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

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

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Numerical simulation of heat transfer mechanisms during femtosecond laser heating of nano-films using 3-D dual phase lag model Presenter: Illayathambi Kunadian – PowerPoint PPT presentation

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Title: 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.
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