Numerical simulation of heat transfer mechanisms during femtosecond laser heating of nano-films using 3-D dual phase lag model

1
Numerical simulation of heat transfer mechanisms
during femtosecond laser heating of nano-films
using 3-D dual phase lag model

• Presenter
• Illayathambi Kunadian
• ikuna0_at_engr.uky.edu
• Co-authors
• Prof. J. M. McDonough
• Prof. K. A. Tagavi
• Department of Mechanical Engineering
• University of Kentucky, Lexington, KY 40506

2
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

3
Classical heat conduction

• Heat flux directly proportional to temperature
  gradient (Fourier's law)
• Incorporation into first law of thermodynamics
  yields parabolic heat conduction equation
• 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)

4
Hyperbolic heat conduction

• Modified heat flux that accommodates finite
  propagation speed of observed thermal waves
  proposed by Vernotte and Cattaneo (1958)
• Combined with equation of energy conservation
  gives hyperbolic heat conduction equation (HHCE)
• 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 pulse-lasers
  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

c is the speed of thermal wave propagation
5
Internal Mechanisms during laser heating
Stage I
Stage II
Energy quanta Phonons
Heated electrons
no temperature rise at time t
temperature rise at time t t
Photon energy at time t
Metal lattice heating by phonon-electron
interactions
Electron-gas heating by photons
D. Y. Tzou, Macro-microscale Heat Transfer, the
lagging behavior
6
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
• eliminating electron-gas temperature (Te)
• eliminating metal-lattice temperature (Tl)
• Where,

(Heating of electrons)
(Heating of metal-lattice)
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
7
Dual phase lag model

• Modified heat flux vector represented by Tzou
  (1995)
• ?T delay behavior in
  establishing the temperature gradient
• ?q delay behavior in heat-flow
  departure
• Taylor expansion gives
• coupled with energy equation gives dual
  phase lag (DPL) equation
• Comparing coefficients of DPL model with those
  of two-step model we can represent microscopic
  properties by

classical diffusion equation
classical wave equation
8
Mathematical formulation

• Volumetric heating in the sample
• Femtosecond laser heating is modeled by energy
  absorption rate
• 
  (1D)
• 
  (3D)
• In presence of DPL equation becomes

Gold Film
where,
z
Laser
L 100 nm

• Laser fluence J 13.4 Jm?2
• Reflectivity R0.93
• Thermalization time tp96fs
• 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
• ?T 90ps
• k 315Wm?1K?1

9
Numerical analysis

•  

z
y
x
100nm
500nm
500nm
500nm
Initial Conditions
500nm
Boundary Condition
10
Numerical analysis

• Explicit finite-difference scheme employed to
  solve the DPL equation

Centered differencing is employed for time
derivative in the source term
Centered differencing approximates second-order
derivatives in space
Stability criterion for 3-D DPL model obtained
using von Neumann eigenmode analysis (Tzou)
Mixed derivative is approximated using
centered difference in space and backward
difference in time
Forward differencing approximates first-order
derivative in time
Centered differencing approximates second-order
derivative in time
11
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.
12
3-D Schematic of femtosecond laser heating of
gold film
200nm laser beam
Work piece-Gold
250nm
500nm
250nm
500nm
100nm
500nm
500nm
500nm
Fig. 2. 3-D schematic of laser heating of gold
film at different locations
13
Temperature distribution predicted by different
models
DPL
Parabolic
DPL
Parabolic
DPL
Parabolic
Parabolic
Hyperbolic
Parabolic
DPL
DPL
Parabolic
Hyperbolic
Fig. 3. Temperature distribution at top surface
of gold film predicted by different models
14
Temperature distribution predicted by different
models
At t 0.3 ps
DPL
Parabolic
Hyperbolic
DPL
Parabolic
Hyperbolic
DPL
Parabolic
Hyperbolic
At t 0.9 ps
Parabolic
Hyperbolic
DPL
DPL
Parabolic
Hyperbolic
Fig. 3a. Temperature distribution at top surface
of gold film predicted by different models
15
Temperature distribution cont.
At t 1.56 ps
DPL
Parabolic
Hyperbolic
DPL
Parabolic
At t 2.23 ps
Parabolic
Hyperbolic
DPL
Parabolic
Fig. 3b. Temperature distribution at top surface
of gold film predicted by different models
16
Grid function convergence test
515111
10110121
20120141
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
17
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

18
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.