Wavy, wavelike, diffusive thermal responses of finite rigid slabs to high-speed heating of laser-pulses

\ PERGAMON

International Journal of Heat and Mass Transfer 31 "0888# 744Ð759

Wavy\ wavelike\ di}usive thermal responses of _nite rigid slabs to high!speed heating of laser!pulses D[ W[ Tang\ N[ Araki Department of Mechanical Engineering\ Faculty of Engineering\ Shizuoka University\ Hamamatsu 321!7450\ Japan Received 01 January 0887^ in _nal form 5 July 0887

Abstract A generalized macroscopic model is introduced in treating the transient heat conduction problems in _nite rigid slabs irradiated by short pulse lasers[ The analytical solution is derived by using Green|s function method and _nite integral transform technique[ Various behaviors of conduction heat transfer\ such as wave\ wavelike\ and di}usion\ are exhibited by adjusting the relaxation parameters[ Then detailed discussions have been given on the interrelations between these behaviors[ The calculated temperature responses by this model are compared with two literature results measured under extremely low temperature and ultra!high speed heating\ respectively[ The calculations show good agreement with the experimental data[ Þ 0887 Elsevier Science Ltd[ All rights reserved[

Nomenclature a thermal di}usivity cp speci_c heat capacity I output intensity of laser L thickness q heat ~ux Q absorbed energy density r spatial vector R re~ectivity of irradiated surface t time T temperature x spatial variable[ Greek symbols d absorption depth of laser irradiation Dh dimensionless absorption depth\ d:1zatq h dimensionless spatial variable\ x:1zatq u dimensionless temperature\ lztq zpðT"x\ t#−T9 Ł: "0−R#I9 za l thermal conductivity j dimensionless time\ t:1tq\ and ratio of relaxation times\ tT:1tq r density t relaxation time[

 Corresponding author

Subscripts L thickness p pulse q heat ~ux T temperature and temperature gradient 9 initial value[

0[ Introduction The development of high!intense and ultra!short lasers has opened some exciting research opportunities in the _eld of heat transfer ð0Ð3Ł[ In treating high!speed laser heating problems\ two di}erent kinds of models have been frequently employed in previous works[ One is the macroscopic thermal wave model\ which was postulated by Cattaneo ð4Ł and Vernotte ð5Ł\ leading to a hyperbolic heat conduction equation and suggesting a _nite speed propagation of heat[ The other includes those from microscopic point of view\ which derived in di}erent forms for various kinds of materials[ The microscopic two!step model ð6Ł and the pure phonon _eld model ð7Ł were developed for evaluating the thermal transport in metals and dielectric solids\ respectively\ which suggest some behaviors of heat conduction\ while neither the macroscopic thermal wave model nor the Fourier thermal di}usion model can do[ To provide a macroscopic description that can capture

9906Ð8209:87:,*see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved PII] S 9 9 0 6 Ð 8 2 0 9 " 8 7 # 9 9 1 3 3 Ð 9

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these microscopic e}ects\ Tzou ð8Ł developed a dual! phase!lag model by generalizing the Fourier|s law in 0884[ In this model\ the microstructural e}ects are lumped into the delayed response in time\ two relaxation parameters were introduced to describe the microscopic interactions[ A generalized form of heat ~ux equation was established\ which could predict all the heat conduction behaviors in the microscopic models[ Almost the same equation\ a heat!~ux equation of Je}reys type\ was given by Joseph and Preziosi ð1Ł in 0878 by applying some ideas for describing shear waves in liquids[ Such a generalized law makes it possible that the practical engineers may describe various heat transport behaviors by adjusting the relaxation parameters in single governing equation[ Inversely\ they can determine the relaxation parameters by _tting the solution of this generalized equation to transient temperature measurements\ and furthermore type of heat conduction by comparing the relaxation parameters[ The present work considers transient heat conduction in a _nite medium exposed to a short laser pulse by introducing the generalized macroscopic conduction model[ The analytical solution of the generalized heat conduction equation is derived by using Green|s function method and _nite integral transform technique[ Cal! culations are performed to exhibit the various type of thermal transports in the medium\ such as wave\ wave! like\ and di}usion[ Especially\ the wavelike behavior is discussed in detail by comparing it with the wavy and the di}usive ones[ To show the adequacy of the present solution\ the calculated temperature responses are com! pared with two literature results measured under extremely low temperature and ultra!high speed heating\ respectively[ The calculations show good agreement with the experimental data[

1[ Generalized macroscopic heat conduction model Almost the same form of generalized heat ~ux equation has been derived by Joseph and Preziosi ð1Ł and Tzou ð8Ł from di}erent macroscopic points of view[ Tzou ð8Ł introduces phase lags to the Fourier|s law\ gives a dual!phase!lag model as q"r\ t#¦tq

$

1q 1  −l 9T"r\ t#¦tT "9T# 1t 1t

%

"0#

where\ tq and tT are the phase lags "the relaxation time# of heat ~ux and temperature gradient\ respectively\ which are the macroscopic description to the microscopic e}ects[ Applying some ideas of Je}reys model for the shear waves in liquids to the description of heat ~ux propagation in solids\ Joseph and Preziosi ð1Ł give the same form of heat ~ux equation as equation "0# but with some parameters from theory of liquids[ They describe the microstructural e}ects by a relaxation function "heat!

~ux kernel#\ and decompose it into a fast and a slow mode\ which are corresponding to two relaxation times\ respectively[ Comparing the Je}reys type of heat ~ux equation with Tzou|s equation shows that the relaxation time of fast and slow mode are corresponding to tq and tT\ respectively[ However\ for Je}reys| presentation\ the time of fast mode should be smaller than that of slow mode\ while for Tzou|s presentation\ either tq or tT may be bigger[ Equation "0# and the conservation equation without heat source lead to a description of transient heat con! duction in the following generalized equation\ 0 1T tq 11 1  91 T¦tT "91 T#[ ¦ a 1t a 1t1 1t

"1#

Comparing equation "1# with microscopic models shows that\ if the two relaxation times are formulated properly by some microscopic quantities for di}erent materials\ this macroscopic model gives exactly the same heat con! duction equation as those in microscopic models[ The comparisons can also provide the microscopic physical pictures to the two new introduced relaxation times[ For dielectric crystals\ from the pure phonon _eld model ð7Ł\ the microscopic heat conduction equation has been derived by Joseph and Preziosi ð1Ł and Tzou ð8Ł\ by com! paring with this equation\ it can be found that tq is the relaxation time for the momentum!nonconserving pro! cesses and tT has the same order of the relaxation time for the normal processes conserving the momentum in the phonon system[ For metals\ by comparing with the microscopic two!step model ð6Ł\ it is found that tq cap! tures the relaxation behavior of electron thermal wave conduction\ while tT captures the e}ect of phononÐelec! tron interaction[ The Fourier|s thermal di}usion law and the CattaneoÐVernotte|s thermal wave law are two special cases of this generalized model for tT  tq  9 and tT  9\ respectively ð8Ł[ In this work\ the generalized model is applied to the heat conduction problem under pulse laser heating and some detail discussions are given about the behaviors predicted by this model[

2[ Analysis of one!dimensional problem under laser! pulse heating Consider a _nite rigid slab of thickness L with an initial temperature distribution T"x\ 9#  T9 "x is the direction along the thickness#\ constant thermal properties\ and insulated boundaries[ From t  t9 its front surface "x  9# is irradiated uniformly by a laser pulse with Gaussian temporal pro_le as follows\ I"t# 

I9 zptp

1

$ 0 1%

exp −

t tp

"2#

where tp is the characteristic time of the laser pulse and I9 is the laser intensity which is de_ned as total energy

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carried by a laser pulse per unit cross!section of the laser beam[ According to ref[ ð09Ł\ the conduction heat transfer in the slab can be modeled as a one!dimensional problem with an energy source Q"x\ t# near the surface\ Q"x\ t# 

0

"0−R#I9

"3#

where d is a characteristic transparent length of irradiated photons called the absorption depth and R is the re~ect! ivity of the irradiated surface[ Then the conservation equation of energy can be given by 1T 1q Q"x\ t#−  cp r [ 1x 1t

"4#

Combining equation "4# with equation "0# in one!dimen!

u"h\ j# 

"0−e−hL:Dh # 1 L p

1h j

g

j

0

ð0−e1"j½ −j# Ł jp −

j9

u  lztq zpðT"x\ t#−T9 Ł:"0−R#I9 za

"8c#

equations "5#Ð"7# become 11 u

1

0 11

x t exp − − d t p tp dzp

and

1j

1

¦1

1u 11 u 0 j 12 u ¦jT ¦ 0−  1 1j 1h 1j 1h1 Dhjp jp1

0

$

×exp −

1

0 1%

h j − Dh jp

1

"9 ¾ h ¾ hL \ j × j9 #

"09#

1"9\ j#:1h  1"hL \ j#:1h  9

"00#

u"h\ j9 #  1u"h\ j9 #:1j  9[

"01#

The problem is solved by using the Green|s function method and the _nite integral transform technique\ the temperature distribution is obtained as follows]

j½ −"j½ :j #1 p e dj½ jp

1

F J F 0−"−0# m e−hL:Dh mph G cos G G hL G 1 N j xzb f when b × 9 s G 1 F J j 1 ½ m0 h j mp 0 G L p G & e− $0¦ 1 jT 0 hL 1 %"j−j½ # sin ðzb"j−j½ #Ł jp − j e−"j½ :jp#1 dj½ G G jp j f j9 ¦j J J F 0−"−0# m e−hL:Dh mph G cos G hL G 1  G xz−b f when b ³ 9 G s j F j ½ G hL jp1 mN J mp 1 0 j −"j½ :j #1 ½ G − $0¦ jT 0 1 % "j−j½ # G ½ p h 1 & e sinh ðz−b"j−j #Ł j − e dj G p L jp j f j9 f

0 1

g

0

1

"02#

0 1

g

0

sional form lead to the following governing equation to the problem\

$

0 1 %

0 1Q ¦ Q"x\ t#¦tq l 1t

"9 ¾ x ¾ L\ t × t9 #[

0 mpDh 1 mp 1 mp \ b − 0¦ jT hL hL 1 hL

0 1

0 1 $

1 1

0 1%

"03# "5#

For the considered situation\ the boundary and initial conditions are taken as 1T"x\ t#:1x= x9  1T"x\ t#:1x= xL  9

"6#

1T"x\ t#:1x= tt9  9\ T"x\ t#= tt9  T9 [

"7#

Introducing h  x:1zatq \ hL  L:1zatq \ Dh  d:1zatq

where x  0¦

1 11 T 0 1T tq 11 T 11 T  ¦tT ¦ a 1t a 1t1 1t 1x1 1x1

1

"8a#

j  t:1tq \ jp  tp :1tq \ j9  t9 :1tq \ jT  tT :1tq "8b#

and N is the point at which b changes from a positive number to a negative one with increasing the value of m[

3[ Wavy\ wavelike\ diffusive behaviors Utilizing equation "02#\ numerical computation has been performed in order to display the various behaviors of temperature responses arising from the pulse surface heating on the _nite rigid slab[ Figure 0 shows the temperature distribution at various instants in a slab with thickness hL  1[9 and absorption depth Dh  9[94 irradiated by laser pulse with charac! teristic duration jp  9[0[ The ratio between the relax!

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Fig[ 0[ Temperature distributions in a _nite rigid slab for various values of relaxation time[ "jp  9[0\ Dh  9[94 and hL  1[9#[ "a# jT  9 "wave#\ "b# jT  9[0 "wavelike#\ "c# jT  9[4 "di}usion#\ "d# jT  0[4 "over!di}usion#[

ation time\ jT  tT:1tq\ is a parameter to control the transition between the di}erent behaviors of heat con! duction[ In Fig[ 0"a#\ the temperature distributions for jT  9\ i[e[ tT  9\ are plotted for di}erent instants[ In this case\ the pulse thermal disturbance propagates in the form of wave\ and clear wave fronts can be seen at di}er! ent positions in the _gure[ Several series of peaks indicate the propagation "j  9[0\ 9[4\ 0[9# and re~ection "j  1[3# of the temperature wave[ Figure 1 shows the temperature responses at both surfaces of a slab with thickness hL  0[9 and absorption depth Dh  9[94 irradiated by laser pulse with characteristic duration jp  9[0[ In the _gure\ the arrivals of propagated and re~ected wave at both surfaces can be seen clearly[ With increasing jT from zero "Fig[ 0"b##\ the sharp wave fronts are smoothed and the portions of pulse ther! mal disturbance are dissipated by the di}usive e}ect of tT[ For all jT ³ 9[4\ i[e[ tT ³ tq\ some wave features such as propagation and re~ection can still be observed[ How! ever equation "09# with jT  9 is not hyperbolic and it does not give a wave solution[ Strictly speaking\ the behavior of temperature response for 9 ³ jT ³ 9[4 should be called wavelike behavior\ a detail discussion about this behavior will be given later[ When jT  9[4 "Fig[ 0"c##\ all the features of wave disappear\ the pulse thermal disturbance transports by di}usion completely[ Figure 0"d# shows the temperature behavior in case of jT × 9[4\ i[e[ tT × tq\ which is called over!di}usion behavior[ Comparing with the classical di}usion behavior "Fig[ 0"c##\ it can be seen that\ a bigger tT

produces high rate thermal di}usion e}ect and results in rapid temperature response in early time[ But it needs a longer time to reach thermal equilibrium than the classi! cal di}usion[ These can also be seen from the temperature responses in Fig[ 1[ Figure 2 shows the details about the e}ect of jT[ It can be seen that\ the increase of jT makes "0# the arrival of wave peaks delayed and the amplitudes attenuated^ "1# the edges of wave front smoothed^ "2# the portion of pulse thermal disturbance dissipated[ When jT is big enough "jT − 9[4#\ the peak of wave front will disappear completely[ The wave features cannot be observed anymore[ Figure 3 shows the comparison between the wavy and wavelike behavior by giving the temperature dis! tributions in a _nite slab with thickness hL  1[9 and absorption depth Dh  9[91 irradiated by a laser with pulse duration jp  9[95[ Conceptually a wavy behavior should have the following features] _nite propagating speed\ wave front\ propagation and re~ection[ For 9 ³ jT ³ 9[4\ i[e[ 9 ³ tT ³ tq\ the temperature responses have almost all the features of wavy behaviors except for the _nite propagating speed which o}ers only the theoretical signi_cance\ and in practical engineering problems the wavelike behavior could be absolutely treated as the wavy behavior[ Some _ne distinctions between these two behaviors\ however\ should be noticed[ By comparing "a# and "b#\ it can be seen that\ the wavelike behavior has smaller amplitude of temperature rise than the wavy one and the increase of jT results in the decrease

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Fig[ 3[ Comparison between wavy and wavelike behavior[ "a# jT  9 "wave#\ "b# jT  9[94 "wavelike#[

Fig[ 1[ Temperature responses at the surface of the medium for various values of relaxation time[ "jp  9[0\ Dh  9[94\ and hL  0[9#[ "a# Front surface\ "b# rear surface[

4[ Comparison with experiments To show the adequacy of the present analysis\ the calculated temperature responses in real coordinate sys! tem are compared with two literature results measured under extremely low temperature and ultra!high speed heating\ respectively[ Figure 4 shows the temperature response at front sur! face of a 9[1 mm Au _lm subjected to 9[985 ps laser!pulse with temporal pro_le in the form of equation "2#\ which is measured at room temperature ð00Ł[ If the following data of relaxation times and thermal di}usivity from ref[ ð8Ł\ tq  9[6327 ps\ tT  78[175 ps\ a  0[1384×09−3 m1 s−0

Fig[ 2[ Comparison of temperature distributions for various values of relaxation time[

of amplitude[ For wavy behavior\ increasing the propa! gation distance only results in the attenuation of the amplitude but not any change in the wide of the portion of pulse thermal disturbance\ while for the wavelike behavior\ the propagation will result in both the attenu! ation of amplitude and the dissipation of the portion of pulse disturbance[

Fig[ 4[ Temperature response at front surface of Au _lm sub! jected to 9[985 ps laser!pulse "at R[T[#[

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are used as the absorption depth is assumed to be 9[994 mm\ the calculation shows very good agreement with the experimental result[ By comparing the two relaxation times\ it is found that the type of heat conduction is over! di}usion[ Figure 5 shows the temperature response at rear sur! face of 7 mm solid He3 slab subjected to 4 ms pulse heat ~ux with temporal pro_le as follows\ I"t# "I9 t:tp1 # exp"−t:tp #

"04#

and the experiment is done at 9[5 K and 43[1 atm ð01Ł[ If the absorption depth of this heating ~ux is d and the energy is absorbed completely by the slab\ the heat source term in equation "5# can be written as Q"x\ t# 

I9 t tp1 d

0

1

x t exp − − [ d tp

"05#

Then the theoretical temperature response in this case of heating can be obtained by solving equations "5#Ð"7# with the same procedure as that in the other case[ If the heating energy is assumed to be absorbed only by the surface "d ³ 9[0 mm# and the relaxation times and thermal dif! fusivity have the following values\ respectively\ tq  50[07 ms\ tT ³ 9[994 ms\ a  1[93 m1 s−0 the calculated curve and experimental data agree with each other[ The rapider attenuation in the experimental curve may be caused by the heat loss\ which is not con! sidered in the analysis[ The theoretical estimation of the relaxation times has not been found in literature\ while the value of thermal di}usivity is estimated to be a  9[953Ð2[90 m1 s−0 for Molar volumes from 00[9Ð 19[8 cm2 mole−0\ by using the reference value of thermal conductivity ð02Ł and Debye temperature ð03Ł and Debye|s formula of speci_c heat capacity[ Comparing the two relaxation times shows that the heat conduction in this case has wavy characteristics[

Fig[ 5[ Temperature response at rear surface of solid He3 sub! jected to 4 ms heat!pulse "at 9[5 K#[

5[ Concluding remarks The analytical solution of the generalized heat con! duction equation under the condition of pulse thermal disturbance is derived[ Corresponding to the actual situ! ation that a _nite rigid slab under pulse laser heating\ various behaviors of heat conduction\ i[e[ the wave\ the wavelike\ the di}usion and the over!di}usion are obtained by adjusting the relaxation parameters in the generalized heat conduction equation[ The temperature responses calculated by this model are compared with two experimental results\ and the calculations show good agreement with the experiments[ This makes it possible to estimate the relaxation parameters by _tting the solu! tion of this generalized equation to transient temperature responses caused by pulse surface heating\ and determine the type of heat conduction by comparing the relaxation parameters in the future work[ References ð0Ł A[M[ Prokhorov\ V[I[ Konov\ I[ Ursu\ I[N[ Mihailescu\ Laser Heating of Metals\ Adam Hilger\ Bristol\ 0889[ ð1Ł D[D[ Joseph\ L[ Preziosi\ Heat waves\ Rev[ Modern Phys[ 50 "0878# 30Ð62^ Addendum to the paper {{Heat Waves||\ Rev[ Modern Phys[ 51 "0889# 264Ð280[ Ý zisik\ D[Y[ Tzou\ On the wave theory in heat con! ð2Ł M[N[ O duction\ ASME J[ Heat Transfer 005 "0883# 415Ð424[ ð3Ł D[Y[ Tzou\ Macro! to Microscale Heat Transfer] The Lag! ging Behavior\ Taylor and Francis\ Bristol\ 0886[ ð4Ł P[ Vernotte\ Les paradoxes de la theorie continue de l|equation de la chaleur\ Comptes Rendus 135 "0847# 2043Ð2044[ ð5Ł C[ Cattaneo\ Sur une forme de l|equation de la chaleur eliminant le paradoxe d|une propagation instantanee\ Comptes Rendus 136 "0847# 320Ð322[ ð6Ł S[I[ Anisimov\ B[L[ Kapeliovich\ T[L[ Perel|man\ Electron emission from metal surfaces exposed to ultra!short laser pulses\ Soviet Phys[ JETP 28 "0863# 264Ð266[ ð7Ł R[A[ Guyer\ J[A[ Krumhansl\ Solution of the linearized phonon Boltzmann equation\ Phys[ Rev[ 037 "0855# 655Ð 679[ ð8Ł D[Y[ Tzou\ A uni_ed _eld approach for heat conduction from macro! to micro!scales\ ASME J[ Heat Transfer 006 "0884# 7Ð05[ ð09Ł D[W[ Tang\ N[ Araki\ The wave characteristics of thermal conduction in metallic _lms irradiated by ultra!short laser pulses\ J[ Phys[ D] Appl[ Phys[ 18 "0885# 1416Ð1422[ ð00Ł S[D[ Brorson\ J[G[ Fujimoto\ E[P[ Ippen\ Femtosecond electron heat!transport dynamics in thin gold _lm[ Phys[ Rev[ Lett[ 48 "0876# 0851Ð0854[ ð01Ł C[C[ Ackerman\ R[A[ Guyer\ Temperature pulses in dielec! tric solids\ Annals of Phys[ 49 "0857# 017Ð074[ ð02Ł Y[S[ Touloukian\ Thermophysical Properties of Matter\ Plenum\ New York\ 0869[ ð03Ł S[B[ Trickey\ W[P[ Kirk\ E[D[ Adams\ Thermodynamic\ elastic\ and magnetic properties of solid helium\ Rev[ Mod! ern Phys[ 33 "0861# 557Ð604[