Nanoscale Heat Transfer using Phonon Boltzmann Transport Equation

Transcription

1 Excerpt fr the Prceedings f the COMSOL Cnference 29 Bstn Nanscale Heat Transfer using Phnn Bltzann Transprt Equatin Sangwk Sihn *,2 and Ajit K. Ry Air Frce Research Labratry, 2 University f Daytn Research Institute *3 Cllege Park, Daytn, OH , sangwk@stanfrdaluni.rg Abstract: COMSOL Multiphysics was used t slve a phnn Bltzann transprt equatin (BTE) fr nanscale heat transprt prbles. One-diensinal steady-state and transient BTE prbles were successfully slved based n finite eleent and discrete rdinate ethds fr spatial and angular discretizatins, respectively, by utilizing the built-in feature f the COMSOL, Cefficient Fr f PDE. A sensitivity study was cnducted with varius discretizatin refineents fr different values f the Knudsen nuber, which is a easure f the nanscale regie. It was fund that sufficient refineent fr angular discretizatin is critical in btaining accurate slutins f the BTE. Keywrds: Nanscale heat transprt, phnn Bltzann transprt equatin, finite eleent ethd, discrete rdinates ethd.. Intrductin Fr the last tw centuries, any heat cnductin prbles have been deled successfully by a Furier diffusive equatin (FDE). The equatin can be derived using a cnservatin law f energy and Furier s linear apprxiatin f heat flux using a teperature gradient. The FDE is a parablic equatin reflecting a diffusive nature f heat transprt. An underlying assuptin is that the heat is effectively transferred between lcalized regins thrugh sufficient scattering events f phnns within a ediu. Therefre, the FDE des nt hld when the nuber f scatterings is negligible, which culd happen when a ean free path f phnn is siilar r a larger rder f agnitude than a dain size f interest, significant bundary scattering ccurs at aterial interfaces causing theral resistance, etc. Anther prble f the FDE is that it adits an infinite speed f the heat transprt, which is cntradictry t the thery f relativity. Therefre, the FDE is inapprpriate fr the heat transfer prbles at sall tie and spatial scales. T reslve the issue f the infinite speed f heat carrier, hyperblic equatinns, such as a Cattane equatin (Telegraph equatin) [] r a relativistic heat equatin [2], are ften used t reflect a wave nature f the heat transprt fr cases f sall speed f heat prpagatin (e.g., lw teperature, lw cnductive ediu, theral insulatr, phase transitin, etc.) and/r high speed f heat flux (e.g., pic- r fetsecnd pulsed-laser heating.) Despite se issues [], these hyperblic equatins can be used t cnsider the finite speed f phnns fr a shrt tie scale. Hwever, they still cannt be used fr sall spatial scale. Fr the nanscale heat transfer analysis, ne needs equatins and ethds fr sall-scale siulatin in ters f bth tie and space. A lecular dynaics (MD) siulatin can be a useful and accurate ethd in this regard. Hwever, the MD siulatin is usually cputatinally expensive, s it is suitable fr systes having a few atic layers r several thusand ats, but nt suitable fr device-level theral analysis. Alternatively, a phnn Bltzann transprt equatin (BTE), a firstrder partial differential equatin fr a phnn distributin functin, has been used fr this purpse [3-5]. The distributin functin is a scalar quantity in the six-diensinal phase space (three space crdinates and three wavevectr crdinates). The phnn BTE is als called an equatin f phnn radiative transfer (EPRT) when the phnn distributin functin is replaced with a phnn intensity functin [3]. The BTE is knwn t be difficult t slve, and thus ften siplified with a relaxatin tie apprxiatin. The BTE can predict a ballistic nature f heat transfer under an assuptin that particle-like behavirs f phnn are uch re significant than its wave-like behavirs, which ake the BTE valid fr structures larger than the wavelength f phnns. The BTE can be slved analytically fr siple geetries [4, 6, 7], r nuerically fr cplex geetries by using either deterinistic ethds (e.g., discrete rdinates ethd, spherical harnics ethd,

2 finite vlue ethd [8], etc.) r statistical ethds (e.g., Mnte Carl siulatin [9, ]). In general, slving the BTE is uch re efficient than the MD, and the predictins agree well with experiental data [, ]. In the present study, we will slve the BTE with the relaxatin tie apprxiatin fr nediensinal heat transprt prbles using the COMSOL Multiphysics. We will cbine the finite eleent (FE) capability f the COMSOL with the DOM t slve bth steady-state and transient heat transfer prbles. The nuerical results will be cpared with analytical nes fr the steady-state prble. A sensitivity study will be cnducted fr varius scale easures t shw the difference between the BTE slutins with the cnventinal FDE nes. Furtherre, we will deterine a suitable discretizatin schee fr spatial and angular spaces in rder t btain reliable transient slutins in using the COMSOL. 2. Use f COMSOL Multiphysics The BTE equatin under a relaxatin tie apprxiatin is written as f f f f + v f = () t t scat τ, where f is the carrier distributin functin, f the equilibriu distributin, v the carrier grup velcity vectr, and τ the relaxatin tie. The EPRT in ters f the phnn intensity is written as I I I + v I = (2) t τ, where the phnn intensity is expressed as I v, r) = v hf v, r) D( ) / 4π, where h is the reduced Planck cnstant, the phnn angular frequency, and D () the density f state per unit vlue. An equilibriu phnn intensity in Eq. (2) is written as I = I dω 4π Ω= 4π. (3) Fr ne-diensinal prbles, Eq. (2) beces I I I I + vcsθ = (4) t x τ, where θ is a plar angle between a phnn prpagatin directin and the glbal heat transfer directin ( x ). A slutin prcess f the -D BTE in Eq. (4) alng with Eq. (3) can be suarized as fllws: With an initial value f I, Eq. (4) is slved fr the phnn intensity field, I, fr a particular phnn prpagatin angle, θ. After btaining the slutins fr all pssible slid angles, 4 π, I can be calculated using Eq. (3). With this updated value f I, the slutin prcess repeats until the cnvergence f I and I. Therefre, the BTE and thus the EPRT are highly nnlinear equatins t slve. Once the phnn intensity is slved, heat flux and an equivalent teperature can be btained by integrating the phnn intensity ver the slid angles, 4 π, such that q = I csθ dω and (5) T u c Ω= 4π = I 4π I dω = c v c v Ω = 4π, (6) where c is the vluetric specific heat. One f the st widely used ethds fr slving the BTE nuerically is the discrete rdinates ethd (DOM). The DOM is a tl t transfr the equatin f radiative transfer int a set f siultaneus partial differential equatins. This is based n a discrete representatin f directinal variatin f intensity. A slutin t the transprt prble can be fund by slving the equatin f radiative transfer fr a set f discrete directins spanning the entire slid angle. The integrals ver the slid angle can be apprxiated by nuerical integratin using Gauss-Legender quadratures. The BTEs in Eqs. (3)-(4) were slved by the DOM using a built-in feature f Cefficient Fr PDEs in COMSOL Multiphysics. The spatial dain ( x ) was discretized int n eleents as a nral FE esh refineent, while the angular dain ( Ω ), referred t as rdinates, was discretized in the directinal crdinates by dividing the angular space int a finite nuber,. The angular dain was discretized at Gaussian quadrature pints within µ = csθ. Fr each plar angle ( θ ), we slved the syste f nnlinear equatins fr the phnn intensity, I x, µ ), and then updated the equilibriu phnn intensity,, using I

3 I fr all angles. Eqs. (3) and (5) can be written in discrete frs as I = I x, µ ) w and (7) 2 q = 2π I x, µ ) µ w, (8) respectively. The weight satisfies w = 2. The slver used fr the steady-state and transient analyses was a built-in direct slver, UMFPACK. Default values were used fr the slver paraeters except that strict tie steps were taken by the slver and a axiu BDF rder was set t. When the relative errr f the calculated value f the equivalent equilibriu intensity between tw iteratin steps was less than a present tlerance, we cnsidered the prble was cnverged and then calculated the equivalent teperature and heat flux using Eqs. (6) and (8), respectively. 5. Results and Discussin We have slved the BTE equatins with the COMSOL under bth steady-state and transient states. Spatial and angular dains were discretized int n eleents and quadrature pints, respectively. A quadratic Lagrange eleent was used fr the spatial FE esh. The ttal degree f freed fr the -D prble beces 2 ( n + 2). Fr the steady-state prbles, it was bserved that the wellcnverged slutins were btained with a relatively carse esh (5 eleents) while using 6 quadrature pints. The sensitivity f esh and quadrature pints fr the transient analysis will be stated later in this sectin. Figure shws steady-state teperature distributins via nndiensinal eissive pwer ( e * ) alng the -D dain fr varius Knudsen nubers ( Kn ). Fr cparisn purpse, an analytical slutin was als btained by using an analgy t the radiatin heat transfer [5] by slving an integral equatin, ξ * * 2e η = η + η η η η ( ) E2( ) e ( ) E( ) d (9), where η = x L is the nndiensinal crdinate, where L is the length f the dain, n E ( = µ 2 exp( x µ ) dµ the expnential n integral, ξ the ptical thickness, which is an inverse f the Knudsen nuber, Kn = Λ L, where Λ is the average ean free path f * + phnns, and e ( η ) = [ e ( η) J q2] [ J q J q2] is the eissive pwer nndiensinalized by a + difference f bundary heat fluxes ( J and J ) at bth ends f the -D dain. In Figure, the analytic and nuerical slutins fr varius Kn (=.,,, ) were drawn with arkers and lines, respectively, which shws an excellent agreeent f the slutins with each ther. While the cnventinal Furier slutin wuld yield the teperature gradient varying fr at η = t at η =, the slutin fr the BTE yields a reduced teperature gradient. The larger the Knudsen nuber, the saller the teperature gradient, which is attributed fr re ballistic phnn transprt as cpared t the diffusive ne. Nndi. eissive pwer, e* q2 Kn=. Kn= Kn= Kn= Nndiensinal crdinate, η Figure. Steady-state nndiensinalized eissive pwer distributin fr varius Knudsen nubers. We studied the effects f refineents in ters f spatial FE eshes and angular quadrature pints n the tie-dependent transient BTE slutins. The transient tie ( t ) was nndiensinalized by the relaxatin tie f phnns ( τ ), s that t * = t τ. Figure 2 shws nndiensinalized teperature distributins at t * =. alng the -D dain fr Kn =. We used three different FE eshes (5, 6 and 2 eleents) and a fixed angular divisin with 6 Gaussian pints ( n = gp 6 ). The three FE eshes yield nearly identical teperature slutins with a decreasing trend fr the ht side ( η = ) t the cld side ( η = ). q

4 Sth slutins were btained except at a regin near the ht bundary. An inset rectangular area in Figure 2 within η. 3 was zed in and repltted in Figure 3, which clearly shws significant wiggles near the ht bundary. Therefre, the FE esh refineent des nt help sthing ut the wiggles at all. The wiggling f the slutin, called a ray effect, is knwn t be attributed t insufficient refineent in the angular dain, and thus cannt be sthed ut with the FE esh refineent in the spatial dain. Nndiensinal teperature, θ eleents 6 eleents 2 eleents Nndiensinal crdinate, η Figure 2. Transient nndiensinal teperature distributin predicted with 5, 6 and 2 finite eleents and 6 Gaussian pints. Nndiensinal teperature, θ eleents 6 eleents 2 eleents..2.3 Nndiensinal crdinate, η t * =. alng the -D dain fr Kn = predicted with six different Gaussian quadrature pints ( n gp = 4, 8, 6, 32, 64, 28 ). We utilized a Matlab-scripting feature f COMSOL t ipleent the large n gp in building the syste f equatins. A fixed spatial divisin with 24 FE eleents was used in these calculatins. An inset rectangular area within η. 3 was agnified and repltted in Figure 5. Althugh a highly refined FE esh (24 eleents) was used, the carse angular divisins alleviate the wiggling f the slutin (e.g., n = gp 4 ). The wiggles near the ht end gradually decrease with the increase f n gp, and thus the ray effect. Therefre, we fund that the spatial and angular refineents are independent f each ther, and thus d nt recend t use the highly refined spatial esh with the carse angular esh. Nndiensinal teperature, θ ngp=4 ngp=8 ngp=6 ngp=32 ngp=64 ngp= Nndiensinal crdinate, η Figure 4. Transient nndiensinal teperature distributin predicted with 24 finite eleents and six Gaussian quadrature pints ( n ) fr angular gp discretizatin. Figure 3. A agnified view f Figure 2 within η.3. T eliinate the ray effect, we ade further refineent in the angular dain by using re quadrature pints. Figure 4 shws nndiensinalized teperature distributins at

5 Nndiensinal teperature, θ ngp=4 ngp=8 ngp=6 ngp=32 ngp=64 ngp= Nndiensinal crdinate, η Figure 5. A agnified view f Figure 5 within η.3. Figure 6 and Figure 7 shw the teperature and heat flux distributins, respectively, at varius tie scales with three different Knudsen nubers. The result successfully reprduced the ne reprted earlier by Chen [4], but this tie using the FE ethd using the COMSOL. The figures shw the heat transfer fr the ht side t the cld side with the finite speed f phnn as the shrt ter (sall t * ) slutins indicate, which cannt be predicted by the FDE. While cnstant teperatures were applied as bundary cnditins at bth sides, we can bserve the teperature jup at these bundaries. The higher the Kn, the larger the jup. The reasn fr the teperature jup at these eissive bundaries was well explained in earlier wrk [7]. Fr a sall Knudsen nuber at Kn =., the teperature slutin appraches that f the steady-state FDE as the tie increases, while fr a large Knudsen nuber at Kn =, the teperature slutin appraches a nearly hrizntal line, which indicates the ballistic heat transfer rather than the diffusive ne, and results in a sall teperature gradient and thus a large theral cnductivity. 6. Cnclusins We have slved the phnn Bltzann transprt equatin fr the nanscale heat transfer prbles with the COMSOL Multiphysics. One-diensinal steady-state and transient prbles were successfully slved using FEM and DOM fr spatial and angular discretizatins, respectively, by utilizing the built-in feature f the COMSOL, Cefficient Fr f PDE. A sensitivity study was cnducted with varius discretizatin refineents fr different values f the Knudsen nuber, which is a easure f the nanscale regie. Significant ray effects were fund with carse angular discretizatins. Hwever, false scattering due t carse spatial discretizatin was nt severe with the COMSOL slutin. It was fund that the spatial and angular refineents were independent f each ther, and it was nt recended t use the highly refined spatial esh with the carse angular esh in slving the BTE with the COMSOL. The present success with the -D siulatin encurages us t apply it fr ultidiensinal cases. (a) Nndiensinal teperature, θ t*=. t*=. t*= t*= (b) Nndiensinal teperature, θ t*=. t*=. t*= t*= (c) Nndiensinal teperature, θ t*=. t*= t*= t*= Nndiensinal crdinate, η Nndiensinal crdinate, η Nndiensinal crdinate, η Figure 6. Teperature distributins at different tie scales. (a) Kn =, (b) Kn = and (c) Kn =..

6 (a) Nndiensinal heat flux, q* t*=. t*=. t*= t*= (b) Nndiensinal heat flux, q* t*=. t*=. t*= t*= (c) Nndiensinal heat flux, q* t*=. t*= t*= t*= Nndiensinal crdinate, η Nndiensinal crdinate, η Nndiensinal crdinate, η Figure 7. Heat flux distributins at different tie scales. (a) Kn =, (b) Kn = and (c) Kn =.. 8. References. C. Bai, et al., On Hyperblic Heat Cnductin and the Secnd Law f Therdynaics, Jurnal f Heat Transfer, 7(2), (995). 2. Y. M. Ali, et al., Relativistic Heat Cnductin, Internatinal Jurnal f Heat and Mass Transfer, 48(2), (25). 3. A. Majudar, Micrscale Heat Cnductin in Dielectric Thin Fils, Jurnal f Heat Transfer, 5(), 7-6 (993). 4. G. Chen, Ballistic-Diffusive Equatins fr Transient Heat Cnductin fr Nan t Macrscales, Jurnal f Heat Transfer, 24(2), (22). 5. G. Chen, Nanscale Energy Transprt and Cnversin: A Parallel Treatent f Electrns, Mlecules, Phnns and Phtns, Oxfrd University Press (25). 6. G. Chen, Nnlcal and Nnequilibriu Heat Cnductin in the Vicinity f Nanparticles, Jurnal f Heat Transfer, 8(3), (996). 7. R. Yang, et al., Siulatin f Nanscale Multidiensinal Transient Heat Cnductin Prbles Using Ballistic-Diffusive Equatins and Phnn Bltzann Equatin, Jurnal f Heat Transfer, 27(3), (25). 8. J. Y. Murthy, et al., Cputatin f Sub-Micrn Theral Transprt Using an Unstructured Finite Vlue Methd, Jurnal f Heat Transfer, 24(6), 76-8 (22). 9. M.-S. Jeng, et al., Mdeling the Theral Cnductivity and Phnn Transprt in Nanparticle Cpsites Using Mnte Carl Siulatin, Jurnal f Heat Transfer, 3(4), (28).. S. Mazuder, et al., Mnte Carl Study f Phnn Transprt in Slid Thin Fils Including Dispersin and Plarizatin, Jurnal f Heat Transfer, 23(4), (2).. G. Chen, Size and Interface Effects n Theral Cnductivity f Superlattices and Peridic Thin- Fil Structures, Jurnal f Heat Transfer, 9(2), (997). 9. Acknwledgeents This wrk was perfred under U.S. Air Frce Cntract N. FA865-5-D-55.