are dependent upon the initial conditions. Any plasma model that does not consider the
mechanisms of plasma inception and early plasma lifetime is therefore incomplete and
open to refinement based on these considerations.
Ultimately, the manner in which a laser-induced plasma forms is likely to have some
effect on the resulting dynamics of plasma-material interaction and will therefore
influence the LIBS response. Consideration of the non-equilibrium behavior in
early plasma formation offers insight into the more complex features of the plasma
environment that are often assumed away.
Toward this end, a series of investigations is implemented to probe and model
the dynamics of plasma formation. An imaging experiment is performed to study the
behavior of early plasma inception events and the subsequent plasma formation at
early times in three different gases. The behavior of initial plasma inception is shown to
vary among the three gases: nitrogen, argon, and helium. Analysis of the differences
in plasma formation characteristics for the three gases suggests that the chemical
properties of the gas influence plasma inception. A theoretical investigation as to why
this is so is carried out.
6.2 Experimental Apparatus and Methods
An imaging study was performed to probe the behavior of laser-induced plasma
formation at its earliest observable lifetimes. The experimental system for this study
is shown in Figure 6-1. For all experiments a Q-switched Nd:YAG laser (Continuum)
129
operating at a fundamental frequency of 1064 nm was used as the laser source.
Furthermore, the laser power was about 400 mJ per pulse, with a 10 ns pulse width
and 1 Hz repetition rate. A 75-mm focal point lens was used to focus the laser beam to
a point to sufficiently cause breakdown within the six-faced sample chamber. Images
were collected with two separate cameras, and Andor ICCD camera and a PI-Max ICCD
camera, both oriented perpendicular to the direction of laser propagation. Each camera
was connected to a laboratory computer and images were recorded with accompanying
software as data sets consisting of two-dimensional arrays of numbers corresponding to
pixel counts across the CCD. The CCD chips on both cameras have a resolution of 1024
by 1024 pixels.
The ICCD camera and laser Q-switch were both triggered from a four-channel
digital delay/pulse generator (Stanford Research Systems, Model DG535). The camera
and laser were each triggered from individual channels of the delay generator with a set
delay between the two triggers so as to capture a specific time in the plasma life. Delay
gaps were adjusted so as to capture plasma images between 1 ns and 108.4 ns after
the laser pulse.
Using the described experimental setup, a series of plasma formation images
were taken over three different days and for three different ambient gases: nitrogen,
argon, and helium. Between 100 and 500 images were taken for each gas on each day
creating an ensemble of plasma formation images of about 1000 images for each gas or
about 3000 images total. In addition to the set of early lifetime data, images were taken
at various stages in the total lifetime up to extinction for comparison.
Lastly, the laser beam profile was measured using an ink-ablation method in order
to position the plasma formation images relative to the beam focal point. Ink was placed
on a series of colorless glass slides and then placed in the path of the laser beam. The
ablated area of the ink by the beam was used to provide an estimate of the beam profile
diameter. The position of each slide was varied along the direction of beam propagation
130
and recorded using the CCD. This provided a plot of the beam profile diameter, not only
in real space, but also in terms of the pixel coordinates of the CCD.
6.3 Data Processing and Analysis
The experimental procedure discussed in the previous section was used to
investigate the plasma inception event for three ambient gases: nitrogen, argon, and
helium, all at atmospheric pressure. A large ensemble of images, around 1000, for
each gas were recorded during the early times of plasma formation. A series of images
over the entire evolution of plasma lifetime of each gas were also taken. Figure 6-2
shows a collection of raw plasma images that were taken at various stages in the total
evolution of the plasma’s lifetime for nitrogen. At early times, less than 100 ns, the
plasma is forming from small, discrete breakdown kernels. The individual kernels grow
and coalesce, forming the full adult plasma around 100 ns. At much later times, on the
order of a few microseconds, the plasma deactivation begins as the excited states begin
to relax and emission fades.
However, it is the earliest plasma lifetimes that are of most interest in the present
study. Figure 6-3 shows another series of plasma life evolution, but over a much shorter
scale than Figure 6-2, better resolving the early progression of formation. In Figure 6-3
(a) the earliest breakdown events are seen clearly as individual and separate kernels.
After only 10 ns the kernels begin to grow together and the adult plasma begins to form
to the left (which is toward the excitation source).
The ensemble of data images for early plasma formation are all taken prior to 10 ns
after the initiation of the laser pulse and therefore resemble Figure 6-3 (a). Each of the
1000 images for each gas show a collection of small, discrete breakdown events that
vary in position and number along the direction of laser propagation. The characteristics
of each image vary somewhat predictably by gas. Each ensemble is processed as a
batch to calculate pertinent characteristics of plasma formation features.
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The large amounts of images collected, however, makes the manual calculation
of the statistical characteristics of each image unwieldy, and therefore an automated
procedure was designed to carry out the process. The fundamentals of the design and
implementation of the techniques of automated peak detection are discussed in detail in
Chapter 2.
6.3.1 Automated Peak Detection
Based on the techniques developed in Chapter 2, an automated detection scheme
was developed for the purpose of calculating the statistical nature of the observed
plasma inception kernels. The algorithms performed several identical steps for each
image based on parameters chosen from the examination of several test cases.
To start, each image was recorded as a two-dimensional array of data representing
the photon count at each pixel on the charge-coupled device (CCD) camera. It is then
assumed that each image resembles Figure 6-3 (a) in that it contains a series of bright
collinear spots whose number and geometrical characteristics are to be extracted. First,
the centerline of laser beam propagation is determined by binning each row of data
and finding the row of maximum count intensity. This operation identifies the pixel row
number of the centerline of the set of collinear spots.
The two-dimensional array of image information can now be condensed into
a one-dimensional profile based on the centerline along the direction of plasma
propagation. The one-dimensional strip used for analysis was taken as the sum of
three row profiles surrounding the centerline. The one-dimensional profile corresponding
to the image shown in Figure 6-3 (a) is shown in Figure 6-4.
After the one-dimensional profile is extracted, it is then analyzed based on the
automated peak detection algorithms described in Chapter 2 and consist of five main
steps: pre-processing, smoothing, baseline correction, peak-finding, and optimization.
The first step implemented in the analysis of raw image profiles consists of several
simple preliminary routines to condition the data to ensure successful and robust
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completion of the analysis. Preliminary tests are conducted on the data to ensure that
it contains valid numerical data values, the appropriate length (1024 elements), and
well conditioned bounds (often spectral data taken from a CCD chip may contain a
few elements of erroneous data near the edge). Other simple conditioning processes
are also implemented, such as ’cosmic ray’ removal. Often CCD pixel outliers may be
observed in random pixels attributed to random cosmic ray events falling onto the CCD,
from improper readout events, or from bleed-over from adjacent saturated pixels. This
often appears as a single bright pixel amid surrounding pixels with significantly less
recorded photon count. Such phenomenon, while rare, may indeed affect the calculated
results. A simple algorithm to remove any cosmic ray event is implemented through a
filter that removes all peaks that have a width of a single pixel.
The data is then sent through a series of smoothing filters in an effort to remove
pixel-to-pixel variation as a source of noise. Smoothing is performed by way of both
second-order and third-order moving-average filters. The smoothing operation is
generally that which requires the most scrutiny and attention from the user as it is
a routine with a tendency to alter working data in a negative way. While insufficient
smoothing produces a data set too noisy to extract meaningful features during the last
stage of analysis, too much smoothing can dampen peak values and, in extreme cases,
even erase entire features. Smoothing parameters are therefore re-evaluated during the
last stage of optimization and investigated manually for a variety of test cases.
Baseline correction was performed using the monotone minimum technique of
Section 2.3.2 which removes a monotonically increasing trend baseline from the data.
The entire data ensemble was well-behaved across the CCD in that baselines were
largely uniform for each image and therefore implementation of the removal routine was
robust.
Once the data was properly smoothed, and the baseline removed, the identification
of important peaks was carried out based on several criteria. A preliminary list of peak
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features was determined based on the area under each feature. Any peak with an
area greater than 1% of the total area is extracted as important. That list is further
refined over several steps. First, peaks with an insufficient width are removed as
false-positives. Second, the absolute magnitude of each peak is compared with the
strongest features. Peaks whose magnitude is a certain multiplier smaller than the
strongest are disregarded as insignificant.
The last step of data processing consists of several simple techniques to ensure
the retrieval of meaningful data. First, for each image, the number of peaks detected
is examined. A threshold value is chosen such that if the number of peaks detected is
above this value, the image must undergo processing again with more stringent filter
parameters. This step has shown to be most necessary in the analysis of helium images
where the low magnitude of features in relation to the noise level greatly increases the
difficulty of finding useful features. Secondly, once a set of peaks have been identified,
the image is processed again, using a slightly different set of filter parameters to
determine how the set of detected features will change. Generally, if slight variation
of the filter parameter produces the same (or similar) set of detected features then
the confidence that those features are truly important is greater. Cases where slight
variation in the filter parameters results in a significantly different set of detected
features are flagged for manual investigation. Lastly, each peak detected is fit to a
Gaussian function in order to determine its full-width at half-maximum (FWHM). The
value of FWHM is used to determine if the feature’s width is proper for its magnitude.
Any features with a FWHM that exceeds a certain threshold are flagged as unlikely
candidates.
The raw data profile along with the resulting processed results for a single case
in nitrogen gas is shown in Figure 6-4. The smoothing routine has removed much of
the high frequency noise, while still retaining the overall physical characteristics of the
inception kernels. The baseline has been removed properly and the algorithm has
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detected the presence of eight peaks in this test case for nitrogen. Manual inspection
of the resulting conditioned profile illustrates that a human user would identify the
same eight peaks as the algorithm as important features suggesting confidence in the
algorithm’s automated results.
A single representative case was examined in argon gas and the results are shown
in Figure 6-5. Note that the algorithm is observed to be successful in the determination
of the largest characteristic peaks, but fails to detect a few of the smaller features. This
does not constitute a failure on the part of the algorithm as the disregarded features may
or may not be important. Note that the spread of pixel range over these peaks is greater
than that of nitrogen gas.
A final test case was examined in helium gas and the results are shown in Figure
6-6. The algorithm detects the major features, although also detects a small peak that
may or may not be a true feature. Such behavior is possible due to the nature of the
shape ratio criterion for peak finding. Also note that a small, single-pixel-wide feature is
completely removed from the conditioned data as a result of the cosmic ray filter.
6.3.2 Plasma Inception Characteristics
With the algorithm developed and functioning properly with confidence for the
aforementioned test cases. The procedure is implemented to the entire ensemble of
3000 images collected in the study. Several metrics for each set of detected features
are chosen to be recorded for each image and the collective statistics for each are
examined. The characteristics recorded for each image are as follows:
1. number of peaks
2. area of each peak
3. full-width at half-maximum of each peak
3. pixel range over all peaks
4. minimum and maximum separation between consecutive peaks
5. number of resolved peaks versus number of combined peaks
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The variation of these characteristics over the three chosen gases: nitrogen, argon, and
helium yield insight into the how changes in chemistry affect plasma formation.
6.4 Experimental Results and Discussion
The experimental procedure outlined in Section 6.2 was carried out and data
analysis was performed on the resulting ensemble as discussed in Section 6.3. As a
result, a set of plasma inception statistics was collected for each gas. The beam profile
was measured and all pixel data was converted to real space for comparison.
Typical results from the ensemble of about 1000 images taken in nitrogen are
shown in Figure 6-7. The figure shows 30 well-conditioned results taken at random, over
all three days, from the ensemble. In Figure 6-7 each collinear set of points represents
an inception image. While each set of inception points were in reality located along the
beam’s centerline, they are shown in the figure displaced above and below for clarity.
The beam profile, as measured, is shown by the dashed line, while a polynomial fit of
the beam profile is shown as a solid line.
Nitrogen kernels, as shown in Figure 6-7, are typically uniformly distributed about
a 3mm region downstream of the beam’s focal point, away from the excitation source.
On average between 7 and 8 plasma inception events were recorded for each image in
nitrogen.
Figure 6-8 shows a similar plot for 30 well-conditioned results taken at random, over
all three days, from the ensemble of images in argon. Each collinear set of inception
points, again represents a single image, and they are displaced above and below the
centerline for clarity. Here, it can be see that the behavior of the distribution of inception
events in argon differs markedly from that of nitrogen. Figure 6-8 shows that peaks are
spread over about 4mm starting at the beam focal point and continuing downstream,
away from the excitation source. Instead of a uniform distribution of inception points,
however, the events are distributed bi-modally. On average between 5 and 6 inception
events were recorded for each image in argon.
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Lastly, Figure 6-9 shows a similar plot for 30 well-conditioned results taken at
random, over all three days, from the ensemble of images in helium. Each collinear
set of inception points, again represents a single image, and they are displaced above
and below the centerline for clarity. While the spatial spread of plasma inception events
in argon and nitrogen were both relatively wide, the spread of events in helium is
significantly smaller, covering an area of only about 2mm downstream from the focal
point. The distribution of events in helium, like nitrogen, was highly uniform about
this region. On average between 4 and 5 plasma inception events were recorded for
each image. The ensemble of images in helium were particularly difficult to produce
consistently acceptable processed results based on the previously discussed automated
routine. Manual inspection of the processed results, therefore, suggests that the average
number of plasma inception events for each image in helium should actually be between
3 and 4.
When comparing the results of the plasma inception characteristics over each
ambient gas, it is interesting to note that most events begin downstream (away from)
of the laser beam focal point. It first glance, this seems counter-intuitive. A plasma is
known to form in ambient gas when the photon density in the laser beam becomes
sufficiently high enough for breakdown to occur. The largest photon density in a focused
beam occurs at the focal point, and it is therefore intuitive that the plasma should form at
the focal point, not downstream from the focal point. However, consider Figure 6-10 that
shows several plasma contours within the beam profile at various times. At the earliest
times, shown in Figure 6-10 (a), individual plasma inception events form downstream of
the beam focal point. At later times, however, in Figure 6-10 (b) and (c), the adult plasma
grows from the initial inception events towards the excitation source and ultimately forms
at the beam focal point in accordance with intuition.
Figure 6-11 shows the final results of the plasma inception study by comparing the
distribution of individual breakdown events for each of the three gases. The minimum,
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average, and maximum locations for breakdown for each gas are shown along with
the beam profile. Nitrogen and argon exhibit a similar range of events spatially, though
nitrogen does so uniformly and argon bi-modally. The range of events in helium are
packed significantly tighter in helium and also uniformly distributed. Note that for each
gas, the distribution of breakdown events is highly repeatable, but vary from gas to gas.
This suggests that the difference in behavior each gas exhibits is due the chemistry of
that gas rather than influences from the laser source or optics.
6.5 Theoretical Considerations and Conclusions
There are two primary mechanisms for the growth of free electrons in the formation
of a laser-induced plasma: cascade ionization and multi-photon ionization (MPI). In
cascade ionization, a free electron impacts an atom, causing ionization, producing an
additional free electron. This leads to a rapid growth of free electron density in a gas and
plasma formation. In multi-photon ionization, multiple photons are all incident on a single
atom at once such that the sum of the photon energies exceeds the ionization energy of
the atom and a free electron is produced.
It is generally thought that both processes play a separate role in laser-induced
plasma formation. The rapid growth of the plasma after the initial breakdown is
commonly attributed to cascade ionization. But for cascade ionization to take place
there must already be free electrons present, or at least a first free electron present to
impact an atom. That first electron is thought to be produced by multi-photon ionization.
Individual plasma inception events, such as that shown in Figure 6-2 (a), may then
correspond to individual instances of multi-photon ionization that create the seed
electrons needed for cascade growth.
Consider the likelihood of multi-photon ionization in gases such as nitrogen, argon,
or helium. The ionization energies for nitrogen, argon, and helium are 1503 kJ/mol, 1520
kJ/mol, and 2372 kJ/mol respectively. Relating these values to the energy in a single
photon from a laser source operating at 1064 nm, it requires 14 simultaneous photons
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to cause ionization in nitrogen and argon, and 21 simultaneous photons to cause
ionization in helium. The likelihood for this to occur can be evaluated by considering the
distribution of photon density in the laser beam along with the probability of photon-atom
interaction.
Figure 6-12 shows a simulation of the distribution of photon density in a laser beam
corresponding to the profile measured in previous sections, with a Gaussian distribution
of energy across its diameter and a pulse energy of 400 mJ. This distribution of photons
exists within an exposure time of 2.9× 10−5ns, the amount of time it takes for a photon of
light to traverse one pixel. The bottommost row of pixels in the figure corresponds to the
beam centerline. The maximum photon density therefore occurs along the centerline at
the point of minimum beam diameter and is about 7× 1015photons/mm3.
An average number of pixels per atom (or molecule) can be calculated by
multiplying the photon density by the volume of a single atom (or molecule). The volume
of an atom (or molecule), however, is not a straightforward property when considering
the sphere of influence a nucleus and its electron cloud exhibits on surrounding
photons. The Van der Waals radius is useful to model an atom (or molecule) as a
hard sphere, but it is unlikely a good estimator of the volume over which a photon must
be within in order to be influenced by the particle. For the purposes of the argument,
a radius of influence of twice the Van der Waals radius will be considered to define the
appropriate volume in which a photon must be to be influenced, or absorbed, by an
atom or molecule. The average number of photons per nitrogen molecule at the peak of
photon density would be about 1.6× 10−3photons/molecule for a single exposure.
A single exposure, however, represents only a small fraction of the time that a
photon, or group of photons may interact with a molecule. For the purposes of this
discussion, an exposure represents the amount of time it takes for a photon to traverse
the length of one pixel in space, about 2.9 × 10−5ns. The amount of time required for
multi-photon ionization to liberate an electron can be estimated to be on the order of
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about 1 × 10−10s (Kulander, 1987). Therefore, from the perspective of the atoms or
molecules present in a single pixel, a photon may linger within the sphere of influence
over a period of about 1000 exposures. This effectively increases the density of photons
that may impinge a particle simultaneously by a factor of about 1000. The peak value,
therefore, for the average number of photons per nitrogen molecule at the peak of
photon density is about 1.6.
Consider further that the arrival of a single or multiple photons to a target, such as
a CCD pixel or particle, is described well by Poisson statistics. The probability that n
photons arrive within a target volume simultaneously is given by:
Pn =µne−µ
n!, (6–1)
where µ is the average number of photons per target volume. The probability of a
multi-photon ionization event in nitrogen over a single pixel can therefore be estimated
by substituting µ = 1.6 and n = 14 in the above equation and multiplying by the number
of particles in a single pixel. This gives the probability of a multi-photon event at the
peak of photon density to be on the order of about 1.
A simulation of the distribution of the probability of a multi-photon ionization event
in nitrogen is shown in Figure 6-13. Here the bottommost row corresponds to the laser
beam centerline. Similar distributions for the probability of MPI events in argon and
helium are shown in Figures 6-14 and 6-15, respectively.
The distribution of MPI probabilities in helium shows a distinctly smaller variation
spatially, than either argon or nitrogen. This is primarily due to the difference in
ionization energy. Not only is it more difficult to free an electron from helium by
multi-photon ionization, but the region in space over which this is possible is smaller
as well. This agrees well with the previous results for the distribution of plasma inception
events.
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6.6 A Note on Spherical Aberration
The differences in the observed spatial distribution of the individual plasma
inception events for each gas has been discussed. While this discussion has focused on
why the spatial distribution varies with the gas, it is still left to be considered as to why it
exists at all. A possible, and likely, explanation of the existence of the spatial distribution
of plasma kernels is due to the presence of spherical abberation on the focusing lens.
As the laser source passes through the focusing lens, light rays are refracted towards
the focal point as described by Snell’s Law. Spherical aberration is essentially due to
the presence of the non-ideal curvature of the lens resulting in an imperfect focal point.
In fact, lenses suffering from spherical aberration, produce, not a single focal point,
but a finite region over which the beam diameter is a minimum. This is caused by the
non-uniform refraction of light rays impinging the lens farther from its center. Lenses
that do not suffer from spherical abberation are known as aspheric lenses and are
characterized by surface profiles that are not simply portions of spheres or cylinders.
Such lenses are more difficult to manufacture and are thus more costly.
The fact that the lens used for the current study was spherical, and therefore suffers
from spherical abberation is one possible explanation behind the existence of the spatial
distribution of plasma inception events. As there is not one single focal point, but a small
range of minimum diameter, then there is an entire region of space where more than
one multi-photon ionization events may take place. However, most LIBS experiments in
the literature use spherical lenses and such a practice is not detrimental.
The fact remains that the distribution of individual plasma inception kernels does
change depending on which ambient gas is being observed. So while the existence of
the distribution of kernels may be due of spherical abberation it is still the gas chemistry
that is responsible for its characteristics.
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Figure 6-1. Schematic of experimental LIBS apparatus for plasma inception study.
142
Figure 6-2. Evolution of laser-induced plasma over its lifetime. (a) Early plasmaformation 20 ns after pulse, (b) Early plasma formation 30 ns after pulse, (c)Early plasma formation 40 ns after pulse, (d) Fully formed laser-inducedplasma 100 ns after pulse, (e) Plasma begins to relax 1µs after pulse, (f)Plasma deactivates and decays 2µs after pulse.
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Figure 6-3. Laser-induced plasma formation in nitrogen at early times. (a) Early plasmainception events at a very short time (∼ 1ns) after pulse, (b) Early plasmaformation 10 ns after pulse, (c) Early plasma formation 20 ns after pulse,(d)Early plasma formation 30 ns after pulse, (e) Early plasma formation 40 nsafter pulse.
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0 50 100 150 200 250 300 350 400 4500
500
1000
1500
2000
2500
pixel
coun
t
Figure 6-4. Line profile across the CCD showing early plasma inception features innitrogen. The upper profile shows the raw, unprocessed signal, while thelower profile shows the processed signal with peaks identified.
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200 300 400 500 600 700 800 9000
1000
2000
3000
4000
5000
6000
pixel
coun
t
Figure 6-5. Line profile across the CCD showing early plasma inception features inargon. The upper profile shows the raw, unprocessed signal, while the lowerprofile shows the processed signal with peaks identified.
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400 450 500 550 600 650 700 750 8000
200
400
600
800
1000
1200
1400
1600
1800
2000
pixel
coun
t
Figure 6-6. Line profile across the CCD showing early plasma inception features inhelium. The upper profile shows the raw, unprocessed signal, while thelower profile shows the processed signal with peaks identified.
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−5 −4 −3 −2 −1 0 1 2 3 4−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x [mm]
y [m
m]
Figure 6-7. Collection of 30 plasma inception images in nitrogen in relation to the laserbeam profile. Each set of inception points occur along the centerline inreality, but are shown displaced for clarity.
148
−5 −4 −3 −2 −1 0 1 2 3 4−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x [mm]
y [m
m]
Figure 6-8. Collection of 30 plasma inception images in argon in relation to the laserbeam profile. Each set of inception points occur along the centerline inreality, but are shown displaced for clarity.
149
−5 −4 −3 −2 −1 0 1 2 3 4−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x [mm]
y [m
m]
Figure 6-9. Collection of 30 plasma inception images in helium in relation to the laserbeam profile. Each set of inception points occur along the centerline inreality, but are shown displaced for clarity.
150
Figure 6-10. In relation to the beam profile, plasma inception events occur past the focalpoint, where the plasma forms at the focal point. (a) Early plasma inceptionevents shortly after the pulse (∼ 1ns), (b) Early plasma formation 20 nsafter pulse, (c) Early plasma formation 40 ns after pulse.
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−5 −4 −3 −2 −1 0 1 2 3 4−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x [mm]
y [m
m]
Nitrogen
Argon
Helium
Figure 6-11. Summary of plasma inception statistics for nitrogen, argon, and helium inrelation to the laser beam profile.
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Figure 6-12. Simulated image of the distribution of photon density across several pixelsof the CCD.
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Figure 6-13. Simulated distribution of the probability of a multi-photon ionization event innitrogen.
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Figure 6-14. Simulated distribution of the probability of a multi-photon ionization event inargon.
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Figure 6-15. Simulated distribution of the probability of a multi-photon ionization event inhelium.
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CHAPTER 7CONCLUSIONS
7.1 Summary
The current study endeavors to understand and quantify the complex plasma-particle
interactions that take place during the laser-induced breakdown spectroscopy of
aerosol systems. Importantly, applications extend to other analytical methods such
as Inductively-Coupled Plasma Atomic Emission Spectroscopy (ICP-AES) and
Laser-Ablation Inductively-Coupled Plasma Mass Spectrometry (LA-ICP-MS), where
plasma-particle interactions in the ICP are analagous to the current study. The study
of the plasma-material interactions is being accomplished through the design and
implementation of a numerical model that takes into account the individual processes of
heat transfer, mass transfer, and vaporization kinetics. Several advancements have been
made toward this goal.
First, the global plasma environment has been modeled by simulating the processes
of heat transfer and mass transfer through diffusion. Based on a prescribed initial
condition and appropriate boundary conditions, the energy equation for conduction is
solved numerically using an implicit finite difference scheme to obtain the temperature
field as a function of plasma radius and time. Mass diffusion is allowed throughout
the plasma environment and the mass transfer equation is solved through a similar
procedure as the energy equation to obtain the concentration field, also as a function
of plasma radius and time, for the various plasma constituents. Once the temperature
and concentration fields are known, several plasma properties are calculated, such as
electron density, ionization state distributions, and emission intensity.
Second, the local plasma-particle interactions are modeled through various
methods to simulate the processes of aerosol particle vaporization and dissociation.
Vaporization is first simulated to occur at a constant prescribed rate as a preliminary
method to investigate the effects of a finite vaporization rate versus an instantaneous
157
rate. Next, vaporization is modeled as a series of distinct steps of melting, evaporation,
and species liberation. Atoms are removed from the aerosol particle at a rate that is
either controlled by heat transfer or mass transfer depending on the current state of the
environment.
Modeling efforts show that the particle vaporization, mass diffusion, and heat
transfer processes that take place, do so over finite time scales. These results show
that while it is often commonplace for researchers to assume that these processes
take place with sufficient rates to be assumed instantaneous, this may not be the
case, especially for early times. Furthermore, finite vaporization and diffusion rates
affect the LIBS response and knowledge of these processes may lead to an increased
understanding of how matrix effects influence the diagnostic. Results suggest that since
the governing processes occur over finite, but rapid, time scales that LIBS observation
should take place at later times to justify the simplifying assumptions and allow time for
the analyte species to diffuse through and equilibrate with the entire plasma.
Lastly, an experimental study has been performed to investigate the earliest times
of plasma existence in order to further the understanding of the physics of plasma
inception. Plasmas were created in several different gases and their behavior at the
earliest observable lifetimes was studied. At early times, plasmas form not from a single
breakdown event, but from several initial breakdown kernels located downstream from
the laser focal point. The number and spatial distribution of initial breakdown events
varies by medium. As time passes the individual breakdown kernels grow and coalesce
toward the laser source, culminating in a fully formed plasma located in the center of the
laser focal point. Spherical abberation of the focal lens and the values of the ionization
energy for the different gases are used to provide an explanation for this behavior.
7.2 Suggestions for Future Research
While there has been much work done towards the fundamental understanding
of the complex plasma-material interactions that govern the LIBS of aerosol systems,
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there are many ways in which the present research may be extended to provide further
insight. Based on the previously discussed results, the following efforts are proposed for
future research:
• Implementation of the solution of the velocity field based either on point-blasttheory, or a full solution of the Navier-Stokes equations. The velocity field maythen be used to determine the importance of convective terms of heat and masstransport. It is desired to also account for compressibility effects, and therefore thepresence of the plasma’s spherical shockwave at early times.
• Evaluation of radiative modes of heat transfer in the global plasma environmentmodel.
• Investigation of the effects of spectrally dependent quantities through theimplementation of line broadening mechanisms and the calculation of line profilefunctions to generate model output that simulates spectra.
• Investigate the effects of electromagnetic forces during the duration of the laserpulse to the formation of the laser-induced plasma. The electric and magneticforce terms act as source functions to drive the hydrodynamic motion during theperiod of time when the laser pulse is active creating a fully magnetohydrodynamicmodel of laser-induced plasma behavior.
• Introduction of a theoretical model of plasma inception, thereby removing thesemi-empirical nature from the current plasma model. The plasma inceptioncharacteristics may explored theoretically through the introduction of the effectof the electromagnetic forces present in the exciting laser pulse to the initialconditions of the system or by the evaluation of a Monte Carlo simulation to theassumption of local thermodynamic equilibrium at early plasma times.
Together with the previously established model, these additions and refinements will
comprise a sophisticated and inclusive description of the processes important to LIBS
of aerosols from which much fundamental knowledge may be gleaned and used for the
benefit of the Laser-Induced Breakdown Spectroscopy research community.
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BIOGRAPHICAL SKETCH
Philip Jackson was awarded bachelor’s degrees in both Aerospace Engineering
and Mechanical Engineering at the University of Florida in 2003. He received a master’s
degree in Mechanical Engineering under Dr. Jill Peterson at the University of Florida in
2005. He is currently a research assistant in the Laser-Based Diagnostics Laboratory at
the University of Florida while pursuing a doctoral degree under Dr. David Hahn.
167
