ESTIMATION OF RELAXATION AND THERMALIZATION TIMES IN MICROSCALE HEAT TRANSFER MODEL

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 4, pp , Warsaw 2013 ESTIMATION OF RELAXATION AND THERMALIZATION TIMES IN MICROSCALE HEAT TRANSFER MODEL Bohdan Mochnack Częstochowa Unversty of Technology, Czestochowa, Poland and Hgher School of Labour Safety Management, Katowce, Poland e-mal: Marek Paruch Slesan Unversty of Technology, Insttute of Computatonal Mechancs and Engneerng, Glwce, Poland e-mal: The energy equaton correspondng to the dual phase lag model(dplm) results from the generalzed form of the Fourer law, n whch the two delay tmes (relaxaton and thermalzaton tme) are ntroduced. The DPLM should be used n the case of mcroscale heat transfer analyss, n partcular when thermal processes are characterzed by extremely short duraton(e.g. ultrafast laser pulse), consderable temperature gradents and very small dmensons(e.g. thn metal flm). In ths paper, the problem of relaxaton and thermalzaton tme dentfcaton s dscussed, at the same tme the heat transfer processes proceedng nthedomanofathnmetalflmsubjectedtoalaserbeamareanalyzed.thesoluton presented bases on the applcaton of evolutonary algorthms. The addtonal nformaton concernng the transent temperature dstrbuton on a metal flm surface s assumed to be known. At the stage of numercal realzaton, the fnte dfference method(fdm) s used. Inthefnalpartofthepaper,anexampleofcomputatonsspresented. Key words: mcroscale heat transfer, laser heatng, dual phase lag model, nverse problem 1. Governngequatons The followng form of the generalzed Fourer law s consdered q(x,t+τ q )= λ T(x,t+τ T ) (1.1) whereqstheuntaryheatflux,λsthethermalconductvty, Tsthetemperaturegradent, τ q stherelaxatontme(themeantmeforelectronstochangetherenergystates),τ T sthe thermalzaton tme(the mean tme requred for electrons and lattce to reach equlbrum). The DPL equaton results from consderatons concernng the parabolc two-temperature model(al-nmr, 1997; Chen and Beraun, 2001; Majchrzak and Poteralska, 2011; Majchrzak et al., 2009), n partcular the energy equatons determnng the heat transfer n the electron gas and the metal lattce are taken nto account. In the case of pure metals, the two-temperature model takes the form c e (T e ) T e = [λ e(t e ) T e ] G(T e T l ) c l (T l ) T l =G(T e T l ) (1.2) wheret e =T e (x,t),t l =T l (x,t)arethetemperaturesofelectronsandlattce,c e (T e ),c l (T l )are thevolumetrcspecfcheats(thermalcapactes),λ e (T e ),λ l (T l )arethethermalconductvtes, G s the couplng factor ths parameter characterzes the energy exchange between a phonon and electrons(ln and Zhgle, 2008).

2 838 B. Mochnack, M. Paruch In Fg. 1(see Majchrzak and Poteralska, 2011) an example of numercal soluton obtaned usng the two temperature parabolc model s shown. The metal flm(t) subjected to a laser pulse has been consdered and the heatng(coolng) curves correspond to the flm surface. DfferencesbetweenT e andt l arevsble,thetmeofbothtemperaturesequalzatoncorresponds to the thermalzaton one. Fg. 1. Changes of surface temperatures Usng a certan elmnaton technque, equatons(1.2) can be substtuted by a sngle equaton contanng the second dervatve of temperature wth respect to tme and a hgher-order mxed dervatvenbothtmeandspace.letusassumethatthevolumetrcspecfcheatsc e andc l are constant values(ths smplfcaton s acceptable when the changes of temperature durng theheatngprocessarenotverybg).so,fromequaton(1.2) 1 tresultsthat T e =T l + c l T l G (1.3) Puttng(1.2) 2 nto(1.1),onehas ( Tl c e +c l 2 T) l G 2 = (λ e T l )+ c [ l G ] λ e ( T T l l) c l (1.4) ths means [ Tl (c e +c l ) +c e G(c e +c l ) 2 T] l c l 2 = (λ e T l )+ c l G [ λ e( T l )] (1.5) Denotng τ T = c l G τ q = 1 G( 1 c e + 1 c l ) 1 (1.6) fnally one obtans [ T(x,t) 2 T(x,t) ] c +τ q 2 = [λ T(x,t)]+τ T [λ T(x,t) ] (1.7) where T(x,t) = T l (x,t)sthemacroscopc lattce temperature(ozskandtzou,1994), c=c l +c e stheeffectvevolumetrcspecfcheatresultngfromtheseralassemblyofelectrons andphononsandλ=λ e (OzskandTzou,1994).

3 Estmaton of relaxaton and thermalzaton tmes n Another approach to the DPLM formulaton s also possble and the followng mathematcal consderatons were presented n Majchrzak et al.(2009). Usng the Taylor seres expansons, the followng frst-order approxmaton of equaton(1.1) can be taken nto account or q(x,t)+τ q q(x,t) T(x,t) ] = λ [ T(x,t)+τ T (1.8) q(x,t)=τ q q(x,t) +λ T(x,t)+τ T λ T(x,t) Ths formula should be ntroduced to the well known macroscopc energy equaton (1.9) c T(x,t) = q(x,t) (1.10) Ths means c T(x,t) =τ q [ q(x,t)]+ [λ T(x,t)]+τ T [λ T(x,t) ] (1.11) Substtutng q by c(t) T/, one obtans the same equaton as equaton(1.7) [ T(x,t) 2 T(x,t) ] c +τ q 2 = [λ T(x,t)]+τ T [λ T(x,t) ] (1.12) In ths paper, the problem of heat dffuson n the presence of volumetrc nternal heat sources Q(x,t)sconsdered.Itcanbeshownthatnthscase,equaton(1.12)mustbesupplemented by addtonal components, n partcular [ T(x,t) 2 T(x,t) ] c +τ q 2 = [λ T(x,t)]+τ T [λ T(x,t) ] Q(x,t) +Q(x,t)+τ q (1.13) It results from the fact that the thermal nteractons between external heatng(laser pulse) and thedomanofthemetalflmaretakenntoaccountbythentroductonofanaddtonalterm supplementng the DPLM, n partcular a functon correspondng to the volumetrc nternal heat sources Q(x, t) s consdered. Ths approach s often used(e.g. Al-Nmr, 1997). The formula determnngthecapactyofthenternalheatsourcestakesaform(1dproblem Chenand Beraun(2001), Chen et al.(2004)) Q(x,t)= µ 1 R [ π t p δ I 0exp x ( δ µ t 2tp ) 2 ] t p (1.14) wherei 0 sthelaserntenstywhchsdefnedasthetotalenergycarredbyalaserpulseper untcross-sectonofthelaserbeam, t p sthecharacterstctmeofthelaserpulse, δsthe characterstc transparent length of rradated photons called the absorpton depth, R s the surfacereflectvty,µ=4ln2.thelocalandtemporaryvalueofqresultsfromthedstancex between the surface subjected to laser acton and the pont consdered. Usng ths approach, the no-fluxboundarycondtonsforx=0andx=lshouldbeassumed. InFg.2thedomanconsderedsshownandtsgeometrcalfeaturesallowsonetotreatthe problemasa1done.

4 840 B. Mochnack, M. Paruch Fg. 2. Doman consdered 2. Numercal soluton basng on fdm(a drect problem) At the stage of numercal modelng, the fnte dfference method has been used. The verson proposed consttutes a generalzaton of the FDM varant proposed by Mochnack and Suchy (1995). So, the followng energy equaton s consdered [ T(x,t) 2 T(x,t) ] c +τ q 2 = x +Q(x,t)+τ q Q(x,t) [λ T(x,t) ]+τ T x x [λ T(x,t) ] x (2.1) ThedfferentalmeshscreatedasaCartesanproductofspatal handtmemeshes.the tmegrdsdefnedasfollows : t 0 <t 1 <...<t f 2 <t f 1 <t f <...<t F < (2.2) whlethespatalmeshsshownnfg.3. Fg.3.Themesh Itsvsblethatthe boundary nodesarelocatedatthedstance 0.5hfromrealboundares(ths type of dscretzaton assures a very smple and exact approxmaton of boundary condtons, Mochnack and Suchy(1995)). The FDM approxmaton of the spatal dfferental operator can be taken n the form (λ T ) f x x =Tf Tf Ψ + Tf Tf Ψ (2.3) whereψ =Ψ =1/harethemeshshapefunctons,whle = h 2λ f 1 + h 2λ f 1 = h 2λ f 1 + h 2λ f 1 (2.4) arethethermalresstancesbetweenthenodeandadjonngthenodes,.thendexf nformula(2.3)showsthatthemplctdfferentalschemewllbeusedhere,atthesametme,

5 Estmaton of relaxaton and thermalzaton tmes n thethermalconductvtesaretakenforthetmet f 1 toobtanthelnearformoffnalfdm equatons.thefdmapproxmatonofequaton(2.1)fortranstont f 1 t f softheform c Tf T f 1 T f 2T f 1 +cτ q () 2 + τ T ( T f Tf ( T f 1 τ T Tf 1 +T f 2 Ψ + Tf Tf ) Ψ Ψ + Tf 1 Tf 1 = Tf Tf Ψ + Tf Tf Ψ Ψ )+Q f +τ q ( Q ) f (2.5) andthelastformulacanbewrttenasfollows(=1,2,...,n)) where A T f +B T f +C T f =D T f 1 +E T f 1 +F T f 1 + τ q Qf ( c τ q Q ) f c A = Ψ ( ) c 1+ τ T B = 1 F = Ψ τ T c ( ) 1+ τq A C C = Ψ ( ) c 1+ τ T D = Ψ τ T c ( E = τq () 2Tf 2 ) D F (2.6) (2.7) Fnally where A T f +B T f +C T f =Gf (2.8) G f =D T f 1 +E T f 1 +F T f 1 + τ q () 2Tf 2 Qf ( c τ q Q ) f c (2.9) Thesameequatonsareacceptedforthenodesclosetoboundares.Itsenoughtoassumethat thethermalresstancesnthedrectons toboundary aresuffcentlybg(e.g ),andthen the non-flux condton s taken nto account. The startng pont of the numercal smulaton processresultsfromthentalcondtons,npartcular T 0 =T 1 =T 0,=1,2,...,N.The system of FDM equatons(2.6) has been solved usng the very effectve Thomas algorthm for a three-dagonal lnear system of algebrac equatons(e.g. Mochnack and Suchy, 1995). 3. Inverseproblem To solve the nverse problem, the least squares crteron s appled S(τ q,τ T )= 1 MF M F (T f T f d )2 (3.1) =1f=1

6 842 B. Mochnack, M. Paruch wheret f d andtf =T(x,t f )arethemeasuredandestmatedtemperatures,respectvely,ms the number of sensors(e.g. Majchrzak et al., 2008; Mochnack and Majchrzak, 2007). The mnmum of functonal(3.1) has been found usng the evolutonary algorthms. So the drect problems have been solved and the results allow one to determne the tme dependent surface temperature(x = 0). Because the temperature hstory resultng from the numercal soluton for thebascnputdatasveryclosetotheexpermentalonesquotednchenandberaun(2001), Tang and Arak(1999), Fg. 4, therefore ths undsturbed numercal soluton s assumed to be the base for the dentfcaton problem soluton( measured surface temperature ). So, the laser parameters determnng capacty of the nternal source functon Q(x, t) and also the thermal conductvtyandvolumetrcspecfcheatofgoldareknown,theparametersτ q,τ T shouldbe determned(fromthepractcalstandpont,theexpermentalestmatonofτ q,τ T snoteasy). Fg. 4. Comparson wth expermental data(chen and Beraun, 2001) Fg. 5.(a) Temperature profles,(b) coolng(heatng) curves InFgs.5aand5b,anexampleofthedrectproblemsolutonsshown.Thelayerssubjected toashort-pulselaserrradatonwhoseparametersarer=0.93(reflectvty),i 0 =13.7J/m 2 (ntensty), t p =0.1ps=10 13 s(tmeoflaserpulse), δ=15.3nm(absorptondepth).the followng parameters of the gold thn flm are assumed: thermal conductvty λ = 317 W/(mK), volumetrcspecfcheatc=2.4897mj/(m 3 K),relaxatontmeτ q =8.5ps,thermalzatontme τ T =90ps.ThentaltemperatureequalsT 0 =20 C(see:Majchrzaketal.,2009;Majchrzak and Poteralska, 2010). UsngthealgorthmpresentedntheprevousSectonontheassumptonthat N=200 and=0.005psthetransenttemperaturefeldhasbeenfound.infg.5a,thetemperature profles are shown, whle Fgs. 4 and 5b llustrate the courses of heatng(coolng) curves at the ponts selected from the doman consdered.

7 Estmaton of relaxaton and thermalzaton tmes n The dentfcaton of delay tmes has been done usng the evolutonary algorthms. In Table 1, the algorthm parameters are collected. The results obtaned are presented n Table 2, and they are qute satsfactory. Table 1. Evolutonary algorthm parameters Number of generatons Number of chromosomes Prob. of Prob. of Prob. of unform nonunform arthmetc mutaton mutaton crossover Prob. of clonng % 30% 50% 10% Table 2. Result of computatons usng the EA Desgn varable Exact value Found value Error[%] τ q τ T Fgures 6-9 show the process of dentfcaton usng the evolutonary algorthm after 1st, 10th,50thand100thgeneratons.Themark showsthepostonoftherealvalue. Fg. 6. The process of dentfcaton evolutonary algorthm after 1st generaton Fg. 7. The process of dentfcaton evolutonary algorthm after 10th generatons

8 844 B. Mochnack, M. Paruch Fg. 8. The process of dentfcaton evolutonary algorthm after 50th generatons Fg. 9. The process of dentfcaton evolutonary algorthm after 100th generatons 4. Fnalremarks The applcaton of evolutonary algorthms for solutons of the dentfcaton problems s(from the numercal pont of vew) tme-consumng. On the other hand, however, the mathematcal and numercal problems connected wth adequate algorthm constructon seems to be essentally smpler n comparson wth the very popular gradent methods. The exactness of estmaton parametersτ q,τ T sverygood,andevennthecaseofrealexpermentaldataapplcatonthe qualty of the obtaned results s also satsfactory. The detals concernng the methods of ndrect measurements of surface temperatures n the case of mcroscale domans can be found n Tang and Arak(1999). The future works wll be connected wth the problems concernng the mcro models of crystallzaton(e.g. Lelto et al., 2012). Acknowledgements Ths work has been supported by project No. 2012/05/B/ST8/01477 sponsored by the Polsh Natonal Scence Centre for Research and Development. References 1. Al-Nmr M.A., 1997, Heat transfer mechansms durng short duraton laser heatng of thn metal flms, Internatonal Journal of Thermophyscs, 18, 5, Chen J.K., Beraun J.E., 2001, Numercal study of ultrashort laser pulse nteractons wth metal flms, Numercal Heat Transfer, Part A, 40, 1-20

9 Estmaton of relaxaton and thermalzaton tmes n Chen G., Borca-Tascuc D., Yang R.G., 2004, Nanoscale heat transfer, Encyclopeda of Nanoscence and Nanotechnology, X, LeltoJ.,ŻakP.L.,GreerA.L.,SuchyJ.S.,KrajewskW.K.,GraczB.,SzuckM., Shrzad A.A., 2012, Crystallzaton model of magnesum prmary phase n the AZ91/SC composte, Compostes Part B: Engneerng, 43, 8, Ln Z., Zhgle L.V., 2008, Electron-phonon couplng and electron heat capacty of metals under condtons of strong electron-phonon nonequlbrum, Physcal Revew B, 77, Majchrzak E., Mochnack B., Greer A.L., Suchy J.S., 2009, Numercal modelng of short pulse laser nteractons wth mult-layered thn metal flms, CMES: Computer Modelng n Engneerng and Scences, 41, 2, Majchrzak E., Mochnack B., Suchy J.S., 2008, Identfcaton of substtute thermal capacty of soldfyng alloy, Journal of Theoretcal and Appled Mechancs, 46, 2, Majchrzak E., Poteralska J., 2010, Two-temperature mcroscale heat treansfer model. Part 1: Determnaton of electrons parameters, Scentfc Research of the Insttute of Mathematcs and Computer Scence, Częstochowa, 9, 1, Majchrzak E., Poteralska J., 2011, Two temperature model of mcroscopc heat transfer, Computer Methods n Materal Scence, 11, 2, Mochnack B., Majchrzak E., 2007, Identfcaton of macro and mcro parameters n soldfcaton model, Bulletn of the Polsh Academy of Scences Techncal Scences, 55, 1, Mochnack B., Suchy J.S., 1995, Numercal Methods n Computatons of Foundry Processes, Polsh Foundrymen s Techncal Assocaton, Cracow 12. Ozsk M.N., Tzou D.Y., 1994, On the wave theory n heat conducton, Journal of Heat Transfer, 116, Tang D.W, Arak N., 1999, Wavy, wavelke, dffusve thermal responses of fnte rgd slabs to hgh-speed heatng of laser-pulses, Internatonal Journal of Heat and Mass Transfer, 42, Manuscrpt receved November 27, 2012; accepted for prnt January 24, 2013