MODELLING AND SIMULATION OF LASER CHEMICAL PROCESSING (LCP) FOR THE MANUFACTURING OF SILICON SOLAR CELLS DISSERTATION

Transcription

1 MODELLING AND SIMULATION OF LASER CHEMICAL PROCESSING (LCP) FOR THE MANUFACTURING OF SILICON SOLAR CELLS DISSERTATION zur Erlagug des aademsche Grades des Dotors der Naturwsseschafte (Dr. rer. at.) a der Uverstät Kostaz Fachberech Phys vorgelegt vo Adreas Fell Frauhofer Isttut für Solare Eergesysteme Freburg m Bresgau 00

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3 Table of cotets Table of cotets... Itroducto.... Motvato.... LCP status of research Lterature revew Ams ad scopes...7 Mathematcal modellg...9. Overvew of physcal effects LCP...9. Optcs..... Complex refracto dex of slco..... Absorpto Reflecto Thermodyamcs Heat trasport Phase chage sold lqud Evaporato Speces trasport Flud dyamcs Basc flud flow Multphase flow Meltg ad soldfcato Chemstry Photochemcal radcal geerato Reactos Fudametals of the smulato methods PDE solvg by fte dffereces...3

4 TABLE OF CONTENTS 3.. Spatal dscretzato Tme tegrato Alteratg drecto explct (ADE) method Fluet Solver bascs User defed code UDS ad UDF Multphase flow Implemetato Heatg ad radcal geerato the lqud jet LCPSm: optcs ad thermodyamcs at the reacto spot Solvg of the heat trasport equato Implemetato of the ADE method Treatmet of the surface geometry Itesty dstrbuto, surface reflectos ad raytracg Sem aalytcal heat sources Surface recesso due to evaporato Dffuso of mpurty atoms slco melt Multlayer mplemetato Grd geerato ad adaptato User settgs ad program structure Verfcato Adapto of Fluet Separate temperature felds Meltg / soldfcato ad desty chage Free surface heat trasfer Verfcato Couplg of LCPSm ad Fluet Couplg algorthm... 88

5 TABLE OF CONTENTS 4.4. Verfcato Smulato results Heatg ad radcal geerato the lqud jet Ifluece of basc parameters slco laser processg Dry laser ablato Slco Slco trde layer Ablato of slco by LCP Evaporato Melt expulso by the lqud jet Coolg effect Dry laser dopg LCP dopg Smulatos wth flat top profle Ifluece of homogeeous testy profle Summary... 7 Deutschsprachge Zusammefassug...7 A Program code ad settgs...3 A. LCPSm: start.m...3 A. LCPSm: grdadapt_d.m...35 A.3 LCPSm: ade_d.m...39 A.4 LCPSm: sms.m...4 A.5 Fluet: Fluet C-code for UDFs...43 A.6 Fluet: solver settgs...47 B Lst of symbols ad abbrevatos...49 C Publcatos...56 Refereces...58 Dasagug...6

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7 Itroducto. Motvato A major part of the curret threats to the huma cvlsato orgates from the world eergy system, whch s based maly o fossl fuels. O the oe had, the lmted amout of fossl ad uclear resources wll lead to a dramatc crease of eergy cost the md term, the effects of whch are already clearly vsble, whch wll have a uforeseeably great mpact o world poverty ad socal stablty. O the other had, the ar polluto by burg fossl fuels affects serously the world s flora ad faua, maly due to clmate chage by the greehouse effect. The greehouse gases emtted up to ow are already hgh eough to possbly heat up the world s average temperature by over degrees, whch s treated as a crtcal value for ohealable clmate chage [] wth strogly creased weather extremes, dessert spreadg, sealevel rse ad so o. Therefore oe of the bggest tass for mad s to strogly reduce ad fally completely elmate the amout of fossls ad uclear power as prmary eergy a very fast way. Besdes eergy savgs the hghest effort has to be made to develop cost effcet reewable eergy usage, because oly the s a rapd worldwde maret troducto possble. The example of Germay showed that wth a sutable poltcal frameset, amely the EEG [], extremely hgh growth rates of stalled reewable power capactes ca be acheved. Ths strog maret growth at the same tme creases research ad developmet efforts, whch leads to a decreasg cost of the stallatos. For wd power, grd party has already bee reached, eve moderately wdy regos. Ths s ow the reaso for ts worldwde rapd maret growth eve wthout hgh poltcal support ad there wll be a sgfcat cotrbuto to world eergy producto the ear future. Aother promsg reewable eergy source s photovoltacs. The potetal of usable solar radato o earth s much greater tha world eergy cosumpto. Also, photovoltacs have show rapd growth rates of 50 % sce 003 [3], as show Fg. -. Curretly, grd party has oly bee reached very suy regos. Therefore poltcal support s stll eeded to esure further cost reductos. I addto to cost reductos by large scale maufacturg, a mportat ey s to troduce ew techologes to crease the lght coverso effcecy of a solar cell, whch drectly mproves the cost per Watt Pea rato of the fal power plat.

8 INTRODUCTION Fg. -: World PV module producto from 990 to 007 tae from [3] Ths wor was doe the feld of crystalle slco solar cells, whch have the bggest maret share, because of techology trasfer from semcoductor dustry ad the hgher effcecy compared to th flm techologes. It s qute well ow, how to maufacture a hghly effcet crystalle slco solar cell, for example the PERC cell [4] or the baccotact cell [5]. But a great deal of ecessary maufacturg steps, maly cosstg of photolthography, are too cost tesve for a dustral applcato. Thus world research actvtes ths feld curretly focus o the developmet of ew cheap processes to troduce hgh effcecy cell structures dustry. Wth ths scope lasers offer a varety of ew promsg applcatos. They wor masless, cotact-free ad are able to process arbtrary geometres wth very hgh speed. A specal laser applcato, called laser chemcal processg (LCP), s curretly beg developed at Frauhofer ISE. Here a laser ad a lqud jet cotag sutable chemcals are combed to trgger dfferet thermochemcal processes at the reacto spot. Wth ths techque, t s for example possble to create local phosphorous dopg udereath the frot metal cotact fgers just oe process step, whch ca sgfcatly crease the effcecy of the solar cell. Ths wor cotrbutes to the developmet of LCP by gag a uderstadg of the process physcs. The use of smulatos eables a better terpretato of expermetal results leadg to more effcet selecto of parameters ad optmzato.

9 INTRODUCTION 3. LCP status of research laser beam wdow focusg les pressure chamber ozzle hole lqud jet processed rego reacto spot Fg. -: Setch of the LCP prcple LCP s based o the LaserMcroJet (LMJ) techology by Syova S.A. ad was developed by Rcherzhage 994 [6, 7]. A har th jet s geerated by pumpg water through a ty sharp edged ozzle. Applyg the correct pressures o the order of 00 bar, a costrcted jet s formed, whch s stable up to 000 tmes the ozzle dameter ad ca ths way act as a optcal wavegude. LMJ s maly used for the cuttg of dfferet materals wth pulsed or cotuous wave (CW) hgh power lasers. The laser heats, melts ad evaporates the wor pece ad the hgh velocty lqud jet carres away the ablated materal. LCP was troduced by Wllee ad Kray [8, 9] by replacg water wth solvets cotag sutable chemcals, whch eables a varety of combed thermal, hydrodyamc ad chemcal processes at the reacto spot. Sce 003 the Frauhofer Isttute for Solar Eergy Systems (ISE) s the oly sttuto developg dfferet LCP applcatos for the maufacturg of crystalle slco solar cells. Possble applcatos rage from mcrostructurg, for example the opeg of passvato layers or edge solato, to waferg, whch ams to cut wafers from a got, to local dopg ad metallzato, whch eables selectve emtter ad local bac surface feld (LBSF) formato. Two examples, the waferg ad the local dopg, wll be dscussed the followg. Waferg Waferg s the process of cuttg wafers out of a crystalle slco got. Ths s usually doe by a mult wre slurry saw (MWSS), where thousads of parallel wres abrasvely cut a

10 4 INTRODUCTION got. The MWSS s able to produce wafers wth a thcess dow to 50 µm, havg a erf loss of at least the same amout. The surface of the wafers s hghly damaged ad cotamated especally by metals from the wre. Therefore wet chemcal etchg ad cleag steps are eeded afterwards whch further creases the overall loss of slco. Ths loss s very crtcal cosderg the hgh prce of crystalle slco. Because LCP the laser s focused over the whole stable jet legth the cetmetre rage, very deep grooves wth hgh aspect ratos ca be processed. Ths maes t sutable for a waferg applcato. Advatages regard to the MWSS are the potetal of smaller erfs ad a better surface qualty, whch overall reduces the loss of slco sgfcatly ad maes addtoal etchg ad cleag uecessary. LCP waferg also has the potetal of slco recyclg by usg sutable reactats, amely chlore, whch would further reduce the loss of slco. The research ths feld s focused o searchg for sutable laser parameters ad chemcal addtves to acheve deep cuts wth good surface qualty. Progress has bee publshed by Hopma, Mayer ad Fell [0, ]. The ablty of 7 cm cuttg depth has bee prove ad the ablato effcecy ad ablato mechasm of dfferet laser systems has bee dscussed detal. Chlore s show to have a very hgh etch rate o molte ad evaporated slco but shows o etchg o sold slco, whch maes t sutable to support the thermal ablato. More recetly, Mayer [] proved a postve fluece o cut qualty ad groove shape by addg chlore to Fluorert FC-770 as a solvet, see Fg. -3. Waferg wth LCP s stll the stage of fudametal research, therefore creasg uderstadg of the process s ecessary for further developmet. For ths the smulato results of the preset wor cotrbute to the uderstadg of dfferet ablato mechasms le evaporato ad melt expulso. lttle Cl much Cl Fg. -3: Ifluece of chlore the lqud jet o the groove shape after LCP le scas; left: low chlore cocetrato; rght: Hgh chlore cocetrato []

11 INTRODUCTION 5 Dopg Dopg wth LCP s already close to a dustral applcato. Most relevat s the selectve emtter formato o p-type solar cells. For hgh effcecy, a lowly doped emtter wth a hgh blue respose s eeded. But a hgh surface dopg s eeded to esure a good cotact resstace to the metal cotacts. The selectve emtter combes these two ssues by havg a lowly doped emtter ad local hgh dopg udereath the frot metal cotact fgers. The hgh dopg also reduces recombato losses the cotact area, whch meas eutralzato of electros ad holes. Therefore the hgh dopg creases the overall effcecy sgfcatly ad cheap processes for the maufacturg of selectve emtters are of great dustral terest. frot metal cotact passvato ad atreflecto layer -doped emtter local hgh dopg p doped base local hgh p dopg passvato layer rear metal cotact Fg. -4: Hgh effcecy p-type slco solar cell structure (PERC) wth selectve emtter ad local bac surface feld (LBSF) Usually selectve emtters are produced by photolthography wth a lot of masg ad etchg steps, whch s much too cost tesve for a dustral applcato. Wth LCP, a techque s proposed that ca acheve local hgh dopg a fast sgle step. For -dopg, phosphorc acd s used as a lqud medum whch acts as a phosphorus dopat source. The laser power melts the slco at the reacto spot, the phosphorus s thermally atomzed ad t the dffuses to the slco melt. Because the phosphorus dffuso coeffcet the slco melt s several orders of magtude hgher tha sold slco, precse local dopg very short melt tmes s possble. Pulsed lasers wth pulse duratos dow to a few aosecods are suffcet to create approprate dopg. Advatages to dry laser dopg processes are the early fte dopg source whch does ot have to be placed a extra step ad eables groovg ad dopg at the same tme. A dsadvatage s that melt flow by the lqud jet ca

12 6 INTRODUCTION create a bad crystal qualty, whereas dry laser dopg wth low power, the melt resoldfcates wthout much movemet. Solar cell results wth LCP dopg have bee publshed [3] ad [4], where t s show that wth low power aosecod laser pulses a proper selectve emtter s produced. For creasg laser power, solar cell effcecy drops rapdly above a fluece threshold of.5 J cm as show Fg. -5. Also, for loger pulse duratos o acceptable solar effceces ca be reached. Here smulatos ca help to expla these pheomea by havg a close loo at the physcal effects tag place durg LCP dopg solar cell effcecy [%] µm ozzle, 35 Hz 60 µm ozzle, 80 Hz 80 µm ozzle, 35 Hz fluece [J/cm²] Fg. -5: Solar cell effceces wth selectve emtter produced wth LCP for dfferet parameters [4] I addto to the selectve emtter, local dopg ca also be appled to create a local bac surface feld (LBSF). As o the frot sde, hgh dopg udereath the metal cotacts decreases cotact resstace ad recombato losses. If the bac sde of the solar cell s cotacted oly locally, a approprate passvato ad reflecto layer ca be troduced, whch reduces bac sde recombato losses ad creases effectve absorpto for log wavelegths. For a stadard p-type solar cell, p-type dopg s eeded for creatg a LBSF. Boro dopg s ow to show the best performace, but because of low dffuso coeffcets, the commoly used furace dffuso taes very log ad s therefore ot dustrally feasble. Wth LCP usg a boro-cotag lqud, fast local p-dopg wth a passvato layer opeg ca be acheved a sgle step. Frst results have prove the ablty of LCP to create local boro dopg [5].

13 INTRODUCTION 7.3 Lterature revew For modellg ad smulato of materal processg usg a lqud jet guded laser the oly wor foud lterature was publshed by L et al. [6]. They preseted a model for laser heatg, meltg, surface coolg ad ablato of slco by the water jet. Rather tha solvg flud flow equatos for melt expulso, they assume that the melt s completely ad stataeously removed. The basc thermal effects of LCP also occur dry laser processg. Therefore a overvew of publshed research ths feld s gve. Because LCP laser pulse duratos have to be greater tha roughly oe aosecod, the reaso for ths s dscussed later ths wor, modellg of shorter laser pulses s excluded ths overvew. The smulato of laser materal teracto started the early eghtes for semcoductor applcatos. Wood, Grgoropoulus et al. preseted models for laser heatg ad meltg of slco combed wth dopat dffuso, see [7-]. The oe dmesoal models were solved by fte dfferece methods. Later addtoal effects le evaporato, gas dyamcs ad plasma yeldg were corporated maly for the applcato of alumum sputterg. Here several publcatos wth fte dffereces methods exst for oe dmesoal models [-5], ad for two dmesoal models wth axal symmetry [6]. Re et al. [7] preseted smlar modellg appled to slco mcromachg. Oe step further was tae the modellg of laser weldg. For ths applcato mostly log pulse or cotuous wave lasers are used, whch cause hgh melt flow. Thus, free surface flow of the melt s corporated here cotrast to the publcatos lsted above. A good overvew of ths feld s gve by Macwood ad Craver [8]. For example, Mazumder et al. developed a fully three dmesoal smulato model for all relevat effects laser eyhole weldg [9]. Mazumder et al. also preseted a good revew of modellg of laser processg up to 996 [30]. For the smulato of melt flow caused by a lqud jet o lterature was foud ad s assumed to be doe for the frst tme the preset wor..4 Ams ad scopes Up to the begg of ths wor, developmet of LCP was maly doe by tral ad error. LCP ows a bg amout of parameters, because of the laser lqud jet couplg. Laser parameters le wavelegth, pulse durato, power ad repetto rate are combed wth the jet parameters pump pressure, ozzle dameter ad flud propertes, ad other process parameters such as worg dstace ad pulse overlap. The quatty already shows that tral ad error s ot a approprate way to search for sutable parameters. Therefore a fudametal uderstadg of LCP physcs s ecessary for effcet parameter locato ad optmzato.

14 8 INTRODUCTION Loog carefully at expermetal results to ga a uderstadg of the process qucly leads to the sght that the quatty ad complexty of the physcal effects s much too great to get coclusve ad correct terpretatos. Also, aalytcal estmatos are oly applcable wth strog smplfcatos, ad eglect the hgh degree of couplg of the dfferet physcal effects, le the teracto betwee thermal meltg ad flud flow. Ths sght was the motvato to start umercal smulato of LCP. Multphyscs smulatos gve the uque possblty to observe arbtrary quattes durg the process at arbtrary pots tme. The preset wor ams to descrbe the coupled physcal effects ot a exact quattatve, but more qualtatve ad effcet way. Ths meas bascally loog at the teracto of dfferet physcal effects, detfyg domatg effects ad observg the fluece of process parameters. Although some effects ca be descrbed precsely, especally because slco s a well ow materal, at very hgh temperatures the appled models are ot fully vald ad the materal propertes ca oly be estmated. Therefore usg smulatos for accurate quattatve forecasts s too challegg at least for moderate to hgh laser power. The frst step for buldg smulato code s to choose proper mathematcal models for the physcal effects. Solvg the resultg equatos s ot trval, because of the hgh degree of teracto ad the extreme scales of the correspodg physcal quattes. For example laser lght absorpto ca occur o a scale of a few aometres, resultg very hgh temperature gradets at the free surface of the melt. Curretly o commercal smulato software s ow to solve all of the relevat effects smultaeously. Therefore self programmed code s ecessary to mplemet adopted solvers ad eable traset couplg of the equatos. However, the commercal software Fluet s used for smulato of flud flow. Basc programmg of a sutable flud flow solver would have bee too extesve for the tmeframe of ths wor, whereas the use of Fluet eables a fast ad effcet problem setup. LCP ca be see as a exteso of dry laser processg, because the basc effects of dry laser processg tae place LCP rather the same way. Ths meas that smulato of LCP ca beeft from prevous wor doe the feld of dry laser smulato, as dscussed above, ad that the smulato code geerated ths wor ca wdely be appled to dry laser processg. The thess cossts of four ma chapters followg ths troducto. I the ext oe the theory of the physcal effects tag place LCP s descrbed ad the correspodg mathematcal models are show. The thrd chapter deals wth the fudametals of the umercal methods used ths wor. A detaled descrpto of the mplemetato of the mathematcal models to program code s gve chapter four. Fally smulato results are show for comparso wth expermets ad for terpretato of LCP results wth respect to solar cell maufacturg.

15 Mathematcal modellg Ths chapter deals wth the mathematcal descrpto of the physcal effects LCP. The effects are summarzed optcs, thermodyamcs, flud dyamcs ad chemstry. Sutable models are tae from lterature ad are appled to the slco case. The materal propertes of slco used the smulatos are also gve wth the descrpto of the correspodg models.. Overvew of physcal effects LCP lqud laser couplg ozzle body total reflecto, absorpto heatg, photochemcal reactos reflecto slco absorpto heatg Fg. -: Effects at the begg of the laser pulse before the oset of meltg ad evaporato 9

16 0 MATHEMATICAL MODELLING vapour / plasma plume Etchg of vapour, plasma sheldg, thermal decomposto evaporato, recol pressure heated zoe melt Fg. -: Effects durg the laser pulse after the oset of evaporato decomposto ad dffuso of mpurty atoms, melt etchg, surface coolg lqud jet drve melt flow heat coducto soldfcato Fg. -3: Effects after the laser pulse ad after reattachg of the lqud to the slco I Fg. - to Fg. -3 the mportat effects occurrg LCP durg oe typcal laser pulse are llustrated. Frst of all the lqud jet s geerated by a sharp edged ozzle ad shows a typcal dameter of 0.83 tmes the ozzle opeg dameter [], cofrmed by Fluet smulatos performed for the preset wor. A laser beam s focused o the ozzle opeg plae ad s guded wth the lqud jet va total reflectos. The laser lght already teracts wth the jet, ether by absorpto heatg or by photochemcal reactos. Ths could be potetally used for geeratg radcals, whch are geerally much more reactve. The jet acts as a optcal multmode wavegude. Therefore the testy dstrbuto the cross secto s ot flat, but shows several terferece peas depedg o ozzle dameter ad couplg optcs [3]. Reachg the surface, the laser lght s partly reflected ad the remag testy s absorbed by the slco ad acts ths way as a volumetrc heat source. Oce the meltg temperature s reached, a phase chage to lqud taes place. Upo further heatg, the slco starts to evaporate. For fast evaporato, a dese vapour plume s produced, whch exerts a

17 MATHEMATICAL MODELLING recol pressure to the melt ad possbly shelds the lqud jet from the slco melt. For eve hgher power, the dese vapour s ozed to a plasma phase, whch shelds sgfcatly the laser power. After the laser pulse, the vapour partly codeses ad s carred away by the lqud jet. Because the melt tme s much loger tha the pulse duratos typcally used LCP, the lqud jet attaches aga to the melt after the laser pulse ad starts expellg t by the lqud jet pressure ad vscous drag. At the terface to the slco melt also evaporato of the lqud taes place resultg a small vapour flm. Ths flm reduces the vscous forces betwee the slco melt ad the lqud ad acts as a thermal sulato layer. Chemcal reactos ca tae place the lqud jet as well as at the reacto spot. I the case of slco etchg by chlore the hghest etch rates are assumed for slco vapour, ad stll sgfcat etchg o slco melt, whereas the etchg of heated sold slco s eglgble [0]. The preset wor deals maly wth low to md laser testes ot much above the meltg threshold, so plasma effects do ot have to be cosdered. I addto, the expaso of the vapour plume ad the resultg recol pressure ad codesato effects are ot yet cosdered. Ths s reasoable at least for farly log pulse duratos, because the lqud jet pressure s exerted much loger o the melt tha the recol pressure ad therefore domates the movemet of the melt. For shorter pulse duratos the low aosecod scale, the laser testy ad therefore the evaporato velocty ad the recol pressure s very hgh, whereas the melt tme s too short for sgfcat expulso by the lqud jet. Ths meas that for hgher power aosecod laser pulses the vapour phase has to be cosdered for a correct modellg of the ablato mechasm.. Optcs The optcal effects ca be categorzed to absorpto ad reflecto effects. They are modelled by geometrcal optcs, because the laser lght ca be well approxmated by drectoal beams. The basc optcal materal property s the complex refracto dex *, where the magary part s called the extcto coeffcet. * = ( - ) Measuremets of the refracto dex for sold slco ca be foud lterature. For lqud slco oly very few data exsts ad therefore a theoretcal model, the Drude model, s used to calculate the optcal propertes. The wavelegths vestgated ths wor are 064 m, 53 m ad 355 m, because these correspod to the Nd:YAG laser wavelegths used expermets.

18 MATHEMATICAL MODELLING.. Complex refracto dex of slco For sold slco, measured data for the optcal propertes has bee tae from lterature [3-34] ad have bee ftted to the expressos show Tab. -. λ [m] [33] [3, 34] e T[ K] 4.7e T[ K ] e T T[ K].5e exp 430K e T[ K].9 Tab. -: Sold slco refracto dex calculated from lterature data For lqud slco o sutable measuremets for the optcal propertes were foud, therefore the Drude model s used as descrbed [35]. Lqud slco ca electrocally be treated as a metal, because the same characterstc free electro gas exsts. The optcal propertes are strogly depedet o the electrcal oes, because comg photos are maly absorbed by the free carrers. Drude gave a expresso for the frequecy depedet complex permttvty of a free electro gas. * ε ω pl γ ω pl f = ( - ) ω γ ω ( ) ( ω γ ) The plasma frequecy ω pl s a materal specfc costat value ad the collso rate γ s a temperature depedet value as proposed [36]. π c ω = λ T γ = γ m T m ( -3 ) Ths results a permttvty depedet o temperature ad frequecy. The requred optcal propertes are related to the permttvty by the complex equato ( -4 ). * * ( ) = ε ( -4 )

19 MATHEMATICAL MODELLING 3 For the calculato of ad by the Drude model data for lqud slco s tae from [37] 6 5 to ω =.50e s ad λ = 4.770e s. p.. Absorpto m Laser lght absorpto occurs due to exctato of atoms or molecules by the photo eergy. For the laser wavelegths used, photos teract most domatly wth electros. Ths meas that oe photo gves all ts eergy to oe electro. For o-free electros ot all eergy levels are allowed, therefore the absorpto coeffcet s strogly depedet o the electroc structure of the materal ad the wavelegth. For example for the semcoductor slco wavelegths ear or above the bad gap are wealy absorbed. Icreasg temperature crease also the amout of free electros, therefore the absorpto coeffcet s strogly creased, whch ca be see [3]. For hgh laser testes,.e. for hgh photo destes, the probablty that more tha oe photo hts a electro s creased. Ths s called multphoto absorpto. The absorpto coeffcet s the depedet o the laser testy. Further crease of the testy leads to more free electros, whch absorb more laser lght, leadg to a further ozato. At a certa testy level ths eds up avalache ozato, called optcal breadow, where suddely all of the lght s absorbed. It s obvous that for LCP the optcal breadow has to be avoded the lqud jet. Therefore oly laser testes ca be used LCP, whch are below the optcal breadow threshold. For the wavelegths 064 m ad 53 m, ths threshold water s at roughly 00GW cm [38]. For a 50 µm jet dameter ad a typcal pulse eergy of mj ths would mea a crtcal pulse durato of roughly oe aosecod. Geerally spoe for LCP applcatos the laser pulse duratos have to be at least the aosecod scale. Because of ths restrcto, multphoto absorpto the slco ca also be eglected. It shows eglbble absorpto coeffcets for the allowed testes, whch ca be derved from [39]. The basc quatty for modellg absorpto effects s the absorpto coeffcet α, whch ca be calculated out of extcto coeffcet ad the wavelegth of the lght. 4π α = ( -5 ) λ Laser lght ca be see as udrectoal lght. Eve after crossg the surface ad dffusg the lght ths approxmato s feasble, because the absorpto legth s much smaller tha the typcal lateral wdth of the spatal testy dstrbuto. Therefore the absorpto equato ca be expressed oe dmeso.

20 4 MATHEMATICAL MODELLING di dz = α I ( -6 ) If the absorpto coeffcet s costat over the z -coordate, equato ( -6 ) ca be solved aalytcally. The soluto show equato ( -7 ) s called the Lambert-Beer law. I ( z) = I ( α z) exp = 0 z ( -7 ) I Fg. -4 the temperature depedet absorpto coeffcet of sold ad lqud slco at varous wavelegths s show absorpto coeffcet [cm ] m 53 m 064 m temperature [K] Fg. -4: Absorpto coeffcet of slco for dfferet wavelegths as used the smulatos

21 MATHEMATICAL MODELLING 5 les ϕ beam ozzle opeg z Fg. -5: Path elargemet by total reflecto the lqud jet I the lqud jet aother effect has to be cosdered. Because the laser lght s coupled uder a agle determed by the couplg optcs, t partly travels ot drectly through the lqud jet as llustrated Fg. -5. Ths results a optcal path elargemet depedg o the agle betwee the actual lght beam ad the z - coordate ϕ beam. The relatve path elargemet δ s calculated by equato ( -8 ). δ = ( -8 ) cos ϕ beam The maxmum agle allowed s determed by the crtcal agle to esure total reflecto. For water as surroudg medum the maxmum relatve path elargemet s δ water, max = I the smulatos the path elargemet s mplemeted by a effectve absorpto coeffcet. α = δ α eff ( -9 )..3 Reflecto Reflecto o the surface plays a mportat role modellg of LCP, because t determes how much eergy s coupled to the materal. The amout of the o reflected laser lght testy s defed by the reflectvty R.

22 6 MATHEMATICAL MODELLING surf ( ) I c I = R ( -0 ) A geeral beam reflecto at a terface betwee materal ad s llustrated Fg. -6. For a geeral agular reflecto, the lght beam has to be dstgushed a vertcally ad horzotally polarzed part. The regardg reflectvty s the calculated by the Fresel formulas ( - ) ad ( - ) [40]. ϕ c b refl * b c * ϕ refr Fg. -6: Beam reflecto at a flat terface betwee materal ad materal * * ( ϕc ) cos( ϕ refr ) * ( ϕ ) cos( ϕ ) * cos R = ( - ) cos c refr R * * ( ϕc ) cos( ϕ refr ) * ( ϕ ) cos( ϕ ) * cos = ( - ) cos c refr The agle of refracto ϕ refr s defed by the refracto dces ad the cdet agle ϕ c. * ϕ = refr s sϕ ( -3 ) * c For a arbtrary cdet beam drecto ad a arbtrary surface ormal drecto the drecto of the reflected beam ca be calculated by the vector equato ( -4 ).

23 MATHEMATICAL MODELLING 7 b refl = bc ( bc ) ( -4 ) I ths case the cdet agle s defed by the scalar product of surface ormal ad cdet beam drecto. ( ) ϕ = cos ( -5 ) c bc I may cases the cdet laser beam ca be treated as perpedcular to the surface,.e. whe the ablated geometry shows o sgfcat steepess. Furthermore, the surroudg medum s our case trasparet, whch meas that the extcto coeffcet of the surroudg medum ca be set to zero. The geeral Fresel formulas the smplfy to the well ow reflectvty equato ( -6 ) depedet of polarzato. ( ) ( ) R = ( -6 ) The resultg perepedcular reflectvty for slco s plotted Fg. -7 for ar ad water as surroudg medum m 53 m 064 m 0.6 reflectvty temperature [K] Fg. -7: Reflectvty of slco for ar (sold le) ad water (dashed le) as surroudg medum

24 8 MATHEMATICAL MODELLING A specal treatmet s eeded for reflecto at small flms, le passvato layers o the frot sde of a slco solar cell. Here mult-reflectos betwee the materal terfaces tae place, whch ca be calculated for perpedcular cdet lght ad a o-absorptve flm materal to a effectve reflectvty [4]. R f * r, f r = * r, f r * f, * f, exp exp ( δ f ) ( δ ) f ( -7 ) r *, f r * f, = f f * f = * f ( -8 ) Here δ f s the optcal path dfferece depedg o wavelegth, flm refracto dex ad flm thcess d f. δ f 4π f d f = ( -9 ) λ f r *, f * r f, d f * Fg. -8: Setch of multple reflectos at th flms.

25 MATHEMATICAL MODELLING 9.3 Thermodyamcs The modellg of thermodyamc effects LCP corporates heat trasport, phase chages ad speces trasport due to evaporato or dffuso..3. Heat trasport As descrbed before secto.. the laser power s domatly absorbed by the electros. Before a uform temperature for the electros ad the lattce s reached, the excted electros have to trasfer eergy to the lattce. Ths relaxato happes o a pcosecod tmescale [35], whch s much shorter tha the used laser pulse duratos of at least a few aosecods. So for smulato of LCP oly oe equal electro-lattce temperature feld eeds to be cosdered. The geeral partal dfferetal equato for traset heat trasport ( -0 ) cossts of a traset, a covecto, a coducto ad a source term. The covecto term taes to accout trasport of heat due to materal movemet le the melt flow. The source term correspods to the absorbed laser power, see equato ( - ). d dt ( c p T ) v ( ρ c p T ) = ( K T ) S h ρ ( -0 ) S h = α I ( - ) For the descrpto of heat trasport the lqud jet the spatal dstrbuto of the temperature the cross secto area s approxmated to be costat, so equato ( -0 ) ca be reduced to the jet drecto,.e. the z -coordate. Ths s feasble because of the hgh legth to wdth rato of the lqud jet o the order of a thousad, whch leads to relatvely hgh lateral heat coducto ad temperature homogezato. Also the testy profle s varyg wth jet legth, whch results lateral homogezato by averagg. Furthermore, the velocty profle of the lqud jet s approxmately flat [4] ad so the velocty s oly depedet o the z -coordate. The heat capacty c p, the desty ρ ad the heat coductvty K are assumed to be costat, so they ca be placed frot of the dervatves resultg equato ( - ). dt dt d T ρ c p ρ c p vz = K S ( - ) h dt dz dz The heat trasport slco s descrbed by the ethalpy-based heat trasport equato ( -3 ), where the ethalpy s the er eergy desty uts of eergy per volume. Ths formulato s used as proposed, for example, by Grgoropoulus et al. []. Durg phase

26 0 MATHEMATICAL MODELLING chages the temperature stays costat whereas the ethalpy s steadly chagg, whch s beefcal for mplemetato of a stable umercal tme tegrato. Furthermore, for the mplemetato of the heat trasport equato slco, o movemet of slco,.e. melt flow, s cosdered, so the covecto term ca be removed. dh dt ( K T ) S h = ( -3 ) I equato ( -3 ) the temperature eeds to be calculated from the ethalpy values. Ths depedecy s defed by the tegral ( -4 ) for arbtrary temperature depedet thermal propertes. ( T ) H = T ρ c dt ( -4 ) p 0 Due to phase chages, ths tegral s defed oly stepwse. For sold slco the temperature depedet heat capacty ad desty s tae from [43]. The data has bee extrapolated to show a zero ethalpy value at the absolute zero pot. Ths depedecy s ot physcally correct, but does ot affect the calculato of the temperature feld above room temperature. Addtoally, ths formulato shows a better umercal covergece tha settg the ethalpy to zero at room temperature. A ft was performed to acheve a secod order polyomal for the temperature ethalpy relato of sold slco wth 7 6 =. e K m J, c c = 5. 5e K m J ad c3 = K. A plot of the relato ( -5 ) s show Fg. -0. T s = c ( -5 ) H c H c3 Aother heat trasport effect s the surface coolg, whch ca occur by radato ad combed coductve ad covectve coolg. At the hot slco surface, the radato heat loss s determed by the Stefa-Boltzma law eglectg the comg radato ( -6 ). J surf T 4 4 ( T T ) = ε σ ( -6 ) surf ev For coductve coolg or combed coductve ad covectve coolg, the surface heat trasfer s approxmately learly depedet o the temperature dfferece of the two eghbourg materals. Ths ca descrbed by the heat trasfer coeffcet ht.

27 MATHEMATICAL MODELLING J surf ( T T ) = ht ( -7 ) surf ev The heat trasfer coeffcet s depedet o the thermal materal propertes, ad for covecto coolg, also o the velocty feld of the coolg flud. Ths meas that for a specfc setup the uow heat trasfer coeffcet has to be derved ether from measuremets or calculatos. I the case of coolg by a lqud mpgg jet, some data ca be foud lterature [44], but ca ot be appled to the LCP case because of dfferet jet types ad dfferet parameter values, especally Reyolds umbers. There s also the possblty to calculate the combed covectve ad coductve surface heat trasfer f the ear surface temperature dstrbuto s ow. For ths the sold sde temperature gradet has to be much smaller tha the flud sde oe, whch s the case for the terface betwee sold or molte slco ad the lqud medum. The the surface heat trasfer s calculated by the flud sde temperature gradet ad flud heat coductvty. J surf dt = K ( -8 ) flud d surf, flud flud sold T ev T dt 0 d v Fg. -9: Setch of combed covecto coducto surface heat trasfer.3. Phase chage sold lqud Meltg ad soldfcato occurs f the ethalpy reaches the meltg pot of the processed materal. I the dyamc descrpto of the phase chage the speed of the movg lqud sold terface s depedet o the temperature dfferece betwee the terface temperature ad the meltg temperature [0], show smplfed equato ( -9 ).

28 MATHEMATICAL MODELLING v sl = Κ L T T ( -9 ) ( ) m sl m T sl exp B Tm Tsl Ths meas, that the phase chage taes place ot exactly at the meltg temperature, but some superheatg or udercoolg s eeded. The etc rate costat Κ for meltg s geerally much hgher tha for soldfcato, because of the hgh actvato eergy for crystallzato. I the smulatos of the preset wor soldfcato veloctes show values ot hgher tha 5 m s. Usg the data reported [45] results a maxmum udercoolg of 75 K. Ths wor ams to approxmately descrbe the physcal effects rather tha gvg hghly exact solutos, so ths amout of udercoolg s feasble to eglect. The superheatg for meltg s eve lower, so for the preset wor the phase chage sold lqud s assumed to tae place exactly at phase chage temperature. I addto, the soldfcato speeds are lower tha the crtcal value for amorphzato of slco of 5 m s reported [45]. Ths esures that the slco soldfes to a crystal rather tha to a amorphous state for ay LCP parameters. Assumg a defed meltg pot, the phase chage ca be descrbed by a ethalpy based approach. Oce the meltg temperature T m s reached, the temperature stays costat whle the ethalpy further - or decreases utl the latet heat of meltg L m has bee overcome. For slco the latet heat of meltg s tae from [46] to L mol, m = 50J mol. Because the varable to be solved for heat trasport s the ethalpy uts of eergy per volume, the latet heat has to be adopted accordg to equato ( -30 ). ρ L = L ( -30 ) mol M I the lqud state the product of heat capacty ad desty of slco shows a costat value of 6 3 ρ c p =.43e J m []. Ths results the lear relato ( -3 ) betwee ethalpy ad temperature wth c ρ c =, show Fg. -0. H m s the 4 = p ad c5 Tm ( H m Lm ) ρ c p ethalpy where meltg starts, ad s calculated by sertg T m equato ( -5 ). T = c H ( -3 ) 4 c 5

29 MATHEMATICAL MODELLING Evaporato For the secod phase chage lqud vapour of slco two dfferet models are used, maly depedet o the speed of evaporato. For low evaporato speed,.e. for relatvely low laser testes, the materal evaporates by so called ormal bolg [47]. Here the evaporato taes place exactly at the bolg temperature, so a ethalpy based descrpto as dscussed above ca be used. Ths meas that the materal s assumed to be fully evaporated f the bolg pot at H v s reached ad addtoally the latet heat of vaporzato L v has bee overcome. Accordg to [47], ths model s vald for laser pulse durato of greater tha roughly oe mcrosecod. For hgher laser lght testes, sgfcat superheatg occurs ad ca o loger be eglected. Therefore a dyamcal descrpto of the phase chage s requred, whch meas a calculato of the evaporato speed. For aosecod laser pulses usually the Hertz-Kudse equato ( -3 ) s used. Here the evaporato rate of atoms s set to the effuso rate derved from statstcal thermodyamcs [48]. p sat & ( -3 ) = π m a B T lv To calculate the surface recesso velocty v lv, equato ( -3 ) has to be dvded by the atom desty of the lqud ρ l m a. I addto t s assumed that the materal does ot evaporate to vacuum, but drectly above the lqud surface a small layer of pure vapour exsts, whch s feasble wth the laser pulse durato. The evaporato speed s the approxmately calculated by the dfferece of saturato pressure ad partal pressure at the surface. I ths wor o vapour dyamcs are modelled ad the partal pressure s set to a exterally calculated value p ext. Ths way a evrometal pressure or a pressure dstrbuto caused by the mpgg jet ca be tae to accout. v lv = ρ l p sat p π m B a ext T lv ( -33 ) Equato ( -33 ) results geerally a creasg evaporato speed wth temperature, because the saturato pressure s strogly depedet o temperature. For the case of slco o measured data for the saturato pressure above the bolg pot exsts, so the Clausus- Clapeyro equato ( -34 ) s used assumg a costat molar latet heat of vaporzato

30 4 MATHEMATICAL MODELLING L mol, v. For slco mol v pot. L, s calculated by the data gve [49] to 400 J mol at the bolg p sat = Lmol, v p exp ( -34 ) 0 R T Tb For reasos of eergy coservato, the latet heat of vaporzato has to be cosdered addtoally as a surface heat flux durg evaporato. qlv = Lv v ( -35 ) I Fg. -0 the etre relato of temperature ad ethalpy for slco as used the preset wor s show. Beyod the bolg pot the horzotal curve correspods to the ormal bolg model, whereas for the Kudse evaporato the further creasg curve llustrates the effect of superheatg temperature [K] sold lqud ethalpy [J/mm³] Fg. -0: Temperature ethalpy relato as used the smulatos; the dashed curve correspods to the superheatg for the Kudse evaporato model ad the sold oe to the ethalpy based model The Kudse evaporato as descrbed above s stll a qute smplfed model, because the fluece of the surroudg vapour s ot cosdered. Especally for hgher laser testes the geerated vapour plume results hgh recol pressures ad drectly flueces the evaporato speed. A good model would be the so called Kudse layer as proposed by Kght [50]. Here t s cosdered that the temperature ad the pressure a small layer above

31 MATHEMATICAL MODELLING 5 the surface are ot thermodyamc equlbrum, whch ca be treated as a jump codto. But usg ths model would requre a coupled solvg of gas dyamcs, whch s ot doe wth the preset wor. Also codesato ca ot be calculated wthout gas dyamcs, because the actual partal pressure at the surface has to be ow..3.4 Speces trasport Two dfferet effects of speces trasport are modelled. Oe s the trasport of chemcal addtves the lqud jet. Here maly the lght duced geerato ad trasport of chlore radcals s of terest. At the reacto spot the dffuso of mpurty atoms the lqud slco s modelled, because ths s the basc mechasm for the dopg process. The geeral traset speces trasport equato ( -36 ) cludes smlar to the heat trasport equato a covecto, a coducto ad a source term. The varable to be solved s the cocetrato of speces partcles C. dc dt v C = D ΔC ( -36 ) S sp I sold slco the dffuso coeffcet s strogly temperature depedet. Usually tme scales of hours are eeded to dffuse a typcal phosphorous emtter for slco solar cells. For lqud slco a crease of the dffuso coeffcet of several orders of magtude taes place, whch s show for phosphorous Fg. - wth data tae from [5] for sold slco ad [9] for lqud slco respectvely. Ths jump s the reaso why eve wth the very short melt tmes below oe mcrosecod produced by laser pulses, a usable amout of mpurty atoms dffuses to the slco. O the other had, ths jump clearly allows for the sold state dffuso to be eglected the modellg of laser dopg. The lqud slco dffuso coeffcet s assumed to be costat. Because lqud phase dffuso s domated by the sze of the mpurty atom [48], whch s temperature depedet, ths approxmato s feasble. The melt s durg ts lfetme domatly at meltg temperature as show later ths wor. Ths further reduces the error of the fal speces dstrbuto troduced by the smplfed temperature depedet dffuso coeffcet.

32 6 MATHEMATICAL MODELLING dffuso coeffcet [cm²/s] temperature [ C] Fg. -: Temperature depedet dffuso coeffcet of phosphorous sold ad lqud slco If flow s ot cosdered, the covecto term equato ( -36 ) ca be removed. Also the materal volume o sources for mpurty atoms are preset. These smplfcatos result the well ow Fc s law ( -37 ), whch s used to calculate mpurty atom dffuso slco melt. dc dt = D ΔC ( -37 ) Dfferet boudary codtos for speces trasport are used ths wor. They ca be dvded to solatg, fte source ad fte source. For the solatg case o speces flux over the boudary s allowed, whch s for example the case at the lqud sold terface slco. dc d surf = 0 ( -38 ) The fte source boudary codto s used at the lqud jet outlet ad at the slco surface f a ulmted dopat source s assumed. C = C surf bc ( -39 )

33 MATHEMATICAL MODELLING 7 I dry laser dopg a phosphorous cotag precursor layer s used as the dopat source, see secto 5.5. Ths has to be modelled by a fte source boudary codto. Here a surface cocetrato depedet o the loadg Θ, whch trasetly decreases by the dopat flux through the surface, s used. Because o sutable physcal model was foud, a lear relatoshp startg from saturato cocetrato slco melt at full loadg ad decreasg to zero at zero loadg s assumed, see equato ( -40 ). Comparg the smulato wth expermetal results shows that ths approxmato descrbes the dopg by a precursor layer source a reasoable way. Θ C = ( -40 ) C surf sat Θ t =0 For dopg wth LCP a fte source boudary codto s used. The surface cocetrato s a free parameter ad s adjusted the smulato to ft the expermetal results. Much better would be a boudary codto whch ca be calculated by the dopat cocetrato the lqud jet. The frst dea, that ths surface cocetrato s equal or proportoal to the dopat cocetrato the lqud jet, gves o sutable results. Here the surface cocetrato of the measured dopg profles should be proportoal to the dopat cocetrato of the solvet, whch s ot the case for the dfferet measured dopg profles. Therefore, the processes at the solvet slco terface has to be modelled more detal. Ths meas cosderg the thermal atomzato of the dopat ad the dopat trasport by dffuso ad covecto the ear terface rego. Ths could ot be doe wth the tmeframe of the preset wor, so the surface cocetrato remas as a free parameter for LCP dopg smulatos. The optcal propertes of slco are geerally flueced by the dopg level. Ths s ot cosdered the preset wor, because o data has bee foud lterature whch covers the eeded cocetrato ad temperature rage. Ths s assumed to troduce a eglgble error, because the laser lght teracts maly wth the lowly doped slco bul, ad the lqud phase the absorpto s evertheless very hgh..4 Flud dyamcs For smulato of flud flow the commercal software Fluet s used. The models ad regardg equatos show ths secto are tae from the Fluet maual [5], where refereces to lterature are cluded.

34 8 MATHEMATICAL MODELLING.4. Basc flud flow The flow to be smulated occurs o a small mcrometer scale, but the Kudse umber stays well below 0.0, eve for extreme assumptos for pressure ad temperature. Ths meas that the mea free path s much lower tha the typcal legth scale, whch allows for cotuum flow equatos to be used. These are the Naver Stoes equatos cosstg of coservato laws for mometum ( -4 ) ad mass( -4 ), where the latter s called the cotuty equato. d dt ( v) ( ρ v v) = p ( τ ) S mom ρ ( -4 ) dρ ρ dt ( v) = S mass ( -4 ) The stress tesor τ accouts for vscosty effects. T τ = μ v v v E ( -43 ) 3 Turbulece ca occur LCP because of the hgh velocty of the lqud jet. It s hard to choose a sutable turbulece model ad parameters, especally because of the traset geometry chage durg expulso of the slco melt. I addto, ths wor s ot amg for exact solutos, therefore turbulece s ot cosdered. The slco melt s treated as a compressble flud, optoally wth a temperature depedet desty. Because o vapour phase s smulated, the recol pressure o the melt surface s ot ow ad ca ot be cosdered. Especally for short laser pulses the recol pressure ca reach very hgh values of up to the order of GPa [5], whch s far above the lqud jet pressure. However, the recol pressure oly taes place durg the evaporato, whereas the lqud jet exerts pressure durg the whole melt durato, whch s sgfcatly loger. Ths meas that for relatvely log pulse duratos ad low pulse eerges the lqud jet duced melt flow domates the melt flow by recol pressure. The results of ths wor dcate that ths s the case for melt duratos greater tha oe mcrosecod..4. Multphase flow Multphase flow meas cosderg the flow of dfferet fluds whch possess dfferet flud dyamc propertes. Multphase models ca be bascally dvded to Euler-Lagrage ad

35 MATHEMATICAL MODELLING 9 Euler-Euler models. I the Euler-Lagrage approach, algebrac equatos of moto are solved for dscrete partcles teractg wth a cotuous flow feld. I the Euler-Euler model, two or more terpeetratg cotuous flow felds are cosdered. I LCP maly the flow of the lqud jet teractg wth the slco melt s of terest, whch correspods to the Euler-Euler model. Slco melt ad lqud solvets are treated as mmscble fluds, therefore a sharp terface exsts ad surface teso has to be tae to accout. I the Fluet smulatos the so called volume of flud (VOF) model s used, whch s a smplfed Euler-Euler approach. The volume fracto a s troduced, whch correspods to the volumetrc occupato of the regardg phase. At every pot space the volume fractos of all phases have to sum to oe. p a q q= = ( -44 ) The flud flow equatos are solved for the mxture phase, where the mxture propertes are defed by the volume fracto weghted average as show equato ( -45 ). p ξ = ξ ( -45 ) mxt a q q= q d dt ( aq q ) ( aq ρ q v) = S a, q ρ ( -46 ) Addtoally the volume fracto equato ( -46 ) has to be solved for each phase q besdes the prmary phase, whch allows tracg of the phase terfaces. The volume fracto of the prmary phase s calculated by equato ( -44 ). The source term of the volume fracto equato correspods to mass sources ad to mass trasfer betwee two phases. Because of solvg oly oe flow feld, the VOF model requres a low computatoal effort ad s predested for the tracg of sharp terfaces. The VOF model further allows the cosderato of surface teso, whch plays a sgfcat role LCP because of the hgh surface teso of slco of N m at the meltg pot [53]..4.3 Meltg ad soldfcato The meltg ad soldfcato model has to accout for the traset chage of a materals aggregate state betwee lqud ad sold. For ths the temperature depedet lqud fracto β s troduced. Fluet usually solves for temperature as the depedet varable ad a sudde

36 30 MATHEMATICAL MODELLING soldfcato at the exact meltg temperature would cause umercal problems. Therefore a mushy rego chagg from sold to lqud. Δ Tm s defed aroud the meltg pot, where the materal s learly ΔTm T Tm β = max m,, 0 ( -47 ) ΔT m To hder movemet the sold state, a very hgh mometum s s added to equato ( -48 ). The s s depedet o the lqud fracto such a way that for a lqud fracto of oe, the value of the s s zero. The olear expresso s optmzed for umercal stablty. S mom ( ) β = e β v ( -48 ).5 Chemstry Chemcal effects modelled the preset wor comprse photochemcal decomposto of molecules ad reacto etcs. The focus s o effects the lqud jet, maly the geerato ad recombato of chlore radcals. At the reacto spot extreme codtos terms of tme scale ad temperature occur. Here o sutable data for surface reacto etcs exsts, whch maes a quattatve smulato mpossble..5. Photochemcal radcal geerato If photo eerges are hgh eough, the laser lght s able to crac molecule bods. Ths ca be used to produce chlore radcals out of molecular chlore the lqud jet. The amout of radcals geerated equals the product of quatum effcecy Φ d ad the amout of photos absorbed by the molecular chlore, whch s the absorbed laser power dvded by the photo eergy [54]. S Cl α Cl I λ = Φ ( -49 ) d hc The absorpto coeffcet of molecular chlore s calculated by the collso cross secto σ c.

37 MATHEMATICAL MODELLING 3 α Cl C σ = ( -50 ) Cl c Data for chlore radcal geerato wth λ = 355m has bee tae from [54] to = Φ d 9 ad σ =.94e cm. c.5. Reactos May reactos ca tae place durg LCP, for example etchg ad oxdato of hot slco, thermal decomposto of dopat sources ad reactos of chemcal actve addtves the lqud jet. At the reacto spot maly terfacal reactos occur, whch are more complex to model tha volumetrc reactos. Addtoally, for the extreme scales of temperature ad tme o sutable lterature data for reacto etcs ca be foud. Therefore o attempt was made wth the preset wor for quattatve modellg of the reacto etcs at the reacto spot. I cotrast to the reacto spot the codtos the lqud jet are much easer. Here the geerato of chlore radcals s modelled as metoed the prevous secto. Oce created, the radcals show a hgh recombato rate, whch ca be descrbed by a basc 0 reacto wth the etc rate costat.e l ( mol ) [54]. Furthermore, the r, Cl Cl = s radcals ca react wth the solvet used as the lqud meda. For the soluto of gases orgac solvets are usually the frst choce. I [55] etc rate costats for the reacto of chlore radcals wth dfferet orgac solvets are show to be the rego of 0..0 ( mol 7 9 r, Cl solv = l s ). Ths s somewhat below the radcal recombato rate, but stll has to be cosdered. I [0] t has bee show that the best sutable solvets are the perflourated carbo compouds, ot least because of ther chemcal stablty. I ths case the reacto of the solvet ad the chlore radcals ca be readly eglected. The recombato s treated as a s of radcal cocetrato, whch s calculated by equato ( -5 ). Cl = N A r, Cl Cl CCl S ( -5 ) The reactos regarded ths wor are exothermal ad have to be cosdered as heat sources. The heat source s calculated by the reacto ethalpy Δ H of reactats ad, as show equato ( -5 ). I case of chlore radcal recombato the reacto ethalpy correspods to the ethalpy of formato of molecular chlore Δ H r, Cl = 43J mol [48]. S h = N A r, C C ΔH ( -5 ) r

38 3 Fudametals of the smulato methods I ths wor essetally two dfferet ways of solvg the mathematcal models are used. Oe way s the programmg of a fte dffereces solver. The advatage over commercal software s the ablty to chage ad exted the program code, whch eables the possblty of basc solver customzato ad adapto to the regardg models. Arbtrary equatos of dfferet ds, le for example the equato of laser lght absorpto ad the heat trasport equato, ca be mplemeted ad smultaeously solved. Ths results a overall hgh problem adapto ad speed of the smulato code. The theory of fte dffereces descrbed ths secto s tae from [56]. The secod way to smulate LCP s by usg the commercal software Fluet by Asys, Ic. Fluet s a powerful tool bascally for solvg flud flow equatos, but has also mplemeted models for other physcs le heat trasport, speces trasport etc. Implemetg a effcet solver code for the complex flud flow equatos by basc programmg requres hgh effort, so usg the hghly developed Fluet solver s a reasoable way for the smulato of flud flow LCP. Fluet also offers the possblty of customzato by addg user defed C code, whch s used for example for the traset couplg of Fluet wth Matlab. 3. PDE solvg by fte dffereces The basc traset equatos to be solved ths wor by fte dffereces show the form of the geeral trasport partal dfferetal equato (PDE) for a scalar Ψ wth dffuso term ad source term S. dψ dt ( ) S = Γ Ψ ( 3- ) Rewrtg the equato Cartesa coordates yelds equato ( 3- ). dψ dt = d dx dψ Γ dx d dy dψ Γ dy d dz dψ Γ S dz ( 3- ) The dffuso term ca be approxmated by fte dffereces for a ow spatal dstrbuto of Ψ at tme t. The sources have also to be computable by Ψ ad t. Equato ( 3- ) ca 3

39 FUNDAMENTALS OF THE SIMULATION METHODS 33 the be solved by choosg proper startg values for Ψ ad t followed by umercal tme tegrato. 3.. Spatal dscretzato Fte dffereces are bascally a method for calculatg dervatves of dscrete quattes. After dvdg the varables to dscrete tervals, dervatves of a geeral scalar ca be approxmated by the eghbourg values. The advatage to the competg fte elemet method s that o certa formulato of the equatos to be solved s requred ad that the mplemetato as program code of lower complexty. The dsadvatage s that fte dffereces wor oly o a structured ad rectagular grd. Sub-grds ad coordate trasformatos ca overcome these lmtatos oly to a certa degree. I ths wor the structured rectagular grd s beefcal, because t allows a easy solver algorthm also for other o-traset models le laser lght absorpto or raytracg. For spatal dscretzato, ths wor a rectagular soluto doma wth rectagular but uequally szed elemets s used to eable adaptve grd szes. The elemets are cosecutvely umbered wth, j ad the x -, y - ad z -drecto respectvely ad have the szes Δ x, Δ y j ad Δ z wth =.. x, j =.. y ad =.. z. The scalar values Ψ, j, are defed the cetrod of each elemet ad correspod to the odes of the used fte dffereces grd. Fluxes betwee the elemets are defed at the elemet faces, detfed by the dces 0. 5 ad y j Ψ, j Δ y j Ψ, j Φ 0.5, j Ψ, j Ψ, j y x x Δ x x 0.5 Fg. 3-: Setch of a exemplary two-dmesoal, oeqdstat, rectagular grd for fte dffereces dscretzato

40 34 FUNDAMENTALS OF THE SIMULATION METHODS Defed by whch eghbourg values are used, dfferet dscretzato schemes, amely forward, bacward ad cetral dffereces exst. A hgher order of dfferece expressos ca be appled, whch mproves the accuracy ad s acheved by tag a hgher umber of eghbourg scalar values to accout. For a secod order expresso for example, the eghbourg values of a eghbour are addtoally tae to accout. I ths wor mostly frst order formulatos were chose, because t s best sutable f some usteadess of a scalar occurs, le e.g. for the temperature at the lqud sold terface. Formulatos of fte dfferece dervatves are show examples for the postve x - drecto oly. If the frst dervatve of a elemet s eeded, the cetral dfferece scheme ( 3-3 ) s used. The gradet x -drecto at y j, x 0. 5, whch s the elemet face betwee elemet, j ad, j, s calculated by the forward scheme ( 3-4 ). dψ dx Ψ = x, j, Ψ x, j,, j, ( 3-3 ) dψ dx Ψ =, j, x 0.5, j, Ψ x, j, ( 3-4 ) Multplcato of the gradet ad the dffusvty results the flux Φ through the correspodg elemet face. dψ Φ 0.5, j, = Γ ( 3-5 ) 0.5, j, dx 0.5, j, Γ Γ Γ, j,, j, 0.5, j, = ( 3-6 ) Γ, j, Γ, j, As proposed [6], the dffusvty at the elemet faces are calculated accordg to equato ( 3-6 ) by so called harmoc averagg rather tha arthmetc averagg, because ths formulato best esures coservato of the scalar. The dffuso term s the calculated by the flux dffereces accordg to equato ( 3-7 ). Ths further esures the coservato of Ψ, whch was detfed to be more essetal laser heatg tha havg a possbly hgher accuracy approxmato by applyg a dfferet dscretzato scheme of the dffuso term.

41 FUNDAMENTALS OF THE SIMULATION METHODS 35 d dx dψ Γ dx, j, Φ = Φ 0.5, j, 0.5, j, Δx ( 3-7 ) I ths wor problems wth axal symmetry have to be solved, therefore cyldrcal coordates eglectg the azmuth coordate are used. For smplcty ad followg the program code, o ew coordate ames are troduced. y s hadled as the azmuth coordate, whch s eglected, ad z as the heght coordate. x has the same drecto of the radal coordate r wth a offset x 0, whch correspods to the cetre of the axal symmetry,.e. where the radus equals zero, r = x x0. Ths allows chagg the cetre of axal symmetry whle eepg the same grd. The formulas for the spatal dervatves the z - coordate are ot affected by gog to cyldrcal coordates, whereas the dffuso term for the x -coordate has to be adopted accordg to equato ( 3-8 ). ( ) Γ Ψ, j, = r Φ , ,, 0.5, 0. 5 r Δx r Φ Φ Φ Δz ( 3-8 ) Boudary codtos are appled by settg sutable values outsde the boudary. For = ad = x, the values at ad do physcally ot exst ad are set to boudary values Ψ bc defed by the boudary codtos. For example a solatg boudary codto, whch correspods to a zero value of the Neuma boudary codto, s appled by settg the outsde values equal to the sde values. Ths boudary codto s also appled o a symmetrcal plae wth a 3d doma. dψ dx = 0.5, j = 0 ( 3-9 ) Ψ = Ψ = Ψ 0, j bc =, j Also Drchlet boudary codtos are used, whch meas a costat scalar value o the boudary surface. Ψ = x 0.5, j Ψ x, j = Ψ surf = Ψ bc = Ψ surf ( 3-0 )

42 36 FUNDAMENTALS OF THE SIMULATION METHODS 3.. Tme tegrato Oce the rght had sde of equato ( 3- ) s expressed by Ψ ad t, the tme dervatve each elemet of the soluto doma ca be calculated. The tme s dvded to dscrete tme steps. After choosg proper startg values, the ew values for Ψ ad t after the frst tme step ca be evaluated by dfferet methods out of the tme dervatve. The ew values are the used for calculato of the ext tme step ad so o. Ths way a defed tme spa ca be solved teratvely, whch s called umercal tegrato. The smplest umercal tegrato method s the Euler explct or forward Euler method. Here the tme dervatve s calculated from the scalar values at tme ad s multpled by the tme step sze. The scalar values at the updated tme are the smply calculated by addg. d Ψ Ψ = Ψ Δt ( 3- ) dt The term explct meas that the updated scalar values ca be explctly calculated from the prevous oes, ad o teratos are eeded. Because of ths, explct tme tegrato s very fast calculatg oe tme step. However, a strct lmtato exsts for the upper sze of the tme step sze to esure covergece of the fte dfferece algorthm. Ths maxmum tme step sze ca be derved from the Cauchy crtero ( 3- ). ( Δx, Δy, Δz) m Δt ( 3- ) exp l 6Γ Here the mmum sze of all grd elemets decdes the lmt of the tme step sze. The depedecy o the grd sze s quadratc, so by usg fer grds, much more tme steps are eeded for the tegrato over a specfc tme spa, whch creases the overall computg tme. A mproved explct tme tegrato method s the Ruge-Kutta method. Here the tme dervatves for several dfferet tme steps are averaged a specal way. I ths wor the fourth order Ruge-Kutta method s used, whch s defed equatos ( 3-3 ). I comparso to the frst order Euler explct method the accuracy of the soluto s mproved sgfcatly, but s the same way restrcted to the maxmum explct tme step sze.

43 FUNDAMENTALS OF THE SIMULATION METHODS 37 Ψ = Ψ dψ Δ dt, t Ψ Ψ = Ψ, dψ Δ dt, t,3 = Ψ dψ dt, Δt ( 3-3 ) dψ 6 dt dψ dt dψ dt dψ dt,,, 3 Ψ = Ψ Δ t Asde from explct tme tegrato, mplct schemes ca be used. Implct meas that the updated scalar values ca ot be explctly calculated, because the tme dervatve s depedet o the uow updated stead of the prevous scalar values. The resultg mplct equatos have to be solved by umercal terato. The advatage of mplct tegrato s that o mathematcal restrcto as to the tme step sze exsts. Especally for small grds much greater tme step szes compared to the explct oes ca be appled, whch ca result a overall lower computato tme. The most smple mplct tme tegrato method s the Euler mplct or bacward Euler method. Equato ( 3-4 ) has to be solved teratvely each tme step. Ψ d Ψ = Ψ Δt ( 3-4 ) dt The frst order methods, both explct ad mplct, show a umercal error because they use for evaluato of the tme dervatve ether the scalar values at the begg or the ed of a tme step. A more accurate approxmato s to use the scalar values the mddle of the tme step, whch ca be calculated by averagg the mplct ad explct tme dervatve, see equato ( 3-5 ). Ths s called the Cra-Ncolso method. It shows o tme step sze lmtato ad the effort for teratve solvg s the same as for the Euler mplct method. Ψ = Ψ dψ dt dψ dt Δ t ( 3-5 )

44 FUNDAMENTALS OF THE SIMULATION METHODS Alteratg drecto explct (ADE) method For fte dffereces a specal tme tegrato method, the alteratg drecto explct method (ADE), exsts, whch combes the advatages of the explct ad mplct tegrato [56]. Wth ths method a explct calculato wthout tme step restrcto s possble. Therefore the ADE method has a bg potetal to speed up the tme tegrato, especally for fe grds where the explct tme step would be very small. The basc ADE formulas for solvg equato ( 3- ) are show below for two dmesos, a costat dffusvty Γ ad a equdstat grd. They are derved from the so-called multlevel dscretzato of equato ( 3- ). Two dfferet scalars U ad V are troduced, whch equal Ψ at tme, j j j V U,,, Ψ = =. 0.5,,,,,,,,,,, Δ Γ Δ = Γ Δ j j j j j j j j j j j S y U U U U x U U U U t U U 0.5,,,,,,,,,,, Δ Γ Δ = Γ Δ j j j j j j j j j j j S y V V V V x V V V V t V V ( 3-6 ) Equatos ( 3-6 ) ca be solved for, j U ad, j V. Startg from = ad = j by cosecutve cremet of ad j, the calculato of, j U each elemet uses the prevously updated values, j U ad, j U. The same s doe for V by startg at x = ad y j =, followg the opposte drecto. Fg. 3- llustrates the meag of the drectos used by the ADE algorthm.

45 FUNDAMENTALS OF THE SIMULATION METHODS 39 y V Ψ, j j x U Fg. 3-: Illustrato of opposte calculato drectos by ADE method Usg both prevous ad updated values for the calculato of the updated oes shows the mxed explct ad mplct character of the ADE method ad explas the abolto of the tme step restrcto. Fally the updated scalar value s calculated by averagg U ad V. Ψ U, j, j = V, j ( 3-7 ) 3. Fluet The reaso for usg the commercal software Fluet s because t s oe of most hghly developed solvers for flud flow avalable. Ths esures o the oe had a accurate ad effcet smulato of flud flow. O the other had, the user terface eables a much faster ad flexble method of grd geerato ad problem setup tha could be accomplshed by basc programmg. Varous addtoal models are avalable ad selectable through the user terface. Especally the ablty of smulatg multphase flow wth free surfaces, cosderg compressble ad compressble fluds at the same tme, s essetally what s eeded for the smulato of LCP. A advatage of Fluet over other commercal solvers s the ablty of addg user defed program code. Wth ths t s for example possble to adjust physcal quattes durg terato, to use user defed fuctos for calculatg materal propertes ad to solve user defed scalar equatos coupled wth flud flow.

46 40 FUNDAMENTALS OF THE SIMULATION METHODS Ths secto explas Fluet s basc abltes ad mplemeted umercal methods that are appled the preset wor. The formato s tae from the Fluet maual [5], where all correspodg lterature refereces ca also be foud. 3.. Solver bascs Fluet solves the flud flow ad addtoal equatos va the fte volume method. Here the soluto doma s made to dscrete volumetrc elemets ad the physcal quattes are defed the cetrod of each elemet. The cocerg coservato laws are tegrated over each elemet volume by usg a terpolato fucto for the varable questo, amely velocty compoets ad pressure the case of flud flow. The resultg algebrac equatos are the learzed ad solved teratvely. The advatage of ths dscretzato method s that the coservato laws are exactly fulflled for each elemet, eve for rough dscretzato. Ths allows for a stable calculato of the occurrece of dscotutes as well, whch exst our case for example at the free surface betwee slco melt ad water. I ths wor the so-called pressure based segregated solver s used, because ths s best to hadle free surface flow. The algorthm starts wth sequetally solvg the mometum equato for the velocty compoets v x, v y ad v z. Afterwards a pressure correcto s calculated by solvg a pressure equato satsfyg the cotuty equato. Ths correcto s used to update the pressure ad velocty feld. Addtoal equatos le heat trasport or user defed scalars are solved afterwards. After checg covergece, all physcal quattes are updated ad the ext terato starts. There s also the possblty to solve all flud flow equatos a coupled way. The advatage would be better covergece behavour, but the system of lear equatos to be solved s greatly creased sze, resultg a overall creased memory requremet ad computg tme. Maly used ths wor s the D axal symmetry solver, whch s avalable for all mplemeted models. Fluet also has the possblty of usg parallel computg. For ths the soluto doma s terally dvded to subdomas solved o dfferet processors. Because of hgh data exchage at the subdoma boudares, a crease of computg speed ca oly be acheved for qute hgh umbers of elemets. Because the LCP smulato model s hghly traset, a large umber of tme steps are eeded. To reach acceptable computg tmes, the umber of spatal elemets has to be qute small, whch s the reaso that parallel computg s ot sutable ad s therefore ot used the preset wor. Fluet offers may optos to customze the solver, le settg dscretzato schemes, uder relaxato factors ad multgrd optos. Explag the theory behd all of these settgs has bee left out of ths thess ad ca be looed up detal the Fluet maual [5]. Sutable settgs for the smulatos doe ths wor are show A.6.

47 FUNDAMENTALS OF THE SIMULATION METHODS User defed code UDS ad UDF As metoed before, Fluet s able to solve so called user defed scalar (UDS) equatos. The geeral equato to be solved correspods to equato ( 3- ), exteded by a covecto term, whch accouts for trasport of the scalar by the flow feld. d dt ( ρ ) ( ρ v Ψ) = ( ρ Γ Ψ) ρ S Ψ ( 3-8 ) The desty s cosdered all terms for mass based scalar trasport, because t ca ot be elmated problems wth chagg desty, e.g. compressble flow. Equato ( 3-8 ) s solved wth the fte volume method the same way as descrbed the prevous secto. Because t s solved each terato together wth the flud flow equatos, the teracto of the UDS wth flud flow s fully cosdered. Multple UDS ca smply be set by the user terface. The user must defe the dffusvty ad the source term. Note that Fluet expects both the dffusvty ad the source term to be already multpled by the desty. A useful feature of the UDS fuctoalty s the possblty to solve the UDS o a per-phase bass. Ths meas that a UDS s oly defed for oe phase multphase flows, ad s ot solved regos where the phase does ot exst. Ths eables, for example, that the UDS s exactly coserved wth a phase, especally at the phase terface, ad o umercal dffuso to the eghbourg phase occurs. Fluet further offers a hgh potetal for customzato by so called user defed fuctos (UDFs). They are wrtte covetoal C code, exteded by a lbrary cotag fuctos ad deftos for access ad mapulato of solver data. For example fuctos to loop over all elemets wth a defed zoe, or to access dfferet physcal quattes of a elemet are provded. Dfferet types of UDFs exst, whch ca be compled ad hooed to the solver by the user terface. Oe type s the adjust-udf, whch ca be used to set ay physcal quatty to user defed values before each terato. Arbtrary materal propertes ca be defed by dfferet property-udfs. Wdely used ths wor s the source-udf, whch source terms are calculated. Through the user terface the source-udfs are hooed to the correspodg physcal quatty, le mass or a UDS. The UDF techology was further used ths wor to couple the Fluet solver wth Matlab by covetoal C code.

48 4 FUNDAMENTALS OF THE SIMULATION METHODS 3..3 Multphase flow Several multphase models are mplemeted Fluet cludg the VOF model, whch s used for the smulato of the free surface lqud jet ad melt flow. As metoed secto.4., the VOF model s best suted for the tracg of sharp terfaces. For ths Fluet provdes the so-called geometrc recostructo scheme. Ths s a dscretzato scheme for the volume fracto, where the terface betwee two phases s always esured to be sharp. Stadard dscretzato schemes le frst order upwd always show sgfcat umercal dffuso, whch meas that the terface would spread out ad several elemets aroud the physcal terface would be occuped by both phases. Especally f terface trasfer mechasms le heat or mass trasfer have to be smulated, a oe elemet sharp terface s requred. Wth the VOF model ad the geometrc recostructo scheme the surface teso s cosdered. Furthermore, t s possble to smulate compressble ad compressble phases at the same tme. Ths allows a smulato cosderg gas dyamcs ad the effect of recol pressure o the melt flow. Ths s ot doe wth the preset wor but s cosdered for further developmet of LCP smulato. To apply user defed code wth a multphase smulato, t s ecessary to uderstad the data structures mplemeted Fluet. The sum of elemets wth a geometrcally defed zoe s called a thread. For multphase flow several threads for oe zoe exst, oe for each phase ad oe mxture thread. The mxture level s orgazed above the phase level. To access elemet data le materal propertes ad flow quattes, the correct thread has to be used. For example the volume fracto values are stored wth each phase thread, whereas the pressure values are oly avalable wth the mxture thread.

49 4 Implemetato As metoed prevously, the mathematcal models are solved by a fte dffereces scheme o the oe had ad by usg Fluet, maly for flud flow, o the other had. I ths chapter detals of the mplemeted codes ad the use of Fluet are gve. I addto, the couplg algorthm of Matlab ad Fluet s descrbed. At the ed of every secto some results are show for verfcato of the correct mplemetato. 4. Heatg ad radcal geerato the lqud jet The heat trasport the lqud jet s descrbed by equato ( - ) the z-drecto. The heat source s determed by the absorpto of laser lght ( -6 ) ad the reacto ethalpes ( -5 ). The Lambert-Beer law ca geerally ot be appled, because the absorpto coeffcet s depedet o the chagg chlore cocetrato accordg to equato ( -50 ). Oly the absorbed laser power ot leadg to radcal geerato acts as a heat source. Furthermore, the speces trasport equato for radcals ( -36 ) has to be solved, whch s expressed oe dmeso. The dffuso term ca be eglected because of the hgh velocty resultg a domatg covecto term. The resultg equatos to be solved are equatos ( 4- ) to ( 4-5 ). di dz = δ ( α α )I ( 4- ) solv Cl dt dt ρ c d T K dz ( α ( Φ ) α ) I N C ΔH = δ Cl d solv A r, Cl Cl Cl Cl p v z dt dz ( 4- ) dc dt hc I δ α Cl λ Cl = Φ d N A r, Cl Cl CCl v dc dz ( 4-3 ) The reacto of chlore wth the solvet s eglected, so the cocetratos of molecular chlore ad chlore radcals have to sum up to the molecular chlore cocetrato at the let C Cl, 0 as equato ( 4-4 ). The let radcal cocetrato s zero. 43

50 44 IMPLEMENTATION C C 0. 5C Cl = Cl,0 Cl ( 4-4 ) For the absorpto equato a arbtrary tme depedet laser testy ca be appled whch s related to the laser power by the jet dameter. I P( t) ( 0.83d ) z= 0 = I ( t) = ( 4-5 ) π ozzle A fte dffereces method wth a equdstat grd programmed Matlab was appled to solve the equatos above. Bascally the rght had sde of equatos ( 4- ) ad ( 4-3 ) are calculated for umercal tme tegrato as descrbed secto 3.. I the Matlab code, all spatally depedet varables le temperature T or chlore absorpto coeffcet α Cl are stored vectors. The dces of the vector correspod to the elemet umber z -drecto. For the costructo of the fte dffereces dervatves, shfted vectors are used, whch s descrbed for the example of the temperature vector T ad s llustrated Fg. 4-. The shfted vectors T z ad T z postve ad egatve z -drecto are calculated by the smple vector assgmets ( 4-6 ) ad ( 4-7 ). T T ( 4-6 ) z, = T T ( 4-7 ) z, = Here oe value each shfted vector s ot defed, amely T z, = z ad T z, =. They are set to the boudary codto value, ths case a costat temperature at the let ad a zero gradet at the ed of the jet. T = z T ( 4-8 ), = T z = z = T = z, ( 4-9 )

51 IMPLEMENTATION 45 0 T bc = T T T Δ z l jet z T z T z T = bc T z T T z Fg. 4-: Setch of shfted vectors wth boudary codto values (grey) Oce the shfted vectors are fully assged, the vectors represetg the frst ad secod spatal dervates are calculated by the vector expressos ( 4-0 ) ad ( 4- ) respectvely. dt dz Tz T = Δz z ( 4-0 ) d T dz T = z Tz T Δz ( 4- ) For the spatal dscretzato of the lqud jet a equdstat grd s used, therefore the deomator equatos ( 4-0 ) ad ( 4- ) smplfes to a costat elemet sze Δ z. The elemet sze s calculated by the lqud jet legth l jet dvded by the umber of elemets z. l jet Δ z = ( 4- ) z The tme tegrato for temperature ad radcal cocetrato s performed by a Matlab bult ODE (ordary dfferetal equato) solver. Here tme step guessg ad dfferet tegrato methods are already mplemeted, whch maes t easy to use. The ODE solver requres a fucto whch calculates the tme dervatves,.e. the rght had sdes of equatos

52 46 IMPLEMENTATION ( 4- ) ad ( 4-3 ). I ths fucto frst the testy dstrbuto s calculated. Ths s doe by startg wth I = I the frst elemet ad teratvely calculatg the testy of the ext elemet by equato ( 4-3 ), a dscrete approxmato of equato ( -6 ). Here the testy s treated as a costat value over each elemet, whch s feasble for small elemet szes. ( ( ) Δz) I = I Cl α ( 4-3 ) solv α Wth the testy dstrbuto ad the curret values for temperature ad radcal cocetrato, the tme dervatves ca be calculated explctly. If o chlore s added to the lqud jet, o speces trasport has to be solved ad the heat source expresso reduces to absorpto by the solvet. The absorpto ca the be descrbed by the Lambert-Beer law ( -7 ). The coductve term has show to be eglgble ad ca therefore be removed. If, addto, a costat laser lght testy over tme s assumed, the steady state equato ( 4-4 ) s acheved. Ths ca be solved aalytcally to equato ( 4-5 ). dt v dz = I δ α solv exp ρ c ( δ α z) p solv ( 4-4 ) T I ( z) = T ( exp( δ α z) ) ρ c p v ( 4-5 ) solv The aalytcal soluto s used to verfy the fte dffereces mplemetato. I Fg. 4- very good agreemet of both curves ca be see, whch proves the proper mplemetato ad a precse umercal soluto of the lqud jet heatg. Two curves were calculated umercally, wth ad wthout a coducto term, ad were observed to be detcal. Therefore the eglectg of the heat coducto for the aalytcal expresso s correct.

53 IMPLEMENTATION temperature [ C] umercal soluto aalytcal soluto jet legth [cm] Fg. 4-: Heatg of a water jet wth d ozzle = 50 µm, v = 50 m/s, P = 80 W ad λ = 064 m 4. LCPSm: optcs ad thermodyamcs at the reacto spot I ths secto the LCPSm program code s descrbed. It teto s to smulate optcal ad thermodyamc effects at the reacto spot for dry laser processg ad LCP of slco ad also of multlayer systems. It cludes absorpto the lqud jet, raytracg ad reflecto at the arbtrary surface geometry, absorpto processed materal, phase chages, evaporato models ad mpurty atom dffuso. Not cosdered are melt flow, desty chages ad vapour effects. LCPSm ca therefore be appled for low laser lght testes causg evaporato wth low recol pressures. The desty chage of slco melt s sgfcat, but the eergy balace s mataed also for the assumpto of costat desty. Therefore the resultg melt depths ad melt tmes after coolg dow to meltg temperature should be correct. For hgh laser testes, the results should ot be tae quattatvely, because temperatures far above the bolg pot are reached ad the mplemeted models ad materal propertes are o loger vald. Nevertheless, the eergy balace remas correct, therefore the results ca feasbly be see as rough approxmatos. LCPSm s mplemeted Matlab by a fte dffereces scheme ad was frst publshed [57]. A o-equdstat grd s used, whch s adapted durg each terato. Ths esures a effcet grd for every tme step, leadg to low computatoal effort ad at the same tme a good spatal resoluto the regos of terest. A setch of a sample dscretzato s show Fg Here t ca be further see that the laser lght travels the postve z -drecto ad that lear le scas are mplemeted the x -drecto. All models are mplemeted as

54 48 IMPLEMENTATION a 3d verso wth or wthout mrror symmetry ad a axal symmetrc d verso. A further teto of LCPSm was to create a rudmetary user terface for fast model swtchg, parameter chagg ad result plottg. I the followg subsectos, detals of the mplemeted code ad the usage of LCPSm are gve. laser beam sca drecto Fg. 4-3: Setch of a exemplary 3d soluto doma wth LCPSm 4.. Solvg of the heat trasport equato I LCPSm bascally the ethalpy based heat trasport equato ( -3 ) s solved by fte dffereces ad umercal tme tegrato. The dscretzato of the heat coducto term s mplemeted accordg to equatos ( 3-4 ) to ( 3-7 ), ad equato ( 3-8 ) for the d axal symmetrc verso. All spatally depedet quattes le ethalpy, temperature or materal propertes are stored d or 3d arrays respectvely. The array dces correspod to the elemet locatos wth the structured grd. Because the materal propertes are defed pecewse the sold ad lqud state, a phase array ph s troduced to easly dffer betwee the aggregate states. The value for ph s determed by the ethalpy each elemet ad s defed to be zero the sold state ad oe the lqud state. Wth the rego of ethalpy based phase chages ph s learly chaged. If a elemet s fully evaporated, the phase array value jumps to three for umercal stablty ssues. I Fg. 4-4 the depedecy of ph o the ethalpy s plotted for the case of ethalpy based evaporato. For the case of Kudse evaporato, ph stays at oe for ethalpes hgher tha H v ad jumps drectly to three f the correspodg elemet s fully evaporated. The phase array value s for example used to calculate the materal propertes accordg to equato ( 4-6 ).

55 IMPLEMENTATION 49 ( ph) ξ sold phξlqud ξ = ( 4-6 ) 3.5 phase array value Hv Hv H vlv ethalpy [J/mm³] Fg. 4-4: Relato betwee ethalpy ad the phase array value ph correspodg to the aggregate states; 0 = sold state, = lqud state, 3 = fully evaporated / empty As descrbed secto 4., shfted arrays are assged for every postve ad egatve coordate drecto. Next, the heat trasport equatos for all elemets ca be expressed by the sgle array equato ( 4-7 ), show for the d axal symmetrc case. Note because of the o-equdstat grd r, x, z, Δ x ad Δ z are ot costat ad are also stored as arrays. The heat coductvty at the elemet faces s calculated accordg to equato ( 3-6 ) to esure coservato of ethalpy. Ths coservatve formulato s especally eeded for the case of large sze dffereces of eghbourg elemets, as ca occur durg the grd adapto. At the lateral soluto doma boudares a costat temperature equal to evrometal temperature s appled, whereas at the top ad bottom a solatg boudary codto s used. Addtoal surface heat flux at the laser affected top surface s mplemeted by heat sources, see secto 4..5.

56 50 IMPLEMENTATION dh dt = r x 0.5 K K z 0.5 x 0.5 T z z z T x x x T K z Δz T r x r Δx z 0.5 x0.5 K T T z z x0.5 z z T T x x S h x x ( 4-7 ) The calculato of the temperature from the ethalpy values s accomplshed usg the explct relatoshp show Fg. -0. Oce the temperature values are ow, all temperature depedet propertes are calculated ad the optcal effects are solved, as descrbed sectos 4..4 ad Wth the resultg heat source array, the rght had sde of equato ( 4-7 ) s fully determed. For umercal tme tegrato the Euler explct, fourth order Ruge-Kutta, Euler mplct, Cra-Ncolso ad ADE methods are programmed rather tha usg Matlab s bult ode solvers. The reaso for ths s that the bult ode solvers use qute complex algorthms for stablty ad accuracy, whch results hghly creased calculato effort compared to the rudmetal tme tegrato methods. All methods besdes the ADE method requre the calculato of the rght had sde each elemet, ad could therefore be drectly mplemeted as descrbed secto 3... For ths the correspodg tme tegrato equatos are used as array equatos. The ADE method had to be adapted to a oequdstat grd ad to cosder temperature depedet materal propertes ad phase chages, whch s descrbed detal secto 4... A adaptve tme steppg s performed LCPSm for lowerg the overall computato tme. The sutable sze of the tme step s determed maly by the explct or mplct character of the tme tegrato method. For the Euler-explct ad Ruge-Kutta method, the tme step sze s restrcted accordg to equato ( 3- ). I the case of the heat trasport equato, the heat capacty s fte durg the phase chages ad s therefore approxmated by the rato of temperature ad ethalpy, resultg equato ( 4-8 ). Here the tme step sze for every elemet s calculated ad the mmum value s used for tegrato. ( Δx, Δy, Δz) H m Δ texp l = ( 4-8 ) 6T K A secod crtero, maly restrctg the mplct tme step sze, s used to allow oly a certa amout of ethalpy chage each elemet. Ths s eeded for reducg the approxmato error ad to esure umercal stablty. Good experece durg the

57 IMPLEMENTATION 5 smulatos was made by lmtg the ethalpy chage to 0 % of the absolute ethalpy. Ths results equato ( 4-9 ), where the deomator s preveted from reachg zero. Note that for the calculato of the mplct or ADE tme step, oe explct calculato of the rght had sde has to be performed to get the curret ethalpy tme dervatves. Δ t mpl = max 0. H dh dt, ( 4-9 ) If the Kudse evaporato model s used, the tme step sze s further restrcted such a way, that ot more tha oe elemet s evaporated durg oe tme step. The formulato equato ( 4-0 ) s appled to every surface elemet ad esures a postve value. Δz Δ t = ( 4-0 ) max ( e, v ) lv 4.. Implemetato of the ADE method As dscussed secto 3..3 the ADE method combes a explct calculato effort wth o tme step lmtato. Because of ths hgh potetal for calculato speed mprovemet, the ADE method s mplemeted LCPSm for the d ad 3d versos. The basc ADE expressos tae from lterature [56] are oly vald for costat materal propertes ad a equdstat grd. Therefore they had to be adapted to the stepwse defed temperature ethalpy relato ad to the o-equdstat grd. For ths the multlevel dscretzato equatos ( 3-6 ) are rewrtte to equatos ( 4- ) ad ( 4- ). The expressos preseted ths secto correspod to the d axal symmetrc verso. The mplemetato of the 3d verso was doe prcple the same way. U, U Δt, J = J 0.5,, 0.5 r r T 0.5 K ( U ) T ( U ), z 0.5, z Δz Δx T ( U ) T ( U ),, x x K, 0.5, S 0.5, ( 4- )

58 IMPLEMENTATION 5 ( ) ( ) ( ) ( ) 0.5, 0.5, 0.5,,, 0.5,,, 0.5, 0.5,, Δ Δ = Δ S z J K z z V T V T x J x x V T V T K r r t V V ( 4- ) Here the heat fluxes betwee the elemets are calculated by expressos ( 4-3 ). ( ) ( ) x x U T U T K r r J =,, 0.5, 0.5.5, 0 ( ) ( ) z z U T U T K J =,, 0.5, 0.5, ( ) ( ),, 0.5, , = x x V T V T K r r J ( ) ( ),, 0.5, 0.5, = z z V T V T K J ( 4-3 ) The heat coductvty ad the heat sources at tme 5 0. are approxmated by the curret values. For the dfferet aggregate states of slco dfferet expressos after solvg for, U ad, V are derved. For sold slco the temperature ethalpy relato ( -5 ) s used, whch results equatos ( 4-4 ) ad ( 4-5 ). ( ) ( ) ( ) 0.5,, 0.5,, 0.5, 0.5, 0.5,, 0.5, 0.5, 0.5 3, 4 Δ Δ Δ Δ Δ Δ Δ Δ Δ = Δ Δ = Δ Δ Δ Δ Δ Δ = U U U U U U U U z z x K U T x x z K U T r r z x S x J z J t z x U C z z x K x x z K r r C C c C c t z x C C c C c C c t z x U ( 4-4 )

59 IMPLEMENTATION 53 ( ) ( ) ( ) V V V V V V V V z z x K V T x x z K V T r r z x S x J z J t z x V C z z x K x x z K r r C C c C c t z x C C c C c C c t z x V Δ Δ Δ Δ Δ Δ Δ Δ Δ = Δ Δ = Δ Δ Δ Δ Δ Δ = 0.5,, 0.5,, 0.5, 0.5, 0.5,, 0.5, 0.5, 0.5 3, 4 ( 4-5 ) For lqud slco the expressos are smpler because of the lear temperature ethalpy relato ( -3 ), resultg equatos ( 4-6 ) ad ( 4-7 ). 4 5, U U U C c t z x C C c U Δ Δ Δ = ( 4-6 ) 4 5, V V V C c t z x C C c V Δ Δ Δ = ( 4-7 ) I the phase chage state, the equatos smplfy further because of the costat phase chage temperature to equatos ( 4-8 ) ad ( 4-9 ). ( ) /, U U v m C C T z x t U Δ Δ Δ = ( 4-8 ) ( ) /, V V v m C C T z x t V Δ Δ Δ = ( 4-9 ) A specal treatmet s eeded whe the ethalpy passes wth a tme step a crtcal ethalpy crt H, whch separates sectos of dfferet temperature ethalpy relatos. For example, f the ethalpy exceeds m H, the expresso has to be swtched from the sold oe to that of the phase chage. I ths case the tme step for the curret elemet s splt to oe whch reaches

60 54 IMPLEMENTATION exactly the crtcal ethalpy Δ tcrt ad the resdual tme step Δ t Δtcrt. Ths way the ethalpy chage s calculated both partal tme steps wth vald expressos. The crtcal tme step for U ad V s calculated by equatos ( 4-30 ) ad ( 4-3 ) settg U V,,, = H. crt Δt Ucrt,, = J 0.5, J, 0.5 r r 0.5 K H K, 0.5 crt 0.5, T Δz Δx ( H ) T ( U ) ( H ) T ( U ) U T crt z, crt z x x,, S, ( 4-30 ) Δt Vcrt,, = r 0.5 r K 0.5, T H crt ( V ) T ( H ), x V Δx, x crt J 0.5, ( 4-3 ) K, 0.5 T ( V ) T ( H ), z z Δz crt J, 0.5 S, The calculato of U ad V s doe by loopg over all elemets the correspodg drecto ad usg the correct expresso or a expresso splttg each sgle elemet. The updated ethalpy values are the calculated by averagg accordg to equato ( 3-7 ). For the ADE method, the mplct tme step sze ( 4-9 ) s used. However, wth ths tme step sze umercal stablty problems are observed the soldfcato regme. Therefore the tme step sze s lmted such a way, that the ethalpy ca pass the meltg ethalpy slghtly every elemet, see equato ( 4-3 ). A value of m H m oly 8 3 e J below the meltg ethalpy has bee observed to show good umercal stablty. Note that equato ( 4-3 ) s oly appled f a elemet passes ad udergoes H m the curret tme step.

61 IMPLEMENTATION 55 8 J H m H e 3 Δt = m ADE ( 4-3 ) dh max, dt 4..3 Treatmet of the surface geometry The materal surface geometry chages durg the process due to evaporato ad melt flow. The surface has to be traced wth the smulato code to apply the boudary codtos. Ths s doe by usg the phase array value ph, whch s assged the value 3 for evaporated or geerally empty elemets. The surface s detfed by all occuped elemets ph 3 wth a empty eghbourg elemet ph 3 ad by all occuped elemets at the x, x, y, y, z, z = soluto doma boudary. The values of the surroudgs are the set to the correspodg boudary values. For example the solatg boudary codto egatve z -surface s set by reassgg the shfted temperature array accordg to equato ( 4-33 ). Correspodg equatos are used for every coordate drecto. Ths meas that the surface geometry s approxmated by a starcase profle due to the rectagular grd. Note that for the ADE method, oly a solatg boudary codto s mplemeted for reasos of cosstet loopg over the ablated rego as well. For the cosderato of surface heat flux wth the ADE method a addtoal surface heat source s used, see secto ( ph = 3 ph == )( T T ) T T! 3 ( 4-33 ) z = z z z z The elemet umbers the z -drecto of the surface elemets are stored a array surf for easy access to the surface geometry wth the code. Ths s doe by coutg the empty elemets the z -drecto for every array pot, j. Note that ths formulato s restrcted to a surface wthout overhags, whch would mea multple surfaces crossed by a le at oe x, y posto. I LCPSm such overhags ca ot occur cotrast to the coupled code wth the cosderato of melt flow, where ths formulato ca ot be used. ( ph == 3) surf,, j = sum, j, ( 4-34 ) =.. z The empty elemets eed ot be solved, but are stll cluded the array expressos. Therefore the tme dervatve the empty elemets s set to zero, whch esures o mpact o the tme tegrato algorthm as for example the tme step sze guessg.

62 56 IMPLEMENTATION Because of the short absorpto legth of laser lght lqud slco, the spatal scale to be cosdered the z -drecto s the rego of aometres. Usg a grd wth such a small spacg would result umercal problems ad a very hgh elemet cout. Therefore a effectve surface elemet sze relatvely rough grd spaces. orgal sze result very small Δ zeff s troduced to trac the surface accurately eve o Δ zeff s vared cotuously betwee 0.5 ad.5 tmes the Δ z. Ths way s chose because a varato betwee 0 ad tmes Δ z would Δ zeff values causg umercal problems. The curret surface geometry s stored the array z surf whch s calculated by expresso ( 4-35 ) cosderato of the effectve elemet szes. A exemplary setch of the effectve elemet szes ad z surf s show Fg Δz = ( 4-35 ) surf,, j z surf,, j z, j, Δz surf,, j eff,, j, surf z x ph = 3 z surf, z surf, Δ z Δ z eff,, Δ z eff,, ph = 0 Fg. 4-5: Setch of d surface treatmet wth effectve elemet szes; grey correspods to occuped elemets ad whte to empty oes; dashed les dcate the approxmato of the actual surface 4..4 Itesty dstrbuto, surface reflectos ad raytracg I LCPSm arbtrary temporal pulse shapes ca be used, for example data by osclloscope measuremets. Also, a aalytcal expresso for a typcal smooth laser pulse [46] s mplemeted for a shape factor of oe accordg to equato ( 4-36 ). The fluece F s related to the maxmum testy by expresso ( 4-37 ), ad the tme costat τ 0 was

63 IMPLEMENTATION 57 calculated to correspod to the full wdth half maxmum (FWHM) pulse durato τ p accordg to equato ( 4-38 ). t t () I t = I exp ( 4-36 ) max τ 0 τ 0 F I max = ( 4-37 ) exp τ 0 ().3 τ 0 = τ ( 4-38 ) p exp () For a rectagular pulse shape the testy equals the fluece dvded by the pulse durato. I LCPSm also a arbtrary spatal beam profle ca be used by terpolatg the put data to the curretly used grd. Further aalytcal expressos are mplemeted for a flat top ad a Gaussa profle. The Gaussa dstrbuto s calculated by equato ( 4-39 ). The testy the elemets further away tha three tmes σ gauss from the spot cetre s set to zero. Ths s a more real approxmato tha havg a mathematcally fte sze of the laser spot ad saves computato tme. For the flat top case, the testy s costat over the spot sze ad zero the elemets outsde the spot. The cetre of the laser spot s determed ether by the soluto doma cetre or by the start posto ad sca speed for le scas. I r I exp σ = gauss ( 4-39 ) The user put s usually the pulse eergy, whch s related to the fluece by the spot area. For the flat top case, the spot dameter equals the lqud jet dameter d = 0. 83d. For the spot jet Gaussa case the spot dameter s defed to d spot = σ. gauss F E p = d spot π ( 4-40 )

64 58 IMPLEMENTATION If the average laser power P s gve, the pulse eergy s determed by the repetto rate f p. P E p = ( 4-4 ) f p Before the laser lght reaches the surface, absorpto the lqud jet s cosdered LCPSm. For ths the Lambert Beer law cludg the path elargemet s cosdered to calculate the testy reachg the surface I c. Here the legth of the lqud jet correspods to the worg dstace,.e. ozzle outlet to orgal materal surface. ( δ ( ) I = exp α ( 4-4 ) c I solv l jet zsurf For shallow ablato craters ad grooves the laser lght ca be approxmated to be perpedcular to the surface. The the testy coupled to the materal s descrbed by equatos ( -6 ) ad ( -0 ) appled to the surface elemets. Th flm reflecto s also mplemeted for perpedcular lght. The equato ( -7 ) s optoally appled f the surface elemets correspod to the orgal surface. Otherwse the th flm s assumed to be ablated ad the stadard reflectvty s used. If the laser lght ca ot be approxmated to be perpedcular, a raytracg algorthm s mplemeted to accout for multple reflectos ad a agular depedet reflectvty. For ths the slope of the surface geometry has to be determed. Durg the smulatos, t has bee observed that for stablty reasos a specal secod order gradet scheme s best sutable for the d case. Here the gradet s calculated from the dfferece betwee slghtly off cetred values accordg to equato ( 4-43 ), llustrated Fg Ths formulato shows a smoother slope ad damps the developg of a observed umercal self focusg effect leadg to ty deep holes. z surf = z surf, x Δx 3 x z surf, x x x z surf, x xx 5 Δx 6 z ( Δx Δx ) x surf, x Δx 3 x x z surf, x x x z x surf, x x ( 4-43 ) From the gradet, the surface ormal vector s calculated.

65 IMPLEMENTATION 59 cos s = ( ta ( z ) surf ( ta ( z ) surf ( 4-44 ) z x laser ray learly terpolated surface z surf, Fg. 4-6: Illustrato of smoothed surface slope ad ray reflecto for the d case For the 3d case a regresso plae s calculated usg the eght surroudg surface pots z surf,, j, z surf,, j, surf,, j z, z surf, j, z surf,, z surf,, z surf, ad z surf,. The,, j, j, j, j correspodg formulas are tae from [58]. Tag to accout eght rather tha the mmum requred four pots has the same reasos for stablty ad accuracy as metoed for the d case. Out of the plae equato the ormal vector ca be easly derved. The comg laser beam s dvded to oe dscrete ray for every grd elemet the x y plae wth the correspodg testy. To save computg tme, the raytracg s performed oly for elemets where the testy s greater tha zero. The ray at, j s assumed to be drected the postve z -coordate. The total testy of the ray s splt to a parallel ad a vertcally polarzed compoet. Every compoet has a tal value equal to the half of the total testy I = I = I c. The frst reflecto occurs at the surface elemet, j, surf,, j resultg a arbtrary ray drecto b calculated by equato ( -4 ). From ths pot the ray s traced by teratvely detfyg the eghbourg elemet to whch the ray travels. If the ray reaches a surface elemet, the ray drecto chages aga by reflecto. The tracg s stopped f the ray reaches the soluto doma boudary. If the mrror symmetry s appled, the y -compoet of the drecto of a ray whch reaches the mrror plae s chaged sg. Ths reflected ray correspods to the ray comg from the other sde of the symmetry plae. I

66 IMPLEMENTATION 60 Fg. 4-7 a exemplary ray propagato s show as calculated by the mplemeted 3d raytracg algorthm wth mrror symmetry. sgle ray Fg. 4-7: Ray propagato o exemplary slco crater surface wth mrror symmetry plae at two dfferet vews; colors wthout meag, just for clearer vew of surface geometry At every reflecto the testy compoets are depedetly decreased by the correspodg reflectvty ( - ) ad ( - ). The the total testy beg drected to the materal at the surface elemet equals the sum of the ot reflected testy compoets. If aother ray touches the same surface elemet, the ew testy s added to the prevous oe. To mprove the accuracy of the raytracer, t s further cosdered that the ray s geerally ot reflected at the mdpot of the surface elemet face. Here the ray s assumed to have the sze of the elemet from whch t orgated. The testy s the portoed to the touched surface elemets correspodg to the covered area. A llustrato for the d case s gve Fg The fal testy at the surface I surf determed by reflectos ad raytracg s approxmated to be drected the z -drecto, cludg for arbtrary refracted ray drectos. Ths s feasble at least for the case of surface melt, because the the absorpto coeffcet s smaller tha the effectve elemet sze Δz eff. Therefore, depedetly of the agle, most of the testy s absorbed the surface elemet. Cosderg a agle of the refracted ray would further oly slghtly chage the affected volume of the surface elemet because of the typcally hgh aspect ratos Δx Δz, Δy Δz. The stuato chages for low absorpto coeffcets, especally for 064 m lght o cool slco, where the ray travels through several elemets. The lmtato could be overcome by extedg the raytracer to the refracted rays, but ths s ot mplemeted wth the preset wor.

67 IMPLEMENTATION Sem aalytcal heat sources Oce the testy at each surface elemet I j, surf,, j, s determed, the absorpto equato ( -6 ) has to be solved. Ths s doe by loopg over the elemet colum at, j startg at the surface elemet = z. For every elemet the absorbed testy s calculated ad the surf,, j.. ot absorbed part s gve to the ext elemet. The absorpto coeffcet s cosdered to be costat over oe elemet, so the Lambert-Beer law ca be appled. Ths gves a much more accurate soluto tha assumg a costat testy over the elemet, case the absorpto coeffcet s sgfcatly smaller tha the elemet sze Δ zeff. Ths s the case especally for the surface elemets, as llustrated Fg Usg equatos ( -7 ) ad ( - ) results the testy decrease ( 4-45 ) ad heat source ( 4-46 ). ( z ) I, j, = I, j, exp, j, Δ eff,, j, α ( 4-45 ) S h eff ( ( α Δz ) I = exp eff ( 4-46 ) Δz Surface heat flux due to radato ( -6 ) or heat trasfer ( -7 ) s mplemeted for the ADE method by heat source terms the surface elemets. The correspodg heat s s geerally calculated by expresso ( 4-47 ) S h,, j, surf,, j J surf,, j, surf,, j = ( 4-47 ) Δz eff,, j, surf,, j 4..6 Surface recesso due to evaporato For both evaporato models the latet heat of evaporato s a decsve factor. I LCPSm o desty chages are cosdered, therefore equato ( -30 ) s appled usg the desty of lqud slco at the meltg pot of.54 g l, whch results L v = J mm³. For the ethalpy based evaporato, a sem aalytcal model s used where the ethalpy dstrbuto the z -drecto of the surface elemets s ot treated as a costat value but calculated by aalytcal equatos. Ths gves a more accurate soluto of the evaporato speed ad of the average ethalpy the surface elemets, cludg for rather large elemet szes. Ths model has bee publshed [57] ad was further mproved wth the preset wor. Durg evaporato the ethalpy dstrbuto ear the surface s domated by the heat

68 6 IMPLEMENTATION sources. Therefore the heat coducto term equato ( -3 ) ca be eglected. At the outer surface the ethalpy equals H L. Defg z = 0 at the outer surface the ethalpy v v dstrbuto alog the surface elemet follows the Lambert-Beer law ( -7 ) ad s therefore descrbed by equato ( 4-48 ). ( H L ) exp( z ) H ( z ) = α ( 4-48 ) v v Ths expresso s oly used for vaporzato. z v s determed by settg equato ( 4-48 ) to z < z, where the ethalpy s greater tha the ethalpy of v H v. z v H v L = l α H v v ( 4-49 ) The elemet szes used LCPSm are greater tha typcal values for z v, so expresso ( 4-48 ) ca be used oly partly. For z > zv the ethalpy s ept costat at H v. Ths s feasble, because here most of the lght s already absorbed ad heat coducto taes place. Ths leads to a low decrease of ethalpy wth the elemet sze, whch s well approxmated by a costat value. A typcal ethalpy dstrbuto accordg to ths model s llustrated Fg z x 0 z v H v Δ z eff H z' Fg. 4-8: Typcal pecewse ethalpy dstrbuto a surface elemet as used durg ethalpy based evaporato To calculate the average ethalpy H of a elemet, the ethalpy dstrbuto s tegrated pecewse from z = 0 to z = Δzeff, whch results expresso ( 4-50 ).

69 IMPLEMENTATION 63 H = H C H = v H C Δz v L α H eff v ( exp( α zv )) H v zv ( 4-50 ) If the average ethalpy step about elemet sze H s above the vaporzato pot H v ad creases durg oe tme Δ H the surface elemet sze s reduced due to evaporato. The ew effectve Δz eff ad the ew average ethalpy balace ( 4-5 ) to the formulas ( 4-5 ) ad ( 4-53 ). H are the derved from the eergy ( Δz Δz )( H L ) H Δz = ( H ΔH ) Δz eff eff ( 4-5 ) v v eff eff Δz ( H L H ΔH ) C eff v v H Δ z eff = ( 4-5 ) Lv C H H ( 4-53 ) H = v Δzeff Note that ths model s oly appled durg evaporato for postve ethalpy chages. After evaporato,.e. f the average ethalpy falls below equally dstrbuted over the effectve elemet sze. H v, the ethalpy s aga assumed to be For the Kudse evaporato o smlar sem aalytcal model has bee derved, because the equatos are more complex tha ca be solved aalytcally. Due to the egatve surface heat flux by the latet heat, the outer surface s ot ecessarly hotter tha the average elemet temperature. Therefore the assumpto of a equally dstrbuted ethalpy the surface elemets s a better approxmato tha the case of ethalpy based evaporato. The chage of the effectve elemet sze wth the Kudse model s calculated by the evaporato velocty ( -33 ) ad the curret tme step accordg to equato ( 4-54 ). The surface heat flux s cosdered by updatg the curret elemet ethalpy correspodg to the latet heat of the evaporated volume, see equato ( 4-55 ). Δ z = Δz v Δt ( 4-54 ) eff eff lv

70 64 IMPLEMENTATION H ( Δzeff Δzeff ) L v = H ( 4-55 ) Note that for the Kudse evaporato the tme step has to be restrcted as dscussed before accordg to equato ( 4-0 ). If the elemet sze Δ falls below Δ durg oe tme step, the elemet s z eff,, j, z, j, assumed to be fully evaporated by settg ph 3. The uderlyg elemet the postve, j, = z -drecto s treated as the ew surface elemet. For coservato of mass ad eergy the elemet sze ad the average ethalpy of the uderlyg elemet have to be updated accordg to equatos ( 4-56 ) ad ( 4-57 ). Δ z eff,, j, = Δz Δzeff,, j, ( 4-56 ) Δz H Δz H, j, eff,, j,, j, H,, = ( 4-57 ) j Δzeff,, j, Because the effectve surface elemet szes are depedetly calculated, eghbourg surface elemets the x - ad y -drectos have geerally dfferet elemet szes. Therefore the heat flux betwee two eghbourg elemets s adjusted by the mmum of both elemet szes. A example for the postve x -drecto s gve equato ( 4-58 ). J eff, 0.5, j, = J 0.5, j, m ( Δz, Δz ) eff,, j, Δz eff,, j, ( 4-58 ) 4..7 Dffuso of mpurty atoms slco melt The dffuso coeffcet of mpurty atoms slco melt ca be approxmated by a costat value as dscussed secto.3.4. Furthermore, o melt movemet s cosdered LCPSm, therefore Fc s Law ( -37 ) ca be appled. The mplemetato s doe the same way for the cocetrato C as descrbed secto 4.. for the ethalpy H. The resultg fte dffereces expressos, show equato ( 4-59 ) for d axal symmetry, are much smpler.

71 IMPLEMENTATION 65 dc dt r = D x 0.5 C x x x C r x r Δx x0.5 C C x x x x C z z z C C C z z z z Δz z ( 4-59 ) If dffuso s swtched o LCPSm, for every elemet two equatos, heat trasport ad dopat dffuso, have to be umercally tegrated. For the mplct methods ad the ADE method ths s doe a segregated way by frst solvg for the ethalpy ad afterwards for the cocetrato. Ths s reasoable because the dopat cocetrato does ot fluece the heat trasport equato. The ADE mplemetato s doe accordg to secto 4... The same tme step sze ca be used as for the sgle heat trasport tme tegrato, because the dffuso occurs o sgfcatly greater tme scales tha the heat trasport. Therefore o error s troduced by way of suffcetly large tme step sze. Smlar to the heat trasport equato, a o solatg boudary codto at the surface s mplemeted by a dopat source term. It s determed by the surface cocetrato accordg to equato ( 4-60 ). S sp,, j, surf,, j C surf,, j, eff,, j, surf,, j, j, surf,, j surf,, j = D ( 4-60 ) Δz C For the fte source boudary codto the surface cocetrato C surf s set to a costat value. I the case of the fte source boudary codto the decrease of loadg over tme has to be cosdered. For ths a loadg array Θ, j s used where values are updated after every tme step accordg to expresso ( 4-6 ). The curret surface cocetrato s the calculated by equato ( -40 ). Θ, j = Θ, j Δt C D surf,, j, surf,, j Δz C eff,, j, surf,, j, j, surf,, j ( 4-6 ) I the sold slco rego the cocetrato chage s set to zero because the sold state dffuso s eglected as dscussed secto.3.4. To hder a umercal dopat s at the sold lqud terface, a solatg boudary codto s appled. Ths s doe a smlar way as for the surface boudary codtos. Equato ( 4-6 ) gves a example for the postve z -drecto.

72 66 IMPLEMENTATION ( ph>.5 ph 0. )( C C ) C 0 5 ( 4-6 ) z = C z z z A specal aalyzg feature was mplemeted LCPSm to get the dopg profle a way that correspods to the secodary o mass spectroscopy (SIMS) measuremet techque. Here the o beam targetg o the sample has a dameter of usually greater tha 0 µm ad gves therefore a dopg profle whch s averaged over the spot sze. I LCPSm a correspodg algorthm was mplemeted whch performs the averagg of the cocetrato over a defed spot sze. It also cosders a arbtrary surface geometry as llustrated Fg The program code s gve A.4. o beam dameter groove cross secto averaged rego Fg. 4-9: Illustrato of SIMS method as mplemeted LCPSm 4..8 Multlayer mplemetato LCPSm focuses o slco processg, but the code s bascally able to calculate other materals by adjustg the materal propertes. Furthermore, a multlayer mplemetato was performed to smulate multple layers of materals. Ths s doe by troducg a VOF model smlar to the flud flow multphase model descrbed secto.4.. For each materal a volume fracto array a s assged. The materal propertes, amely the ethalpy temperature relato ad the optcal propertes, are calculated every tme step accordg to equato ( -45 ). Also, the costat materal propertes le the latet heat are stored arrays assged to the actual materal property. The tme tegrato s the performed o the mxture level. I LCPSm up to three layers wth tme depedet terfaces at user defed z -values ca be appled, see Fg The elemet faces are esured to ft exactly the terface posto. I

73 IMPLEMENTATION 67 ths way, o elemet are dfferet materals shared. Especally the pecewse defed temperature ethalpy relato ca the be drectly appled stead of havg a more complex formulato for mxed elemets. Ths actually s o restrcto, because o flow,.e. chage of the materal dstrbuto over tme, s cosdered LCPSm. z x materal a =, a = 0, a3 = 0 materal a = 0, a =, a3 = 0 materal 3 a = 0, a = 0, a3 = Fg. 4-0: Exemplary materal dstrbuto for three layers as used LCPSm For the upper layer the th flm reflectvty ( -7 ) s mplemeted for optoal use ad taes to accout the curret thcess of the layer. Wth ths wor oly a d solver for the multlayer model s mplemeted. A mplemetato 3d would be qute straght forward, but was ot doe wth the tmeframe of the preset wor Grd geerato ad adaptato Besdes the ADE mplemetato, a adaptve grd possesses great potetal to reduce the computg tme compared to a overall fe grd. Laser processg shows large dffereces spatal scales. For example, dopg has to be calculated o the scale of several tes of aometers, whereas the soluto doma should be several hudred mcrometers. Therefore adaptg the grd to the regos of terest greatly reduces the overall elemet cout. The grd adaptato s appled every tme step. The basc strategy dffers betwee the pulse ad the pulse pause, whch s typcally may tmes loger tha the pulse. Durg the pulse a fer grd s used, whch s stored at the pot of full soldfcato. Also, the dopat dstrbuto s stored because t does ot chage durg the pulse pause. After soldfcato a much coarser grd s allowed as well wth a coarseg of the surface geometry. Ths creases the possble tme step sze ad reduces the elemet cout, resultg hghly

74 68 IMPLEMENTATION creased computato speed. Before the begg of the ext pulse, the stored fe grd ad dopat dstrbuto s loaded ad the curret ethalpy values are terpolated to the fe grd. Ths strategy shows a very hgh computato speed the pulse pause whle eepg the precse surface geometry ad dopat dstrbuto. The basc values to set whch determe the allowed rage of elemet szes are Δ y m, ymax Δ xm, xmax Δ, Δ, Δ zm ad Δ zmax. A typcal elemet at the surface has a hgh aspect rato, because the spatal scales of terest the vertcal z -drecto are much smaller tha the lateral drectos. Therefore especally for the explct tegrato method, domatly the overall computg tme ad accuracy. I the pulse pause the reasos dscussed above. Δ zm flueces Δ zm s creased for The startg grd s already adapted to the problem setup. I the lateral drectos the mmum elemet szes are set at the edge of the laser spot dameter. I the vertcal drecto the mmum elemet sze s appled at the upper surface. The eghbourg elemet szes are teratvely creased by a factor of two utl the maxmum allowed elemet szes ad the soluto doma boudary s reached. The resultg grd s llustrated Fg. 4-. If a tal dopg profle s appled, t s esured that the vertcal grd spaces are small eough to suffcetly resolve the dopg profle. If the multlayer model s used, Δ zm s further appled at each sde of the terface posto as llustrated Fg z x Δ x m d Δ x spot max Δ z m Δ z max Fg. 4-: Typcal startg grd of LCPSm aroud the laser spot cetre

75 IMPLEMENTATION 69 The adapto algorthm checs the ethalpy, heat source ad cocetrato dffereces betwee eghbourg elemets. If they are above a crtcal value, the elemets are cosdered for refemet, ad f they are below aother crtcal value they are cosdered for coarseg. Because of the structured grd the adapto s appled to a elemet row for d ad a elemet plae for 3d. The algorthm s explaed the followg for the elemet szes the z - drecto for the d case. For every z -value, the maxmum dfferece to the eghbourg elemets s evaluated accordg to expresso ( 4-63 ) for the ethalpy case. Ths s doe the same way for the heat source ad cocetrato. ( H H ) ΔH ( 4-63 ) max, = max =.. x,, If oe dfferece s above the crtcal value, Δ Cmax > ΔC crt, ad f the elemet sze z Δ H max > ΔH crt or S h, max > ΔS h, crt Δ or Δ s greater tha two tmes Δ zm, a refemet step s performed for both rows at ad the same way. The elemet sze s dvded such a way that the ew sze s ot greater tha the eghbourg elemet sze the egatve z -drecto, see expressos ( 4-64 ) ad ( 4-65 ) for the row. The reaso for ths s that typcally the gradets to be resolved crease for decreasg dstace to the surface affected by the laser. Δz Δz ew = m, Δz ( 4-64 ), Δ ( 4-65 ) z, ew = Δz Δz, ew Oce the elemet szes are determed, values for ethalpy ad cocetrato have to be assged to the ew elemets. For the cocetrato, smply the value of the old elemets s assged. For the ethalpy a more accurate terpolato s appled. Here the slope of the ethalpy betwee the elemets ad s used to set the ew ethalpy values accordg to equatos ( 4-66 ) to ( 4-68 ). Ths formulato esures coservato of eergy ad gves a better approxmato to the actual ethalpy dstrbuto. H H H,, = ( 4-66 ), 0.5 z z

76 70 IMPLEMENTATION H,, ew Δz, ew = H, H ( 4-67 ), 0.5 H,, ew Δz, ew = H, H ( 4-68 ), 0.5 If all dffereces for ethalpy, heat source ad cocetrato are below a crtcal value, the elemet rows ad are cosdered for coarseg. Several restrctos for coarseg are mplemeted. At the begg of the laser pulse o coarseg s allowed to esure catchg of the laser spot wth a sutable grd. Mergg of surface elemets wth empty elemets s ot allowed durg the pulse. Durg the pulse pause t s allowed, but oly to a certa amout to eep roughly the surface geometry. Furthermore, the merged elemet sze must be smaller tha two tmes the eghbourg elemet szes ad smaller tha Δ zmax. For the explct tme tegrato methods, a coarseg step s performed oly oce for every 50 tme steps. Ths s doe to hder grd oscllatos as observed for small tme step szes. For the mplct ad ADE methods, every secod tme step coarseg was show to be practcal. If all restrctos are checed, the elemet rows or plaes are merged. The ethalpy ad cocetrato values are calculated by sze weghted averages accordg to equato ( 4-70 ) for the ethalpy case. Δz = Δz Δz ( 4-69 ), ew H H Δz H Δz,,,, ew = ( 4-70 ) Δz, ew The same algorthm s performed for the lateral coordate drectos. The oly dfferece s that the ethalpy s equally dstrbuted durg a refemet step, rather tha usg the slope method descrbed above. Ths s because especally for flat top laser profles, qute sharp jumps of ethalpy are observed the lateral drecto. Therefore the approxmato of a cotuous slope of ethalpy s bad ad leads to umercal problems. The programmg of a cosstet ad stable adapto algorthm was ot straghtforward, especally because of the arbtrary surface geometry ad the coarseg of the surface

77 IMPLEMENTATION 7 geometry durg the pulse pause. Care had to be tae to the loopg orders ad to esure cosderato of all rows to be refed. For more detals the program code s gve A.. The refemet ad coarseg thresholds of ethalpy, source ad cocetrato were derved emprcally by comparg the smulato results for dfferet settgs. A good choce for the refemet thresholds was observed to be Δ e 5 W ³, Δ H 5 8 crt = e J m³ ad S h, crt = m Δ C crt = 3e 8 m 3. The coarseg thresholds are set to 0.3 tmes the refemet thresholds User settgs ad program structure A rudmetary text based user terface s mplemeted by summarzg all parameters ad other applcable settgs le graphcal output to oe fle amed start.m. Each temperature depedet materal property s set a extra fle, because ths fle has to be loaded durg every tme step. The costat materal propertes ad the parameters for the temperature ethalpy relato are set oe extra fle. For the multlayer smulato, separate fles for each materal are used. A example for the fle start.m ca be looed up A.. I the frst secto the models to be cosdered ca be swtched. Ths s amely d or 3d wth correspodg symmetres, dopat dffuso, evaporato model, raytracg ad laser type. The laser type s ether a dry laser or LCP, where the latter taes to accout absorpto the lqud jet ad the pressure duced by the lqud jet mpgemet o a flat surface. The ext secto provdes settgs for the laser parameters le wavelegth, pulse durato, spot dameter etc. I case of dffuso swtched o, a tal dopg profle from a data fle ad the parameters for the boudary codtos ca be appled. Furthermore, the dmesos of the soluto doma ad the grd adaptato parameters are set. The tme tegrato method ca be swtched betwee the mplemeted models for the evaporato ad o evaporato regme depedetly. The smulato tme s the physcal tme the smulato wll ru, determed ether by the tme spa tself, the sca legth or the umber of pulses. A flexble fucto for graphcal output has bee programmed, whch s cotrolled also by settgs start.m. Bascally a fgure wth four axes s plotted, where every plot ca be defed depedetly. Predefed surface ad curve plots are assged by a umber for each axs. For every plot the axs scales ca be set, otherwse the whole soluto doma s plotted. Multple curves ca be plotted o oe axs by assgg multple umbers to oe plot. Next, the cut coordates ad the surface plot drecto are set to determe the posto ad oretato of the plots the soluto doma. The graphcal output s performed at defed tme tervals durg the smulato, where also the curret grd ad soluto values are stored. Optoally, vdeos of each axs ca be stored wth the frames correspodg to the

78 7 IMPLEMENTATION output tme tervals. A exemplary output of a 3d smulato cludg dopat dffuso s gve Fg. 4-. Fg. 4-: Exemplary output of LCPSm correspodg the settgs of start.m as show A. At the ed of start.m a parameter sweep s mplemeted. Here ay parameter defed above ca be used for sweepg ad the results are automatcally stored separate folders. For a better uderstadg of LCPSm a program flow chart s show Fg. 4-3 for the stadard ADE solver.

79 IMPLEMENTATION 73 H, C, t H C, Fg. 4-3: Program flow chart of LCPSm for the case of the ADE solver 4.. Verfcato To verfy the basc solver for the heat trasport equato, a aalytc soluto for a smplfed case s derved. Assumg tme ad spatally depedet costat laser testy, a oe dmesoal steady state problem wth a costat surface recesso speed v lv s acheved. For a movg frame of referece wth z = 0 fxed at the surface, the heat trasport equato ca be expressed by equato ( 4-7 ). I addto, a costat absorpto coeffcet ad heat coductvty are assumed here. K d H dh = vlv I surf α exp ρ c dz dz 0 p ( α z) ( 4-7 ) For ethalpy based evaporato the surface recesso speed s gve by v = I L, ad the ethalpy value at the surface equals the sum of the ethalpy of vaporzato ad the latet heat H z = 0 = H v L. I a defed rego z < zv, the ethalpy s above H v of vaporzato, ( ) v ad the temperature s costat at the vaporzato temperature. Therefore the heat coducto term ca be eglected. Ths gves a smple expoetal soluto of equato ( 4-7 ). lv surf v

80 74 IMPLEMENTATION H ( z z ) = ( H L ) ( α z) < exp ( 4-7 ) v v v Ths expresso s equal to equato ( 4-48 ), therefore z v s calculated accordg to equato ( 4-49 ). For z > zv the soluto s gve by the sum of two expoetal fuctos. H C ( z > z ) = ( H C ) = H v ρ c v exp K exp( α zv ) K α L ρ c I v v p surf p surf ( ) ( z z ) ( ( )) v C α z zv H L exp v I v ( 4-73 ) I Fg. 4-4 the comparso of d smulato results ad the aalytcal soluto s show. The laser spot dameter the smulato s chose hgh eough to esure that o lateral effects fluece the ethalpy dstrbuto at the laser spot cetre. Typcal propertes for lqud slco 6 are used wth α = e cm ad K = 70W m K. A very good agreemet ca be see, provg the correct mplemetato ad a good umercal accuracy of the basc code for dfferet grd szes as well. Note that the symbols left of the sharp bed at z = zv do ot correspod to elemet cetres, but are calculated by equato ( 4-7 ) for llustrato of the semaalytcal ethalpy dstrbuto aalytc soluto smulato, Δzm = 50 m smulato, Δzm = 0 m ethalpy [J/mm³] dstace to surface [µm] Fg. 4-4: Comparso of aalytc soluto ad smulato results for two dfferet grd settgs ad a ot reflected laser testy of 55 MW/cm²; dotted le correspods to the oset of sold lqud phase chage

81 IMPLEMENTATION 75 The dfferet tme tegrato methods are compared Fg. 4-5 for detcal problem setup. A 53 m laser pulse wth Gaussa beam profle, a pulse durato of 50 s ad a pulse eergy of 83 µj o pure slco s smulated wth the d axal symmetrc solver. The tme tegrato s vared for two grd settgs Δ z m = 00m ad Δ z m = 30m. The results agree very well, whch proves the correct mplemetato ad the grd depedecy of all methods. Fg. 4-6 shows the great fluece of the mmum grd sze o computg tme for the same settgs, ad a speed advatage of the ADE method by a factor of 0 to 0. For other settgs, speed mprovemets of up to 00 tmes ca be observed. Because of ths, the ADE method s set as the stadard tme tegrato method LCPSm. maxmum melt depth [µm] ADE, Δzm = 00 m euler explct, Δzm = 00 m cra colso, Δzm = 30 m tme [µs] Fg. 4-5: Maxmum melt depth over tme for dfferet tme tegrato methods ad grd settgs

82 76 IMPLEMENTATION 0 3 euler explct cra colso ADE computg tme [m] mmum elemet sze [m] Fg. 4-6: Depedecy betwee computg tme ad grd sze for dfferet tme tegrato methods for a typcal laser pulse Further verfcato s doe by reproducg measured ad calculated melt duratos preseted by Lowdes et al. [59]. Oe smulato was performed usg the optcal propertes suggested [59], maly a costat reflectvty of 0.7 of molte slco, ad the other oe wth the models as descrbed ths wor. A secod set of measuremets s show as derved from [60]. The results Fg. 4-7 show a good agreemet wth the data of Lowdes et al., usg the reported settgs. For the stadard settgs, the smulato s stll well wth the measured data. The dffereces betwee the curves ear the meltg threshold could be due to dfferet grds used, because here the melt depth s smaller tha the mmum elemet sze of LCPSm. Ths s also the reaso for the step at roughly molte. 0 m, where the surface elemet s fully

83 IMPLEMENTATION 77 melt durato [µs] Austo et. al. (smoothed) Lowdes et. al. LCPSm, settgs accordg to Lowdes et. al. LCPSm, stadard settgs fluece [J/cm²] Fg. 4-7: Comparso of smulato results wth measured data from lterature, [59] ad [60] To verfy the mplemetato of the dffuso equato, the phosphorous dffuso by a ulmted source slco melt was smulated ad compared wth the aalytcal soluto ( 4-74 ). The results by the aalytcal soluto ad the ADE solver Fg. 4-8 show excellet agreemet ad prove the correct mplemetato off the dffuso equato LCPSm. ( ) z C z = C surf erf ( 4-74 ) D t

84 78 IMPLEMENTATION 0 0 phosphorous cocetrato [cm 3 ] aalytc soluto smulato dstace from surface [µm] Fg. 4-8: Comparso of aalytcal ad smulated dopat profle wth ulmted dopat source slco melt after 7 µs 4.3 Adapto of Fluet The stadard models of Fluet for solvg multphase flow together wth heat trasport show some lmtatos applyg them to the LCP process. The ma lmtato s that o defed codtos ca be set at the free surfaces. Furthermore, oe temperature feld s shared by all phases, whch leads to sgfcat umercal accuraces for hgh temperature dffereces at the free surfaces. Ths s the case for example at the terface betwee the slco melt ad the lqud jet. Ths secto shows how these lmtatos are overcome by usg the possblty of Fluet to solve user defed scalars (UDS) ad addg user defed C-code. Results have bee publshed [6] Separate temperature felds As metoed above, wth the stadard heat trasport ad VOF model of Fluet the temperature dffuses umercally at the free surfaces f the temperature gradet s ot hghly resolved by the grd. Ths s the case at the terface betwee slco melt ad the lqud jet, where wth the scale of mcrometers a temperature dfferece above 000 K occurs. The result s umercal coolg ad soldfcato of the melt surface ad ths way melt flow s stopped at a very early pot tme. Tryg to prevet ths umercal heat trasfer by adjustg the materal propertes at the terface, especally by applyg a very low coductvty, dd ot produce satsfactory results.

85 IMPLEMENTATION 79 To solve ths problem, oe UDS for each phase correspodg to the temperature of the regardg phase s appled. As dscussed secto 3.., ths esures that the temperature s coserved wth the phase boudares, cludg for large dffereces betwee the separate temperature felds. For solvg the heat trasport equato ( -0 ) by the UDS equato ( 3-8 ), the geeral dffusvty has to be set to Γ = K c p ad the heat sources have to be dvded by the heat capacty. A dffusvty UDF s used for adjustg the temperature depedet dffusvty durg each terato. The UDS uder relaxato factor Fluet has to be set to 0.9, whch has bee observed to be a sutable value for good covergece behavour Meltg / soldfcato ad desty chage I Fluet a meltg / soldfcato model s mplemeted by a ethalpy based phase chage approach. However, t requres the stadard temperature to be solved, whch s ot the case f separate temperature felds va UDS are solved stead. Therefore the meltg soldfcato model s addtoally mplemeted by UDF usg the slco UDS temperature feld. For ths a mometum source UDF for radal ad axal mometum s appled accordg to equatos ( -47 ) ad ( -48 ), usg the UDS for the slco phase rather tha the stadard Fluet temperature. For a fast covergece ad umercal stablty, ad also for relatvely large tme step szes, t s mportat to assg also the dervatve of the mometum sources accordg to equato ( 4-75 ). Ths user defed meltg / soldfcato model, combed wth the UDS temperature felds, s umercally ot as stable as the stadard models. Especally a hgh grd qualty s requred to hder accuraces ad dvergece of the pressure feld at the free surface. Therefore ths wor oly rectagular structured grds are used the rego of the free surface. ds mom 3 dv ( ) 5 β = e β 0.00 ( 4-75 ) If heat trasport s smulated Fluet by UDS temperature felds, the latet heat of meltg has to be cosdered durg the phase chage. Ths s doe by mplemetg a user defed source to the UDS. The phase chage taes place wth a defed temperature terval as metoed secto.4.3. If durg oe tme step the temperature chage of a elemet ΔT = T T Δ Tm tersects wth ths phase chage terval, the latet heat has to be partly released as a heat source accordg to the amout of tersecto. The smplest formulato s to use a equally dstrbuted temperature specfc latet heat as llustrated Fg The absolute latet heat to be appled s calculated by the tegral over the tersected temperature

86 IMPLEMENTATION 80 rego. However, ths smple formulato was observed to be umercal ustable because of the sudde oset of the source term whe the temperature frst reaches the phase chage terval. Therefore a susodal dstrbuto of the temperature specfc latet heat was chose such a way, that the tegral over the whole temperature terval equals the latet heat. Δ Δ Δ = m m m m m m T T T T T L L cos π ( 4-76 ) Wth the tersecto terval s T, s T the heat source s calculated by the tegral of equato ( 4-76 ), dvded by the curret tme step sze. Further the expresso has to be multpled by the volume fracto ad dvded by the specfc heat to ft to the temperature UDS. Δ Δ Δ Δ Δ Δ Δ = m m m s m m m s m s s p m h h T T T T T T T T T T T c a t T L S s s π π π ( 4-77 ) T temperature L m /L m /ΔT m equal dstrbuto, ustable susodal dstrbuto, stable T m ΔT m / T m ΔT m / m Fg. 4-9: Normalzed temperature specfc latet heat dstrbuto for the UDS phase chage calculato

87 IMPLEMENTATION 8 Fluet s able to cosder temperature depedet desty chages wth compressble flow whe solvg of the temperature. Usg the UDS temperature felds, the adjustg of the desty s also doe by Fluet but o volume chage s calculated. Ths correspods to a umercal mass source whch s compesated by a UDF mass s accordg to expresso ( 4-78 ). S mass ρ ρ = ( 4-78 ) Δt Free surface heat trasfer Itroducg the depedet temperature felds results o heat trasfer betwee the phases. To cosder physcal heat trasfer wthout umercal dffuso, the model accordg to Fg. -9 s appled to the free surface betwee sold or molte slco ad the lqud meda. Ths s possble, because the lqud sde temperature gradet s satsfactorly resolved by a typcally used grd. To apply equato ( -8 ), the drectoal dervatve of the lqud sde temperature has to be evaluated. Addtoally, the heat flux has to be multpled by the surface area desty,.e. the surface area per volume, to yeld heat source terms that ca be gve to Fluet. The mplemeted algorthm for the free surface heat trasfer model s explaed the followg. A loop over all faces of the curret elemet s performed to detfy the eghbourg elemets. The elemet s treated as a surface elemet, f the lqud volume fracto s greater tha 0.53 ad at least oe eghbourg elemet shows a slco volume fracto greater tha The the face wth the hghest scalar product of volume fracto gradet ad face are vector A face ( a) face s determed to detfy the eghbourg elemet located closest the drecto of the surface ormal. The surface s ow approxmated to be located parallel to the elemet face, whch eds up the stuato llustrated Fg. 4-0 for a d rectagular grd.

88 8 IMPLEMENTATION slco temperature lqud eghbourg elemet elemet face curret elemet possble terface locato Fg. 4-0: Setch of approxmated volume fracto ad temperature dstrbuto at the phase terface The temperature of the surface s approxmated to equal the temperature of the slco the eghbourg elemet ad the lqud temperature s defed the cetre of the curret elemet. For calculatg the lqud sde temperature gradet, the dstace betwee surface ad curret elemet cetre has to be used, whch s calculated by the volume fractos accordg to the deomator expresso ( 4-79 ). Here t s further cosdered that the actual dstace s decreased f the face ormal s ot the drecto of the surface ormal by usg the scalar product of the ormalzed vectors. To hder a zero deomator, the lqud volume fracto has to be greater tha 0.5, whch explas the threshold value of 0.53 as metoed above. dt d = T slco, b T lqud ( x xb ) ( y yb ) ( alqud, b alqud 0.5) Aorm ( a) orm ( 4-79 ) For calculato of the heat sources, expresso ( 4-79 ) has to be multpled by the lqud heat coductvty ad the surface area desty, whch equals the orm of the volume fracto gradet a. For umercal reasos, the volume fracto gradet s ot oe elemet sharp but s spread over the eghbours of the elemet cotag the actual terface. Therefore the volume fracto gradet of the curret ad the eghbourg elemet s added, whch has bee observed to approxmate the real surface area desty satsfactorly. The heat source term for the lqud temperature feld s the gve by equato ( 4-80 ). For the slco temperature feld, the heat source s appled to the eghbourg elemet ad has to be adjusted by the elemet volumes to esure coservato of eergy.

89 IMPLEMENTATION 83 S dt ( a a ) h lqud = K ( 4-80 ), lqud b d V, = ( 4-8 ) V S h slco, b S h, lqud b The algorthm s mplemeted by source UDFs ad s appled wth Fluet to the UDS temperature felds. Detals ca be looed up A.4. The basc algorthm for the terfacal heat trasfer ca be also adapted to arbtrary terfacal models le mass trasfer due to surface reactos or dffuso, whch s ot possble wth the stadard models of Fluet Verfcato Eve the very small spatal ad temporal scales of LCP Fluet produces accurate smulato results for the basc flud flow. Ths ca be derved from publshed smulato results for smlar scales. For example [6] the lqud jet geerato dfferet ozzles comparable to the LCP setup was succesfully smulated by Fluet. To verfy the free surface heat trasfer betwee the UDS temperature felds a steady test case was smulated. A jet mpggmet o slco wth a slghtly agled surface ad a costat temperature of 000 K s calculated such a way that the heght dfferece of the surface at the outer soluto doma boudares s the rage of several elemet depths. Ths way the heat trasfer s calculated for dfferet volume fractos of water at the surface elemets. As a referece, the jet mpgemet o a eve wall wth a temperature of 000 K s smulated usg the stadard heat trasport model of Fluet. The axal symmetrc smulato setups are llustrated Fg. 4-. I both setups a 50 µm wde water jet wth a velocty of roughly 50 m s s used. The mmum grd sze of the rectagular grd rego as llustrated Fg. 4-7 s 00 m axal ad 50 m radal drecto.

90 84 IMPLEMENTATION water jet 300 K 0 µm water jet 300 K eve wall at 000 K slco at 000 K Fg. 4-: Jet mpgemet setups for heat trasfer calculatos; Left: referece setup for Fluet s stadard temperature model; Rght: slghtly ageled slco surface wth the soluto doma for UDS temperature felds ad UDF surface heat trasfer The calculated heat trasfer coeffcets at the surface are compared Fg. 4-. The mplemetato of the free surface heat trasfer shows o accurate results compared to the referece setup. The ma reaso for ths s that the temperature dstrbuto s oly roughly resolved as show Fg Therefore the UDS dervatves show sgfcat approxmato errors depedg o the actual volume fracto values, whch cause the hgh spatal varatos of the resultg surface heat flux. But evertheless the average heatg of the water jet agrees wth a error well below 0 %, amely 7 K for the UDS setup ad 7.5 K for the referece setup. Therefore t s feasble to use ths mplemetato to get a feelg of the amout ad the fluece of the lqud jet coolg.

91 IMPLEMENTATION 85.5 heat trasfer coeffcet [MW/(m²K)] mpgg jet o wall, stadard models mpgg jet o agled surface, UDS UDF radal coordate [µm] 0 water volume fracto Fg. 4-: Heat trasfer coeffcets calculated at the surface for the two setups llustrated Fg temperature [K] dstace to surface [µm] Fg. 4-3: Exemplary temperature dstrbuto at the water slco terface at a radal dstace of 30 µm; marers correspod to elemet cetrods Because of the small temporal ad spatal scales ad the resultg rough dscretzato of the temperature ad velocty gradets, t s mportat to proof that the soluto s depedet o the grd. For ths, the melt expulso by a exemplary LCP laser pulse s smulated o two dfferet grds. A rectagular grd s used for both cases the rego of possble melt expulso. For the rough grd the elemet sze axal drecto s set to 00 m ad the

92 86 IMPLEMENTATION fe grd to 50 m. As startg values the steady state soluto of a 50 µm wde ad 50 m s fast water jet mpgg o the flat slco surface was used. Wth LCPSm the temperature dstrbuto observed at the ed of a 00 s, 53 m, 58 µj laser pulse wth a 50 µm wde water jet was calculated ad exported to Fluet. The the melt expulso was smulated utl full resoldfcato, as show Fg Note that o free surface heat trasfer was cosdered. waterjet 5 µm t = 0 t = µs lqud slco sold slco Fg. 4-4: Start values (left) ad fal crater geometry (rght) of the exemplary melt expulso smulato; creasg red value correspods to creasg temperature of sold slco The fal cotour of the expelled crater s smlar for the rough ad the fe grd. However, some dfferece ca be see Fg Ths meas that the soluto s ot fully grd depedet. But the melt expulso s a hghly ustable ad sestve process ad so the exact predcto of the real surface cotour s mpossble ayway. Therefore the smlarty of these results gves the cocluso that the adapted Fluet code gves good approxmatos for the melt expulso by the lqud jet.

93 IMPLEMENTATION m grd 50 m grd axal coordate [µm] radal coordate [µm] Fg. 4-5: Slco surface posto after µs melt expulso for dfferet elemet szes Besdes evaporato of slco also evaporato of the lqud jet meda s ot yet cosdered wth the developed Fluet code. Ths meas that o vapour flm at the hot slco surface s formed. Ths vapour flm s assumed to sgfcatly reduce the vsous forces betwee the lqud jet ad the slco melt. However, the pressure forces exerted by the lqud jet mpgmet domate the vsous forces by far. To proof ths fact, detcal melt expulso smulatos wth dfferet vscostes of the lqud jet, 0.8mPa s ad.mpa s, where carred out, showg o dffereces of the resultg slco surface after resoldfcato. Therefore the decrease of vscous forces due to a vapour layer would ot fluece the fal smulato results ad s thus feasble to eglect. The evaporato of the lqud jet meda at the slco surface addtoally chages the pressure forces by recol pressure ad compressblty effects. However, t s assumed that the vapour layer forms smoothly at the begg of the laser pulse ad that t shows a quas steady state behavour durg the relatve log melt durato at costat surface temperature after the laser pulse. Ths s dcated by the quas steady state behavour of the lqud jet temperature durg smulatos cludg for surface heat trasfer. Furthermore, the thcess of the flm should be small the order of well below oe mcro, as ca be deduced from the sharp temperature drop Fg So pressure chages are also feasble to egelect as a frst approxmato. 4.4 Couplg of LCPSm ad Fluet As metoed before, LCPSm s a effcet smulato program for the thermodyamc ad optcal effects laser processg but eglects ay flud dyamcal effects. For ths Fluet was

94 88 IMPLEMENTATION adapted to cope wth melt expulso ad by the lqud jet. However, Fluet s mplemeted optcs models are ot sutable for the correspodg optcal effects laser processg. A prcpal problem s that the smulato of laser lght absorpto ad slco meltg requres a much smaller tme step sze tha the smulato of melt flow, but Fluet s ot able to use dfferet tme step szes for dfferet models. For these reasos a traset couplg of LCPSm ad Fluet was programmed [6]. Ths way each physcal effect s solved separately ad effcetly wth sutable tme step szes ad grds Couplg algorthm Wth the coupled code, LCPSm calculates all effects as usual. However, surface recesso due to evaporato s ot calculated,.e. assumg a costat materal dstrbuto durg each tme step. Surface recesso correspods to a movemet of the surface ad has to be cosdered wth Fluet. Wthout cosderato of a vapour phase, ths would mea a mass s leadg to correct results for flud flow. Therefore the coupled code s curretly lmted to laser pulses whch cause o evaporato. Fluet calculates the flud flow ad heat covecto by UDS temperature felds for slco ad water. For the slco UDS the ethalpy rather tha the temperature s used as the scalar, order to allow a drect exchage of data wth LCPSm. Furthermore, terfacal heat trasfer ca be calculated wth Fluet. The heat coducto s calculated by LCPSm rather tha by Fluet, because the fast laser heatg process has to be resolved by the smaller tme step szes of LCPSm. Fluet s used as the ma program resposble for talzato ad cotrollg of the couplg. It starts by settg the tal values ad performg the frst tme step. The the data, amely the curret tme, the curret tme step sze, the slco dstrbuto a slco ad the slco ethalpy, s exported to text fles va a UDF. Afterwards, Fluet s paused ad LCPSm s started. LCPSm reads ad terpolates the data ad performs tme tegrato over the curret Fluet tme step sze by several smaller tme steps. The resultg ethalpy s exported to Fluet ad the ext tme step s calculated by Fluet. Both LCPSm ad Fluet watch a fle cotag a executo flag wth a edless loop utl the rght flag s recogzed ad each chages the flag after calculato respectvely. Ths woraroud s eeded, because there s o possblty for exteral cotrol of Matlab ad Fluet durg the ru. Completely startg ad closg Matlab or Fluet durg each tme step would mea wastg most of the overall computg tme. A flow chart explag the couplg algorthm s gve Fg. 4-6.

95 IMPLEMENTATION 89 Fg. 4-6: Program flow charts for the couplg algorthm of Fluet ad LCPSm; data exchage llustrated by red arrows The bg advatage of the coupled code s the possblty to use dfferet tme step szes ad eve dfferet grds for optcs ad thermodyamcs o the oe had, ad for flud flow o the other had. Ths eables the smultaeous smulato of laser heatg ad melt expulso. A dsadvatage s that every tme step data exchage ad terpolato have to be performed. For typcal settgs, however, t has bee observed that stll the domatg part of the computg tme s spet for solvg rather tha data exchage. Keepg md the hghly customzato ad adaptato to the problem of the coupled code, ths meas that the regardg effects are solved very effcetly. Specal care has to tae to the data terpolato betwee the dfferet grds of Fluet ad LCPSm. Here t has to be esured that the arbtrary slco surface s always oe elemet sharp. For the ethalpy export to Fluet, the empty elemets of LCPSm ear the surface are set to the surface ethalpy values. Ths hders a terpolato error by Fluet leadg to sgfcatly wrog values at the surface. To mmze the error by the multple terpolatos, data exchage s oly performed a predefed rego, where lqud slco could be preset. For the regos where oly sold slco s preset, o Fluet smulato s ecessary, whereas the rego where o slco s preset at all, LCPSm s ot appled. The dfferet regos

96 90 IMPLEMENTATION are llustrated Fg I the tersectg rego, where the data terpolato occurs, a detcal rectagular grd s used both by Fluet ad LCPSm. Fluet data exchage oly o detcal grd LCPSm Fg. 4-7: Illustrato of coupled Fluet ad LCPSm soluto domas wth rego of detcal grd for d axal symmetry 4.4. Verfcato The grd depedecy of the coupled code s already prove by the grd depedecy of Fluet ad LCPSm. What has to be verfcated s the suffcet degree of couplg. To proof ths, a exemplary melt expulso smulato was carred out usg dfferet couplg tme step szes. Smulatos were performed wth Fluet tme step szes of 0.s ad 0.3s. Addtoally a data exchage oly every two Fluet tme steps was appled. I Fg. 4-8 the expelled slco volume over tme s show. The dfferet curves agree wth a error, whch s ot sgfcat for the terpretato of the results wth ths wor. Also cosderg the hgh physcal sestvty of the expulso process to small chages, ths agreemet proves the error troduced by the dscrete couplg to be small.

97 IMPLEMENTATION expelled volume [µm³] s x 0. s 0.3 s tme [µs] Fg. 4-8: Expelled slco volume over tme for a 53 m, 300 s, 50 µj laser pulse ad a 50 m/s, 50 µm jet smulated wth the coupled code for dfferet couplg tme step szes

98 5 Smulato results I ths chapter the applcato of the developed smulato code to several processes of practcal terest s preseted. The am s to crease the basc uderstadg of the processes ad the fluece of the mportat parameters. Furthermore, the smulatos are compared to measuremets, order to ad the correct uderstadg ad terpretato of expermetal results. 5. Heatg ad radcal geerato the lqud jet To correctly calculate the absorpto of the laser lght the lqud jet, t s ecessary to ow the effectve path elargemet due to teral reflecto as descrbed secto... For ths a glass plate was placed uder the lqud jet at varyg dstaces to the ozzle outlet, ad the trasmtted laser power was measured by a laser power meter. A 064 m laser ad water as lqud were used, because uder these crcumstaces measurable absorpto occurs. The resultg power dstace curve was ftted to a expoetal fucto. Here the expoetal factor equals the effectve absorpto coeffcet. The path elargemet s the rato of the measured effectve ad theoretcal absorpto coeffcet accordg to equato ( -9 ) ad was observed to be roughly δ =.. Ths value s used for the smulatos ths secto ad s used as the stadard path elargemet LCPSm. A commo LCP system uses laser lght wth a wavelegth of 064 m ad water as lqud medum because of overall low cost. I ths case however, the absorpto legth s 7 cm whch s the rage of the lqud jet legth, resultg sgfcat absorpto. Therefore smulatos were preformed accordg to secto 4. to detfy the lmts of the system. 9

99 SIMULATION RESULTS 93 relatve testy jet legth [cm] temperature [ C] pulsed, smulato cw, aalytc soluto jet legth [cm] Fg. 5-: Exemplary testy loss (left) ad heatg (rght) the water jet I Fg. 5- the average testy loss ad the temperature dstrbuto over the jet legth for pulsed ad CW laser power are show. A water jet produced by a 50 µm ozzle wth a velocty of 50 m s was used ad the appled laser power was 80 W wth a repetto rate of 3 Hz ad a pulse durato of µs for the pulsed case. Ths correspods to the stadard 064 m laser used at Frauhofer ISE for LCP. It ca clearly be see that ot oly s the power loss a problem, but also the bolg temperature of water ca be reached, whch would lead to jet breaup. The temperature peas duced by the sgle pulses are equalzed over the jet legth because of heat coductvty ad overlappg of several pulses. Therefore the jet heatg by laser pulses s well approxmated by cosderg oly the average power ad applyg the aalytcal equato ( 4-5 ). Ths was observed to be true as well for shorter pulse duratos dow to s. As a cocluso t s mportat that for choosg a well suted LCP system, absorpto ad heatg the lqud jet should be cosdered. Ths s especally true for the waferg applcato, where cuts wth hgh power lasers are performed over several cetmetres jet legth rage, as dscussed [0]. Aother effect vestgated here s the possblty to geerate chlore radcals by lght duced dssocato of chlore molecules. As metoed before these radcals are geerally much more reactve ad should therefore crease the etch rate o slco. Ther presece could mprove the speed ad qualty of etchg asssted processes e.g. the waferg applcato. Oe possble approach for ths dea s based o couplg a laser to a lqud jet cotag chlore for both slco heatg ad radcal geerato at the same tme. The laser wavelegth has to be short eough to allow for dssocato of a molecule by the eergy of a sgle photo. Ths s the case for ultra volet (UV) lght, see [63].

100 94 SIMULATION RESULTS Ths setup was smulated by the d smulato code as descrbed secto 4.. CW laser lght wth a wavelegth of ad a velocty of propertes of Fluorert FC770 wth 355 m ad a power of 5 W a lqud jet wth a dameter of 4 µm 50 m s was assumed. As a low reactve ad absorptve solvet the 3 ρ = 793g m ad c p 038 J g K = were appled for the smulatos. The let molecular chlore cocetrato was vared betwee ad mol l. 0.0mol l I Fg. 5- the resultg testy ad radcal dstrbuto over the jet legth are show. For the hghest molecular chlore cocetrato the laser testy drops to zero wth the frst few mllmetres. The radcals geerated ths rego ca ot be further trasported sgfcatly by the jet flow because of the hgh radcal recombato rate. So for worg dstaces of several cetmetres, early o laser lght or radcals are preset. For the low chlore cocetrato, the laser lght s trasmtted sgfcatly to several cetmetres, but the geerated radcal cocetrato s far too low for practcal relevace. There exsts o optmum of the molecular chlore cocetrato betwee where both sgfcat laser power ad radcal cocetrato are observed the several cetmetres rage. Cosderg that curret LCP maches the mmum worg dstace s at least oe to two cetmetres for techcal reasos, the smulato results show that the couplg of a UV laser for radcal geerato s ot applcable. The alteratve of a sde UV lght s also challegg, because the lght has to be drected very close to the reacto spot to hder recombato of the radcals. For the cuttg of deep grooves these ssues become eve more problematc. relatve testy mol/l 0. mol/l mol/l jet legth [cm] radcal cocetrato [mmol/l] mol/l 0. mol/l mol/l jet legth [cm] Fg. 5-: Smulated testy drop ad radcal cocetrato for dfferet molecular chlore cocetratos wth the lqud

101 SIMULATION RESULTS Ifluece of basc parameters slco laser processg Ths secto ams to ga a uderstadg of the fluece of the basc laser parameters wth the restrcto to low to medum laser power. As metoed earler, for hgh laser power the models are ot suffcetly vald. The parameters of greatest fluece pulsed laser processg are the fluece, the pulse legth ad the wavelegth. There are further parameters le repetto rate, spatal beam profle ad pulse shape. But a geeral fluece of these parameters ca hardly be obtaed because t strogly depeds o the dvdual case. The results of most terest for low to medum power laser processes are the melt depth ad the melt durato. Furthermore, the meltg ad evaporato thresholds are determed so that the relevat parameter regos, for laser dopg processes for example, ca be detfed. For the parameter study, sgle pulses have bee smulated wth the d axal symmetrc solver of LCPSm. A flat top profle wth a relatvely large dameter was used, so that lateral effects do ot fluece the result ear the spot cetre. Thus the smulato results correspod to a d soluto. I Fg. 5-3 ad Fg. 5-4 the smulato results for the dfferet wavelegths are show. The d colour plots show the fluece of the fluece ad the pulse legth over several orders of magtude o the melt durato ad melt depth. I addto, the evaporato threshold s dcated, whch correspods to the fluece value where the bolg temperature s overcome. For flueces sgfcatly above the evaporato threshold, the mplemeted models lose ther valdty ad the results should be tae oly as rough approxmatos. Note that the colour spes occurred due to covergg problems ad are ot physcal. It has bee observed that the wavelegths 355 m ad 53 m show early the same results ad are therefore ot dfferetated. Ths ca be explaed by the relatvely hgh absorpto coeffcets for sold slco both cases. Thus the o-reflected part of the pulse eergy s deposted ear to the surface wth a dstace smaller tha the fal meltg depth. Although the absorpto coeffcet for 355 m s much hgher tha for 53 m, see Fg. -4, the same amout of absorbed eergy s avalable for surface meltg ad evaporato ad therefore o sgfcat dffereces are observed. The observatos usg the laser wavelegth of 064 m dffer cosderably from the wavelegths 355 m ad 53 m. Here the absorpto coeffcet s relatvely low for low temperatures, whch leads to deposto of the pulse eergy wth a depth much greater tha the fal meltg depth. Ths eergy s ot fully avalable for meltg ad evaporato at the surface ad therefore much hgher flueces are eeded for the oset of meltg. Because the eergy s more wdely deposted for the wavelegth of 064 m, the resultg temperature

102 96 SIMULATION RESULTS gradets after the pulse are lower. Ths results lower resoldfcato speeds, whch explas the somewhat hgher melt duratos ad melt depths for equal pulse duratos pulse durato [µs] 0 0 pulse durato [µs] fluece [J/cm²] fluece [J/cm²] Fg. 5-3: Smulated melt duratos [µs] for 53 m (left) ad 064 m (rght) sgle laser pulses; dashed le correspods to the evaporato threshold µ pulse durato [µs] 0 0 pulse durato [µs] µ 4 µ µ fluece [J/cm²] fluece [J/cm²] 0 Fg. 5-4: Smulated melt depths [µm] for 53 m (left) ad 064 m (rght) sgle laser pulses I Fg. 5-5 the meltg ad evaporato thresholds are plotted agast the pulse durato, as they were observed by the sgle pulse smulatos. The thresholds were evaluated also for water wth a pressure of 40bar as a typcal LCP evromet. The meltg thresholds are lowered because the hgher refracto dex decreases the reflectvty at the slco surface, see Fg. -7. Because of the hgher creased surroudg pressure the bolg pot s creased accordg to equato ( -34 ), whch explas the creased evaporato thresholds. As explaed above, usg the wavelegth of 064 m requres more eergy to acheve oset of meltg. The meltg threshold remas smlar for all pulse duratos. I cotrast to ths, the meltg threshold for a shorter wavelegth s much lower ad strogly depedet o pulse durato. The reaso for ths s the creasg amout of eergy coducted away from the

103 SIMULATION RESULTS 97 surface rego wth tme. Oce the surface s molte, the absorpto coeffcet for all wavelegths s hgh eough to allow full eergy desposto wth the molte rego. Ths explas the smlar absolute fluece dfferece betwee the meltg ad the evaporato threshold for both wavelegths, whch s observed whe plottg Fg. 5-5 a lear scale. However, the relatve fluece dfferece s much lower for the wavelegth of 064 m, whch maes t dffcult practse to apply the correct laser power. Because of ths ad the hgher power eeded, 064 m s ot the frst choce for effcet laser processes f accurate cotrol of the meltg behavour s requred. 0 fluece [J/cm²] m, ar 53 m, water 40 bar 064 m, ar 064 m, water 40 bar pulse durato [µs] Fg. 5-5: Thresholds for meltg ad evaporato by sgle pulse rradato for dfferet evrometal codtos; the upper les correspod to the evaporato threshold ad the lower les to the meltg threshold Gog to hgher flueces tha used the above smulatos, the observed dffereces betwee the wavelegths decrease. The reaso for ths s that the part of the fluece eeded to create a surface melt, o whch all wavelegths behave smlarly, becomes eglgble. Ths s also cofrmed by expermetal vestgatos []. Here the ablato effcecy for 064 m s small for relatvely low laser power, but creases for hgher power ad approaches smlar values as for 53 m. Ofte le scas by overlappg pulses are used laser processg. Uder these codtos, the fluece of the pulse o pulse dstace ad the repetto rate becomes a ssue of terest. As a dscusso bass, a exemplary result for the maxmum temperature, whch equals the surface temperature, over tme s show Fg Although the meltg tme s

104 98 SIMULATION RESULTS much loger tha the pulse durato, the temperature drops early to room temperature wth a few mcrosecods. Typcal laser pulse frequeces are below 00 Hz. Ths meas that the tme betwee the laser pulses are loger tha 0 µs, so that for most cases the laser pulses ca be assumed to be thermally depedet. Ths agrees also wth expermetal results. For example, Rodofl showed that LCP dopg le scas produce detcal results for pulse frequeces of Hz ad 35 Hz f the pulse overlap ad pulse eergy are ept costat [64]. temperature [K] tme [µs] 00 testy [MW/cm²] Fg. 5-6: Coolg curve after a 53 m, 30 s, 5 J/cm² laser pulse rradato Specal care ths case has to be tae whe usg the wavelegth of 064 m. There the absorpto coeffcet depeds strogly o temperature. That meas that the amout of lost pulse eergy as dscussed above s very sestve to the tal temperature of the affected slco volume. I Fg. 5-7 a sgfcat lowerg of the meltg threshold for hgher tal temperatures of the slco bul ca clearly be see. Thus, for a 064 m laser wth hgh repetto rates, the remag heat of the prevous pulse s able to fluece the mpact of the followg pulse.

105 SIMULATION RESULTS fluece [J/cm²] temperature [K] Fg. 5-7: Smulated depedecy betwee meltg threshold ad tal slco temperature for a 064 m, 30 s laser pulse 5.3 Dry laser ablato 5.3. Slco The frst quattatve comparsos of smulato ad expermet were performed for dry laser scas. A 064 m laser wth a maxmum average power of 70 W ad a pulse legth of 70 s was used to cut les at dfferet sca speeds, average power ad repetto rate. After breag the processed wafer, cross secto scag electro mcroscope (SEM) pctures were made to measure the groove depth. The Gaussa wdth of the laser spot could ot be determed precsely, ad was therefore vared to ft the expermetal data, resultg σ gauss = 30 µm. Smulatos were carred out for the same laser parameters usg the ethalpy based evaporato model. Because ths s hardly allowed for the appled pulse duratos, the resultg traset results have to be tae wth care. The hgh laser power used lmts the valdty of the results further. However, just by loog at the evaporato depth the smulato should be a feasble approxmato, because the eergy balace s ept. Early results were publshed [57], where a d axal symmetrc verso model was used for the smulatos. Ths method troduces a error heat coducto, because the groove geometry s ot axally symmetrc. However, the heat trasport tag place o a spatal scale greater tha the groove depth s stll well descrbed by the axal symmetrc model.

106 00 SIMULATION RESULTS cuttg depth [μm] % overlap 80% overlap 93% overlap pulse eergy [mj] Fg. 5-8: Measured (squares) ad smulated (les) groove depth for dfferet pulse eerges ad pulse overlaps Fg. 5-9: Scag electro mcroscope (SEM) pcture of a groove cross secto compared to the smulated groove shape (red le) correspodg to a pulse eergy of 4.4 mj ad a pulse overlap of 93 % Nevertheless the results show good agreemet wth the expermetal data as llustrated Fg. 5-8 ad Fg Also, the evaporated cotour s good agreemet wth the SEM pctures ad follows bascally the Gaussa testy dstrbuto. A lot of partcles ca be see lyg o the groove surface. They cosst of redeposted slco from the vapour ad the lqud phase. Ths dcates that melt flow ad vapour dyamcs play a sgfcat role durg the ablato process, whch s ot cosdered wth the smulatos. Despte all of these smplfcatos, the good agreemet wth the smulato results allows the cocluso, that the

107 SIMULATION RESULTS 0 dry laser ablato depth qute learly follows the locally deposted eergy,.e. the fluece. But ths ca oly be stated f pulse eergy ad pulse legth do ot dffer greatly from the parameters vestgated here. From the smulato experece ad lterature [47] t ca be assumed that ths lear relato stays true for hgher pulse legths f the process parameters assure stayg well above the meltg threshold, whereas for shorter pulse legths vapour dyamcs sgfcatly fluece the process results Slco trde layer A topc of great terest curretly solar cell maufacturg s the local opeg of atreflecto ad passvato coatgs by laser ablato. At Frauhofer ISE a process s vestgated where a 355 m laser wth pulse legths aroud 30 s s used to ablate a atreflecto slco trde (SN x ) layer. Sgle pulse expermets were performed by Korz [65] o a 80 m ad 350 m thc SN x layer o a slco wafer. The pulse eergy was vared betwee 0.4 µj µj. Wth a cofocal mcroscope the cotour ad the depth of the ablato crater was determed. For the smulato of the sgle pulses the d axal symmetrc multlayer model was used. Most crtcal was the selecto of sutable materal propertes, because precse data for the requred temperature rage s ot gve curret lterature. Some temperature depedet data were foud [66] for the heat capacty ad the heat coductvty, whch were ftted ad used as extrapolatos up to the bolg pot. [ K] c p = 59.6T K = 0.3T [ K] J g K W m K ( 5- ) The extcto coeffcet was tae from [67] to = 0.. The real part of the refracto dex was measured at Frauhofer ISE as =. ad the desty as e g m. Because SN x s amorphous, o defed meltg pot exsts, ad t ca be treated as a lqud at ay temperature. Therefore o phase chage s cosdered by settg the latet heat of meltg to zero. No data for the bolg pot ad the latet heat of vaporzato has bee foud lterature. Therefore the average bod eergy of SN x was calculated ad assumed to equal the ethalpy eeded for full evaporato startg from zero Kelv. The bolg pot was estmated to be 3000 K, smlar to pure slco. From these assumptos the ethalpy at the bolg pot was calculated accordg to equato ( -4 ). The latet heat of vaporzato s the derved by the dfferece betwee the average bod eergy ad the calculated ethalpy at

108 0 SIMULATION RESULTS the bolg pot to be L v = 0 3.7e J m. Wth ths calculato, the eergy balace s somewhat depedet of the ot precsely ow bolg temperature. The Gaussa wdth of the laser spot was derved from expermets to σ =. µm. gauss 0 I Fg. 5-0 the results are compared, showg good agreemet betwee the smulated ad measured curves despte of the rough approxmatos as dscussed above. Note that the parameters were ot chaged from the orgal settgs to ft the smulato results. Ths leads to the cocluso that the ablato mechasm s well descrbed by the smulato models. Ths meas that the SN x s drectly thermally ablated, ad other mechasms le lft off or drect heatg play a mor role. crater volume [µm³] smulato expermet pulse eergy [µj] crater depth [µm] smulato expermet pulse eergy [µj] Fg. 5-0: Comparso of smulato ad expermet for sgle pulse ablato of a 350 m SN x layer 5.4 Ablato of slco by LCP I cotrast to dry laser ablato, addtoal effects have to be cosdered for ablato by LCP. The mpgemet of the lqud jet results creased surroudg pressure at the surface ad melt expulso. Ths secto shows smulatos wth LCPSm for ablato by evaporato o the oe had, ad wth Fluet to accout for melt expulso o the other had to mprove the uderstadg of the LCP ablato mechasm Evaporato Smulatos were carred out wth LCPSm to calculate the groove depth for dfferet LCP parameters. As descrbed secto 5.3., the d axal symmetrc solver was used to get approxmated results wth a acceptable computg tme. A flat top profle wth the lqud jet dameter s assumed. Ths s feasble for groovg because of the averagg effect due to the pulse overlap. Furthermore, the creased evrometal pressure ad refractve dex caused by the lqud jet are tae to accout. As dscussed before, LCPSm does ot allow

109 SIMULATION RESULTS 03 for the smulato of melt flow. Therefore evaporato s the oly cosdered ablato mechasm for the smulato results show ths secto. The ablato process was vestgated for the hgh power laser commoly used at Frauhofer ISE for the waferg applcato. Ths laser has a wavelegth of average power of 064 m, a maxmum 80 W, a repetto rate of 3 Hz ad a pulse durato of approxmately µs. A parameter study was performed where the ozzle dameter ad the sca speed was vared such a way that a costat pulse overlap of 75 % was acheved. Here the pulse overlap s defed as the rato of the lqud jet dameter ad the pulse to pulse dstace. The resultg groove depths were measured from mcroscope mages of the groove cross secto expermet smulato wthout raytracg smulato wth raytracg groove depth [µm] fluece [J/cm²] Fg. 5-: Comparso of measured groove depths wth the smulato results; dashed le correspods to the lear ft of the smulatos wthout cosderato of raytracg I Fg. 5- the measured ad smulated groove depths are plotted over the laser fluece. Wth raytracg swtched off,.e. multple reflectos are gored, the smulated groove depth shows a lear relatoshp to the laser fluece. Ths meas that usg dfferet lqud jet dameters results the same groove depth f the sca speed ad the laser power are chose such way that a detcal pulse overlap ad laser fluece s acheved. For the expermetal data o lear relatoshp ca be perceved, but a sgfcat curvature s observed. Furthermore, the measured groove depths for early detcal flueces vary stroger tha could be explaed by measuremet accuraces. Ths dcates that eve at the same laser fluece, the lqud jet dameter has a sgfcat fluece o the groove depth. Nevertheless,

110 04 SIMULATION RESULTS the ablato threshold agrees very well wth the smulatos at approxmately ca be derved from Fg J cm as Oe possblty to expla the dffereces of smulated ad expermetal values s the ot cosdered effect of agular depedet multple reflecto. These effects should be depedet o the groove geometry ad therefore o the lqud jet dameter. Therefore the same smulatos were carred out usg the raytracg feature of LCPSm ad the results are also plotted Fg. 5-. It ca be see that some scatterg s observed, whch correspods to the expected depedecy o the lqud jet dameter. A slght curvature s acheved as well. Although the cosderato of raytracg gves qualtatvely better agreemet of the smulated results wth the measuremets, the correlato wth the expermetal results s stll low. Ths meas that other physcal effects, le melt flow ad vapour dyamcs, play a sgfcat role. I the low fluece rego the lqud jet s assumed to expel the molte slco, whch results deeper grooves tha predcted by evaporato oly. For creasg flueces, redeposto of melt ad recodesato of slco vapour taes place ad could expla the decreasg slope Fg. 5-. Therefore the ext step was the vestgato of melt flow Melt expulso by the lqud jet The ablato mechasm of LCP especally for comparably log laser pulse duratos s assumed to be domated by lqud jet drve melt expulso. Thus, ths effect wll be vestgated here. The frst smple model for melt expulso was mplemeted LCPSm by addg a addtoal costat surface recesso speed of.4 m s f the surface s molte, derved from a ft to the expermetal data. For ths the Kudse evaporato model s used ad the addtoal surface recesso speed s added equato ( 4-54 ). As you ca see Fg. 5-, ths smple model gves already a much better agreemet wth the expermetal results, but s stll ot satsfactory. Ths dcates that melt flow duced by the lqud jet plays rather a decsve role ad wll therefore be vestgated more detal.

111 SIMULATION RESULTS expermet smulato wth smple melt expulso model groove depth [µm] fluece [J/cm²] Fg. 5-: Measured ad smulated groove depths wth smple melt expulso model as descrbed the text; LCP parameters as descrbed prevous secto Geerally, the lqud s ot able to accelerate the slco melt stataeously due to mometum forces, meag that t taes some tme utl sgfcat melt expulso occurs. To uderstad the depedece of the ablato mechasm o the laser parameters, t s mportat to ow the tme scale for melt expulso. For ths, a smple melt expulso smulato was carred out usg Fluet. Frst, the steady state jet mpgemet o a flat slco surface was calculated. Next, a exemplary meltg volume wth a depth of 0 µm at the slco surface was stataeously set to a lqud phase havg the propertes of slco melt. Wth ths state the traset smulato was started as show Fg No heat trasport s cosdered ad therefore o resoldfcato of the melt occurs. However, by just loog at the start of the melt expulso, where o sgfcat soldfcato taes place, ths smplfcato s feasble.

112 06 SIMULATION RESULTS water jet 5 µm t = 0 t =.5 µs lqud slco Fg. 5-3: Expulso of a typcal molte volume by the lqud jet wthout cosderato of heat trasport ad resoldfcato The expelled melt volume was plotted agast tme ad was observed to follow a expoetal crease as ca be see Fg The expoetal factor represets the characterstc tme for sgfcat melt expulso ad s evaluated by a expoetal ft. The smulato was performed for a ozzle sze of 00 µm ad for dfferet jet veloctes. I Fg. 5-5 the resultg characterstc tme s plotted versus the jet velocty, correspodg well to a recprocal relatoshp. Ths meas that the speed of melt expulso pretty much learly scales wth the jet velocty rather tha wth jet pressure. If t s ecessary to hder melt flow by the lqud jet, the meltg tme caused by oe laser pulse should be sgfcatly lower tha the characterstc expulso tme. 0.5 smulato result *exp(t/0.34) expelled volume fracto tme [µs] Fg. 5-4: Rato of expelled volume to tal lqud volume over tme for a lqud jet velocty of 00 m/s

113 SIMULATION RESULTS characterstc expulso tme [µs] smulato versely proportoal ft jet velocty [m/s] Fg. 5-5: Characterstc tme for slco melt expulso derved from a expoetal ft to the smulated expelled volume over tme To further uderstad the ablato by melt expulso, smulatos wth cosderato of laser lght absorpto ad resoldfcato were performed. Here the coupled code as descrbed secto 4.4 was appled to two dfferet sgle laser pulses that ca be geerated the LCP systems avalable laboratory. The frst pulse has a wavelegth of durato of 53 m ad a pulse 5 s ad s called short pulse the followg. The secod pulse, called log pulse, has a pulse durato of.3µs ad a wavelegth of 064 m. Because the coupled code s ot able to cosder evaporato, the pulse eergy was chose just below the evaporato threshold. The same water jet propertes wth a velocty of 40 m s ad a dameter of 83 µm, correspodg to a ozzle dameter of 00 µm, were used both cases. A flat top testy profle wth the lqud jet was assumed. 5 µm t =. µs t =. µs t = 3.6 µs Fg. 5-6: Melt expulso dyamcs for the log pulse setup smulated wth the coupled code; creasg red value correspods to creasg temperature of sold slco

114 08 SIMULATION RESULTS I Fg. 5-6 smulato results at dfferet pots tme for the log pulse are show. The coolg of the slco melt due to heat coducto leads to resoldfcato aroud the edges of the reacto spot. Moreover, some droplets detachg from the slco melt ad tae away by the lqud jet ca be see, whch correspods to the slco partcles the waste water observed expermet. Full resoldfcato s reached after 5.3µs, whch s sgfcat lower tha the melt durato of above 0 µs as predcted wthout cosderg melt flow, see Fg The reaso for ths s that heat s spread over a large volume by the melt flow, thus creasg the overall coolg speed. Ths meas that the melt duratos ad melt depths show Fg. 5-3 ad Fg. 5-4 are overestmated for the case of LCP usg pulses whch cause a melt durato above the characterstc expulso tme of Fg The fal cotour the smulato after resoldfcato for both laser pulses s show Fg Because of the smaller melt durato for the short pulse, oly lttle movemet of the melt taes place ad results a hump loog le a small wave wth a sze of a few mcrometers. The log pulse duces much more melt expulso ad eds up a hump wth a lateral sze o the order of 0 µm. These hump szes are good agreemet wth the expermetally observed melt flow structures show Fg Here LCP les wth overlappg pulses correspodg to the smulated pulses were processed. The reaso for the vrtually smaller laser spot sze of about oe thrd of the lqud jet dameter s the homogeeous testy profle wth the jet. It shows somewhat hgher testes ear the cetre ad decreasg testes to the edges. The homogeetes the testy profle are also the reaso for the radomly located melt waves observed for the short pulse, rather tha havg oe rg as the smulato result. Fg. 5-7: 3d vew of smulated slco surface after full resoldfcato for the log pulse (left) ad the short pulse (rght) setup

115 SIMULATION RESULTS 09 Fg. 5-8: Scag electro mcroscope (SEM) pctures of LCP processed slco surfaces by overlappg laser pulses correspodg to the log pulse (left) ad the short pulse (rght) setup; ote the dfferet scales I cocluso, the sze of the expermetal melt flow structures ca be explaed by the melt expulso smulatos. For aosecod laser pulses the melt flow duced by the lqud jet s oly small. Ths s assumed to troduce mor crystal damage ad s therefore sutable for LCP applcatos where a good qualty has to be acheved, e.g. the dopg processes. The loger the pulse duratos, whch result loger melt duratos, the hgher s the amout of melt expulso. For a pulse durato of aroud µs, sgfcat melt movemet was observed the smulatos ad s detfed as the ma ablato mechasm for low pulse eerges. Ths agrees wth the relatvely large expermetal groove depths for low flueces Fg. 5- ad Fg. 5-. Because meltg ad melt removal requres less eergy tha evaporato, the melt expulso ablato mechasm s more effcet ad therefore sutable for waferg applcatos Coolg effect It s ofte metoed that a bg advatage of the lqud jet guded laser s the hgh stu coolg effect, whch reduces the thermal mpact ad therefore produces clea cuts wth low damage the processed materal [7, 68]. Especally for slco solar cell maufacturg crystal damage caused by thermal stress s a very mportat ssue laser processg as t ca serously decrease the solar cell effcecy. Therefore t s worthwhle to have a closer loo at the coolg effect LCP. A hgh drect coolg effect taes place f the ablato mechasm s domated by melt removal. Here the heat s removed together wth the melt ad s ths way trasported away from the reacto zoe. Ths s a advatage to dry laser processg, where a hgh fracto of the expelled ad evaporated slco s redeposted aroud the reacto spot. Thus for log pulse durato or CW, LCP s assumed to troduce less, but stll sgfcat, remag heat ad thermal stress.

116 0 SIMULATION RESULTS Aother coolg mechasm s radato from the slco melt surface. Ths effect s cluded to the smulatos accordg to equato ( -6 ) ad was observed to be ot relevat at all. The secod drect coolg mechasm s the heat trasfer to the lqud jet at the melt surface. The hgh speed mpgg jet shows a very hgh heat trasfer coeffcet aroud 6 0 W ( m K), as derved from the Fluet smulatos, see for example Fg. 4-. Besdes, the heat coducto to the slco bul s also qute fast because of the hgh heat coductvty of slco. Typcally the temperature below the surface chages about 000 K over a spatal scale of a few mcrometers. Ths results a correspodg heat trasfer 7 coeffcet for heat coducto to the slco bul of aroud 0 W ( m K), whch s somewhat larger tha the surface heat trasfer coeffcet, but ca be o the same order of magtude. To estmate the fluece of ths drect coolg effect, melt expulso smulatos were carred out wth cosderato of surface heat trasfer as descrbed secto The same parameters as for the log pulse of the prevous secto were appled. Oly the pulse eergy was slghtly creased to have comparable values for the fal melt durato ad melt depth. Ths was acheved by creasg the pulse eergy by approxmately 9 %. Ths small crease shows that oly a small amout of the totally absorbed heat s trasferred to the lqud jet. Evaporato of the lqud was ot cosdered the smulatos. Because the lqud s cotact wth the slco at melt temperature, a small vapour flm s supposed to be formed real world codtos, actg as a thermal sulato layer. Therefore the coolg effect s eve smaller tha predcted by the smulatos. Nevertheless coolg taes place ad flueces the melt expulso behavour. I Fg. 5-9 the smulato results for the log pulse cludg free surface heat trasfer s show. The ma dfferece to the prevously show smulato s that o smooth sold hump forms at the edge of the reacto spot. Because of the coolg of the melt, resoldfcato taes place before reattachg to the slco surface. Ths results several overlappg layers of resoldfed slco. These layers correspod to the melt flow structures observed expermets, as show detal Fg For sgfcatly shorter laser pulses wthout otable melt expulso, the drect coolg effect s eve smaller because of shorter coolg tme ad s therefore eglgble.

117 SIMULATION RESULTS 5 µm t =. µs t =.64 µs t = 3.6 µs Fg. 5-9: Melt expulso dyamcs for the log pulse wth cosderato of free surface heat trasfer Fg. 5-0: SEM pcture of the flow structure detal correspodg to the left mage of Fg Dry laser dopg A promsg process uder developmet at Frauhofer ISE s local dopg by dry laser processg [69]. Here the phosphorous slcate glass (PSG) remag after the hgh temperature emtter dffuso process acts as a fte dopat source. The laser locally melts the slco surface ud the phosphorous dffuses out of the PSG to the melt. The laser testes must ot be too hgh to hder evaporato of the dopat source layer. Because the laser testes are oly slghtly above the meltg threshold ad o melt movemet by exteral forces taes place, dry laser dopg s supposed to be well approxmated by the models mplemeted LCPSm. Expermets ad smulatos wth LCPSm for dry laser dopg were performed by Wühterch [70]. A 355 m laser wth a pulse eergy of 3.8 µj ad a pulse durato of 40 s was used to process le scas wth dfferet sca speeds. The Gaussa wdth of the laser profle the focal plae was hard to determe precsely the expermet ad was therefore used as a ft parameter for the smulatos. I LCPSm the fte source boudary codto s most approprate to the dopg process out of the PSG. Here the tal loadg could also ot be measured ad was used as a secod ft parameter. Note that the same values for the ft parameters were used for all smulatos.

118 SIMULATION RESULTS I Fg. 5- the dopg profles acheved by LCPSm ad by secodary o mass spectroscopy (SIMS) measuremets of the expermetal laser les are compared. The very good agreemet proves that the dry laser dopg process s well descrbed by laser meltg, lqud phase dffuso ad recrystallzato ad that o other dopat trasport processes, such as covecto, are sgfcat. The good agreemet of the surface cocetrato values usg the same tal loadg for both pulse to pulse dstaces shows further that the smple fte source model s a feasble approxmato of the real process. Thus, the smulato ca be used to detfy a sutable rego for the process parameters, f a defed surface cocetrato ad dopg depth are requred. 0 SIMS measuremet smulato dopat cocetrato [/cm³] µm 4 µm depth [µm] Fg. 5-: Comparso of smulato result ad SIMS measuremet of slco samples processed by dry laser dopg wth two dfferet pulse to pulse dstaces (4µm ad 0 µm) 5.6 LCP dopg The stadard laser used for LCP dopg produces laser pulses wth a wavelegth of pulse duratos of aroud uses a fluece of aroud 53 m, 5 s ad a repetto rate of 35 Hz. The best process at preset J ad a sca speed of mm s 0.5 cm 50. Uder these codtos, melt duratos are assured whch are suffcetly short to avod sgfcat melt expulso by the lqud jet as ca be derved from sectos 5. ad Therefore the eglectg of melt flow seems feasble as a frst approxmato for ths process ad LCPSm ca be used. As dscussed secto.3.4, o physcal boudary codto for dffuso at the melt surface s

119 SIMULATION RESULTS 3 appled. Therefore the surface cocetrato C surf wth the fte source boudary codto s set to the surface cocetrato of the correspodg measured dopg profles Smulatos wth flat top profle Due to the spatal averagg of the homogeeous testy profle by the hgh pulse overlap appled the stadard dopg process, t s feasble to use a flat top testy profle as a frst approxmato. To smulate several overlappg pulses wth practcable computg tmes, the d axal symmetrc verso of LCPSm was used. Ths s correct for the heat trasfer wth shallow grooves, as dscussed secto The dopat dstrbuto after several pulses s ot axally symmetrc. However, most of the dffuso taes place z -drecto ad the lateral dffuso affected by the axal symmetrc formulato plays oly a mor role, especally for the flat top case. Thus, the results for temperature ad dopat dstrbuto are well approxmated by the axal symmetrc model. To cosder evaporato, the Kudse model was chose due to the relatvely short pulse durato. I Fg. 5- two dfferet dopg profles of LCP les acheved by SIMS measuremets are show as dopat cocetrato values over the depth from the surface. Oe SIMS profle correspods to the best LCP solar cell processed so far wth 0.4% effcecy [3]. The dopg profle produced wth the same laser parameters, but a hgher fluece s also gve. Furthermore, the smulato result for the 0.4% parameters s plotted. Both the smulated ad the measured profle for the 0.4% parameters show a smlar smooth decrease of dopat cocetrato wth µm to.5µm. Ths dcates that the dopat trasport process s well descrbed by laser meltg ad lqud phase dffuso as mplemeted LCPSm. The dffereces betwee the smulated ad the measured profle are due to the homogeeous testy profle wth the lqud jet, as dscussed detal secto For the hgher pulse eergy, a dfferet shape of the dopg profle s acheved. The dopg depth s creased ad a strogly varyg curvature s observed, whch s ot achevable by smulatos wth LCPSm. Ths meas that for the hgher pulse eergy, other trasport processes besdes lqud phase dffuso must play a sgfcat role. For LCP dopg expermets wth hgher pulse durato, smlar o smooth dopg profles were measured. Here t s obvous that due to sgfcat melt flow duced by the lqud jet, covectoal trasport of the dopat taes place. Ths s assumed to be the reaso for the dfferet curve shape comparso to the pure lqud phase dffuso process. For the shorter pulse duratos but hgher pulse eerges, t s assumed that melt flow duced by evaporato ad recol pressure leads to covectoal trasport processes.

120 4 SIMULATION RESULTS 0 phosphorous cocetrato [/cm³] smulato, 0.4 % solar cell SIMS, 0.4 % solar cell SIMS, hgh pulse eergy depth [µm] Fg. 5-: Dopg profles after LCP dopg derved from SIMS measuremets ad smulato results The terpretato of the dopat trasport mechasms ca be appled to the measured solar cell effceces. For the LCP dopg parameters usg hgh pulse eerges or hgh pulse duratos, o reasoable solar cell effceces have yet bee acheved at Frauhofer ISE. Recombato the space charge rego was detfed as the prmary reaso for ths, see [4]. Furthermore, Rodofl showed that for the hgh flueces a great fracto of the phosphorous s electrcally actve [64]. Ths leads to the cocluso that melt movemet ad covectoal dopat trasport reduces the overall dopg qualty wth respect to solar cell effcecy. Cosderg the small pulse to pulse dstace of.4 µm for the stadard sca speed of 50 mm s, the fal le cossts oly of the outer regos of a sgle pulse mpact as llustrated Fg The outer regos are much more affected by melt flow tha the er rego ad are therefore assumed to show a worse dopg qualty. Usg lower pulse overlaps, the er rego of a sgle pulse mpact becomes more domat ad should ths way sgfcatly crease the overall dopg qualty. Upublshed expermets at Frauhofer ISE showed a mprovemet of the electrcal characterstcs of LCP processed slco samples usg hgher pulse eerges wth lower pulse overlap ad cofrm ths theory.

121 SIMULATION RESULTS 5 bad dopg qualty good dopg qualty hgh overlap low overlap Fg. 5-3: Illustrato of the fluece of the pulse overlap o the surface qualty 5.6. Ifluece of homogeeous testy profle Some expermetal observatos usg low pulse eerges could ot be descrbed by the smulatos wth the flat top testy profle. For example, mcroscope pctures of LCP processed slco samples show structures wth a sze much smaller tha the lqud jet dameter, see Fg Furthermore, expermets a LCP dopg effect s observed eve for average flueces sgfcatly below the meltg threshold. For example [64] a chage electrcal propertes starts at aroud threshold of 0.J cm, although smulatos predct a meltg J for the correspodg pulse durato. The reaso for ths was 0.35 cm assumed to be the homogeeous testy profle wth the lqud jet cross secto ad was therefore vestgated more detal. 50 µm 50 µm Fg. 5-4: Mcroscope mages of LCP processed slco samples usg the stadard dopg parameters wth low flueces; left: 9 sgle pulses; rght: four les produced by hgh pulse overlap Frst, measuremets of the testy profle were carred out. For ths am a glass plate was placed such a way that the lqud jet mpges ormally oto t. The plate was maged oto a CCD camera usg a mcroscope objectve. Very low laser testes had to be used to

122 SIMULATION RESULTS 6 prevet damage to the camera. It s assumed that the optcal propertes of the couplg optcs ad the lqud jet are smlar for the laser testes these measuremets ad those durg LCP dopg. Therefore the testy dstrbuto should be the same ad the measured profle ca be used for all pulse eerges. I Fg. 5-5 the camera mage for the stadard dopg laser wth a ozzle dameter of 60 µm s show. The observed dameter of 53.9 µm agrees well wth the expected value of 0.83 tmes the ozzle dameter. Because the gray values G are lear to the laser lght testy, they ca be scaled by settg the area tegral equal to the curret laser power. Ths results the ormalzed testy dstrbuto ( 5- ), where x = 0 ad y = 0 correspods to the spot cetre. The pxel area s the correspodg object area rather tha the physcal pxel sze. Ths matrx s saved to a Matlab fle ad ca be used LCPSm by terpolato o the curret grd ad multplcato wth the curret average testy. I ( x, y ) = G ( x, y ) I d spot π G pxel Apxel ( 5- ) ormalzed testy all pxels y [µm] x [µm] Fg. 5-5: Itesty profle wth the lqud jet for the stadard LCP dopg parameters; rght: orgal camera mage; left: calculated ormalzed testy The ormalzed testy dstrbuto as calculated ad used for the smulatos s show Fg A specled dstrbuto s observed, whch s caused by the terferece of several wavemodes due to the multmode wavegude characterstcs of the lqud jet. It ca be see that the testy s locally creased up to a factor of fve compared to the average testy. Ths meas that local meltg ad evaporato starts sgfcatly earler tha predcted by a flat testy profle ad explas the early oset of the dopg effect ad the sub-spot sze