FLUCTUATIONS IN THERMAL FIELD OF WORKPIECE DUE TO …

The Pennsylvania State University

The Graduate School

College of Engineering

FLUCTUATIONS IN THERMAL FIELD OF WORKPIECE

DUE TO MULTIPLE PULSES

IN ULTRA-HIGH PRECISION LASER NANO-MACHINING

A Thesis in

Engineering Science and Mechanics

by

Hana H. Krechel

© 2014 Hana H. Krechel

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

December 2014

ii

The thesis of Hana H. Krechel was reviewed and approved* by the following:

Edward W. Reutzel

Associate Professor of Engineering Science and Mechanics

Thesis Advisor

Vladimir V. Semak

Associate Professor of Engineering Science and Mechanics

Timothy F. Miller

Energy Science and Power Systems Division Head, Applied Research Laboratory

Graduate Faculty in Mechanical Engineering

Judith A. Todd

Professor of Engineering Science and Mechanics

P. B. Breneman Department Head

*Signatures on file in the Graduate School

iii

ABSTRACT

Laser drilling and nano-machining have become widely used techniques for industrial

manufacturing. Ultra-short pulse lasers, in the picosecond to femtosecond range, are desirable for micron

and sub-micron level precision. However, based on experimental observations and recent research, the

highly precise and consistent drill depth required in some applications has not been achieved at this level.

Despite significant efforts to keep all material and machining properties stable, micrometer level pulse-to-

pulse fluctuations are apparent. The work presented here attributes these fluctuations to a fluctuating

temperature field.

It is hypothesized and confirmed in simulation that fluctuations in drill depth in surface texturing

and surface machining are due to transient temperature effects from multiple pulses despite maintaining

all machining properties constant. The effects are numerically simulated and studied by extending the

current single-pulse model. It is observed that transient temperature effects are a significant contributing

factor to observed fluctuations since the material and machining properties have been modeled as

perfectly stable. The temperature field itself creates instabilities for multiple laser pulses, and can be

correlated to instabilities in drill depth and melt depth.

This unexplored area of research can be of benefit to industry. Once fluctuations have been fully

modeled and understood in relation to drill depth and melt depth, it will be possible to refine the precision

and consistency of high repetition rate ultra-short pulse laser machining at the nanometer level.

iv

TABLE OF CONTENTS

List of Figures…………………………………………………….……………………………...…………v

Nomenclature…………………………………………………….…………………………..…………….vi

Introduction……………………………………………………………………….………………………...1

CHAPTER 1. Transient Heat Transfer Processes……………….………………………………………….3

Transient Heat Conduction………………….…………………….……………….………………3

Convective Cooling……………………………………..…………………………………………4

CHAPTER 2. Thermal Processes in Laser-Material Interaction……………………….……...…………...5

Energy Absorption..………………………………….…………………………………………….5

Conductive Heat Transfer………………………………...………………..………………………6

Evaporative Cooling……………….………………………………………………………………6

Critical Temperature………………………….……………………………………………………7

CHAPTER 3. Description of Modeling Process………………………………………….……………..….9

Finite Difference Approximation……………………………...……….…………………………..9

Simulation of Laser Energy Deposition………………………..…………………………………10

Assumptions………..….…………………………..………………………………………..…….12

Sample of Results Obtained………………….…………….…………………………….……….13

CHAPTER 4. Thermal Instabilities in Laser Nano-Machining…………………..…….………………....17

Description of Instabilities…………………….……………….…………………………………17

Results Found………………………………….…………………………………………….……23

One Pulse per Each Location…………………………………………………………….23

Two Pulses per Each Location…………………………………………………………...24

Four Pulses per Each Location…………………………………………………………..25

Factors Affecting Fluctuations……………………………………..……………..………………26

Changing Substrate Thickness…….……………………………………………………..26

Changing Pulse Repetition Rate…………..………………………….…………….……28

Changing Heat Diffusion Coefficient………………………………………….………...30

CHAPTER 5. Conclusion………………………….……………………………………………………...33

Summary and Discussion………………….………………………………………………..……36

Works Cited……………………………………………………………….………………………………38

Appendix A: Nontechnical Abstract…………….…..……………………….……………………………39

Appendix B: Time Step Independence……………………………..……………………………………..40

v

LIST OF FIGURES

Figure 3-1: Simulation Model…………………...……………………..…………………….……………12

Figure 3-2: Temperature versus time for twelve pulses of a laser……………………...…………………14

Figure 3-3: Maximum temperature versus time for twelve pulses of a laser………………………..….…15

Figure 4-1: Temperature versus depth for a single pulse of a laser…..……….…………………….….…18

Figure 4-2: Temperature versus depth for the first and second pulse of a laser…………………….…….19

Figure 4-3: Change in temperature versus depth…………………………….……………………………20

Figure 4-4: Heat deposition for the first pulse of a laser…………………………………….……………21

Figure 4-5: Heat deposition for two pulses of a laser at the same location……………………………….21

Figure 4-6: Heat deposition for three pulses of a laser at the same location………..…………………….21

Figure 4-7: Heat deposition for four pulses of a laser at the same location……………………....……….22

Figure 4-8: Heat deposition for five pulses of a laser at the same location……………………………….22

Figure 4-9: Maximum temperature reached at the point of laser pulse, one pulse per location….....…….23

Figure 4-10: Maximum temperature reached at the point of laser pulse, two pulses per location……..…24

Figure 4-11: Maximum temperature reached at the point of laser pulse, four pulses per location.……….25

Figure 4-12: Thickness of .01 meters……………………..……………………………..………………..26



Figure 4-15: Pulse Repetition Rate of ~30 kHz..….……………..……………………..…….…………...28

Figure 4-16: Pulse Repetition Rate of ~40 kHz ……….……………..……………………...……………29

Figure 4-17: Pulse Repetition Rate of ~50 kHz ……….……………..…………….……..………………29

Figure 4-18: Heat Diffusion Coefficient of .05 centimeters squared per second…………………..……..31

Figure 4-19: Heat Diffusion Coefficient of .1 centimeters squared per second…………..………………31

Figure 4-20: Heat Diffusion Coefficient of .2 centimeters squared per second………………..………....32

Figure 5-1: Surface temperature of 4 laser pulses in the same location…..…………..…….…………….33

Figure 5-2: Background temperature of 4 laser pulses in the same location…...……………..…………..34

Figure 5-3: Maximum temperature of 4 laser pulses in the same location………………….…………….34

Figure B-1: Time Steps of .2 µs……..…………………………………………………………………….40

Figure B-2: Time Steps of .02 µs…………………………..……………………………………………...41

vi

NOMENCLATURE

A Surface Area

Cp Specific Heat for Constant Pressure

h Heat Transfer Coefficient

H Enthalpy

I Laser Beam Intensity

i Location Step

J Mass Flux

K Thermal Conductivity

k Imaginary Part of the Refractive Index

n Real Part of the Refractive Index

n Time Step (in Finite Difference Equations)

Qab Thermal Energy Absorbed

Qin Thermal Energy input

Q Thermal Energy

r Stability Condition

t Time

Tbackground Background Temperature

Tcritical Critical Temperature

Tenv Temperature of the Environment

Ts Surface Temperature

T Temperature

u Arbitrary Function

x Width

y Length

z Depth

vii

zcr Critical Depth

α Arbitrary Constant

γ Absorption Coefficient

ΔHv Change in Enthalpy for Liquid to Vapor Transition

ΔT Change in Temperature

Δx Change in x

Δy Change in y

Δz Change in z

ρ Density

1

INTRODUCTION

Ultra-short pulse laser drilling and nano-machining have become widely used techniques for

industrial manufacturing. Among the many advantages of laser drilling over mechanical or chemical

drilling are high speed processing and micron (and sub-micron level) precision. However, researchers

have reported that the desired level of precision is not always achieved when multiple pulses are required.

Refinements in the efficiency and precision of the laser drilling and laser micromachining process are of

great interest to both industry and academia. The work presented here proposes a solution to further

improve upon the pulse-to-pulse precision of laser drilling and micromachining in industry.

Micrometer level pulse-to-pulse fluctuations in drill depth and melt depth have been

experimentally observed using picosecond to femtosecond lasers. In general, high quality laser machining

becomes a discussion of “hole quality,” which scientist and engineers go to great lengths to optimize,

using stabilization measures in their drilling strategies [1]. Improvements in efficiency and productivity of

laser-material interaction has been attributed to proper control of the laser beam and the workpiece [1]. In

cases for both long and short pulses, any small inaccuracies in pulse-to-pulse drill depth and melt

production have been attributed to instabilities in machining properties or imperfections in the material

itself, along with instabilities in other physical properties such as melt flow or vaporization front [2, 3].

However, the research presented here demonstrates that there is another contributing factor to

inaccuracies for ultra-short (picosecond to femtosecond range) pulses: a fluctuating temperature field.

Existing models of laser-material interactions, such as the Two-Temperature Model (TTM), have

been problematic when it comes to predicting melt at higher laser intensities [4, 5]. Significantly more

material is removed than predicted. Past research in this area has proposed a correction to the TTM that

accounts for this excessive removal of material. Essentially, the ponderomotive force on electrons in the

skin depth region accelerates the electrons deeper into the metal, where energy, in the form of heat, is

deposited [4, 5]. Inclusion of the ponderomotive force correction to the TTM accurately predicts drill depth

and melt depth of laser nano-machining and has been fully comprehended and modeled for a single laser

pulse [4, 5].

The research presented here further enhances this improved model by expanding its application to

account for multiple laser pulses. This newly-created model simulates the temperature field that is

generated when material is irradiated by multiple pulses (as opposed to a single pulse) on a longer time

scale (as opposed to the time scale for a single pulse) and in three dimensions. There are multiple heating

and cooling effects that define the temperature near the source point of a laser pulse. The input of energy

from the laser, conductive heat transfer, convective heat transfer, and evaporative cooling are some of the

competing heat transfer effects that influence the temperature field. This simulation shows that the

2

thermal field itself near the laser interaction zone at the surface of the workpiece is not steady when

multiple pulses are applied to the same point. Fluctuations in the thermal field are shown to be dependent

on factors such as thickness of the substrate, heat diffusion coefficient of the workpiece, and pulse

repetition rate of the laser. The results presented here will help to improve the precision provided by ultra-

short pulse laser nano-machining in applications including micro-machining during production of micro-

electronic devices like integrated circuits and in micro-drilling of cooling holes in turbine blades.

Simulation of thermal field variation that occurs when machining with multiple laser pulses may

be used to help optimize and stabilize industrial laser processing. Although equipment stability and spatial

variation in material properties will play a role in material removal during laser processing, understanding

fluctuations of the thermal field and fully calculating the effect that it has on melt depth and material

removal will lead to improved precision in the field of laser drilling and nano-machining.

3

Chapter 1

TRANSIENT HEAT TRANSFER PROCESSES

In their most basic form, transient heat transfer equations can be used to describe heating and

cooling as a function of time and space. The fundamental thermodynamic processes outlined in this

chapter are necessary for understanding and interpreting the more specific laser heating and cooling

processes described in Chapter 2.

Transient Heat Conduction

Internal heat conduction is defined as the transfer of energy within a body due to a temperature

gradient, and heat spontaneously flows from a hotter body to a cooler body. A body desires to reach

thermal equilibrium over an extended period of time. In non-steady-state conduction, or transient

conduction, temperature can be determined by solving equations that are functions of time and space

coordinates [6].

Heat conduction can be described using the heat equation, or Fourier’s law. Equation 1.1

represents an arbitrary function, u, represented by Fourier’s law:

(1.1)

In this equation, α is a constant, and x, y, z, and t are space and time coordinates. When this equation is

applied to heat transfer, it is generally written as:

(1.2)

In Equation 1.2, T is time and space dependent temperature, α is thermal diffusivity, and ∇2 denotes the

Laplace operator. This differential equation can be readily solved numerically through computer

simulations.

4

Convective Cooling

Convection is a general term for heat transfer through a moving fluid. It is the mode of energy

transfer between a solid surface and the adjacent liquid or gas [6].

Convective cooling can be described using Newton's law of cooling when h, the heat transfer

coefficient, is assumed constant, as it is in this simulation. Newton’s law states that the rate of heat loss of

a body is proportional to the temperature difference between the body and its surroundings.

(1.3)

In Equation 1.3, Q is thermal energy lost through convection, A is the heat transfer surface area, and h is

the convective heat transfer coefficient, assumed independent of temperature. Convective cooling is a

necessary boundary condition and can also be solved numerically through computer simulations.

These theories and equations are well-known and are utilized in the formulation of boundary

conditions, explained in Chapter 3.

5

Chapter 2

THERMAL PROCESSES IN LASER-MATERIAL INTERACTION

Modern theories of laser-material interaction can describe thermodynamic interactions in enough

detail to explain the interaction of the laser energy with the electrons and ions [4, 5]. Relaxation times for

electrons and ions are in the picosecond range, and would require much smaller time steps in order to

accurately simulate the physical phenomena [7]. However, local thermodynamic equilibrium can be

established within the workpiece (over space and time) when the diffusion time scale of the system is

much larger than the relaxation time scale of relevant energy carriers. In this study, the time steps of the

numerical simulation are in the nanosecond (or longer) range, so that thermal equilibrium is achieved.

Therefore, it is appropriate to use Fourier heat conduction to model the temperature interactions of the

laser.

Energy Absorption

The energy deposited in the workpiece due to the pulse of a laser is highly dependent on the

material’s optical properties. Consider Qab, the volumetric thermal energy intensity absorbed by the

material [7]:

(2.1)

In Equation 2.1, I is the laser beam intensity, γ (gamma) is the absorption coefficient of the

workpiece, and x, y, z, and t represent the space and time coordinates. The surface reflectivity, R, is given

by [7]:

(2.2)

In Equation 2.2, n is the real part of the refractive index and k is the imaginary part. In this

equation, n and k are known to be dependent on temperature.

6

In a more general equation, the volumetric energy absorption is given with a temperature-

dependent absorption coefficient, where Ts is the surface temperature [7]:

(2.3)

These equations assume uniform material properties, with the laser-material interaction occurring

in a vacuum. Scattering occurs in actual applications, but is assumed to be negligible here in order to

simulate ideal laser machining conditions.

Conductive Heat Transfer

As noted previously, for time steps longer than characteristic relaxation times of energy carriers

(as modeled in this simulation), Fourier heat conduction sufficiently describes the heat transfer processes.

Fourier’s law was given in general terms in the previous chapter and is given here specifically for the heat

conduction equation. Hence, the transient (time-dependent) temperature field can be solved through the

heat conduction equation [7]:

(2.4)

In Equation 2.4 ρ, Cp, T, and K represent density, specific heat for constant pressure, temperature,

and thermal conductivity of the substrate. Although technically these properties are temperature

dependent, in this work, they are assumed constant to develop approximate solutions. This is assumption

generally accepted in thermodynamics calculations [6]. The energy deposited in the workpiece due to the

pulse of a laser is represented by Qab.

Evaporative Cooling

In laser-material thermal interactions, evaporative cooling is another significant heat transfer

phenomenon that must be accounted for. As a material is heated above the boiling point, the material is

transformed into a vapor. As evaporation occurs, the material that is removed from the interaction zone is

at a higher temperature than the material that remains and effectively removes a portion of the initially

deposited energy. Additionally, vaporization absorbs process energy, and thus the material left behind is

7

at a lower temperature. In laser-material interaction this accounts for a significant amount of surface

cooling and surface temperature “saturation.”

The boundary condition that describes evaporative cooling is given in Equation 2.5 [8].

(2.5)

This boundary condition applies to the top surface, and has a significant effect as the laser

deposits high amounts of energy at the surface. Here, K is thermal conductivity, ∂T/∂z|z = 0 is the

temperature gradient at the surface, J is mass flux due to evaporation, and ΔHv is change in enthalpy for

liquid to vapor transition. Essentially, the larger the temperature gradient due to laser energy deposited

into material, the greater the energy loss due to material evaporation. Therefore the higher the initial

temperature reached during the laser pulse, the greater the cooling effect due to evaporative cooling.

Critical Temperature

The critical point and the so-called critical temperature is a well-defined concept in

thermodynamics, and is defined as the point at which saturated liquid and saturated vapor states are equal

[6]. However, a direct application of this concept to laser-material interactions is not straightforward. The

difficulty arises due to evaporation that occurs as a result of ultra-short pulse ultra-high intensity laser-

material interaction. Evaporation resulting from a pulsed laser is known to expel both large and small

clusters of material [9].

This process of laser ablation has been extensively investigated both experimentally and

theoretically, and ongoing research is attempting to develop an understanding of the transition from a

superheated liquid to exploded material [10]. When the surface of a material reaches a temperature higher

than the critical temperature, at sufficiently high laser intensity, ablative material removal is induced. This

process is commonly known as phase explosion or explosive boiling. Explosive boiling has been

described as the transition from a superheated liquid to a mixture of liquid and vapor, in which the

material heated above the critical temperature is expelled from the surface in a mixture of vapor and

liquid droplets. The plume created due to this process is a mixture of individual atoms/molecules, clusters

of material, and liquid droplets.

8

Once the critical temperature is reached, the near-surface of the material becomes overheated, and

is no longer thermodynamically stable. This leads to phase explosion of the material, and the material is

expelled from the surface. Therefore, for the purpose of this work, the critical temperature of the material

is viewed as the point at which the material can be removed from the simulation. Specifically, for longer

time scale, multi-pulse laser-material interaction modeling, it can be assumed that material heated above

the critical temperature undergoes a phase explosion and is expelled from the surface, and may thus be

discounted from further calculations in the model. Thus, the laser-material interaction process can be

simulated on a macro scale without detailed representation of molecular and fluid dynamics.

9

Chapter 3

DESCRIPTION OF MODELING PROCESS

Finite Difference Approximation

Finite difference methods provide an approximate solution for a continuous partial differential

equation, in this case, the heat equation. This technique involves creation of a mesh of points, called

nodes; where at each node a solution is computed [11]. The nodes are separated in space by a designated

Δx, Δy, and Δz, creating computational blocks, and are separated in time by a designated Δt.

To compute a solution, it is required to specify initial conditions at t = 0 and boundary conditions

for all faces, edges, and corners. In this case, convective cooling boundary conditions are applied, and

heat conduction within the material was calculated. The evaporative cooling boundary condition was also

applied to the top surface.

There are various numerical methods that can be used to find approximate solutions to the heat

equation [11]. The Forward-Time Centered-Space method is illustrated using the one-dimensional heat

equation as an example, but can be easily expanded to three dimensions.

Consider the one-dimensional heat equation

(3.1)

where 0 ≤ x ≤ L, and t ≥ 0. This equation describes transient heat conduction in a slab of material with a

thickness of L.

The Forward-Time Centered-Space approximation to the heat equation is given in Equation 3.2,

where i represents the location step and n represents the time step [11].

(3.2)

A so-called stability condition in numerical analyses can be used to ensure that computational

errors do not grow within the sequence of the numerical procedure [11]. For this method, the stability

condition is [11]:

(3.3)

10

Note that in three dimensions, this stability condition also applies to y and z. Application of this

condition ensures that as Δt grows, errors from any source (rounding, truncation, etc.) do not grow [11].

Numerical errors generated when applying this numerical solution are proportional to the time step and

the square of the space step.

It should also be noted that the Forward-Time Centered-Space method is an explicit solution

method. An explicit solution method calculates the state of a system at a later time based on the current

(known) state of a solution. In comparison, implicit solution methods calculate the state of a system based

on the current state and a later state. Implicit solution methods require extra computation in each iteration,

but can utilize larger time steps. When using an explicit solution method, accuracy and numerical stability

are maintained by using small time steps; however, computation time is often reduced because explicit

methods require one fewer computation for each time step [11].

A solution for the thermal field can be developed using the Forward-Time Centered-Space

explicit solution method to the Fourier equation and applying appropriate boundary conditions. The

thermal energy from a laser pulse was applied as an energy flux on the top surface at the designated time

and location. Evaporative cooling (temperature dependent) was included on the top surface boundary

condition.

Simulation of Laser Energy Deposition

In order to model the input of energy from the laser, Equation 2.1 is rewritten as:

(3.4)

In Equation 3.4, Δx, Δy, and Δz describe the size of one computational block. Volumetric thermal

energy absorbed is related to change in temperature in the computational cell through the Equation 3.5:

(3.5)

In Equation 3.5, Cp is specific heat at constant pressure and ρ is density.

As briefly described above and further explained in the next chapter, during the laser process, any

material heated above the critical temperature is expelled from the surface, leaving material with a surface

temperature equal to the critical temperature, and exponentially cooler temperature with increasing depth.

This specific depth at which the temperature of the material is the critical temperature (prior to material

expulsion) will be referred to as zcritical, or zcr. Material that is removed from the surface (in the range 0 < z

11

< zcr) does not contribute to the heating process in the remaining material, since energy contained in the

expelled material is removed from the surface, and is thus not absorbed into the material.

Therefore, in order to simulate the input of thermal energy from the absorbed laser energy, Qin, it

must be averaged over the height of one computational block, starting at the critical depth. Hence,

(3.6)

This average over the computational block, beginning at the critical depth, was used to calculate

change in average temperature of the computational block due to volumetric thermal energy absorbed

from the pulse of a laser. In other words, the very high thermal gradients that would actually be present

near the surface of the material are averaged over the size of the computational block for the purpose of

this simulation. This “smearing” effect can be reduced when simulation is performed with smaller Δx, Δy,

and Δz steps; however, maintaining large computational blocks provides a means to reduce the

computational cost when modeling large scale thermal effects.

Figure 3-1 describes the simulation model. The material is modeled with a specified size,

expressed by x, y, and z. The values for x and y were in the range of 1 to 2 meters, and 0.01 to 0.05

meters for depth, z. This piece of material was then broken down into a mesh of computational blocks of

a specified size Δx, Δy, and Δz. These variables ranged from 0.001 to 0.005 meters. The input of thermal

energy from the laser was modeled as instantaneous in one computational block at the specified time and

location.

12

Figure 3-1: Simulation Model

Assumptions

There were several assumptions made in creating this model. These assumptions are not believed

to degrade the qualitative correctness of the model, but are recognized to result in quantitative

inaccuracies. All simplifications and assumptions are prescribed to maintain qualitative integrity of the

process over the long time steps used in this model.

First, consider the simulation of a laser pulse and the laser energy deposition, described in the

preceding section. In order to model the thermal energy added from a laser, the beam was modeled as

being square (the size of one computational block). The heat input was also assumed to be delivered

instantaneously. These assumptions do not affect the qualitative accuracy of this model because

characteristic relaxation time for energy transfer of laser pulses is in the picosecond range [7]. Heat transfer

occurring at this time scale need not be considered when time steps are in the nanosecond and longer

range.

13

Sample of Results Obtained

The simulation calculated the temperature of every computational block in the workpiece at every

time step. In order to interpret these results, research focused almost exclusively on the temperature near

the point at which the laser interacted with the material. Therefore, for each time step, the simulation

produced the temperature where the laser was located. When the laser translated to a different location on

the surface, the temperature was calculated at that new location. Therefore, results are given in time (as

the time step moved) and with a moving reference frame (following the laser).

For example, consider a laser translating across the surface of the material. This laser pulsed one

time in each computational block in a straight line in the y-direction. The resulting temperatures are

shown in Figure 3-2. This graph shows temperature in the vertical axis and time in the horizontal axis.

These temperatures represent the temperature averaged over the surface computational block after the

energy from the expelled material has been eliminated from the simulation. Since the laser is moving

between pulses, a second axis is included showing the Δy location step. In this particular simulation, the

laser traversed in a straight line along the y direction. Each pulse occurred at Δy distance away from the

previous pulse; therefore, the temperature in Figure 3-2 is plotted at that new location. As a general rule,

the temperature plotted in all subsequent graphs is the temperature at the surface of the material wherever

the laser beam is currently located.

14

Figure 3-2: Temperature versus time for twelve pulses of a laser

Figure 3-2 shows twelve pulses of a laser, and each pulse of the laser occurred in a straight line across the

surface of the workpiece, one pulse occurring every Δy = .5 millimeters. The temperature (i.e.

temperature averaged over the computational block) peaks during the laser pulse, and then the

temperature cools down. The peak occurs at the new location of the laser pulse, and then that new

location cools down as well before the laser moves to the next location.

In order to achieve the overall objective of examining the stability of the thermal field, these

temperature graphs were examined by focusing attention on the maximum temperature reached for each

pulse. Maximum temperature is important, especially when making the results useful to industry. For

example, the maximum temperature can be used to calculate drill depth or amount of material removed [5].

0

1

2

3

4

5

6

7

8

9

10

11

300

320

340

360

380

400

420

440

460

0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03

Surf

ace

Tem

per

ature

(K

)

Time (s)

Surface Temperature of 12 Laser Pulses

Δy L

oca

tio

nS

tep (

.5m

m)

15

In order to more closely examine the maximum temperature achieved, Figure 3-3 plots the maximum

temperature reached for the same 12 laser pulses. Again, these maximum temperatures are moving in both

time and space. Time is plotted on the horizontal axis, and the location of each maximum is determined

by the location of the laser pulse, plotted in the Δy location step function. Note the peak temperature rises

steadily throughout this simulation.

Figure 3-3: Maximum temperature versus time for twelve pulses of a laser

0

1

2

3

4

5

6

7

8

9

10

11

463

463.2

463.4

463.6

463.8

464

0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03

Max

imum

Tem

per

ature

(K

)

Time (s)

Maximum Temperature of 12 Laser Pulses

Δy L

oca

tio

n S

tep

(.5

mm

)

16

In the next chapter, the model was applied to a case in which a laser is pulsed multiple times in

the same location prior to moving to the next location. In order to examine the stability of the thermal

field, research focused on the maximum temperature reached in the laser-material interaction zone.

Therefore, graphs for maximum temperatures reached are plotted in the same manner, in the sense that the

maximum surface temperature was calculated at the beam (averaged over the computational block after

the energy from the expelled material has been removed), and when the beam moves (after a given

number of pulses in the same location), the temperature was calculated at that new location. For clarity,

the graphs shown in Chapter 4 do not include the second axis for Δy location; however, it is understood,

as a general rule, that maximum temperature plotted is at the location of the laser beam.

17

Chapter 4

THERMAL INSTABILITIES IN LASER NANO-MACHINING

Description of Instabilities

A model for multi-pulse laser drilling or micromachining has been developed, and

simulations of multiple pulses along the surface of a material have revealed thermal fluctuations. These

thermal fluctuations become more interesting when multiple pulses of a laser are repeated at one location

on the surface of the material. The temperature at this spot is not uniform at each pulse, nor is it

consistently increasing or decreasing. In fact, thermal instabilities create fluctuations in the temperature.

These fluctuations are the results of competing heating and cooling effects.

Before any laser irradiation, the temperature of the workpiece is at the prescribed background

temperature. The first pulse of the laser deposits energy in the material. The surface temperature of the

material rises to a temperature higher than the original background temperature, and temperature

exponentially decreases as the depth, z, increases. This is shown in Figure 4-1, a graph of temperature

versus depth. As described in Chapter 2, when a material is heated above the critical temperature, a

thermal explosion occurs, and the material is expelled. The remaining material has a lower temperature,

starting at the critical temperature and exponentially decreasing with depth.

The first laser pulse generates a rise in temperature, relative to background temperature. This

pulse deposits a certain amount of energy throughout the volume of the computational area, and the

change in temperature is proportional to this volumetric energy deposition.

18

Figure 4-1: Temperature versus depth for a single pulse of a laser

The shaded region in Figure 4-1 represents the energy deposited in the material. Since material in

the range 0 < z < zcr is expelled due to thermal explosion above critical temperature, that energy is not

deposited in the material. After cooling, the deposited energy leads to an increase in substrate temperature

(as compared to the background temperature prior to the first pulse).

Now, consider a second pulse at the same location, which irradiates the substrate that is at a

temperature greater than the original background temperature. As seen in Figure 4-2, the graph of

temperature versus depth for the second pulse is the same as the first, except slightly shifted up due to the

increase in background temperature. Since critical temperature remains the same, total energy deposited

in the material is reduced, because more energy is lost due to material expulsion resulting from the

thermal explosion above the critical temperature. Less energy deposited in the material for the second

pulse leads to a decrease in maximum surface temperature reached (as compared to the first pulse).

Background temperature influences the amount of energy deposition for a laser pulse.

19

In Figure 4-2, the shaded region represents the energy deposition of the second pulse, which,

when compared to Figure 4-1, is shown to be less than the first pulse.

Figure 4-2: Temperature versus depth for the first and second pulse of a laser

This effect can also be explained by considering the change in temperature, ΔT, which is a

reflection of absorbed energy from the laser pulse. Equation 4.1 represents the temperature at the surface

of the material.

(4.1)

The temperature, T, can be found by summing the change in temperature and the background

temperature. At the point of the surface immediately after the pulse, this is critical temperature, since all

material at a temperature higher than critical temperature has been expelled. Hence, at the new surface:

(4.2)

20

Since background temperature increases for the second pulse (Tbkg2 > Tbkg1), this leads to a

smaller change in temperature as compared to the first pulse. Therefore, the second pulse of the laser

deposits less energy into the material than the first pulse at the same location, because the initial

temperature of the substrate prior to the laser pulse has increased, as shown in Figure 4-3.

Figure 4-3: Change in temperature versus depth

Background temperature influences the amount of laser energy imparted to the substrate. A

higher background temperature results in lower laser energy deposition. This effect would continue for

subsequent pulses, except that the heat conduction and surface convection serve to transport heat from the

interaction zone and lower the substrate temperature near the surface. As described in prior chapters, the

heat from the laser diffuses radially through the material, away from the source point of the laser beam.

See Figures 4-4 through 4-8.

21

Figure 4-4: Heat deposition for the first pulse of a laser

If the second pulse is close behind the first pulse, the heating effects from the first pulse remain,

causing further penetration of the thermal field, diffusing radially outward from the source point of the

laser, as can be seen in Figure 4-5.

Figure 4-5: Heat deposition for two pulses of a laser at the same location

Eventually, this heat reaches the bottom surface of the material and starts to conduct and dissipate

further radially outward from the source point of the laser. The top and bottom boundary provide a

convective cooling boundary condition. Later repetition of pulses in the same location allow more time

for the heat to diffuse radially away from the source point, shown in Figures 4-6 and 4-7.

Figure 4-6: Heat deposition for three pulses of a laser at the same location

22

Figure 4-7: Heat deposition for four pulses of a laser at the same location

Soon after, heat dissipates far enough away from the source point of the laser that it leads to a

change in background temperature at the surface of the workpiece. This effect is observed in later pulses

repeated at the same location. See Figure 4-8.

Figure 4-8: Heat deposition for five pulses of a laser at the same location

Ultimately, heat conduction and diffusion lead to a change in background temperature, which

leads to a change in the amount of laser energy imparted to the substrate. However, the heat conduction

and dissipation effect described above occurs with a significant time delay (relative to the laser pulse),

due to the time that is required for heat to move through the material. Heat conduction and dissipation

throughout the material have an effect in later pulses as more pulses are repeated and more time passes.

Since the two processes described in this section are occurring at different rates, temperature fluctuations

are seen, highly dependent on the material properties and material geometry.

23

Results Found

Figures 4-9 through 4-11 represent the maximum temperature (averaged over the computational

block at the surface, with volume of 4x10-4 cm3) that is reached at the source point of the laser pulse. The

pulse duration and pulse energy ranges used in the simulations were typical of industrial ultra-short pulse

laser machining. The temperatures are graphed in both time and space as described above, though the step

function showing y-location is emitted for clarity. The laser deposits energy in one computational block

then moves Δy distance along the y-axis to the next computational block. Four cases are presented below,

with increasing number of pulses occurring at each location before the laser moves to the next location.

One Pulse per Each Location (Figure 4-9)

This baseline simulation example shows that if no pulses are repeated at the same location, no

thermal fluctuations at the surface of the material were discovered. The temperature at the location of the

laser beam increases slightly for each pulse due to general heating throughout the material from the pulses

before it.

Figure 4-9: Maximum temperature reached at the point of laser pulse, one pulse per location

463.8

463.85

463.9

463.95

464

464.05

464.1

464.15

464.2

464.25

464.3

3.50E-03 3.70E-03 3.90E-03 4.10E-03 4.30E-03 4.50E-03 4.70E-03 4.90E-03

Max

imum

Tem

per

ature

(K

)

Time (s)

Maximum Surface Temperature for Laser Pulse Repeated 1 Time

at Each Location

24

Two Pulses per Each Location (Figure 4-10)

As described above, the second pulse repeated at the same location reaches a lower maximum

temperature at the surface than the first pulse. This is because the higher background temperature for the

second pulse leads to less energy deposition as compared to the first pulse. Note that this graph is also

plotted in both time and space for each pair of pulses in the same location; Figure 4-10 follows the laser

as it pulses twice in one location, then moves forward one space step along the y-axis. The step function

showing y location is omitted, but it is understood that the maximum temperature plotted is at the location

of the laser for the two pulses in the same location. The first pulse is the higher temperature, and the

second pulse is the lower temperature. One example pair of pulses in the same location is circled for

clarity.

Figure 4-10: Maximum temperature reached at the point of laser pulse, two pulses per location

465.5

465.55

465.6

465.65

465.7

465.75

465.8

465.85

465.9

465.95

466

3.50E-03 3.70E-03 3.90E-03 4.10E-03 4.30E-03 4.50E-03 4.70E-03 4.90E-03

Max

imum

Tem

per

ature

(K

)

Time (s)

Maximum Surface Temperature for Laser Pulse Repeated 2

Times at Each Location

25

Four Pulses per Each Location (Figure 4-11)

As described above, the more pulses that are repeated at the same location eventually lead to a

change in background temperature due to heat conduction and diffusion, which leads to a change in

energy deposition. In this example, heat conduction causes an effect by the fourth pulse. Again, note that

this graph is plotted in both time and space for the four pulses in the same location; Figure 4-11 follows

the laser as it pulses four times in one location, then moves forward to the next laser pulse location along

the y-axis. The step function showing y location is omitted, but it is understood that the maximum

temperature plotted is at the location of the laser for the four pulses in the same location, then moves Δy

along the y-axis before the next four pulses. The maximum temperature reached at the surface of the

material at the location of the laser pulse is highest after the first pulse, then lower after the second pulse,

slightly lower after the third pulse, then slightly higher after the fourth pulse. An example group of four

pulses in one location is circled for clarity.

Figure 4-11: Maximum temperature reached at the point of laser pulse, four pulses per location

470.2

470.25

470.3

470.35

470.4

470.45

470.5

470.55

470.6

470.65

470.7

3.50E-03 3.70E-03 3.90E-03 4.10E-03 4.30E-03 4.50E-03 4.70E-03 4.90E-03

Max

imum

Tem

per

ature

(K

)

Time (s)

Maximum Surface Temperature for Laser Pulse Repeated 4

Times at Each Location

26

Factors Affecting Fluctuations

All results given in this section are plotted as the maximum surface temperature at the source

point of the laser (averaged over that computational block and after excluding energy removed in the

expelled material) for four laser pulses repeated in the same location.

Changing Substrate Thickness

Substrate thickness is the first important factor affecting thermal fluctuations. While keeping all

other laser processing properties the same, it was predicted that the thinner the workpiece, the faster the

heat will conduct to the bottom boundary (a convective cooling boundary), and the sooner an effect due to

conduction would be observed. Such an effect has been seen through simulation of multiple thicknesses

of material. Decreasing the thickness of the workpiece decreases the time it takes to observe an effect in

maximum temperature at the surface due to heat dissipation (due to its effect on background temperature).

An effect from heat conduction and dissipation can be seen in the earlier repetitions of pulses.

While maintaining all other machining properties, the thickness of this sample was varied from

0.01 to 0.02 to 0.04 meters. Four pulses were repeated in each location. Results found appear to support

the hypothesis. An example group of four pulses in one location is circled for clarity.

Figure 4-12: Thickness of .01 meters

469.8

469.9

470

470.1

470.2

470.3

470.4

470.5

2.00E-03 2.20E-03 2.40E-03 2.60E-03 2.80E-03 3.00E-03 3.20E-03 3.40E-03

Tem

per

ature

(K

)

Time (s)

Thickness of .01 meters

27



469

469.1

469.2

469.3

469.4

469.5

469.6

469.7

2.00E-03 2.20E-03 2.40E-03 2.60E-03 2.80E-03 3.00E-03 3.20E-03 3.40E-03

Tem

per

ature

(K

)

Time (s)


469

469.1

469.2

469.3

469.4

469.5

469.6

469.7

2.50E-03 2.70E-03 2.90E-03 3.10E-03 3.30E-03 3.50E-03 3.70E-03 3.90E-03

Tem

per

ature

(K

)

Time (s)


28

As can be seen, substrate thickness has an effect on thermal fluctuations, specifically in the later

repetitions of pulses. This data suggests that thinner material demonstrates an earlier effect (a fluctuation

to a higher temperature) due to heat conducting to the bottom convective cooling boundary more quickly.

In Figure 4-12, an effect can be seen by the third or fourth pulse. This effect becomes less prevalent and

eventually non-existent when the thickness of the material is increased.

Changing Pulse Repetition Rate

The pulse repetition rate has an effect on thermal fluctuations. While keeping all other machining

properties the same, the pulse repetition rate was varied. Higher pulse repetition rates exaggerate the

cooling effect between the first and second pulses due to varying background temperatures. When there is

less time for the surface of the material to cool between pulses, a higher background temperature leads to

a decrease in energy deposited to the substrate.

While maintaining all other laser processing properties, the pulse repetition rate was varied from

approximately 30 kHz to 40 kHz to 50 kHz. Four pulses were repeated in each location. Results appear to

support the hypothesis. An example group of four pulses in one location is circled for clarity.

Figure 4-15: Pulse Repetition Rate of ~30 kHz

468.2

468.3

468.4

468.5

468.6

468.7

468.8

468.9

2.00E-03 2.20E-03 2.40E-03 2.60E-03 2.80E-03 3.00E-03 3.20E-03 3.40E-03

Tem

per

ature

(K

)

Time (s)

Pulse Repetition Rate of ~30 kHz

29



469.8

469.9

470

470.1

470.2

470.3

470.4

470.5

2.00E-03 2.20E-03 2.40E-03 2.60E-03 2.80E-03 3.00E-03 3.20E-03 3.40E-03

Tem

per

ature

(K

)

Time (s)


472.5

472.6

472.7

472.8

472.9

473

473.1

473.2

2.00E-03 2.20E-03 2.40E-03 2.60E-03 2.80E-03 3.00E-03 3.20E-03 3.40E-03

Tem

per

ature

(K

)

Time (s)


30

As can be seen, the pulse repetition rate of the laser has an effect on thermal fluctuations,

specifically in the first two pulses. The temperature difference between the maximum surface temperature

reached for the first and second pulses increases as pulse repetition rate is increased. This is because

background temperature for the second pulse is higher since there is less time for heat to dissipate

between pulses, leading to decreased laser energy deposition.

Changing Heat Diffusion Coefficient

The heat diffusion coefficient of the workpiece also has a significant effect on thermal

fluctuations. The lower the heat diffusion coefficient, the slower heat is conducted throughout the

material. When the heat diffusion coefficient is too low, heat conduction does not have an effect on

thermal fluctuations because there is not enough time for heat to diffuse throughout the material; heat

moves very slowly. When the heat diffusion coefficient is higher, heat moves very quickly and easily

throughout the material. This causes a change in background temperature due to heat conduction and

diffusion in the later repetitions of pulses (as described earlier in this chapter). However, when the heat

diffusion coefficient is too high, heat is moving too quickly for any fluctuations to occur.

While maintaining all other machining and material properties, the heat diffusion coefficient was

varied from 0.05 to 0.1 to 0.2 centimeters squared per second. Four pulses were repeated in each location.

Results found support the hypothesis. An example group of four pulses in one location is circled for

clarity.

31

Figure 4-18: Heat Diffusion Coefficient of .05 centimeters squared per second


486.7

486.8

486.9

487

487.1

487.2

487.3

487.4

3.50E-03 3.70E-03 3.90E-03 4.10E-03 4.30E-03 4.50E-03 4.70E-03 4.90E-03

Tem

per

ature

(K

)

Time (s)

Heat Diffusion Coefficient of .05 centimeters2 per second

480.9

481

481.1

481.2

481.3

481.4

481.5

481.6

3.50E-03 3.70E-03 3.90E-03 4.10E-03 4.30E-03 4.50E-03 4.70E-03 4.90E-03

Tem

per

ature

(K

)

Time (s)


32


As can be seen, heat diffusion coefficient has an effect on thermal fluctuations. When the heat

diffusion coefficient is too low, fluctuations due to heat conduction and dissipation are not prevalent

because the heat does not move throughout the material quickly enough for an effect to be seen. Figure 4-

18 shows maximum temperature reached at the surface of the material for four pulses repeated in the

same location is decreasing for each pulse. When the heat diffusion coefficient is too high, fluctuations in

the maximum temperature reached at the surface are also not prevalent because heat is diffusing away

from the source point of the laser too quickly for a substantial increase in substrate temperature. Figure 4-

20 shows nearly consistent maximum surface temperatures for four laser pulses. However, in Figure 4-19,

fluctuations in maximum surface temperature are prevalent. These three results suggest an optimal range

for heat diffusion coefficient to reduce thermal fluctuations.

550.1

550.2

550.3

550.4

550.5

550.6

550.7

3.50E-03 3.70E-03 3.90E-03 4.10E-03 4.30E-03 4.50E-03 4.70E-03 4.90E-03

Tem

per

ature

(K

)

Time (s)


33

Chapter 5

CONCLUSION

In conclusion, background temperature affects energy absorption in ultra-short pulse laser nano-

machining, and thus impact material removal. Background temperature is influenced by energy deposition

from earlier pulses and heat conduction and diffusion. Higher background temperature at the surface

implies lower energy deposition from a laser pulse. Consider the surface temperature of four laser pulses

in the same location, graphed in Figured 5-1.

Figure 5-1: Surface temperature of 4 laser pulses in the same location

The simulation models four pulses of a laser in the same computational block. The increase in

temperature for each pulse is dependent on the energy deposition for each pulse. This is dependent on the

background temperature prior to each pulse.

300

320

340

360

380

400

420

440

460

480

500

2.32E-03 2.35E-03 2.38E-03 2.41E-03

Tem

per

ature

(K

)

Time (s)

Surface Temperature of 4 Pulses in Same Location

34

To illustrate this point, the background temperature for the same four pulses is graphed in Figure

5-2.

Figure 5-2: Background temperature of 4 laser pulses in the same location

Maximum surface temperature of the material for the same four pulses is graphed in Figure 5-3.

Figure 5-3: Maximum temperature of 4 laser pulses in the same location

306.1

306.2

306.3

306.4

306.5

2.32E-03 2.35E-03 2.38E-03 2.41E-03

Bac

kgro

und

Tem

per

ature

(K

)

Time (s)

Background Temperature

470.04

470.06

470.08

470.1

470.12

2.32E-03 2.35E-03 2.38E-03 2.41E-03

Max

imum

Tem

per

ature

(K

)

Time (s)

Maximum Temperature

35

A higher background temperature clearly correlates with a lower maximum temperature for laser

pulses. Therefore, a higher background temperature results in decreased deposition of laser energy to the

substrate. Competing heating and cooling effects induced by multiple pulses of a laser create a fluctuating

background temperature, therefore the laser energy deposition fluctuates, hence the maximum

temperature at the surface of the material fluctuates as well (for distinct pulses). The fluctuation of

maximum temperature at the surface of the material creates pulse-to-pulse fluctuations in drill depth and

melt depth in ultra-short pulse laser nano-machining.

36

Summary and Discussion

A practical model for extended time duration laser nano-machining, with time scale sufficient to

model bulk thermal effects in the substrate, has been developed. This model shows that multiple pulses in

laser nano-machining contribute to a non-uniform, fluctuating temperature field. The fluctuations are

shown to be dependent on factors such as depth of the workpiece, heat diffusion coefficient of the

workpiece, and pulse repetition rate of the laser. It should also be noted that these fluctuations are

averaged over the size of one computation block, and therefore are predicted to be higher in magnitude at

the surface of the material.

Although the thermal field has been proven to be unstable due to the competing thermal effects

caused by multiple pulses, there are many other possible sources of instability. For example,

imperfections in the material or equipment instabilities in both the laser itself and the machining surface

would also be expected to produce thermal instabilities. Therefore, in practice, observed fluctuation

effects likely result from multiple factors.

Explicit industrial applications of these results would require further development. Surface

temperature near the pulse of the laser is directly related to laser induced drill depth and melt depth,

which are of concern in industry. The fluctuations seen in the temperature field can be expected to lead to

fluctuations in drill depth and melt depth for multiple pulses. Refining the results shown here and

applying them to other physical thermodynamic effects can be of value to industry, possibly assisting in

the fine-tuning of the precision of laser drilling and surface texturing.

This model has shown many interesting results which can be expanded upon further in the future.

For future research in this area, other texturing patterns should be examined. These results only simulate a

straight line of pulses, and expanding this idea to other patterns might apply to a wider realm of industrial

applications. Also, examining other physical factors that affect fluctuations (and the magnitude and

importance of these effects) would aid in further understanding this subject. Lastly, and perhaps most

importantly, these simulations should be calculated with a finer mesh (smaller computational blocks).

With a finer mesh, the fluctuations that have been discovered at the surface of the material may be

exaggerated because they will not be averaged over a large computational block. The smaller the mesh,

the less averaging over depth occurs, and more accurate results would be expected. This is essential to

obtaining quantitatively accurate results.

37

Further, the thermal fluctuations discovered and their periodicity seem to suggest a thermal

resonance. Future researchers might examine the effects described above in terms of a resonance; the

effects described in the above chapters could be viewed in terms of oscillations and the resonance induced

by forcing functions and feedback functions. If a thermal resonance is discovered, the thermal resonance

frequency would be of great interest to the industrial field of laser nano-machining.

In conclusion, multiple pulses of a laser in the same location have been shown to yield a non-

uniform thermal field. These thermal fluctuations are the result of competing heating and cooling effects,

and contribute to undesirable fluctuations in drill depth and melt depth during laser nano-machining.

Fully understanding the fluctuations effects described here will help improve the precision of laser nano-

machining in industry.

38

WORKS CITED

[1] Beck, Thomas. "Laser drilling in gas turbine blades." Laser Technik Journal 8.3 (2011): 40-43.

[2] Anisimov, Sergei I., and Viktor A. Khokhlov. Instabilities in laser-matter interaction. CRC press,

1995.

[3] Semak, V. V., J. G. Thomas, and B. R. Campbell. "Drilling of steel and HgCdTe with the

femtosecond pulses produced by a commercial laser system." Journal of Physics D: Applied

Physics 37.20 (2004): 2925.

[4] Likhanskii, Alexandre. "Study of Plasma Phenomena at High Electric Fields in Applications for

Active Flow Control and Ultra-Short Pulse Laser Drill." Diss. Pennsylvania State University,

2009. Print.

[5] Moore, Russel. Ponderomotive Force Corrections in Laser Ablation. Thesis. Pennsylvania State

University, 2013. Print.

[6] Cengel, Yunus A.. Introduction to Thermodynamics and Heat Transfer. ed. New York: McGraw-

Hill, 1997. Print.

[7] Grigoropoulos, Costas P. Transport in Laser Microfabrication: Fundamentals and Applications.

Cambridge: Cambridge UP, 2009. Print.

[8] Semak, Vladimir V., et al. "Transient model for the keyhole during laser welding." Journal of

physics D: Applied physics 32.15 (1999): L61.

[9] Wu, Chengping, and Leonid V. Zhigilei. "Microscopic mechanisms of laser spallation and

ablation of metal targets from large-scale molecular dynamics simulations." Applied Physics A

114.1 (2014): 11-32.

[10] Zhigilei, Leonid V., et al. "Atomic/Molecular-Level Simulations of Laser–Materials

Interactions." Laser-Surface Interactions for New Materials Production. Springer Berlin

Heidelberg, 2010. 43-79.

[11 Anderson, Dale A., John C. Tannehill, and Richard H. Pletcher. Computational Fluid Mechanics

and Heat Transfer. 2nd ed. Washington: Taylor & Francis, 1997. Print.

[12] Recktenwald, Gerald W. Finite-Difference Approximations to the Heat Equation. Tech. 2004.

Print.

39

APPENDIX A

Nontechnical Abstract

Laser drilling and nano-machining have become widely used techniques for industrial

manufacturing. Some of the many advantages of laser drilling over mechanical or chemical drilling

include speed and micron and sub-micron level precision, along with a high level of productivity.

However, experimental observations have shown that the desired detailed precision is not always

achieved for multiple pulses. Variations in properties such as drill depth and melt depth from pulse-to-

pulse have been observed. Often, attempts to correct this are limited to controlling the laser beam and the

workpiece. The research presented here suggests an additional factor inducing fluctuations: thermal field

fluctuations.

The laser-induced thermal field is not stable, nor is it steadily increasing or decreasing, due to

competing heating and cooling effects. The thermal field has been numerically simulated, fluctuations

have been discovered, and a potential explanation for these fluctuations is presented in this work.

This unexplored area of research will be of benefit to industry. Fluctuations in the thermal field

can be correlated to predictable changes in drill depth and melt depth. Once fluctuations have been fully

modeled and understood in relation to drill depth and melt depth, considerable advancements can be made

in refining the accuracy of ultra-high precision laser machining at the nanometer level.

40

APPENDIX B

Time Step Independence

Two experiments were conducted to confirm time step independent solutions. It is necessary to

confirm that time steps utilized for data acquisition provide the necessary accuracy to obtain qualitative

proof of temperature field fluctuations. Therefore the size of the time step was chosen in order to provide

adequate accuracy while maintaining reasonable computational times. Figures B-1 and B-2 show example

temperature data points obtained from performing the simulation using times steps of 0.2 µs and 0.02 µs.

Figure B-1: Time Steps of .2 µs

402.5

402.7

402.9

403.1

403.3

403.5

2.50E-03 2.60E-03 2.70E-03 2.80E-03 2.90E-03 3.00E-03

Tem

per

ature

(K

)

Time (s)

Time Steps of .2 µs

41

Figure B-2: Time Steps of .02 µs

The two solutions presented display nearly identical trends, though the time steps differ by a

factor of ten. Most importantly, the amplitudes and frequencies of fluctuations in the temperature field do

not vary significantly between the two solutions. It was concluded that using the ten times larger time step

of 0.2 µs was adequate for all cases in this study.

402.5

402.7

402.9

403.1

403.3

403.5

2.50E-03 2.60E-03 2.70E-03 2.80E-03 2.90E-03 3.00E-03

Tem

per

ature

(K

)

Time (s)

Time Steps of .02 µs

Documents

FLUCTUATIONS IN THERMAL FIELD OF WORKPIECE DUE TO …