Predictive modeling of solidification during laseradditive manufacturing of nickel superalloys: Recentdevelopments, future directions

Supriyo Ghosh‡Materials Science and Engineering Division, National Institute of Standards andTechnology, Gaithersburg, MD 20899, USA

E-mail: supriyo.ghosh@nist.gov

Abstract. Additive manufacturing (AM) processes produce parts with improved physical,chemical, and mechanical properties compared to conventional manufacturing processes. InAM processes, intricate part geometries are produced from multicomponent alloy powder,in a layer-by-layer fashion with multipass laser melting, solidification, and solid-state phasetransformations, in a shorter manufacturing time, with minimal surface finishing, andat a reasonable cost. However, there is an increasing need for post-processing of themanufactured parts via, for example, stress relieving heat treatment and hot isostaticpressing to achieve homogeneous microstructure and properties at all times. Solidificationin an AM process controls the size, shape, and distribution of the grains, the growthmorphology, the elemental segregation and precipitation, the subsequent solid-state phasechanges, and ultimately the material properties. The critical issues in this process arelinked with multiphysics (such as fluid flow and diffusion of heat and mass) and multiscale(lengths, times and temperature ranges) challenges that arise due to localized rapid heatingand cooling during AM processing. The alloy chemistry-process-microstructure-property-performance correlation in this process will be increasingly better understood throughmultiscale modeling and simulation.

1. Introduction

The production of metallic parts via additive manufacturing processes (for recent reviews,see [1], [2], and [3]) such as laser powder bed fusion and direct metal laser sintering is growingrapidly to achieve the ever-increasing demand for improved strength and resistance to creepand fatigue for aerospace, defense, and medical applications [4–6]. Ni−based superalloysin this context possess excellent mechanical properties at elevated temperatures, makingthem essential to the above sectors. However, there is a lack of confidence in the qualityof additively manufactured parts due to the inherent multiscale and multiphysics problems

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associated with processing. On a macroscopic scale, i.e. millimeters, as the laser rastersacross the powder bed, local regions begin to melt resulting in a molten pool of certaindimensions. The molten pool then undergoes a solidification, remelting and subsequentadditional solidification processes driven by the resultant complex thermal history due torepeated passes of the laser. Temperature is highest at the top surface and varies alongthe depth, width and length of the melt pool. Therefore, the melt pool represents differentlocal solidification conditions. As these local conditions vary, the microstructures withinthe solidified puddle also vary in different locations. The uncertainties in the quality ofthe parts arise precisely due to those location specific complex microstructures and due tounpredictable microstructure evolution paths, leading to beneficial or detrimental phasesand microstructural anisotropies due to segregation and orientations [1,5,7–9]. These issueshave received little attention, but they are important to consider during AM solidification.

The new classes of AM materials are substantially different than those produced bytraditional casting, mechanical working and final machining mechanisms. This is due to therapid solidification processing (RSP) during AM, leading to nonequilibrium segregation of thealloy elements in the molten pool, making the quantification of the resulting microstructuresdifficult. The prediction of material properties therefore becomes a serious hurdle, resultingin a lack of confidence in the quality of the final part. Some 47 % of manufacturers surveyedindicated that the uncertain quality of the final product was a barrier to adoption of additivemanufacturing [10–12]. A predictive multiscale modeling framework could optimize the AMprocess parameters to improve the likelihood of producing qualified parts.

Due to the high temperature and small volume of the molten pool, in situ measurementsof the solidification conditions are difficult. Numerical simulations of the laser depositionprocess is a viable alternative to obtain the local solidification conditions in the meltpool. A finite element analysis (FEA) method, which includes heat transfer, fluid flow,Marangoni convection and other hydrodynamic effects [12, 13], can simulate realistic meltpool shapes as well as temperatures. Energy balance equations on the macroscale are solvedin ABAQUS [14]/ANSYS [15]/COMSOL [16] environment to obtain the temperatures andgeometries in the melt pool (refer to Fig. 1a). Laser processing parameters such as thelaser power, speed, and size control the dimensions of the melt pool. These dimensionsare important as they determine the density of the final parts through the solidificationmicrostructures as well as through the cooling rates; for example, with increasing laser poweror reduced laser speed, the melt pool becomes deeper [10, 12]. Solidification conditionstherefore change along the melt pool solid-liquid boundary and influence the resultingmicrostructure formation.

For the microstructure simulations, phase-field models [19–21] have been the mostpopular choice. In these models, a scalar-valued order parameter field φ is introducedto distinguish the phases present in a microstructure: if liquid is defined by φ = −1and solid by φ = 1, then the interface can be taken as the φ = 0 contour. Therefore,

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Figure 1: (a) Typical melt pool shapes are shown for the consideration of different physicswithin the melt pool (reproduced from [10], with permission from Elsevier). (b) Typicalsolidification boundary is illustrated from which the local solidification conditions areestimated (reproduced from [17], open access). Melt pool solidify into columnar (on theside) and equiaxed (bottom) dendritic microstructures. (c) Phase-field simulation predictsa typical columnar dendritic morphology. The colors represent spatial solute concentrationvariation. Solute enriched droplets pinch off from the dendrite roots, where the secondarysolid phases form. (d) Typical snapshot of a solid-liquid interface in thermal equilibriumduring MD simulation is shown (reproduced from [18], with permission from Elsevier).

explicit tracking of the interface is no longer needed, and one can simulate the complexsolid-liquid interface in an efficient way. Phase-field equations of motion are essentially time-dependent partial differential equations, which are solved by efficient numerical methods toobtain the steady state composition, temperature and order parameter maps through thesolution of diffusion of heat, composition, and order parameter fields. The microstructuressimulated in this way correspond only to a particular location in the melt pool. During thecourse of phase-field simulations, melt pool often solidifies into columnar and/or equiaxeddendritic microstructures. These microstructures often contain defects such as microporosityand solidification and liquation cracks, microstructural anisotropies due to alloy chemistry,segregation and orientations of the solidifying dendrites, and residual stresses due toshrinkage during terminal solidification. These complex features within the microstructurecollectively influence the properties and performance of the solidified material. These effectshave yet to be explored for AM solidification. Simulation of these aspects may requireconsiderable computation time. Fortunately, phase-field models are found to scale well in

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parallel environments (MPICH [22], OpenMP [23] and CUDA [24]) to run the simulationcodes faster. The phase-field model parameters are approximated to behave quantitativelyonly at small-valued solidification conditions [25]. Better models are therefore needed totreat larger values of the melt pool solidification conditions, appropriate for AM.

For more accurate modeling of the melt pool solidification, realistic solid-liquid interfaceproperties are also needed. Molecular dynamics (MD) simulation can provide the informationregarding the interfacial energy anisotropy and the interfacial kinetic coefficient for metallicalloys in AM solidification regime. Such an efficient combination of finite element, phase-field, and molecular dynamics approaches could potentially capture the predictive AMmicrostructural evolution. The present overview is aimed at the following outstanding issuesthat arise due to the multiscale, multiphysics nature of the solidification in the melt pool.

2. Macroscale: Finite element analysis

FEA simulations determine the actual solidification conditions in the melt pool. Usinga suitable commercial software, a FEA model generate the global temperature historyduring laser irradiation on a single/multi layer(s) of powder of finite thickness depositedon a solid substrate of the same base alloy [26, 27]. Ni−based superalloys have beencommonly used for the powder as well as the substrate properties in these simulations.Bulk material properties, such as latent heat, density, and specific heat, were estimatedfrom CALPHAD-based thermodynamic calculations [28–30]. To reduce computational time,the FEA mesh elements that interact with the laser beam were finely meshed, and acoarse mesh was used in the far-field. Heat input from the laser was approximated in theliterature by a moving point, line or plane heat source [31], a double ellipsoidal volumetricheat source [32], or a Gaussian source [7, 33]. Both convective and radiative heat losseswere considered in these studies. The temperature distribution as a function of time wasobtained by solving equations for the conservation of energy. The computed results aregenerally visualized using the appropriate visualization module of the FEA software or usinga custom MATLAB [34]/PARAVIEW [35] program. The temperature gradient at eachFEA element was estimated by the magnitude along the Cartesian directions using thetemperature values from the neighboring elements. The trailing edge of the melt pool inexperiments represents the solid-liquid boundary (refer to Fig. 1b), which was approximatedby the melting temperature isotherm. Solidification begins at this boundary and theresulting columnar microstructures grow roughly perpendicular to this boundary [26, 27].The curvature of this boundary introduces an angle, which correlates the beam speed withthe local solidification velocity. The solidification boundary represents different temperaturegradients and solidification rates. Typically in simulations, the temperature gradient variesbetween ≈ 105 K m−1 and 107 K m−1 and the solidification rate varies between ≈ 0.01 m s−1

and 0.5 m s−1 for a laser scan speed on the order of 1 m s−1 [26, 27]. Temperature gradient

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times the solidification velocity is the cooling rate. These local parameters were providedas inputs to the phase-field model in order to simulate location specific microstructures. Awide range of transient nonequilibrium physical phenomena take place within the moltenpool (refer to Fig. 1a). Some of these phenomena are fluid flow, Marangoni convection,keyhole mode melting, gravity forces, and recoil pressure due to any evaporation heatlosses [10–12]. A more realistic result can be obtained when all of these complex phenomenaare considered. However, this will bring an additional cost, e.g. long computational timeand more computational resources. Implementing heat transfer, fluid flow, and Marangoniconvection in the melt pool could be a suitable first approach [7,10,12]. The resulting mass,momentum, and energy conservation equations can be solved at each discrete element toobtain the transient thermal profiles. For validation of the model, single-track laser scansimulations [36] can be used to generate melt pools with certain dimensions along with thetemperature profiles which can be compared with in situ thermography measurements [27].The solidification conditions, for example the cooling rate varies between 103 K s−1 and106 K s−1 in different studies due to different approximations of the melt pool physics. Anunified modeling approach is therefore needed to quantify the contribution of each melt poolphysics and their interactions.

3. Nanoscale: Molecular dynamics

Crystal-melt free energy calculations can be performed using suitable interatomic potentialsfor AM alloys of interest (Ref. [18] and the references within). Molecular dynamicssimulations, via the LAMMPS [37] software package, are generally conducted to assess thesolid-liquid interfacial free energy and kinetic coefficient via a capillary fluctuation techniquewhich monitors the amplitude of atom fluctuations in the interface position [18] (refer toFig. 1d). Interface properties data are still rare for the solidification conditions relevantto additive manufacturing regime for multicomponent alloys. Atomistic simulations werealso used to estimate the diffusivity of the liquid and the partitioning of the solute acrossa solid-liquid interface during nonequilibrium rapid directional solidification [38, 39]. MDSimulations can help to estimate the parameters that characterize a solid-liquid interface inAM regime. Recently, MD simulations were performed on a stable aluminum solid-liquidinterface under rapid solidification conditions and it was found that the interfacial free energyincreased by a factor of 1.25 as the temperature gradient increased by a factor of 3, while theinterface anisotropy parameter remained independent of the solidification conditions [40].The interfacial properties of multicomponent alloys under non-equilibrium conditions needto be calculated in MD for use in the phase-field model.

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4. Mesoscale: Phase-field

Phase-field models use the FEA simulated melt pool solidification conditions and the MDsimulated solid-liquid interface information to simulate the solidification microstructures.AM alloys of interest, for example Inconel 718, are typically multicomponent alloys with overa dozen elements. A three component analog, for example Ni−Cr−Nb, may describe the alloy718 effectively. It is noteworthy to mention that the design of a ternary AM alloy depends onthe microstructural features of interest; two classes of alloys are possible which include eitherthe positive segregation elements (equilibrium partition coefficient > 1) such as Ti, Mo, andNb in Inconel 718 or the negative segregation elements (equilibrium partition coefficient <1) such as Fe and Cr. In the former class of alloys, the γ-dendrite will be lean in Ti, Mo,and Nb and the liquid will be enriched with these elements as the solidification proceeds,while solute partitioning occurs in the opposite direction in the later. A multicomponentphase-field formalism [21, 41–44] can be adopted which combines the chemical bulk freeenergy, the interfacial free energy, and the driving forces for phase transformations such aslowering of the chemical potential. The resultant free energy functional is then minimizedusing standard variational derivatives with respect to the microstructure field variables, i.e.composition, temperature, and order parameter. The resulting time-dependent equationsare known as the Cahn-Hilliard equation [45] which describes the temporal evolution ofthe conserve quantity composition and the Allen-Cahn equation [46] or the time dependentGinzburg-Landau equation which describes the temporal evolution of the non-conservedquantity order parameter [19–21]. These partial differential equations are solved in 2D/3D ona uniform/adaptive mesh, using the finite difference/finite volume method, and explicit/semi-implicit time stepping scheme. Simulations often begin with either a thin layer or a circularsolid seed in the supercooled liquid at the bottom of the simulation box with relevantinitial and boundary conditions and with random, small amplitude perturbations. Stableperturbations grow with time and break into steady state dendritic microstructures (refer toFig. 1c). For equiaxed mode of solidification, nucleation mechanisms need to be incorporatedin the phase-field equations [47]. Software packages such as MATLAB [34], PARAVIEW [35],GNUPLOT [48] are generally used to visualize and analyze those complex morphologies inthe following facets, which need further attention.

4.1. Dendrite properties

The size, shape and distribution of dendrites are different at different locations in themelt pool. Phase-field simulations for AM solidification conditions often result in columnardendrites with primary arms with average spacing ≈ 1.0 µm and secondary sidearms withaverage spacing ≈ 0.5 µm [1, 26]. These spacings control the material properties, such asyield strength and tensile strength. A prediction of these spacings from the microstructuresimulations and subsequent comparison with the experiments and dendrite growth theories

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are therefore essential. This comparison can be used, for example, as a standard referencespacing data for Ni−based superalloys for AM solidification. Fourier analysis and othermethods [49] can be used to extract the dominant spacings in the microstructure. Theseresults will be significant, since current understanding is still largely limited to low-velocitycasting solidification regime. Moreover, dendrite spacing varies significantly in the presenceof melt pool convection, and thus needs to be studied in AM regime [50].

4.2. Microstructural anisotropies

Rapid solidification processing leads to nonequilibrium partitioning of solute atoms in thesolid and liquid. As a result, solute gets segregated in the volume of liquid that solidifiesin the spaces between the already solidified dendrites and gets enriched by ≈ 2 to 5 timesof the nominal composition of the alloy element with the progress of solidification [26, 27].This is known as microsegregation [51]. The presence of multiple elements in the liquidmakes the elemental segregation in the interdendritic regions complex. Moreover, theorientation/texture of the growing dendrites is often different in different locations withinthe melt pool, depending on the direction of the temperature gradient/heat flow. Anisotropyin orientations (the change in preferred orientation with respect to the growth direction),leading to intrinsic anisotropy in the mechanical properties, is therefore natural to consider.An anisotropic gradient energy surface tension [52,53] in the phase-field free energy functionaland a rotation matrix representation [52,53] for the normal vector components of the solid-liquid interface can suitably describe the orientation selection during solidification [52, 53].Simulations can also be performed in single/bicrystals in which the misorientation anglesand the convergent/divergent growth conditions between columnar dendrites are consideredas solidification variables. In this context, the competition and the transition betweenthe columnar and the equiaxed growth modes of dendrites were also modeled usingphase-field [47] or stochastic analyses [54], but in the low-velocity limit. A quantitativedescription of these phenomena in the high-velocity limit will be a significant step towardsmicrostructure and property control, since this knowledge is required to perform the solid-state homogenization heat treatment. A material with uniform properties will therefore bedesigned and controlled.

4.3. Solidification defects

A prediction of the solidification defects such as microporosity, solidification cracking, andresidual stresses in the semisolid mushy zone during the late stages of solidification is veryimportant, which have yet to be explored for AM solidification. The mushy zone in columnardendrites is a two-phase solid and liquid coexistence region between the fully solid and thefully liquid states where majority of the solidification defects form. Those defects arise dueto random growth of the solid dendrites toward each other and finally coalesce, leading to

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insufficient feeding of the liquid to accommodate shrinkage. As the fraction of solid in themushy zone increases to ≈ 0.6 to 0.98 [49, 55], the liquid is not able to flow freely andcompensate for shrinkage, resulting in microporosity. The semisolid mushy zone thereforebecomes weak and ruptures when stressed. This phenomenon is known as hot tearing. Thisbehavior can be quantified by extracting the fraction of solid and liquid from either thesimulation data or the Scheil solidification approximation [49, 55] and making a correlationwith the Euler characteristics [56] of the solidifying sections by extracting the connectiontopology between the microstructure order parameter, i.e. the coalescence behavior [57].One can also determine the mean and Gaussian curvatures at each point of the solid-liquid interface in this purpose [49, 54, 58]. Further, the mushy zone can be extracted fromthe microstructure and can be provided as an input to the volume-of-fluid (VOF) basedmethods which couple Darcy’s law and mass-conservation continuity equations [49, 55, 59].The residual liquid fraction in the mushy zone, primary and secondary dendrite arm spacings,and local solidification conditions can be used for the calculation of criteria functions of linearor low-order polynomial forms for the size, distribution, and growth of the pores. The pore-microstructure interactions can also be modeled in this regard using stochastic approachesfor nucleation of pores in combination with continuum solutions for diffusion [60]. Oncethese defect formation mechanisms are analyzed, approaches could be prescribed in order toreduce/eliminate them.

4.4. Precipitation of solid phases

Solid-state phase transformations follow the solidification process when precipitation of thesecondary solid phases takes place. The simulated microstructures and the representativemicrostructure variables (composition, temperature, and order parameter) can be used asinputs for the simulation of subsequent solid-state phase changes. As the columnar/equiaxeddendrites grow, as a direct consequence of microsegregation, they leave behind solute enrichedpockets in the interdendritic regions (refer to Fig. 1c) in a mechanism similar to the Plateau-Rayleigh instability [26, 27, 61]. Secondary solid phases, such as γ′, γ′′, δ, and Laves inInconel 718 [62], are expected to form in these pockets following an eutectic or non-eutectictype of reaction beyond a threshold solute composition and below a certain temperature.On average, the volume fraction of these phases decreases with increasing cooling rate inthe melt pool and typically varies between ≈ 2 % and 20 % in experiments [63–66]. Thesize and distribution of the secondary phases precipitation in the matrix determine thetensile strength, fracture toughness, and fatigue properties of the solidified material [67]. Afiner size and discrete distribution of those phases are beneficial compared to a coarser andcontinuous distribution, when resistance to deformation is considered [63–66]. Determinationof the influence of the solidification conditions on the size, distribution, and volume fractionof the secondary phases can be significantly different in AM solidification compared to

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casting solidification and is therefore essential. A proper implementation of the thresholdcomposition and temperature boundary conditions in the phase-field model could predictthese solid-state phases [61]. Note that these phases form in the as-deposited microstructuresas a function of the solute content or the microsegregation. Following solidification, the as-deposited microstructures often undergo the solution heat treatment, such as annealing,as a function of time and temperature to dissolve the desired/undesired secondary phasescompletely/partially in the matrix in order to modify the distribution of these phases forimproved material properties.

4.5. Solid state modeling

Phase-field simulations can also predict the stress evolution during solidification processesby a mechanical coupling of the phase and temperature fields using the stress equilibriumequation with elasto-plastic constitutive equations, the numerical solutions of which revealthe residual stress distribution in the dendritic microstructures [68–70]. Such stressdistribution can be effectively engineered by the laser processing parameters, which determinethe melt pool solidification conditions and ultimately the morphology of the microstructure.Depending on the alloy and the ultimate application of the as-built part, it may be necessaryto perform post-solidification thermal processing via homogenization heat treatment andhot isostatic pressing to obtain the desired mechanical properties for optimum performance.Moreover, the formation and growth of the secondary solid precipitates (such as carbides,intermetallics and Laves) need to be modeled [20, 70] to predict the processing window(for example, as a function of time and temperature during heat treatment) of each phases,depending on whether the precipitate is either beneficial or harmful in terms of yield strength,tensile strength, fracture toughness and fatigue life requirements [71]. The mechanicalinteractions, interface anisotropies and orientations, defect densities and diffusion andgrowth kinetics are significantly different in solids compared to solidification microstructureevolution [20,70,72]. The discussion of these topics is beyond the scope of the present article.The present review is limited only to the as-deposited state of the microstructures.

5. Summary

Additive manufacturing has the potential to become the technology of the future. There isan increasing demand for the predictive simulation of AM microstructures to achieve bettermaterial properties than traditional casting and metal forming routes. Future work is stillrequired to address the process and microstructural challenges during AM to improve theconfidence in the quality of the material in service. In this context, the present report reviewsthe multiscale modeling of multicomponent solidification and solid-state transformations asfar as they relate to the solidification models, considering Inconel 718 as an example alloy.The numerical results are also needed to benchmark with experiments and other simulations,

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since AM microstructural features are far-from-equilibrium. In this way, the processing-microstructure-property correlation of AM materials can be improved which will enable theAM industry to reliably use this novel application to manufacture parts of predictable qualityand behavior.

Acknowledgements

I thank Dilip Banerjee, William Boettinger, Jonathan Guyer, Trevor Keller, Greta Lindwall,Li Ma, Kevin McReynolds, and Nana Ofori-Opoku from National Institute of Standards andTechnology, Marianne Francois and Chris Newman from Los Alamos National Laboratory,and Balasubramaniam Radhakrishnan from Oak Ridge National Laboratory for usefuldiscussions.

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