Modeling of Ti-W Solidification Microstructures Under Additive Manufacturing Conditions

Transcription

1 Materals Scence and Engneerng Publcatons Materals Scence and Engneerng 2017 Modelng of T-W Soldfcaton Mcrostructures Under Addtve Manufacturng Condtons Matthew R. Rolchgo Iowa State Unversty, Mchael Mendoza Iowa State Unversty, Peyman Samm Iowa State Unversty Davd A. Brce Iowa State Unversty, Peter C. Collns Iowa State Unversty, See next page for addtonal authors Follow ths and addtonal works at: Part of the Materals Scence and Engneerng Commons The complete bblographc nformaton for ths tem can be found at mse_pubs/267. For nformaton on how to cte ths tem, please vst howtocte.html. Ths Artcle s brought to you for free and open access by the Materals Scence and Engneerng at Iowa State Unversty Dgtal Repostory. It has been accepted for ncluson n Materals Scence and Engneerng Publcatons by an authorzed admnstrator of Iowa State Unversty Dgtal Repostory. For more nformaton, please contact dgrep@astate.edu.

2 Authors Matthew R. Rolchgo, Mchael Mendoza, Peyman Samm, Davd A. Brce, Peter C. Collns, and Rchard Lesar Ths artcle s avalable at Iowa State Unversty Dgtal Repostory:

3 Modelng of T-W Soldfcaton Mcrostructures Under Addtve Manufacturng Condtons MATTHEW R. ROLCHIGO, MICHAEL Y. MENDOZA, PEYMAN SAMIMI, DAVID A. BRICE, BRIAN MARTIN, PETER C. COLLINS, and RICHARD LESAR Addtve manufacturng (AM) processes have many benefts for the fabrcaton of alloy parts, ncludng the potental for greater mcrostructural control and targeted propertes than tradtonal metallurgy processes. To accelerate utlzaton of ths process to produce such parts, an effectve computatonal modelng approach to dentfy the relatonshps between materal and process parameters, mcrostructure, and part propertes s essental. Development of such a model requres accountng for the many factors n play durng ths process, ncludng laser absorpton, materal addton and meltng, flud flow, varous modes of heat transport, and soldfcaton. In ths paper, we start wth a more modest goal, to create a multscale model for a specfc AM process, Laser Engneered Net Shapng (LENS ), whch couples a contnuumlevel descrpton of a smplfed beam meltng problem (couplng heat absorpton, heat transport, and flud flow) wth a Lattce Boltzmann-cellular automata (LB-CA) mcroscale model of combned flud flow, solute transport, and soldfcaton. We apply ths model to a bnary T-5.5 wt pct W alloy and compare calculated quanttes, such as dendrte arm spacng, wth expermental results reported n a companon paper. DOI: /s z The Author(s) Ths artcle s an open access publcaton I. INTRODUCTION THE Laser Engneered Net Shapng (LENS ) Process s an addtve manufacturng (AM) technque for the fabrcaton of metallc parts, orgnally developed at Sanda Natonal Laboratores n the 1990s. Parts are bult n a layer-by-layer fashon as powder s njected nto a melt pool created by a focused laser beam, whch s moved n a specfc pattern to buld the desred part. LENS and other energy deposton-based AM processes have gathered nterest because of ther potental for producng parts wth advantageous mcrostructural features, [1] unque structures, [2] and/or mproved mechancal propertes. [3] Addtonal abltes of such deposton processes nclude buldng of compostonally or functonally graded parts, [4] fully dense parts, and use n part repar. [5] Parts made from ttanum alloys n partcular, owng to ts low densty, hgh strength, and corroson resstance, have the potental to be produced MATTHEW R. ROLCHIGO, MICHAEL Y. MENDOZA, DAVID A. BRICE, and BRIAN MARTIN are wth the Department of Materals Scence and Engneerng, Iowa State Unversty, 2220N Hoover Hall, Ames, IA, PEYMAN SAMIMI, PETER C. COLLINS, and RICHARD LESAR are wth the Department of Materals Scence and Engneerng, Iowa State Unversty and also wth the Center for Advanced Non-Ferrous Structural Alloys (CANFSA), Iowa State Unversty, 2220N Hoover Hall, Ames, IA, Contact e-mal: lesar@astate.edu Manuscrpt submtted January 15, Artcle publshed onlne May 15, 2017 va LENS for applcatons rangng from bomedcal to aerospace. [6] One of the bggest challenges n the use of AM processes s, however, the detals of the relatonshps between processng parameters and the propertes of the fnal part. Whle extensve research has been done n modelng of the molten pool and mcrostructure produced n weldng operatons, dfferent process parameters, materal propertes, and forces actng on the molten pool n laser and electron beam-based deposton processes necesstate model extensons for AM condtons. The development of such a model for a gven addtve process s aded by a detaled understandng of smlartes and dfferences between weldng and AM condtons, and how the condtons affect development of the mcrostructure n the soldfed materal. The purpose of ths paper s to present the frst step n the development of a multscale computatonal model of AM processes, wth the goal of predctng detals of the as-soldfed mcrostructures and propertes n metallc alloys. We focus on the LENS process for a specfc alloy system. There s a wde range of physcal phenomena occurrng durng the deposton and rapd soldfcaton of alloys created va LENS and smlar beam-based deposton processes. As the laser s moved across the sample, a melt pool s created that nvolves not only the melted powder feedstock but also prevously deposted layers. Heat transport through varous modes n sold, lqud, and gas are mportant n determnng the 3606 VOLUME 48A, JULY 2017 METALLURGICAL AND MATERIALS TRANSACTIONS A

4 temperature feld and molten pool dmensons. [7] Owng to the nherent asymmetry of the beam meltng problem (partcularly when beam moton s accounted for), there wll be a sgnfcant locaton dependence of the thermal gradent and coolng rate; these temperature felds play a large role n ntal mcrostructure formaton and thus on the propertes of the fnal part. Much of our detaled understandng of these felds come from applcatons of the fnte element method (FEM), whch has been used extensvely to model them. [8 13] For soldfcaton n the LENS process n partcular, Hofmester [8] used a combnaton of thermal magng and fnte element modelng to estmate coolng rates and thermal gradents n the molten pool for varous sets of process parameters. Smlarly, Wang and Felcell [10] used a 2D fnte element model to calculate temperature as a functon of tme n dfferent regons of the molten pool for the deposton of a thn plate structure wth a movng beam, a work later expanded to 3D by the same authors. [11] Whle the role of flud transport s sometmes neglected n the calculaton of these temperature felds, the melt pool created n beam-based meltng problems s hghly dynamc. [14] Large temperature gradents, capllary forces, and Marangon convecton play mportant roles n ts evoluton as well as n the evoluton of the correspondng temperature felds. [15,16] For example, the role of Marangon convecton n weldng has been studed extensvely [17 19] and several FEM-based models of coupled flud and heat transport n weld pools that nclude Marangon and buoyancy forces have been developed. [20 22] Such models typcally used mult-layer meshes and solved the ncompressble Naver Stokes equatons for flud flow n the melt pool. Later models expanded on ths work to nclude materal addton, wth the correspondng forces on the molten pool arsng from powder njecton. [23 25] More recently, We et al. [26] modeled a beam-based meltng and soldfcaton problem wth flud flow, heat transport, and soldfcaton n order to study the effects of eptaxy on the gran growth and texture of multple deposted layers. Acharya et al. [27] modeled a smlar problem for scannng laser eptaxy use n part repar, nvestgatng the role of process parameters and flud transport on soldfcaton condtons. Addtonally, Morvlle et al. [28] used COM- SOL Multphyscs to smulate the flud flow, meltng, and transent temperature condtons encountered durng drect laser metal deposton. Another approach has been based on the use of the Lattce Boltzmann method (LB), whch s an attractve method for solvng ths partcular flud transport problem (relatve to Naver Stokes algorthms) because of ts ease of handlng complex geometres, [29,30] ease of modelng the changng boundary condtons found n soldfcaton [31,32] and ts ablty to model coupled flud, heat, and solute transport problems. [7,31,33] Addtonally, the method s computatonally effcent and easly parallelzed. [34,35] Ko rner et al. [7,36] used a thermal Lattce Boltzmann model for the combned heat transport, flud flow, and phase change problem, for T-6Al-4V n a powder bed-based AM process. The ntal soldfcaton s hghly dependent on the temperature feld and s mportant to the development of the as-soldfed mcrostructure. By nfluencng the gas lqud nterface along wth local coolng rates and thermal gradents, flud flow can nfluence mcrostructural changes such the formaton of stray grans, [37] the orentaton of grans at the soldfcaton front, [38] and the development of porosty and surface roughness. [39] Calculated thermal gradent G and soldfcaton rate R values from ether mathematcal models or heat transport FEM-based models (wth and wthout consderaton of flud flow) can be used to emprcally determne mcrostructural quanttes such as dendrte arm spacng, [40] or map pars of G and R nto zones of columnar, equaxed, and mxed gran morphology. [41 43] Cellular automata (CA) models of soldfcaton, frst developed by Rappaz and Gandn, [44] have successfully been coupled wth process-scale models to smulate gran growth [43,45] as well as the growth of dendrtc colones wthn grans. [46] The CA used by Yn and Felcell, [46] orgnally developed by Beltran-Sanchez and Stefanescu [47] based on the work of Rappaz, has been used to successfully model bnary alloy soldfcaton by others as well. [48 50] It has also been expanded nto 3D and used for ternary alloy growth. [51] The coupled LB- CA model has recently been appled to model combned flud flow, solute transport, and soldfcaton problems for bnary and ternary alloy systems. [33,52 58] Use of a CA for soldfcaton has the advantage of computatonal effcency over alternatve approaches, such as the phase feld method, but dscretzaton error and grd effects are sgnfcant drawbacks. Correctons to mnmze the grd effects that tend to occur n such models have been proposed [59 61] and utlzed by many soldfcaton models. Combnatons of the phase feld and cellular automata methods can be used to model multscale soldfcaton as well; for example, Tan and Shn [62] consders soldfcaton of both grans (usng a CA) and dendrtc colones (usng phase feld) n 2D at the approprate length and tme scales. The present work uses a multscale approach to model mcrostructure n a bnary alloy (T-5.5 wt pct W) system, whch was chosen because of ts the potental for sgnfcant gran growth restrcton based on ts phase dagram. [63] We employed the commercal software package COMSOL Multphyscs to model the macroscale process as a smplfed beam meltng problem, consderng the effects of flud flow, heat transport, and temperature-dependent materals propertes. At the mcroscale, nformaton passed from COMSOL smulatons s used to determne ntal condtons, thermal gradents, coolng rates, and flud flow veloctes to be passed to a LB-CA model. The focus of ths paper s on the applcaton of the LB-CA model to the combned flud flow, solute transport, and soldfcaton for domans wthn the COMSOL-smulated LENS melt pool. In Secton II, we present background nformaton on soldfcaton that motvates our choce of the model, whch s descrbed n Secton III. Results are summarzed n Secton IV and comparsons to experment are gven n Secton V, accompaned by a dscusson of the valdty of the model. Secton VI presents conclusons and needs for future work. METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 48A, JULY

5 II. BACKGROUND The development of the as-soldfed mcrostructure n beam-based soldfcaton processes depends on the temperature feld, partcularly the local thermal gradent G and coolng rate _T, as well as materals propertes of the soldfyng alloy. Assumng that the rate of heat transport n the sold s much faster than that of the lqud and that the thermal gradent ahead of the nterface s very large, there wll be a postve temperature gradent ahead of the soldfcaton front and the latent heat released on soldfcaton wll be conducted back through the sold. Ths stuaton leads to constraned growth as descrbed by Kurz and Fscher, [64] wth a neglgble thermal component to the undercoolng ahead of the front. For bnary alloy growth of an alloy wth composton C 0, these condtons lead to the development of a solutal boundary layer arsng from the dfference n lqudus and soldus composton at a gven undercoolng relatve to the ntal composton s lqudus temperature T C 0 L. The dfference between the local temperature n the melt and the lqudus temperature at the local solute concentraton s referred to as the consttutonal undercoolng. Perturbatons of the orgnally planar nterface can more effectvely conduct away ths boundary layer than a planar boundary and, provded a regon of consttutonal undercoolng exsts ahead of the soldfcaton front, the planar nterface wll break down nto long cells as the perturbatons grow nto the undercooled lqud. The surfaces of these cells may n turn become unstable and break down nto secondary and/or ternary arms, dependng on the soldfcaton condtons. [65] The rad of the perturbatons that form cells or dendrtes are lmted by the sold lqud surface energy, whch serves to locally depress the lqudus temperature and works aganst soldfcaton n regons of hgh curvature. We can represent ths effect n a model by ncludng an nterfacal component to the total undercoolng. The dendrte tp radus represents a balance (for a gven set of condtons) between a sharp tp (whch can most effectvely conduct away the solutal boundary layer beng formed n ts wake) and a blunt tp (whch has less sold lqud nterfacal area). Gven the rapd rates of soldfcaton n beam meltng processes, the process of planar front breakdown and development of cells and dendrtes s very common. In the case of the T-W system wth low solute concentraton (see Fgure 1(a) for ts phase dagram, [66] ), the solute partton coeffcent k, whch descrbes the rato of the lqudus to the soldus slope, s greater than 1; the equlbrum concentraton of solute at a gven undercoolng for the sold phase wll be larger than that for the lqud phase. Therefore, as the sold grows and absorbs solute, the solutal boundary layer wll be depleted of solute relatve to the ntal alloy concentraton. The regon of the phase dagram focused on n ths paper s such that the lqudus and soldus curves can be approxmated as lnear, as shown n Fgure 1(b). As detaled by Rappaz and Gandn, [44] the soldfcaton morphologes depend on the nterplay between the local thermal gradent G and coolng rate. For the planar front to become unstable and break down nto cells, G must be small enough for development of the regon of consttutonal undercoolng ahead of the front and the soldfcaton rate must be fast enough such that the more effcent solute dffuson around the perturbaton tps overcomes the nterfacal energy penalty of ther curvatures, allowng them to grow nto cells rather than meltng back nto the lqud. At even faster soldfcaton rates (large _T) and smaller G, the surfaces of the cells themselves are unstable and a columnar Fg. 1 (a) The equlbrum phase dagram for the T-W system. As a beta-somorphous element, W exhbts mutual solublty wth the hgh-temperature BCC phase of T. Ths combned wth the large meltng pont dfference between the two elements leads to a large range of temperatures over whch soldfcaton occurs. The alpha HCP phase appears at low temperatures. (b) The regon of nterest n the phase dagram for the soldfcaton problem (hghlghted n (A) wth blue). Over ths relatvely narrow range of temperature and dlute W concentratons, the equlbrum lqudus and soldus are approxmated as lnear (Color fgure onlne) VOLUME 48A, JULY 2017 METALLURGICAL AND MATERIALS TRANSACTIONS A

6 dendrtc mcrostructure s formed. At the fastest soldfcaton rates and smallest G, nucleaton n the undercooled zone domnates over growth of the orgnal grans and a transton to an equaxed dendrtc mcrostructure occurs. Kurz and Fscher [64] detal the typcal ranges of G and _T values n bnary alloy soldfcaton that leads to dfferent morphologes. Addtonally, the cells and dendrtes that form tend to be fner wth ncreasng coolng rate. [64,67,68] Soldfcaton durng the LENS process occurs under locally hgh coolng rates and large thermal gradents; expermental and modelng work on LENS n Hofmester et al. [8] shows G on order of 10 5 K/m and _T between 10 3 and 10 4 K/s, wth the larger values closer to the melt pool bottom. The multscale model of LENS soldfcaton n Yn and Felcell [46] yelded smlar values, whle the soldfcaton maps by Bontha et al. [42] show thermal gradents and soldfcaton veloctes on par wth or larger than those n the prevous works. These condtons would be expected to heavly favor planar nterface breakdown and the development of cells and dendrtes, [65] and expermental work on rapd soldfcaton of β-t alloys has shown the domnant mcrostructural feature to be long columnar dendrtc grans and/ or cellular structures heavly textured wth the 001 drectons algned wth the heat flow drecton. [69 71] Whle these large thermal gradents typcally gve rse to a strong dendrtc or cellular mcrostructure, reducton n G and _T near the top of the melt pool may gve rse to mxed columnar and equaxed structure. [42] As shown later, the current model s capable of smulatng all of these morphologes (cellular, dendrtc, and equaxed) under approprate condtons. Whle the prmary drver n the development of soldfcaton mcrostructures are the effects descrbed above, there are other phenomenon that must also be taken nto account to best model molten pool development and soldfcaton n the LENS process. Specfcally, the effects of eptaxal growth [26] as well as the movement of the laser beam [9,13,72] can have a sgnfcant effect on the dendrte orentatons. Eptaxal growth leads to preferred ntal growth n prescrbed drectons, whch may not algn wth the large thermal gradents. The movng beam leads to an ansotropc melt pool where G and _T vary sgnfcantly based on melt pool locaton. [73,74] Reheatng and remeltng of prevous layers, the dfferent condtons encountered n each layer, and sold-state phase transformatons n prevously deposted layers wll also play roles n mcrostructural development. [1] Whle these factors are not fully accounted for n the present work, the basc formalsm descrbed next s capable of modelng these effects. III. APPROACH Owng to the dfferent physcal phenomena at vared length and tme scales of ths problem, as descrbed above, a multscale approach s needed. In the long term, our goal s to couple flud flow, heat flow, and solute transport to a soldfcaton model to more completely descrbe the process. In ths paper, we focus on a frst step toward ths goal: the couplng of a mcroscale model of soldfcaton wth a contnuumlevel smulaton of the melt pool, whch wll provde boundary condtons for the mcroscale. At the contnuum scale, the commercal software package COMSOL Multphyscs s used to model heat absorpton wth heat and flud transport for a smplfed beam meltng problem. At the mcroscale, a LB-CA model smulates solute and flud transport durng soldfcaton. A. Process Modelng The sets of conservaton equatons for energy, momentum, and mass equatons are coupled and solved n two dmensons usng the flud flow and heat transfer modules of COMSOL Multphyscs. The flud flow module accounts for buoyancy and the Marangon effect assumng lamnar flow and a Newtonan flud, whle the heat transport module consders conducton, convecton, and radaton as modes of transport. Lnear varaton of surface tenson wth temperature through the thermocapllary coeffcent γ, as well as temperaturedependent model nput values for densty, thermal conductvty, and heat capacty for both lqud and sold β T-5.5 wt pct W are consdered n the model. They are estmated as descrbed n Secton IV. The Marangon effect was ncorporated n a smlar manner as n the COMSOL smulatons of Morvlle et al., [28] beng modeled wth the relaton g@u=@y ¼ c@t=@x, n whch η s the dynamc vscosty of the flud (equvalent to the knematc vscosty multpled by the flud s the varaton n flud velocty n the drecton perpendcular to the free surface, and T/ x s the temperature gradent along the free surface. [75] Three values of the beam power were used n the COMSOL smulatons: 183 W, 259 W, and 367 W (referred to from ths pont on as low, mddle, and, hgh ). The beam dameter was 500 μm for each power, wth a Gaussan dstrbuton of energy. The model smulates a statonary beam meltng problem, wth a fxed molten pool geometry (estmated based on SEM mages) that s vared as a functon of nput power. [63] Owng to these assumptons, calculated values for temperature and flud flow felds are used here only to estmate the thermal gradents, coolng rates, and flud flow felds at short tmes near the melt pool walls after turnng off the energy source. These estmated values wll serve as ntal and boundary condtons for the mcroscale flud flow and soldfcaton model, detaled n Sectons III B and III C. B. The Lattce Boltzmann Method The Lattce Boltzmann Method (LBM), an evoluton of the lattce gas cellular automata, provdes an alternatve to solvng the full Naver Stokes equatons n flud dynamcs problems. [29,76] LBM models the flud as a seres of fcttous flud partcle dstrbuton functons f ðx; tþ at each grd pont x and tme t, whch removes the nose nherent to the partcle-based lattce gas model. [29,77] Beng based on both mcroscale partcle behavor whle satsfyng mesoscale conservaton and evoluton equatons for flud, the LBM provdes a way METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 48A, JULY

7 to brdge the two scales [78,79] and has accurately smulated benchmark flud flow problems (e.g., Couette flow). [29,76,80] In the two-dmensonal applcatons descrbed here, an LBM tme step conssts of solvng the dscrete Boltzmann equaton for the dstrbuton functons on a square lattce va successve collson and propagaton steps. In the present work, the D2Q9 lattce s used (see Fgure 2). A second set of dstrbuton functons, represented by g ðx; tþ, s used to model the dstrbuton of solute. These dstrbuton functons undergo collson and propagaton steps on the same LB lattce as the flud and are coupled to the flud partcle dstrbuton functons through the macroscopc velocty u. The detals of the LBM to solve coupled flud and solute transport problems are well documented n the lterature [52 56] and are shown brefly n the Appendx. The LBM for flud and solute transport was verfed by Zhou for several coupled flud and solute transport problems n 1D and 2D. [81] We note that the LBM s also capable of smulatng turbulent flow, though t typcally requres very fne grds and/or alternatons to some of the equatons. [29] Coupled transport of flud flow, heat, and solute would not be possble to smulate accurately n the present model for molten ttanum alloys, whch s lmted to a sngle grd spacng and tme step, owng to the dsparate rates of heat and solute transport. The thermal Lattce Boltzmann method for coupled heat and ncompressble flud flow felds has been used prevously to smulate convecton problems [29,82 84] and been coupled to cellular automata (CA) to smulate the sold lqud phase change for pure materals. [7,31,32,36,79] If t s assumed that the flud s ncompressble, vscous heat dsspaton s neglgble, and no work s done by the external pressure, an addtonal set of dstrbuton functons can be ntroduced to the model for heat transport. These are analogous to those for solute (the heat s advected as a passve scalar), but nclude a heat dffusvty, nternal energy densty, and a heat source Fg. 2 The D2Q9 lattce used n the Lattce Boltzmann model. The veloctes are denoted by the vectors e wth the magntudes gven n Eq. [A1]. term n place of solute dffusvty, concentraton, and the solute source term, respectvely. The ncluson of temperature-dependent force terms for effects such as capllarty, wettng, or buoyancy allows the couplng of the energy evoluton equatons nto those for flud. [7,85] Use of the thermal Lattce Boltzmann method wth a second set of dstrbuton functons to model the nternal energy densty has been shown to agree wth analytcal solutons for natural convecton, ld-drven convecton n square cavtes, and other benchmark flud dynamcs problems. [78,82,85,86] Though the present work s focused on mcrostructure evoluton and not the process-scale melt pool dynamcs, we are explorng such a model for consderaton of the coupled flud flow and heat transport problem at the approprate scale. Ths approach has a notable advantage over the present COMSOL model n that t enables drect trackng of the sold lqud nterface, allowng for drect couplng to the mcrostructure evoluton model. C. Cellular Automata (CA) for Soldfcaton In our approach, soldfcaton of an ntally lqud doman s calculated on the same grd and at the same tme step as the LBM calculatons. Three cell types are consdered: sold, lqud, and nterface. Sold cells act as mpenetrable boundares for the flud flow, whle flud flow s allowed wthn nterface cells. Lqud cells cannot drectly border a sold cell a layer of nterface cells s always present. Once an nterface cell has completely soldfed, t s changed to sold and neghborng lqud cells become new nterface cells. As n the orgnal CA of Rappaz, the amount of soldfcaton n an nterface cell over a gven tme step s calculated as a functon of the local undercoolng. [44] In ths CA, however, the calculaton s not gven by an analytcal functon of soldfcaton velocty wth undercoolng but rather by assumng that the drvng force for free energy s domnated by local nterface knetcs. The drvng free energy, G K, s proportonal to an nterfacal velocty V for the cell. The fundamental equatons for these quanttes are [87] DG K RT I ¼ ð1 keqþ m eq L T T m þ meq L C L T I and DG K V ¼ fv 0 ½2Š RT I n whch R s the gas constant, T I s the local nterface temperature n the cell (determned va the macroscale temperature felds as functons of tme), and C L the local solute concentraton n the cell (as calculated va LBM). T T m, keq, and m eq L are the meltng pont of pure solvent (ttanum here), the equlbrum lqudus slope, and the equlbrum solute partton coeffcent, respectvely, as determned from the phase dagram (Fgure 1). The parameters m eq L and k eq are assumed to be constant wth low solute concentratons, as ndcated n Fgure 1 (b). The change n sold fracton Δf s s related to the nterface velocty V, Δx, Δt, and the angle between the nterface normal and the grd axs drecton va ½1Š 3610 VOLUME 48A, JULY 2017 METALLURGICAL AND MATERIALS TRANSACTIONS A

8 Df s ¼ VD cos ð u ÞDt Dx Ths equaton for ΔG K can be rewrtten n terms of the local equlbrum solute concentratons C eq L and Ceq as DG K ¼ C eq S Ceq L RT þ C Lð1 k eq Þ I These equlbrum concentratons, n turn, are functons of the cell s local temperature and the phase dagram, as descrbed below. An addtonal term, related to the local nterface curvature j (calculated va the algorthm gven n Beltran-Sanchez and Stefanescu, [88] ) Gbbs Thomson coeffcent, and a functon characterzng the crystal ansotropy (the functon for a crystal wth fourfold symmetry used by Jelnek et al. [35] s used here) s added to the expresson for C eq L to account for the local depresson of the lqudus arsng from the sold lqud nterfacal energy, C eq L ¼ T I T T m m eq þ L j ð 1 d cosð4ð/ hþ Þ m eq L n whch h s a gven gran s preferred orentaton, and d a coeffcent characterzng the surface energy ansotropy. The solute concentraton of the sold formed s gven by C eq S ¼ keq C eq L If a non-zero change n sold fracton s calculated, the approprate amount of solute must be added or wthdrawn from the cell va the LBM solute dstrbuton functons. Ths s calculated by solvng the solute balance n the cell; note that f f s approaches or reaches 1, the solute must nstead be added or removed from cells that stll contan apprecable lqud. S ¼ Df sðc L C eq S Þ ½7Š 1 f s Df s If k eq > 1 (as n the case of the T-W system), there s an addtonal constrant that soldfcaton cannot occur f the amount of wthdrawn solute leads to a subzero concentraton. To reduce the deleterous effects of grd ansotropy on the growng colones, the approach taken by Zhan et al. [61] s used n whch soldfcaton of clusters of cells are evolved n tandem, effectvely expandng the local neghborhood of the soldfcaton from the typcal Moore or Von Neumann neghborhoods and allowng the colones that are msalgned wth the grd to mantan ther preferred orentatons. To account for nucleaton of new grans, a probablstc approach, smlar to that of Rappaz, s taken n whch the change n local undercoolng over a gven amount of tme s related to the probablty that nucleaton has occurred n the cell. If the probablty s larger than a randomly generated number, a new gran s nucleated. The equaton used for the densty of nucle as a functon of undercoolng, as well as the equaton for S ½3Š ½4Š ½5Š ½6Š the nucleaton probablty, s descrbed n more detal n Yn and Felcell. [46] However, for the COMSOL smulatons of LENS soldfcaton at the melt pool wall, the thermal gradent s large enough that such nucleaton dd not occur at a rate comparable to that of columnar growth from eptaxal colones at the wall. The resultant mcrostructures are thus domnated by the growth of eptaxal dendrtes nucleated along the boundary. IV. RESULTS AND DISCUSSION Because of sparse thermochemcal data on the T-W system, many of the materals constants had to be approxmated. For the estmaton of specfc heat, densty, and vscosty as functons of temperature as requred for COMSOL, the values for a dfferent molten ttanum system wth more avalable data (T-6Al-4V) were used. The assumpton s that such propertes for the T-5.5W system would be smlar, whch s reasonable f the materal s treated as homogenous contnuum as n the COMSOL smulatons. For the soldfcaton model, reasonable values from the prevous soldfcaton models or order-of-magntude estmates were made when parameters were unknown for the T-5.5W system. The constants used n the soldfcaton model and sources for the values and/or equatons used to obtan them are descrbed n Table I, along wth temperaturedependent equatons for T-6Al-4V propertes. Please see Fgure 1(b) for defntons of the parameters used to characterze the phase dagram. The phase dagram and soldfcaton model n general are performed usng mole fracton of W; the results are converted to weght percent of W for the fgures. All soldfcaton model results used the T-W phase dagram, though for the multscale smulatons, the temperature and flud flow data are from COMSOL results for T-6Al-4V. A. Model Examples To test the soldfcaton model, the thermal gradent G was held constant at 100,000 K/m and coolng rate Ṫ vared to examne dendrte tp undercoolng as a functon of tp velocty. The mesh sze was set to 0.4 μm, and the tme step to 0.64 μs. For each Ṫ, the model was run untl the rate of advance of the tp was near the steady state Ṫ/G (addng rows to the smulatons as necessary), at whch the nterface temperature was determned. Due to dscretzaton, these temperatures and veloctes were not exactly constant and error bars are ncluded to account for the uncertanty n the exact values. At larger Ṫ, the steady-state dendrte tp veloctes are larger and a correspondng larger drvng force for soldfcaton s necessary to mantan the steady state. Snce ths requres a larger undercoolng, the tp temperature decreases wth ncreasng coolng rate as shown n Fgure 3, whch shows the same trends as the analytcal model of Kurz, Govanola, and Trved for constraned alloy growth. [89] However, the present work used dfferent equatons, model parameters, and was lmted to two dmensons, so the exact dependence of tp velocty and undercoolng acheved here and that METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 48A, JULY

9 Table I. Parameters Used n the Macro- and Mcroscale Models Parameter Name Symbol Value Unts Source COMSOL parameters Sold heat capacty (298 K to 1268 K) (25 C to 995 C) S c p T K/kg K ( C) 94 Sold heat capacty (1268 K to 1923 K) (995 C to 1650 C) S c p T K/kg K ( C) 94 Sold densty (298 K to 1268 K) (25 C to 995 C) ρ S T kg/m 3 94 Sold densty (1268 K to 1923 K) (995 C to 1650 C) ρ S T kg/m 3 94 Sold thermal conductvty (298 K to 1268 K) (25 C to 995 C) k S T W/m K ( C) 94 Sold thermal conductvty (1268 K to 1923 K) (995 C to 1650 C) k S T W/m K ( C) 94 Lqud heat capacty L c p 831 K/kg K ( C) 94 Lqud densty ρ L T kg/m 3 94 Lqud thermal conductvty k L T W/m K ( C) 94 Thermocapllary coeffcent γ N/m K ( C) 28 LB-CA Parameters Lqudus slope m L 59.4 K ( C)/mol. pct 66 Partton coeffcent k 3.56 none 66 Intal lqud solute concentraton C mol. pct Pure solvent meltng pont T m 1943 (1670) K ( C) 66 Gbbs Thomson coeffcent Γ m K ( C) 52 Degree of surface energy ansotropy δ 0.6 none 35 Fracton of stes for growth f 0.01 none 87 Upper lmt crystal growth velocty V m/s 87, 95 Lqud densty ρ 4865 kg/m 3 95 Knematc vscosty υ m 2 /s 95 through 97 Solute dffuson frequency factor D m 2 /s 98 Solute dffuson actvaton energy Q 62.5 kj/mol. 98 Fg. 3 The present model predcton for the dendrte tp velocty undercoolng relatonshp of a sngle T-5.5 weght percent dendrte at steady state durng constraned soldfcaton. At larger soldfcaton veloctes, a larger undercoolng relatve to the equlbrum lqudus temperature for T-5.5W s necessary to drve soldfcaton. The thermal gradent was held at 100,000 K/m for all data ponts. calculated by the analytcal model s not drectly comparable. For many soldfcaton veloctes, somewhat smaller undercoolng s needed to acheve the same dendrte tp velocty here relatve to the analytcal predcton. Ths suggests that the choces for f and V 0 n Eq. [2] may be large, leadng to a small overestmaton of local soldfcaton rates To valdate the combned solute transport, soldfcaton, and flud flow model, we compared the case of a sngle nucle growng n a melt at constant undercoolng (20 K below T C 0 L ) wth and wthout flud flow. The mesh sze was set to 0.4 μm and the tme step to 0.64 μs, wth a 301 by 301 grd and 40,000 total tmes steps of smulaton tme. The left, top, and bottom boundares were held at constant solute concentraton C 0. The rght boundary s held at a constant concentraton gradent to mmc the fact that the doman s sem-nfnte and at some pont far away from the dendrte, there s no concentraton gradent. A constant velocty boundary condton on the left boundary mantaned a flud flow feld from left to rght, dvergng to the left of the growng dendrte and convergng on ts rght. In Fgure 4, we compare a growng dendrte for the case wthout flud flow (a) and that wth flud flow (b). In the fgure, we see that flud flow has a clear mpact on the dendrte symmetry; the arm growng opposte the drecton of the flud flow has solute suppled to t va the flud flowng from the left boundary, and the solutedepleted boundary regon that formed as the sold grew was advected away by the flow. Thus, the arm growng opposte the drecton of the flud flow grows more rapdly than the other arms. The opposte arm has ts growth stunted, as the solute-depleted regon tends to buld up near the convergence regon of the flud. The arms transverse to the drecton of flow grew somewhat faster as well, though not nearly as much as the arm opposte the flow drecton. Ths effect s confrmed by other smlar LB-CA models for dendrtc growth n bnary alloys. [52,53,55] B. Mcrostructure Smulaton In ths secton, we nvestgate the varety of mcrostructures the present model s capable of smulatng. At relatvely small thermal gradents and sold VOLUME 48A, JULY 2017 METALLURGICAL AND MATERIALS TRANSACTIONS A

10 Fg. 4 Soldfcaton of a sngle dendrte n a melt of constant undercoolng (20 K below the lqudus temperature for T-5.5W) after s. (a) In the absence of flud flow, solute transport s lmted to dffuson. The dendrte s symmetrcal as the dendrte arms grow nto the lqud. (b) Wth flud drven at m/s from left to rght, solute transport va both dffuson and advecton alters the symmetry of (A) as the arms grow at dfferent rates dependng on ther orentaton relatve to the flud flow drecton. The color bar corresponds to the concentraton of W n wt pct. fcaton veloctes, nucleaton of new grans ahead of the prmary soldfcaton front becomes sgnfcant. Homogeneous nucleaton of new grans s modeled as a Gaussan process that s a functon of undercoolng and s characterzed by three parameters [46,90] : the maxmum nucleaton densty N max, the mean nucleaton undercoolng ΔT N, and the standard devaton of the nucleaton dstrbuton T. These parameters are typcally unknown and dffcult to predct, partcularly when heterogeneous nucleaton stes are actvated much more easly. [91] They are often roughly estmated; for example, [90] used ΔT N =5K,ΔT = 0.5 K, and N max =10 12 m 3 for Al-3 wt pct Cu. [92] used a Gaussan dstrbuton for both surface and volume nucleaton of Al-7 wt pct S, placng ΔT N at 10 K. Expermental observatons by Basak and Das [93] estmated ΔT N to be near 0.2 ΔT m n a pure metal. In the present calculatons, the ΔT N and ΔT σ values from Boettnger [91] and N max of m 3 were used. As shown by Dong and Lee, [90] the choce of parameters wll have a sgnfcant mpact on the locaton of the columnar to equaxed transton wthn the soldfcaton maps n Bontha et al. [42,65] Unmelted partcles and other melt nhomogenetes, whch are partcularly an ssue wth hgh meltng pont alloyng addtons such as tungsten, would be expected to play a sgnfcant role n nucleaton and wll be addressed n future work wth a process model that consder such factors. To model eptaxal growth of multple dendrtc colones, lqud domans were ntalzed wth the bottom wall 5 K below T C 0 L. Heterogeneous nucleaton stes coverng ths wall are all ntally actve, wth randomly chosen sold fractons and orentatons near or algned wth the heat flow drecton. Domans were chosen to have constant G and V (where V ¼ T _ G ). Wth V = m/s and G = 50,000 K/m, a relatvely coarse columnar dendrtc mcrostructure results (as shown n Fgure 5(a)). At a faster soldfcaton rate (V = m/ s) and the same thermal gradent, a transton to a mcrostructure domnated by grans nucleated ahead of the ntal columnar dendrtc front s observed (see Fgure 5(b)). There were no exact values for V or G at whch ths transton occurs, rather a gradual trend toward nucleated grans domnatng the mcrostructure over the eptaxal columnar grans as G was reduced and V ncreased. Ths observaton s on par wth that observed by Dong and Lee [90] n drectonal soldfcaton modelng. Under the condtons encountered n AM processes, ths columnar to equaxed transton may occur near the top of the molten pool n the latter stages of soldfcaton. However, owng to the lmtatons of the present macroscale model, the present work s lmted to the early-stage soldfcaton near the molten pool bottom. The more rapd soldfcaton rate and larger thermal gradent near the molten pool walls lead to fner columnar dendrtes as shown n Fgure 5(c), or cellular structures as shown n Fgure 5(d). Nucleaton at the small length scale of these structures dd not occur n the model. Growth of both cells and dendrtes under AM condtons has been observed expermentally, [93] and the transton from dendrtes to cells at fast coolng rates has been modeled va CA prevously. [46] C. Multscale Modelng of LENS Mcrostructure The COMSOL smulatons for T-6Al-4V, despte usng three dfferent values for power (183 W ( low ), 259 W ( mddle ), and 367 W ( hgh )) and melt pool sze, yelded the same general flud flow pattern as shown n Fgure 6. The general pattern of the flow conssts of two large convecton cells on the left and rght sdes of the melt pool, wth some more mnor flud flow n the central porton of the melt pool just below the energy source. The formaton of smlar convecton patterns, wth two man cells drven by surface forces and slower flud flow below the regon of energy METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 48A, JULY

11 Fg. 5 Varatons of mcrostructure dependng on soldficaton condtons. (a) represents coarse columnar dendrtes (coolng rate 10 K/s, thermal gradent 50,000 K/m) and (b) A mxture of columnar and equaxed dendrtes nucleated from the bulk lqud (coolng rate 50 K/s, thermal gradent 50,000 K/m). Under the more rapd soldficaton condtons and larger thermal gradents as would typcally be encountered n the LENS process, more fine columnar dendrtes as n (c) (coolng rate 2500 K/s, thermal gradent 250,000 K/m) and columnar cells as n (d) (coolng rate 4000 K/s, thermal gradent 250,000 K/m) would be more typcal. The color bar corresponds to the concentraton of W n wt pct. Fg. 6 COMSOL smulaton results for temperature (a) (n K) and flud velocty magntude (b) (n m/s) under condtons representatve of the LENS process. The black outlne represents the fixed molten pool geometry consdered. Due to the effects of the strong Marangon convecton, the temperature field s not symmetrc about the regon where beam absorpton occurred. Ths leads to sgnficant dfferences n thermal gradent and flud flow pattern n Doman A, represented by the red square, and Doman B, represented by the blue square (Color figure onlne). absorpton, has been seen n two-dmensonal models of weldng and AM processes and arses prmarly from the Marangon effect.[18,23,27,28] The sze and relatve velocty of the flud wthn these convecton cells were observed to have sgnficantly larger surface velocty at hgher beam powers because of the larger thermal gradents present under those condtons. For the mcrostructure results presented here, we consder two regons at the melt pool walls that are algned wth the maxmum temperature gradents (the 0.1 mm squares hghlghted n Fgure 6 for the low-power case). The analogous regons to those of Fgure 6 are used for smulaton of the soldficaton for the each of the three power levels, though the exact X and Y coordnates wll 3614 VOLUME 48A, JULY 2017 vary as the sze of the melt pool changes wth beam power. For regon A, just underneath the large convecton cell, the flud flow s parallel to the wall. To model ths regon, the flud s ntalzed everywhere to the COMSOL value for the magntude of the flud flow n the regon for the gven power. Snce there s addtonal soldficaton occurrng on both sdes of ths locaton, perodc boundary condtons are used for the LB flud flow, solute flow, and the soldficaton CA. The top boundary s held at a flud velocty n the postve X drecton from COMSOL. Immedately above regon B, flud flow s perpendcular to the nterface, dvergng to the left and rght near the wall. For ths regon, the top boundary s held at a constant velocty n the METALLURGICAL AND MATERIALS TRANSACTIONS A

12 Fg. 7 Local temperature as a functon of tme followng energy source shutoff at the bottom (sold lne) and top (dashed lne) of Doman A for the COMSOL smulaton performed at low power. The thermal gradent and coolng rate across the doman depend on poston and tme. negatve Y drecton as determned wth COMSOL, whle the left and rght boundares are subject to zero velocty gradent condtons to allow the flud to flow out ether sde. The left and rght boundares are also subject to a zero-concentraton gradent boundary condton. In both cases, the top boundary s held at a constant concentraton C 0, provdng a source of solute for the growng sold. The soldfcaton at other regons along the soldfcaton boundary was not modeled here. Whle the local thermal gradents and coolng rates vary as a functon of laser power for a gven locaton, they also vary strongly from the effects of convecton n the melt pool. Because of the large convecton cell, the thermal gradents ahead of the soldfcaton front at locaton A are small relatve to the rest of the molten pool. The coolng rates are also relatvely slow as heat from the top of the melt pool s advected nto ths regon by the flud flow from the adjacent cell. At locaton B, the thermal gradent s very large as a result of the heat beng advected downward and the fast heat conducton out through the sold. The coolng rate s ntally slow owng to the latent heat release and the advected heat from above, but as the Marangon convecton weakens followng shutoff of the heat source, more rapd coolng occurs. As shown n the table below, the expected trend of decreasng thermal gradent and coolng rate wth ncreasng laser power generally holds for locaton A, but at locaton B the slowest coolng rate occurs for the mddle power. At hgher powers, ncreased flud flow advects heat toward the sdes and away from the bottom melt pool wall. Usng the temperatures predcted by COMSOL at multple tmes followng shutoff of the power source, we approxmated the temperature as a functon of tme at the tops and bottoms of the two soldfcaton regons followng laser shutoff. It was observed that for all three nput powers and both domans, coolng started slowly (generally n the frst 0.02 seconds) and then tended to occur at a constant rate. The thermal gradent was at ts largest at the tme of laser shutoff, then tendng to decrease or reman constant. For ths reason, parabolc functons were ft to the frst 0.02 seconds for the top and bottom temperatures, and lnear functons after the 0.02 second mark. It was assumed that the temperature gradents over these small domans are lnear, and cells between the top and bottom walls were assgned temperatures nterpolated from the two wall values dependng on ther relatve locatons n the doman. Fgure 7 shows an example of these functons for the low-power case n regon A. The modeled mcrostructures for regon A under the low, mddle, and hgh powers (along wth the total tme taken for 37.5 and 97.5 pct doman soldfcaton) are shown n Fgure 8. Chosen values for the length and tme scales n the model were Δx = mand Δt = seconds. As expected, the tme to soldfcaton ncreased wth ncreasng power because of the decrease n coolng rate and correspondng decrease n steady-state dendrte tp velocty. Wth the decreased coolng rate, the solute concentratons wthn the dendrtes ncreased as well; the smaller tp veloctes requred smaller local undercoolng values, whch correspond to hgher values for C S eq. A fner mcrostructure (smaller dendrte arm spacng) was formed n the lowpower case relatve to the mddle and hgh powers another functon of coolng rate. Relatve to control smulatons wth no flud transport and solute transport n the lqud by dffuson only, there was neglgble dfferences n soldfcaton tme, dendrte arm spacng, or concentraton profles n the lqud and sold, largely because there s lttle to no vertcal movement of the flud, and the horzontal movement of the flud wth the perodc boundary condton does not effectvely mx the solutal boundary layer ahead of the sold. Ths condton s somewhat smlar to that encountered by the tps perpendcular to the drecton of the flud flow n the example n Fgure 4. In addton, the velocty at the soldfcaton front tself tends toward zero as t was drven from the top boundary rather than from the sdes. As a result, the regon of flud that s movng fast enough to effectvely mx the solute s always ahead of, not at, the soldfcaton front. The doman B mcrostructures at 99 pct soldfcaton are shown n Fgure 9 alongsde ther control counterparts that ncluded no flud transport. Chosen values for length and tme scales were x = m and t = s n these calculatons, whch results n the same relaxaton parameters for flud and solute as those n doman A. These structures are more sgnfcantly affected by the flud flow as t s ntally flowng toward the front, whch leads to a scenaro smlar to that of the dendrte tp facng the flud flow shown n Fgure 4; solute s suppled to the front by the ncomng solute-rch flud as the soldfcaton process consumes t. For the low-power case, the flud flow s not strong enough to have an effect and for the hgh-power case, the local undercoolng near steady-state s already near the equlbrum soldus value. Under ths condton, addng more solute wll not ncrease the drvng force. Because of ths effect and the relatvely small flud velocty, the dfference between the low-power mcrostructure and ts control counterpart s small METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 48A, JULY

13 Fg. 8 Model results for solute concentraton profile n Doman A mcrostructure. Arrows represent flud velocty, for cells that have not completely soldfied. At partal doman soldficaton (a-c), more tme s needed to reach the same soldficaton threshold wth ncreasng nput power. At complete doman soldficaton (d-f), ths s agan true; t s also seen that a larger solute concentraton n the dendrte arms s present as well wth ncreasng nput power. The color bar corresponds to the concentraton of W n wt pct. Fg. 9 Model results for solute concentraton profile n Doman B mcrostructure. (a-c) represent the mcrostructure wthout flud flow for the low, mddle, and hgh power, whle (d-f) represent the mcrostructure wth flud flow at the same power. The most notable dfference s the tendency for more solute to get ncorporated nto the dendrte arms, as flud from the top boundary s drven toward the soldficaton front allowng some mxng of the solutal boundary layer. The color bar corresponds to the concentraton of W n wt pct. and only vsble at the doman bottom (correspondng to early soldficaton, pror to reachng larger undercoolng values). The 99 pct soldficaton threshold s 3616 VOLUME 48A, JULY 2017 reached at approxmately the same tme for calculatons wth and wthout flud flow. The hgh-power case, wth ts faster flud flow, shows a larger regon near the bases METALLURGICAL AND MATERIALS TRANSACTIONS A

14 Fg. 10 Solute concentraton profle for Doman B at the mddle nput power. (a-c) represent the solute concentraton feld as well as the flud velocty feld for varous tmes, whle (d-f) represent the same tmes on a control run done n the absence of flud flow. The early soldfcaton occurs faster and at hgher solute concentratons wth flud flow, as t allows mxng of the solutal boundary layer wth the addton of relatvely solute-rch flud from the top boundary. The color bar corresponds to the concentraton of W n wt pct. Table II. Thermal Parameters from the Macroscale Smulatons Intal Thermal Gradent (K/m) ( C/m) Thermal Gradent After 0.06 s (K/m) ( C/m) Overall Mean Coolng Rate Over 0.06 s (K/s) ( C/s) Locaton A: low power 500, , Locaton A: mddle power 480, , Locaton A: hgh power 470, , Locaton B: low power 960, , Locaton B: mddle power 760, , Locaton B: hgh power 930, , Flud Velocty at Boundary (m/s) of the dendrtes wth larger concentraton than ts control counterpart. However, t too has the majorty of soldfcaton takng place near the lmt of consttutonal undercoolng and the 99 pct threshold s reached only slghtly faster wth the ncluson of flud flow. The mddle power, wth the slowest soldfcaton rate, s examned more closely at three dfferent tmes n the soldfcaton process n Fgure 10. The flud, ntally movng n the drecton of the front, s pushed back as t begns to advance and forced out the sde. The flud goes from a prmarly vertcal flow to more horzontal, and the flud flow at the wall tends toward zero. However, the tme perod n whch the flud flow was supplyng solute from the constant concentraton top boundary to the front was long enough that a sgnfcant dfference n early soldfcaton rate and sold concentraton s seen. Table III. Predcted Dendrte arm Spacng Under Dfferent AM Processng Condtons Smulaton Mean Dendrte Arm spacng (μm) Doman A low power 9.09 Doman A mddle power Doman A hgh power Doman B low power 5.26 Doman B mddle power 9.09 Doman B hgh power 5.26 D. Comparson to Experment By plottng solute (W) concentraton at the tops of the dendrte arms, and dvdng the doman wdth by the number of concentraton peaks, the prmary dendrte METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 48A, JULY