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Alloy casting solidification processes involve many physical phenomena such as chemistry variation, phase transformation, heat transfer, fluid flow, microstructure evolution, and mechanical stress.¹ Simulation technologies are applied extensively in casting industries to understand the effects of these phenomena on the formation of defects and on the final mechanical properties of the castings. As of today, defect prediction is still one of the main purposes for casting solidification simulation. In this paper, we will first present the commonly used microstructure simulation methods, then discuss the predictions of the major defects of a casting, such as porosity, hot tearing, and macrosegregation. The modeling of casting solidification can be chained with later stages of heat treatment such that the resultant microstructure, defects, and mechanical state will be used as the initial conditions of the subsequent processes, ensuring the tracking of the component history and maintaining a high level of accuracy across metallurgical stages.

The purpose of solidification micromodeling is to predict the microstructure developed during the casting processes. Understanding the solidification process and the resultant microstructure will greatly facilitate casting design and quality control and eventually to predict mechanical properties.^2-6 There are several ways of simulating the microstructure formation during alloy solidification, such as the deterministic method and the stochastic method. For the deterministic method, the density of grains that have nucleated in the bulk liquid at a given moment during solidification is described as a deterministic function (e.g. a function of undercooling). The stochastic method uses probabilistic means to predict the nucleation and growth of grains, including the stochastic distribution of nucleation locations and the stochastic selection of grain orientation, and so on.

Deterministic Micromodeling
Modeling of solidification processes and microstructural features has benefited from the introduction of averaged conservation equations and the coupling of these equations with microscopic models of solidification. When conservation equations are averaged over the liquid and solid phases, the interfacial continuity condition automatically vanishes and average entities (e.g. mean temperature or solute concentration) appear. Rappaz et al.^7–9 proposed a model using averaging methods to predict the growth of equiaxed grains under isothermal conditions. Rappaz and Boettinger² studied the growth of an equiaxed multicomponent dendrite. In their study, for each element, the supersaturation is described as Equation 1, where j=1, n is the solute element, c_l,j* is the tip liquid concentration, C_o,j is the nominal concentration, and k_j is the partition coefficient. The Peclet number is defined as Pe_j = Rv/2D_j, where D_j is the diffusion coefficient, R is the tip radius, and v is the tip velocity.

The Ivanstsov function is defined as: Iv(Pe) = Pe × exp(Pe) × E₁(Pe), where E₁(Pe) is the first exponential integral.

Assuming growth at the marginal stability limit, the dendrite radius is calculated by Equation 2, where Γ is the Gibbs–Thomson coefficient, and m is the liquidus slope.

Please refer to the original Reference 2 for details about the derivation of the equations above. The secondary dendrite arm spacing (SDAS) is calculated by Equation 3 and Equation 4. Based on the equations, the solidification of multicomponent casting can be predicted. The averaged microstructure, such as grain size and secondary dendrite arm spacing, can be calculated based on the chemistry and cooling conditions.

An experiment of an A357 sand casting with chills on some locations is performed to validate the micromodeling above. Figure 1 illustrates the experimental setup. Six cast iron chills are used, and they are coated with a light layer of sand to prevent blows from gasses and condensation in this model. With such a setup, cooling conditions dramatically varied for different locations throughout the casting.

Based on the micro-model presented, the solidification process is simulated. As mentioned earlier, because of the use of chills, the solidification times are very different across the casting. For the chilled section, the solidification time is as fast as 29 seconds. For a location which is not chilled but with similar wall thickness, the solidification time is 242 seconds, which is one order of magnitude longer. Because of the different solidification rates, the microstructure varies in the casting. For this prediction, it is assumed that only equiaxed dendrites are formed in the primary phase, which is the case based on the experimental observation. Figure 2 shows the primary dendritic grain radius. For a faster cooling rate, the dendrite grain size is smaller.

The secondary arm spaces were measured in the experiment. For comparison, Figure 3 shows the measured and predicted secondary arm spacing at two locations, one on the chilled section and the other one on an unchilled section. The exact values can be found in Table I. For the chilled section, the measured SDAS is 0.00132? and corresponding prediction is 0.0014?, a difference of only 6%. For the unchilled section, the SDAS is larger but the difference between the measured and predicted ones is even smaller, being 4.3%. Based on Equation 3, the secondary dendrite arm spacing is proportional to the one third power of the solidification time. This can be verified by the values of the secondary arm spacing and solidification time at those two locations numerically as well. The solidification time is around 9 times different, which corresponds to roughly three times difference of secondary arm spacing

There is a large amount of eutectic phase formed during solidification for this A357 alloy. The current micromodeling can predict the microstructure of the eutectic phase as well. Because of the limitation of the paper length, these results are not presented here.

Based on the microstructure, the ascast room temperature yield strength can be calculated, as shown in Figure 4. With the finer microstructure in the chilled section because of the faster cooling, the yield strength is higher. But the difference is relatively small compared to that of SDAS and solidification time. The yield strength difference between the chilled section, 83.8 MPa, and the unchilled section, 74.8 MPa, is less than 10 MPa, around 10%. Even though most in service castings are heat treated, the prediction of as-cast microstructure and mechanical properties are still important. First, the as-cast microstructure and properties affect the final structure and properties. And more importantly, as-cast properties are critical for stress analysis during the casting process.

Stochastic Method
Beside the deterministic micromodeling, other models have been studied extensively as well, such as the stochastic method. One example is the cellular automaton (CA) model. The algorithm of CA models is to describe the discrete spatial and/or temporal evolution of complex systems by applying deterministic or probabilistic transformation rules to the sites of a lattice. In a CA model, the simulated domain is divided into a grid of cells, and each cell works as a small independent automaton. Variables and state indices are attributed to each cell, and a neighborhood configuration is also associated with it. The time is divided into finite steps. At a given time step, each cell automaton checks the variables and state indices of itself and its neighbors at the previous time step, and then updates the results at the present step according to the pre-defined transition rules. By iterating this operation with each time step, the evolution of the variables and state indices of the whole system is obtained.

Several CA models have been developed over the years for the prediction of microstructure formation in casting.^10–14 The CA model is usually coupled with a finite element heat flow solver, such as the Cellular Automaton—Finite Element (CAFÉ) model developed by Rappaz and colleagues. CAFÉ is particularly well suited to tracking the development of a columnar dendritic front in an undercooled liquid at the scale of the casting.^12,13 The CA algorithm can be used to simulate nucleation and growth of grains. This model can be used to predict cellular to equiaxed transition (CET) in alloys. Although these models do not directly describe the complicated nature of the solid/liquid interface that defines the dendritic microstructure, the crystallographic orientation of the grains as well as the effect of the fluid flow can be accounted for to calculate the undercooling of the mushy zone growth front. Two- and three-dimensional CAFÉ models were successfully applied to predict features such as the columnar-to-equiaxed transition observed in aluminum-silicon alloys,¹² the selection of a single grain and its crystallographic orientation due to the competition among columnar grains taking place while directionally solidifying a superalloy as is shown in Figure 5,13 as well as the effect of the fluid flow on the fiber texture selected during columnar growth.¹⁴ Coupled with the macrosegregation that has been developed,¹⁵ it thus provides an advanced CAFÉ model to account for structure formation compared to purely macroscopic models developed previously, e.g. References 16–18.

While the CA methods produce realistic-looking dendritic growth patterns and have resulted in much insight into the CET, some questions remain regarding their accuracy. For example, the independence of the results on the numerical grid size is rarely demonstrated. Furthermore, the CA techniques often rely on relatively arbitrary rules for incorporating the effects of crystallographic orientation while propagating the solid–liquid interface. It is now well accepted that dendritic growth of crystalline materials depends very sensitively on the surface energy anisotropy.^19,20

An alternative technique for investigating microstructure formation during solidification is the phase field method. Phase field models were first developed for simulating equiaxed growth under isothermal conditions.^21,22 A desirable extension of the model was to study the effect of heat flow due to the release of latent heat. A simplified approach was proposed in which the temperature was assumed to remain spatially uniform at each instant, and a global cooling rate was imposed with consideration of the heat extraction rate and increase of the fraction of solid.²³ The attempt to model non-isothermal dendritic solidification of a binary alloy was made by Loginova et al. by solving both the solute and heat diffusion equations and considering the release of latent heat as well.²⁴ Besides equiaxed growth in the supersaturated liquid, the phase field model was also applied to the simulation of directional solidification, under well-defined thermal conditions.^25,26 The phase field model has also been used to simulate the competitive growth between grains with different misorientations with respect to the thermal gradient.²⁷ Further development in phase field models includes the extension into three dimensions,^28,29 and multi-component systems.^30,31

Usually a regular grid composed of square elements is used in the phase field models,^22,23 but an unstructured mesh composed of triangular elements has also been used, which enables the phase field method to be applicable in a domain with complex geometrical shape and also in a large scale. From a physical point of view, the phase field method requires knowledge of the physical nature of the liquid–solid interface. However, little is known about its true structure. Using Lennard–Jones potentials, molecular dynamics simulations of the transition in atomic positions across an interface have suggested that the interface width extends over several atomic dimensions.³² At present, it is difficult to obtain usable simulations of dendritic growth with interface thickness in this range due to the limitations of computational resources. Thus the interface width will be a parameter that affects the results of the phase field method. It should be realized that in the limit as the interface thickness approaches zero the phase field equations converge to the sharp interface formulation.^33,34 In contrast to CA models which adopt a pseudo front-tracking technique, phase field models express the solid–liquid interface as a transitional layer which usually spreads over several cells.

The diffusion equation for heat and solute can be solved without tracking the phase interface using a phase-field variable and a corresponding governing equation to describe the state in a material as a function of position and time. This method has been used extensively to predict dendritic, eutectic, and peritetic growth in alloys, and solute trapping during rapid solidification.³⁵ Figure 6 shows one such example. The interface between liquid and solid can be described by a smooth but highly localized change of a variable between fixed values such as 0 and 1 to represent solid and liquid phases. The problem of applying boundary conditions at an interface whose location is an unknown can be avoided. Phase field models have recently become very popular for the simulation of microstructure evolution during solidification processes.^36–40 While these models address the evolution of a solid–liquid interface using only one phase field parameter, interaction of more than two phases or grains, and consequently the occurrence of triple junctions, needs to be included into the multiphase field approach.^41–44

The deterministic model is capable of tracking the evolution of the macroscale or average variables, e.g. average temperature and the total fraction of solid, but it cannot simulate the structure of grains. CA models can simulate the macro-scale and meso-scale grain structures, but they have difficulty resolving the microstructure. The phase field method can well reproduce the microstructure of dendritic grains. However, with the current computational power, phase field models can only work well on a very small scale (up to hundreds of μm). The typical scale of laboratory experiments is 1 cm, and the scale of an industry problem can be up to 1 m. Both of them are beyond the capability of the phase field method. In industry, for larger castings, deterministic micromodeling still is a main player.

Defects reduce the performance and increase the cost of castings. There are many kinds of casting defects. Those defects are dependent on the chemistry of alloys, casting design, and casting processes. Defects can be related to thermodynamics, fluid flow, thermal, and/or stress. For most cases, all of those phenomena are intertwined. It is necessary to consider every aspect in order to minimize defect formation.

In this section, we will discuss some common casting defects in foundries, which include porosity, macrosegregation, and hot tearing.

Porosity
Porosity formed in castings leads to a decrease in the mechanical properties.^45–51 This porosity may be a combined result of solidification shrinkage and gas evolution. They can occur simultaneously when conditions are such that both may exist in a solidifying casting. One of the most effective ways to minimize porosity defects is to design a feeding system using porosity prediction modeling. In such a way, the model can determine the location of porosity so that the feeding system can be redesigned. This process is repeated until porosity is minimized and not likely to appear in the critical areas of the castings.

There are many models which can predict the shrinkage porosity from the pressure drop during inter-dendritic fluid flow.^49–57 A comprehensive model should calculate the shrinkage porosity, gas porosity, and pore size.

As for many casting defects observed in solidification processes, the mushy zone is the source of porosity. The basic mechanism of porosity formation is pressure drop due to shrinkage and gas segregation in the liquid.^46,47,49 The liquid densities of many alloys are lower than that of the solid phase. Hence solidification shrinkage happens due to the metal contraction during the phase change. The dynamic pressure within the liquid decreases because of the contraction and sometimes cannot be compensated by the metallostatic pressure associated with the height of the liquid metal. The decrease of pressure lowers the solubility of gas dissolved in the liquid. If the liquid becomes supersaturated, then bubbles can precipitate.⁵⁸ Most liquid metals can dissolve some amount of gas. The solubility of gases in the solid phase is usually much smaller than that in the liquid phase. Normally the rejected gases during solidification do not have enough time to escape from the mushy zone to the ambient air. Being trapped within the interdendritic liquid, the gas can supersaturate the liquid and eventually precipitate in the form of pores if nucleation conditions are met. The formation of bubbles requires overcoming the surface tension.⁵⁹ Homogeneous nucleation is very difficult. In castings, nucleation of pores can be expected to occur primarily on heterogeneous nucleation sites, such as the solid–liquid interface and inclusions.^59,60

Generally speaking, there are two ways to predict the level of porosity in castings. One is a parametric method derived from first principles by using a feeding resistance criteria function combined with macroscopic heat flow calculations.^61–64 Parametric models are easy to apply to shaped castings and have been mainly directed at the prediction of centerline shrinkage. Another approach is a direct simulation method.^53–57,59 They usually derive governing equations based on a set of simplifying assumptions and solve the resulting equations numerically. By combining the cellular automata technique, some models can not only predict the percentage porosity but also the size, shape, and distribution of the pores.^54–56

In order to predict porosity defects in casting processes accurately, the following factors should be considered: macroscopic heat transfer, interdendritic fluid flow, gas redistribution by diffusion and convection, microstructure evolution, and porosity growth.

Pores will form in a solidifying alloy when the equilibrium partial pressure of gas within the liquid exceeds the local pressure in the mushy zone by an amount necessary to overcome surface tension. Hence gas porosity develops as shown in Equation 5, in which P_g is Sievert pressure, P_a is ambient pressure, P_m is metallostatic pressure, P_d is pressure drop due to the friction within the interdendritic liquid, P₈ = 2σ/r is surface tension, σ is surface tension, r is pore radius. Volume fraction of gas porosity can be calculated based on Equation 6, where f_v is volume fraction of gas porosity, α is gas conversion factor.

A set of castings with different initial hydrogen content using an iron chill plate was simulated and compared with experiment for an A319 casting.⁴⁵ The initial pouring temperature was 750°C. Initial hydrogen contents were 0.108 ppm, 0.152 ppm, and 0.184 ppm. The experiments and simulation results were taken at different distances from the chill end. The comparison of the value of percentage porosity against local solidification time and hydrogen content between simulation and experiment is shown in Figure 7.

It shows that increasing solidification time and hydrogen content increase considerably the percentage of porosity. Numerical simulation results give excellent agreement with the measurements of percentage of porosity.

Macrosegregation
Macrosegregation is another defect for a lot casting processes. Modeling and simulation of macrosegregation during solidification has experienced explosive growth since the pioneering studies of Flemings and co-workers in the mid-1960s.^65–67

Beckerman did a comprehensive review of recent macrosegregation models and their application to relevant casting industries.⁶⁸ There are numerous factors that can cause macrosegregation during casting solidification processes. Those include thermal and solute induced buoyancy, forced flow, solid movement, and so on. Macrosegregation models have been applied extensively in the casting industry, such as steel ingot castings, continuous and DC castings, nickel-based superalloy single-crystal castings (freckle simulation), and shape castings as well.

Freckles have been the subject of intense research efforts for about 30 years due to their importance as a defect in alloy casting.^69,70 They represent a major problem in directionally solidified superalloys used in the manufacture of turbine blades.^69–73 Upward directional solidification provides an effective means of producing a columnar microstructure with all the grain boundaries parallel to the longitudinal direction of the casting. In conjunction with a grain selector or a preoriented seed at the bottom of the casting, directional solidification is used to make entire castings that are dendritic single crystals. During such solidification the melt inside the mushy zone can become gravitationally unstable due to preferential rejection of light alloy elements into the melt. Since the mass diffusivity of the liquid is much lower than its heat diffusivity, the segregated melt retains its composition as it flows upward and causes delayed growth and localized remelting of the solid network in the mush. At the end, a pencil-shaped vertical channel forms in the mushy zone. Highly segregated liquid flows upward as a plume or solutal finger into the superheated melt region above the mushy zone. This flow is continually fed by segregated melt flowing inside the mushy zone radially toward the channel. At the lateral boundaries of the channel, dendrite arms can become detached from the main trunk, and those fragments that remain in the channel are later observed as freckle chains.

The complex convection phenomena occurring during freckle formation represent a formidable challenge for casting simulation.^68,74–76 In 1991, Felicelli et al. simulated channel formation in directional solidification of Pb-Sn alloys in two dimensions.⁷⁷ Since then, numerous studies have been performed to simulate and predict freckling in upward directional solidification.^78–87 Neilson and Incropera performed the first three-dimensional simulations of channel formation in 1993.⁸¹ Three-dimensional simulations have also been performed by Poirier, Felicelli, and coworkers for both binary and multicomponent alloys.^87,88

Figure 8 illustrates the freckle formation for a Pb-10%Sn binary alloy directional solidified in a simple geometry. A temperature gradient is imposed initially in order to simulate a directional solidification system. Cooling is achieved by lowering the temperatures of the upper and lower walls of the cavity at a constant rate such that the overall temperature gradient is maintained over the height of the cavity. The lateral walls of the cavity are taken as adiabatic. Figure 8a shows the final composition of Sn. In order to see the freckles clearly, a cut-off view of the same results is shown in Figure 8b. There are five freckles for this case. Four of them are in each corner of the domain, and one is in the middle.

Hot Tearing
Hot tearing is one of the most serious defects encountered in castings. A lot of studies have revealed that this phenomenon occurs in the late stage of solidification when the fraction of solid is close to one. The formation and propagation of the hot tearing have been found to be directly affected by the cooling history, the chemical composition, and mechanical properties of the alloy, as well as the geometry of the casting. Various theories have been proposed in the literature on the mechanisms of hot tearing formation. Detailed reviews on the theories and experimental observations of the formation and evolution of the hot tearing can be found in References 89 and 90 and the references therein.

Most of the existing hot tearing theories are based on the development of strain, strain rate, or stress in the semisolid state of the casting. For strainbased theory, the premise is that hot tearing will occur when the accumulated strain exceeds the ductility.^91–93 The strain rate-based theories suggest that hot tearing may form when the strain rate, or strain rate related pressure, reaches a critical limit during solidification.^94,95 The stress-based criteria, on the other hand, assume that hot tearing will start if the induced stress in the semi-solid exceeds some critical value.^96,97 Although these theories were proposed independently as distinct theories, they can, indeed, be considered as somewhat related due to the relationship between strain, strain rate, and stress. It is such a relationship that motivates the development of a hot tearing indicator which uses the accumulated plastic strain as an indication of the susceptibility of hot tearing.⁸⁹ This considers the evolution of strain, strain rate, and stress in the last stage of solidification. A Gurson type of constitutive model, which describes the progressive micro-rupture in the ductile and porous solid, is adopted to characterize the material behavior in the semi-solid state. The proposed hot tearing indicator has been validated by different alloy castings.

The constitutive model used to describe the material behavior in the semi-solid state is the Gurson model,^98–100 which was originally developed for studying the progressive micro-rupture through nucleation and growth of micro-voids in the material of ductile and porous solid.

When the material is considered as elastic-plastic, the yield condition in the Gurson model is of the form shown in Equation 7, Where F(σ) = (3(s–x) : (s–x)/2)^1/2 is the Mises stress in terms of the deviatoric stress s = σ – (trσ)I/3. κ represents the plastic flow stress due to isotropic hardening, χ denotes back stress due to kinematic hardening. The Gurson coefficient G_u is defined as Equation 8, in which q₁ is a material constant. Here, f_u = 1/q₁,f_c is the critical void volume fraction and f_F is the failure void volume fraction. Their values should be different for different materials.

The evolution of the void volume fraction is described by the nucleation of the new void and the growth of the existing void, i.e., Equation 9.

In this study, the nucleation of the void is assumed to be strain controlled and is written as Equation 10, where Equation 11 is defined as our hot tearing indicator (HTI). t_C represents time at coherency temperature and t_S denotes time at solidus temperature. It is observed that the hot tearing indicator is, in fact, the accumulated plastic strain in the semi-solid region and it corresponds to the void nucleation. Therefore, it should provide a good indication for the susceptibility of the hot tearing during solidification. The value of the hot tearing indicator is determined by the finite element analysis.¹⁰¹ For materials described by viscoplastic or creep models, a yield condition does not exist.

Cao et al. performed some experiments to study the hot tearing formation during solidification of binary Mg-Al and ternary Mg-Al-Ca alloys in a steel mold,^102,103 which is shown in Figure 9. A hot cracking susceptibility (HCS) was introduced which is a function of maximum crack width, crack length factor, and the crack location. It was found that it is easier to have cracks at the sprue end than at the ball end. It is less likely to have a crack in the middle of the rod. Also, the longer rod is easier to crack. Figure 10 shows the simulated results of hot tearing indicator for a Mg-2%Al alloy casting. The computed hot tearing indicator agrees very well with the experiments.

From experiment as well as from simulation, it can be seen that the hot tearing is less severe as the Al content increases from 2% to 4% and then to 8% at the same location for the same casting with the same conditions. The susceptibility rises sharply from pure Mg, reaches its maximum at Mg-1% Al, and decreases gradually with further increase in the Al content.

The hot tearing indicator is calculated at the end of the sprue for the longest rod with a different alloy composition. For comparison, the hot tearing indicators as well as a crack susceptibility coefficient (CSC), which is defined as the temperature difference between fraction of solid at 0.9 and at the end of solidification, are shown in Figure 11. As in the experiment, the susceptibility of hot tearing rises sharply from pure Mg, reaches its maximum at Mg-1% Al, and decreases gradually with further increase in the Al content. Similarly, different ternary Mg alloys have different hot tearing susceptibilities. The alloy chemistry, casting geometry, and cooling conditions all contribute to the formation of hot tearing and they are included in the model directly or indirectly.

Experimentally, Li systematically studied the effects of various casting conditions, such as mold temperature, pouring temperature, and grain refinement, on hot tearing of different cast aluminum alloys.¹⁰⁴ In Li's study, an instrumented constrained rod mold was developed such that it can be used to simultaneously measure the load/ time/temperature during solidification. The set-up is shown schematically in Figure 12. Figure 13 gives the dimensions of the studied casting. A detailed description of the experimental setup for quantitatively measuring hot tearing onset and contraction during solidification of aluminum alloys can be found in Reference 104.

In one of the experiments, three mold temperatures were used, 200°C, 300°C, and 370°C. The cracks in the hot spot region of alloy M206 for those three mold temperatures are shown in Figure 14.

The results clearly suggest that the mold temperature has a significant effect on hot tearing susceptibility of this 206 alloys. The hot tearing susceptibility decreases with increasing mold temperature.

Based on the experimental casting condition, the simulations were performed. Figure 15 shows the hot tearing indicator distribution for those three mold temperatures. Table II lists the actual values of the indicator. Increasing mold temperature can reduce hot tearing formation for this situation. It can be concluded that modeling can predict the hot tearing susceptibility for this aluminum alloy casting very well. As there are additional experimental results in Li's work,²³ further simulations are under investigation to validate the current model, such as the effect of melting temperatures and grain refinement.

Modeling of casting and solidification has been used extensively in foundries to solve routine production problems. Casting defects such as those related to filling, solidification, stress, and microstructure can be predicted. As always, better understanding and accurate material properties, including mold materials, lead to improved predictive capabilities. Further development efforts should emphasize the enhancement of the accuracy of predicting various casting defects. Further coupling of heat treatment simulation with casting simulation can then predict the final mechanical properties of the part in service.

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Jianzheng Guo and Mark Samonds are with ESI US R&D, 6851 Oak Hall Ln, Suite 119, Columbia, MD 21045. Dr. Guo can be reached at jianzhengguo@gmail.com.