Microstructure modelling for metallic additive manufacturing: a review

The microstructure of metals depends on the additive manufacturing (AM) process and the process parameters. However, experimentation on different process parameters for different materials is costly and time-consuming. To overcome these challenges, numerical simulations provide insights that allow prediction of microstructure formed under different process parameters. Microstructure modelling requires coupling of macro-scale thermal model or experimentally measured temperature profiles with either a meso-scale or micro-scale microstructure model. In this review, the commonly used AM techniques for metals are introduced, followed by a discussion on the microstructure of parts fabricated using these processes. This review then presents the latest models used in simulating different microstructural aspects of metal AM parts. In addition, the models were compared and the potential and challenges in microstructure modelling were discussed. ARTICLE HISTORY Received 27 June 2019 Accepted 4 October 2019


Introduction
Additive manufacturing (AM) is a group of manufacturing methods that can produce highly customised parts catering to the low volume markets (Chua and Leong 2017). According to ISO/ASTM 52900:2017, AM is the general term for those technologies that create physical objects by successive addition of materials based on geometrical representations. AM involves the process of joining materials to make parts from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing and formative manufacturing methodologies. AM is able to build complex geometries without the need for moulds which potentially reducing part count (Gibson, Rosen, and Stucker 2010;Atzeni and Salmi 2012). The aerospace and healthcare industries can use AM to their advantage with metals like stainless steel, titanium, cobalt chrome and nickel alloys (Vandenbroucke and Kruth 2007;Brunette et al. 2012;Cantor, Assender, and Grant 2015). While AM can achieve complex geometry, the manufacturing process is highly complex with numerous factors affecting the properties of the part. An important feature of metal is its microstructure. For a given metal, there can be a variety of microstructural features that affects its mechanical properties. The size of grains, microsegregation of alloying elements, phases within the metal and size of dendrites relates to the tensile strength and ductility (Warren and Boettinger 1995;Lu et al. 2010). During the AM process, the microstructure is formed in-situ and would therefore depend largely on the process parameters and material used. The process parameters are dependent on the metal AM method used.

Additive manufacturing methods for metals
There are two commonly used techniques in AM of metals, powder bed fusion (PBF) and directed energy deposition (DED). PBF process includes selective laser sintering (SLS), selective laser melting (SLM) and electron beam melting (EBM). SLM and EBM are used for the fabrication of metals.
Both processes start from digital files. 3D computeraided design (CAD) files are sliced into layers and the data are then sent to the machine for printing. The next step is a set of cycles that melts the powders layer by layer until the final product is completed. In each cycle, a fixed thickness of powder is deposited onto a substrate and a heat source is applied to the powder in a raster scanning pattern. The heat source could either be laser or electron beam. This melts the layer of powder, bonding it to the substrate or to the previously melted layer. Lastly, the platform is lowered and the cycle repeats.
DED processes, on the other hand, can use wire or powder as the feedstock, and energy sources could use laser, electron beam and electricity. As DED machines can have 5 axis of movement, the layer added to the CAD file may need to be sliced in a computer-aided manufacturing (CAM) software. Energy source and feedstock could be applied to the feedstock coaxially or off-axis. The feedstock fed as an energy source is applied to it. The feedstock material is melted and bonds with the substrate thereby forming a layer. Examples of SLM and coaxial laser direct energy powder deposition setups are shown in Figure 1.
While both PBF and DED processes build the part layer by layer, there can be distinct differences between the two. PBF machines are currently limited to build up layers in a single direction. DED machines can have 5+ axis systems allowing layers to be built in any direction provided there is no clashing of the part and tool head. This allows DED to repair parts in multiple directions while PBF is limited to repairing in a single direction. Parts produced by PBF can have better resolution due to the smaller powder size and smaller energy spot size as compared to DED machines.

Microstructure of additive manufactured metals
AM processes have been used to fabricate certain alloys like 316L stainless steel, Ti-6Al-4V and Inconel 625/718 which are common alloys used in marine and aerospace applications. The microstructure of metals determines the mechanical properties of the part such as yield strength, ductility and hardness (Peel et al. 2003;Ziętala et al. 2016;Guo et al. 2017;Shuai et al. 2018). Due to their wider applications, most of the modelling works done have been on these materials, however, they have the potential to be applied for new alloys. New alloys that have been processed by AM includes, but not limited to, Al-Cu alloys Martinez, Todd, and Mumtaz 2019), Al-Cu-Mg (Nie et al. 2018), Al-Cu-Mg-Si , Al-Zn-Si-Mg-Cu (Casati et al. 2019), Cu-Sn alloys (Mao et al. 2018;Tan et al. 2019), Cu-10Zn ) and Cu-Cr (Uchida et al. 2019). AM process predominantly produces columnar grains with a mix of equiaxed grains that consist of cellular dendrites ( Figure 2) Frazier 2014;Roehling et al. 2017). The columnar grains typically grow along the build direction causing anisotropy and difference in performance with varying build orientations (Lewandowski and Seifi 2016). By varying the process parameters like the energy sources and feed types can lead to differences in grain structure (Frazier 2014). Ti-6Al-4V had columnar grains when printed with multiple tracks using electron beam-wire-DED and laser-powder-DED, however, when printed using laser-powder-DED for a single track the grains do not grow in a columnar fashion. The Ti-6Al-4V microstructure was finer in the electron beam-wire-DED as compared to the laser-powder-DED Dehoff et al. 2015). Ti-6Al-4V was shown to exhibit near equiaxed prior-beta grains that quickly grow to large columnar grains after the initial layers, indicating that varying cooling rates and temperature gradient can induce changes in microstructure (Tan et al. 2015). The cooling rate and temperature gradient also affect dendritic formation. Cellular dendrites are formed when the cooling is fast, while slower cooling rates allow formation of secondary dendrite arm (Kou 2003). High cooling rate and low temperature gradient can cause an increase in mechanical properties, like in the case of Inconel 718 where the finer distributed laves phases are formed, leading to lesser chance of hot cracking (Nie, Ojo, and Li 2014). As the grain structure and dendrites affect the overall mechanical properties, it is important to know how different printing techniques and process parameters affects the grain Figure 1. Schematic of selective laser melting and coaxial laser directed energy powder deposition. structure (Gifkins 1976;Baufeld, Van der Biest, and Gault 2010).
The microstructure of metals depends largely on the thermal gradient and cooling rate during solidification. By varying the process parameters, it is possible to alter the grain structure. Raghavan et al. used two different scanning strategies, one with line scans and another spot scanning. Line scans tend to cause columnar grains while spot scanning together with controlling of the current, duration and distance between each spot lead to more equiaxed grains being formed (Raghavan et al. 2016). Dehoff et al. additionally alternate line and spot scanning within the same part to achieve different areas with columnar and equiaxed grains. The sequence in which the boundary area, equiaxed area and columnar area is printed first also matters as it affects the temperature gradient (Dehoff et al. 2015). In both cases, EBM is used and had the powder bed pre-heated to near melting temperature of the material, this which helps lower the overall temperature gradient allowing equiaxed grains to form.
It is shown that changing the parameters and scanning strategy can result in tailored microstructure (Körner et al. 2014;Parimi et al. 2014;Dehoff et al. 2015;Wei, Mazumder, and DebRoy 2015;Raghavan et al. 2016). Tailored microstructure can help in applications that require texture or the lack of it like in single crystal turbine blades and provide possibilities in grain boundary engineering.
Transition of columnar to equiaxed grains depends largely on the temperature gradient as well as the cooling rate. These values can be found experimentally or from computer simulations. Experimental methods can only investigate microstructure after the entire part is printed. In-situ monitoring of the AM process have limited capabilities and are restrained to monitor surface effects, and provide no information beyond the surface (Clijsters et al. 2014;Huang et al. 2015;Everton et al. 2016;Lu and Wong 2018).
Computer simulations of the AM process help in understanding the evolution of temperature beyond the top surface. Simulations use the temperature results from thermal models to predict microstructure on varying length scales. As AM process involves building layer upon layer, the melt tracks experience multiple remelting. Experimental data can only give the grain structure after the entire build, while simulation provides access to the evolution of the grain structure throughout the process.
The aim of the paper is to review the different methods of microstructure modelling used specifically for AM metals. The different microstructure models are compared, and their strengths and limitations are explored.

Modelling for metal additive manufacturing
Modelling of microstructure requires coupling models at different length scales. First, a macro-scale thermal model is required to get the thermal history of the part during the AM process. The thermal model can come from simulation or from experimental data. Experimental data, however, would only capture the surface and a numerical model would be needed to translate it to a 3D melt pool. Simulation of the thermal model can range from a single melt track to multiple melt tracks and across multiple layers. Second, data from the thermal model is used as input to the microstructure model. Microstructure models can be further categorised based on another length scale, meso-scale and micro-scale. Meso-scale simulation refers to models that simulate multiple grains, while micro-scale simulation refers to models that simulate the dendritic formation.

Thermal modelling in additive manufacturing
Thermal numerical models can investigate the temperature the material would experience during the AM process. Thermal models can predict the temperature profile, fluid flow, porosity and be coupled to microstructure models or thermo-mechanical models to get the microstructure or residual stress and deformation (Vastola, Pei et al.;Denlinger, Heigel, and Michaleris 2015;Yang et al. 2018). The temperature profile could also be used to predict the amount of precipitate formed, columnar grain growth orientation and columnar to equiaxed transition (CET) Yin et al. 2018;. Microstructure modelling uses thermal models to find the temperature history and shape of the scanned tracks as input. This section reviews the thermal models used in conjunction with microstructure models.
2.1.1. Thermal models without fluid flow Finite difference method (FDM) and finite element method (FEM) have been coupled with various microstructure models, typically simulating for large volumes of multiple tracks and layers (Mercelis and Kruth 2006;Zeng et al. 2015;Chen, Guillemot, and Gandin 2016;Zinovieva, Zinoviev, and Ploshikhin 2018). FEM/FDM solves for the heat equation partial differential equation (PDE): In Equation (1), r is the density, c p is the specific heat capacity, T is the temperature, t is the time, k is the thermal conductivity and Q s is the heat source added to the model. Heat loss by radiation and convection is sometimes added to the top surface as boundary conditions (Loh et al. 2015).
Simulating for PBF, FEM models often use elements of different materials. The top elements are assigned as the powder material and the remaining elements are the substrate material (Figure 3(a)). The powder element often has the density based on the porosity of the powder bed multiplied by the substrate density. The thermal conductivity is similarly dependent on the porosity of the powder bed (Chen, Guillemot, and Gandin 2016;Foroozmehr et al. 2016;Chiumenti et al. 2017). Elements that are assigned as powder material would change to substrate material once the temperature of the element goes above the liquidus temperature. Simulating for DED differs slightly from PBF models. Material has to be added while the heat source moves. FEM models do this by first placing dead/inactive elements above the substrate, as the heat source moves, the dead/inactive elements are set to be alive/active ( Figure 3b) (Fallah et al. 2011). This simulates the material being fed and added to the melt  pool. Dead/inactive elements work by having its thermal conductivity set to near zero such that the temperature of the element would not change.
Meshing for FEM plays an important role in getting accurate temperature profiles and melt depths. Hexahedral element is commonly used as it provides better accuracy and works well with simple geometry used in simulations (Lindgren et al. 1997;Yin et al. 2018). AM processes have localised areas of large temperature changes in the area of the heat source. Chiumenti et al. suggested that in order to have better estimation of the temperature, the size of the element have to be smaller than the energy source spot size, Foroozmehr et al. used eight by eight elements within the laser spot (Foroozmehr et al. 2016;Chiumenti et al. 2017). Having high resolution throughout the entire mesh can lead to long computational time. As such, most models simulate a single melt track. In order to simulate a larger volume, Patil et al. used adaptive mesh refinement (AMR) to have higher resolution in and around the melt pool and progressively coarser mesh beyond the melt pool leading to a decrease in computing time by a hundred times . This allows solving for a larger area allowing investigations on scanning strategies involving simulating for multiple tracks. Denlinger used a similar method and found it to be 432 times faster than using static mesh (Denlinger 2018).
2.1.2. Thermal models with fluid flow in continuum scale or powder scale considering actual powder particles Computational fluid dynamics (CFD) methods like the finite volume method (FVM) have been used to get the shape of the track, melt pool and the temperature history. FVM methods are used to solve the Navier-Stokes equation which can simulate effects like vaporisation and Marangoni effect seen in AM (Khairallah et al. 2016;Andani et al. 2017). This would lead to higher accuracies in the temperature distribution in the melt pool and also higher accuracies in geometry (Khairallah et al. 2016). In order to simulate the fluid flow within the melt, the mass and momentum equation have to be included alongside the energy equation (Gürtler et al. 2013;Yuan and Gu 2015): Equation (2) is the energy equation for heat conduction where H is the enthalpies and v is the velocity of fluid. Equation (3) is the mass conservation equation where M s is the mass source. In PBF where powders are already laid before simulating, the mass source can be considered as zero while in DED where powders are added while running would require the mass source term. Equation (4) is the momentum equation where m is the dynamic viscosity and F is a force term like gravitational force that may be included. The increase in the number of PDEs leads to an increase in computing power required. Most FVM models only simulate a single melt track in 3D (Fallah et al. 2012;Lee and Zhang 2015;Khairallah et al. 2016;Panwisawas et al. 2017). CFD of PBF can use varying types of powder distribution, either by powder packing density or using numerical models like discrete element method (DEM) (Lee and Zhang 2015;Zielinski et al. 2017). Numerical methods of packing powder led to irregular powder distribution that is closely resembling to actual powder deposition in PBF processes. CFD-DEM method has been used for DED processes as well, taking into account gas flow, powder flight and heating as well as interactions with the melt pool (Pinkerton 2015). The amount of details required of the simulation causes it to be even more computationally expensive as DEM is simulated together with CFD and gas flow has to be taken into account as well. While CFD can lead to more realistic tracks, literature of using CFD for DED processes is few. The Lattice Boltzmann method (LBM) is another CFD method that is used to couple with microstructure models. LBM does not solve the Navier-Stokes equation, instead, it solves discrete Boltzmann equations to model the fluid flow (Rausch et al. 2017). LBM can simulate the powder distribution as well as the fluid flow and evaporation while FVM would require coupling with DEM to simulate the powder distribution (Khairallah et al. 2016;Liu and Wang 2017). LBM becomes comparatively more expensive when changing from 2D to 3D, with most 2D models being able to simulate multiple layers for a single track, a 3D model however, can simulate multiple tracks in a single layer (Körner, Bauereiß, and Attar 2013;Rai, Markl, and Körner 2016;Degenhardt 2017). LBM has been used to model PBF processes however no literature was found to have used LBM for DED processes at the time of writing.
CFD models have an average cell size of 5 µm and average domain of 150 × 1000 × 150 µm (Körner, Bauereiß, and Attar 2013;Lee and Zhang 2015;Zielinski et al. 2017). In comparison, FEM models have element size ranging from 25 µm to 5 mm and domain size ranging from 0.4 × 1 × 0.4 mm to 250 × 250 × 325 mm (Loh et al. 2015;Foroozmehr et al. 2016;Chiumenti et al. 2017). The model and element size can differ by three orders of magnitude. The number of cells for CFD also range in the millions while FEM elements tend to be less than a million. While CFD models can simulate multiple physical phenomena, the complexity and small cell size limit the total domain to be less than a millimetre in length, limiting the simulation to few short tracks. FEM models have many assumptions, reducing accuracy but increasing the size of the domain to accommodate multiple longer tracks and multiple layers.

Microstructure modelling in additive manufacturing
There are similarities between the melting of a single track in AM and welding. Welding typically adds material between two metal parts thereby joining them while AM adds material to a substrate or previously printed layer. The solidification of both welding and AM are very similar since both use a moving heat source to melt the added material leading to a moving melt pool. Since welding research is more developed, microstructure modelling techniques in welding can be applied and translated to AM (Seufzer 2014;Rolchigo et al. 2017).
There are three microstructure models that are reviewed, phase-field modelling (PFM), kinetic Monte Carlo (MC) and cellular automata (CA). CA and MC are considered meso-scale models as they simulate multiple grains mainly looking at the overall size and aspect ratio of the grains. PFM is considered a micro-scale model as it can simulate on the sub-grain level, getting the solute concentration, precipitates and dendrites shape. It also able to simulate multiple grains using multi-phase field models (Miyoshi et al. 2019).
The coupling between macro thermal model and the microstructure model can be done in two ways. The models can be weakly coupled, where the thermal model is simulated first and the thermal history is used as input to the microstructure model. Another way is to strongly couple them and simulate both at the same time, where information is shared and interacts between both models. Although strongly coupled simulations would be able to yield more accurate results, it requires longer computational time (Gandin et al. 1999). A schematic of the different microstructure models is shown in Figure 4 showing the boundary conditions and assumptions of the model.
2.2.1. Phase field modelling PFM has been shown to be able to model solid, liquid and material phases within the same model. Microstructure evolution like grain coarsening and dendrite growth in forced convection could be simulated as well (Gránásy, Pusztai, and Warren 2004;Singer-Loginova and Singer 2008;Sahoo and Chou 2014). PFM is thermodynamically consistent, however, the interface width has to be thin in PFM increasing difficulty in modelling thereby limiting the size of the model (Elder et al. 2001). A large model with several interfaces will be too computationally expensive to run, especially if it is in 3D. While PFM provides the ability to simulate multiple phenomena with high accuracy, computational resources will be limiting factors (Francois et al. 2017).
2.2.1.1. Phase field modelling theory. PFM simulates microstructures by having all variables to be continuous across the interface. The interface can be a fraction of both solid and liquid. Mathematically, the description of the interface or otherwise called the phase field variable, f, can be represented as f [ [−1, 1]. During the solidification process, the system of a fixed volume would tend towards a lower free energy in order to achieve stability (Boettinger et al. 2002;Kim and Kim 2005;Steinbach 2009): where F is the total free energy of the system, f is the thermodynamic potential, c is the concentration, vg(f) is the double well potential, e 2 |∇f 2 | is the phase-field gradient energy and V is the volume. The phase-field gradient energy is dependent on the gradient of the interface and therefore would be zero in fully solid and liquid regions. The thermodynamic potential is the sum of the free energies of the liquid and solid states. Over time, the system would decrease the total free energy of system. In order to solve the transient problem, the Allen-Cahn and Cahn-Hilliard equations can be used (Boettinger et al. 2002;Chen 2002;Kim and Kim 2005): where M f and M c are the motilities of the phase field and concentration field at the interface respectively. In a multi-phase problem like in solidification of certain steels, there can be liquid, ferrite and austenite, each of these phases has its own variable, f i ( x, t) where i denotes the state that the field is. The sum of the phases in a multi-phase system has to reach unity, Boettinger et al. 2002  2D PFM to investigate the CET of magnesium alloy during welding (Montiel et al. 2012). FEM and PFM have been coupled to simulate AM processes. The temperature gradient and cooling rates were found using FEM and used as a boundary condition in PFM. Gong and Chou simulated in 2D for EBM of Ti-6Al-4V using FEM and PFM (Gong and Chou 2015). The thermal model shows that faster scanning speeds led to higher cooler, the cooling rate is used to find the distance between seeds for the PFM models too. The PFM model showed columnar prior-beta grains which grew faster with higher cooling rates and experimental results showed similar prior-beta grain sizes. However, the grain size was partially affected by the predetermined distance between seeds. Sahoo and Chou found the grain size differently, simulating from a single dendrite and using the primary dendrite arms spacing as the grain size (Sahoo and Chou 2016). Sahoo and Chou simulated the EBM process using a thermal model in 3D and a PFM model in 2D. A single dendrite was simulated and its primary dendrite arms grew in the columnar fashion and was compared to the experimental results showing columnar grains. The primary dendrite spacing and arm width were found to get smaller as scanning speed increases. The simulated dendrite arm spacing was larger than experimental results by approximately 0.5 µm. Within a melt pool, the solid-liquid interface can have different temperature gradient and solidification velocity at the bottom and trailing edge of the melt pool. Fallah et al. simulated for laser-powder-DED of Ti-Nb using 3D FEM to find the localised temperature gradient and solidification velocity which is input to 2D PFM models (Fallah et al. 2012). The thermal model found that the temperature gradient is smaller and the solidification velocity faster near the surface as compared to the bottom. PFM models for different temperature gradient and solidification velocity were simulated and the dendrite spacing was compared to experimental results. The PFM model was accurate except at the extreme ends where temperature gradient or solidification velocity is highest. Different PFM model or boundary conditions may need to be applied to get more accurate results. CFD can simulate the Marangoni flow which enhances cooling and thus can get more accurate melt pool shape and temperature profile, Acharya et al. simulated with SLM process for Inconel 718 using 2D PFM with the domain shaped like the melted track found using 3D CFD (Acharya, Sharon, and Staroselsky 2017). Although CFD was used, the PFM model had some differences with experimental results. The primary dendrite spacing and segregation of solutes tally with experimental results, however, the simulated results show secondary dendrites growing while experimental results show little to no secondary dendrites. These PFM models reduced the complexity by simplifying the alloys to binary alloys. However, PFM can also consider multi-phase and multi-components. Kumara et al. simulated with data from Thermo-Calc to incorporate multiple elements for Inconel 718. The model assumed a fixed cooling rate with no temperature gradient and simulated to study Laves phases during solidification and heat treatment ).

Kinetic Monte Carlo
Kinetic MC modelling is a completely probabilistic simulation of the grain structure. MC is able to simulate grain structures over large volumes due to the low computational resource required. MC models differ from PFM models which simulate the solute concentration and dendrite shape, MC only models the grains therefore simulating on a different length scale. MC models provide the shape and size of grains.
2.2.2.1. Kinetic MC theory. The MC method is based on an atomistic simulation and the minimisation of the overall system energy. Each site on the lattice is given a number or orientation and if the neighbour of a site has a different orientation, it would add to the overall system energy. If the neighbour has the same orientation, there is no change in the system energy. Each site has a probability of changing orientation, if the orientation contributes to a decrease in the overall energy system, then it is accepted. If the change in orientation causes an increase, a random number would generate for that site and compare to an acceptance probability and if the random number is smaller than the acceptance probability, it is accepted. These are shown in the following (Yang et al. 2000;Rodgers, Madison, and Tikare 2017): where DE is the overall system energy, J is a scaling factor of the energy system, d is the Kronecker delta function, lattice assigned number or spin, I j, I k are the number/spin of the site and neighbour, k B is the Boltzmann's constant and T 0 is the ambient temperature of the process. This happens within a MC time step, every time step the lattice is taken as-is and every site is recalculated and compared with the previous time step. In general, if the current step has an increase in overall energy, it is rejected and the previous time step configuration is kept. In order to simulate solidification, some modifications to the MC method have to be made. The grain boundary mobility is dependent on the temperature at the site and follows the Arrhenius equation (Yang et al. 2000;Rodgers, Madison, and Tikare 2017). Equation 9 is then modified to: where M 0 is the Arrhenius pre-exponential factor, Q act is the activation energy, R gas is the gas constant and T is the site temperature. MC method requires relatively lower computational resources and is able to simulate both 2D and 3D (Stefanescu 2015). It is able to simulate multiple laser passes and layers due to the low computation cost (Rodgers, Madison, and Tikare 2017).

Application of kinetic MC.
MC methods have been used to simulate microstructure for casting, sintering and welding. It is also able to model the change of microstructure during recrystallisation and grain growth (Zhong 2011;Liu et al. 2014). Mishra and DebRoy used MC to predict grain sizes in the heataffected zone (HAZ) due to solidification and grain growth in Ti6Al4 V during welding (Mishra and DebRoy 2004a;Mishra and DebRoy 2004b). The models showed potential to be used for simulating repairs using AM, especially to treated metal, which leads to HAZ. Thermal models are also coupled with MC to predict grain structures in welding (Yang et al. 2000; Elmer, and DebRoy 2017). Wei et al. capture the aspect ratio and grain sizes at varying welding speeds of aluminium alloys using CFD and MC. However, the model was not validated against experimental data (Wei, Elmer, and DebRoy 2017). Rodgers et al. predicted the 3D microstructure in a single pass electron beam welding using MC and later increasing the number of scans extending it to EBM and electron beam powder DED (Rodgers et al. 2016;Rodgers, Madison, and Tikare 2017). The model was able to simulate a large volume capturing the crystal growth bias towards the scanning direction. While the model was able to capture the patterns of the experimental EBSD data, the aspect ratio and sizes are not fully captured. Sun et al. used different melt pool geometry in MC for the solidification of 316L and found equiaxed grains growing in the middle of the track for fast scanning speed (Sun et al. 2018). The results support experimental data that columnar grains being broken off each layer at faster scanning speed. However, the grain size and aspect ratio also differ from experimental data. While the MC method has been used extensively for welding, the use of it for AM is few. MC method, however, has the least computational cost and has the potential to model over large volumes.

Cellular automata
CA simulation is a probabilistic model with deterministic elements in it. The nucleation rate and dendrite direction are probabilistic while the growth of dendrites and preferential crystallographic direction is taken from theory. Similar to MC, CA simulates the grain structure rather than the dendrites. Having more equations to solve, CA uses more computational resources than MC but is able to capture grain angles and grow accordingly. This can lead to more accurate results.
2.2.3.1. Cellular automata theory. CA uses certain rules to change the state of a cell and in the case of solidification, the states are solid, liquid and mushy. CA requires a mesh of equally divided cells, usually square (2D) or cube shape (3D), with each cell interactive with its neighbouring cells by the rules. The CA model would require rules that govern the transition from solid to liquid, liquid to solid, setting of crystallographic orientation, speed and orientation of capturing neighbouring cells for crystal growth. CA follows the rules of solidification theory as well as some probability elements for nucleation. CA models are often coupled with macro thermal models to help in the rules of solid-liquid transition. CA models are like MC models in that both do not calculate the dendritic structures like secondary arm growth but instead simulate the grain structure allowing it to capture a larger scale. The changing of cell state from liquid to solid depends not only on temperature but the probability of nucleation as well. CA models use a probabilistic approach to determine nucleation, as undercooling, DT, drives the solidification, the continuous grain density, n, is given by (Gandin and Rappaz 1994): A Gaussian distribution can be used to determine if a cell forms a nucleus: where n max is the maximum nucleation density, DT is the undercooling of the liquid, DT mean is the mean undercooling when nucleation occurs and DT s is the standard deviation of the undercooling. Capturing of neighbouring cells by crystal growth is determined by solidification theory. One of the growth kinetics models is the Kurz-Giovanola-Trivedi (KGT) model. The KGT model assumes that the solid-liquid interface would grow at near absolute stability limit and the velocity can be found as a function of the undercooling (Kurz, Giovanola, and Trivedi 1986): where V is the solute supersaturation, C * is the liquid solute concentration at the tip of the dendrite, C 0 is the initial solute concentration, D is the solute diffusion coefficient, K is the partition coefficient, R is the radius of the tip of the dendrite, m is the slope of the liquidus, G is the Gibbs-Thomson coefficient, G c is the solute gradient, G is the thermal gradient and DT is the undercooling temperature. Solving equations 13-16 together would give a correlation between the undercooling temperature and the velocity of the dendrite growth.
2.2.3.2. Application of cellular automata. CA was developed for casting and has been adapted for welding and laser cladding (Rappaz and Gandin 1993;Zhan et al. 2008;Yin and Felicelli 2010; CA has been used to simulate the SLM process which was coupled with varying thermal macro models. Zinoviev et al. coupled FDM and CA to simulate SLM of 316L, getting the 2D grain structure at the traverse cross section of a melt track for multiple layers (Zinoviev et al. 2016). Although the model was able to capture the columnar growth, the experiment data displayed small amounts of equiaxed grains and some columnar grains being cut off by another columnar grain. This could be due to the model being only 2D and lack of nucleation that could happen in the experiment. Zinoviev et al. similarly coupled FDM and CA for 3D grains, however, the grains grew purely columnar which can be different from experimental data that had equiaxed grains (Zinovieva, Zinoviev, and Ploshikhin 2018). This leads to inaccurate grain structure as only columnar grains will grow from bottom to top since no nucleation occurs which differs from experimental data found. Koepf et al. simulated in 3D for multiple layers with multiple tracks per layer for EBM of Inconel 718 (Koepf et al. 2018). The thermal solution was found analytically and CA was used to simulate the grain structure and the simulated grain size and aspect ratio matched the experimental results well through the multiple layers. Similar to the models mentioned above, there are equiaxed grains mixed within the columnar grains which affects the overall grain structure to a certain extent. However, the cause for these equiaxed grains may not be fully understood and can be difficult to capture in the CA models. One issue that multiple tracks and layer CA models must deal with is the large memory required deal to the small cell sizes. Both CA and thermal models would require parallelisation and the thermal solution may need to be simplified. Koepf et al. used an analytical solution instead of a purely numerical simulation to reduce the computational resource. Koepf et al. used numerical simulation for the thermal model and reused the temperature profile at varying regions in the entire model, rotating by 90°at certain regions (Koepf et al. 2019). This reduced the computational memory and storage required for the entire thermal profile. Lopez-Botello et al. simulated the microstructure along the longitudinal cross section using 2D FEM and CA models for SLM of AA-2024(Lopez-Botello et al. 2017. The model predicted columnar growth with grain sizes close to experiment data. Nucleation is captured based on experimental data with the probability of nucleation depending on the grain size. Although this method managed to capture the scattered grains, it requires experiments first for tuning of parameters. Rai et al. coupled CA with Lattice Boltzmann to find the microstructure of different laser scanning pattern (Rai, Markl, and Körner 2016). Although LBM is more accurate than FEM, it is computational resource heavy and difficult to translate to 3D. The grain sizes matched the experimental data that's compared to other literature, however, the grain orientation could not be captured fully as it is simulated in 2D and cannot be compared to the EBSD results. Panwisawas et al. coupled CFD and CA models to predict the microstructure of Ti-6Al-4V using SLM (Panwisawas et al. 2017). While CFD provides greater detail in terms of melt pool shape and the flow in the melt is accounted for, the volume simulated is limited to a single scan track. This limits the study and single scan tracks may not be able to represent the overall grain structure.
CA has been modified to consider the momentum and species transport equations called modified cellular automata (MCA) (Zhu, Lee, and Hong 2004). MCA solves the additional equations by finite difference, which is an additional step CA does not have (Wang, Lee, and Mclean 2003). Unlike CA, MCA simulates on the micro-scale also getting the dendrite shape and solute concentration in a given melt pool that is solidifying. MCA has been shown to be able to model for multiple solute elements, CET and also include effects like convection in the melt pool (Zhu, Lee, and Hong 2004;Michelic, Thuswaldner, and Bernhard 2012;Chen, Xu, and Liu 2014). While having different equations and methods as PFM the results found were similar (Zaeem, Yin, and Felicelli 2013). However, MCA has not been applied to AM as of the time of writing.

Comparison of microstructure modelling methods
A comparison of the length scale and accuracy of the different models is shown in Figure 5. The smaller the length scale, the smaller the simulation time step needed, this to large amount of data created. Higher accuracy models tend to have more equations and variables to solve which requires more computational resources. In general, accurate models have small length scales, short time steps thus limiting the volume the model can simulate. CFD models tend to only simulate a single melt track while FEM models simulate multiple melt tracks. PFM models simulate a few dendrites while CA and MC models simulate the grain structure.
Comparing the models, PFM not only gives realistic dendrite shapes, but it also accounts for the solute distribution between the dendrite arms. When combined with thermodynamic databases, TTT and CTT curves, PFM can have not only very accurate details of solute distribution and dendrite shapes and sizes but also predict the composition of the various metals and intermetallic and transformation and growth of precipitates (Ferreira et al. 2017;Mullis, Bollada, and Jimack 2018;Böttger et al. 2019). All the PFM models for AM were in 2D and only simulated a few dendrites. This is most likely being due to the computational resources required thus making 3D models time-consuming and expensive to run. MCA has been modelled in 3D and have been shown to be more efficient and faster than PFM while having the similar results, however, simulations are also done on small number of dendrites (Wang, Lee, and Mclean 2003;Zaeem, Yin, and Felicelli 2013). In comparison, CA and MC were used to simulate multiple tracks and layers only capturing the grains thus only producing results like grain sizes, aspect ratio and CA grain angles. A comparison of all the models is shown in Table 1.

Challenges
Modelling for microstructure requires thermal models as they provide the temperature, cooling rate and temperature gradient for inputs to the microstructure model. Having more accurate cooling rates and temperature gradients can lead to better predictability of microstructure. However, the length scale of the models is either a single track scan or multiple tracks and multiple layers, with CFD being used most often to simulate a single line and FEM for multiple tracks and layers. This most likely happens due to the computational resource limitation. FEM models have to use bulk material properties and the powder bed, the porosity of the powder bed is used to estimate the bulk properties. However, finding the porosity of the powder bed is difficult, the density of a powder bed can be estimated by the apparent density. The apparent density can be found using a Scott volumeter, but during the recoating of the powder during a PBF process the underlying surface is uneven and actual distribution of the powder is unknown, this will cause the density of the powder bed to vary. FEM models have codes developed to allow mesh refining and coarsening or having lumped heat per layer to reduce computational time and increase the volume simulated (Papadakis et al. 2014;Patil et al. 2015). However, the FEM codes were written to help improve simulation time for AM processes and commercial FEM software currently unable to reproduce it at the time of writing. CFD models, on the other hand, are not replicating the volume, FEM models are making but instead focus on the different phenomena in the melt pool. This leads to more accurate details which would be better for the microstructure model as better growth kinetics can be predicted. A challenge thermal model face in getting accurate temperature is an accurate input of the heat source. FEM models tend to take a heat distribution either as a 2D input on the top surface or as a 3D volumetric heat (Loh et al. 2015;Yin et al. 2018). This heat input often uses estimated power and absorptivity. CFD models can use ray tracing to model laser PBF which leads to higher accuracy (King et al. 2015). However, in all models, the emissivity and absorptivity can change with temperature and this is difficult to capture in experiments.
Thermal models are considered macro scale in comparison with microstructure models which are micro-or meso-scale. MC and CA both are better suited to simulate large volumes as they require lesser calculations while PFM and MCA require more calculations and simulates a smaller volume. The choice of models comes down to the length scale required. Apart from the length scale, each microstructure models also face challenges.
The dendrite shape and solute concentration that micro-scale models predict can be used to investigate new materials to be used in AM. Having an estimate on element phases and solute concentration can help in knowing if defects like hot cracking are likely to form or segregation of alloying elements (Keller et al. 2017;Gao et al. 2018). However, PFM and MCA also face certain challenges. Currently, both models only model the solidification portion without the melting portion. Instead, seeds are placed in predefined positions. While this may provide an estimate of the Primary dendrite arm spacing (PDAS), the actual size of the dendrite is dependent on the given spacing from each seed. This leads to bias based on the seed position placed and changing of material or process parameters may require changes in seed positioning or experiments to provide better positioning. Another challenge is the computational resources required in micro-scale modelling (Glicksman 2010). This limits the volume of the simulation within a single grain or a few grains. Due to the nature of AM, depending on the scanning strategy, tracks can be re-melted several times. For PBF processes, tracks are often overlapping causing a single track to be melted at the sides and the next layer would also cause the top portion of the previous layer to melt. The grain structure of the metal would then depend on the melting and growing of grains from previously printed tracks. In order to track the grain structure, microstructure models would have to be able to simulate a large enough volume to investigate different scanning strategies. In this aspect meso-scale MC and CA models would work well as it has been shown to be able to model in 3D as well as multiple tracks and layers. A challenge for MC models is to get results closer to experimental data, although the trends of the grain structure are captured, the aspect ratio differs (Rodgers, Madison, and Tikare 2017;Sun et al. 2018). CA models face challenges in getting the scattered mix of equiaxed grains found in experimental data (Rai, Markl, and Körner 2016;Zinoviev et al. 2016;Zinovieva, Zinoviev, and Ploshikhin 2018). Most CA models depend on nucleation models used in casting or do not implement nucleation, similar to welding CA models. This leads to grains that grow indefinitely in the build direction which differs from experimental data. New nucleation methods may have to be used to capture nucleation in CA models.
Meshing plays an important role in numerical models and inappropriate meshing can lead to inaccurate results or alter results causing mesh anisotropy. PFM models have a very thin interface which requires very small cell sizes at the interface, however, the small cell size also reduces the time step of the model creating large amounts of data and requires more computational time for the same amount of time length . Some models use adaptive meshing to reduce the number of total cells (Francois et al. 2017). Kinetic MC and CA also face a similar problem of having smaller cell sizes requiring small time steps leading to large amounts of data and computational resource. CA also can face mesh anisotropy and to counter this some models use hexagonal cells, however, majority of the models use the decentred square method developed by Gandin and Rappaz which allows square cells while eliminating mesh anisotropy (Gandin and Rappaz 1997;Dezfoli et al. 2017;Panwisawas et al. 2017).

Potentials
Most PFM models reviewed in this paper reduced the alloy to a binary alloy thus only finding the solute concentration or precipitate of a single solute. Microsegregation occurs in as-built AM parts and causes non-homogenous microstructure (Ghosh 2018). Simulating multicomponent alloy could help in studying how segregation occurs and possible workarounds, however, it would increase the computational time as the equations would have to solve the summations of each component (Nestler, Garcke, and Stinner 2005). MC models do not face this issue as it does not account for solutes. CA models use solidification theory which can account for multiple elements, however, most model also reduces it to binary or tertiary alloys (Zineviva, Columnar to equiaxed transition during alloy solidification).
While this paper reviews the microstructure modelling done for AM processes, the fundamental concepts can be also applied to post-processing heat treatments of AM parts as the thermal cycles also affect the microstructure despite the different heating and cooling rates. Numerous works have been done on post-process heat treatment of AM parts and how they affect the microstructure (Kajima et al. 2018;Kim et al. 2018;Kong et al. 2018;Kurzynowski et al. 2018;Salman et al. 2019). Using Inconel 718, Deng et al. applied heat treatment to the material produced by SLM, aiming to homogenise the microstructure and optimise the mechanical properties. While they observed that the heat treatments can precipitate the strengthening γ ′ /γ ′′ phase, and dissolve the Laves phase and precipitate the δ phase depending on the heat treatment temperature, the mechanical properties are also improved due to the removal of residual stresses ). Li et al. have similar observations, however, the anticorrosive ability of the material decreases after heat treatment .
In this review, the focus is on the microstructural evolutions during the AM processes. The results from these models have the potential to be fed and linked to mechanical properties prediction of the AM parts. Recently, Ge et al. used an integrated modelling of process-structureproperty relationship in laser AM for duplex titanium alloy. It is found that the formation of larger grain sizes and more α phase are due to higher scanning speed. This higher composition of α phase is then linked to higher mechanical properties (Ge et al. 2019). At current stage, most of the work done has linked the thermal simulations to the residual stresses in AM parts (Ding et al. 2011;Heigel, Michaleris, and Reutzel 2015;Yang et al. 2016;Zhao et al. 2017). The main challenge to link the different models together to achieve the microstructure to mechanical properties relation is the exchange of information between the individual model. Yan et al. have proposed to use a fixed format, flat-file database as a solution ). However, more efficient approaches for information exchange have to be developed (Smith et al. 2016).

Conclusion
Numerical modelling can help in predicting microstructure without the need for experiments. The grain structure, microsegregation, precipitate and dendrite size can be predicted using varying microstructure models. In order to get it, two models are usually coupled together, thermal model and microstructure model. Two types of thermal models and three types of microstructure modelling for AM are reviewed. The macro thermal models predict the temperature, cooling rate and temperature gradient which are used as input for microstructure models.
Of the two types of thermal models, FEM is most often used model for multiple tracks and layers, while CFD is used for simulating limited number of tracks. FEM models ignore the flow of the fluid reducing the amount of computation required and thus are able to simulate a larger volume. CFD models often focus on the physics of the melt pool, requiring more computation resources. FEM models being able to simulate larger volumes can be used in microstructure modelling to simulate the grain structure where grains often grow over multiple layers. CFD models have more accurate cooling rate and temperature gradient which can be used to get more accurate microstructure.
There are two different scales that microstructure modelling can take place: micro-scale and meso-scale. Micro-scale models simulate the dendrites and the solute composition while meso-scale simulates the grain structures. PFM and MCA are able to simulate the micro-scale getting the segregation of alloying elements and finding the PDAS which relates to the local strength and hardness. It, however, requires large computational resource limiting its volume to simulate a few dendrites. MC and CA are able to simulate a larger volume getting the grain structure. The grain size and aspect ratio which relates to the strength and ductility of the part is found in both models. MC models, however, have poorer predictions in grain size as compared to CA models. PFM and MCA can be useful in finding how new materials would form in the AM process and can facilitate the addition of new materials for AM. The overall mechanical properties would also depend on the size and aspect ratio of grain structure which would require MC or CA to model. Depending on the type of mechanical properties to be predicted, simulation has to be done on different length scales.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This research is supported by the National Research Foundation, Prime Minister's Office, Singapore under its Medium-Sized Centre funding scheme.

Notes on contributors
Joel Heang Kuan Tan is a research student at the Singapore Centre for 3D Printing. He received his Bend(Hons) in Mechanical Engineering from Nanyang Technological University, Singapore. His research interest is in additive manufacturing of metals, using simulation to relate and predict process parameters to resulting microstructure and mechanical properties.
Dr Swee Leong Sing is a Research Fellow at the Singapore Centre for 3D Printing. He received his B.Eng. (Hons) in Aerospace Engineering and Ph.D. in Mechanical Engineering from the Nanyang Technological University, Singapore. His research interest is in additive manufacturing for biomedical applications, focusing on materials creation and development, qualification and certification of additive manufactured medical devices. His doctoral thesis focuses on laser-based powder bed fusion processing of a novel titanium alloy for medical applications. Based on Web of Science (Clarivate Analytics), he has co-authored 28 peer-reviewed articles in the field of additive manufacturing or 3D printing. He currently has a Hindex of 15 and is also the co-inventors for two patents on the powder bed fusion process.
Associate Professor Wai Yee Yeong received her B.Eng. (Hons) and Ph.D. in Mechanical and Aerospace Engineering from the Nanyang Technological University (NTU), Singapore. Prior to joining NTU as a faculty member in 2013, she has worked in the manufacturing industry at supervisory functions in research and development, with key focus in medical device industry. She is now serving as Associate Chair (Students) in the school. Her portfolio also includes serving as Programme Director (Aerospace and Defence) at Singapore Centre for 3D Printing (SC3DP), Deputy Director (Technical) for HP-NTU Digital Manufacturing Corporate Lab, and Co-Director of NTU Institute for Health Technologies. Her main research interest is in 3D printing, bioprinting and the translational of the advanced technologies for industrial applications. Her current research topics include 3D printing of new materials, hybrid electronic-mechanical structures and bioprinting for tissue engineering. She has coauthored 2 textbooks (published by World Scientific and Elsevier, respectively), published more than 130 peer-reviewed technical papers, with a current H-index of 29. To-date, she has filed 2 PCT patents in new alloy and new bioink formulations, and 2 Singapore provisional patents on 3D printing.