ikuna0@engr.uky.edu
Department of Mechanical Engineering
University of Kentucky, Lexington, KY 40506
Advanced Computational Fluid Dynamics
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• 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
Advanced Computational Fluid Dynamics
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• 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
 t
   2
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)
Advanced Computational Fluid Dynamics
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• 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 )
 t
 q ( r

, t )
  k

T ( r

, t )
• Combined with equation of energy conservation gives hyperbolic heat conduction equation (HHCE)

 t
2
T
2


1 q


T t
 c
2  2
T c
1
2


 q c is the speed of thermal wave propagation
• 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
Advanced Computational Fluid Dynamics
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Stage I Stage II
Heated electrons
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
Advanced Computational Fluid Dynamics
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Anisimov et al. (1974) proposed two-step model to describe the electron temperature and lattice temperature during the short-pulse laser heating of metals
C e
C l

T e
 t

T l
 t



G

( T e
 q

G(T e

T l
)

T l
) (Heating of electrons)
(Heating of metal-lattice) eliminating electron-gas temperature ( T e
)
G

 4
( n e
V s k
B
)
2 k
1
C
E
2
 2
T l
 t
2


1 e

T l
 t


C
2
E e

(

 t
2
T l
)
  2
T l eliminating metal-lattice temperature ( T l
)
1
C
E
2
Where,
 2
T e
 t
2

1
 e

T e
 t

 e
C
E
2

(

 t
2
T e
)
  2
T e
G = Phonon-electron coupling factor
V s
= speed of sound n e
= number density of free electrons per unit volume k
B
= Boltzmann constant k = Thermal conductivity
C e
= Heat capacity of electrons
C l
= Heat capacity of metal lattice
 e

C e k

C l
C
E
 kG
C e
C l
Advanced Computational Fluid Dynamics
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•
Modified heat flux vector represented by Tzou (1995)

T
 q ( r

, t
  q
)
  k

T ( r

, t
 
)
T
~ delay behavior in establishing the temperature gradient
•
 q
~ delay behavior in heat-flow departure
Taylor expansion gives
 q ( r

, t )
  q

 q ( r

,
 t t )
  k  
T ( r

, t )
 
T

[

T ( r

, t )]
 t
• coupled with energy equation gives dual phase lag (DPL) equation

 q

 t
2
T
2

1


T
 t
 
T

(

 t
2
T
(
C
C e l
C e

)
•
Comparing coefficients of DPL model with those of two-step model we can represent microscopic properties by
 
C e k

C l

T

C l
G
 q

G C l

)
 2
T
G

 4
( n e
V s k k
B
)
2 n e
 

T

G
 
0

 q

 2
T
 t
2
 q

1

 
T

T
 t

0
 
T

 q
 2
T
 t
2

1



T t

(
 2
T )
 t
  2
T
 
T


(
 2
T )
 t
  2
T
 classical diffusion equation classical wave equation
Advanced Computational Fluid Dynamics
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
D

Volumetric heating in the sample
S ( z , t )

S
0 e

 z
I ( t ) where,
S
0

0 .
94 J 
 t p

R



I ( t )

I
0 e
 a t t p
Gold Film
Laser
Femtosecond laser heating is modeled by energy absorption rate
S ( z , t )

0 .
94 J 
  t p

R

 
 exp





 z

1 .
992 t t p

2 t p




(1D)
S ( r

, t )

0 .
94 J 
  t p

R

 
 exp




 x
2  r
0
2 y
2

 z

1 .
992 t t p

2 t p



 (3D)
S (
 r t

 q
 2
T
 t
2

1


T
 t
 
T

(
 2
T )
 t
  2
T

1 k
S
  q

S
 t
L = 100 nm
• Laser fluence J = 13.4 Jm  2
• Reflectivity R=0.93
• Thermalization time t p
=96 fs
• Depth of laser penetration  = 15.3nm
• Radius of laser beam r
0
= 100nm
• S
0 is the intensity of laser absorption
•
•
•
•
• I(t) is the intensity of laser pulse
 = 1.2
 10  4 m 2 s  1
 = 8.5
ps
 q
T
= 90 ps k = 315 Wm  1 K  1 z
Advanced Computational Fluid Dynamics
C F D

 q ( r

, t )
  q

 q ( r

, t )
 t
  k  
T ( r

, t )
 
T

[

T ( r

, t )]
 t q
1
  q
 q
 t
1   k

T
 x
 
T

 t

T
 x q
2
  q
 q
 t
2   k




T
 y
 
T

 t



T
 y




 q
3
  q
 q
 t
3   k

T
 z
 
T

 t
  
 q

S

C

T
 t
500nm
( r

, t )

T
 z




 q
 2
T
 t
2

1


T
 t
 
T

(
 2
T )
 t
  2
T

1 k
S
  q

S
 t x z
100nm y
500nm
500nm
500nm
Initial Conditions
T ( r

, 0 )

T o


T  t
( r , 0 )

0
Boundary Condition

T
 r

0 , r
 x , y , z
Advanced Computational Fluid Dynamics
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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
 2
T
 x
2

1
 x
2
[ T i n
1 , j , l

2 T i , n j , l

T i n
1 , j , l
]

S
 t

1
2
 t
[ S i n
,

1 j

S i n
,

1 j
]
Mixed derivative is approximated using centered difference in space and backward difference in time
Stability criterion for 3-D DPL model obtained using von Neumann eigenmode analysis (Tzou)
 3
T
 t
 x
2

1
[ T
 t
 x
2

T i n


1
1 , j , l i n

1 ,
 j , l

2 T i , n j

1
, l
2 T i , n j , l

T
 i n


1
1 , j
T
, l i n

1 ,
] j , l
 t ( 2
 t
 x
2
(
 t


4

T
2
 q
)
)

 t ( 2
 t
 y
2
(
 t


4

T
2
 q
)
)

 t ( 2
 t
 z
2
(
 t


4

T
2
 q
)
)

1
Forward differencing approximates first-order derivative in time

T
 t

1
 t
[ T i , n j

1
, l

T i , n j , l
]
 t

 b
 a b
2 
4 ac
2 a
 
2

(
 y
2  z
2   x
2  z
2   x
2  y
2
)
Centered differencing approximates secondorder derivative in time 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
 2
T
 t
2

1
(
 t )
2
[ T i , n

1 j , l

2 T i , n j , l

T i , n j

1
, l
] c

2
 x
2  y
2  z
2  q
Advanced Computational Fluid Dynamics
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Laser
Gold Film
X
L = 100 nm
Fig. 1. Normalized Temperature change at front surface of a gold film of thickness 100nm:  = 1.2
 10  4 m 2 s  1 , k = 315 Wm  1 K  1 , 
T
= 90 ps,  q
= 8.5
ps .
Advanced Computational Fluid Dynamics
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200nm laser beam
Work piece-Gold
500nm
500nm
250nm
250nm
500nm
500nm
Fig. 2. 3-D schematic of laser heating of gold film at different locations
Advanced Computational Fluid Dynamics
C F D

DPL
DPL DPL
Parabolic
Parabolic
Hyperbolic
DPL
DPL
Parabolic
Parabolic Hyperbolic
Fig. 3. Temperature distribution at top surface of gold film predicted by different models
Advanced Computational Fluid Dynamics
C F D

At t = 0.3
ps
At t = 0.9
ps
DPL
DPL DPL
Parabolic Hyperbolic
DPL
DPL
Parabolic
Parabolic
Hyperbolic
Hyperbolic
Fig. 3a. Temperature distribution at top surface of gold film predicted by different models
Advanced Computational Fluid Dynamics
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At t = 1.56
ps
At t = 2.23
ps
DPL
DPL
Parabolic
Parabolic
Hyperbolic
DPL Parabolic
Parabolic
Hyperbolic
Fig. 3b. Temperature distribution at top surface of gold film predicted by different models
Advanced Computational Fluid Dynamics
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312
310
308
306
304
302
300
51×51×11
101×101×21
201×201×41
0 50 100 150
Radial distance (nm )
200
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
250
Advanced Computational Fluid Dynamics
C F D

• 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
Advanced Computational Fluid Dynamics
C F D

• 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.
Advanced Computational Fluid Dynamics
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