ROMAI J., v.9, no.2(2013), 173 179 THE INTERVAL LATTICE BOLTZMANN METHOD FOR TRANSIENT HEAT TRANSFER IN A SILICON THIN FILM Alicj Piseck-Belkhyt, Ann Korczk Institute of Computtionl Mechnics nd Engineering, Silesin University of Technology, Gliwice, Polnd licj.piseck-belkhyt@polsl.pl, nn.korczk@polsl.pl Abstrct In this pper the numericl modelling of het trnsfer in two-dimensionl crystlline solid is considered. It is ssumed tht some prmeters (the relxtion time nd the boundry conditions) ppering in the mthemticl model of the problem nlyzed re given s intervl numbers. The problem discussed hs been solved using the intervl form of the lttice Boltzmnn method using the rules of the directed intervl rithmetic 2]. Keywords: intervl lttice Boltzmnn method, directed intervl rithmetic, het trnsport. 2010 MSC: 97U99. 1. INTRODUCTION Microscle het trnsfer is new field of study which rises from the need to understnd the mechnisms of the energy trnsport in thin films. Microscle het trnsfer cn be defined s the study of therml energy trnsfer with tking into ccount individul crriers. The clssicl diffusion-bsed model for trnsient het trnsfer cnnot be used for nlyzing smll length scle. In semiconductors nd dielectric mterils, the het trnsfer is crried minly by vibrtions in the crystl lttice clled phonons. The more common method to nlyze this kind of phenomen is the ppliction of the Boltzmnn trnsport eqution. In the mthemticl model describing this process some prmeters s the relxtion time re estimted experimentlly, so it seems nturl to tke the vlues of these prmeters s intervl numbers 5]. This ssumption is closer to the rel physicl conditions of the problem nlyzed. In the pper the intervl lttice Boltzmnn method for solving non-stedy trnsient het trnsfer with intervl thermophysicl prmeters hs been presented. 2. DIRECTED INTERVAL ARITHMETIC Let us consider directed intervl ā which cn be defined s set D of ll directed pirs of rel numbers defined s 173
174 Alicj Piseck-Belkhyt, Ann Korczk ā =, +] := { ā D, + R } where nd + denote the beginning nd the end of the intervl, respectively. The left or the right endpoint of the intervl ā cn be denoted s s, s {+, }, where s is binry vrible. This vrible cn be expressed s product of two binry vribles nd is defined s follows + + = = + + = + = An intervl is clled proper if +, improper if + nd degenerte if = +. The set of ll directed intervl numbers cn be written s D = P I, where P denotes set of ll directed proper intervls nd I denotes set of ll improper intervls. Additionlly subset Z = Z P Z I D should be defined, where Z P = { ā P 0 +} Z I = { ā I + 0 } For directed intervl numbers two binry vribles re defined. The first of them is the direction vrible { +, i f τ (ā) = +, i f > + nd the other is the sign vrible { +, i f σ (ā) = > 0, + > 0, i f < 0, + < 0, ā D\Z For zero rgument σ (0, 0]) = σ(0) = +, for ll intervls including the zero element ā Z, σ (ā) is not defined. The sum of two directed intervls ā =, +] nd b = b, b +] cn be written s The difference is of the form ā + b = + b, + + b +], ā, b D ā b = b +, + b ], ā, b D The product of the directed intervls is described by the formul σ( b) b σ(ā), σ( b) b σ(ā)] ā, b D\Z σ(ā)τ( b) b σ(ā), σ(ā)τ( b) b σ(ā)], ā D\Z, b Z ā b = σ( b) b σ( b)τ(ā), σ( b) b σ( b)τ(ā) ], ā Z, b D\Z ( min b +, + b ), mx ( b, + b +)], ā, b Z mx ( P b, + b +), min ( (ā ) b (ā +, + b )], ā, ) b Z I 0, ZP, b Z I ZI, b Z P
The Intervl Lttice Boltzmnn Method for trnsient het trnsfer in silicon thin film 175 The quotient of two directed intervls cn be written using the formul ā/ b = σ( b) /b σ(ā), σ( b) /b σ(ā)], ā, b D\Z σ( b) /b σ( b)τ(ā), σ( b) /b σ( b)τ(ā) ], ā Z, b D\Z In the directed intervl rithmetic re defined two extr opertors, inversion of summtion D ā =, +], ā D nd inversion of multipliction 1/ D ā = 1/, 1/ +], ā D\Z So, two dditionl mthemticl opertions cn be defined s follows ā D b = b, + b +], ā, b D nd ā/ D b = σ( b) /b σ(ā), σ( b) /b σ(ā)], ā, b D\Z σ( b) /b σ( b), σ( b) /b σ( b) ], ā Z, b D\Z Now, it is possible to obtin the number zero by subtrction of two identicl intervls ā D ā = 0 nd the number one s the result of the division ā/ D ā = 1, which ws impossible when pplying clssicl intervl rithmetic 2]. 3. THE INTERVAL LATTICE BOLTZMANN METHOD In dielectric mterils nd semiconductors the het trnsport is minly relized by qunt of lttice vibrtions clled phonons. Phonons lwys move from the prt with the higher temperture to the prt with the lower temperture. During this process phonons crry energy. This process cn be described by the Boltzmnn trnsport eqution (BTE) f t + v f = f 0 f τ r + g e f where f is the phonon distribution function, f 0 is the equilibrium distribution function given by the Bose-Einstein sttistic, v is the frequency-dependent phonon propgtion speed, τ r is the frequency-dependent phonon relxtion time nd g e f is the phonon genertion rte due to electron-phonon scttering. The BTE is one of the fundmentl equtions of solid stte physics. In order to tke dvntge of the simplifying ssumption of the Debye model, the Boltzmnn
176 Alicj Piseck-Belkhyt, Ann Korczk trnsport eqution cn be trnsformed to n equivlent phonon energy density eqution 1] e e 0 + v e = e + q v where e is the phonon energy density, e 0 is the equilibrium phonon energy density nd q v is the internl het genertion rte relted to n unit of volume. This eqution must be supplemented by the dequte boundry-initil conditions. The dependence between phonon energy nd lttice temperture cn be clculted from the following formul using the Debye model 9η k b e (T) = Θ 3 D Θ D /T 0 τ r z 3 exp(z) 1 dz T 4 where Θ D is the Debye temperture of the solid, k b is the Boltzmnn constnt, T is the lttice temperture while η is the number density of oscilltors. The intervl lttice Boltzmnn method (ILBM) is discrete representtion of the intervl Boltzmnn trnsport eqution 6]. The ILBM discretizes the spce domin considered by defining lttice sites where the phonon energy density is clculted. The lttice is network of discrete points rrnged in regulr mesh with phonons locted in lttice sites. Fig. 1. Two dimensionl 4-speed (D2Q4) lttice Boltzmnn model There re few possibilities for sptil position of the prticles. In the pper the D2Q4 lttice, which hs two dimensions nd four velocities, hs been pplied. Phonons cn trvel only to neighboring lttice sites by bllisticlly trvelling with certin velocity nd collide with other phonons residing t these sites (see Fig. 1).
The Intervl Lttice Boltzmnn Method for trnsient het trnsfer in silicon thin film 177 The discrete set of propgtion velocities in the min lttice directions cn be defined s c 1 = (c, 0) c 2 = (0, c) c 3 = ( c, 0) c 4 = (0, c) In the ILBM it is needed to solve four intervl equtions supplemented by the boundry nd initil conditions llowing to compute intervl phonon energy in different lttice nodes. The intervl Boltzmnn trnsport equtions for two-dimensionl problem tke the following form 1, 3, 4] ē 1 + c ē 1 x = ē 1 ē 0 1 τ r, τ + r ] + q v ē 2 + c ē 2 y = ē 2 ē 0 2 τ r, τ + r ] + q v ē 3 c ē 3 x = ē 3 ē 0 3 τ r, τ + r ] + q v ē 4 c ē 4 y = ē4 ē 0 4 τ r, τ + r ] + q v This set of equtions must be supplemented by the boundry conditions x = 0, 0 y L : e(0, y, t) = e(t b1 ) x = L, 0 y L : e(l, y, t) = e(t b2 ) y = 0, 0 x L : e(x, 0, t) = e(t b3 ) y = L, 0 x L : e(x, L, t) = e(t b4 ) nd the initil condition of the following form t = 0 : ē (x, y, 0) = ē (T 0 ) where T b1 = T b1, T + b1 ], T b2 = T b2, T + b2 ], T b3 = T b3, T + b3 ] nd T b4 = T b4, T + b4 ] re the intervl boundry tempertures, T 0 is the initil temperture. The totl intervl energy density is defined s the sum of discrete intervl phonon energy densities in ll the lttice directions ē = 4 d=1 After subsequent computtions the intervl lttice temperture is determined using the formul describing the reltion between intervl phonon energy nd intervl lttice temperture D/ T f +1 = 4 ē( T f ) Θ 3 9η k b ē d Θ D / T f 0 z 3 exp(z) 1 dz
178 Alicj Piseck-Belkhyt, Ann Korczk 4. RESULTS OF COMPUTATIONS As numericl exmple the het trnsport in silicon thin film of the dimensions 200 nm 200 nm hs been nlyzed. The following input dt hve been introduced: the relxtion time τ r = 6.36675, 6.69325] ps, the Debye temperture Θ D = 640 K, the boundry conditions T b1 = 780, 820] K nd T b2 = T b3 = T b4 = 292.5, 307.5] K, the initil temperture T 0 = 300 K. The lttice step x = y = 20 nm nd the time step t = 5 ps hve been ssumed. Clcultions were crry out t the three internl nodes (40, 20) - 1, (160, 40) - 2 nd (100, 100) - 3 (see fig. 2). Fig. 2. Discretized domin Figures 3 nd 4 present the courses of the temperture function t chosen internl nodes for the het source q v = 0 nd q v = 10 18 W/m 3 respectively. Fig. 3. The intervl heting curves for q v = 0
The Intervl Lttice Boltzmnn Method for trnsient het trnsfer in silicon thin film 179 Fig. 4. The intervl heting curves for q v = 10 18 W/m 3 5. CONCLUSIONS The intervl lttice Boltzmnn method is n effective tool to numericl simultion of the het trnsfer in crystlline solids. The generliztion of LBM llows one to find the numericl solution in the intervl form nd such n informtion my be importnt especilly for the prmeters which re estimted experimentlly, for exmple the relxtion time. In the pper the Boltzmnn trnsport eqution with the intervl vlues of the relxtion time nd the boundry conditions hs been considered. References 1] R. A. Escobr, S. S. Ghi, M. S. Jhon, C. H. Amon, Multi-length nd time scle therml trnsport using the lttice Boltzmnn method with ppliction to electronics cooling, Journl of Het nd Mss Trnsfer, 49, 2006, 97-107. 2] S. M. Mrkov, On Directed Intervl Arithmetic nd its Applictions, Journl of Universl Computer Science, Vol. 1, 1995, 514-526. 3] A. Piseck Belkhyt, The intervl lttice Boltzmnn method for trnsient het trnsport, Scientific Reserch of the Institute of Mthemtics nd Computer Science, 1(8), Czestochow, 2009, 155-160. 4] A. Piseck Belkhyt, A. Korczk, Modelling of trnsient het trnsport in one-dimensionl crystlline solids using the intervl lttice Boltzmnn method, 20th Interntionl Conference on Computer Methods in Mechnics CMM-2013, Poznn, 2013, MS11-3. 5] S. Nrumnchi, J. Y. Murthy, C. H. Amon, Simultion of unstedy smll het source effects in sub-micron het conduction, Journl of Het Trnsfer, 123, 2003, 896-903. 6] M. Kviny, A. J. H. McGughey, Integrtion of moleculr dynmics simultions nd Boltzmnn trnsport eqution in phonon therml conductivity nlysis, IMECE 2003-41899.