Lamellar Spacing Modelling for LPBF Aluminum Parts

Ci a ion: Anglada, E.; Ga cía, J.C.; A ue, M.; Cea solo, X.; Ga mendia, I. Lamella Spacing Modelling o LPBF Aluminum Pa s. J. Manu . Ma e . P ocess. 2022,6, 164. h ps://doi.o g/ 10.3390/jmmp6060164 Academic Edi o : S e en Y. Liang Recei ed: 18 No embe 2022 Accep ed: 15 Decembe 2022 Published: 17 Decembe 2022 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). Manu ac u ing and Ma e ials P ocessing Jou nal o A icle Lamella Spacing Modelling o LPBF Aluminum Pa s E a Anglada 1, JoséCa los Ga cía1, Ma io A ue 2, Xabie Cea solo 2and Iñaki Ga mendia 3,* 1TECNALIA, Basque Resea ch and Technology Alliance (BRTA), Mikele egi Pasealekua, 2, E-20009 Donos ia-San Sebas ián, Spain 2IMH, Azkue Auzoa 1, E-20870 Elgoiba , Spain 3Mechanical Enginee ing Depa men , Enginee ing School o Gipuzkoa, Uni e si y o he Basque Coun y UPV/EHU, Plaza de Eu opa, 1, E-20018 Donos ia-San Sebas ián, Spain *Co espondence: [email p o ec ed] Abs ac : The high cooling a es eached du ing me al addi i e manu ac u ing (MAM) gene a e mic os uc u es e y di e en om hose ob ained by o he con en ional manu ac u ing me hods. The e o e, esea ch abou he modeling o his ype o mic os uc u e is o g ea in e es o he MAM communi y. In his wo k, he p edic ion o he lamella spacing o an AlSi10Mg sample manu ac u ed by lase powde bed usion (LPBF), is p esen ed. A mul iscale app oach is used, combining a CALPHAD (Compu e Coupling o Phase Diag ams and The mochemis y) model o p edic he ma e ial p ope ies, wi h a mac oscale model o he sample manu ac u ing and wi h a mic oscale model o p edic he mic os uc u e. The manu ac u ing and me allog aphic cha ac e iza ion o he sample is also included. The esul s p o e ha he mul iscale s a egy ollowed is a alid app oxima ion o simula e his ype o manu ac u ing p ocess. In addi ion, i is shown ha he use o a gene ic simula ion so wa e ocused on me al cas ing p ocesses can be use ul in p edic ing he lamella spacing o he mic os uc u e manu ac u ed by LPBF. Finally, he ela ionship be ween he cooling a e and he esul ing lamella spacing has been es ablished o his AlSi10Mg unde he speci ic manu ac u ing condi ions conside ed. Keywo ds: LPBF; mic os uc u e; simula ion 1. In oduc ion Addi i e manu ac u ing (AM) is a dis up i e echnology ha makes i possible o manu ac u e geome ies ha a e almos impossible o achie e by o he manu ac u ing me hods. This ac , oge he wi h i s good adap abili y o cus omized p oduc ion and wi h he possibili y o being in eg a ed in o in elligen p oduc ion sys ems, makes AM an essen ial ing edien o Indus y 4.0, whe e i may become a key echnology [ 1 ]. Al hough he e a e s ill some doub s abou i s applicabili y in mass p oduc ion, he u iliza ion o AM in he indus y is on he ise in di e en sec o s such as ae ospace, biomedical, au omo i e, e c. In ac , he o e all AM ma ke o me al and polyme sys ems was alued a 8.33 billion EUR in 2021 wi h a p ojec ed g ow h o compound annual g ow h a e (CAGR) o 18.2 % un il 2026. Rela ed o he me al AM (MAM) ma ke alone, i was alued a 2.50 billion EUR in 2021 wi h an expec ed CAGR o 25.5% un il 2026 [2]. The main ypes o p ocesses used in MAM a e powde bed usion (PBF), di ec ene gy deposi ion, ma e ial je ing, and binde je ing. Conside ing powe bed usion, he wo main echnologies o his g oup a e lase powde bed usion (LPBF), also known as selec i e lase mel ing (SLM), and elec on beam mel ing (EBM). The LPBF, one o he mos ex ended in his indus y, basically consis s o he deposi ion o a me allic powde laye on a pla o m and he selec i e mel ing o he powde by he lase . Once he laye has been selec i ely mel ed, he pla o m descends, and a new laye o me allic powde is deposi ed o be selec i ely mel ed; his p ocess is epea ed un il he comple e pa is manu ac u ed laye by laye [3]. J. Manu . Ma e . P ocess. 2022,6, 164. h ps://doi.o g/10.3390/jmmp6060164 h ps://www.mdpi.com/jou nal/jmmp J. Manu . Ma e . P ocess. 2022,6, 164 2 o 14 An impo an cons ain o MAM is ela ed o he esul an mic os uc u e o he manu ac u ed pa , as i is di ec ly co ela ed wi h he ma e ial ' s mechanical pe o mance. The high empe a u es eached du ing he p ocess, oge he wi h he ex emely as hea ing and cooling a es, may p omo e he appea ance o ce ain mic os uc u es, which a e de imen al o he mechanical p ope ies. Fo example, he ine mic os uc u e ha esul s om high cooling a es can be de imen al o c ack p opaga ion esis ance in a igue beha io [4]. In ac , cu en ly, he e a e only a limi ed numbe o alloys ha can be p ocessed by MAM since he es p esen a poo pe o mance ela ed o ho c acking and/o low mechanical p ope ies [5–9]. The e o e, mic os uc u e p edic ion is a e y use ul ool no only o suppo he p ocess pa ame e ’s op imiza ion bu also o help in he de elopmen o new alloys mo e sui able o be used in his manu ac u ing p ocess. Se e al au ho s ha e s udied di e en app oaches o he modeling o he LPBF. Fo example, Bida e e al. s udy he in e ac ion be ween he lase beam and he powde bed by means o FEM analysis, including he dynamics o he gas and plasma phases o s ainless s eel 316 L [ 10 ]. Acha ya e al. de eloped a model combining a CFD model wi h a phase- ield model o p edic he mic os uc u e o he Inconel 718 [ 11 ]. Zhang e al. use a 3-D hea ans e ini e elemen model o p edic ing mel pool dimensions in he LPBF o s ainless s eel 17-4PH [ 12 ]. Denlinge e al. model he he mal beha io o he Inconel 718 du ing he LPBF manu ac u ing by means o a 3D Lag angian ansien he mal analysis [ 13 ]. The e a e also wo ks de o ed o he pa icula case o aluminum alloys, al hough he numbe is lowe . Fo example, Azizi e al. s udied he LPBF o AlSi by means o a phase- ield o mula ion [ 14 ], and Rehman e al. s udied he spa e o ma ion and splashing-induced de ec s in AlSi10Mg LPBF [ 15 ]. Many o he s ha e also made in e es ing con ibu ions [16–22]. The mo i a ion o he p esen wo k is o con ibu e o he LPBF modeling by p esen ing an al e na i e app oach o ackle his ype o simula ion. In addi ion, he expe imen al esul s by hemsel es a e also an in e es ing con ibu ion o o he esea che s in e es ed in his ype o manu ac u ing me hod. The esea ch hypo hesis is he possibili y o p edic ing he mic os uc u e o an LPBF pa h ough a mul iscale and mul i-so wa e app oach, being one o he so wa e speci i- cally de eloped o o he ypes o me allic manu ac u ing p ocesses, whe e cooling a es a e e y di e en om hose expe imen ed in LPBF. To alida e his hypo hesis, he man- u ac u ing p ocess simula ion was ca ied ou , combining h ee di e en models. One he modynamic model, one model a he mac oscale le el ep esen a i e o LPBF manu ac- u ing, and one model a he mic oscale le el ocused on he mic os uc u e e olu ion. The models ha e been adjus ed, aking as a e e ence he me allog aphic s udy o expe imen- ally manu ac u ed samples. The main no el y o he app oach is he mul iscale modeling: he so wa e used o p edic he mic os uc u e was no de eloped o he LPBF mic os uc u es. This app oach may be use ul o hose esea che s ha ha e no a ailable speci ic so wa e ools o his manu ac u ing p ocess bu a e in e es ed in hei modeling. The wo k con ibu ion is in ended o p o ide an al e na i e app oach o his ype o modeling a he han closing a speci ic esea ch gap in his opic. 2. Ma e ials and Me hods In o de o be able o alida e he simula ion models de eloped in his wo k, sample pa s ha e been manu ac u ed and cha ac e ized o e alua e hei mic os uc u e expe imen ally. The alloy s udied is he AlSi10Mg. I s chemical composi ion has been analyzed by op ical emission spec ome y, ob aining he composi ion shown in Table 1. J. Manu . Ma e . P ocess. 2022,6, 164 3 o 14 Table 1. Alloy chemical composi ion. Al Si Fe Zn Mg Cu Ti Mn Ni Base 10.7 0.21 <0.01 0.26 <0.01 0.02 <0.01 <0.01 2.1. Samples Manu ac u ing The sample pa s we e buil using a Samylabs ALBA300 machine (Figu e 1), which has a ibe lase wi h a maximum powe ou pu o 250 W and 1080 nm wa eleng h. The lase spo ocalized in he subs a e co esponds o a diame e o 71 µm (1/e2 me hod). J. Manu . Ma e . P ocess. 2022, 6, 164 3 o 15 Table 1. Alloy chemical composi ion. Al Si Fe Zn Mg Cu Ti Mn Ni Base 10.7 0.21 <0.01 0.26 <0.01 0.02 <0.01 <0.01 2.1. Samples Manu ac u ing The sample pa s we e buil using a Samylabs ALBA300 machine (Figu e 1), which has a ibe lase wi h a maximum powe ou pu o 250 W and 1080 nm wa eleng h. The lase spo ocalized in he subs a e co esponds o a diame e o 71 μm (1/e2 me hod). Figu e 1. Samylabs ALBA300 machine a IMH Campus. The build chambe was illed wi h a gon (pu i y le el 99.999%) be o e ope a ion and a cons an shielding gas low was used du ing he build o spa e emo al ollowing [23]. The oxygen con en in he build chambe was main ained below 0.43 ol% (4300 ppm) du ing he p ocess ollowing [24,25]. The samples we e buil di ec ly on EN AW-5083 aluminum subs a e wi hou p e- hea ing because he equipmen in which he expe imen was ca ied ou does no ha e a subs a e hea e . The 20 mm hick subs a e pla e unc ioned as a hea sink o allow hea o be conduc ed away om he samples. The manu ac u ing pa h was gene a ed by SamyS udio 5.1 so wa e using a powde laye hickness o 25 µm. Di e en pa ame e s we e se o he co e and he edge o he pa o imp o e pa densi y and su ace quali y (Figu e 2). The co e was buil by a bidi- ec ional meande ha ching pa e n wi h 67 deg ees clockwise u n be ween laye s. The edge o he pa was buil by a single con ou pa h. Figu e 2. Manu ac u ing pa h s a egy: laye pa h and pa pa h. Figu e 1. Samylabs ALBA300 machine a IMH Campus. The build chambe was illed wi h a gon (pu i y le el 99.999%) be o e ope a ion and a cons an shielding gas low was used du ing he build o spa e emo al ollowing [ 23 ]. The oxygen con en in he build chambe was main ained below 0.43 ol% (4300 ppm) du ing he p ocess ollowing [24,25]. The samples we e buil di ec ly on EN AW-5083 aluminum subs a e wi hou p e- hea ing because he equipmen in which he expe imen was ca ied ou does no ha e a subs a e hea e . The 20 mm hick subs a e pla e unc ioned as a hea sink o allow hea o be conduc ed away om he samples. The manu ac u ing pa h was gene a ed by SamyS udio 5.1 so wa e using a powde laye hickness o 25 µ m. Di e en pa ame e s we e se o he co e and he edge o he pa o imp o e pa densi y and su ace quali y (Figu e 2). The co e was buil by a bidi ec ional meande ha ching pa e n wi h 67 deg ees clockwise u n be ween laye s. The edge o he pa was buil by a single con ou pa h. J. Manu . Ma e . P ocess. 2022, 6, 164 3 o 15 Table 1. Alloy chemical composi ion. Al Si Fe Zn Mg Cu Ti Mn Ni Base 10.7 0.21 <0.01 0.26 <0.01 0.02 <0.01 <0.01 2.1. Samples Manu ac u ing The sample pa s we e buil using a Samylabs ALBA300 machine (Figu e 1), which has a ibe lase wi h a maximum powe ou pu o 250 W and 1080 nm wa eleng h. The lase spo ocalized in he subs a e co esponds o a diame e o 71 μm (1/e2 me hod). Figu e 1. Samylabs ALBA300 machine a IMH Campus. The build chambe was illed wi h a gon (pu i y le el 99.999%) be o e ope a ion and a cons an shielding gas low was used du ing he build o spa e emo al ollowing [23]. The oxygen con en in he build chambe was main ained below 0.43 ol% (4300 ppm) du ing he p ocess ollowing [24,25]. The samples we e buil di ec ly on EN AW-5083 aluminum subs a e wi hou p e- hea ing because he equipmen in which he expe imen was ca ied ou does no ha e a subs a e hea e . The 20 mm hick subs a e pla e unc ioned as a hea sink o allow hea o be conduc ed away om he samples. The manu ac u ing pa h was gene a ed by SamyS udio 5.1 so wa e using a powde laye hickness o 25 µm. Di e en pa ame e s we e se o he co e and he edge o he pa o imp o e pa densi y and su ace quali y (Figu e 2). The co e was buil by a bidi- ec ional meande ha ching pa e n wi h 67 deg ees clockwise u n be ween laye s. The edge o he pa was buil by a single con ou pa h. Figu e 2. Manu ac u ing pa h s a egy: laye pa h and pa pa h. Figu e 2. Manu ac u ing pa h s a egy: laye pa h and pa pa h. J. Manu . Ma e . P ocess. 2022,6, 164 4 o 14 The co e pa ame e s used o he manu ac u ing o he samples we e 800 mm/s scanning speed, con inuous 250 W lase powe and 0.15 mm ha ch wid h. These p ocess pa ame e alues we e se by he p e ious Box–Behnken Design o Expe imen s analysis op imized o minimizing po osi y. Mo e in o ma ion abou Box-Behnken Design o Ex- pe imen s can be ound a [ 26 ]. The con ou pa h was buil by 800 mm/s scanning speed, con inuous 200 W lase powe and 0.07 mm con ou dis ance, which we e de e mined based on p e ious wo ks. The sample geome y was designed in o de o a oid addi ional suppo s uc u es, o ule ou any e ec hey migh ha e on he inished pa , and o acili a e he manual emo al o pa s om he subs a e (Figu e 3). J. Manu . Ma e . P ocess. 2022, 6, 164 4 o 15 The co e pa ame e s used o he manu ac u ing o he samples we e 800 mm/s scan- ning speed, con inuous 250 W lase powe and 0.15 mm ha ch wid h. These p ocess pa- ame e alues we e se by he p e ious Box–Behnken Design o Expe imen s analysis op imized o minimizing po osi y. Mo e in o ma ion abou Box-Behnken Design o Ex- pe imen s can be ound a [26]. The con ou pa h was buil by 800 mm/s scanning speed, con inuous 200 W lase powe and 0.07 mm con ou dis ance, which we e de e mined based on p e ious wo ks. The sample geome y was designed in o de o a oid addi ional suppo s uc u es, o ule ou any e ec hey migh ha e on he inished pa , and o acili a e he manual emo al o pa s om he subs a e (Figu e 3). Figu e 3. Manu ac u ed samples o me allog aphic cha ac e iza ion. 2.2. Me allog aphic Cha ac e iza ion Me allog aphic samples we e p epa ed in h ee di e en planes (Figu e 4) o check he ela ionship o mic os uc u e–di ec ion, assuming ha he componen s manu ac- u ed by LPBF show a high aniso opic empe a u e dependence. Figu e 4. Cu ing planes o ca y ou he mic os uc u e inspec ion. Samples we e p epa ed acco ding o he classical me hod [27]: esin embedded, pol- ished, and e ched (2 mL o HF, 3 mL o HCl, 5 mL o HNO3, and 190 mL o wa e ). The obse a ion o mic os uc u es was pe o med by means o ligh (b igh and da k ields) and elec onic mic oscopy. 2.3. Modeling A mul iscale app oach has been used o model he mic os uc u e e olu ion. Fi s , he he mo-physical ma e ial p ope ies ha e been calcula ed based on he alloy's chemi- cal composi ion. Nex , he sample manu ac u ing p ocess has been simula ed by means o a mac oscale simula ion, aking in o accoun he p e iously calcula ed ma e ial p op- Figu e 3. Manu ac u ed samples o me allog aphic cha ac e iza ion. 2.2. Me allog aphic Cha ac e iza ion Me allog aphic samples we e p epa ed in h ee di e en planes (Figu e 4) o check he ela ionship o mic os uc u e–di ec ion, assuming ha he componen s manu ac u ed by LPBF show a high aniso opic empe a u e dependence. J. Manu . Ma e . P ocess. 2022, 6, 164 4 o 15 The co e pa ame e s used o he manu ac u ing o he samples we e 800 mm/s scan- ning speed, con inuous 250 W lase powe and 0.15 mm ha ch wid h. These p ocess pa- ame e alues we e se by he p e ious Box–Behnken Design o Expe imen s analysis op imized o minimizing po osi y. Mo e in o ma ion abou Box-Behnken Design o Ex- pe imen s can be ound a [26]. The con ou pa h was buil by 800 mm/s scanning speed, con inuous 200 W lase powe and 0.07 mm con ou dis ance, which we e de e mined based on p e ious wo ks. The sample geome y was designed in o de o a oid addi ional suppo s uc u es, o ule ou any e ec hey migh ha e on he inished pa , and o acili a e he manual emo al o pa s om he subs a e (Figu e 3). Figu e 3. Manu ac u ed samples o me allog aphic cha ac e iza ion. 2.2. Me allog aphic Cha ac e iza ion Me allog aphic samples we e p epa ed in h ee di e en planes (Figu e 4) o check he ela ionship o mic os uc u e–di ec ion, assuming ha he componen s manu ac- u ed by LPBF show a high aniso opic empe a u e dependence. Figu e 4. Cu ing planes o ca y ou he mic os uc u e inspec ion. Samples we e p epa ed acco ding o he classical me hod [27]: esin embedded, pol- ished, and e ched (2 mL o HF, 3 mL o HCl, 5 mL o HNO3, and 190 mL o wa e ). The obse a ion o mic os uc u es was pe o med by means o ligh (b igh and da k ields) and elec onic mic oscopy. 2.3. Modeling A mul iscale app oach has been used o model he mic os uc u e e olu ion. Fi s , he he mo-physical ma e ial p ope ies ha e been calcula ed based on he alloy's chemi- cal composi ion. Nex , he sample manu ac u ing p ocess has been simula ed by means o a mac oscale simula ion, aking in o accoun he p e iously calcula ed ma e ial p op- Figu e 4. Cu ing planes o ca y ou he mic os uc u e inspec ion. Samples we e p epa ed acco ding o he classical me hod [ 27 ]: esin embedded, pol- ished, and e ched (2 mL o HF, 3 mL o HCl, 5 mL o HNO3, and 190 mL o wa e ). The obse a ion o mic os uc u es was pe o med by means o ligh (b igh and da k ields) and elec onic mic oscopy. 2.3. Modeling A mul iscale app oach has been used o model he mic os uc u e e olu ion. Fi s , he he mo-physical ma e ial p ope ies ha e been calcula ed based on he alloy ' s chemical J. Manu . Ma e . P ocess. 2022,6, 164 5 o 14 composi ion. Nex , he sample manu ac u ing p ocess has been simula ed by means o a mac oscale simula ion, aking in o accoun he p e iously calcula ed ma e ial p ope ies. A e wa ds, he empe a u e p o iles ob ained as a esul o he mac oscale modeling ha e been used o de ine bounda y condi ions ha , applied o he mic os uc u e e olu ion model, p oduce equi alen cooling a es in he ma e ial (see Figu e 5). J. Manu . Ma e . P ocess. 2022, 6, 164 5 o 15 e ies. A e wa ds, he empe a u e p o iles ob ained as a esul o he mac oscale model- ing ha e been used o de ine bounda y condi ions ha , applied o he mic os uc u e e o- lu ion model, p oduce equi alen cooling a es in he ma e ial (see Figu e 5). Figu e 5. Flow cha o he mul iscale models' connec ion. 2.3.1. The mo–Physical Ma e ial P ope ies Calcula ion Ma e ial p ope ies ha e been calcula ed by a CALPHAD (Compu e Coupling o Phase Diag ams and The mochemis y) based modeling. The CALPHAD me hodology is based on compu a ional he modynamics models, which use expe imen al in o ma ion collec ed o bina y and e na y subsys ems o ex- apola e he p ope ies o highe o de alloys (mul icomponen sys ems), p edic ing as- pec s like he phases o some he mo-physical p ope ies, such as, o example, he densi y o he en halpy among o he s [28]. In his case, comme cial CALPHAD so wa e [29] has been used o calcula e he ma- e ial p ope ies o he alloy. A mic oseg ega ion model known as back di usion has been used o p edic he ma e ial p ope ies, assuming a cooling a e acco ding o alues ound in he bibliog aphy o he LPBF p ocess (750 ×·103 K/s) [30] and he chemical composi ion p e iously measu ed (Table 1). The back di usion model [31] can be conside ed a balance be ween he well-known le e ule model (which conside s a comple e mixing o he so- lu e in he solid, ha is a e y good di usion) [32] and he Scheil model (which assumes no di usion a all in he solid phase) [33]. The solid ac ion cu e has been ob ained o- ge he wi h he densi y, he he mal conduc i i y, and he en halpy, all o hem empe a- u e-dependen (Figu e 6). Figu e 5. Flow cha o he mul iscale models' connec ion. 2.3.1. The mo–Physical Ma e ial P ope ies Calcula ion Ma e ial p ope ies ha e been calcula ed by a CALPHAD (Compu e Coupling o Phase Diag ams and The mochemis y) based modeling. The CALPHAD me hodology is based on compu a ional he modynamics models, which use expe imen al in o ma ion collec ed o bina y and e na y subsys ems o ex ap- ola e he p ope ies o highe o de alloys (mul icomponen sys ems), p edic ing aspec s like he phases o some he mo-physical p ope ies, such as, o example, he densi y o he en halpy among o he s [28]. In his case, comme cial CALPHAD so wa e [ 29 ] has been used o calcula e he ma e- ial p ope ies o he alloy. A mic oseg ega ion model known as back di usion has been used o p edic he ma e ial p ope ies, assuming a cooling a e acco ding o alues ound in he bibliog aphy o he LPBF p ocess (750 ×· 10 3 K/s) [ 30 ] and he chemical compo- si ion p e iously measu ed (Table 1). The back di usion model [ 31 ] can be conside ed a balance be ween he well-known le e ule model (which conside s a comple e mixing o he solu e in he solid, ha is a e y good di usion) [ 32 ] and he Scheil model (which assumes no di usion a all in he solid phase) [ 33 ]. The solid ac ion cu e has been ob ained oge he wi h he densi y, he he mal conduc i i y, and he en halpy, all o hem empe a u e-dependen (Figu e 6). J. Manu . Ma e . P ocess. 2022,6, 164 6 o 14 J. Manu . Ma e . P ocess. 2022, 6, 164 6 o 15 Figu e 6. The mo-physical p ope ies ob ained. The so wa e used o model he LPBF p ocess is no able o wo k wi h empe a u e- dependen p ope ies, he e o e, conside ing he as solidi ica ion a es and he punc ual hea sou ce, he a e age alue a solid s a e has been used o he densi y and he he mal conduc i i y. The empe a u e co esponding o 50% o he solid ac ion has been as- sumed as he mel ing poin . Ano he es ain o he so wa e is he equi emen o use he speci ic hea (𝐶𝑝) alue ins ead o he en halpy alue (𝐻). The e o e, he speci ic hea cu e has been calcula ed om he en halpy cu e a solid-s a e, ollowing Equa ion (1), whe e (T) ep esen s he empe a u e, and i s a e age alue has been used in he simu- la ion. 𝐶𝑝(T)=𝜕H 𝜕T (1) 2.3.2. Sample Manu ac u ing Modeling The sample manu ac u ing has been modeled by means o comme cial so wa e spe- ci ically de eloped o he LPBF simula ion [34]. The sample geome y is a small cube wi h a py amidal suppo whose dimensions can be seen in Figu e 7 (le ). Figu e 7. Sample geome y (le ), mesh ( igh ). Figu e 6. The mo-physical p ope ies ob ained. The so wa e used o model he LPBF p ocess is no able o wo k wi h empe a u e- dependen p ope ies, he e o e, conside ing he as solidi ica ion a es and he punc ual hea sou ce, he a e age alue a solid s a e has been used o he densi y and he he mal conduc i i y. The empe a u e co esponding o 50% o he solid ac ion has been assumed as he mel ing poin . Ano he es ain o he so wa e is he equi emen o use he speci ic hea (Cp) alue ins ead o he en halpy alue (H) . The e o e, he speci ic hea cu e has been calcula ed om he en halpy cu e a solid-s a e, ollowing Equa ion (1), whe e (T) ep esen s he empe a u e, and i s a e age alue has been used in he simula ion. Cp(T)=∂H ∂T(1) 2.3.2. Sample Manu ac u ing Modeling The sample manu ac u ing has been modeled by means o comme cial so wa e speci ically de eloped o he LPBF simula ion [ 34 ]. The sample geome y is a small cube wi h a py amidal suppo whose dimensions can be seen in Figu e 7(le ). J. Manu . Ma e . P ocess. 2022, 6, 164 6 o 15 Figu e 6. The mo-physical p ope ies ob ained. The so wa e used o model he LPBF p ocess is no able o wo k wi h empe a u e- dependen p ope ies, he e o e, conside ing he as solidi ica ion a es and he punc ual hea sou ce, he a e age alue a solid s a e has been used o he densi y and he he mal conduc i i y. The empe a u e co esponding o 50% o he solid ac ion has been as- sumed as he mel ing poin . Ano he es ain o he so wa e is he equi emen o use he speci ic hea (𝐶𝑝) alue ins ead o he en halpy alue (𝐻). The e o e, he speci ic hea cu e has been calcula ed om he en halpy cu e a solid-s a e, ollowing Equa ion (1), whe e (T) ep esen s he empe a u e, and i s a e age alue has been used in he simu- la ion. 𝐶𝑝(T)=𝜕H 𝜕T (1) 2.3.2. Sample Manu ac u ing Modeling The sample manu ac u ing has been modeled by means o comme cial so wa e spe- ci ically de eloped o he LPBF simula ion [34]. The sample geome y is a small cube wi h a py amidal suppo whose dimensions can be seen in Figu e 7 (le ). Figu e 7. Sample geome y (le ), mesh ( igh ). Figu e 7. Sample geome y (le ), mesh ( igh ). J. Manu . Ma e . P ocess. 2022,6, 164 7 o 14 The manu ac u ing p ocess has been modeled conside ing he expe imen al condi ions p e iously de ailed in Sec ion 2.1. The model includes he powde laye s deposi ion and he hea ans e phenomena, including he powde usion by he lase and he cooling. Re e ing o he bounda y condi ions, he con ec ion coe icien has been ixed o a alue equi alen o na u al con ec ion (12.7 W/m 2 K). The powe abso p ion is included as a pe cen age, and i has been ixed a 11%. The p ocess condi ions co espond o hose used du ing he pa manu ac u ing (Table 2). Table 2. P ocess pa ame e s. Ha ch Space (mm) Lase Diame e (mm) Veloci y (mm/s) Lase Powe (W) Powde Laye Thickness (mm) 0.15 0.071 800 200 0.025 The main phenomenon conside ed in he simula ion is hea ans e , he e o e, he main equa ion sol ed is he hea ans e equa ion shown in (2). Q=ρ·Cp∂T ∂ − ∇·(k·∇T)(2) The model is sol ed by he ini e elemen me hod. Al hough ou in e es is ocused on he he mal beha io du ing he LPBF manu ac u ing, he he mo-mechanical sol e has been used due o i s be e s abili y when compa ed wi h he s abili y o he he mal sol e alone, included in he so wa e. In ac , using he he mal so wa e alone, con e gence p oblems we e expe ienced ha p e en ed i s use in his case. Re e ing o he ma e ial p ope ies, he mel ing poin , he densi y, he he mal con- duc i i y, and he speci ic hea alues p e iously calcula ed ha e been used (Table 3). The emissi i y alue co esponds o he de aul alue included in he so wa e da abase o his alloy. Fo he mechanical p ope ies (Young’s modulus, Poisson a io, e c.), he de aul alues included in he da abase o his alloy ha e been used. Table 3. Ma e ial p ope ies. Densi y (kg/m3)Speci ic Hea (J/kgK) Conduc i i y (W/mK) Mel ing Poin (K) Emissi i y 2608.79 1005.11 151.55 846.82 0.18 The geome y has been disc e ized by a egula mesh o med by oxel elemen s wi h 0.25 mm side (Figu e 7 igh ); his size is 10 imes bigge han he powde laye hickness, bu i has no been possible o employ smalle elemen s due o so wa e limi a ions. The empe a u e– ime cu e o e e y node has been ex ac ed and p ocessed o ob ain he solidi ica ion cooling a e (SCR), which has been calcula ed ollowing Equa ion (3), whe e T ep esen s empe a u es exp essed in K and ep esen imes exp essed in seconds. The SCR is exp essed in K/s. SCR (K/s)=Tliquidus(K)−Tsolidus(K) liquidus(s)− solidus (s)(3) The SCR dis ibu ion in he manu ac u ed pa has been analyzed by applying clus e - ing echniques. Mo e p ecisely, a KMeans algo i hm has been used o classi y he cooling a es in 5 di e en clus e s. KMeans is a classi ica ion algo i hm ha clus e s he da a by ying o sepa a e he samples in n g oups o equal a iance, minimizing a c i e ion known as he ine ia o wi hin-clus e sum-o -squa es [35]. J. Manu . Ma e . P ocess. 2022,6, 164 8 o 14 2.3.3. Mic os uc u e P edic ion The inal objec i e is o p edic he lamella spacing o he mic os uc u e. Conside ing ha he magni ude o de expec ed o he lamella spacing is mic ons, a educed model based on a ep esen a i e olume elemen (RVE) has been used o equilib a e he use o a mesh smoo h enough wi h a easonable calcula ion ime. The RVE used is a cube 0.25 mm side, whose heigh is equi alen o 10 powde laye s (0.025 mm/laye ). I has been d awn and disc e ized by a ini e elemen mesh o med by e ahed al elemen s 0.025 mm side, gi ing a mesh o med by 1479 nodes and 6786 elemen s (Figu e 8le ). J. Manu . Ma e . P ocess. 2022, 6, 164 8 o 15 2.3.3. Mic os uc u e P edic ion The inal objec i e is o p edic he lamella spacing o he mic os uc u e. Conside - ing ha he magni ude o de expec ed o he lamella spacing is mic ons, a educed model based on a ep esen a i e olume elemen (RVE) has been used o equilib a e he use o a mesh smoo h enough wi h a easonable calcula ion ime. The RVE used is a cube 0.25 mm side, whose heigh is equi alen o 10 powde laye s (0.025 mm/laye ). I has been d awn and disc e ized by a ini e elemen mesh o med by e ahed al elemen s 0.025 mm side, gi ing a mesh o med by 1479 nodes and 6786 ele- men s (Figu e 8 le ). Figu e 8. RVE model (le ). Hea ex ac ion condi ion ( igh ). Comme cial so wa e speci ically de eloped o me al cas ing simula ions [36] has been used o p edic he mic os uc u e. As a consequence, some assump ions mus be made in o de o se up he case o be ep esen a i e o he LPBF p ocess. The idea is o ep oduce only he solidi ica ion p ocess. Fo his eason, he ini ial empe a u e o he alloy has been ixed, equal o liquidus empe a u e (585 °C), and a hea ex ac ion bound- a y condi ion has been applied o he op su ace o he RVE (Figu e 8 igh ). The hea ex ac ion bounda y condi ion has been adjus ed o p oduce an SCR in he RVE simila o he alues p edic ed in he LPBF p ocess simula ion, ollowing equa ion (4). Conside ing a olume equal o 1.5625 ·10−11 m3, an a e age alue in he solidi ica- ion ange o he densi y and he speci ic hea (𝜌 = 2507 kg/m3,𝐶𝑝 =1132.8 J/kgK) and a SCR equal o 698,062 K/s (see Sec ion 3), a alue equal o 30.98 W is ob ained. 𝑄 (W) = 𝑉(m3)· 𝜌(kg m3 ⁄ ) ·𝐶𝑝(J/kgK) · 𝑆𝐶𝑅(K/s) (4) The main go e ning equa ions o he physics in ol ed in alloy cooling and solidi i- ca ion, o his case, a e he hea ans e Equa ion (2) and he nuclea ion and g ow h o he g ains. The so wa e sol es he nuclea ion and g ow h o he g ains ollowing Equa- ions (5) and (6) p oposed by Old ield [37], whe e 𝑁𝑛𝑢𝑐𝑙𝑒𝑖 is he numbe o nuclei, 𝐴𝑒 and 𝑛 a e he nuclea ion cons an s, T is he empe a u e, (𝑑𝑅_𝑛𝑢𝑐𝑙𝑒𝑖)/𝑑 is he g ow h, and 𝜇𝑔 is he g ow h coe icien . N𝑛𝑢𝑐𝑙𝑒𝑖 = 𝐴𝑒·(∆T)𝑛 (5) 𝑑𝑅𝑛𝑢𝑐𝑙𝑒𝑖 𝑑 = 𝜇𝑔·(∆T)2 (6) Ma e ial p ope ies we e calcula ed by he CALPHAD me hod p e iously men- ioned, bu he alues o he nuclea ion and g ow h cons an s (𝐴𝑒, 𝑛, 𝜇𝑒) mus be adjus ed depending on he alloy, he cooling a e, he me al ea men , e c. In his case, an op imi- za ion algo i hm has been used o adjus hem. The model adjus men can be conside ed an op imiza ion p oblem, whe e he objec- i e is o minimize he di e ences be ween he alues p edic ed by he simula ion and he Figu e 8. RVE model (le ). Hea ex ac ion condi ion ( igh ). Comme cial so wa e speci ically de eloped o me al cas ing simula ions [ 36 ] has been used o p edic he mic os uc u e. As a consequence, some assump ions mus be made in o de o se up he case o be ep esen a i e o he LPBF p ocess. The idea is o ep oduce only he solidi ica ion p ocess. Fo his eason, he ini ial empe a u e o he alloy has been ixed, equal o liquidus empe a u e (585 ◦ C), and a hea ex ac ion bounda y condi ion has been applied o he op su ace o he RVE (Figu e 8 igh ). The hea ex ac ion bounda y condi ion has been adjus ed o p oduce an SCR in he RVE simila o he alues p edic ed in he LPBF p ocess simula ion, ollowing equa ion (4). Conside ing a olume equal o 1.5625 · 10 −11 m3 , an a e age alue in he solidi ica ion ange o he densi y and he speci ic hea ρ=2507 kg/m3,Cp =1132.8 J/kgK and a SCR equal o 698, 062 K/s (see Sec ion 3), a alue equal o 30.98 W is ob ained. Q(W)=Vm3·ρkg/m3·Cp(J/kgK)·SCR(K/s)(4) The main go e ning equa ions o he physics in ol ed in alloy cooling and solidi ica- ion, o his case, a e he hea ans e Equa ion (2) and he nuclea ion and g ow h o he g ains. The so wa e sol es he nuclea ion and g ow h o he g ains ollowing Equa ions (5) and (6) p oposed by Old ield [ 37 ], whe e Nnuclei is he numbe o nuclei, Ae and n a e he nuclea ion cons an s, T is he empe a u e, (dR_nuclei)/d is he g ow h, and µg is he g ow h coe icien . Nnuclei =Ae·(∆T)n(5) dRnuclei d =µg·(∆T)2(6) Ma e ial p ope ies we e calcula ed by he CALPHAD me hod p e iously men ioned, bu he alues o he nuclea ion and g ow h cons an s (Ae,n,µe) mus be adjus ed de- pending on he alloy, he cooling a e, he me al ea men , e c. In his case, an op imiza ion algo i hm has been used o adjus hem. The model adjus men can be conside ed an op imiza ion p oblem, whe e he objec i e is o minimize he di e ences be ween he alues p edic ed by he simula ion and he e e ence alues measu ed expe imen ally. This is achie ed modi ying he alues assigned J. Manu . Ma e . P ocess. 2022,6, 164 9 o 14 o he a iables o be co ela ed (in his case, he nuclea ion and g ow h cons an s) un il he alue p edic ed (in his case, he lamella spacing) ma ches he alues measu ed ex- pe imen ally. This minimiza ion can be made by manually modi ying he alues assigned o he a iables [ 38 – 41 ] o by means o op imiza ion algo i hms. Se e al au ho s ha e explo ed he use o di e en ypes o op imiza ion algo i hms o model adjus men s [ 42 – 48 ] in di e en ields. In his case, he NEWUOA algo i hm included in he Py hon Pa allel Global Mul iobjec i e Op imize [ 49 ] was used o pe o m he adjus men . NEWUOA is a de e minis ic local op imiza ion algo i hm de eloped by M.F.D. Powell is well known o his wo k in nume ical analysis, especially in op imiza ion [50]. 3. Resul s and Discussion 3.1. Me alog aphycal Cha ac e iza ion The mic os uc u e in planes YZ and XZ, as de ined in Figu e 4, shows a s uc u e e y simila o hose ob ained in a ypical welding p ocess. The di e en weld seams can be obse ed in Figu e 9. J. Manu . Ma e . P ocess. 2022, 6, 164 9 o 15 e e ence alues measu ed expe imen ally. This is achie ed modi ying he alues as- signed o he a iables o be co ela ed (in his case, he nuclea ion and g ow h cons an s) un il he alue p edic ed (in his case, he lamella spacing) ma ches he alues measu ed expe imen ally. This minimiza ion can be made by manually modi ying he alues as- signed o he a iables [38–41] o by means o op imiza ion algo i hms. Se e al au ho s ha e explo ed he use o di e en ypes o op imiza ion algo i hms o model adjus men s [42–48] in di e en ields. In his case, he NEWUOA algo i hm included in he Py hon Pa allel Global Mul iobjec i e Op imize [49] was used o pe o m he adjus men . NEWUOA is a de e minis ic local op imiza ion algo i hm de eloped by M.F.D. Powell is well known o his wo k in nume ical analysis, especially in op imiza ion [50]. 3. Resul s and Discussion 3.1. Me alog aphycal Cha ac e iza ion The mic os uc u e in planes YZ and XZ, as de ined in Figu e 4, shows a s uc u e e y simila o hose ob ained in a ypical welding p ocess. The di e en weld seams can be obse ed in Figu e 9. Figu e 9. Mic os uc u e on he XZ (le ) and YZ ( igh ) planes (×100). In he plane XY, a di e en pa e n can be obse ed (Figu e 10). This is mo e ela ed o he welding pa h along he lase beam mo emen on he uppe su ace o he sample (diagonally in e laced). Figu e 10. Mic os uc u e on he XY plane ×50 (le ), ×100 ( igh ). To ha e a close iew o he mic os uc u e, samples we e obse ed in a scanning elec on mic oscope. The images (Figu e 11) showed a needle-like s uc u e (eu ec ic ype) qui e ypical in as -cooled me als. These needles appea ine in he co e o he weld seam and sligh ly coa se in he bo de o adjacen seams. Figu e 9. Mic os uc u e on he XZ (le ) and YZ ( igh ) planes (×100). In he plane XY, a di e en pa e n can be obse ed (Figu e 10). This is mo e ela ed o he welding pa h along he lase beam mo emen on he uppe su ace o he sample (diagonally in e laced). J. Manu . Ma e . P ocess. 2022, 6, 164 9 o 15 e e ence alues measu ed expe imen ally. This is achie ed modi ying he alues as- signed o he a iables o be co ela ed (in his case, he nuclea ion and g ow h cons an s) un il he alue p edic ed (in his case, he lamella spacing) ma ches he alues measu ed expe imen ally. This minimiza ion can be made by manually modi ying he alues as- signed o he a iables [38–41] o by means o op imiza ion algo i hms. Se e al au ho s ha e explo ed he use o di e en ypes o op imiza ion algo i hms o model adjus men s [42–48] in di e en ields. In his case, he NEWUOA algo i hm included in he Py hon Pa allel Global Mul iobjec i e Op imize [49] was used o pe o m he adjus men . NEWUOA is a de e minis ic local op imiza ion algo i hm de eloped by M.F.D. Powell is well known o his wo k in nume ical analysis, especially in op imiza ion [50]. 3. Resul s and Discussion 3.1. Me alog aphycal Cha ac e iza ion The mic os uc u e in planes YZ and XZ, as de ined in Figu e 4, shows a s uc u e e y simila o hose ob ained in a ypical welding p ocess. The di e en weld seams can be obse ed in Figu e 9. Figu e 9. Mic os uc u e on he XZ (le ) and YZ ( igh ) planes (×100). In he plane XY, a di e en pa e n can be obse ed (Figu e 10). This is mo e ela ed o he welding pa h along he lase beam mo emen on he uppe su ace o he sample (diagonally in e laced). Figu e 10. Mic os uc u e on he XY plane ×50 (le ), ×100 ( igh ). To ha e a close iew o he mic os uc u e, samples we e obse ed in a scanning elec on mic oscope. The images (Figu e 11) showed a needle-like s uc u e (eu ec ic ype) qui e ypical in as -cooled me als. These needles appea ine in he co e o he weld seam and sligh ly coa se in he bo de o adjacen seams. Figu e 10. Mic os uc u e on he XY plane ×50 (le ), ×100 ( igh ). To ha e a close iew o he mic os uc u e, samples we e obse ed in a scanning elec on mic oscope. The images (Figu e 11) showed a needle-like s uc u e (eu ec ic ype) qui e ypical in as -cooled me als. These needles appea ine in he co e o he weld seam and sligh ly coa se in he bo de o adjacen seams.