V. R. Voller University of Minnesota

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1 Micro-macro Modeling of Solidification Processes and Phenomena, in Computational Modeling of Materials, Minerals and Metals Processing (eds M. Cross, J.W. Evans, and C. Bailey), TMS Warrendale, V. R. Voller University of Minnesota

2 MICRO-MACRO MODELING OF SOLIDIFICATION PROCESSES AND PHENOMENA Vaughan R. Voller University of Minnesota 500 Pillisbury Drive Minneapolis, MN Abstract The wide range of length and time scales found in solidification processes are outlined and discussed. Methods for Direct Microstructure Simulation (DMS) are introduced. Key features in sharp interface and phase field models are presented. Concepts in micro-macro solidification modeling are covered. Basic details of microstructure and segregation models are provided. A description of a recent segregation micro-macro model is presented in detail.

3 Introduction A solidification process involves the containment of a liquid region, which, due to cooling, transforms over time to form a fully solid product. The development of the solid microstructure and the segregation of solute components during this transformation are controlled by phenomena that occur across a wide range of length and time scales. A successful model of a given solidification process needs to account for these phenomena. Standard models of solidification employ a discrete grid of node points and associated volume elements in the special domain, with a discrete time step to track the transient changes of nodal values, describing the composition and microstructure. Through the use of such models significant progress has been made towards a complete understanding of solidification process and phenomena (see the conference proceedings in Refs [1-9]). As solidification models become more sophisticated, however, they all run into the problem of resolving the full range of disparate space and time scales. Two approaches can deal with this problem 1. Direct Microstructure Simulation (DMS) where the grid size and time step are chosen to scale with the smallest length and time scales of the given problem. 2. Micro-macro modeling where the grid size and time step scale with the macroscopic process scale and phenomena at smaller scales are accounted for via sub-grid modeling and volume averaging. Currently full DMS of solidification is beyond the reach of available computer power and the application and development of micro-macro models is an important and critical research area. The current paper starts with a discussion of the various length and time scales associated with solidification phenomena. This is followed by a discussion of some of the methods used in DMS of solidification, in particular sharp interface and phase field models. A part of this discussion is directed at determining the lead-time required to achieve a DMS of a complete casting. This discussion justifies the need to develop micro-macro models and a large section of the paper is focused on the basic framework for a micro-macro model; examples of microstructure and segregation modeling are used to illustrate this discussion. The paper concluded with a brief outline of the details in a recent micro-macro model of segregation. Solidification Scales Before we examine some of the computational methods open to us for resolving the various scales that occur in a solidification process it is worthwhile to identify what the scales are. Three alternative and complementary views of solidification scales are presented. Structure Length Scales The various length scales associated with a dendritic solidification structure are schematically illustrated in Figure 1. Taking each one in turn for the largest to the smallest. Macro-Messo Scales: The macroscale is the process scale which is typically ~0.1m. A computational model of the heat and mass transfer at this process scale will usually invoke a volume averaging over the solid and liquid regions [10]. The grid size in this calculation will be a Representative Elemental Volume (REV) over which the average is taken (~10 s mm).

4 ~ 0.1 m Macro- chill A Casting Messo- The REV ~10 mm Nucleation Sites columnar equi-axed The Grain Envelope ~ mm ~100 m m The Secondary Arm Space ~10 mm Messo- Micro- Micro- The Tip Radius ~1 nm Sub-Microφ = 1 φ = 1 The Diffusive Interface Figure 1: Structure length scales

5 10 3 casting heat and mass tran. grain formation Time Scale (s) growth solute diffusion 10-7 nucleation 10-9 interface kinetics Length Scale (m) Figure 2: Length and time scales of solidification phenomena (after Dantzig [11]) Messo-Micro Scales: In the micro-macro modeling of solidification process and phenomena the concept of a grain envelope a surface encompassing the microstructure that will, on complete solidification, form a grain is useful in attempting to characterize the sub-grid, local scale of the REV. The grain envelope can also be considered to be at the messoscale, but its length scale (~mm) is smaller than the REV, which would be expected to contain several grains. The secondary arm spacing is also used to characterize the local behavior in the REV. The arm spacing is a microscopic scale ~100 µm. At this scale, solute diffusion in to the solid (backdiffusion) and, in rapid solidification, solute pushing ahead of the solid-liquid front, are important phenomena. Micro-Sub-micro Scales: Direct Miciroscale Simulation (DMS) [8] is focused on understanding the microscopic evolution of the solid-liquid front. If a so called sharp interface model is used (see Eqs. (1)-(3)) then the numerical codes need to be resolved below the scale of the dendrite tip radius (~ 10µm). Below the scale of the tip radius, although artificial, the concept of a finite thickness diffusive interface (~ nm) is a useful devise for developing phase field models of the growth of solidification structures.

6 Time and Length Scales of Solidification Phenomena The schematic in Fig. 1 only shows the length scales associated with solidification microstructure. In modeling a solidification process, however, it is also important to understand the time scales associated with solidification phenomena. Dantzig [11] has presented a chart that identifies both the time and length scales of key solidification phenomena. This chart, reproduced in Figure 2 provides a compliment to the schematic of length scales given in Fig 1. Volume Averaging The concept of a volume average [10] is useful in developing macro scale process models that can capture micro scale phenomena. Volume averaging filters the local variations in a parameter at one scale and obtains an average and representative value for use at a larger scale. Figure 3 schematically illustrates how this averaging process can work in representing the solid volume fraction. Consider a sample box moving from through the mushy region from the full liquid to the full solid. As the box moves changes in its solid volume fraction are recorded. The box has a fixed height equal to half the primary arm spacing, but a variable width. When the box is thin (approaching the tip radius ~10 µm) micro variations in the solid fraction will be recorded, curve 1 in Fig. 3. If the width of the box scales to the secondary arm space (~100 µm) then the same trend in the solid fraction variation will be followed but higher frequencies will be filtered out, curve 2 in Fig. 3.This filtering process continues when a relatively wide box, approaching the size of a grain envelope (~ 1 mm) is used. In this case solid fraction fluctuations are eliminated and a smooth change through the mushy region is recorded, curve 3 in Fig. 3. Sample boxes moved through mush ~1mm liquid solid ~100m m ~10mm mushy-region 1 Solid fraction Curve 3 Curve 1 Curve Distance across mush Figure 3: Variation of solid fraction with measurement scale.

7 Direct Microstructure Simulation An ultimate objective of solidification models is to provide, using a term first coined by Thomas and Beckermann [8], Direct Microstructure Simulation (DMS) of the microstructure formed in a casting, i.e., develop and implement numerical simulation that can span the length and time scales illustrated in Figs Although we are a long way from having the computer power to achieve this recent progress has been made in DMS at the messo-length scale of the grain envelope. Below two promising areas in DMS are outlined in detail and a third level set methods is briefly discussed. Direct Numerical Simulation (DNS) of the Sharp Interface Model Direct numerical simulation attempts to fully resolve the solid-liquid interface in a domain no smaller than the messo scale of the grain envelope. There is a close analogy with Direct Numerical Simulation (DNS) of turbulence [12] where the small eddies in the flow are resolved in a domain which is no smaller than the size of the largest eddies. As an example of DNS consider the classic example of the dendritic solidification of a pure material in an undercooled melt, often referred to as the sharp interface model. Neglecting flow the heat transfer is described by the diffusion equation u = D 2 u (1) t where D is the thermal diffusivity. This microscopic equation is applied separately at each point in the solid and liquid regions. When u is the dimensionless temperature c(t Tm ) u = (2) L c, specific heat, T m melting temperature, and L latent heat, the boundary conditions on the sharp front separating the solid and liquid regions can be written as [13-15] v u n i u u = D D (3) n n 0 s n l = d κ βv (4) where n is the unit normal on the interface pointing from the solid (s) to the liquid (l) and v n is the speed of the front along this normal. The first of these conditions is the Stefan condition [16]. The second is the Gibbs-Thomson condition [17]; d 0 is the capillary length ( σ T m c/ L ), σ the surface tension, κ, neglecting anisotropy, is the mean curvature and, β is a kinetic coefficient. A number of DNS s of the sharp interface model have been attempted. In relatively early work Sullivan and Lynch [18] use a deforming finite element front tracking method; a problem with this approach is the requirement of re-meshing to avoid large grid distortions. Problems with deforming elements are avoided in a number of fixed grid solutions. Tacke [19] uses a fixed grid enthalpy algorithm and calculates a local curvature in terms of the predicted nodal solid 2

8 fraction. Juric and Tryggavson [20] use an embedded interface method with particles to track the solid-liquid front over a fixed grid on which the thermal field is resolved; a similar method is employed by Tonhardt [21]. Phase Field Models The phase field model [13-15, 21-32] is a general tool for the study of microstructure development in growth processes that are defined by a phase separation, e.g., solidification. The key concept is the introduction of a diffuse interface, of thickness ε, between the phases. The interface is characterized by an order parameter, φ, that changes smoothly from -1 to 1 (or 0 to 1). A separate governing equation for this parameter is obtained on minimizing a free energy functional [26]. If carefully established the resulting evolution equation for φ naturally take account of interface and growth phenomena, e.g., Gibbs-Thomson undercooling and anisotropic growth. Further extensions can also be made to account for solute transport [24,25] and fluid flow [29,30]. Closely following the ideas and presentations in the work of Karma and Rappel [13,14] and Almgren [15] a basic phase field model, applicable to the solidification of an undercooled pure liquid with an isotropic surface energy is φ τ t 2 2 g( φ) f ( φ) = ε φ λ φ φ (5) = D t u 2 h( φ) q( φ)u + t (6) where τ is the characteristic time for attachment of atoms at the interface, λ is a dimensionless parameter that controls the strength of the coupling between the phase and diffusion field, g(φ) is a symmetric double well potential with minimums at ± 1; a traditional choice is φ 2 φ 2 g( φ) = 2 ( 2 1), f(φ) and h(φ) are internal energy interpolations across the interface and q(φ) is an extra interpolation that accounts for different diffusivities between the solid and liquid. Although phase field models are clearly in the class of DMS models they differ from the Direct Numerical Simulations of the sharp interface model, outline immediately above, in that the introduction of the diffusive interface of thickness ε and the evolution equation for the order parameter can be considered to be sub-grid models that capture the microscopic behavior at the solid-liquid interface in a model constructed with fixed and relatively large numerical elements. This concept is supported by the fact that an asymptotic analysis [23] shows that the interface thickness, ε, and the parameters λ and τ can be related to the capillary number, d 0, and the kinetic term β in the Gibbs-Thomson condition, Eq. (4), in such a way that the sharp interface model is recovered in the limit of a vanishing interface thickness, i.e, in the limit ε 0. Use of a phase field method requires a grid size that is small enough to allow 3-4 grids to span the diffusive interface [26]. Clearly assuming a thicker, ε, diffusive interface will lead to the use of a larger grid size and more efficient calculations. The critical question, however, is how does the thickness of the interface influence the predictions. In particular, how does the thickness of the interface affect the ability of the phase field model to match sharp interface predictions, such as dendrite tip velocities, [13]. Some investigations of the phase field s ability to match sharp interface characteristics place a very severe restriction on the interface thickness. For example, Braun [32] shows that ε needs to be less than the capillary length d 0 to recover sharp

9 interface morphological stability results [17]. A typical size for the capillary length is ~ 5 x m [25] which is approaching atomic dimensions, e.g., the lattice size for aluminum is ~ 4 x m [33]. With this restriction two and three-dimensional DMS in macroscopic domains, using basic phase field models, is beyond the reach of current computer resources and even DMS in messo domains stretches current computer power to the limit. There are, however, a number of situations and techniques that can reduce the demand on computer resources. 1. At high undercooling the interface speed, v n is large so the kinetic term β, in the Gibbs- Thomson condition, dominates the growth and the restriction imposed by the capillary number is reduced. A high interface speed also leads to a smaller diffusion length (D/v n ), which allows for a smaller solution domain (grain envelope). Hence problems with large undercoolings are computationally more attractive since a larger grid size can be used in a smaller domain [26]. 2. Adaptive grids can be employed [21,27,28,31] to ensure that a high intensity of grids are only employed in the vicinity of the rapid change in the phase field variable. Provatas and co-workers [27] have shown that the CPU time for using such approaches, in twodimensions, scales with the arc-length of the solid-liquid interface and not quadratically with the grid size. 3. Analysis of the phase field by Kamma and Rappel [13,14] and the recent extension by Almgren [15] account for the variations in temperature across the interface. The asymptotic analyses for this case indicates how the phase field parameters (τ, ε, and λ) can be related to the terms in the Gibbs-Thomson condition (β, d 0 ) and integral constraints can be imposed on the interpolation functions (f(φ), q(φ) and h(φ)) such that the convergence to the sharp interface model (ε 0) is second order. This places a much-reduced restriction on the size of the interface thickness and sharp interface results can be matched by a phase field method using larger grid sizes that scale to the microstructure features, e.g., the dendrite tip radius. Furthermore, in the case of equal diffusivities between the solid and the liquid the phase field parameters can be modified so that the model can be applied to the physically relevant case of equilibrium at the solid-liquid front, i.e., the limit of β 0. Level Set Methods The phase field is an example of a so-called level set method [34]. In the general level set method a moving front between physical domains is marked by the zero of a level set defined by an order parameter [ 1 φ 1]. The parameter φ is evolved by an advection equation, resetting the level set between solution steps prevents excessive smearing. A recent level set solution of the sharp interface model, Eqs. (1-3), is reported by Kim et al. [35]; in this work interfaces properties, related to undercooling and anisotropy are accounted for in the advection and resetting of the level set. Examples of DMS Many example of the wonderful and colorful results generated by DMS can be found on the world wide web. A search on the key words - dendrite -phase field- on google.com uncovered nearly 1700 sites. A very small selection of sites from this group are listed in Table 1 (all last accessed in May 2001)

10 Authors Subject Site Table I DMS Web Page Examples B. T. Murray, R. J. Braun and J. Soto. Adaptive Mesh Phase- Field Computations of Dendritic Growth J. Rappaz Dendrite formation using a phase field model X. Tong, C. Beckermann & A. Karma D.M. Anderson N. Provatas, N.Goldenfeld, and J. Dantzig Bill Boettinger, J. Warren, and N. Provatas Phase-field simulations of dendritic growth with convection Diffuse- Interface and Phase-Field modeling Modeling Solidification using Phase- Field Equations Solved by Adaptive Grid Methods Phase Field Models of Binary Alloys Numerical issues With reference to the discussion above it can be concluded that a DMS in a solidification process (~ 0.1m say) that is to resolve interface kinetics and nucleation needs to have a time step on the order of ~ 10-6 seconds and, assuming the use of a basic phase field model, a space resolution (volume element size) approaching the capillary length scale (~10-9 m say). Such a model will require 8D N R = 10 (7) nodes in the grid; where D (=1,2 or 3) is the dimension of the problem.

11 Can such grid sizes be achieved using current (2001) technology? And if not, how long will it take before computers are powerful enough to handle such grid sizes? Recently the author [36] analyzed the grid sizes (number of nodes) used in the various solidification models reported in a well known proceedings of solidification and casting modeling [1-9] dating back twenty years to It was found that the largest grid sizes used in a given year Y are bound by the line N (Y 1980) A 6000 x 2 = (8) This line shows that grid sizes are doubling every 18 months (consistent with Moore s law for computer power). This equation indicates that one dimensional calculations that meet the requirement in Eq. (7) are currently available; N A (2001) = If the rate in Eq. (8) were to be maintained then grid sizes for a two dimensional DMS solidification models will come on line in 2040 and three dimensional DMS models will be presented at the centenary anniversary meeting of the casting and welding conference in The above estimates are somewhat conservative, e.g., they neglect the potential improvements of the revised asymptotic analyses of the phase field model [13-15] and the use of adaptive grids [21,27,28,31]. A more relaxed estimate for the lead-time for DMS can be obtained by assuming a grid size of 1 µm. In this case a solidification process scaling to 0.1 m would need 10 5 nodes in each direction. With this scenario two-dimensional DMS will be obtained in ten years (2012) and three-dimensional DMS will follow in An Overview Micro-Macro Models Since a DMS of a solidification process, that resolves all the length and time scales, will not be possible for sometime the development of modeling techniques that can bridge across the disparity in scales is an important and critical research area. Following the ground-breaking work of Rappaz and co-workers who developed models of the structure development in equiaxed eutectic and dendritic alloys [37] such models are usually referred to as micromacro models. The basic concept in a traditional micro-macro approach is to model a macroscopic process (~ 0.1 m) by employing a numerical grid that resolves at the messo scale and captures the microscopic behavior via sub-grid models. In the simplest case these sub-grid models are basic constitutive relationships derived from limiting solutions, e.g, the Gulliver- Scheil and lever microsegregation rules [17]. Once again, looking at methods used in turbulence modeling, a direct analogy can be made between micro-macro models of solidification and large-eddy models of turbulence. In the later the element sizes are on the order of the large-eddies and sub-grid models are employed to account for (in an average sense) the higher order modes associated with the smaller eddies [12]. General Micro-Macro Models Figure 4 shows a schematic of the basic components in a general micro-macro model. The macro domain is covered by a grid of elements, any numerical discreteization will suffice, the finite element grid, in Fig. 4 is shown purely for illustration purposes. The thermodynamic state

12 in an REV (V) around a node (vertex) on this grid is provided by the definitions of a volumeaveraged enthalpy, H, a mixture concentration for each solute component, C k, and a solid fraction, g and [ c T L] [ ρh] = ρs ( φ)cpst + ρl(1 φ) pl + dv (9) V [ ρc] = ρ ( φ)c +ρ (1 φ)c dv (10) k V s ks [ ρ ] = ρ φ V s l kl g dv (11) where the subscripts (s) and (l) stand for solid and liquid values respectively, T is temperature, c p specific heat, ρ density, L latent heat, C is concentration, and φ is a microscopic phase marker that takes a value of 0 in a full liquid region and a value of 1 in a full solid region (c.f. the phase field marker). The rate of change of the thermal and solutal fields is given by [* ] d * dt = q n ds (12) where [*] is [ρh] or [ρc], q is the heat or mass flux entering the volume V across its surface S, and n is the outward pointing normal on that surface. If an appropriate volume averaging is used the flux in Eq. (12) can be approximated in terms of macroscopic variables stored at the node points of the domain grid. For example, assuming a fixed solid with a dendritic morphology in a messo grid volume, a locally well mixed liquid concentration and constant properties, the messo scale heat and mass fluxes in Eq. (12) can be written as q H = ρu (1 φ)(ct + L) + K T (13) l and q C = ρu (1 φ) C (14) l lk respectively, where u l is the velocity of the solid phase and it has been assumed that solute diffusion at the messo scale can be neglected. Heat and mass equations in the form of Eq. (12) are the governing macroscopic scale equations. At a given time step of a numerical solution these equations will provide predictions for the nodal mixture quantities of enthalpy and concentration, the LHS of Eqs. (9) and (10). In order to advance to the next time step the mixture solid fraction, Eq. (11), and the mass and heat fluxes, Eqs. (13) and (14), need to be calculated. In a general setting, along with basic information such as phase equilibrium data, this calculation requires a sub-grid description of nucleation, growth and mass diffusion (microsegregation) processes occurring at a microscopic scale that represents the phenomena at the solid-liquid interface. This requires the specification of a subgrid domain, e.g., an equiaxed grain envelope, or secondary arm spacing. In this way the predictions of the macroscopic quantities on the macro-domain grid are coupled to the behavior of the underlying microscopic phenomena.

13 Specific examples of how this coupling is achieved in a general micro-macro model are outlined below. To facilitate a well-focused discussion it is assumed throughout that the properties specific heat, density etc., are constant. Microstructure Predictions The classic micro-macro model developed by Rappaz and co-workers [37] focused on the prediction of equiaxed dendritic and eutectic microstructures. With reference to Fig. 4 the key elements in the prediction of equiaxed dendritic structures are 1. The solution of the heat equation to obtain the nodal temperature field on the macro domain grid. 2. The identification of a grain envelope as a sub-grid domain. Under the assumption of a constant mixture composition [C] = C 0 and a uniform domain temperature, T domain = T(t) (the calculated nodal temperature) a microsegregation model on this domain leads to a evaluation of the growth of the average grain within a numerical grid volume. 3. A statistical nucleation model to estimate the number of grains in a numerical grid. 4. The use of a micro-scale time step t m, in recognition of the fact that nucleation and growth events occur over a shorter time scale than the heat transfer (see Fig. 2); typically ~ 10 micro time steps are taken for every macro heat transfer time step. 5. The combination of the nucleation and growth models provides an estimate of the liquid fraction, g, in the numerical grid volume. The evaluation of this value provides the bridge between the micro and macro calculations. A recent variation of this structure model is the so-called CAFE model [38]. In this model, similar to the earlier work, a Finite Element (FE) grid is used to solve for the nodal temperatures and a statistical nucleation model is used. Unlike the early work, however, no specific sub-grid domain (such as a grain envelope) is specified to model the growth instead Celluar Automator (CA) rules are used. The main advantage is that the CA rules can be constructed such that the model is not restricted to equiaxed solidification, in particular the important features of columnar to equiaxed transition can be identified and tracked. Macro REV nucleation Micro-structure Dt m growth Segregation coarsening arm space Figure 4. Two examples of Micro-Macro Modeling

14 Macrosegregation Models In addition to the final microstructure, solidification research is also focused on determining the distribution of alloying components during solidification, i.e., macrosegregation.. At the macroscopic process scale, macrosegregation models [39-46] solve discrete equations describing the transport of heat, mass and momentum. Within a given time step, the models are closed by the specification of the diffusion controlled solute segregation (microsegregation) occurring at the solid-liquid interface. In a dendritic alloy, the domain for this sub-grid model is a representative secondary arm space (see Fig. 4). The most basic sub grid model in this context is to assume complete mass diffusion in both the solid and the liquid fractions of the arm space, the lever rule. An alternative and often more realistic assumption for the microsegregation model is to assume zero mass diffusion in the solid, the Gulliver-Scheil rule [17]. More sophisticated microsegregation treatments are based on aspects of the many existing microsegregation models [47-61]; these sub-grid models can account for finite solid diffusion (back-diffusion) and the coarsening of the microstructure. Perhaps the most sophisticated model proposed to date for the treatment of the local scale segregation in a macrosegregation model is due to Wang and Beckermann [61]. These authors consider two length scales for the sub-grid treatment of the microsegregation, the secondary arm space (interdendrtic liquid) and the grain envelope (extradendritic liquid). In this way they are able to take account, at the macro scale, of back-diffusion into the solid and finite liquid diffusion in the vicinity of the solid-liquid interface. This last feature is important if the solidification process is rapid or if the transition from columnar to equiaxed grains needs to be tracked. An Example of a Micro-Macro Segregation model To fully illustrate the steps in micro-macro model a recently propose micro-macro coupling for segregation modeling is presented [62]. This model can be used to track the macrosegregation during the solidification of a multi-component alloy. The version presented here is restricted to an alloy that is undergoing primary solidification resulting in a single solid phase. In addition a fixed microstructure is assumed (no coarsening) such that the sub-grid domain is a onedimensional secondary arm space of fixed length λ. Progress of the solidification in this arm space is tracked in terms of the dimensionless space and time variables, ξ = x/λ and τ =t/t f, where t f is a characteristic solidification time. The Macroscopic scale At the macroscopic scale the thermodynamic sate of an alloy casting can be described by the mixture enthalpy [ρh] and mixture solute concentrations [ρc] k These variables are defined in terms of mixture quantities averaged over the (REV). k g [ ρc] = ρ C dξ +ρ (1 g) C (15) s 0 sk l lk [ H] = ρc T +ρ (1 g) L ρ (16) p where g is the solid fraction in the representative secondary arm space. On comparison with Eq. (9) and Eq. (10) it is observed that the sub-grid model domain of the secondary arm space has been used to define the volume averages in the REV. In a numerical simulation, changes in time of the nodal mixture quantities on the LHS of Eqs. (15) and (16) will be obtained from a macro scale calculation that solves the transport equation l

15 given in Eq. (12). In this setting the role of the sub-grid micro-scale modeling is to retrieve the values on the RHS of Eqs. (15) and Eqs. (16) for subsequent use in further macro-scale calculations. At time τ the mixture quantities, [ρh] and [ρc] k, and liquid fraction, solute concentrations, and temperature in the arm space are known. Next time step values of the mixture quantities, [ρh] new and [ρc] i new, can be calculated at time τ new = τ + τ from Eq. (12), and looking toward an iterative solution a current estimate of the temperature can be obtained from Eq. (16), T c [ ρh] new p c ρl (1 g )L = (17) ρc where the superscript c indicates the most current estimate of the new time step values. As outlined below, the micro-macro sub-grid model, which operates in the secondary arm spacing domain, is used in this iterative solution to obtain the value of g c ; these calculations are based on the solute balance and the thermodynamics of the solid-liquid phase change. The Solute Balance At time τ the diffusion of solute into the solid of the arm spacing the back diffusion is given by C sk q back = ρsαk (18) ξ where α = D s t f /λ 2 is a mass diffusion Fourier number and D s is the mass diffusion coefficient. The value of the back diffusion is small and can be considered to be constant over the small time step τ. In this way the solute balance at the new time can be written as new k g [ ρc] = ρ C + τq + ρ (1 g) C (19) s 0 sk where items on the left hand side are all evaluated at time t = τ and C k * is the average concentration of the liquid in the arm space that changes phase over the time step τ. Comparing Eq. (15) and Eq. (19) leads to Thermodynamics C new [ ρc] [ ρc] back * k k back i (1 g) ρl l τq + (1 g) ρl Clk = (20) Over the time step τ the solid from the phase change of the C k * liquid forms in the vicinity of the solid-liquid interface. Hence, if τ is small, the current liquid concentrations C lk new can be calculated assuming an equilibrium solidification without loss of accuracy. In the most general case, following the proposal of Boettinger et al [63], one can imagine inputting the current values of T c and C k * into a in a thermodynamic subroutine that returns equilibrium values of the liquid and solid concentrations that match the input values. With these values the solute balance * k

16 C * * * * new i f Csk + (1 f ) C lk = (21) * can be written, where C sk is the concentration in the solid at the solid-liquid interface. With respect to Eq. (21) it is critical to note that this balance is only operating on the fraction of the arm space that is in the liquid state at time τ, i.e., (1-g) and the fraction f * is the mass fraction of this volume that changes to solid over the time step. The associated volume fraction is given by g * s s * ρlf = (22) * ρ ( ρ ρ )f Under the conditions of a binary alloy, equal densities, the prescription of a constant cooling rate (T(τ) = constant), a straight liquidus line (with slope m), and a constant partition coefficient κ, the proposed thermodynamic treatment will give where the current liquid concentration is given by C l = C 0 + mt. l * * 1 C g = 1 (23) 1 κ Cl From Eq. (22) or (23) the solid volume fraction in the entire arm space can be updated as g c c c ( g + g g ) = g + ω (24) where the product g = g * g is the volume fraction of solid formed in the time step and ω is an under-relaxation (~0.2 works well in cases tested to date). The Back Diffusion The sub-grid model is closed by an iterative solution through Eqs (17), (20) and (22), with an appropriate thermodynamic treatment based on the C k * and T c values. Following convergence of this iteration, an estimate for the back-diffusion at the next time step needs to be calculated. Voller [64] has shown that the approximation new * sk * old sk τq = ρ β g (C C ) (25) back s i where 0 β 1 is a measure of the level of back-diffusion, can work well. When a constant temperature drop controls the cooling of the domain an appropriate average value is [65] 2α β = 1+ 2α (26) Further modifications can be made in this term to account for arm coarsening [66]. Alternative treatments for the back-diffusion include a complete numerical solution in the solid fraction of the arm space [67] and approximations based on assuming a solid concentration profile, e.g., Wang and Beckermann [56].

17 An Example Calculation Consider a binary alloy with, equal densities, the prescription of a constant cooling rate (T(τ) = constant), a straight liquidus line, a constant partition coefficient κ = 0.1, and a fixed mixture [C] = C o = 1. In this case Eq. (23) determines the thermodynamics and the back-diffusion can be calculated from Eqs. (25) and (26). The solidification is assume to terminate when the eutectic concentration is reached C l = C eut = 5. At this point the remaining liquid fraction gives the volume fraction of eutectic that will form. Figure 5 shows the prediction of the fraction eutectic for the test problem across a range of values for the Fourier number α. In this calculation C l is incremented is 100 steps of These predictions are compared with predictions obtained with the full numerical model, based on a finite difference calculation, reported by Voller [67] and available for down load at (last access May 2001). Note in these results 1. The close agreement between the full numerical and approximate solutions. 2. The accurate predictions in the limit states of zero (Gulliver-Scheil) and complete (lever) mass diffusion in the solid. This result demonstrates that a sequence of calculations that assume equilibrium throughout the liquid fraction of the arm space (the basic approximation adopted in our thermodynamic treatment) is a good approximation for the non-equilibrium conditions imposed by the Gulliver-Scheil Scheil limit Eutectic Fraction Numerical Approximate 0.11 lever limit Fourier Number Figure 5. Eutectic fraction predictions against Fourier number [62]

18 Conclusions In the future if computer power continues to increase Direct Microstructure Simulation of solidification processes will be possible. Calculations made in this paper, however, indicate that, due to the up to 8 decades (per dimension)) in length and time scales that such a simulation will require, it may be at least 50 years before such calculations become commonplace. In the meantime researchers needs to focus attention on 1. The continued development of DMS techniques on domains that can be accommodated by current computer technologies (e.g., grain envelopes). This is required so that as computer power becomes available appropriate models are ready to go on line. Further, such models will provide valuables information for the design of averaging techniques for use in micro-macro models. 2. The development of micro-macro models that allow for the average treatment of micro scale phenomena in macro scale models. Although such models may have a limited life expectancy and will continually be replaced over time by more general DMS they provide significant insights into the behavior of solidification phenomena that point the way towards building better solidification process. References 1. H.D Brody, D. Apelian, Modeling of Casting and Welding and Processes, TMS, Warrendale, J.A. Dantzig and J.T. Berry, Modeling of Casting and Welding Processes-II, TMS, Warrendale, S. Kou and R. Mehrabian, Modeling of Casting and Welding Processes-III, TMS, Warrendale, A.F. Giamei and G.J. Abbaschian, Modeling of Casting and Welding Processes-IV, TMS, Warrendale, M. Rappaz, M.R. Ozgu and K.W. Mahin, Modeling of Casting, Welding and Advanced Solidification Processes-V, TMS, Warrendale, T.S. Piwonka, V.R. Voller and L. Katgerman, Modeling of Casting, Welding and Advanced Solidification Processes - VI, TMS, Warrendale, M.Cross and J. Campbell, Modeling of Casting, Welding and Advanced Solidification Processes-VII, TMS, Warrendale, B.G. Thomas and C Beckermann, Modeling of Casting, Welding and Advanced Solidification Processes-VIII, TMS, Warrendale, P.R. Sahm, P.N. Hansen and J.G. Conley, Modeling of Casting, Welding and Advanced Solidification Processes-IX, Shaker Verlag, Aachen, C. Beckermann and C.Y. Wang, Multipjase/-Scale Modeling of Alloy Solidification, Annual Review of Heat Transfer, 6, (1995),

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20 28. N. Provatas, N. Goldenfeld and J. Dantzig, Efficient Computation of Dendritic Microstructures using Adaptive Mesh Refinement, Phys. Rev. Letters, 80, (1998) C. Beckermann, H.-J. Diepers, I. Steinbach, A. Karma, and X. Tong., Modeling of Melt Convection in Phase-Field Simulations of Solidification, J. Comp. Phys., 154 (1999), X. Tong, C. Beckermann and A. Karma, Velocity and Shape Selection of Dendritic Crystals in a Forced Flow, Phys. Rev. E, 61 (2000), R49-R C.M. Elliott and A.R. Gardiner, Double Obstacle Phase Field Computations of Dendritic Growth, Center for Mathematical Analysis and Its Applications, Research Report, 96/19, R.J. Braun, G.B. McFadden, and S.R. Coriell, Morphological Instability in Phase-Field Models of Solidification, Phys. Rev. E., 49, (1994), M. H. Richman, An Introduction to the Science of Metals, (Blaisdell, Walthan. MD, 1967). 34. J.A. Sethian, Level Set Methods, (Cambridge University Press, 1996). 35. Y-T Kim, N. Goldenfeld, and J, Dantzig, Computations of Dendritic Microstructure using Level Set Method, Phys. Rev. E., (2000), V.R. Voller, Simulation Grid Sizes and Moore s Law, in preparation. 37. M. Rappaz, Modeling of Microstructure Formation in Solidification Processes, Int. Mater. Rev., 34 (1989), Ch.-A. Gandin, T. Jalanti and M.Rappaz, Modeling of Dendritic Grain Srructures, Modeling of Casting, Welding and Advanced Solidification Process VIII, ed. B.G. Thomas and C. Becermann (Warrendale, PA, The Minerals metals & Materials Society, 1998), W.D. Bennon and F.P. Incropera, The evolution of macro-segregation in statically cast binary ingots, Metall. Trans. B,18 (1987) C. Beckermann and R. Viskanta, Double-Diffusive Convection during Dendritic Solidification of a Binary Mixture, Physico. Chem. Hydrodyn,10 (1988) V.R. Voller, A.D. Brent and C. Prakash, The Modelling of Heat, Mass and Solute Transport In solidification systems, Int. J. of Heat and Mass transfer, 32 (1989) C.R. Swaminathan and V.R. Voller, Towards a General Numerical Scheme for Solidification Systems, Int, J. Heat and Mass Transfer, 40 (1997) M.J.M. Krane, F.P. Incropera and D.R. Gaskell, Solution of Ternary Metal Alloys I. Model Development, Int J. Heat and Mass Transfer, 40 (1997)

21 44. M.C. Schneider and C. Beckermann, Formation of Macrosegregation by Multicomponent Thermosolutal Convection during Solidification of Steel, Metall Trans. A, 26 (1995) S.D. Felicelli, J.C. Heinrich and D.R. Poirier, Simulation of Freckles During Vertical Solidification of Binary Alloys, Metall. Trans. B, 22 (1991) G. Amberg, Computation of Macrosegregation in An Iron-Carbon Cast, Int. J. Heat and Mass Transfer, 34 (1991) T. Kraft and Y.A. Chang, Journal of Metals, 49 (1997) H.D. Brody and M.C. Flemings, Solute Redistribution During Dendritic Solidification, Trans. Met Soc. AIME, 236 (1966) T.W. Clyne and W. Kurz, Solute Redistribution During Solidification with Rapid Solid State Diffusion, Metallurgical Transactions, 12A (1981) S. Kobayashi, Solute Redistribution During Solidification wth Diffusion In Solid Phase: A Theoretical Analysis, Journal of Crystal Growth, 88 (1988) I. Ohnaka, Mathematical Analysis o Solute Redistribution During Solidification with Diffusion In Solid Phase, Transactions ISIJ, 26 (1986) A. Roosz, E. Halder and H.E. Exner, Numerical Calculation of Microsegregation in Coarsened Dendritic Microstructures, Mater. Sci. Technol., 2 (1986) T.P Battle and R.D. Pehlke, Mathematical Modeling of Microsegregation in Binary Metallic Alloys, Metallurgical Transactions, 21B (1990) A.J.W. Ogilvy and D.H. Kirkwood, Appl. Sci. Res., 44 (1987) V.R. Voller and S. Sundarraj, Modelling of Microsegregation, Mater. Sci. Technol., 9 (1993) C.Y. Wang and C. Beckermann, Unified Solute Diffusion Model for Columnar and Equiaxed Dendritic Alloy Solidification, Material Science and Engineering 171 (1993) H. Yoo and C.-J. Kim, A Refined Solute Diffusion Model for Columnar Dendritic Alloy Solidification, Int. J. Heat Mass Transfer, 41(1998) V.R. Voller, A Semi-analytical model of Microsegregation in a Binary Alloy, Journal of Crystal Growth, 197 (1999) L. Nastac and D.M. Stefanescu, An Analytical Model for Solute Redistribution During Solidification of Planar, Columnar and Equiaxed Morphology, Metallurgical Transactions., 24A (1993) T. Himemiya and T. Umeda, Solute Redistribution Model of Dendritic Solidification Considering Diffusion in both the Liquid And Solid Phases, ISIJ International, 38 (1998)

22 61. C.Y. Wang and C. Beckermann, A Multiphase Solute Diffusion Model for Dendritic Alloy Solidification, Metall. Trans. A, 24 (1993), V.R. Voller, A General Method for Coupling Macro and Micro Phenomena During the Solidification of an Alloy, EUROMECH 408, ed., P. Ehrhard and D. Riley (Kluwer, Netherlands, 2001). 63. W.J. Boettinger, U.R. Kattner and D.K. Banerjee, Analysis of solidification path and microsegregation in muticomponent alloys, Modeling of Casting, Welding and Advanced Solidification Process VIII, ed. B.G. Thomas and C. Becermann (Warrendale, PA, The Minerals metals & Materials Society, 1998), V.R. Voller, On a General Back-Diffusion Parameter, J. Crystal Growth, in press. 65. V.R. Voller, A Semi-Analytical Model of Microsegregation in a Binary Alloy, J. Crystal Growth, 197, (1999), V.R. Voller and C. Beckermann, A unified model of microsegregation and coarsening, Metallurgical Transactions, 30A (1999) V.R. Voller, A model of microsegregation during binary alloy solidification, Int. J. Heat Mass Transfer, 43, (2000),

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