Theoretical approach to algorithm for metrological comparison of two photothermal methods for measuring of the properties of materials

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1 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN Thortical approach to algorithm for mtrological comparison of two photothrmal mthods for masuring of th proprtis of matrials R. Ivanov I. Morno J.M. Rivra Juárz Unidad Académica d Física Univrsidad Autónoma d Zacatcas E. Marin Cntro d Invstigación n Cincia Aplicada y Tcnología Avanzada Instituto Politécnico Nacional G. Gutiérrz Juárz Instituto d Física Univrsidad d Guanajuato J.L. Pichardo Molina Cntro d Invstigacions n Óptica Univrsidad d Guanajuato E mail: Abstract An algorithm dsignd for th comparison of photothrmal mthods is dscribd, basd on th knowldg of th valu to uncrtainty ratio of th masurd thrmal paramtrs of th invstigatd objct. This comparison allows us to slct th bst mthod for th prcis dtrmination of thrmal proprtis of matrials. Th usfulnss of th algorithm is dmonstratd through numrical simulations. ywords: ffusivity, mthod, photopyrolctric, photothrmal, uncrtainty. Introduction Th objctiv of a photothrmal xprimnt is to masur th main thrmal proprtis of a sampl, i.., th thrmal diffusivity α and th thrmal ffusivity. On branch of ths xprimntal mthods is th ara of th photopyrolctric tchniqus [1]. Thir advantags (high snsitivity, simplicity, and low cost) hav raisd a grat intrst in th last yars [,3]. Th sourcs of xprimntal rrors and nois in ths tchniqus ar vry divrs: tmpratur fluctuations, capacitiv pick-up of lctromagntic filds (5/6 Hz), intnsity fluctuations of 1

2 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN optical xcitation sourc, optical misalignmnt, xtrnal mchanical nois, lock in amplifir uncrtainty and nois tc. [4]. A larg part of ths advrs influncs can b cancld or substantilly diminishd by normalization procdurs, th us of stabilizd lasr or lctrical, acoustical and thrmal shilding. Unfortunatly, no procdur can cancl th influnc of th lock in amplifir uncrtainty and nois. Thrfor, w will concntrat on this sourc of xprimntal rror. For a photopyrolctric xprimnt, th signal that givs information is th voltag (or currnt) gnratd btwn th two facs of th pyrolctric causd by th chang of its avrag tmpratur. Th valu of this voltag (currnt), apart from th thrmal paramtrs of th sampl and of th pyrolctric itslf, dpnds on th tchniqu which is usd (for xampl: back photopyrolctric tchniqu BPPE [5], and front photopyrolctric tchniqu FPPE [6]) and of th paramtrs of th xprimntal arrangmnt frquncy, dimnsions of th cll, lasr powr. In this papr, w will dvlop our calculations for th idal voltag mod only, i.., for voltag signal. Howvr, our mthodology can also b usd for th currnt mod, for which can b substitutd by I in all formulas. By mans of photopyrolctric xprimnts, ffusivity can b masurd in a thrmally thick sampl rgim [1]. For ach on of th two tchniqus, BPPE and FPPE, according to th stablishd thortical modl, w hav th function (,p 1,,p n ), whr p i indicats th i th paramtr of th xprimntal arrangmnt. On of th first stps whn planning a photopyrolctric xprimnt is to dtrmin th optimum valus of ths xprimntal paramtrs. Aftr this stp, th voltag dpnds xclusivly on th thrmal ffusivity of th sampl: =(). Th objctiv of this work is to propos th paramtr that xprsss th valu to uncrtainty ratio rgarding th thrmal ffusivity for photopyrolctric tchniqus, and to us it to compar th mtrological prformanc of both tchniqus FPPE and BPPE. Thus, th paramtr bcoms a quantitativ indicator for idntifying th bst photopyrolctric tchniqu.

3 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN Thortical modl For our simulation w will considr th charactristics of opration of th lock in usd in many laboratoris, th modl R83/85 (tanford Rsarch ystms) [7]. Th snsitivity rangs of th lock in R83 ar displayd as If magnitud and th optimum rang (minimum possibl rang) for voltag of th lock in is always usd during th masurmnt of th voltag, it can b writtn as: < <. 5 (1) A lock in amplifir R83/85 masurs th voltag amplitud with an uncrtainty Δ ( ), which dpnds on th opration rang: Δ ) = f ( ). Th ( function is monotonic (if th voltag of th opration rang grows, th uncrtainty also grows). This function is known for th R83/85 lock in, and for non vry small rangs it is: f ( () 1 ). 1 For small rangs th noiss gnratd in th lock in bgin to influnc th total uncrtainty. Ths noiss can b of two typs: th combination of both Johnson's nois and 1/f nois, and th nois that is gnratd in th A/D convrtr. As a consqunc of th prsnc of th first two noiss th input nois of th lock in is about 5 (n rms / Hz). W work at low frquncis and for this rason ar forcd to us a low pass filtr with high tim constants τ (for f=hz τ>3s). This is why th quivalnt nois bandwith (ENBW), qual to 1/4 τ, is smallr than.8hz. Consquntly, th influnc of th nois gnratd in th A/D convrtr prvails, 3

4 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN which can b rprsntd as an quivalnt gnrator of 1n of amplitud connctd to th input of th lock in [7]: f ( ) = 1n (3) Th total uncrtainty is: f ( = + n (4) ) (.1 ) (1 ) Rgarding th masurd voltag, for ach photopyrolctric tchniqu a wll known, stablishd modl xists: = () (5) Driving this quation and taking modulus from both sids, w obtain: Δ = Δ (6) whr th subscript indicats that th drivativ dpnds on. Aftr a normalization procdur and masurmnt, th valus of th voltag amplitud, lock in rang, and uncrtainty of th voltag f( ) ar known. Th valu of th thrmal ffusivity is calculatd by mans of numrical fitting, using th formula = ( ), implicitly containd in formula (5). This valu is masurd with a dtrmind uncrtainty Δ that corrsponds mainly to th uncrtainty in th masurmnt of th voltag amplitud f( ). To apprciat th fficincy of th indirct masurmnt tchniqu, th trm «snsitivity» is usd in 4

5 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN mtrology [8]. For our cas, th snsitivity of th two tchniqus FPPE and BPPE can b xprssd as:, and th drivd = (7) ar rprsntd with complx numbrs. For th rlationship btwn th modulus of ths quantitis, w can writ: = (8) Th modulus of snsitivity givs us information rgarding th prcntag of changin th masurd voltag if th thrmal ffusivity changs by 1%. For photopyrolctric tchniqus, paramtr can b dfind using th formulas (4), (6) and (8): = = = = = (9) Δ Δ f ( n ) f ( ) (.1* ) + (1 ) On can s that dosn't hav a dimnsion and dpnds on th lock in snsitivity rang. Howvr, it discontinuously dpnds on th voltag according to formula (1). For this rason, it is dsirabl to find an intrval for th valu of whos limits dpnd only on th voltag amplitud : For > 5

6 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN = < = (.1 ) + (1n ) (.1 ) + (1n ) max (1) For <. 5 = > = (.1 ) + (1n ) (.5 ) + (1n ) min (11) Finally w hav: > > (1) max min For most practical cass, > 5μ. In ths cass, formulas (1) and (11) ar simplifid considrably: = max 1 (13) = min 4 (14) In formulas, (13) and (14), w can s th narrow rlationship btwn valu to uncrtainty ratio and th snsitivity of th mthod for a photopyrolctric masurmnt. If w compar th mtrological proprtis of two tchniqus, tchniqu 1 and tchniqu, w hav: 6

7 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN , i i i = max, i > i > min, i =, i (.1i ) + (1n ) (.5i ) + (1n ) (15) whr subscript i=1, and rprsnts th numbr of th tchniqu. Th location of ths intrvals on th axis is shown in Figur1 Figur 1 For cas a), w hav > 1. Consquntly, mthod has bttr charactristics than mthod 1. For cas b), w hav th opposit conclusion that tchniqu 1 is bttr. For cas c), this procdur can not lad us to a dfinitiv conclusion. Thrfor, it is ncssary to compar 1 and using formula (4), taking into account th chang in th lock in rangs for spcific valus of voltag. Numrical simulation W now apply our algorithm to compar th mtrological proprtis of th FPPE and BPPE tchniqus, using pur watr as th sampl. W suppos w can chang virtually th valu of β without significant chang in th thrmal proprtis of th watr. W calculat th valus of min y max in function of th absorption cofficint β for both tchniqus in th rgim of idal voltag, for svral sampl thicknsss l m ( l m = mm 1mm, stp mm), and svral frquncis f ( f = Hz, 5Hz, 1Hz, Hz). For th lowst frquncy, th minimum thicknss of th sampl is thirtn tims highr than th dpth of th thrmal wav pntration (μ.15mm). For this rason, th rsults don't dpnd on th thrmal diffusivity α, but only on th thrmal ffusivity of th sampl. Th othr paramtrs of th simulation hav valus that ar normally usd in th xprimnts, i..: th pyrolctric thicknss is 8μm, th maximum powr of th lasr is 1mW and th lctric capacitanc of th pyrolctric is 1nF. Th rsults ar shown in Figs. 5 for both configurations. 7

8 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN Figur Figur 3 Figur 4 Figur 5 Whn comparing th mtrological proprtis of th two tchniqus, w find som procsss which occur simultanously, and thir ffcts can b sn in Figs. 5. For tchniqu BPPE, whn th valu of th optic absorption cofficint β incrass, th intnsity of th lasr bam that impacts th pyrolctric diminishs, thrby causing th gnratd lctric signal to b smallr. Thus, th valus of th drivativ modulus and cofficint ar diminishd. Convrsly, for FPPE th lctric signal gnratd in th pyrolctric dosn't dpnd on th optic proprtis of th sampl, and for this rason th valu of cofficint isn t affctd by th β chang. Whn th sampl is practically transparnt (β<4 m - 1 ), at all frquncis th valus of for tchniqu FPPE ar considrably lowr than thos for tchniqu BPPE. In addition, at th highst frquncy, max is smallr in tchniqu FPPE than min in tchniqu BPPE. Howvr, with a frquncy incras from Hz to Hz, th valu of for tchniqu BPPE is gradually diminishd by 18.5%, whil th valu of cofficint for tchniqu FPPE is diminishd by 6.7%. uch a drastic diffrnc in th bhavior of both tchniqus lis in th principl of its opration. For tchniqu BPPE, all light that rachs th black surfac of th pyrolctric is transformd into hat, which thn simultanously hats th pyrolctric and th sampl. If th frquncy incrass, th distribution of hat btwn both matrials changs, and this chang is vry slight. For tchniqu FPPE, howvr, th lasr bam only hats th pyrolctric and gnrats a thrmal wav insid. To rach th sampl, this wav must cross th pyrolctric. During its trajctory toward th sampl, th thrmal wav diminishs its amplitud by =,71 tims 8

9 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN whn travling distanc is qual to th pntration dpth μ. Whn frquncy f incrass, th valu of μ diminishs. Thus, th thrmal wav amplitud diminishs rapidly and only a small part ntrs insid th sampl. Consquntly, th signal gnratd in th pyrolctric dpnds littl on th thrmal proprtis of th sampl and th snsitivity of this tchniqu is low, rndring th valu of as low. For tchniqu BPPE, dpndnc in function of th sampl thicknss can b xplaind with th hlp of th Br law I = I xp( βl ), whr I is th intnsity of m th lasr bam that impacts th sampl surfac, and I is th light intnsity absorbd on th pyrolctric surfac. Whn thicknss l m incrass, th prcntag of absorbd light lost during th sampl hating procss incrass as wll. As a rsult, it dos not rach th pyrolctric. Th pyrolctric thn bcoms coolr, th gnratd lctric signal is smallr and th valu of is smallr. Th hat dissipatd in th sampl dosn't contribut significantly to th altrnating hating of th pyrolctric (synchronizd with frquncy f) bcaus th sampl is thrmally vry thick and th thrmal wav is absorbd insid th sampl until it is almost zro. Conclusions W hav dmonstratd proposd paramtr as a vry usful and innovativ tool to compar th bhavior of two important pyrolctric tchniqus (FPPE and BPPE). Th main advantag of paramtr lis in incorporating th principl mtrological proprtis of th pyrolctric tchniqus into a singl valu. Thus, th comparison btwn th two tchniqus is rducd to a simpl algbraic opration to rlativly locat th valus for ach tchniqus on an axis. This algorithm can also b applid to othr photothrmal tchniqus if th natur of masurmnt uncrtaintis is known. Paramtr can also b usd to optimiz th main paramtrs of th xprimnt. 9

10 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN Rfrncs A. Mandlis, Ptr Hss, Progrss in Photothrmal and Photoacoustic cinc and Tchnology, v.iii: Lif and Hart cincs, (PIE Optical Enginring Prss, Washington, 1997), pp A. Matvinko, A. Mandlis, Rv. ci. Instum. (U) 76:1 (5), pp M. Chirtoc, I. Chirtoc,. Pittois, C. Gloriux, J. Thon, Rv. ci. Instum. (U) 74:1 (3), pp C. Wang, A. Mandlis, Rv. ci. Instum. (U) 71:5 (), pp Mandlis, M.M. Zvr, J. Appl. Phys. (U),57:9 (1985), pp A. Cruz Ora, E.H. Bntour, P. Jam, M. Chirtoc, c. Gloriux, G. Pitsi, J. Thon, Rv. ci. Instum. (U) 74:1 (3), pp Manual, Amplifir Lock-in R83/85, < J. Mandl, Th tatistical Analysis of Exprimntal Data, (Dovr Publications, Inc., N.Y,1984), p Figur captions Figur1. Rlativ intrval location for valus of paramtr for both pyrolctric tchniqus (1 and ) Figur. Paramtr vs. optical absorption cofficint β for photopyrolctric tchniqus FPPE and BPPE; frquncy f=hz; thicknss of sampls l m =mm; 4mm; 6mm; 8mm; 1mm. Figur 3. Paramtr vs. optical absorption cofficint β for photopyrolctric tchniqus FPPE and BPPE; frquncy f=5hz; thicknss of sampls l m =mm; 4mm; 6mm; 8mm; 1mm. Figur 4. Paramtr vs. optical absorption cofficint β for photopyrolctric tchniqus FPPE and BPPE; frquncy f=1hz; thicknss of sampls l m =mm; 4mm; 6mm; 8mm; 1mm. 1

11 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN Figur 5. Paramtr vs. optical absorption cofficint β for photopyrolctric tchniqus FPPE and BPPE; frquncy f=hz; thicknss of sampls l m =mm; 4mm; 6mm; 8mm; 1mm. Figur 1 Figur 1 BPPE Paramtr - sampl thicknss l m (mm) Watr with ink, Frquncy f= Hz 8 6 max FPPE BPPE min FPPE β, m -1 11

12 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN Figur max (BPPE) Paramtr - sampl thicknss l m (mm) Watr with ink, Frquncy f=5 Hz max (FPPE) 4 min (BPPE) min (FPPE) β, m -1 Figur 4 Paramtr - sampl thicknss l m (mm) Watr with ink, Frquncy f=1 Hz 8 max (BPPE) max (FPPE) min (BPPE) min (FPPE) β, m -1 1

13 Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN Figur Paramtr - sampl thicknss l m (mm) Watr with ink, Frquncy f= Hz 8 7 max (BPPE) min (BPPE) 1 max (FPPE) min (FPPE) β, m -1 13

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