Laser Heating Applications

310 Pages·2012·3.945 MB·English

CHAPTER ONE
Introduction to Laser Heating
Process
Contents
References 5
Advanced processing technologies are demanding focus areas of the mate-
rials and manufacturing disciplines. Lasers are considered to be advanced
material processing tools filling the gap in the advanced manufacturing
systems because of their precision, low cost, localized processing, and high
speed of operation. In laser-machining applications, a laser beam is used as
aheatsource,increasingtemperaturerapidlytothemeltingandevaporation
temperature of thesubstrate material. Since thearrangements of the optical
setting for the laser beam are very precise, the localized heating can be
controlled easily. With recent advancement in laser technology and
computation power, laser-machining application has become almost an
integral part of the aerospace, power, electronics, and sheet metal forming
industries. However, in laser-machining operations, the physical processes
are complicated in nature and they require a deep understanding of the
process to secure improved end-product quality.
Lasermachiningcanbecategorizedintotwogroupsbasedonthetypeof
processing being involved during the machining such as drilling, cutting,
welding, alloying, and others. The laser processing can be pre- or post-
treatmentoperationssuchasduplextreatmentforcoatingsandscribingafter
coatings. In order to optimize the laser-machining process and reduce the
experimentaltimeandcost,themodelstudiesreceiveconsiderableattention.
Inaddition,themodelstudiesgiveinsightintothephysicalprocessesthattake
place during the heating process, being easier to accomplish as compared to
experimental studies. The measurement of physical properties during laser–
workpieceinteractionisdifficultandcostlysincetheprocessisinvolvedwith
hightemperature,shortduration,andlocalizedheating.Fromthemodeling
point of view, laser machining can be classified into two categories: i) laser
conduction limited heating, and ii) laser nonconduction limited heating. In
the laser conduction limitedheating situation, the substrate surface isheated
up to the melting temperature of the substrate material; in this case, the
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2 LaserHeatingApplications
substrate remains in the solid state during the process. One of the laser
conductionlimitedapplicationsisthelaserquenchingofthesurfaces.Inthe
lasernonconductionlimitedheatingsituation,thesubstratesurfaceundergoes
a phase change during the processing, i.e., melting and subsequent evapo-
ration result.The laser drilling, cutting, and welding are typical examples of
laser nonconduction limited heating situation.
Whenahigh-powerlaserbeamisfocusedontothesubstratesurface,the
beam energy is partially absorbed by the substrate material. Depending on
the focused beam diameter at the surface, laser power intensity (combining
thelaseroutputenergyandpulselength),andreflectivityofthesurface,the
substratematerialundergoessolidheating,melting,andevaporation.Inthe
case of evaporation process, the evaporating front detaches from the liquid
surface,generatingarecoilpressureacrossthevapor–liquidinterface.Asthe
evaporation of the surface progresses, the recoil pressure increases consid-
erably while influencing the evaporation rate. As the heating progresses
further,theliquidsurfacerecessestowardthesolidbulk,formingthecavity
in the substrate material. Depending on the pulse length and power
intensities, the liquid ejection from the cavity occurs, which is particularly
true for the long pulse lengths (wms pulse lengths); however, the surface
ablation without liquid ejection takes place for short length pulses (wns
pulselengths).Moreover,theliquidejectionimprovesthematerialremoval
rates from the cavity. In the case of laser short-pulse processing, the recoil
pressure increases substantially due to high rates of momentum exchange
duringtheevaporationprocess.Inthiscase,highpressureatthevapor–liquid
interfaceactsasapressureforcegeneratingsurfaceindentionandhighstress
levels at the liquid–solid interface. This, in turn, results in a pressure wave
propagatingintothesubstratematerial.Dependingonthemagnitudeofthe
pressure wave, the plastic deformation through dislocations in the surface
region of the substrate material takes place. The depth of the deformed
region is limited with the interaction of the loading (plastic wave) and
unloading(elasticwave)waves,i.e.,astheloadingphaseiscompleted(when
theevaporationiscompleted,therecoilpressurediminishes),theunloading
wave (elastic wave) from the liquid–solid interface initiates. Since the
unloading wave travels faster than the loading wave, both waves meet at
some depth below the surface. Since the wave motion in the substrate
material is complicated, a comprehensive investigation is required for the
understanding of the physical insight into the process.
Short pulse heating of metallic surfaces results in thermal separation of
theelectronandthelatticesubsystems.Thermalcommunicationinbetween
IntroductiontoLaserHeatingProcess 3
bothsubsystemsgivesrisetononequilibriumenergytransportintheheated
region. The collisional process taking place between excited electrons and
thelatticesubsystemgovernstheenergytransferfromtheelectronsubsystem
tothelatticesubsystem.Thisprocesscontinuesuntilthethermalequilibrium
is established between the subsystems. When the heating duration is
comparable to electron relaxation time, nonequilibrium energy transfer
takes place through the collisional process while dominating over the
diffusional energy transfer in the solid. In this case, the Fourier heating
model fails to give a physical insight into the heat transfer in the substrate
material.Consequently,theelectronkinetictheoryapproachincorporating
the electron–lattice site collisions between the lattice and electron subsys-
tems becomes essential to account for the formulation energy transport in
thesolids.Moreover,theclosed-formsolutionforthegoverningequationof
the physical problem becomes fruitful, since it provides the functional
relationbetweentheindependentvariables,suchastimeandspace,andthe
dependent variable, such as temperature. Although the analytical approach
giving the approximate solution is possible, the solution is limited in time
andspacescalesduetotheassumptionsmadeintheanalysis.Consequently,
the general form of analytical solution for the nonequilibrium energy
transport in the metallic substrates due to short-pulse heating becomes
essential.
The word laser is an acronym for “light amplification by stimulated
emission of radiation.” Albert Einstein in 1917 showed that the process of
stimulated emission must exist [1] but it was not until 1960 that Maiman
[2] first achieved laser action at optical frequencies in ruby. The basic
principles and construction of a laser are relatively straightforward, and it is
somewhat surprising that the invention of the laser was so long delayed. In
thetimewhichhaselapsedsinceMaiman[2]firstdemonstratedlaseraction
in rubyin 1960,theapplicationsoflasers have multipliedtosuch anextent
that almost all aspects of our daily lifeare touched upon by lasers. They are
used in many types of industrial processing, engineering, meteorology,
scientificresearch,communications,holography,medicine,andformilitary
purposes. It is clearly impossible to give an exhaustive survey of all of these
applications.
In considering the various properties of laser light, one must always
rememberthatnotallofthedifferenttypesoflasersexhibittheseproperties
to the same degree. This may often limit the choice of laser for a given
application.Therearecertaindistinctivespatialprofilesthatcharacterizethe
crosssectionsoflaserbeams.Thespatialpatternsoflasersaretermedtransverse
4 LaserHeatingApplications
modes and are represented in the form TEM , where m and n are small
mn
integers. The term TEM stands for transverse electromagnetic. The transverse
modes arise from considerations of resonance inside the laser cavity and
represent configurations of the electromagnetic field determined by the
boundary conditions in the cavity.
The notation TEM can be interpreted in rectangular symmetry as
00
meaning the number of nulls in the spatial pattern that occur in each of
twoorthogonaldirections,transversetothedirectionofbeampropagation.
TheTEM modehasnonullsineitherthehorizontalorverticaldirection.
00
The TEM mode has one null in the horizontal direction and none in the
10
verticaldirection.TheTEM modehasonenullasonepassesthroughthe
11
radiation pattern either horizontally or vertically. In addition, there are
solutions of the boundary conditions which allow cylindrical symmetry.
The mode denoted by TEM represents a superposition of two similar
01
(cid:2)
modes rotated by 90 (rectangle) about the axis relative to each other. In
many cases, a superposition of a number of modes can be present at the
sametime,sothattheradiationpatterncanbecomequitecomplicated.Itis
desirable to obtain operation in the TEM mode for the machining
00
operation. This transverse mode has been called the Gaussian mode. The
Gaussian intensity distribution I(x) as a function of the radius from the
center of the beam is given by
(cid:1) (cid:3)
x2
IðxÞ ¼ I0exp (cid:3) r2 (1.1)
0
whereI istheintensityofthebeamatthecenter,xistheradialdistance,and
0
r is the Gaussian beam radius, i.e., the radius at which the intensity is
0
reduced from its central value by a factor e2. The total power is given by
P ¼ pr2I (1.2)
0 0
The beam divergence angle q of a Gaussian beam is
2 l
q ¼ (1.3)
pr
0
The spatial profile of the Gaussian TEM mode is desirable, since its
00
symmetry and the beam divergence angle are smaller than for the higher-
order transverse modes.
Spectrometric examination of temporal laser output modes reveals that
the output power consists of very narrow spectral lines. The two mirrors
IntroductiontoLaserHeatingProcess 5
of the laser form a resonant cavity and standing wave patterns are set up
between the mirrors. The standing waves satisfy the condition
l
L ¼ m (1.4)
2
or
c
v ¼ m (1.5)
2L
where c is the speed of light and L is the optical path length between the
mirrors, in which case the wavelength l would be the vacuum wavelength
andmisanintegerinEqn(1.4).Asmallchangeinlresultsindifferentvalues
of m and each value of m satisfying Eqn (1.5) defines the temporal mode of
the cavity.
Equation(1.5)showsthatthefrequencyseparationdvbetweenadjacent
modes (dm ¼ 1) is given by
c
dv ¼ (1.6)
2L
As Eqn (1.6) is independent of m, the frequency separation of adjacent
axialmodesmustbethesameirrespectiveoftheiractualfrequencies.Hence,
themodesofoscillationofalasercavitywillconsistofaverylargenumberof
frequencies, each given by Eqn (1.5) for different valuesof m and separated
by a frequency difference given by Eqn (1.6).
The most common method for providing single-mode operation
involves construction of short laser cavities, so that spacing between modes
(c/2L,wherecisthespeedoflightandListheopticalpathlengthbetween
themirrors)becomeslargeandlasingactionoccursinonetemporalmode.It
hasthedisadvantagethattheshortcavitylimitstheoutputpowerextracted.
Further frequency stabilization is obtained by vibration isolation, tempera-
turestabilization,andcontrolofthemirrorspacingaccordingtotheoutput
power.
REFERENCES
[1] Einstein A.Onthe quantum theoryof radiation. PhysZ1917;18(6):121–8.
[2] Maiman TH. Stimulated optical radiation inruby. Nature 1960;187(4736):493–4.
CHAPTER TWO
Conduction-Limited Laser Pulsed
Laser Heating: Fourier Heating
Model
Contents
2.1. IntroductiontoHeatGenerationDuetoAbsorptionofIncidentLaserBeam 7
2.2. TemperatureFieldDuetoLaserStepInputPulseHeating 10
2.2.1. InsulatedBoundaryConditionattheSurface 11
2.2.1.1. StepInputPulseHeatingwithoutCoolingCycle 11
2.2.1.2. StepInputPulseHeatingIncludingHeatingandCoolingCycles 14
2.2.1.3. ExponentialPulseHeating 18
2.2.2. ConvectiveBoundaryConditionattheSurface 23
2.2.2.1. StepInputPulseHeatingIncludingHeatingandCoolingCycles 23
2.2.2.2. ExponentialPulseHeating 31
2.3. ThermalEfficiencyofHeatingProcess 35
2.4. ResultsandDiscussion 38
2.4.1. StepInputPulseHeatingwithoutCoolingCycle:InsulatedBoundary
ConditionattheSurface 39
2.4.2. StepInputPulseHeatingIncludingHeatingandCoolingCycles:
InsulatedBoundaryConditionattheSurface 40
2.4.3. ExponentialPulseHeating:InsulatedBoundaryConditionatthe
Surface 42
2.4.4. StepInputPulseHeatingIncludingHeatingandCoolingCycles:
ConvectiveBoundaryConditionattheSurface 44
2.4.5. ExponentialPulseHeating:ConvectiveBoundaryConditionatthe
Surface 45
2.4.6. ThermalEfficiencyofHeatingProcess 47
References 50
2.1. INTRODUCTION TO HEAT GENERATION DUE
TO ABSORPTION OF INCIDENT LASER BEAM
The absorption of laser light takes place through photon interaction with
bound and free electrons in the material structure, whereby these electrons
are raised to higher energies. The further conversion of this energy
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8 LaserHeatingApplications
takes place through various collision processes involving electrons, lattice
site phonons, ionized impurities, and defect structures. The mean collision
timeinthisenergytransferenceisoftheorderof10(cid:2)12–10(cid:2)14s[1].During
the pulses at which pulse duration is 10(cid:2)9 s, the absorbing electrons
have had time to undergo many collisions. This leads to the equilibrium
energy transport in the lattice system, i.e., the laser energy is instanta-
neously converted to internal energy gain at the point at which absorption
takes place. The concept of equilibrium energy transport allows
a conventional heat transfer analysis (the Fourier heating analysis) to be
considered. In a laser-machining operation, two regimes are of interest,
whichcorrespondtolow-andhigh-intensityradiation. Thetermslowand
high are purely relative and apply when the machining process is
conduction-limitedandnonconduction-limited,respectively.Ananalysisis
to be performed which predicts the temperature profiles for the conduc-
tion-limited case and the material removal rates due to evaporation in the
nonconduction-limited case.
Instudyingthetemperatureprofilesduetoabsorptionofalaserbeamby
asubstratematerial,itisobviouslyofcrucialimportancetoknowhowmuch
oftheincidentradiationwhichreachesthetargetsurfaceactuallypenetrates
the material. It should be noted that the form of dependence of absorption
on temperature greatly affects the form of the resulting temperature profile
in the material. The electromagnetic theory of light, which is based on the
solution of the Maxwell equations, generally leads to results which are in
qualitative agreement with experimental results. However, there are defi-
ciencies in this method which are due to the phenomenological nature of
theanalysis.Forexample,theHagen–Rubensrelationshipdoesnotholdon
the infrared and is even less valid in the visible region of the spectrum. In
order to explain more accurately the interaction of electromagnetic waves
with matter, it is necessary to take a more fundamental view of the inter-
actionprocess.Thiscanbeachievedusingtheideasofquantumtheory,but
there are great difficulties associated with the practical use of the results
which are obtained.
Thelaserintensityinaone-dimensionalsolid,atanypointinit,isgiven
by the Beer–Lambert law:
dI
¼ (cid:2)dI (2.1)
dx
where x is the depth along the substrate material in units of m. The surface
ofthesubstratematerialliesatx¼0.I¼I(x)isthelaserintensityatadepth
Conduction-LimitedLaserPulsedLaserHeating:FourierHeatingModel 9
x in units of W/m2. d is the absorption coefficient of the material in units
of m(cid:2)1. This may be constant or variable along the depth of the substrate
material.
To find the intensity as a function of x, we integrate Eqn (2.1):
dI
¼ (cid:2)dðxÞdx
I
ZI ZL (2.2)
dI
0 ¼ (cid:2) dðxÞdx
I
I0 0
Intheaboveintegration,xactsasadummyvariableandmaybechangedto
anotherdummyvariable,sayu.Also,tofindtheintensityatanypointx,we
may replace L by x in the above integration:
ZI Zx
dI
0 ¼ (cid:2) dðuÞdu
I
I0 0
(cid:2) (cid:2) Zx
0ln(cid:2)(cid:2)(cid:2)I(cid:2)(cid:2)(cid:2) ¼ (cid:2) dðuÞdu (2.3)
I0
0
R
x
(cid:2) dðuÞdu
0I ¼ I0e 0
Hence, the intensity at any point x is given as
R
x
(cid:2) dðuÞdu
I ¼ I0e 0 (2.4)
For a constant absorption coefficient (d ¼ constant), Eqn (2.4) yields
I ¼ I0e(cid:2)dx (2.5)
When the substrate material absorbs laser beam energy, it generates heat,
which can be defined through
dI
S ¼ (cid:2) W=m3 (2.6)
dx
10 LaserHeatingApplications
Substituting Eqn (2.4) into Eqn (2.6) and using the Leibniz rule yield
8 9
R
< x =
d (cid:2) dðuÞdu
S ¼ (cid:2)dx:I0e 0 ;
8 9
R
x < Zx =
(cid:2) dðuÞdu d
¼ (cid:2)I0e 0 dx:(cid:2) dðuÞdu;
0 (2.7)
9
R (
x Zx =
(cid:2) dðuÞdu
¼ I0e 0 0:duþdðxÞ:1(cid:2)dð0Þ:0;
0
R
x
(cid:2) dðuÞdu
¼ I0dðxÞe 0
For a constant absorption coefficient (d ¼ constant), Eqn (2.7) yields
S ¼ I0de(cid:2)dx (2.8)
Equation (2.8) gives volumetric heat generation in the substrate material
when subjected to laser heating.
2.2. TEMPERATURE FIELD DUE TO LASER STEP INPUT
PULSE HEATING
Twomajorheatingconditionscanbeidentifiedinthelaserheatingprocess,
namely,conduction-limitedandnonconduction-limitedheatingcases.Inthe
case of conduction-limited heating, heat conduction is a major mechanism
governing the energy transport process inside the substrate material during
the laser heating process, while phase change (melting and evaporation)
occurs in the nonconduction-limited heating case. In general, laser surface
heat treatment and quenching are involved with a conduction-limited
heating process. The temperature field associated with the conduction-
limited heating can be obtained through using the Fourier heating law.
Since the thickness of the substrate material is considerably larger than the
absorptiondepthofthemetallicsubstrateandthelaser-irradiatedspotsizeat
theirradiatedsurfaceissmall,theconsiderationofone-dimensionalheating
can be sufficient to model the heating situation. This consideration enables
Conduction-LimitedLaserPulsedLaserHeating:FourierHeatingModel 11
us to obtain the closed-form solution for temperature distribution in the
laser-irradiatedregionwhileusingtheanalyticalmethods.Consequently,an
analytical method incorporating the Laplace transformation method is
adoptedtoobtaintheclosed-formsolutionfortheheatconductionequation
including the volumetric heat source due to absorption of the laser beam.
TheFourierheatequationisintheformofapartialdifferentialequation,
andtheclosed-formsolutiondependsontheinitialandboundaryconditions
aswellasthetemporalbehaviorofthevolumetricheatsource.Consequently,
closed-form solutions for the insulated and the convective boundary
conditionsatthesurfacearepresented.Inaddition,threelaser-pulsetypesare
incorporated, namely step input pulse with no cooling period, step input
pulse with cooling period, and time exponentially varying pulse.
2.2.1. Insulated Boundary Condition at the Surface
2.2.1.1. Step Input Pulse Heating without Cooling Cycle
Laserheatingpulsecanbeconsideredassteppulsewithnocoolingcycle,in
which case heating terminates once the laser pulse ends (Figure 2.1).
Therefore,temperaturebehaviorafterthepulseendingcannotbepredicted
from the resulting closed-form solution (Eqn (2.2)).
TheFourierheattransferequationforalaserstepinputheatingpulsecan
be written as
v2T þI1de(cid:2)dx ¼ 1vT (2.9)
vx2 k a vt
where
(cid:3) (cid:4)
I1 ¼ 1(cid:2)rf I0
where I is the step input pulse intensity, x is the distance, t is the time,
1
kisthethermalconductivity,aisthethermaldiffusivity,distheabsorption
coefficient,r isthereflectioncoefficient,andI isthepeakpowerintensity.
f 0
y
sit
n
nte Unit Step Function
I
Time (t)
Figure2.1 Laserstepinputpulse.