laser processing of materials - teil 6 von 8 - temperature distributions - skript - prof dr frank...

7/31/2019 Laser Processing Of Materials - Teil 6 Von 8 - Temperature Distributions - Skript - Prof Dr Frank Mcklich - Dr Andr

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Functional MaterialsFunctional Materials Saarland UniversitySaarland University

Laser processing of materials

Prof. Dr. Frank Mcklich

Dr. Andrs Lasagni

Lehrstuhl fr FunktionswerkstoffeSommersemester 2007

Temperature distributions


Contents:

1. Definitions

2. Diffusion length

3. Temperature distribution in Bulk materials

i. Heat diffusion equation

ii. Constrain conditions

iii. The error functioniv. General solution

v. Formulae for different conditions

vi. Examples of laser heating

4. Temperature distribution in Thin Films

i. Heat diffusion equation

ii. Constrain conditions

iii. Formulae for different conditions

iv. Cooling of thin-filmsv. Heat transfer to Substrate

vi. Lateral Heat-transfer in the film

Temperature distributions


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Temperature distributions in bulk materials

LASER BEAM

txhp heat

heat

: light penetration depth

xhp: heat affected zone

t (diffusion length)

k: thermal diffusivity

t: interaction time

The temperature rise is basically

controlled by , xhp and r0.



EXAMPLE

light penetration depth ()for metals:

10-6 - 10-7 mts

Thermal diffusivity (k)

for metals:10-4 - 10-5 m/s

For t = 10 ns

Xhp ~ 0.1-1m

For t = 1ms

Xhp ~ 100-300m

LASER TYPE:


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How to calculate the temperature

as function of the time and depth?

t

T

cqx

T

Kx p

=+

.

q

T (,y,z,t) = T0

Temporal condition:

T (x,y,z,0) = T0

0),,,0( =

tzyx

TK

(no heat lost (no radiation))

x

q=0 q=0

xeRqxq

= )1()( 0.

=1/: absorption coefficient

R: reflectivity;

K: thermal conductivity

q0: Power density (laser-light

flux density [W/cm])

cp: specific heat



In one-dimensional case, for uniform surface irradiation, and

constant K, cp, :

++

+

+

+=

kt

xkterfcxkt

K

qR

kt

xkterfcxktKqR

eK

qR

kt

xierfckt

K

qRTtxT

x

2)exp(

2

)1(

2)exp(

2)1(

)1(

2

)1(2),(

20

20

000

Where: erfc(u) is the complementary error function and ierfc(u)

its integral


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Complementary error function:

Integral of the complementary

error function:



However, different simplified equations can be used in different

situations (, xhp and r0): (q = q0 (1-R))

metalspolymers


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Example: Laser heating of stainless steel AISI 304:

t = tp/2

t = tp

Melting point

Melting point

Temperature at different intensities before laser is turned off

tp = 10-3 s


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t = 2 tp

t = 10 tp

Melting point

Melting point

Temperature at different intensities after laser is turned off

tp = 10-3 s




tp = 10-3 s

Melting point


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Temperature distributions in thin films

q0

0),,,0(/ = tzyxTK

x

=1/: absorption coefficient

for h>=> we can neglect the wavereflected from the film substrate

In case of thicker films it is

necessary to consider that heat

release is not uniform in depth

h

However, heat release does not

follow light-absorption law given that

the diffusion length >> (up to h ~ 5m)

(1)

(2)

xeAqxq 1)()( 10

.=



t

Tctxq

x

TK p

=+

2,1

2,12,1

.

2,12

2,12

),(

h

Aqq 10

.

1 =

)](exp[)1( 21102

.

2 hxARqq =

(this means that

T1=C along x)

Considering that r0>>(1)1/2 =>

q0

0),,,0(/ = tzyxTK

x

h (1)

(2)(1-R1-A1) = D1 (x = h)

xeAqxq 1)()( 10

.=

Observation: R1+A1+D1 = 1

T1(x)=Cte


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Considering:

00/)(1 == xatxtT

0)()( 021 === tatTxTxT

hxattTtT == )()( 21

hxatx

tTk

x

tTk =

=

)()( 22

11

== xatTtT 02 )(

q0

0),,,0(/ = tzyxTK

x

h (1)

(2)

xeAqxq 1)()( 10

.=



D1 = 0 (Transmission)

D1

> 0 ; A1

>0

(A1+D1 = 1 R1)

D2 = 0

R1+A1+D1 = 1

D1 > 0 ; A1>0

(A1+D1 = 1 R1)

D2 > 0

D1 > 0 ; A1~0 (D1 = 1 R1)

D2 = 0

D1 > 0 ; A1~0 (D1 = 1 R1)

D2 > 0

= laser pulse duration Thin-film temperature-time dependence


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q0 = 1012 W/m

Film thickness = 5m

Substrate: Glass

Numerical calculations

Films temperature Films temperature



Cooling of thin-films

Cooling of the film depends on the heating regime

For the case of an opaque film on any substrate, the temperature at the film

after the laser interaction time () is given by:

nstk

herfTtT 100

)()()(

2

1 =

hc

AqT

p11

101 )(

=

2

2

102

2)(

k

K

AqT =

Example:

considering: = 10ns, h=100nm, k2 = 6E-3 cm/s (glass),

the time tto cool the film up to 0.1T1 is 100 ns!

Question: why are such cooling-times extremely short?


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Heat transfer to Substrate

1 - During laser irradiation, a more or less significant part of the thermal

energy is drained through the substrate2 - Only a part of the substrate with depth lp is heated up:

lp ~ (2 )1/2

3 - The energy efficiency of the treatment can be described as:

= 1 Qd/QaQa = energy absorbed

Qd = energy dissipated

4- The energy efficiency can be written in terms of thermal properties of

the substrate and the film:

with:2

1+= tkc

hc

p

p

222

11

=



Heat transfer to Substrate

Qd = 0 Qd = Qa

With the rise of (lower), thefilm-heating efficiency drops

quickly!

If:2

22

22

11

4 k

h

c

c

p

p

then >

Consequently: laser thin-film

treatment should be carried out in a

pulsed regime at short times. This

provides lower energy loss and lower

risk of substrate damage!

E.g.: 100 nm Cu-film on quartz

substrate : < 36 ns


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Lateral Heat-transfer in the film


2

0r

hr02h

Generally, the lateral heat flow weakly

affects the film heating due to:

2

0r hr02>

q

q

From 3D analysis it can be proved that

if:

Then the lateral heating is negligible =>

10 2 kr >

Strong lateral

heat flow

T0c: temperature at the center

without considering lateral heat flow

Trc: temperature at the centerconsidering lateral heat flow



Lateral Heat-transfer in the film

Example:

Cu-Film

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0.00 10.00 20.00 30.00 40.00 50.00

][2 1 mk

Laser interaction time [ns]

10 2 kr >

10 2 kr liquid expulsion, vaporization

Liquid expulsion

Vapor/plasma

plume

Vapor consist on: clusters, molecules,

atoms, ions, and electrons

The higher the laser-light intensity the

higher the density of species

The energy required to remove an atom

from a solid can be estimated from:

Ha [J/atom] = HV[J/g] / Ns

HV: enthalpy of vaporization

Ns=L/M: atom number density(L=Avogadro Number; M=atomic weight)


Plume

P

PS

PL

Laser ablation - vaporization

However the plume will absorb and scatter the incident laser radiation!

Considering only evaporation:

( )( )[ ]Lsinput PRPPE = 1Energy input:P: laser power

Ps: power absorbed by the vapor plume

PL: energy looses (heat conduction, radiation, convection, reaction

enthalpies, etc.)

R: reflectivity

: dwell time of the laser beam

( ) ( )mvplmpsmvvap TTcTTcLLhAE +++= 0Energy to required to vaporize a volume A.h:

Lv: latent heat of vaporization; Lm: latent heat of melting

Cps, Cpl: specific heat of solid and liquid, respectively

Tm: melting point; Tv: vaporization point; T0: initial temperature

: density

h: ablated depth; A: ablated area

h


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Plume

P

PS

PL

When plasma is produced, an important part of the energy is absorbed

by the plume and the

calculation is more

complicated



( )( )( ) ( )[ ]

mvplmpsmv

ls

TTcTTcLLA

PRPPh

+++

0

1


Combining both equations:

( )( )

v

ls

H

Rh

1

: laser fluence (J/cm)

This equation is only valid if: l(optical penetration depth) bulk heating is minimized => PL ~ 0

( )( )

vH

Rh

1 Ablation rate is proportional to Laser fluence(relative low energy densities, without considering

liquid expulsion)


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One-dimensional model (surface temperature ):

Considerations:

We ignore any liquid layer

Tvs is the temperature at the solid-vapor interface and vvs is its velocity Any attenuation of incident laser light by the plume is ignored

( )

+

+= mv

vsp

s LLv

PR

cTT

)1(

2

1)1( 0

=

s

vvs

Tvv

exp)2( 0

we need to solve equations (1) and (2) simultaneously

B

a

v

B

vv

k

H

k

E

=

vo is in the order of the sound velocity within the solid

is the activation energy for vaporization and can be

replaced by the enthalpy of vaporization per atom/moleculev


Laser ablation

Influence of liquid layer:

Vapor

Recoil force

Liquid

expulsion

( )ssatrec Tpp ~

The recoil pressure (prec) is originated

because of the momentum conservation

of the evaporated species. This is in the

order of the saturated vapor pressure at

Ts (psat) and increases nonlinearly with

P (W/cm)

The melt-ejection flux (J)

is given by:

(w = laser-beam radius)

4/11~ recm p

wJ

laser processing of materials - teil 6 von 8 - temperature distributions - skript - prof dr frank...

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