Analysis of thermal processes occurring in heated multilayered metal films using the dual-phase lag model

Transcription

1 Arc. Mec., 69, 4 5, pp , Warszawa 2017 Analyss of termal processes occurrng n eated multlayered metal flms usng te dual-pase lag model E. MAJCHRZAK, G. KAŁUŻA Insttute of Computatonal Mecancs and Engneerng Slesan Unversty of Tecnology Konarskego 18a Glwce, Poland e-mals: ewa.majcrzak@polsl.pl, grazyna.kaluza@posl.pl A multlayered tn metal flm subjected to an ultra-sort laser pulse s consdered. A matematcal descrpton of te dscussed process s based on te system of te dual-pase lag equatons supplemented by approprate boundary and ntal condtons. Specal attenton s devoted to te deal contact condtons at te nterfaces between te layers, wc n te case of te dual-pase lag model must be formulated n a dfferent way tan n te macroscopc Fourer model. To solve te problem te explct sceme of te fnte dfference metod s developed. In te fnal part of te paper te example of computatons s sown. Key words: multlayered tn metal flms, dual-pase lag equaton, FDM. Copyrgt c 2017 by IPPT PAN 1. Introducton One of te models descrbng te termal processes occurrng n tn metal flms subjected to an ultra-sort laser pulse s te dual-pase lag equaton 1 6] T(x, t) 2 ] ] T(x, t) (1.1) x Ω : c + τ q 2 = λ T(x, t) + τ T λ T(x, t) ] Q(x, t) + Q(x, t) + τ q, were T(x, t) s te temperature, c s te volumetrc specfc eat, λ s te termal conductvty, τ q s te relaxaton tme, τ T s te termalzaton tme and Q(x, t) s te source functon connected wt te laser eatng. Ts equaton can be derved wen te followng formula s ntroduced n place of te classcal Fourer law q(x, t) = λ T(x, t): (1.2) q(x, t + τ q ) = λ T(x, t + τ T ), were T s te temperature gradent and q s te eat flux.

2 276 E. Majcrzak, G. Kałuża Usng te Taylor seres expansons te followng frst-order approxmaton of equaton (1.2) can be accepted ] q(x, t) T(x, t) (1.3) q(x, t) + τ q = λ T(x, t) + τ T. In te case of multlayered domans, te system of equatons of type (1.1) for eac subdoman sould be taken nto account and on te contact surfaces te approprate boundary condtons sould be assumed. In te case of deal contact tese boundary condtons take te followng form: { Te (x, t) = T (1.4) x Γ e : e+1 (x, t), e = 1, 2,...,E 1. q e (x, t) = q e+1 (x, t), Usng te relatonsp (1.3) between te eat flux and temperature gradent, te second part of boundary condton (1.4) sould be properly formulated. It sould be noted tat n te lterature dfferent ways of modelng of deal contact condton (1.4) are presented. For example, at te stage of computatons, te FDM grd n wc te nodes are located not on te contact surface but at a certan dstance from ts surface s ntroduced 7 9]. Anoter approac s to use te staggered grd were te temperature nodes and eat flux nodes are dstngused separately, e.g., 10, 11]. Fnally, n several artcles (e.g. 12, 13]), te condton of deal contact s formulated n te same manner as n te case of macroscopc Fourer model, wc s, of course, a sgnfcant smplfcaton. In ts paper, te second part of nterfacal condton (1.4) s expressed n terms of temperature. Ts allows to ntroduce te spatal grd wt te nodes located on te contact surface. Te am of te study s also to compare te results of computatons for dfferent ways of deal contact condton modelng. Tus, n Secton 2 te matematcal model of termal processes n te multlayered tn metal flm subjected to te ultrasort laser pulse s presented. Secton 3 presents te boundary condton of deal contact, wc takes nto account te relatonsp (1.3) between te eat flux and temperature gradent. Te explct sceme of te fnte dfference metod s presented n Secton 4. Te FDM algortm consttutes a bass for an n ouse computer program used at te stage of numercal modelng (Secton 5). In te fnal part of te paper te conclusons are formulated. 2. Governng equatons A multlayered tn flm of tckness L = L 1 + L L E, as sown n Fg. 1, wt an ntal temperature dstrbuton T(x,0) = T p s consdered. Te constant termal propertes of successve layers and te deal termal contact between te layers are assumed.

3 Analyss of termal processes Fg. 1. Multlayered tn metal flm. Te temperature dstrbuton n te successve layers s descrbed by te system of equatons (1D problem) 7, 8]: T e (x, t) (2.1) L e 1 < x < L e : 3 T e (x, t) + τ Te a e x 2 2 T e (x, t) 2 T e (x, t) + τ qe 2 = a e x Q e (x, t) + τ qe Q e (x, t), e = 1, 2,...,E, c e c e were a e = λ e /c e (λ e s te termal conductvty of e-t layer, c e s te volumetrc specfc eat), τ qe s te relaxaton tme, τ Te s te termalzaton tme, T e s te temperature, x s te spatal coordnate and t s te tme. A front surface x = L 0 = 0 s rradated by a laser pulse and te source functon Q 1 (x, t) connected wt te laser eatng s defned as follows 14]: (2.2) Q 1 (x, t) = β π 1 R 1 t p δ 1 I 0 exp xδ1 β (t 2t p) 2 were I 0 s te laser ntensty, t p s te caracterstc tme of a laser pulse, δ 1 s te absorpton dept, x δ 1 < L 1, R 1 s te reflectvty of te rradated surface and β = 4 ln2. For e = 2, 3,...,E: Q e (x, t) = 0. Introducng te source functon (2.2) allows to assume for x = 0 and x = L te no-flux condtons, namely 14]: (2.3) q 1 (0, t) = 0, q E (L, t) = 0. Te boundary condtons on te contact surfaces between subdomans ave te form of contnuty ones, ts means tat t 2 p ], (2.4) x = L e : { Te (x, t) = T e+1 (x, t), q e (x, t) = q e+1 (x, t), e = 1, 2,...,E 1. Te ntal condtons (t = 0) are also gven

4 278 E. Majcrzak, G. Kałuża (2.5) t = 0 : T e (x,0) = T p, T 1 (x, t) x δ 1 : = Q 1(x,0), t=0 c 1 T e (x, t) x > δ 1 : = 0. t=0 One can see tat te ntal eatng rate s defned n dfferent ways for x δ 1 and for x > δ 1. It results from te assumpton tat te nternal eat source connected wt te laser radaton acts only n te layer of tckness δ 1 correspondng to te absorpton dept. 3. Interfacal condton From te numercal pont of vew, t s convenent to express te deal contact condton (2.4) n terms of temperature. For ts purpose, te followng relatonsp between te eat flux and temperature gradent must be appled (cf. formula (1.3)): (3.1) q e (x, t) + τ qe q e (x, t) = λ e Te (x, t) x 2 ] T e (x, t) + τ Te. x Te dependence (3.1) s dfferentated wt respect to tme and ten multpled by τ qe+1 ] q e (x, t) (3.2) τ qe+1 q e (x, t) + τ qe Te (x, t) 2 ] T e (x, t) = λ e τ qe+1 + τ Te. x x Addng te sdes of equatons (3.1) and (3.2) gves (3.3) q e (x, t) + (τ qe + τ qe+1 ) q e(x, t) 2 q e (x, t) + τ qe τ qe+1 2 Te (x, t) 2 ] T e (x, t) Te (x, t) = λ e + τ Te λ e τ qe+1 x x x In a smlar way one obtans (3.4) q e+1 (x, t) + (τ qe + τ qe+1 ) q e+1(x, t) Te+1 (x, t) = λ e+1 x λ e+1 τ qe 2 q e+1 (x, t) + τ qe τ qe ] T e+1 (x, t) + τ Te+1 x Te+1 (x, t) x 2 ] T e (x, t) + τ Te. x 2 ] T e+1 (x, t) + τ Te+1. x

5 Analyss of termal processes Takng nto account te second part of condton (2.4) t can be seen tat te left-and sdes of equatons (3.3) and (3.4) are te same. Tus te rgt-and sdes of tese equatons are created equal (3.5) λ e T e (x, t) x = λ e+1 T e+1 (x, t) x λ e (τ Te + τ qe+1 ) 2 T e (x, t) x λ e τ Te τ qe+1 3 T e (x, t) 2 x λ e+1 (τ Te+1 + τ qe ) 2 T e+1 (x, t) x 3 T e+1 (x, t) λ e+1 τ Te+1 τ qe 2. x Te fnal form of te above desgnated condton (3.5) corresponds to te formula presented n 15]. It sould be noted tat n te case of Neumann boundary condtons (2.3) one as (cf. formula (3.1)) (3.6) 4. Metod of soluton T1 (x, t) 2 ] T 1 (x, t) + τ T1 x x TE (x, t) 2 ] T E (x, t) + τ TE x x x=0 x=l = 0, = 0. At te stage of numercal computatons te fnte dfference metod s proposed. A geometrcal mes s sown n Fg. 2. Fg. 2. Geometrcal mes. Let T f = T(x, f t) were t s te tme step, x = ( s te mes step) and f = 0, 1,...,F. Takng nto account te ntal condtons (2.5) one as T 0 = T p, x δ 1 : T 1 = T p + tq 1 (x,0)/c 1, x > δ 1 : T 1 = T p. For transton t f 1 t f (f 2) te approxmate form of Eq. (2.1) resultng

6 280 E. Majcrzak, G. Kałuża from te ntroducton of adequate dfferental quotents can be proposed (4.1) T f T f 1 T f 2T f 1 + T f 2 + τ q t ( t) 2 ( T f a τ T f 1 2T t 2 + T f 1 1 T f 2 +1 After te matematcal manpulatons one as (4.2) T f T f 1 +1 = a f 2 2T t 2 2T f c Q f 1 = t2 + 2τ q 2 2a t( t + τ T ) 2 T f 1 ( t + τ q ) + a t( t + τ T ) f 1 2 (T+1 ( t + τ q ) + T f 1 1 ) a tτ T 2 ( t + τ q ) ( t) 2 ( ) + Q f 1 Q f 1 ] + τ q. c ( t + τ q ) 2 + T f τ q c + T f 1 1 ) ( ) Q f 1. τ q 2 2a tτ T 2 T f 2 ( t + τ q ) (T f T f 2 1 ) Ts formula allows to calculate te temperatures at te nternal nodes. Let m e, e = 1, 2,..., and E 1 s te node located on te nterfacal surface x = L e, as sown n Fg. 2. Te approxmate form of boundary condton (3.5) can be taken n te followng way: (4.3) λ e T f m e T f m e 1 λ e τ Te τ qe+1 1 ( t) 2 = λ e+1 T f m e+1 T f m e ence λ e+1 τ Te+1 τ qe 1 ( t) 2 λ e (τ Te + τ qe+1 ) 1 t ( T f me T f m e 1 λ e+1 (τ Te+1 + τ qe ) 1 t ( T f m e+1 T f m e ( T f me T f m e 1 2 T m f 1 e T m f 1 e T f 1 m e 1 + T m f 2 e ( T f m e+1 T f m e 2 T f 1 m T f 1 e+1 m e + T f 2 T f 1 m e 1 T f 2 m e 1 m T f 2 e+1 m e ) ) T f 1 m T f 1 ) e+1 m e ), (4.4) T f m e = A e1 T f m A e1 + A + A e2 e 1 T f m e2 A e1 + A e+1 e2 + A e3(tm f 1 e A e4(t f 1 m T f 1 e+1 T f 1 m )+A e 1 e5(tm f 2 e T f 2 m ) e 1 A e1 + A e2 m e ) + A e6 (T f 2 m T f 2 e+1 m e ), A e1 + A e2

7 Analyss of termal processes were (4.5) A e1 = λ e ( t) 2 + t(τ Te + τ qe+1 ) + τ qe+1 τ Te ], A e2 = λ e+1 ( t) 2 + t(τ Te+1 + τ qe ) + τ qe τ Te+1 ], A e3 = λ e t(τ Te + τ qe+1 ) + 2τ qe+1 τ Te ], A e4 = λ e+1 t(τ Te+1 + τ qe ) + 2τ qe τ Te+1 ], A e5 = λ e τ qe+1 τ Te, A e6 = λ e+1 τ qe τ Te+1. In turn, for te nodes 0 (x = 0) and n (x = L) (cf. boundary condtons (3.6)) one as T f 1 T f ( 0 1 T f 1 + τ T f 0 T1 T f 1 1 T f 1 ) 0 = 0, t (4.6) Tn f T f ( n 1 1 T f n T f n 1 + τ TE T n f 1 T f 1 ) n 1 = 0, t t means tat (4.7) τ T1 T f 0 = T f 1 (T f 1 1 T f 1 0 ), t + τ T1 Tn f = T f n 1 + τ TE (Tn f 1 T f 1 n 1 t + τ ). TE Because te explct sceme of te fnte dfference metod s proposed ere, terefore te stablty crtera sould be formulated (see 16, 17]). 5. Results of computatons As an example, a double-layered tn flm (gold and cromum) subjected to a laser pulse s consdered. Te tcknesses of layers are equal to 50 nm. Te ntal temperature s equal to T e0 = 300 K. In Table 1 te termopyscal parameters of materals are collected 18, 19]. Table 1. Termopyscal parameters. Gold Cromum Volumetrc specfc eat MJ/(m 3 K)] c 1 = c 2 = Termal conductvty W/(m K)] λ 1 = 315 λ 2 = 93 Relaxaton tme ps] τ q1 = 8.5 τ q2 = Termalzaton tme ps] τ T1 = 90 τ T2 = 7.86 Reflectvty R 1 = 0.95 R 2 = Absorpton dept nm] δ 1 = 15.3 δ 2 = 15.3

8 282 E. Majcrzak, G. Kałuża Fg. 3. Temperature story at te rradated surface: 1 n = 100, t = ps, 2 n = 1000, t = ps. Te computatons are performed for te laser ntensty I 0 = 40 W/m 2 and te caracterstc tme of te laser pulse t p = 0.1 ps (cf. formula (2.2)). In Fg. 3 te temperature story at te rradated surface for n = 100, t = ps and n = 1000, t = , respectvely, s sown. Because te results are grd ndependent for n 1000, so n te furter calculatons n = 1000 nodes s assumed. Fg. 4. Temperature dstrbuton for 0.2 and 0.3 ps (1 DPL model, 2 DPL wt te macroscopc boundary condtons). In Fgs. 4 6 te temperature profles for tmes 0.2, 0.3, 0.4 and 0.5 ps are sown. Te curves marked as 1 correspond to te soluton of DPL equaton (2.1) wt te boundary condtons (3.5), (3.6), wle te curves marked as 2 llustrate te soluton of te same equaton supplemented by te classcal macroscopc

9 Analyss of termal processes Fg. 5. Temperature dstrbuton for 0.4 ps (1 DPL model, 2 DPL wt te macroscopc boundary condtons). Fg. 6. Temperature dstrbuton for 0.5 ps (1 DPL model, 2 DPL wt te macroscopc boundary condtons). contnuty condtons (n equatons (3.5), (3.6): τ qe = 0 and τ Te = 0). It can be seen tat over tme te temperature dfferences ncrease especally n te vcnty of te contact surface. Fgure 7 llustrates te temperature courses on te contact surface between te layers for te DPL model (curve 1) and te DPL model wt te macroscopc boundary condtons (curve 2). After 0.5 ps te maxmum temperature on te contact surface s equal to K and K, respectvely. In Fg. 8, te temperature story at te rradated surface for bot cases s presented. Te dfferences between te temperatures are small (temperatures for tme 0.5 ps are equal to K and K, respectvely), wc sows tat n te case of te Neumann boundary condton (cf. equaton (3.6)) te assumpton tat τ T1 = 0 s acceptable.

10 284 E. Majcrzak, G. Kałuża Fg. 7. Temperature courses on te contact surface (1 DPL model, 2 DPL wt te macroscopc boundary condtons). Fg. 8. Temperature courses on te rradated surface (1 DPL model, 2 DPL supplemented by te macroscopc boundary condtons). Fg. 9. Temperature courses at ponts 1 40 nm, 2 50 nm, 3 60 nm, sold lnes one layer, symbols two layers.

11 Analyss of termal processes To ceck te correctness of te deal contact modelng te followng problem as been analyzed. In te frst verson, te layer of tckness L made of gold s consdered, wle n te second verson two layers of tcknesses L/2 made of te same materal (gold) are taken nto account. Between tese layers te condton correspondng to te deal termal contact (3.5) s assumed. It turned out tat te same solutons were obtaned, wat s sown n Fg. 9. So te analytcal form of condton (3.5) and ts numercal realzaton are defntely correct. It sould be noted tat wen usng te dual-pase lag equaton for numercal modelng of termal processes occurrng n te eated metal flms te pyscal anomales can take place 20 22]. Tey are connected wt te values of te delay tmes τ q and τ T 23]. Wen τ T > τ q (cf. Table 1) te DPL model mgt volate te second law of termodynamcs and ts case s defned as over-dffuson 20, 24]. Suc penomena, owever, can be explaned by te non-equlbrum entropy producton teory 25]. Te nterpretaton of ts penomenon, from a matematcal pont of vew, can be also found n 20]. On te oter and, te termalzaton tme τ T and te relaxaton tme τ q are te ntrnsc propertes of te materal n queston and n te lterature one can fnd a lot of papers connected wt te ultrafast pulse-laser eatng on metal flms related to te case τ q < τ T, e.g., 5 11, 14 19]. Moreover, te calculatons sow very good agreement wt te expermental results 7, 24]. 6. Conclusons A dual-pase lag model descrbng te termal processes occurrng n te multlayered tn metal flm subjected to te ultrasort laser pulse s consdered. Te deal contact condtons at te nterfaces between te layers are formulated accordng to te formula descrbng te dependence between te eat flux and temperature gradent n wc te pase lag tmes are taken nto account. To solve ts problem te explct sceme of te fnte dfference metod s proposed. Te results of computatons sow tat te condton of deal contact assumed n te same manner as n te case of macroscopc Fourer model s a sgnfcant smplfcaton. Presented n ts paper, formulaton of deal contact condton supplementng te dual-pase lag model can be used n te case of oter numercal metods, e.g. te general boundary element metod 26 28] or analytcal metods 29 31]. Acknowledgements Te paper and researc were fnanced wtn te project 2015/19/B/ST8 /01101 sponsored by te Natonal Scence Centre (Poland).

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