Title: Numerical simulation of heat transfer mechanisms during femtosecond laser heating of nano-films using 3-D dual phase lag model
1Numerical 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
2Contents 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
3Classical 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)
4Hyperbolic 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
5Internal 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
6Two-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
7Dual 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
8Mathematical 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
9Numerical analysis
z
y
x
100nm
500nm
500nm
500nm
Initial Conditions
500nm
Boundary Condition
10Numerical 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
111D 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.
123-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
13Temperature 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
14Temperature 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
15Temperature 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
16Grid 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
17Conclusions
- 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
18Conclusions 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.