We propose a model of equiaxed eutectic solidification that couples the macroscopic level of heat diffusion with the microscopic level of nucleation and growth of the eutectic grains. The heat equation with the source term corresponding to the latent heat release due to solidification is calculated numerically by means of an implicit finite difference method. In the time stepping scheme, the evolution of solid fraction is deduced from a stochastic model of nucleation and growth which uses the local temperature (interpolated from the FDM mesh) to determine the local grain density and the local growth rate. The solid-liquid interface of each grain is tracked by using a subdivision of each grain perimeter in a large number of sectors. The state of each sector (i.e. whether it is still in contact with the liquid or already captured by an other grain) and the increase of radius of each grain during one time step allows one to compute the increase of solid fraction. As for deterministic models, the results of the model are the evolution of temperature and of solid fraction at any point of the sample. Moreover the model provides a complete picture of the microstructure, thus not limiting the microstructural information to the average grain density but allowing one to compute any stereological value of interest. We apply the model to the solidification of gray cast iron

Charbon, Ch.

LeSar, R.

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LA-UR- . Title: Au th or(s): Submitted to: Los Alamos N A T I O N A L  L A B O R A T O R Y  A FULLY COUPLED 2D MODEL OF EQUIAXED EUTECTIC SOLIDIFICATION Ch. Charbon - CMS R. LeSar - CMS Annual meeting of Japan Institute of Metals Honolulu, Hawaii December 1995 Los Alarms Natlonal Laboratoly, an affirmatwe actlordequal opportunty employer, is operated by the Universlty of Califomla for the U S Department of Energy under contract W-7405ENG-36 By acceptance of this article, the publisher recognizes that the U S Government retains a nonexcluswe. royaily-free license to publish or reproduce the published form of thts contribution. or to allow others to do so, for U S Government purposes The Los Alamos Natlonal Laboratory requests that the publisher ldenefy this artlcle as work performed under the ausplces of the U S Department of Energy DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. . A FULLY COUPLED 2D MODEL OF EQUIAXED EUTECTIC SOLIDIFICATION Ch. Charbon and R. LeSar Los Alamos National Laboratory Center for Materials Science Mail Stop G755 Los Alamos, New Mexico 87545, USA Abstract We propose a model of equiaxed eutectic solidification that couples the macroscopic level of heat diffusion with the microscopic level of nucleation and growth of the eutectic grains. The heat equation with the source term corresponding to the latent heat release due to solidification is calculated numerically by means of an implicit finite difference method. In the time stepping scheme, the evolution of solid fraction is deduced from a stochastic model of nucleation and growth which uses the local temperature (interpolated from the FDM mesh) to determine the local grain density and the local growth rate. The solid-liquid interface of each grain is tracked by using a subdivision of each grain perimeter in a large number of sectors. The state of each sector (i.e. whether it is still in contact with the liquid or already captured by an other grain) and the increase of radius of each grain during one time step allows one to compute the increase of solid fraction. As for deterministic models, the results of the model are the evolution of temperature and of solid fraction at any point of the sample. Moreover the model provides a complete picture of the microstructure, thus not limiting the microstructural information to the average grain density but allowing one to compute any stereological value of interest. We apply the model to the solidification of gray cast iron. Introduction Eutectic alloys are an important class of metals. Due to their low melting point, good castability and useful mechanical properties, they are widely used in industrial applications. Motivated by this industrial interest, scientists have long studied these alloys in order to provide models describing the microstructural characteristics (grain density, interlamellar spacing, etc.) of a casting as a function of the alloy and of the cooling conditions. Equiaxed eutectic solidification occurs usually by the heterogeneous nucleation of grains in the undercooled liquid and their subsequent growth. Due to a perfect isotropic growth velocity, the grains grow with a spherical solid-liquid interface until they impinge on their neighbors. From this simple description, three main phenomenon have to be accounted for in a model of equiaxed eutectic solidification: nucleation, growth and impingement. In the present study, a new model is presented, which couples the heat balance computation by a finite difference method (FDM) with a stochastic model of nucleation and growth. The aim of the model is to predict the evolution of the microstructure in a more accurate way than do deterministic models, to avoid rebounding or oscillations in cooling curve, and to produce a picture of the microstructure, thus allowing one to study any stereological parameter of interest. The Model The principle of the model described in the following paragraphs is very similar to the one developed by Gandin et ai [ l ]  for the modeling of dendritic solidification. As in the model described by these authors, the basic coupling between the microscopic scale of the grain growth and the macroscopic scale of the heat flow is accomplished by using in the macroscopic heat balance the solid fraction evolution predicted by a stochastic model of nucleation and growth. Macroscopic heat diffusion Defining the enthalpy as In order to describe heat diffusion, we use the heat equation in its enthalpy formulation. where H is the volumetric enthalpy, T the temperature, pcp the volumetric specific heat, L the volumetric latent heat and f, the volumetric solid fraction, we can write the heat balance as: aH div(K gradT) = - . at We solve Eq.(2) with an implicit finite-difference method [2]. The system is discretized in space with an orthogonal mesh of rectangular cells and time is discretized by a division in time steps of length At. Following Rappaz [3], we use an iterative Newton method in order to take into account the non-linear relationship between temperature and enthalpy. From the solution of Eq.(2), we find the change in enthalpy during a time step at each node of the mesh. Provided that the increase in solid fraction at each node, Af,, is known, the new temperatures are given by where i is a subscript indicating the different nodes of the FDM mesh. The role of the stochastic method is to provide the increment of solid fraction at each node during each time step. The quantity Af, being known, the new temperatures can be calculated and Eq.(2) can be solved for the next time step. The coupling between the macroscopic heat diffusion and the microscopic microstructural evolution takes place through Eq.(3). Microstructure evolution The basic idea of the two-dimensional stochastic method is to use the temperature field predicted by the macroscopic heat balance to drive the nucleation and growth of individual grains at the microscopic scale. The temperatures are used explicitly at the microscopic scale. This approach is very similar to the three dimensional method used by Charbon et al. [4] to study the influence of grain movement on eutectic solidification. For both the nucleation and the growth of the grains, it is necessary to know the temperature at any point of the sample and not only at the mesh nodes. We determine the temperature at any point P by a bilinear interpolation of the temperature of the four surrounding mesh nodes. Nucleation We use an empirical nucleation law similar to that used by Oldfield [ 5 ] ,  which gives the grain density as a function of the undercooling, AT = T - TeU,, ~ = A , A T ~  , (4) where T,,, is the equilibrium eutectic temperature, n is the grain density and A, and b are empirical parameters. At the beginning of a calculation, we randomly choose N, coordinates (xn, y,) that define possible nucleation sites. The maximum number of grains, N,, is determined by N, = AnATl,S , where S is the surface of the sample and ATm, is the maximum undercooling, a parameter chosen to be greater than the undercooling reached at any point in the liquid during the solidification process. The value of AT, is not known prior to the computation and thus a test is performed during the computation to determine whether this condition is satisfied. If the undercooling ever exceeds the set value of AT-, the calculation must be restarted with a larger value. For each nuclei, the activation undercooling, ATg, is chosen randomly in the interval [0, At,,,] in such a manner that Eq.(4) is satisfied. A nucleation site is activated only if its position reaches an undercooling greater than ATg while it is still liquid (i.e. it has not been captured by a previously-nucleated grain). As soon as a nucleation site is activated, it is considered a new grain, k, with an initial infinitesimally small radius, rk. Growth At each time step, the grain radii are updated according to the Jackson and Hunt relationship: 2 = r; + Ar; = r; + v:At = r; + A, (AT;) At , where the local undercooling of grain k, ATk, is determined from the undercooling at the position of the nucleation site of the grain. In the present model, it is assumed that the grains are small compared to the thermal gradient, so that the temperature of each grain is considered uniform. For that reason, each grain is characterized by one radius (i.e., it is assumed that the solid-liquid interface of the grains remain circular during growth). Determination of the solid fraction increments While the grains have circular growth, their final shape is determined by the impingement between growing grains. To determine this impingement, we first divide the perimeter of each grain in a large number, N,, of equal angular sectors characterized by an angle 68, 2rc 6 e = - ,  Ns (7) where the angular position of the axis of symmetry of each sector, j ,  is given by the angle Oj, e, = (j-1)Se . (8) Just after the nucleation of a new grain, k, all the sectors are in contact with the liquid. As solidification proceeds, the portion of the perimeter of the grain which is in contact with the liquid decreases due to the impingement with neighboring grains. In order to determine the state of each of these sectors, i.e., whether they are in contact with the liquid (and thus can participate in further solidification) or if they have been captured by another grain, a loop over the N,,, nearest neighbors is performed. A sector, j ,  is considered captured if it is no longer in contact with the liquid, i.e., if the tie 66 sector of annulus defined by the two angles ej - - and 8, + - and by the two radii rk and 2 2 rk+Ark is intersected by one of the N, neighbors. If a sector is still in contact with the liquid, during growth it generates an increase in solid fraction equal to This microscopic solid fraction increment is then split in four contributions which are attributed to the four nearest nodes surrounding the solid sector. The weighting factors for each contribution are the same as those used to determine the temperature of any given point other than a mesh node. At each node of the finite difference mesh, the microscopic contributions, Sf,, are added and give rise, at the end of the loop over all grains and all sectors, to the macroscopic increases of solid fraction at each node of the FDM mesh during the time step, Afs ,. It is this macroscopic Afs,, that is used in the heat balance Eq.(3) to determine the new temperatures. Represen tation of the microstructure During the computation, the radii of the different sectors as they are captured by an other grain are stored. Knowing the coordinates of the center of each grain, it is then possible to draw the shape of each grain as a polygon with N, edges. The quality of the drawing is naturally a function of the number of sectors and the duration of the time step. Results and Discussion We consider here a simple geometrical and thermal situation. A rectangular plate (0.03 x 0.1 1 m) of liquid metal initially at a uniform temperature, Tin,, is solidified by a constant heat flux applied on one of its small sides. The three other sides are considered perfectly insulated. The simulation was performed with the data listed in Table 1, which were taken from the Ph.D. thesis of Zou [6] and correspond to gray cast iron. TABLE 1 : Values of the different parameters used in the simulation. Thermal conductivity, K 40. [Wm-'l Specific heat, pc, 6 . 3 7 ~ 1 0 ~  [Jm-3K'] Latent heat, L 1 . 4 4 ~ 1 0 ~  [Jm"] Eutectic temperature, Tu,  1054. ["Cl Growth law Nucleation law I Dimensions v(AT)=4.~10-~ AT2 [ms-'1 Mesh 13 x45  evenly-distributed nodes Heat flux, Q 400000 [wm-21 Time step, NDdtr At 1500x 1.0 [SI Number of sectors, N, 40 sectors In Figure 1, we show the final microstructure of the ingot. There are 37932 grains. The grain size is plotted as a function of position in Figure 2. As expected, the grain size increases from the bottom of the sample (position of the chill) to the top (adiabatic condition). The grain size is an average over a 1 cm centered on the position indicated on the horizontal axis of the plot. A transition in grain density occurs around the position y=0.08m. Due to the model of instantaneous nucleation, nucleation is stopped as the recalescence is reached and no new grains appear until a temperature below the minimum of the recalescence is reached. In other words, no nucleation occurs during the whole eutectic plateau. Only a few grains are nucleated in the remaining liquid after the eutectic plateau is passed. Where no recalescence is noted, nucleation continues regularly as long as liquid is present. The grain density as a function of the location is nicely fitted by a power relationship (n(y)=6.033~10-~exp[6.137 l(y)]). This simple relationship is certainly a consequence of the simple thermal situation and nucleation law used here. Only the three last points of the graph clearly deviate from this fit. They correspond to the locations where a recalescence was observed. The model also yields the time evolution of the temperature and solid fraction. As shown in Figure 3, the shape of the cooling curves evolves from being s-shaped curves near the cooled surface, to classical eutectic cooling curves exhibiting the typical recalescence and eutectic plateau near the top of the sample. The appearance of a recalescence is characteristic of low thermal gradients. It is an extraordinary feature of the stochastic model coupled with the FDM to reproduce this transition since neither the enthalpy method nor the latent heat method were able to produce such a result. In Figure 4 we show the temperature profiles at different times as a function of the position in the sample. We also show (as a dotted line) the minimum temperature of the recalescence. Note that the temperature levels off to this value the at later times as the solidification front approaches the top of the sample. Conclusion A new model has been presented which couples the evolution of the microstructure (stochastic model) with the heat diffusion (finite difference method). This model gives a realistic picture of the microstructure during and in the end of solidification. Moreover, the model predicts cooling curves which exhibit a realistic appearance of the recalescence. Recalescences are modeled only when the thermal gradient is small and disappear when it is large. Further experiments are necessary to go toward more quantitative comparisons. Further developments of the model will include 3-dimensional modeling of the solidification as well as fluid flow to account for grain movement in the mushy zone due to natural or forced convection. References Ch.-A. Gandin and M. Rappaz, Acta metall. mater. 42, 2233 (1994) D. R. Croft and D. G. Lilley, ‘Heat Transfer Calculations Using Finite Differences Equations’, 1977, London, Applied Science. [31 [41 M. Rappaz, International Materials Review 34, 93 (1989) Ch. Charbon, A. Jacot and M. Rappaz, Acta metall. mater. 42,3953 (1994) VI  W. Oldfield, Trans. ASM 59, 945 (1966) J. Zou, PhD Thesis, Ecole Polytechnique F6dCrale de Lausanne, No 765, (1988) i Y Figure 1 Final simulated microstructure. The total number of grains is equal to 37932 y = 6.0332e-48 * eA(6.1371x) I<= O.S??7X2 1.2 4 110-7 - ai .Y cn 9 - c .3 I 0 0.02 0.04 0.06 0.08 0.1 0.12 Position, y Im] Figure 2 Grain size as a function of the position. The grain size is an average over a lcm wide slice. Time, t [s] Figure 3 Cooling curves calculated at different locations, from the cooling chill to the top of the sample. The distance between two succesive positions is equal to 1 cm. - 1300 c) e 1200 ? 1100 3 1000 & 900 6 800 ’ 700 0 0.02 0.04 0.06 0.08 0.1 Position, y [m] Figure 4 Temperature profiles at different times. 

A fully coupled 2D model of equiaxed eutectic solidification

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A fully coupled 2D model of equiaxed eutectic solidification

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