M. Rappaz

A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes

Abstract A new algorithm based upon a 2-dimensional Cellular Automaton (CA) technique is proposed for the simulation of dendritic grain formation during solidification. The CA model takes into account the heterogeneous nucleation, the growth kinetics and the preferential growth directions of the dendrites. This new CA algorithm, which applies to non-uniform temperature situations, is fully coupled to an enthalpybased Finite Element (FE) heat flow calculation. At each time-step, the temperature at the cell locations is interpolated from those at the FE nodal points in order to calculate the nucleation-growth of grains. The latent heat released by the cells and calculated using a Scheil-type approximation is fed back into the FE nodal points. The coupled CA-FE model is applied to two solidification experiments, the Bridgman growth of an organic alloy and the one-dimensional solidification of an Al-7wt% Si alloy. In the first case, the predicted boundaries between grains are in good agreement with experiment, providing the CA cell size is of the order of the dendrite spacing. For the second experiment, the quality of the coupled CA-FE model is assessed based upon grain structures and cooling curves. The columnar-to-equiaxed transition and the occurrence of a recalescence are shown to be in good agreement with the model.

Solidification Microstructures: Recent Developments, Future Directions

The status of solidification science is critically evaluated and future directions of research in this technologically important area are proposed. The most important advances in solidification science and technology of the last decade are discussed: interface dynamics, phase selection, microstructure selection, peritectic growth, convection effects, multicomponent alloys, and numerical techniques. It is shown how the advent of new mathematical techniques (especially phase-field and cellular automata models) coupled with powerful computers now allows the following: modeling of complicated interface morphologies, taking into account not only steady state but also non-steady state phenomena; considering real alloys consisting of many elements through on-line use of large thermodynamic data banks; and taking into account natural and forced convection effects. A series of open questions and future prospects are also given. It is hoped that the reader is encouraged to explore this important and highly interesting field and to add her/his contributions to an ever better understanding and modeling of microstructure development.

Modeling of equiaxed microstructure formation in casting

A general micro/macroscopic model of solidification for 2-D or 3-D castings, valid for both dendritic and eutectic equiaxed alloys, is presented. At the macroscopic level, the heat diffusion equation is solved with an enthalpy formulation using a standard FEM implicit scheme. However, instead of using a unique relationship between temperature and enthalpy (i.e., a unique solidification path), the specific heat and latent heat contributions, whose sum equals the variation of enthalpy at a given node, are calculated using a microscopic model of solidification. This model takes into account nucleation of new grains within the undercooled melt, the kinetics of the dendrite tips or of the eutectic front, and a solute balance at the scale of the grain in the case of dendritic alloys. The coupling between macroscopic and microscopic aspects is carried out using two time-steps, one at the macroscopic level for the implicit calculation of heat flow, and the other, much finer, for the microscopic calculations of nucleation and growth. This micro/macroscopic approach has been applied to one-dimensional and axisymmetric castings of Al-7 pct Si alloys. The calculated recalescences and grain sizes are compared with values measured for one-dimensional ingots cast under well-controlled conditions. Furthermore, the influence of casting conditions on temperature field, undercooling, grain size, and microstructural spacings is shown to be predicted correctly from axisymmetric calculations with regard to the expected experimental behavior.

A 3D cellular automaton algorithm for the prediction of dendritic grain growth

Two- and three-dimensional (3D) Cellular Automaton (CA) algorithms are proposed for modelling the growth of dendritic grains from the liquid phase. These CA growth algorithms are validated for simple thermal situations by comparing the predicted grain shapes with those deduced from analytical models. The insight obtained by the 3D approach is demonstrated by studying the extension of a single dendritic grain in a squared platform (i.e. at a section change of a casting mould) under various conditions. In particular, the effects of crystallographic orientation, thermal gradient, velocity of the isotherms and growth kinetics are shown. This 3D CA growth algorithm, coupled with finite element heat flow calculations, will become a major tool for the prediction of dendritic grain structures in solidification processes.

A simple but realistic model for laser cladding

A model which takes into account the main phenomena occurring during the laser-cladding process is proposed. For a given laser power, beam radius, powder jet geometry, and clad height, this model evaluates two other processing parameters, namely, the laser-beam velocity and the powder feed rate. It considers the interactions between the powder particles, the laser beam, and the molten pool. The laser power reaching the surface of the workpiece is estimated and, assuming this power is used to remelt the substrate with the clad having been predeposited, the melt-pool shape is computed using a three-dimensional (3-D) analytical model, which produces mmediate results, even on personal computers. The predictions obtained with this numerical model are in good agreement with experimental results. Processing engineers may therefore use this model to choose the correct processing parameters and to establish cladding maps.

Solute diffusion model for equiaxed dendritic growth

Abstract A new approach to the modeling of the equiaxed solidification of dendritic alloys is proposed. It is assumed that, in metallic alloys, microstructure formation is primarily controlled by solute diffusion (i.e. there is complete “thermal mixing” at the scale of one grain), and that the dendrite interface is an iso-concentrate at all times. The evolution of one dendritic grain is therefore modelled as follows: (i) complete mixing of solute within the interdendritic liquid; (ii) no back-diffusion in the solid; (iii) spherical solute diffusion in the liquid around the grain envelope; (iv) overall solute balance; (v) overall thermal balance; (vi) growth velocity, υg, of dendrite tips governed by the kinetic equation derived for the isolated dendrite case. By using an explicit finite difference scheme to solve these coupled equations, the concentration profiles, cooling curve, fraction of solid and evolution of dendritic grain envelope can be calculated. The intitial conditions used to start the calculation are provided by two parameters related to nucleation: the initial undercooling and the density of gains. The effect of nucleation and thermal conditions on equiaxed growth are studied. The theoretical predictions of recalescence and of the distribution of an interdendritic eutectic phase are in good agreement with experimental observations.

A thermal model of laser cladding by powder injection

A two-dimensional (2-D) finite element model is presented for laser cladding by powder injection. The model simulates the quasi-steady temperature field for the longitudinal section of a clad track. It takes into account the melting of the powder in the liquid pool and the liquid/ gas free surface shape and position, which must conform to the thermal field in order to obtain a self-consistent solution. The results for an idealized problem, where there is almost no melting of the substrate material, demonstrate the linear relationship between the laser power, the processing velocity, and the thickness of the deposited layer. The calculated clad heights agreed well with the experimental values for the conditions where a cobalt-based hard-facing alloy is clad onto mild steel with a linearly focused laser source.

Analysis of solidification microstructures in Fe - Ni - Cr single crystal welds

A geometric analysis technique for the evaluation of the microstructures in autogenous single-crystal electron beam welds has been previously developed. In the present work, these analytical methods are further extended, and a general procedure for predicting the solidification microstructure of single-crystal welds with any arbitrary orientation is established. Examples of this general analysis are given for several welding orientations. It is shown that a nonsymmetric cell structure is expected in transverse micrographs for most welding geometries. The development of steady-state conditions in the weld pool is also examined in terms of the weld pool size, its shape (as revealed by the dendritic growth pattern), and the size of the dendritic cells. It is found that steady state is established within a few millimeters of the beginning of the weld. Furthermore, steady state is achieved faster in welds made at higher welding speeds. A general analysis of the three-dimensional (3-D) weld pool shape based on the dendritic structure as revealed in the two-dimensional (2-D) transverse micrographs is also developed. It is shown that in combination with information on the preferred growth direction as a function of the solidification front orientation, the entire dendritic growth pattern in single-crystal welds can be predicted. A comparison with the actual weld micrographs shows a reasonable agreement between the theory and experiment. Finally, the theoretical analysis of the dendrite tip radius is extended from binary systems to include the case of ternary systems. The theoretical dendrite trunk spacing in a ternary Fe-Ni-Cr alloy is calculated from the dendrite tip radius and is compared with the experimental values for several weld conditions. Good agreement between experiment and theory is found.

Development of microstructures in Fe−15Ni−15Cr single crystal electron beam welds

A detailed analysis of the microstructures produced in an autogenously welded single crystal of Fc−15Ni−15Cr was performed in order to investigate the relationship between growth crystallography and solidification behavior. Electron beam welds were made at various speeds on the (001) surface of single crystals in either the [100] or [110] directions. A geometrical analysis was carried out in order to relate the dendrite growth velocities in the three 〈100〉 directions to the weld velocities for the different crystallographic orientations examined. From this analysis, the preferred dendrite trunk directions were determined as a function of the solidification front orientation based upon a minimum velocity or minimum undercooling criterion. A thorough examination of the weld microstructures and a comparison with the geometrical relationships developed in this work permitted a three-dimensional reconstruction of the weld pool shape to be performed. In addition, the dendrite spacings were measured, and the variation in spacings as a function of growth velocity was compared with theoretical predictions. It was found that the range of velocities over which dendritic growth is expected agreed with the experimental findings, and, furthermore, the change in dendrite spacing with growth velocity varied as predicted by theory. These results clearly demonstrate the effect of crystallography on the micro-structural development during weld pool solidification. The results also show that the resultant microstructures and pool shapes can be explained by geometrical analysis in conjunction with existing solidification models.

A Pseudo-Front Tracking Technique for the Modelling of Solidification Microstructures in Multi-Component Alloys

Abstract A two-dimensional model for the simulation of microstructure formation during solidification in multi-component systems has been developed. The model is based on a new pseudo-front tracking technique for the calculation of the evolution of interfaces that are governed by solute diffusion and the Gibbs–Thomson effect. The diffusion equations are solved in the primary solid phase and in the liquid using an explicit finite volume method formulated for a regular hexagonal grid. Volume elements located in the liquid phase undergo a transition to interfacial (or mushy) cells before being incorporated in the solid phase. This layer of interfacial elements, which always separates the solid from the liquid sub-domains, permits to handle the displacement of the interface in agreement with the flux condition at the interface. The interface curvature is obtained from the field of the signed distance to the interface, as reconstructed with a PLIC (piecewise linear interface calculation) technique. The concentrations at the solid–liquid interface are calculated using thermodynamic data provided by the phase diagram software Thermo-Calc [Sundman et al. CALPHAD 1987;9:153] . Different coupling strategies between the microstructure model and Thermo-Calc have been developed, in particular a computationally-efficient direct coupling using the TQ-interface of Thermo-Calc . After testing the accuracy of the model with respect to curvature calculation, comparisons are made with predictions obtained with the marginal stability theory, a one-dimensional front-tracking method and two-dimensional phase-field simulations of dendritic growth in binary alloys. The model is then used to describe the formation of several grains in an Al–1%Mg–1%Si alloy, as a function of the heat extraction rate and inoculation conditions. It is shown that the model is capable of reproducing the transition between globular and dendritic morphologies.