Transcript mytalk - University of Kentucky

Numerical simulation of heat transfer mechanisms
during femtosecond laser heating of nano-films
using 3-D dual phase lag model
Presenter:
Illayathambi Kunadian
[email protected]
Co-authors:
Prof. J. M. McDonough
Prof. K. A. Tagavi
Department of Mechanical Engineering
University of Kentucky, Lexington, KY 40506
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Contents of this talk
•
Overview of different models
• Classical heat conduction model
• Hyperbolic heat conduction model
• Two-step models (parabolic and hyperbolic two-step models)
• Dual phase lag heat conduction model
•
Mathematical formulation
•
Numerical analysis
• Stability criterion
•
Numerical results
• 1D problem (short-pulse laser heating of gold film)
• 3D problem (short-pulse laser heating of gold film at different locations)
• Grid function convergence tests
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Classical heat conduction
•
Heat flux directly proportional to temperature gradient (Fourier's law)
 

q (r , t )  kT (r , t )
•
Incorporation into first law of thermodynamics yields parabolic heat conduction equation
T
  2T
t
Anomalies associated with Fourier law
•
Heat conduction diffusion phenomenon in which temperature disturbances propagate at
infinite velocities. Assumes instantaneous thermodynamic equilibrium
•
Heat flow starts (vanishes) simultaneous with appearance (disappearance) of temperature
gradient,violating causality principle
Fourier's law fails to predict correct temperature distribution
•
Transient heat flow for extremely short periods of time (applications involving laser
pulses of nanosecond and femtosecond duration)
•
High heat fluxes
•
Temperatures near absolute zero (heat conduction at cryogenic temperatures)
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Hyperbolic heat conduction
•
Modified heat flux that accommodates finite propagation speed of observed thermal
waves proposed by Vernotte and Cattaneo (1958)
 

q (r , t   q )  kT (r , t )

q(r , t )



 q(r , t )  kT (r , t )
t
•
Combined with equation of energy conservation gives hyperbolic heat conduction
equation (HHCE)
 2T 1 T
c is the speed of thermal wave
1 q

 c 2 2T

2
propagation
 q t
t
c2 
•
HHCE suffers from theoretical problem of compatibility with second law of
thermodynamics
•
•
•
predicts physically impossible solutions with negative local heat content
Neglects energy exchange between electrons and the lattice, applicability to short pulselasers becomes questionable
No clear experimental evidence of hyperbolic heat conduction even though wave
behavior has been observed
•
•
Earliest experiments detecting thermal waves performed by Peshkov (1944) using superfluid liquid
helium at temperature near absolute zero
He referred to this phenomenon as “second sound”, because of similarity between observed thermal
and ordinary acoustic waves
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Internal Mechanisms during laser heating
Stage I
Heated
electrons
Stage II
Energy quanta;
Phonons
no temperature
rise at time t
temperature
rise at time t + τ
Photon
energy at
time t
Electron-gas heating by photons
Metal lattice heating by
phonon-electron interactions
* D. Y. Tzou, Macro-microscale Heat Transfer, the lagging behavior
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Two-step models
Anisimov et al. (1974) proposed two-step model to describe the electron temperature and
lattice temperature during the short-pulse laser heating of metals
Te

   q  G(Te  Tl )
t
T
Cl l  G (Te  Tl )
t
eliminating electron-gas temperature (Te)
Ce
(Heating of electrons)
(Heating of metal-lattice)
1  2Tl
1 Tl  e ( 2Tl )

 2
  2Tl
2
2
 e t C E
t
C E t
eliminating metal-lattice temperature (Tl)
1  2Te
1 Te  e ( 2Te )
2




Te
t
C E2 t 2  e t C E2
Where,
e 
k
Ce  Cl
CE 
 4 (neVs k B ) 2
G
k
G = Phonon-electron coupling factor
Vs = speed of sound
ne = number density of free electrons per
unit volume
kB = Boltzmann constant
k = Thermal conductivity
Ce = Heat capacity of electrons
Cl = Heat capacity of metal lattice
kG
CeCl
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Dual phase lag model
•
Modified heat flux vector represented by Tzou (1995)
 

q (r , t   q )  kT (r , t   T )
T ~ delay behavior in establishing the temperature gradient
•
•
•
q ~ delay behavior in heat-flow departure
 


q
(
r
,
t
)

[

T
(
r
, t )]




Taylor expansion gives q (r , t )   q
 k T (r , t )   T

t
t


coupled with energy equation gives dual phase lag (DPL) equation
 q  2T 1 T
( 2T )

T
  2T
2
 t
 t
t
Comparing coefficients of DPL model with those of two-step model we can represent
microscopic properties by
4
Cl Ce
k

(neVs k B ) 2
Cl
q 

G
T 
Ce  Cl
G (Ce  Cl )
G
k
ne   G     q   T  0
T  0
 q  2T 1 T
( 2T )
2





T  classical diffusion equation
T
2
 t
 t
t
 q  2T 1 T
( 2T )
2





T  classical wave equation
T
 t 2  t
t
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Mathematical formulation
Gold Film
Volumetric heating in the sample

z
S ( z, t )  S0e  I (t )


where, S 0  0.94 J 1  R 
 t p 
I (t )  I 0e
a
t
tp
z
Laser
Femtosecond laser heating is modeled by energy absorption rate
1  R   z 1.992 t  2t p
S ( z , t )  0.94 J 
 exp  
tp
 t p   




1  R   x 2  y 2 z 1.992 t  2t p

S (r , t )  0.94 J 
 
 exp 
2
t


tp
r
0
 p  
L = 100 nm
(1D)
 (3D)




In presence of S (r , t ) DPL equation becomes
 q  2T 1 T
( 2T )
1
S 
2





T

S




T
q
 t 2  t
t
k
t 
•
•
•
•
•
•
•
•
•
•
•
Laser fluence J = 13.4 Jm2
Reflectivity R=0.93
Thermalization time tp=96fs
Depth of laser penetration  = 15.3nm
Radius of laser beam r0= 100nm
S0 is the intensity of laser absorption
I(t) is the intensity of laser pulse
 = 1.2104m2s1
q = 8.5ps
 = 90ps
T
k = 315Wm1K1
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Numerical analysis
 

q (r , t )
[T (r , t )]
 


q (r , t )   q
 k T (r , t )   T

t
t


q1   q
q1
 T
  T 
 k    T  
t
t  x 
 x
 T
q2
  T 
q2   q
 k    T  
t
t  y 
 y
q3   q
q3
 T
  T 
 k    T  
t
t  z 
 z
500nm
T 

q  S  C
(r , t )
t
 q  2T 1 T
( 2T )
1
S 
2





T

S




T
q
 t 2  t
t
k
t 
z
y
x
100nm
500nm
500nm
500nm
Initial Conditions

T (r ,0)  To
T 
(r ,0)  0
t
Boundary Condition
T
 0,
r
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Numerical analysis
Explicit finite-difference scheme employed to solve the DPL equation
Centered differencing approximates second-order
derivatives in space
Centered differencing is employed for time
derivative in the source term
 2T
S
1

[ Sin, j 1  Sin, j 1]
t 2t

x 2
1
x
[T n
2 i 1, j ,l
 2Tin, j ,l
 Tin1, j ,l ]
Mixed derivative is approximated using centered
difference in space and backward difference in time
3T
tx
2

1
tx
[T n
 2Tin, j ,l  Tin1, j ,l
2 i 1, j ,l
 Tin1,1j ,l  2Tin, j,1l  Tin1,1j ,l ]
Stability criterion for 3-D DPL model obtained
using von Neumann eigenmode analysis (Tzou)
t (2t  4 T )
x 2 (t  2 q )

t (2t  4 T )
y 2 (t  2 q )

t (2t  4 T )
z 2 (t  2 q )
Forward differencing approximates first-order
derivative in time
 b  b 2  4ac
t 
2a
T 1 n 1
 [Ti, j ,l  Tin, j ,l ]
t t
a  2 (y 2z 2  x 2z 2  x 2y 2 )
Centered differencing approximates secondorder derivative in time
 2T
1

[Tin, j,1l  2Tin, j ,l  Tin, j,1l ]
t 2 (t ) 2
1
b  x 2y 2z 2  4y 2z 2 T  4x 2z 2 T
 4x 2y 2 T
c  2x 2y 2z 2 q
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1D problem: Geometry and results
Gold Film
Laser
X
L = 100 nm
Fig. 1. Normalized Temperature change at front surface of a gold film of
thickness 100nm:  = 1.2104 m2s1, k = 315 Wm1K1, T = 90 ps, q = 8.5 ps.
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3-D Schematic of femtosecond laser heating of gold film
200nm laser
beam
Work piece-Gold
250nm
500nm
100nm
250nm
500nm
500nm
500nm
500nm
Fig. 2. 3-D schematic of laser heating of gold film at different
locations
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Temperature distribution predicted by different models
DPL
Parabolic
DPL DPL
Parabolic
Parabolic
Parabolic
Hyperbolic
Parabolic
DPL
DPL
Parabolic
Hyperbolic
Fig. 3. Temperature distribution at top surface of gold film predicted by different models
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Temperature distribution predicted by different models
At t = 0.3 ps
DPL
Parabolic
DPL DPL
Parabolic
Parabolic
DPL
Parabolic
Hyperbolic
Hyperbolic
Hyperbolic
At t = 0.9 ps
DPL
Parabolic
Hyperbolic
Hyperbolic
Fig. 3a. Temperature distribution at top surface of gold film predicted by different models
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Temperature distribution cont.
At t = 1.56 ps
DPL
Parabolic
DPL
Hyperbolic
Parabolic
At t = 2.23 ps
DPL
Parabolic
Hyperbolic
Parabolic
Fig. 3b. Temperature distribution at top surface of gold film predicted by different models
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Grid function convergence test
Temperature (K)
312
51×51×11
310
101×101×21
308
306
304
201×201×41
302
300
0
50
100
150
200
250
Radial distance (nm )
Fig. 4. Temperature plots of front surface of gold film at t = 0.3 ps in radial
direction using different grids: 515111, 10110121 and 20120141
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Conclusions
•
•
•
•
•
•
•
DPL model agrees closely with experimental results in one
dimension compared to the classical and the hyperbolic models
Energy absorption rate used to model femtosecond laser heating
modified to accommodate for three-dimensional laser heating
Simulation of 3-D laser heating at various locations of thin film
carried out using pulsating laser beam (~ 0.3 ps pulse duration) to
compare different models
Stability criterion for selecting a numerical time step obtained
using von Neumann eigenmode analysis: x = y = z = 5nm
 t = 3.27 fs
Different grids (515111, 10110121 and 20120141) were
used to check convergence in numerical solution
Classical and hyperbolic models over predict temperature
distribution during ultra-fast laser heating, whereas DPL model
gives temperature distribution comparable to experimental data
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Conclusions cont.
•
Compared to experimental data large difference in diffusion
model due to negligence of both micro structural interaction in
space and fast transient effect in time.
•
Hyperbolic model redeems difference between experimental and
diffusion, but still overestimates transient temperature because
it neglects micro structural interaction in space.
•
DPL model incorporates delay time caused by phonon-electron
interaction in micro scale
– Time delay due to fast transient effect of thermal inertia absorbed
in phase lag of heat flux
– Time delay due to finite time required for phonon-electron
interaction to take place absorbed in phase lag of temperature
 gradient
transient temperature closer to experimental observation.
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