Modeling and Simulation of Microstructure Evolution in Solidifying Alloys (Mathematics and Its Applications) [1 ed.] 1402078315, 9781402078316, 9781402078323

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MODELING AND SIMULATION OF MICROSTRUCTURE EVOLUTION IN SOLIDIFYING ALLOYS

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MODELING AND SIMULATION OF MICROSTRUCTURE EVOLUTION IN SOLIDIFYING ALLOYS

by

Laurentiu Nastac

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

CD-ROM available only in print edition eBook ISBN: 1-4020-7832-3 Print ISBN: 1-4020-7831-5

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Dedication

To my lovely wife, Mihaela To my fantastic kids, Gabriel and Michael To others...

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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Contents

1. Length-Scales and Generations of Modeling Methodologies for Predicting the Evolution of the Solidification Structure 1.1 Introduction 1.2 Length Scales in Modeling of the Solidification Structures 1.3 Generations of Modeling Techniques, Modeling Capabilities and Limitations

1 1 1 2

2. Deterministic Macro-Modeling: Transport of Energy, Momentum, Species, Mass, and Hydrodynamics During the Solidification Processes 5 2.1 Introduction 5 2.2 A Macroscopic Model for Calculating Energy, Momentum, Mass 6 and Species Transport 11 2.3 References Appendix: Methods for Coupling HT-SK Models 13 A2.1 Introduction 13 13 A2.2 HT-SK Model 13 A2.2.1 HT Model A2.2.2 Solidification Kinetics Model 15 15 A2.2.3 Stability Criterion 16 A2.2.4 Coupling of HT-SK Models

3. Deterministic Micro-Modeling: Mathematical Models for Evolution of 23 Dendritic and Eutectic Phases 3.1 Introduction

viii

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys 3.2 A Microscopic Model for Predicting the Evolution of the Fraction of Solid 3.3 Theoretical Analysis 3.3.1 Comparison between Hemispherical and Parabolic Growth 3.3.2 Comparison between Calculated and Experimental Growth Velocities of Dendrite Tip for Succinitrile 3.4 References

4. Stochastic/Mesoscopic Modeling of Solidification Structure 4.1 Introduction 4.2 Mesoscale Model for Dendritic Growth 4.3 Solution Methodology 4.4 Algorithm 4.5 Results and Discussion 4.6 References 5. Solute Transport Effects on Macrosegregation and Solidification Structure 5.1 Analytical Modeling of Solute Redistribution during Unidirectional Solidification 5.1.1 Introduction 5.1.2 Mathematical Formulation and Analytical Solution 5.1.3 Model Validation 5.1.4 Size of Initial Transient Region 5.1.5 Solid/Liquid Interface Instability of Dilute Binary Alloys 5.2 Numerical Modeling of Segregation 5.3 References 6. Micro-Solute Transport Effects on Microstructure and Microsegregation 6.1 Introduction 6.2 Dendrite Coherency and Grain Size Evolution 6.3 Deterministic Modeling of Microsegregation 6.3.1 Introduction 6.3.2 Models based on the “Closed System” Assumption 6.3.3 An analytical Model for Estimation of Microsegregation in Open and Expanding Domains 6.3.4 Partition Coefficient Evaluation 6.3.5 Predictions of Microsegregation in commercial alloys 6.3.6 Microsegregation Index (MSI) 6.4 Deterministic Modeling of Secondary Phases

23 26 26 26 28 29 29 32 34 40 41 51

53 53 53 54 58 60 61 65 72

75 75 75 78 78 79 82 91 95 97 99

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys 6.5 References 7. Probabilistic (Monte Carlo) Modeling of Solidification Structure 7.1 Fourth Generation of Solidification Modeling 7.2 Results and Discussion 7.3 References Appendix: Monte Carlo Program 8. Modeling and Simulation of Solidification Structure in Shaped and Centrifugal Castings 8.1 Shaped Castings 8.1.1 Prediction of Grain Structure and of Columnar-toEquiaxed Transition in Steel Castings 8.1.2 Gray-to-White Transition in Cast Iron 8.1.3 Modeling of Solidification Structure in Al-based Alloy Castings 8.1.4 Modeling of Solidification Structure in RS5 Alloys 8.2 Centrifugal Castings 8.2.1 Introduction 8.2.2 Model Description 8.2.3 Results and Discussion 8.2.4 Summary of the Parametric Studies 8.3 References 9. Modeling and Simulation of Ingot Solidification Structure in Primary and Secondary Remelt Processes 9.1 Introduction 9.2 Description of a Modeling Approach for Simulation of Remelt Ingots 9.2.1 A Deterministic Macroscopic Model for Calculation of Mass and Energy Transport in Cast Ingots 9.2.2 A Stochastic Mesoscopic Model for Simulation of Structure Evolution in Solidifying Ingots 9.2.3 Computational Aspects for Modeling of Remelt Ingots 9.2.4 Primary and Secondary Dendrite Arm Spacings in Commercial alloys (Deterministic Modeling) 9.2.5 A Stochastic Model for Modeling Secondary Phases During Solidification of Alloy 718 Ingots 9.3 Simulation Results for Some Commercial Applications 9.3.1 Modeling Parameters 9.3.2 Global Comparison of VAR, ESR, and PAM Processes

ix

107 109 109 111 112 113

115 115 115 117 128 128 136 136 138 143 145 148

151 151 153 153 156 164 165 166 170 170 173

x

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

9.3.3 VAR Process Modeling 9.3.4 ESR Process Modeling 9.3.5 PAM Process Modeling 9.3.6 Process Optimization 9.3.7 Alloy Systems and Solidification Maps 9.3.8 Prediction of Primary and Secondary Dendrite Arm Spacings 9.3.9 Stochastic Modeling of Secondary Phases 9.3.10 Experimental Technique for Composition Measurements and for Estimating the Secondary Arm Spacing in Ti-17 Alloy 9.4 References Appendix: Model Analysis A9.1 Sensitivity Analysis A9.1.1 Melt Rate Effects A9.1.2 Mesh Resolution - Columnar Growth A9.1.3 Mesh Resolution - Equiaxed Growth A9.1.4 Time Step A9.1.5 Impingement Factor - Columnar Growth A9.1.6 Impingement Factor - Equiaxed Growth A9.1.7 Nucleation - Columnar Growth A9.1.8 Nucleation - Equiaxed Growth A9.1.9 Summary A9.2 Grain Growth Analysis A9.2.1 Growth Parameter m - Columnar A9.2.2 Growth Parameter n - Columnar A9.2.3 Time Step A9.2.4 Growth Parameters for High Melt Rate A9.2.5 Growth Parameters for Base Melt Rate A9.2.6 Summary A9.3 CET Analysis A9.3.1 Melt Rate A9.3.2 Thermal Gradient A9.3.3 Nucleation - Columnar A9.3.4 Nucleation - Equiaxed A9.3.5 Time Step A9.3.6 Comparison with Experiments A9.3.7 Summary

175 177 181 183 184 188 192 203 207 211 211 211 212 212 212 218 219 219 222 222 226 226 227 227 230 232 233 234 235 235 237 238 238 238 245

10. Practical Techniques with Simulation Examples for Controlling the 247 Solidification Structure 10.1 Introduction 247 10.2 Electromagnetic Stirring 248

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys 10.2.1 Mathematical Formulation 10.2.2 Solution Methodology and MHD Model Validation 10.2.3 Results and Discussion for PAM-Processed Ti-6-4 Ingots (Experiments and 3D Computed Results with and without EMS) 10.3 Micro-Chilling in Steel Castings 10.3.1 Use of Steel Powder (Micro-Chills) for Efficient Superheat Removal 10.4 Ultrasonic Vibration 10.4.1 Ultrasonic Vibration Effects in Fluids 10.4.2 Modeling of Ultrasonic Vibration in Fluids 10.5 Modeling of Electromagnetic Separation of Phases to Produce In-Situ Composites 10.6 References

xi

249 251 252 253 259 259 262 262 264 265 267

Subject Index

269

About the Author

281

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Preface

The purpose of this monograph was to cover both the fundamentals and the state-of-the-art techniques used in mathematical modeling and computer simulation of the microstructure evolution during the solidification processes. Selected simulation results published in the last decade on the evolution of the solidification structures (both macro- and microstructures) will be presented throughout the book. The book is intended for graduate students and seniors interested in the science and engineering of solidification technology. It can also be used as a reference book for engineers in industry as well as researchers in academia and research institutes. A considerable research effort was done during the last decade in this area. Although a significant number of technical papers on solidification structure modeling were published in technical journals and conference proceedings in the last decade, just few book chapters have attempted to provide a systematic introduction to the modeling and simulation of solidification structure evolution. This book describes in detail some of the most commonly recognized state-of-the-art techniques in this field. The aim of the present book is to describe in a clear mathematical language the physics of the solidification structure evolution of cast alloys. The concepts and methodologies presented here for the net-shaped casting and the ingot remelt

xiv

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

processes can be applied, with some modifications, to model other solidification processes such as welding and deposition processes. Modeling and simulation of solidification structure evolution requires complex multi-scale computational areas, from computational fluid dynamics macroscopic modeling through mesoscopic to microscopic modeling, as well as strategies to link various length-scales emerged in modeling of microstructural evolution. Another aim of the book is to provide simulation examples of the solidification structure modeling in some crucial commercial casting technologies as well as to provide practical techniques for controlling the structure formation during the solidification processes. The simulation codes (PC versions) for 2D (includes fluid flow computations) and 3D mesoscale dendrite growth models (for 2D and 3D simple casting geometries) written in Visual Fortran 90 are provided in a CD attached to the book. The reader can use the dendrite growth codes to compute and visualize on the computer screen the evolving dendritic morphologies for his choices of material and process parameters. Finally, the author would like to thank his friends and colleagues that made writing this book possible, particularly, Prof. Doru Stefanescu for being my mentor and Ph.D advisor and Dr. Roxana Ruxanda for her valuable comments on this book. Laurentiu Nastac

1

LENGTH-SCALES AND GENERATIONS OF MODELING METHODOLOGIES FOR PREDICTING THE EVOLUTION OF THE SOLIDIFICATION STRUCTURE

1.1 INTRODUCTION Process modeling has become a viable tool to optimize the casting and solidification processes and is currently being applied to the remelt processes. Solidification structure is usually generated as part of a casting, welding or remelt process. Very often the solidification structure is the final structure of the component or it is the main contributor to the mechanical behavior of the final component. Accordingly, the mechanical properties of the manufactured parts, which are a direct outcome of the solidification structure, can be tailored through a controlled solidification process.

1.2 LENGTH-SCALES IN MODELING OF SOLIDIFICATION STRUCTURES A comprehensive modeling approach to simulate solidification phenomena in these processes should include computations for macroscopic mass, heat transfer, fluid flow, electromagnetic, and species transport to

2

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

provide temperature, velocity, and concentration fields as outputs. From the macroscopic temperature field the pool profile and pool size, as well as the shape and size of the mushy region, can be determined. In addition, from the concentration field, macrosegregation-related defects can be obtained. The approach should also include microscopic computations to predict the evolution of microstructure in solidifying ingots. The micromodel should compute the grain size and columnar-to-equiaxed transition (CET), as well as microsegregation. Lastly, the approach should also include computations at the dendrite tip length scale (mesoscopic scale) for prediction of dendritic morphology and microsegregation patterns. Figure 1-1 shows the length scales for this modeling approach.

1.3 GENERATIONS OF MODELING TECHNIQUES, MODELING CAPABILITIES AND LIMITATIONS To simulate the microstructure evolution during the solidification processes several modeling techniques were developed in the last 50 years. Their general classification is presented in Table 1-1.

Chapter 1. Length Scales and Generations of Modeling Methodologies

3

The first generation of modeling was started during the middle of this century, simultaneously with computer developments. It is a pure deterministic approach and it can be used to qualitatively predict the final solidification structure. The scale is 1 mm-1 m (macroscale). The second generation of modeling includes the calculation of solid fraction evolution at the micro-scale based on the solidification kinetics approach. This is a totally deterministic approach. The scale is (mesoscale). It cannot accurately predict the formation of the solidification structure. The second generation of computer models (so called macro transport-solidification kinetics codes) is commonly used to: (1) evaluate micro- and macro-segregation, (2) predict microstructural features in terms of amount and size only, (3) predict location and amount of shrinkage and gas pores, and (4) estimate of material properties. The third generation of modeling involves the calculation of structure evolution (including the calculation of the fraction of solid evolution) at the micro-scale based on probabilistic approaches. The transport of mass and energy is calculated at the macro-level with deterministic models. Because of geometrical complexity, the dendrite tip and interdendritic microsegregation is probabilistically calculated. The overall approach is stochastic. The scale is (microscale). The third generation of computer models is composed of two main components: (1) a deterministic macroscopic approach for macroscopic calculations and (2) a stochastic microscopic approach. This generation of modeling is applied to calculate both the formation of the solidification structure (amount, size, and morphology) and the fraction of solid evolution. The fourth generation of modeling involves the calculation of structure evolution at the micro-scale based on the probabilistic/stochastic approaches. The transport of mass and energy is calculated at the micro-level with a “direct” Monte Carlo approach. The overall approach is probabilistic. The scale is (microscale). The significance of the probabilistic approaches is that the simulated structures can be directly compared with the actual structures from experiments at two different scales: grain characteristics can be visualized at the macro-scale, while the amount, size, and distribution of secondary phases can be viewed at the micro-scale. A description of modeling techniques and their predictive capabilities is the aim of the present book. A summary of predictive capabilities is presented in Figure 1-2. Also, some recent developments and results obtained with the third and fourth generations of modeling techniques are summarized in this book.

4

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Another goal of this book is to demonstrate that probabilistic approaches are comprehensive and more accurate than deterministic approaches in predicting the solidification characteristics of various processed alloys. The major problem of deterministic models is shown in Figure 1-3, where it can be seen that the geometry of the micro-volume element in deterministic models is incapable of describing adequately the physics involved in nucleation and growth of microstructures. On the contrary, the macrovolume element in the stochastic/probabilistic models is usually divided in cubic micro-meshes with a typical micro-mesh size of thus, the mathematical models involved at this scale can correctly be solved and then linked with the macro-models. At present, the probabilistic models are mature enough to be used effectively by the manufacturing industry for process development as well as parametric design and optimization studies on microstructure management and control.

2

DETERMINISTIC MACRO-MODELING: TRANSPORT OF ENERGY, MOMENTUM, SPECIES, MASS, AND HYDRODYNAMICS DURING THE SOLIDIFICATION PROCESSES

2.1 INTRODUCTION

Understanding the precise role of each flow mechanism and their effects on heat, mass, and solute transport involves solving the conservation equations for mass, momentum, energy, and species in the geometry of interest. This is the first step required to predict the phase evolution and the associated phenomena in solidifying alloys such as segregation and shrinkage. The magnitude of thermosolutal convection of each particular system can be estimated by using the thermal and solutal Rayleigh numbers:

6

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where the subscripts T and S denote solutal and thermal quantities, respectively, g is the acceleration due to gravity, is the characteristic length scale, v is the kinematic viscosity, is the thermal diffusivity, is the coefficient of thermal expansion, is the coefficient of solutal expansion, is a characteristic temperature difference, and is a characteristic concentration difference. The Rayleigh number compares buoyancy forces to viscous forces. For small Rayleigh numbers, viscous forces should dominate over buoyancy forces, and the conduction regime is maintained with small and stable disturbances in the fluid. Negligible thermosolutal convection effects on macrosegregation should be encountered in this regime. Above a critical value for the Rayleigh number, buoyancy forces become important, disturbances grow, and a convection regime is established. Here, the effects of thermosolutal convection on macrosegregation become significant. For a binary alloy-system, the critical Rayleigh number is about 2000. Below this value, insignificant macrosegregation occurs [3]. In this chapter, a comprehensive macroscopic model for calculating energy, momentum, mass, species transport in solidifying alloys, is described.

2.2 A MACROSCOPIC MODEL FOR CALCULATING ENERGY, MOMENTUM, MASS, AND SPECIES TRANSPORT Using the assumptions of Newtonian, incompressible, and laminar flow, the governing equations for macroscopic transport in cylindrical coordinates (2-D axisymmetric geometry) are as follows [3]: (1) Conservation of mass:

(2) Conservation of momentum:

Chapter 2. Deterministic Macro-Modeling

where u and w are the velocity components, P is the pressure, average density of the control volume, is the viscosity, and friction drag sources in r and z directions, respectively, and buoyancy term given by the Bousinesq approximation as [4-8]:

7

is the are the is the

where is the average liquid concentration, is the reference temperature (equal to the liquidus temperature of the alloy, and is the reference concentration (initial melt concentration). The friction drag sources and are related to the rheology of the system. For equiaxed solidification, these sources can be calculated as follows:

and

where M denotes either eutectic (e) or dendritic (D) solidification, and and are the solid velocities. For eutectic solidification, the drag coefficients are:

where and

is the shape factor for nonspherical particles, is the liquid fraction, is the volume-surface mean diameter. The Equations (2-6) are

termed Ergun’s equations [7, 9, 10] and are appropriate for systems in which the liquid phase flows through the porous mush in the two-phase solid-liquid region. For dendritic solidification, the coefficient in Eq. (2-5) is much smaller than and it is neglected. In this case, the coefficient (here in Eq.

8

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

(2-5) is typically calculated at the level of secondary dendrite arm spacing as [1,2, 11]:

where

and Here, is the fraction of solid, is the solid/liquid (S/L) interface velocity of spherical instabilities, is one half of the average spacing between instabilities (at time t), the initial instability radius at time

is the instability radius at time t,

is

is a stability constant of the order of

is the intrinsic volume average liquid concentration,

is

the undercooling, is the liquid diffusivity, is the Gibbs-Thomson coefficient, is the liquidus slope, and k is the partition ratio. The Equations (2-1)-(2-8) represent the continuum conservation equations for phase change and convection and constitute the relative motion model [12, 13]. For systems in which there is no relative motion between liquid and solid, e.g. dendritic systems after the coherency, a viscosity function based on the solid fraction is used. In this case, the drag source terms defined in Eq. (2-5) are omitted. It will be shown later that the interphase transport terms in the energy and species equations (last two terms in Eq. (2-12) and Sc = 0 in Eq. (2-15)) are also dropped. This is the no-relative-motion model [12, 14]. For multiphase (eutectic, equiaxed dendritic, and columnar dendritic) solidification, both models should be used to describe the pressure drop in the mushy zone. Therefore, in such cases, a hybrid model as that described by Oldenburg and Spera [12] becomes appropriate. The hybrid model [12] uses permeability and viscosity functions to represent the complex behavior of the flow within the dilute mush (at very small solid fraction). At coherency-fraction of solid [1, 2, 11], the mush begins to behave more like a solid than a liquid. Coherency-fraction of solid is dependent on the morphology of the solid phase and can vary from about 0.05 (star dendrite) to maximum 0.8 (globulitic crystals) [11]. To ensure a smooth transition between the dilute and concentrated mush, two switching functions are used as follows [12]:

9

Chapter 2. Deterministic Macro-Modeling

where

is the coherency-fraction of solid. Thus, before coherency, the

interphase transport terms (last two terms in Eq. (2-12) and Sc = 0 in Eq. (215)) are turned off and the viscosity is related to the solid fraction through the following relation [12]:

After coherency, the interphase transport terms are turned on, and the permeability is a function of solid fraction through Ergun’s equations, i.e.,

and An alternative approach for the description of the rheology of the mushy zone was developed by Beckermann and Viskanta [14]. They used a twophase model that required the specification of both effective liquid and solid viscosities over the entire solidification interval. They made the following assumptions: (1) for very small solid fraction, i.e., the effective viscosities are described by:

and

and (2) for flow

through a rigid solid structure, such as packing of equiaxed crystals: and The transition in the macroscopic solid viscosity was obtained by using Krieger’s model (referred to in [14]) for the mixture viscosity of concentrated suspensions, i.e.:

and then solving for (3) The energy transport equation:

with

10

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

and

where T is the temperature, K is the thermal conductivity, is the density, is the specific heat, is the source term associated with the change of equiaxed and columnar phases, which describes the rates of latent heat evolution during the liquid/solid transformation, L is the latent heat of fusion,

and

are the solid fractions of the equiaxed, columnar,

and eutectic phases, respectively, is the liquid fraction, u and w are the superficial velocity components, and and are the solid velocities of the equiaxed and eutectic phases. The governing equations are written in terms of superficial velocities defined as:

Note that, for columnar solidification or after the occurrence of coherency in equiaxed solidification, the solid velocities are null. The continuum thermophysical properties (K, and are weighted by the solid fraction as follows:

(4) The species transport equation: Neglecting solid diffusion at the macro-scale level, the conservation of species can be written as follows:

with

Chapter 2. Deterministic Macro-Modeling

where

is the liquid diffusivity,

11

is the average liquid concentration,

is the average solid concentration, and is the average concentration within the elemental volume and is defined as:

where and are the concentration profiles in the solid and liquid, respectively, and can be obtained through the microsegregation model described in Ref. [11, 15]. The above governing equations are in the conservative form as recommended by Patenkar’s [16] for the numerical solution of heat, mass, and fluid flow problems. To solve these equations, it is required to know the competitive evolution of the solid fractions encountered in the present solidification system, i.e., eutectic, dendritic equiaxed, and dendritic columnar structures. The solidification-kinetics models for the calculation of the evolution of these solid fractions are presented in Chapter 3.

2.3 REFERENCES 1. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 4061-4074, 1996. 2. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 4075-4083, 1996. 3. L. Nastac, Numerical Heat Transfer, Part A, Vol. 35, No. 2, 1999, pp. 173-189. 4. D. R. Poirier, and J. C. Heinrich, in the Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes-VI, T. S. Piwonka (ed.), TMS, pp. 227-234, 1993. 5. B. Gebhart, Buoyancy-Induced Flows and Transport, Hemisphere Pub. Corp., 1988. 6. J. Szekely, Fluid Flow Phenomena in Metals Processing, Academic Press, 1979. 7. J. Szekely, J. W. Evans, and J. K. Brimacombe, The Mathematical and Physical Modeling of Primary Metals Processing Operations, John Wiley & Sons, 1988. Edition, Oxford Science 8. D. J. Tritton, Physical Fluid Dynamics, Publications, 1988.

12

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

9. G. H. Geiger and D. R. Poirier, Transport Phenomena in Metallurgy, Addison-Wesley Publishing Company, 1973. 10. R. W. Fahien, Fundamentals of Transport Phenomena, McGraw-Hill Book Company, 1983. 11. L. Nastac, Simulation of Microstructure Evolution during Solidification Processes, Ph.D. Thesis, The University of Alabama, Tuscaloosa, 1995. 12. C. M. Oldenburg and F. J. Spera, Numerical Heat Transfer, vol. 21B, pp. 217-229, 1992. 13. W. D. Bennon and F. P. Incropera, International Journal of Heat and Mass Transfer, vol. 30, no. 10, pp. 2161-2170, 1987. 14. C. Beckermann and R. Viskanta, Applied Mechanical Reviews, vol. 46, no. 1, pp. 1-27, 1993. 15. L. Nastac and D. M. Stefanescu, Met Trans, vol. 24A, pp. 2107-2118, 1993. 16. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, 1980. 17. M. V. K. Chari and S. J. Salon, Numerical Methods in Electromagnetism (Academic Press, USA, 2000). 18. P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Edition (Cambridge University Press, UK, 1996). 19. K. A. Hoffmann, S. T. L. Chiang, M. S. Siddiqui, M. Papadakis, Fundamental Equations of Fluid Mechanics (Engineering Education System, USA, 1996). 20. Fluent 6.0 User’s Guide Supplement, Magnetohydrodynamics Analysis, Fluent Inc., March 2002, Lebanon, NH, USA. 21. D. M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Met. Trans., Vol. 21A, pp. 997-1005, 1990. 22. L. Nastac and D. M. Stefanescu, Micro / Macro Scale Phenomena in Solidification, HTD-Vol. 218/AMD- Vol. 139, ASME, Aneheim, CA, pp. 27-34, 1992. 23. D. M. Stefanescu and C. Kanetkar, in the proceedings of the State of the Art of Computer Simulation of Casting and Solidification Processes, H.Fredriksson editor, Les Edition de Physique, Courtaboeuf, France, pp. 255-266, 1986. 24. M. Rappaz M. and P. Thevoz, Acta Metall., vol.35, pp. 1487-1497, 1987. 25. M. Rappaz, International Materials Reviews, Vol.34, No.3, pp 93- 123, 1989. 26. Ph. Thevoz Ph., J. L. Desbiolles, and M. Rappaz, Met Trans, Vol. 20A, pp. 311- 322, 1989.

Chapter 2. Deterministic Macro-Modeling

13

APPENDIX: METHODS OF COUPLING HT-SK MODELS A2.1 Introduction

The present section is a discussion of possible schemes for coupling the macroscopic heat transfer (HT model) with the microscopic solidification kinetics (SK model). The main problem in coupling SK and HT codes consists of incorporating the latent heat evolved during solidification, which is calculated through the SK model, in the HT model. The advantage of the HT-SK codes is that, unlike other traditional methods, the solidification path described by the evolution of solid fraction is not imposed a priori. This is of major importance since physically, the solidification path is the result of solidification conditions, and therefore should not be imposed. For example, the solidus temperature is not a material constant and depends upon the solidification path. Unfortunately, HT-SK codes cannot be validated against analytical solutions. The only alternative is to validate them against experimental data. The local evolution of the thermal field in castings can be represented through cooling curves. From these curves information on undercooling, recalescence and local solidification time can be extracted through Computer Aided-Cooling Curve Analysis (CA-CCA) [21]. Such validation was already performed for various types of cast irons and aluminum alloys. For instance, for the case of eutectic lamellar graphite iron, at normal cooling rates, some intergranular compounds, especially carbides and ternary eutectics can be sometimes produced at the end of solidification. Only the HT-SK models allow prediction of such events. Traditionally, the end of solidification is predicted through CA-CCA by determining the time of occurrence of the second minimum on the first derivative curve. This is incorrect [2], and a 5-10% error may occur in the evaluation of the solidification time and of the solidus temperature. A2.2 HT-SK Model A2.2.1 HT Model

A 2D cylindrical coordinates (axisymmetric geometries) heat transfer code used in this work is based on the Control Volume Method. The energy equation can be written as:

14

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where T is the temperature, t is the time, L is the latent heat of fusion,

is

the solid fraction, is the rate of latent heat evolved, and K, and are the thermal conductivity, density, and specific heat as functions of temperature, respectively. In the case of the Specific Heat Method (SHM) one has to include the evolution of latent heat in a modified specific heat term, as follows:

Solving these equations requires knowledge of initial temperature and boundary conditions. Details can be found in [22]. For higher values of the Biot criterion in the mold, i.e. for higher cooling rates, sometimes is a higher risk to omit the latent heat evolved during solidification. Therefore, the Temperature Recovery Method (TRM), schematically shown in Fig. 2-1, can be applied in the modeling of solidification in order to correctly recover the latent heat, especially for the higher cooling rates existing at the metal-mold interface.

In this model, for the first time step, the TRM was used. The first temperature under the equilibrium temperature, TE, which is is calculated from Eq. (2-17) without including the latent heat. This is accomplished by writing an integral energy balance for an elemental volume:

Chapter 2. Deterministic Macro-Modeling

15

From this equation the corrected temperature, is calculated as a function of TE, and Then, the temperature in the volume element is reset as A2.2.2 Solidification Kinetics Model

The heat source term

The term

in Eq. (1) can be calculated as:

can be calculated with appropriate nucleation and growth

laws as follows:

where N is the number of grains per unit volume (grain density), R is the grain radius and V is the growth velocity of the grain. It can be expressed as:

where the growth constant, can be either calculated or determined from experiments. The interface temperature, can be calculated from the overall energy balance within a grain of spherical shape [23]. Eq. (2-21) is valid under the same assumptions as proposed in [21]. It has the advantage that it accounts for the grain impingement, being weighted by the factor Then, solidification kinetics described by Eqs. (220)–(2-22) are coupled to the heat transfer equation described by Eq. (2-17). A2.2.3 Stability Criterion

Stability requirements are always associated with the explicit numerical analysis. The general stability criterion for cylindrical coordinates is derived

16

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

in Appendix A in [22]. When the Control Volume Method (CVM) with the heat source term in the implicit form is used, i.e. Specific Heat Method (SHM) or Enthalpy Method (EM), the required stability criterion is given by Eq. (A12) in [2]. When the heat source term is used explicitly, i.e. Latent Heat Method (LHM) or Micro-Enthalpy Method (MEM), the temperature derivative term is introduced as follows:

Inserting Eqs. (10) and (11) in (12) and then differentiating, the following equation is obtained:

Finally, the stability criterion is obtained by introducing Eq. (2-24) in Eq. (A12) from Appendix A in [22]:

where the terms L, R, U, and D are described in Appendix A in [22]. This stability criterion can be used not only with CV-FDM, but also with FEM forward difference or explicit Euler scheme (conditionally stable). A2.2.4 Coupling of HT-SK Models

The coupling between the macroscopic heat flow and the microscopic growth kinetics can be achieved through various schemes. Basically, the two different coupling schemes which are illustrated in Fig. 2-2 can be used. The most straightforward methods are those described in Fig. 2-2a. The Latent Heat Method has been used to calculate the solidification of various cast irons and Al alloys [21, 23]. The Enthalpy Method can be used in a similar manner. Formulating Eq. (1) with FDM or FEM, the variation of fraction of solid, between t and (time increment at macro level) at all nodes is computed based on the microscopic model of solidification, Eq. (2-21). The variation or can be derived explicitly or implicitly from macroscopic heat flow equations, while the variation is given explicitly at each node, at time t, in order to compute the new temperature field

Chapter 2. Deterministic Macro-Modeling

17

The source term Eq. (2-20), in the heat flow equation is coupling directly the heat transfer and solidification kinetics models. The disadvantage of this coupling scheme is that the time step increment necessary to solve the heat flow equation is limited by the microscopic phenomena. It has to be much smaller than the recalescence period in order to properly describe the microscopic solidification and to obtain a better accuracy in the prediction of local solidification times.

Eq. (2-25) can be successfully used, not only to satisfy the required stability condition for the explicit schemes, but also to correctly describe the microstructural features. Unfortunately, this condition is more severe then the classic stability criterion for explicit schemes (see Eq. (A12) in [22]). The Micro-Latent Heat Method (MLHM), and the Micro-Enthalpy Method, illustrated in Fig. 2-2b are based on the LHM and EM, respectively. The MEM has been first introduced in [24]. The MLHM has been presented for the first time in [22]. The essence of the MLHM and of the MEM is to

18

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

compute heat flow and microstructural evolution at two different scales. At the macro-level the heat flow is computed through one of the standard equations, without including the source term, A large time step, obtained from the general stability criterion, is used. Thus, the macro-enthalpy change, or the macro-temperature chance, can be obtained. Each of the macro-time steps is then divided into a number of small time steps at the micro-level, Again, the stability criterion, Eq. (2-25), is used. Once the variations of enthalpy, or temperature, at all cells are known, the solidification path can be independently computed. Assuming that the heat removal is made at a constant rate during the macro-time steps the enthalpy change, or temperature change, at the microlevel, are obtained after including the source term i.e. nucleation and growth kinetics. Finally, the new micro-temperature, and microstructural features can be obtained. The main advantage of this two time-step procedure is saving of computational time (CPU). Although the basic mechanisms of nucleation and growth are taken into consideration through solidification kinetics, the computing time is only 10% longer than that for standard heat flow calculation [25, 26]. Both MLM and MEM can be further modified as shown in Fig. 2-3. The macroscopic part remains unchanged. The microscopic part is changed. Maintaining the same assumption that the heat extraction per unit volume is constant during one macro-time step, SK calculation can be accomplished by simply integrating the microscopic models into the macroscopic heat flow calculation. The macro-variations or are not divided by the number of micro-time steps. Since the time step given by the classic stability criterion is small, it is not necessary to divide the heat extraction rate into other small values in order to maintain or to improve the convergence of the solution. The Specific Heat Method assumes an equivalent specific heat, Eq. (2), which requires precludes the possibility that

insuring that

This requirement

takes both negative and positive values,

required to predict recalescence. As shown in Fig. 2-4 some minor differences exist when using different coupling methods. MLHM and LHM are very close. Typically, HT-SK models produce larger maximum undercoolings than measured experimentally. Consequently, since LHM shows a smaller maximum undercooling than the other two methods, it can be considered more accurate. Also, since LHM does not include the

Chapter 2. Deterministic Macro-Modeling

19

additional assumption on the heat extraction, it is mathematically more accurate.

An overall comparison between the predictions of different parameters for the three basic coupling methods and the experiment is shown in Table 21, where TER is the temperature of eutectic recalescence and is the maximum undercooling. It is evident that all three methods give very close results. Nevertheless, as shown in Table 2-2, the CPU-time for the LHM scheme is about five times larger than that for MLHM or MEM. The difference in CPU-time between the MMLHM or the MLHM and the standard heat transfer method without SK is only 3 to 6% for the explicit FDM formulation. The MLHM, MMLHM, MEM, or MMEM can be used, in order to save CPU-time, not only with explicit FDM or FEM formulation, but also with implicit formulation. The bouncing effect phenomenon, often

20

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

reported by other investigators [26] in HT-SK calculation was basically nonexistent when the correct stability criterion was used.

Summary. (i) The Latent Heat Method is the most accurate method that can be used in HT-SK codes. The drawback of this method is a much longer computational time. (ii) Assuming a constant heat transfer throughout the micro-solidification path, the Micro-Latent Heat Method and MicroEnthalpy Method can be used with enough accuracy instead of the Latent Heat Method or Enthalpy Method. For example, in the case of eutectic gray iron, for cooling rates between 0.5°C/s and 10°C/s a maximum error of 0.3% was calculated in the prediction of recalescence temperature and solidus temperature, with respect to the LHM. A max. 0.5% error in the prediction of recalescence rate, solidification time, and temperature of eutectic undercooling was also obtained. On the other hand, these methods decrease five times the CPU time. (iii) The Specific Heat Method, especially in its explicit form, should not be used to calculate the evolution of latent heat during phase change in conjunction with HT-SK codes. The main disadvantage is that it cannot predict recalescence during eutectic solidification and, consequently, it results in higher error in predicting microstructural features. (iv) A stability criterion for explicit schemes within the phase transformation region, which allows prediction of recalescence, was proposed. This criterion precluded the occurrence of bouncing effects.

Chapter 2. Deterministic Macro-Modeling

21

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3

DETERMINISTIC MICRO-MODELING: MATHEMATICAL MODELS FOR EVOLUTION OF DENDRITIC AND EUTECTIC PHASES

3.1 INTRODUCTION The most common alloys used in practice, such as aluminum alloys, magnesium alloys, superalloys, steel alloys, and titanium alloys usually solidify with a dendritic structure. Below is a detailed description of a comprehensive deterministic model that can be used to predict dendritic growth kinetics in such alloys.

3.2 A MICROSCOPIC MODEL FOR PREDICTING THE EVOLUTION OF THE FRACTION OF SOLID The coupling between macro-transport and solidification kinetics is accomplished through the fraction of solid evolution that is described at the microscopic scale [1, 2]. This must be done for dendritic columnar, dendritic equiaxed, and eutectic equiaxed growth. For dendritic growth, the model

24

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

developed by Nastac and Stefanescu [1, 3] is used. For dendritic growth (both columnar and equiaxed), the evolution of the fraction of solid is expressed as [1, 3]:

where

is the liquid fraction,

is the solid fraction of the phase p, p

denotes a dendritic columnar, a dendritic equiaxed, or an eutectic phase, the geometrical factor is 3 for dendritic equiaxed growth and 2 for dendritic columnar growth, is the average growth velocity of the envelope (see Eqs. (3-2)–(3-5) in the following paragraphs), is the equivalent dendrite envelope, is the shape factor of the dendrite envelope, is the average growth velocity of the solid instability,

is the radius of the solid

instability, and is the shape factor of the instability. With the present model, three grain growth morphologies can be simulated: equiaxed dendritic, columnar dendritic, and eutectic. Nucleation and growth competition of the three grain morphologies controls the distribution and amount of phases. For the case of columnar dendritic solidification, the growth kinetics of the dendrite tip, is calculated with [3, 5]:

where is the liquid diffusivity, is the Gibbs-Thomson coefficient, k is the partition ratio, is the solidification interval, is the S/L interface undercooling, d is the mesh size, and is the instantaneous cooling rate. For equiaxed dendritic solidification, the model developed by Nastac and Stefanescu [1, 2, 3, 6, 7] is applied. Thus, the growth velocity of the tip is described by:

25

Chapter 3. Deterministic Micro-Modeling

where m is the liquidus slope, liquid thermal conductivity,

is the density, L is the latent heat,

is the

is the liquid interface concentration, and

The melt undercooling for the system under consideration can be calculated based on the following definition/assumption:

where

is the equilibrium liquidus temperature,

volume-averaged liquid concentration,

and

is the intrinsic

is the bulk temperature

defined as the average temperature in the volume element.

and

are calculated with the microsegregation model described in Ref. [4, 8].

The growth of equiaxed eutectic grains is calculated by [8-11]:

Note that the Eq. (3-1) is valid until impingement of the growing grains occurs. The first term in Eq. (3-1) involves calculations at the dendrite length scale, while the second term in Eq. (3-1) includes calculations at the instability length scale and describes both formation and coarsening of instabilities [1, 3]. The position of the equivalent dendrite envelope, is calculated until where the final grain radius is:

Here, and are the volumetric and surface grain density, respectively, and x and z are the coordinates of the microelement within the macrosystem. Note that the solution of the envelope growth velocity in Eq. (3-1) requires solute calculations at the micro-scale level for both interface and intrinsic volume average liquid concentrations. They are obtained by a complete analytical solution of the diffusion field in both liquid and solid phases. The microsegregation model was developed by Nastac and Stefanescu and is described in detail in Ref. [4].

26

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

3.3 THEORETICAL ANALYSIS 3.3.1 Comparison Between Hemispherical and Parabolic Growth

Although there is a significant difference between the hemispherical approximation and the parabolic solution in terms of Péclet (Pe) number [13], small discrepancies are observed when growth velocities vs. either tip radius or solutal Péclet number are plotted. The results are presented in Fig. 3-1. The range of cooling rates used in this analysis is between 0.1 to (that corresponds to the variation of the Péclet number from 0.001 to 1.0 and of the tip radius from 0.2 to Usually, the range of growth velocities encountered in castings is between 0.1 to This corresponds to a variation of Péclet number from 0.001 to 0.50 and of tip radius from 1 to The maximum solutal Pe number used in the present model is 0.5. For this range of Pe numbers the error is max. 5%. Because the Ivantsov’s solution of the tip velocity gives tremendous difficulties in multi-scale coupling and in solving the interface liquid concentration in parabolic coordinates, the hemispherical approximation for the tip was adapted in the present model. 3.3.2 Comparison Between Calculated and Experimental Growth Velocities of Dendrite Tip for Succinonitrile

In many contemporary casting solidification models, dendrite kinetics is calculated assuming a parabolic dendrite tip, while the diffusion field is calculated using spherical coordinates. This is not consistent. A more correct approach would be to use a hemispherical dendrite tip in conjunction with spherical coordinates for diffusion calculation. However, there is an ongoing discussion on the relative merits of the parabolic tip over the hemispherical tip. To verify calculation accuracy or lack of it when using a hemispherical tip, a classic experiment performed on succinonitrile was selected [10]. This particular experiment was conducted isothermally, in a large bath (infinite domain). Since Eqs. (3-3) and where R is the tip radius, describe the non-isothermal solidification into a closed system, two changes were made in these equations. First, the intrinsic volume average concentration of the liquid phase was assumed to be equal to consistent with the infinite domain. Second, the interface liquid concentration was obtained from the hemispherical approximation [1]. The thermophysical parameters of succinonitrile used in calculations are listed in [12, 13, 14]. The growth velocity for succinonitrile-0.07 mole % impurity (assumed to be acetone) calculated with the present modified model is compared with experimental values in Fig. 3-2. The model compares favorably with the

Chapter 3. Deterministic Micro-Modeling

27

experimental data, in particular in the region of moderate velocities, which are typical for castings. The discrepancy shown in Fig. 3-2 at low undercooling may probably be diminished by including thermal convection calculations [15].

28

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

3.4 REFERENCES 1. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 40614074, 1996. 2. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 40754083, 1996. 3. L. Nastac, Simulation of Microstructure Evolution during Solidification Processes, Ph.D. Thesis, The University of Alabama, Tuscaloosa, 1995. 4. L. Nastac and D. M. Stefanescu, Met Trans, vol. 24A, pp. 21072118, 1993. 5. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, vol. 5, no. 4, pp. 391-420, 1997. 6. L. Nastac and D. M. Stefanescu, AFS Trans, Vol. 104, pp. 425-434, 1996. 7. L. Nastac and D. M. Stefanescu, in the Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes-VI, T. S. Piwonka (ed.), TMS, pp. 209-218, 1993. 8. L. Nastac, Mathematical Modeling of Equiaxed Dendritic Solidification-Second Generation of Computer Models, MS Thesis, The University of Alabama, Tuscaloosa, 1993. 9. D. M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Met Trans, vol. 21A, pp. 997-1005, 1990. 10. M. Rappaz, International Materials Reviews, vol. 34, no. 3, pp. 93123, 1989. 11. L. Nastac and D. M. Stefanescu, Micro/Macro Scale phenomena in Solidification, ASME, HTD-Vol. 218/AMD-Vol. 139, pp. 27-34, 1992. 12. J. Lipton, M. E. Glicksman, and W. Kurz, Met. Trans., Vol. 18A, 341-345, 1987. 13. W. Kurz and D. J. Fisher, Fundamentals of Solidification , 2nd ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1986 14. M. E. Glicksman, R. J. Schaefer, and J. D. Ayers, Met Trans, Vol. 7A, 1747-1759, 1976. 15. R. Ananth and W. N. Gill: J. of Crystal Growth, Vol. 108, pp. 17389, 1991.

4

STOCHASTIC/MESOSCOPIC MODELING OF SOLIDIFICATION STRUCTURE The stochastic/mesoscopic modeling is the most powerful and practical approach for simulating the evolution of microstructure during the solidification of castings. The description of a comprehensive stochastic mesoscopic model and typical results showing the capabilities of this model are presented in this chapter.

4.1 INTRODUCTION Dendritic growth is perhaps the most observed phenomenon in solidification of cast alloys. It is also the most studied phenomenon in solidification science. Nevertheless, because of its complexity, it is not yet well understood and further research would be required. In the last two decades, experimental techniques were developed to study dendritic solidification. The soundest experiments for studying the evolution of dendrite morphology in transparent materials were performed by Glicksman and his collaborators [2, 3] (see Fig. 4-1). Several analytical models starting with LGT’s model for equiaxed dendritic growth [4] (see complete list of references in [5]) were developed to calculate various microstructural features of solidifying materials including solidification interface morphology transitions (e.g., stable-to-unstable solid/liquid interface or planar-to-cellular-to-dendritic, columnar-to-equiaxed, etc.) and dendrite morphology parameters (e.g., dendrite tip radius, dendrite tip velocity, primary dendrite arm spacing, and secondary dendrite arm spacing). Also, numerical models were developed and used for simulation of dendritic growth in material processing. They include models based upon deterministic techniques such as “phase field method” or spline mathematics

30

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

approaches for simulation of dendrite morphology and microsegregation patterns, solidification-kinetics deterministic models for calculating the evolution of fraction of solid and dendritic grain size, and probabilistic/stochastic approaches such as Monte Carlo and Cellular Automaton (CA) techniques for simulating the evolution of dendritic morphologies [1, 6–18] (see also the list of references in [1, 6–18]. Other valuable tracking methods of the S/L interface, particularly deterministic approaches that can be used for modeling of dendritic growth are described in details in [19, 20]. The significance of stochastic/probabilistic approaches is that the evolution of simulated microstructures can be directly visualized and compared with the actual microstructures from experiments at two different scales: dendrite grain characteristics such as grain size and location and size of the columnar-to-equiaxed transition can be visualized at the micro-scale, while dendrite morphology (including dendrite tip, various dendrite arm spacings, microsegregation patterns) can be viewed at the mesoscale [9]. Stochastic approaches are mostly used because of their capabilities in simulating (a) the heterogeneous nucleation of grains which is continuous and of probabilistic nature; (b) the crystallographic effects, that is, the growth anisotropy (grain selection/preferential growth) and the probabilistic nature of grain extension; and (c) the nucleation and growth competition of various phases and morphologies (columnar and equiaxed grains, solidification defects, etc.).

Chapter 4. Stochastic/Mesoscopic Modeling

31

Phase Field Method is a popular approach that uses an implicit front tracking method to track a “diffuse” L/S interface. Although the curvature and anisotropy are directly included in the model, the method is computationally intensive and mesh dependent. Also, it would be difficult to account for all physical phenomena by using this method. The entropy form S and the evolution equation for were postulated in [22] as:

where s is the thermodynamic entropy density, is a function of the interface thickness is the phase field variable, e is the internal energy density, is the mesh size, and c is the concentration of the solute B in the solvent A. Examples of simulation results for Cu-Ni alloys are presented in Figure 4-2. Interesting features such as solute redistribution and growth competition can be seen in Fig. 4-2.

Level Set Method was developed for pure substances by Chen et al. [25]. This method uses an implicit Eulerian approach to track the sharp L/S interface. It requires curvature and anisotropy computations and it still computationally intensive but mesh independent. The equation of motion is [25]:

32

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where F is the speed function and is the level set function. At any time t, the front is at zero level set of The surface tension effects are shown in Figs. 4-3a and 4-3b for a hypothetical pure substance [25]. The initial seed was an irregular pentagon. The role of anisotropy can be revealed when comparing Figs. 4-3b (isotropic surface tension) and 4-3c (anisotropic surface tension). It can be seen from Fig. 4-3 that the surface tension has indeed a strong stabilizing effect of the L/S interface.

4.2 MESOSCALE MODEL FOR DENDRITIC GROWTH The mathematical representation of the dendritic solidification process of a binary alloy is considered in a restricted 2-D domain as shown in Fig. 4-4. Here, is the interface normal vector, is the mean curvature of the interface, and the curve represents the solid/liquid (S/L) interface which evolves in time and has to be found as part of the solution. The solidification of binary alloys is governed by the evolution of the temperature (T (x,y,t)) and concentration (C (x,y,t)) fields that have to satisfy several boundary conditions at the moving S/L interface as well as the imposed initial and boundary conditions on the computational domain. The equations that describe the physics of the solidification process are presented in the following paragraphs [27, 28, 29].

Chapter 4. Stochastic/Mesoscopic Modeling

Temperature (T) in

33

(heat transfer equation):

where t is time, is the density, K is the thermal conductivity, is the specific heat, L is the latent heat of solidification, is the liquid fraction, is the solid fraction, is the convective velocity in y direction, and x and y are the domain coordinates.

Concentration (C) in In the liquid phase

where

and

respectively.

(solute diffusion equation):

are the interdiffusion coefficients in the liquid and solid, in Eqs. (4-3) and (4-4) is used for studying

convection effects on dendritic growth and segregation.

34

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Local equilibrium at the S/L interface on interface):

(here, “*” means at

Solute conservation at the S/L interface:

where is the normal velocity of the interface and n denotes the normal to the S/L interface that is pointing into the liquid (see Fig. 4-4). The interface temperature (T * ) is defined as (assuming local equilibrium with both phases):

where

is the equilibrium liquidus temperature of the alloy,

is the

liquidus slope of the phase diagram, is the mean curvature of the S/L interface, is the Gibbs-Thomson coefficient, and is a coefficient used to account for growth anisotropy where is the growth angle (i.e., the angle between the normal and the x-axis) and is the preferential crystallographic orientation angle. In Eq. (4-8), second term in the right side is the constitutional undercooling and the last term in the right side is the curvature undercooling (that reduces the total undercooling at the dendrite tip, that is, has a stabilizing effect on the S/L interface). The interface temperature is also affected by the kinetic undercooling. The kinetic undercooling is not accounted for in this model since its effect becomes significant only at very high solidification velocities (i.e., in the rapid solidification regime).

4.3 SOLUTION METHODOLOGY The solidification process is governed by Eqs. (4-3) to (4-8) and a stochastic model for nucleation and growth. The numerical procedures for calculating the nucleation and growth, temperature and concentration fields as well as the growth velocity of the S/L interface during dendritic solidification are described in details below.

Chapter 4. Stochastic/Mesoscopic Modeling

35

A. Stochastic Model for Nucleation and Growth. The structure of the stochastic model is similar to that described in Ref. [1]. It consists of a regular network of cells that resembles the geometry of interest. The model is characterized by (a) geometry of the cell; (b) state of the cell; (c) neighborhood configuration; and (d) several transition rules that determine the state of the cell. In this work, the geometry of the cell is a square. Each cell has three possible states: “liquid”, “interface”, or “solid”. The selected neighborhood configuration is based on the cubic von Neumann’s definition of neighborhood, that is the first order configuration and it contains the first four nearest neighbors. Solidification behavior depends to a great extend on the transition rules. In this model, the change of state of the cells from “liquid” to “interface” to “solid” is initiated either by nucleation or by growth of the dendrites. At the beginning of the simulation, all cells are liquid, and their state index is set to zero. As nucleation proceeds, some cells become “interface” cells, and their index is changed to an integer larger than zero, n. The cells in contact with the mold wall are identified with a different reference index, m. The index is transferred from the parent cell to adjacent cells, as they become “solid” cells through growth. The integer accounts for the preferential growth of cubic crystals in the direction (for 2dimensions). For graphical representation, each integer has a color associated with it, and each cell is a pixel on the computer screen. Both crystallographic orientation and random location of the new dendrites are chosen randomly among 256 orientation classes that are the first 256 colors used for graphical representation. In 2-D calculations, the probability, that a newly nucleated dendrite has a principal growth direction in the range is given by

where takes into account the four-fold symmetry of the cubic crystal (see integral in Eq. (4-9)). The number of equiaxed and columnar dendrites, and that can nucleate in the volume of the liquid and at the surfaces (boundaries) of the mold (computational domain) during one time step, is calculated by using the nucleation site distributions, and respectively. Thus, assuming no solid movement in the liquid, nucleation rate is given by [1]:

where N is the number of nuclei, is the bulk undercooling, and is a nucleation parameter. The nucleation probabilities for “liquid” cells located

36

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

in the bulk of the liquid and at the surface of the mold nucleate during each time step are given by

where

and

and

to

are the number of cells in the bulk and at the mold

surface, respectively. During the time step calculation both the mold-surface cells and bulk cells are scanned and a random number, rand, is generated for each of them The nucleation of a “liquid” cell will occur only if or It is assumed that a nucleus formed at a particular location will grow based on the growth velocity of the S/L interface given by Eq. (4-7) and the neighborhood configuration rule previously described. As nucleation succeeds, a nucleated “liquid” cell will become an active “interface” cell (e.g., n or m > 0), and will grow until its solid fraction becomes one. Thereafter, the “interface” cell would capture the neighboring cells if a randomly generated number, rand, is smaller than the capture probability, defined as follows:

where takes values from to At the capturing time, the state index of the “interface” cell is transferred to the captured neighboring cells that will become “interface” cells. Then, the “interface” cell becomes a “solid” cell. Further, the same procedure is used until all “liquid” or “interface”cells become “solid” cells. B. Calculation of the Temperature Field. An implicit finite difference scheme (based on the successive over-relaxation method) is used to calculate the temperature of each cell as described by Eq. (4-3). This scheme does not impose any restriction on the time step used in calculations. An initial temperature higher than the equilibrium melting temperature of the alloy was assigned for all cells and convective boundary conditions (through the use of surface heat transfer coefficients) were used for surface cells. C. Calculation of the Time Step. Time step used in calculations is given by

Chapter 4. Stochastic/Mesoscopic Modeling

37

where a is the mesh size (uniform and constant for both x and y directions) and is the maximum growth velocity obtained by scanning the growth velocities of all “interface” cells during each time-step (see Eq. (414)). Equation (4-13) satisfies the conditions for both the explicit tracking scheme of the moving S/L interface and for explicitly calculating the concentration fields in both the solid and liquid phases. D. Calculation of the Concentration Fields in the Liquid and Solid Phases. An explicit finite difference scheme is used for calculating the concentration fields in the liquid and solid phases. This scheme is stable for the time-step condition shown in Eq. (4-13). Zero-flux boundary conditions were used for cells located at the surface of the geometry. The solution algorithm includes the “interface” cells by multiplying the concentration in the liquid by the liquid fraction and the concentration in the solid by the solid fraction of the particular interface cell. Also, during each time-step calculation and for each “interface” cell, the previous values of the liquid and solid concentrations are updated to the current values of the interface liquid and solid concentrations calculated with Eqs. (4-6), (4-7), and (4-14) to (4-17). E. Calculation of the Growth Velocity of the S/L Interface. For “interface” cells, the values of the interface velocities in the x and y directions are obtained from Eqs. (4-6) and (4-7). For the finite difference form is as follows (similar for

F. Calculation of the Fraction of Solid Evolution. Knowing the velocity components in both x and y directions, the solid fraction increment is calculated with [6, 7]:

Then, the solid fraction and growth velocity Eq. (4-15) as:

are computed based on

38

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where superscripts “n” and “o” denote new and old solid fraction values, respectively. G. Calculation of the Interface Liquid Concentration. from Eq. (4-8) as

The procedures for calculating

and

is calculated

are described below.

H. Calculation of the Interface Curvature. The average interface curvature for a cell with the solid fraction is calculated with the following expression:

where N is the number of neighboring cells. In the present calculations, N = 24, that includes all the first and second order neighboring cells. Equation (4-18), a modification of the method proposed in Ref. [21], is a simple counting-cell technique that approximates the mean geometrical curvature (and not the local geometrical curvature). The values of the curvatures calculated with Eq. (4-18) vary from a maximum of 1/a to zero for convex surfaces and from zero to a minimum of –1/a for concave surfaces. The curvature can also be computed with [26-29]:

where the unit normal

is derived from a normal vector

which is the gradient of From Eq. (4-19), the curvature in nonconservative form can be rewritten as [25]:

39

Chapter 4. Stochastic/Mesoscopic Modeling

where the partial derivatives of

are computed using centered finite

difference approximations. is computed from the first 8 neighboring cells only at gridpoints adjacent to the S/L interface. For

3-D

mesoscale

modeling,

the

3-D

curvature

in

nonconservative form can be calculated as:

where

is computed form the first 26 neighboring cells only at

gridpoints adjacent to the S/L interface. I. Calculation of the Anisotropy. The anisotropy of the surface tension (see Eq. (4-8)) is calculated from [6, 7]:

where is calculated with Eq. (4-9) and accounts for the degree of anisotropy. For four-fold symmetry, [22]. J. Reduction in the Mesh Anisotropy. Because of mesh anisotropy dendrites will grow aligned with the axis of the mesh or at 45 degrees independently of the initial crystallographic orientation, which can be called anisotropy in growth direction. Furthermore, dendrites aligned with the axis would have narrower tips than predicted by classic theories, which can be termed anisotropy in growth kinetics. To reduce the mesh anisotropy, Sanchez and Stefanescu [32] proposed the following formulation:

40

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

4.4 ALGORITHM The flowchart for the stochastic model is presented in Fig. 4-5, where T is the new temperature distribution, is the equilibrium liquidus temperature of the alloy, C and the new concentration distribution, is the current time step used in calculations, and IC and BC stand for initial and boundary conditions. The simulation software was written in Visual Fortran 90. The output of the model consists of screen plotting at any chosen time of C, T, or color indexes of all cells. Also, the final values of C, T, and color indexes of all cells are saved at the end of computations on the computer disk.

Chapter 4. Stochastic/Mesoscopic Modeling

41

4.5 RESULTS AND DISCUSSION Thermophysical properties of the alloys used in simulations are presented in Table 4-1. The grid size of the domain is unless otherwise specified. This grid size is fine enough to approximately resolve the dendrite tip radius that, in the present solidification conditions, is typically larger than Newton cooling boundary conditions were applied at the boundaries of the computational domain for multidirectional solidification simulations. For directional solidification simulations, Newton cooling boundary condition was applied only to the bottom of the computational domain. The surface heat transfer coefficient and convective velocity used in the present simulations is

and

respectively, unless

otherwise specified. Zero-flux solute boundary conditions were applied at the boundaries of the computational domain (i.e., a closed system was assumed). The initial melt temperature was assumed to be the liquidus temperature of the alloy under consideration (i.e., no superheat), unless otherwise specified. Also, an initial concentration equal to was assumed everywhere on the computational domain.

First, the effects of constitutional undercooling on the solute redistribution and on dendrite morphology were analyzed. A constitutional undercooling parameter, defined as is used here. Thus, using the material properties in Table 4-1, we can calculate the parameter A as follows: for Al-7%Si, A = 280, for Pb-10%Sn, A = 52, for IN718-5%Nb, A = 57, and for Fe-0.6%C, A = 93. A comparison of simulated equiaxed dendritic morphologies for the above alloy systems (shown at approximately similar total solid fractions) is presented in Fig. 4-6 for a domain, where the legend indicates the solute concentration levels.

42

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Chapter 4. Stochastic/Mesoscopic Modeling

43

Coarsening of secondary dendrite arms and growth of tertiary dendrite arms (or branching of secondary arms) can be observed in Figs 4-6c and 4-6f for Pb-10%Sn and IN718-5%Nb alloys, respectively. The secondary and tertiary dendrite arm spacings of IN718-5%Nb alloy are a bit smaller than those of Pb-10%Sn alloy. Note also the evolution of C segregation in the liquid phase for the Fe-0.6%C alloy in Figures 4-6g and 4-6h. Overall, a small effect of the constitutional undercooling parameter (A) on the morphology of equiaxed dendrites was observed. Nevertheless, it was observed from time-evolution of these systems that alloys with a smaller A (e.g., Pb-10%Sn and IN718-5%Nb) have faster dendritic growth than alloys with larger A (e.g., Al-7%Si and Fe-0.6%C). The morphological evolution of a single columnar dendrite in IN7185%Nb alloy is presented in Fig. 4-7. Solidification starts first with dendritic growth until the tip of the dendrite reaches the top of the simulated domain. Then, coarsening and some branching of secondary dendrite arms take place concomitantly with the dendritic growth process. The effects of the variation (one order of magnitude variation) in the (i.e., surface tension) on the evolution of Nb solute redistribution patterns and dendritic morphologies in IN718-5%Nb alloy system are shown in Fig. 4-8. Insignificant branching (i.e., growth of secondary/tertiary dendrite arms) takes place for the case of higher In Fig. 4-9, the competition between nucleation and growth of multiple columnar dendrites assuming unidirectional solidification of IN718-5%Nb alloy is presented. The strong growth competition from the sample bottom (10 dendrites) to 1/3 of the sample height (5 dendrites) to the sample top (2 dendrites) can be observed.

44

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Simulated microstructure (dendritic morphologies and CET formation) and Nb segregation patterns in multidirectional solidification of IN718-5 wt.% Nb alloy are shown in Fig. 4-10. Here, the competition of equiaxed and columnar morphologies controls the formation of CET. Last, the simulated microstructure (columnar cellular/equiaxed dendritic morphologies and CET formation) in unidirectional solidification of IN718-5 wt.% Nb alloy is presented in Fig. 4-11. To study the effect of the heat extraction rate (i.e., increase in the G/V ratio, where G is the thermal gradient in the mushy region and V is the growth velocity) a one order of magnitude increase in the value of h was used (i.e., In this case, columnar cellular growth occurs as opposed with previous cases where columnar dendritic growth was observed to take place. Note also from Fig. 4-11 the sharp CET that takes place at approximately one half of the sample height where the thermal gradient in the mushy region is about 3000 K/m. The CET is sharp because of faster growth of equiaxed dendrites that have nucleated in the undercooled liquid ahead of the columnar front. A comparison of the curvature models described by Eqs. (4-18) and (419) is provided in Fig. 4-12 for a Pb-10%Sn alloy. The dimensional curvatures, plotted in Fig. 4-12 as a function of the total solid fraction in the computational domain, were computed as the average of all cell curvatures in the solid phase and then normalized by cell size. Although large

Chapter 4. Stochastic/Mesoscopic Modeling

45

differences exist between the two curvatures at the onset of solidification, both curvatures converge to the same value thereafter.

46

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Figure 4-13 shows the evolution of a simulated equiaxed dendrite and of Nb concentration in IN718 alloy. The curvature model described by Eqs. (419) and (4-20) was used to compute the curvature undercooling. Note the evolution of Nb segregation in the liquid phase for the IN718-5 wt.% Nb alloy in Figs. 4-13a to 4-14c. Simulated microstructures (columnar/equiaxed morphologies and CET formation) and Sn segregation patterns during the unidirectional solidification of Pb-10 wt.% Sn alloy cast in microgravity conditions [30] are presented in Fig. 4-14. Note from Fig. 4-14 the sharp CET that takes place at approximately 2/3 of the sample height where the thermal gradient in the mushy region is less than about 3000 K/m. The CET is sharp because of faster growth of equiaxed dendrites that have nucleated in the undercooled liquid ahead of the columnar front. The influence of convection on dendritic growth and segregation patterns in IN718-5 wt.% Nb is shown in Fig. 4-15a. A convective velocity, mm/s, was used in computations and only the left-side wall was cooled off Note from Fig. 4-15a that the columnar dendrites grow opposing the convective flow direction. This is because convective flow changed the temperature and concentration fields in the liquid around each growing dendrite. Thus, solute is depleted at the S/L interface (i.e., dendrite tip) and since smaller solute content gives higher growth velocity of the S/L

Chapter 4. Stochastic/Mesoscopic Modeling

47

interface, the dendritic growth will be preferentially in the upstream direction. Interestingly, the deflection of the dendrite growth direction is also controlled by the strong competitive growth of the primary arms and secondary arms which are growing preferrentially in the upstream direction, as shown in Fig. 4-15a. This phenomenon is described in details in [31]. Also, SCN dendrites (see experiments in [3] and Fig. 4-1b) not growing parallel to gravity are influenced by the asymmetric flow field. It was also demonstrated in [2] that the fluid flow direction with respect to the dendrite growth direction affects both the morphology and stability of dendritic crystals. Figure 4-15b shows the effect of growth morphologies (columnar vs. equiaxed) on Nb segregation during unidirectional solidification of IN718 alloy. As expected, segregation of Nb is reduced during equiaxed growth, at least during the initial transient. In practice, alloy innoculation for enhancing equiaxed solidification and reducing equiaxed grain size can be addressed to decrease alloy segregation in castings. The legends in Figs. 4-6 to 4-10 and 4-13 to 4-15 display solute compositions in both the liquid and solid phases using the following nondimensional quantities: and

Thus, the solute concentration in the liquid

phase, can be displayed simultaneously with the solute concentration in the solid phase, The legend in Fig. 4-15 represents either Sn composition or dendrite color indexes. The legend in Figs. 4-11, 4-14, and 4-15 shows 256 color indexes (as 16 classes, where each class contains 16 colors) that are used to display dendritic morphologies.

48

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Figure 4-16 presents a simulation of equiaxed solidification of Al4wt.%Cu alloy with the improved model [32] that accounts for the reduction in the anisotropy of the mesh in the crystallographic orientation and in growth kinetics. The simulation is performed in a square domain, with three nucleated grains with different crystallographic orientations. Note that each dendrite can grow in its preferred crystallographic growth direction. Figure 4-17 shows the cooling curve recorded at the center of the domain and the evolution of the total solid fraction of the entire simulation domain presented in Fig. 4-16. Typical features for equiaxed solidification such as undercooling, recalescence, and final temperature drop at the end of solidification are correctly predicted by the model.

Chapter 4. Stochastic/Mesoscopic Modeling

49

50

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The computer memory and CPU-time requirements in the present stochastic calculations are discussed briefly below. A RAM memory size of 120 bytes/cell is needed in the present stochastic calculations. For example, to simulate the solidification microstructure and segregation patterns of the geometry shown in Fig. 4-11 (10mm x 20mm), a grid resolution of 500x1000 mesh size) was used. The RAM and CPU-time required to obtain the simulated results presented in Fig. 4-11 are 60 Mbytes and about 6 CPU hours on a PC-150 MHz (about 30 CPU minutes on a PC2GHz), respectively. The CPU-time for the computations is very competitive when compared, for example, with computations based on the phase-field approach. To simulate a single equiaxed dendrite with the phase-field

Chapter 4. Stochastic/Mesoscopic Modeling

51

approach, 7500 CPU seconds on the Cray Y-MP4E/232 was required [22]. The CPU-time of the current computations is reasonably close to classical stochastic CA models for simulating dendritic/cellular grains [9, 24].

4.6 REFERENCES 1. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, Institute of Physics Publishing, Vol. 5, No. 4, pp. 391-420, 1997. 2. M. E. Glicksman, E. Winsa, R. C. Hahn, T. A. Lograsso, S. H. Tirmizi, and M. E. Selleck, Met Trans, Vol. 19A, pp. 1945-1953, 1987. 3. M. A. Chopra, M. E. Glicksman, and N. B. Singh, Met Trans, Vol. 19A, pp. 3087-3096, 1988. 4. J. Lipton, M. E. Glicksman, and W. Kurz, Met Trans, Vol. 18A, pp. 341345, 1987. 5. W. Kurz and D. J. Fisher, Fundamentals of Solidification, 3rd ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1989. 6. U. Dilthey, V. Pavlik, and T. Reichel, Mathematical Modeling of Weld Phenomena 3, Eds. H. Cerjak and H. k. D. H. Bhadeshia, The Institute of Materials, pp. 85-105, 1997. 7. U. Dilthey and V. Pavlik, Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes-VIII, Eds. B. G. Thomas and C. Beckermann, pp. 589-596, 1998. 8. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 27A, pp. 4061-4074 and pp.4075-4084; 1996, Met Trans, Vol. 28A, pp. 1582-87; 1997, AFS Trans pp. 425-34, 1996. 9. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, J. of Metals, TMS, pp. 3035, March 1998. 10. J. A. Spittle and S. G. R. Brown, Ada Metall. Vol. 37, No. 7, 1989, pp. 1803-10 and J. Materials Science, Vol. 30, pp. 3989-3994, 1995. 11. H. W. Hesselbarth and I. R. Goebel, Acta Metall., Vol. 39, No. 9, pp. 2135-43, 1991. 12. M. P. Anderson, D. J. Srolovitz, G. S. Crest, and P. S. Sahni, Acta Metall., Vol. 32, No. 5, pp. 783-91, 1984. 13. G. S. Crest, D. J. Srolovitz, and M. P. Anderson, Acta Metall., Vol. 33, No. 3, pp. 509-20, 1985. 14. M. Rappaz and Gh. A. Gandin, 1993 Acta Metall., Vol. 41, No. 2, 345-60, 1993. 15. N. H. Packard, Proceedings of the First International Symposium for Science on Form, University of Tsukuba, Japan, November 26-30, 1985, Eds. S. Ishizaka, Y. Kato, R. Takaki, and J. Toriwaki, KTK Scientific Publishers, 1987.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

16. H. Pang and D. M. Stefanescu, Solidification Science and Processing, Eds. I. Ohnaka and D. M. Stefanescu, TMS, 1996. 17. D. M. Stefanescu and H. Pang, Canadian Metallurgical Quarterly, Vol. 37, No. 3, pp. 229-240, 1998. 18. I. Steinbach, F. Pezzolla, B. Nestler, R. Prieler, G. J. Schmitz, and J. L. L. Rezende, Physica D, Vol. 94, pp. 135-147, 1996. 19. W. Shyy, H. S. Udaykumar, M. M. Rao, and R. W. Smith, Computational Fluid Dynamics with Moving Boundaries, Taylor & Francis, 1996. 20. H. S. Udaykumar and W. Shyy, Int. J. Heat and Mass Transfer, Vol. 38, No. 11, pp. 2057-2073, 1995. 21. R. Sasikumar and R. Sreenivisan, Acta Metall., Vol. 42, No. 7, pp. 23812386, 1994. 22. J. A. Warren and W. J. Boettinger, Acta Metall., Vol. 43, No. 2, pp. 689703, 1995. 23. L. Nastac, Numerical Heat Transfer, Part A, Vol. 35, No. 2, pp. 173-189, 1999. 24. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, Proceedings of the International Symposium on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, Eds. A. Mitchell and P. Aubertin, pp. 145-165; 1997, Proceedings of the Fourth International Special Emphasis Symposium on “Superalloy 718, 625, 706, and Derivatives”, Ed. E. A. Loria, pp. 55-66, 1997. 25. S. Chen, B. Merriman, S. Osher, and P. Smereka, J. Comp. Physics, Vol. 135, pp. 8-29, 1997. 26. D. B. Kothe, R. C. Mjolsness, and M. D. Torrey, RIPPLE: A Computer Program for Incompressible Flows with free surfaces, Los Alamos National Lab., LA-10612-MS, Los Alamos, NM, 1991. 27. L. Nastac, Acta Materialia, Vol. 47, No. 17, pp. 4253-4262, 1999. 28. L. Nastac, Pacific Rim International Conference on Modeling of Casting and Solidification Processes (MCSP-4), Ed. C. P. Hong, Yonsei University, Seoul, Korea, September 5-8, 1999. 29. L. Nastac, Proceedings of the “Modelling of Casting, Welding, and Advanced Solidification Processes IX”, Engineering Foundation, Aachen, Germany, August 20-25, 2000. 30. L. Nastac and S. Sen, Influence of Gravitational Acceleration on the Segregation and Solidification Structure of Dendritic Alloys, NASA proposal, NRA-98-HEDS-05, 1999. 31.] K. Murakami, T. Fujiyama, A. Koike, and T. Okamoto, Acta Metall., Vol. 31, pp. 1425-1432, 1983. 32. L. Beltran-Sanchez and D. M. Stefanescu, Proceedings of the “Modelling of Casting, Welding, and Advanced Solidification Processes X”, TMS, Destin, FL, USA, pp. 75-82, May 25-30, 2003.

5

SOLUTE TRANSPORT EFFECTS ON MACROSEGREGATION AND SOLIDIFICATION STRUCTURE Solute transport during solidifying alloys is by diffusion and convection. Analytical and numerical modeling of macrosegregation and its effects on the solidification structure are discussed in this chapter.

5.1 ANALYTICAL MODELING OF SOLUTE REDISTRIBUTION DURING UNIDIRECTIONAL SOLIDIFICATION 5.1.1 Introduction

The importance of investigating solute redistribution during the dilute alloy solidification is broadly discussed in the literature [1-12]. One of the most important applications of this investigation would be the mathematical modeling of the equiaxed and columnar solidification [3-5, 7, 8]. This includes the instability of the solid/liquid interface. There are several other mass and heat transfer processes that involve the calculation of solute redistribution during directional solidification. For instance, this is the case of the power-down process used for making turbine blades, high temperature gradient liquid metal cooling furnace [6], various laboratory and industrial directional solidification processes, continuously cast processes which include the remelting processes [7, 8], Czochralski crystal growth technique

54

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

and floating zone techniques used for producing ingot diameters less than 10 mm.

5.1.2 Mathematical Formulation and Analytical Solution

The equation that describes the transfer of solute in the liquid region during the unidirectional solidification with an axially moving boundary is:

Here, t is time, z is the axial coordinate in the moving coordinate system, is the liquid concentration, is the diffusion coefficient of solute in the liquid phase, and W is the solid/liquid interface velocity in the z-direction. Equation (5-1) is solved in a semi-infinite domain together with the following boundary and initial conditions:

where

is the initial liquid concentration, k is the equilibrium distribution

coefficient, and and are the interface liquid and solid concentrations, respectively. Note that the flux balance at the moving boundary (Eq. 5-2) is used to correctly describe the time-evolution of the interface solid (or liquid) concentration. Equations (5-1) to (5-5) represent the mathematical formulation of the transient unidirectional solutal transport in the liquid phase with an axially moving boundary. To be in line with the quasi-steady state theory, this formulation also assumes no convection (fluid flow), constant moving frame velocity, no diffusion in the solid phase (e.g., no back-diffusion) and local equilibrium at the solid/liquid interface (Eq. 5-5). The assumption related to

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

55

the absence of convection (or dominant diffusive solutal transport) is particularly valid for either process involving a horizontal moving frame velocity or microgravity experiments. The solutions for the liquid concentration profile, and interface solid concentration,

calculated at z = 0, are as follows [12]:

Here, erf and erfc are the error and complementary error functions, respectively. Rearranging terms and introducing the coordinate z = Wt in Eq. (5-7), where z represents the distance measure from the beginning of the sample, we found the same solution as that obtained by Smith, Tiller, and Rutter (Eq. (10) in [13]). The liquid concentration gradient at the solid/liquid interface is calculated as

56

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where is the time dependent concentration gradient at the solid/liquid interface. The transient and steady-state solutal boundary layer thicknesses, and are calculated with:

A “local boundary layer” can also be defined as:

where

is dimensionless time.

A comparison between

and

in Fig. 5-1 for k = 0.1. Small differences exist between

is shown and

at

any and k. However, at any and k, the ratio between the equivalent and local boundary layers for both the transient state and the steady state equals 2. Which boundary layer has a greater fundamental significance? Perhaps, based upon their definitions (e.g., Eq. (5-10) and (5-11)), the “local boundary layer” should be more correct than the “equivalent boundary layer” for studying local phenomena, such as interactions among diffusion fields of various dendrite tips or time evolution of various dendrite arm spacings.

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

57

The variation with time of the interface solid concentration is plotted in Fig 5-2 for different values of k. It can be seen that large differences exists between the transient and the quasi-steady solutions. The unsteady species transport tends to quasi-steady state with time, that is, for and The error between the unsteady state and quasisteady state calculations also decreases with time. The use of the conventional quasi-steady state solution for provides good accuracy (less than 5 % errors) (see Fig. 5-3). For instance, for k = 0.1,

at at and at The transient interface concentration gradient varies from at

to

at

The variation with time of the dimensionless interface liquid concentration gradient is plotted in Fig. 5-3 for different values of k. The absolute values of the interface liquid concentration gradients, can be found by multiplying the values of in Fig. 5-3 by The variation of the interface liquid concentration gradient during the unsteady state is significant, in particular for small k (k < 0.2). For instance, for k = 0.1, the interface liquid concentration gradient decreases with time (from until almost one order of magnitude. Also, the interface liquid concentration

58

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

gradient is positive for k > 1 and negative for k < 1, with and It is noteworthy to mention that for k > 1, the absolute value of the interface liquid concentration gradient decreases with time whereas for k < 1, it increases with time. At steady state, the higher k, the closer to one is

5.1.3

Model Validation

The validation of the analytical model described by Eq. (5-7) is performed against two well-performed experiments [20]: (1) Kagawa and Okamoto (14) used a floating zone melting technique to measure the redistribution of Si during the directional austenite-graphite eutectic solidification of Fe-C-0.56 wt.% Si alloys; and (2) Favier et al. [15] measured the redistribution of Bi during the directional solidification of Sn0.5 at.% Bi alloys by using the Mephisto-USMP1 microgravity experiments. The comparisons between the calculated and experimental results are presented in Figs. 5-4 and 5-5. The thermophysical parameters used in calculations are also shown in these figures. Si segregates negatively (k > 1) during the directional austenite-graphite eutectic solidification of Fe-C-0.56 wt.% Si alloys (see Fig. 5-4), while on the contrary Bi redistributes positively (k < 1) during the directional solidification of Sn-0.5 at.% Bi alloys (see Fig. 5-5). Both the and k are key parameters in modeling the solute redistribution during the directional solidification. They have opposite effects on the solute redistribution in the

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

initial transient state, that is, the higher

the smaller

59

and the higher k

the higher Therefore, results using at least two sets of values for k are presented in Figs. 5-4 and 5-5.

and

The calculated results compare closely with these well-performed experiments implying that: (i) the assumptions used in developing the mathematical formulation were correctly chosen, in particular the assumption related to the dominant diffusive solutal transport (i.e., noconvection) in the liquid phase; and (ii) the analytical solution presented in Eqs. (5-6) and (5-7) is accurate.

60

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

5.1.4 Size of the Initial Transient Region

The initial transition effects during the directional solidification in the absence of convection and solid diffusion occurs over a length that can approximately be calculated as:

where

is the length of the initial transient region and

is the time

to reach quasi-steady state. Equation (5-12) shows that both k and W have a destabilizing effect on the solid/liquid interface, that is, can be increased by decreasing both the value of k (related to the alloy type) and W. For example, for the case of Al-1.0 wt.% Cu, [3, 19] and [3], changes from Taking to 105 mm, when W varies from to From Eq. (5-12), the length of the initial transient region is inverse proportional with k, and therefore, for a

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

61

higher k a smaller time is required for reaching steady state. Furthermore, transposing the present theory to the unconstrained spherical growth theory (applicable for equiaxed solidification) it can be shown that the equiaxed grains would solidify during the initial transient region and the steady state growth region would rarely be attained. This is because the equiaxed grain size is usually less than and, as shown above, can reach In both the calculated and experimental results shown in Fig. 5-4 for FeC-0.56 wt.% Si alloy, steady-state is reached at approximately 20 mm from the base, while for Sn-0.5 at.% Bi alloy (Fig. 5-5), the initial unsteady state length is approximately 5 mm. Similar values for the length of the initial unsteady state are obtained with Eq. (5-12) for both cases presented in Figs. 5-4 and 5-5. 5.1.5 Solid/Liquid Interface Instability of Dilute Binary Alloys

The concept of constitutional undercooling (CS) [21] can be used to estimate the growth conditions where stability or instability can be expected for the solid/liquid interface of a dilute binary alloy during the unidirectional solidification at constrained velocity W. The CS criterion simply states that the presence of constitutional supercooling would correspond to morphological instability and its absence to morphological stability [2]. The appropriate equation that described mathematically the demarcation between the presence and the absence of constitutional supercooling is

where is the thermal gradient (see Refs. [1, 3, 22] for its definition) at the solid/liquid interface and is described by

Assuming that the initial unsteady interface concentration gradient calculated with Eq. (5-8), can replace in Eq. (5-13) the quasi-steady state gradient calculated with Eq. (5-14), the CS criterion can be extended to include the initial unsteady growth state. It was shown in [11]

62

that

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

can vary from

at

to

at

Further, a critical velocity from stable to unstable growth conditions, can be defined based on Eq. (5-13) as [12]:

Equation (5-15) represents an extended CS stability criterion of a planar solid/liquid interface during the solidification of dilute binary alloys in the absence of convection and solid diffusion. It shows that, during the initial stage of growth and for selected solidification conditions, an alloy that would solidify with an unstable solid/liquid interface based on the steady state CS criterion (of Chalmers et al. [21]), it would probably solidify with a planar interface based on the present calculations. A CS stability diagram based on Eq. (5-15) is presented in Fig. 5-6 for Al-1.0 wt.% Cu alloy. Experimental data obtained under microgravity conditions [23] and terrestrial conditions [23, 24] are also shown in Fig. 5-6. The experimental values were normalized to 1.0 wt.% Cu. The solid/liquid interface becomes more unstable under microgravity solidification conditions than under terrestrial conditions as no chemical mixing occurs in the liquid bulk because of reduced convection. In Fig. 5-7 the effect of the equilibrium partition coefficient, k, on interface instability is plotted. As expected, at small growth rates as encountered under normal solidification conditions, the increase of W would destabilize the solid/liquid interface. Also, a value of k close to one would provide a more stable interface. This is also shown by Eqs. (5-8), (5-12), (513), and (5-15). The effects of convection on interface instability during the solidification of alloys can be expressed through the variation of both k and The variation of the effective partition coefficient, with the growth rate was developed by Burton et al. [25] based on the classical boundary layer theory. Therefore, can be expressed as [25]: An effective liquid diffusion coefficient, was calculated by Saques and Horsthemke [26] for spatially periodic

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

hydrodynamic flows. Their main result is

63

where d is the

diameter of the tube and is the average flow velocity. Their result clearly demonstrates that the contribution from the fluid flow can be significant for liquid alloys, where is typically of the order of Note that CS stability criterion discussed above was developed based on thermodynamic considerations and a more consistent kinetic analysis of the instability phenomenon is considered by the theory of morphological stability [1-3]. The morphological stability condition of an interface in binary dilute alloys was first derived by Mullins and Sekerka [1]. They showed that for the quasi-steady state growth:

where

is the wavelength,

is the Gibbs-Thomson coefficient,

and f is the magnitude of the maximum value of a frequency-dependent stability function as defined by Eq. 21 in [1]. In fact, f includes the curvature effects on the modified constitutional supercooling criterion (CS).

64

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

A dimensionless parameter similar to [3], p. 198, as: S = f ( A * , k ) , where

is defined in It can be seen from

Eq. (5-16) that f [3], or S as shown in Ref. [3], has a stabilizing effect on growth (see also [4]). As also discussed in Refs. [1-3], for practical purposes, since (e.g., S > 0.9 ), it is probably safe to take f = 0 in Eq. (5-16) that is, Eq. (5-16) will be similar to Eq. (5-13). A thorough discussion on tests of the onset of interface instability is presented by Sekerka [2]. It is noteworthy to mention that Hecht and Kerr [27] have investigated the stable/unstable interface transition in Bi-Sn alloys and have found, contrary to previous findings, that the interface was more stable than predicted by either steady-state stability analysis or steady state CS. An important outcome of the stability analysis is the theoretical calculation of the dendrite tip radius at the limit of morphological stability. The dendrite tip radius, R, can be approximated as [2, 3]: where interface, and

is the wavelength of instability of the solid/liquid is a stability constant of the order of

By using

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

65

instead of a more accurate evolution of the dendrite tip radius would be predicted. Again, during the initial transient state, the magnitude of the dendrite tip radius is higher than that predicted by the steady-state theory, denoting a more stable solid/liquid interface. Warren and Langer [28] have attempted a transient analysis to predict dendritic spacings during the directional solidification. Their calculation for matches the experiments much closer than obtained from the steadystate Mullins and Sekerka linear stability analysis (see Fig. 4 in [28]), suggesting again a more stable solid/liquid interface in the unsteady state. Summary: The analytical model presented in this chapter can be successfully used to calculate the solute redistribution during the initial unsteady unidirectional solidification of dilute binary alloys. It was shown that the results obtained with the analytical model presented in Eq. (5-5 to 5-8) compare very well with experimental results for Fe-C-0.56 wt.% Si alloys, directionally solidified by using a floating zone melting technique, and with Sn-0.5 at.% Bi alloys, directionally solidified by using Mephisto-USMP1 microgravity experiments. One of the most important applications of this analytical model is the possibility of obtaining accurate values of some thermophysical properties of dilute binary alloys in the absence of convection, such as the diffusion coefficient of solute in the liquid phase and the equilibrium distribution coefficient, if the solute concentration profile in the solid is known from directional solidification experiments. An extended CS stability criterion of a planar solid/liquid interface during the solidification of dilute binary alloys in the absence of convection and solid diffusion has been derived (e.g., Eq. 5-15). It shows that, during the initial stage of growth and for selected solidification conditions, an alloy that would solidify with an unstable solid/liquid interface based on the steady state CS criterion, would probably solidify with a planar interface based on the present calculations. This may have important implications in solidification processing of binary alloy systems. This theory can further be used to study the morphological transition from the initial unsteady (unperturbed) growth to the dendritic (perturbed) growth during the dendritic solidification.

5.2 NUMERICAL MODELING OF SEGREGATION The numerical model presented in chapter 2 has been extensively validated against experimental data for cast iron [30], vacuum arc remelted (VAR) and electroslag remelted (ESR) alloy 718 ingots [31], Pb-26.5 wt. %

66

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Sn and Sn-16 wt.% Pb VAR ingots alloys [31-33], Sn-15 wt.% Pb and Al4.4 wt.% Cu ESR ingots [31, 34], and Pb-10 wt.% Sn unidirectional solidified ingots [31, 35]. Data required to perform the present simulations are shown in Tables 5-1 to 5-3. The sample geometry and boundary conditions are presented in Fig. 5-8. The sample dimensions are similar to those used in the existent NASA’s Isothermal Casting Furnace. Numerical calculations have been performed for thermal and solutal Rayleigh numbers corresponding to gravitational acceleration of 1 g, 0.1 g, 0.05 g, and 0.01 g. A gravity acceleration of 0.01 g is a typical value attainable in the NASA’s KC-135 aircraft. Some numerical results are presented in Fig. 5-9 and 5-10 for Pb-10 wt.% Sn alloy. Simulated Sn concentration (defined as the average total Sn concentration within the elemental volume, i.e., in Eq. 2-15) contours are shown in Fig. 5-9 for a normal gravity environment (1 g).

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

67

For a low gravity environment (below 0.05 g), the macrosegregation tendency for Sn is very low. In this case, Sn concentration varies between 9.97 to 10.05 wt.% Sn. However, for a normal gravity environment (1 g), macrosegregation of Sn is large (see Fig. 5-9), that is, Sn concentration changes from 6.2 to 19.189 wt.% Sn (eutectic composition). Freckles, fingers (which are caused by plumes), and channels enriched in Sn as well as isolated pockets poor in Sn are seen in Fig. 5-9. Here fingers are defined as

68

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

continuous extensions of the boundaries, while pockets are discrete configurations. Some of the pockets originate from plumes, which break away from the boundaries (see the pockets beneath the plumes in Fig. 5-9). Freckles are long channels formed at the center and edges of the sample. Macrosegregation of Sn in freckles is high. Two large symmetrical Benard cells are formed for this geometry where the ratio between the height and width of the ingot sample is 2.

The studied case is a typical mode of convection which can develop when the liquid is compositionally buoyant and light but statically and thermally stable, i.e. when both and and for sufficiently low thermosolutal Rayleigh numbers. See also [37, 38] for the modes of convection that may develop during the upward solidification (cooling a liquid alloy from below). In this case, the net thermosolutal Rayleigh number (Ra in Eq. 1-1) is approximately for a gravitational acceleration of 1

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g and for a gravitational acceleration of 0.01 g. The selected characteristic length scale was the average mushy zone thickness that is about For the present simulation conditions, no finger-type convection develops for a gravity level below 0.05 g The effect of thermosolutal convection on the local liquid thermal gradients and liquidus isochrones are illustrated in Fig. 5-10a and 5-10b for both the normal gravity (1 g) and low gravity (0.01 g) environments. The liquid/mush interface (represented in Fig. 5-10a and 5-10b by liquidus isochrones) and local liquid thermal gradients are perturbed under 1g condition. No alteration of these solidification parameters is seen in Fig. 510b for a gravity acceleration of 0.01 g.

For a gravity level of 0.01 g, negligible convection takes places in this system. As insignificant back-diffusion occurs in the present case, we expect to see large longitudinal Sn segregation. However, for a gravity level of 0.01 g, negligible longitudinal Sn segregation exists. The rationalism of this phenomenon is explained in the following paragraphs. The initial transition effects during the directional solidification in the absence of convection and solid diffusion occurs over a length that can be calculated with the analytical

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

model developed by Nastac [11, 12, 20] as (i.e., Eq. 5-12). Here, can be increased by decreasing k and W. For Pb-10 wt. % Sn alloy, k = 0.31 and Using an average which is based on the geometry and process parameters shown in Fig. 5-8, equals Above this value, no macrosegregation occurs. The present numerical results for a gravity level below 0.01 g confirm this phenomenon. Note that the variation of Sn concentration during the initial transient cannot be captured by the present numerical model. This is because the length of the initial transient is considerably smaller than the mesh size In Fig. 5-10c, simulated solidification macrostructure is presented for a gravity level of 1 g. A similar appearance for macrostructure was obtained for low gravity environment (0.01 g). The stochastic model used for simulating these macrostructures is presented in details in [7, 8, 29, 39]. It uses the temperature history results obtained with the present numerical model. The resolution used in the stochastic calculations was of 1000x2000 with a uniform mesh size of The surface grain density used in calculations was of The grain size in Fig. 5-10c varies from approximately (measured at 2 mm from the sample bottom) to (measured at 5 mm from the sample top). Overall, the calculated grain size for a gravity level of 0.01 g was about 15 % larger than that obtained under normal gravity conditions (1 g). In the case presented in Fig. 5-10c, the solidification macrostructure is columnar dendritic. The columnar to equiaxed transition (CET) does not take place for the current solidification conditions, i.e., the temperature gradients are larger than and the average solid/liquid interface is greater than (see Fig. 5-10a). In this system, CET occurs for temperature gradients smaller than when the average solid/liquid interface velocity is less than (see [40] for the experimental results in this alloy system). Also, the columnar grain structure shown in Fig. 5-10c is, unlike the microstructure, relatively insensitive to fluid flow and local solutal gradients. The growth direction of the columnar grains is mostly controlled by the direction of the thermal gradients normal to the S/L interface and the grain size (e.g., nucleation and grain growth kinetics) is determined by the cooling rates in the mushy region and the grain selection mechanism (see also [7, 8, 39]). Summary: The magnitude of thermosolutal convection and therefore, macrosegregation intensity are directly related to the gravity level. In Pb-10 wt.% Sn alloy, the mode of convection is solutally unstable and thermally stable. For the geometry and process parameters presented in Fig. 5-8 and Table 5-1 to 5-3, the critical threshold value for gravity level is 0.01 g. This corresponds to a critical Rayleigh number of about Below this value, insignificant macrosegregation occurs during the unidirectional

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solidification of Pb-10 wt.% Sn alloy. This observation is in line with other numerical results [41]. For 1 g and sufficiently low thermosolutal Rayleigh number (approximately freckles, fingers (which are caused by plumes), and channels enriched in Sn as well as isolated pockets poor in Sn develop. Similar appearance for solidification macrostructures simulated in both normal and low gravity conditions was observed. However, the grain size for a gravity level of 0.01 g was overall about 15 % larger than that simulated under normal gravity conditions (1 g). The experimental hardware that can be used for the experimental validation of this work is the existent NASA’s Isothermal Casting Furnace (ICF) that has a temperature range of 373 K to 1623 K, and quenching rate capabilities of 1 to 50 K/s (the sample diameter is 10 mm). Typical ICF experiments are conducted under a low gravity environment (0.01 g for 25 s) during parabolic flights on board a NASA KC-135 aircraft (see [42, 43] for the experimental procedure). Lastly, an example of numerical calculation of macrosegregation in ESR alloy 718 ingots is presented in Fig. 5.11 [31]. The numerical model formulation [32] is similar with that in chapter 2. However, it does not include solidification-kinetics. As shown in Fig. 5-11, the numerical results are in line with the experimental measurements.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

REFERENCES

1. W. W. Mullins and R. F. Sekerka, J. Applied Physics 35, No. 2, p. 444, 1964. 2. R. F. Sekerka, Crystal Growth, ed. P. Hartman, North-Holland Publ. Co., p. 403, 1973. 3. W. Kurz and D. J. Fisher, Fundamentals of Solidification, Trans Tech Publications, 1986. 4. L. Nastac and D. M. Stefanescu, Met Trans 27A, p. 4061 and p. 4075, 1996. 5. L. Nastac and D. M. Stefanescu, Modelling Simul. Mater. Sci. Eng. 5, p. 391, 1997. 6. M. McLean, Directionally Solidified Materials For High Temperature Service, The Metals Society, p. 118, 1983. 7. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, Proceedings of the International Symposium on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, AVS, eds. A. Mitchell and P. Aubertin, p. 145, 1997. 8. L. Nastac, S. Sundarraj, and K. O. Yu, Proceedings of the Fourth International Symposium on Superalloys 718, 625, 706 and Derivatives, TMS, ed. E. Loria, p. 55, 1997. 9. A. V. Catalina and D. M. Stefanescu, Met Trans 27A, p. 4205, 1996. 10. L. Nastac and D. M. Stefanescu, Met Trans 24A, p. 2107, 1993. 11. L. Nastac, International Communications in Heat and Mass Transfer 5, No 3, p. 407, 1998. 12. L. Nastac, J. of Crystal Growth, Vol. 193, No. 1-2, pp. 271-284, 1998. 13. V. G. Smith, W. A. Tiller, and J. W. Rutter, Can. J. Physics 33, p. 723, 1955. 14. A. Kagawa and T. Okamoto, Metal Science, p. 519, 1980. 15. J. J. Favier, P. Lehmann, J. P. Garandet, B. Drevet, and F. Herbillon, Acta Mater. 44, No. 12, p. 4899, 1996. 16. D. R. Poirier and G. H. Geiger, Transport Phenomena in Materials Processing, TMS, p. 450, 1994. 17. J. D. Verhoeven and E. D. Gibson, Met Trans 2, p. 3021, 1971. 18. J. D. Verhoeven, E. D. Gibson, and R. I. Griffith, Met Trans 6B, p. 475, 1975. 19. J. L. Murray, Binary Alloy Phase Diagrams 1, ed. T. B. Massalski, ASM Intern., p. 141, 1990. 20. L. Nastac, Scripta Materialia, Vol. 39, No. 7, pp. 985-989, 1998. 21. J. W. Rutter and B. Chalmers, Can. J. Physics 31, p. 15, 1953. 22. W. A. Tiller, K. A. Jackson, J. W. Rutter, and B. Chalmers, Acta Metall. 1, p. 428, 1953.

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23. J. J. Favier, J. Berthier, Ph. Arragon, Y. Malmejac, V. T. Khryapov, and I. V. Barmin, Acta Astronautica 9, No. 4, p. 255, 1982. 24. M. H. Burden and J. D. Hunt, J. Crystal Growth 22, p. 99, 1974. 25. J. A. Burton, R. C. Primm, and W. P. Slichter, J. Chem. Phys. 21, p. 1987, 1953. 26. F. Sagues and W. Horsthemke, Physical Review A 34, No.5, p. 4136, 1986. 27. M. V. Hecht and H. W. Kerr, J. Crystal Growth, 7, p. 136, 1970. 28. J. Warren and J. S. Langer, Physical Review E 47, No. 4, p. 2702, 1993. 29. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, vol. 5, no. 4, pp. 391-420, 1997. 30. L. Nastac, D. M. Stefanescu, and L. Chuzhoy, Proceedings of the Modeling of Casting, Welding and Advanced Solidification ProcessesVII, M. Cross, M. and J. Campbell (eds.), TMS, pp. 533-540, 1995. 31. J. Chou, L. Nastac, S. Sundarraj, Y. Pang, and Kuang-O Yu, Experimental Evaluation and Computer Model Verification of Secondary Remelt Ingot Structures, Aeromat '98, Tysons Corner, VA, June 1998. 32. S. Sundarraj, L. Nastac, Y. Pang, and K. O. Yu, Proceedings of the Modeling of Casting, Welding and Advanced Solidification ProcessesVIII, C. Beckermann and B. G. Thomas (eds.), TMS, pp. 297-304, 1998. 33. S. D. Ridder, S. Kou, and R. Mehrabian, Met Trans, vol. 12B, pp. 435447, 1981. 34. S. Kou, D. R. Poirier, and M. C. Flemings, Electric Furnace Proceedings, pp. 221-228, 1977. 35. J. R. Sarazin and A. Hellawell, Met Trans, vol. 19A, pp. 1861-1871, 1988. 36. S. D. Felicelli, J. C. Heinrich, and D. R. Poirier, Met Trans, vol. 22B, 847-859, 1991. 37. A. W. Woods, J. of Fluid Mechanics, vol. 239, pp. 429-448, 1992. 38. H. E. Huppert, J. of Fluid Mechanics, vol. 212, pp. 209-240, 1990. 39. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, J. of Metals, TMS, pp. 30-35, March 1998. 40. R. B. Mahapatra and F. Weinberg, Met Trans, Vol. 18B, pp. 425-432, 1987. 41. J. C. Heinrich, S. Felicelli, P. Nandapurkar, and D. R. Poirier, Met Trans, vol. 20B, pp. 883-891, 1989. Pacific 42. J. L. Torres, D. M. Stefanescu, S. Sen, and B. K. Dhindaw, Rim International conference on Modeling of Casting and Solidification Processes, Beijing, pp. 190-197, 1996. 43. L. Nastac, S. Sundarraj, S. Sen, Thermosolutal Effects on Columnar-toEquiaxed Transition during Solidification of Castings, NASA Proposal (NRA-96-HEDS-02), 1997.

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44. L. Nastac, Numerical Heat Transfer, Part A, Vol. 35, No. 2, pp. 173-189, 1999.

6

MICRO-SOLUTE TRANSPORT EFFECTS ON MICROSTRUCTURE AND MICRO SEGREGATION

6.1 INTRODUCTION Microstructure is crucial in controlling the mechanical behavior of the final component. In this chapter, computer models and predictions of important microscopic features, such as dendrite coherency, grain size, microsegregation, and evolution of secondary phases, are presented.

6.2 DENDRITE COHERENCY AND GRAIN SIZE EVOLUTION Dendrite coherency occurs when dendrite tips of adjacent grains come into contact, i.e. when It is one of the most important parameters used to establish the rheology of a particular system. Since it is dependent on the evolution of the dendrite envelope, all factors that augment the evolution of envelope fraction, such as topology and movement of envelope, should reduce the dendrite coherency. The purpose of the following analysis is to evaluate the influence of some process and material parameters on the onset of dendrite coherency. The alloy selected for this analysis is a Fe-0.6 wt.% C. The data used in calculation are given in Table 6-1. These are typical data found for example in [1].

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The evolution of envelope fraction and internal fraction of solid during solidification of the Fe-0.6% C alloy for a cooling rate of 4 °C/s was calculated using the model for the growth of the star dendrite, described in [2]. The results are presented in Figs. 6-1 and 6-2. From Fig. 6-1a it can be seen that coherency is calculated to occur for an envelope fraction of one. For this particular cooling rate coherency occurs at 0.55 fraction solid, when the internal fraction of solid becomes equal to the fraction of solid. Figure 6-1b shows the dendrite coherency during solidification of the same alloy as a function of cooling rate. It is seen that, for the range of cooling rates selected for this analysis, the onset of coherency moves to higher fraction solid as the cooling rate decreases. To evaluate the influence of the diffusion model on the onset of coherency, the interface liquid concentration was calculated using Scheil, equilibrium, and microsegregation model described in [3]. Then, the model has been used to obtain data on the onset of coherency and on the solidus temperature, for these three different assumptions on micro-diffusion calculations. The results presented in Table 6-2 indicate that equilibrium calculation results in higher coherency, while Scheil predicts lower coherency. The model, as expected, predicts an intermediate coherency because it accounts for back diffusion.

In Fig. 6-2, the complex influence of the cooling rate on equiaxed dendritic growth is presented. The cooling rate was calculated immediately above the liquidus temperature. In Fig. 6-2a it is seen that, as the cooling rate

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77

increases, recalescence increases first, and then disappears. The cooling rate also affects grain size and the onset of coherency, as shown in Fig. 6-2b. The model has been incorporated into a commercial macro transport code for modeling of casting solidification (PROCAST). Computation details and experimental validation performed on Inconel 718 castings are described in [2]. Some of the results for IN718 are presented in Fig. 6-3.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

6.3 DETERMINISTIC MODELING OF MICROSEGREGATION 6.3.1. Introduction Assessment of microsegregation occurring in solidifying alloys is important, since it influences mechanical properties. This is especially true for cast crystalline materials. Also, a comprehensive theoretical treatment of dendritic growth requires an accurate evaluation of the solutal field

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(microsegregation) during solidification. Models for calculation of microsegregation differ by the problem they tackle as well as by the approach. Most models are restricted to the case of planar (plate) solidification (Fig.6-4a), or columnar solidification (Fig.6-4b), with the volume element over which the calculation is performed being selected between the primary or the secondary dendrite arms. One dimensional (1D) Cartesian or cylindrical coordinates solution has been proposed. When equiaxed solidification is considered (Fig.6-4c), a 3D problem (or at least 1D spherical coordinates) must be considered.

The majority of the current models are based on the “closed system” assumption, i.e. no net mass enters or leaves the domain during solidification. Some of the noteworthy models are shortly presented in the following section. 6.3.2. Models based on the “Closed System” Assumption The earliest description of solute redistribution during solidification by Scheil [4] involves several assumptions such as: negligible undercooling during solidification, complete solute diffusion in liquid, no diffusion in solid, no mass flow into or out of the volume element, constant physical properties, and fixed volume element (no dendrite arm coarsening). The equation that describes Scheil’s model is:

where is the interface concentration of solute in the solid phase, is the effective partition coefficient, is the solid fraction, and is the initial solute concentration.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

However, the diffusion of solute into the solid phase can affect microsegregation significantly, especially toward the end of solidification [5, 6, 7]. The models described in [5, 6] were used to explain microsegregation in Al-Cu and Al-Si alloys at cooling rates of up to 200 K/s [8]. Ohnaka [9] proposed models for plate (Fig. 6-4a) and “columnar” dendrites (Fig. 6-4b). Complete mixing in the liquid and parabolic growth was assumed. On the basis of an assumed profile, an equation for solute redistribution in the solid that includes the equation developed by Clyne and Kurz [7] was derived:

where is the interface concentration, is the primary dendrite arm spacing, n = 1 for plate, n = 2 for columnar, and is the local solidification time. Prior knowledge of the final solidification time is required. Note that for this equation reduces to the Scheil equation, and for it becomes the equilibrium equation. Ogilvy and Kirkwood [10] further developed the model developed by Brody and Flemings [6] to allow for dendrite arm coarsening in binary and multicomponent alloys. For binary systems the basic equation was:

Here, X is the distance solidified. Thus, The end term represents the increase in the size of the element due to arm coarsening, which brings in liquid of average composition that requires to be raised to the composition of the existing liquid. This equation was solved numerically under the additional assumptions of constant cooling rate and liquidus slope. Also, a correction factor for fast diffusing species was added. Kobayashi [11] obtained exact analytical solutions for the “plate” and “columnar” dendrite models. Diffusion in solid was calculated but complete liquid diffusion was again assumed. Solidification rate and physical properties, including partition coefficients, were considered constant. Linear solidification where is the final solidification time) was also assumed, which means that the motion of the interface was prescribed. The first order approximate solution derived by Kobayashi reduces to Ohnaka’s solution when applied to the interface. Calculations with the second order approximate solution were very close to the exact solution. The equation is:

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with

and The model has been further extended to include a thermal model of solidification, multicomponent alloys, and temperature dependence of diffusivity. Note that again for and Eq. (6-4) reduces to the Scheil equation. Also, for it becomes the equilibrium equation. One of the major disadvantages of Kobayashi’s solution is the large number of terms (20,000 for a required for convergence [13]. Matsumiya et al. [12] developed a 1D multicomponent numerical model in which both diffusions in liquid and solid were considered. Toward the end of solidification, especially for small partition ratios, lower liquid concentration than the analytical models was predicted. Yeum et al. [13] proposed a finite difference method to describe microsegregation in a “plate” dendrite that allowed the use of variables k, D, and growth velocity. However, complete mixing in liquid was assumed. Battle and Pehlke [14] developed a 1D numerical model for “plate” dendrites that can be used either for the primary or for the secondary arm spacing. Diffusion was calculated in both liquid and solid, and dendrite arm coarsening was considered. Further complications arise when multicomponent systems are considered. Chen and Chang [15] have proposed a numerical model for the geometrical description of the solid phases formed along the liquidus valley of a ternary system for plate dendrites. Constant growth velocity, variable partition ratios and the Brody-Flemings model for diffusion were used. A complete analytical model for ‘Fickian’ diffusion with temperatureindependent diffusion coefficients and zero-flux boundary condition in systems solidifying with plate, columnar or equiaxed morphology (Fig.6-4) was developed by Nastac and Stefanescu in [3]. The model takes into account solute transport in the solid and liquid phases and includes overall solute balance. The overall solute balance in integral form rather then the flux condition at the interface (time-derivative form) has been used for three reasons. First, it is more conservative than the interface mass balance, second, the solution does not require a prescribed movement of the interface, and third, unlike term-by-term differentiation of Fourier series, term-by-term integration is always valid. This model allows calculation of liquid and solid

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interface composition during and after solidification, for volume elements with plate, cylindrical or spherical geometry. Thus, it can be used to predict solute redistribution (microsegregation) for planar, columnar or equiaxed morphologies. The model in [3] and all other existing analytical microsegregation models are based on the “closed system” assumption, i.e. no net mass enters or leaves the domain during solidification. Such assumption may lead to erroneous calculations in many cases. Few models tackle coarsening and coalescence phenomena that occur during solidification. An analytical microsegregation model for open and expanding domains is summarized in the next section. The purpose of such a model was to establish the impact of the “open system” assumption (mass transport in and out of the element) and of the “expanding system” assumption (coarsening and coalescence) on microsegregation. 6.3.3. An Analytical Model for Estimation of Microsegregation in Open and Expanding Domains

Assumptions, governing equations and boundary conditions

Consider a macro-volume element within the solidifying metal. It can be for example, the mesh size of a macro-heat transfer model for casting solidification. Within this macro-element the temperature is assumed uniform and is obtained from the solution of the thermal field. The macroelement is further subdivided in a number of spherical micro-elements (Fig. 6-5a). Within each of these elements of radius a spherical equiaxed grain of radius

is growing until the whole volume is filled. If the particular

case of SG iron is considered, a graphite spheroid of radius within the spherical austenite grain of radius

The

is growing aggregate

solidifies by simultaneous growth of the graphite and austenite phases. The assumed geometry and the schematic solute concentration profiles developed in the solid and liquid phases for an element are presented in Fig. 6-5b. Micro-diffusion transport within the element starts concomitantly with solidification. Coarsening and/or coalescence are allowed to take place during solidification. The problem to solve is to calculate the composition profiles in both the solid and the liquid during solidification. Then, the microsegregation ratio

Chapter 6. Micro-Solute Transport Effects on Microstructure and Microsegregation

can be calculated. At the nucleation temperature,

83

the first solid

formed has a solute concentration k where k is the partition ratio. The liquid in the vicinity of the solid/liquid interface is either enriched in (if k < 1), or depleted of (if k>1) solute. The final liquid fraction will have a solute concentration that depends on the diffusion coefficients in the solid and liquid phases, growth velocity of the grain, coarsening and coalescence, as well as macro-convective flow through macrosegregation. The main assumptions of the model are as follows: (1) Solute transport in both phases is by diffusion with diffusion coefficients independent on concentration. Therefore, the double boundary problem must be solved for and The solute concentrations in the solid (S) and liquid (L) phases must satisfy Fick’s second law:

where, and are the diffusion coefficients in the solid and liquid phases, respectively, and m is an exponent (m = 2 for spherical geometry, m = 1 for cylindrical geometry, and m = 0 for plate geometry). (2) The material is incompressible and the densities in both phases are constant. (3) The solid-liquid interface is planar and under local equilibrium:

where the superscript * denotes values at the interface. From Eq. (2) two boundary conditions, unknown a priori, are obtained:

(4) There is solute flow into or out of the volume element considered (“open system”). Thus microsegregation calculation can include the contribution of mass transport by convective flow, and the effects of coarsening and coalescence. The overall mass balance for an open system can be written in integral form as:

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where

is the local mass average concentration over the volume

element, and is the volume of the element over which the mass balance is computed. Note that the volume of the element is varying in time. Coarsening and coalescence will impose an increase of this volume (“expanding system”). Assuming, for the sake of notation simplification, that the densities of the solid and liquid phases are not only constant, but also equal:

Then Eq. (6-9) is used to couple the concentration fields in both the solid and liquid phases. The boundary conditions for the finite open system are:

where is coarsening velocity. In the boundary condition for the liquid phase, the first term on the right hand side represents the influence of coarsening and the second term is the contribution of convective fluid flow on microsegregation. Due to the rapid liquid diffusion it is computationally convenient to include it in the flux boundary condition at rather than in the flux balance [16]. The flux boundary condition described by Eq. (610) is obtained by differentiating in time Eq. (6-9) and applying the Leibnitz’s integral formula for differentiation. In non-dimensional form, the flux boundary condition at is:

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(5) The initial concentration in the liquid, is constant for each volume element (micro-scale), but it is variable at the macro-scale level (casting). Finally, the solution of the coupled double boundary value problem can be obtained by solving Eqs. (1) with the boundary conditions described by Eqs. (6-6), (6-7) and (6-10). The liquid and solid solutal fields are coupled through Eq. (6-9). The following initial conditions are used:

where

is time when the local solidification starts.

Analytical solution

The solution of ‘Fickian’ diffusion for the interface solid concentration of a spherical element during solidification consists of the following equations:

with

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where:

where is the root of the equation The mathematical assumptions involved in derivation of Eqs. (6-13)-(6-16) are in line with those described in [3, 17]. They are valid for For the case of “closed system” and no coarsening, the microsegregation model (Eq. 6-13) reduces to that derived in [3]. Since no assumption on the evolution of the fraction of solid was used in the present derivation, any transformation kinetics model can be used to calculate the movement of the interface. Here, the fraction of solid is calculated through the heat transfer-transformation kinetics model for SG iron [18]. The radius of the austenite-liquid interface (austenite shell) is given by:

where

is the growth velocity of the austenite phase,

coefficient of carbon in austenite,

is the diffusion

is the concentration of carbon in

the austenite matrix at the graphite/austenite interface, is the concentration of carbon in the austenite matrix at the liquid/austenite interface, and

is the concentration of carbon in the liquid at the

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liquid/austenite interface. and are the radii of the graphite spheroid and of the austenite shell, respectively. They have been obtained in the previous time step of the numerical calculation. Coupling micro- and macro- segregation

The general methodology for coupling between micro- and macrosegregation (open system) is described in [19] for equilibrium (lever rule) and non-equilibrium (Scheil) cases. For limited diffusion in both solid and liquid phases, the link between micro- and macro- segregation done through use of the following equation:

where is the average macroscopic velocity over the volume element, and is the superficial velocity. Their origin is the shrinkage flow, the buoyancy flow (thermosolutal convection), and the relative motion flow of the liquid/solid interface. Eq. (6-18) assumes no diffusion at the macro-scale level. It is similar with that derived by Beckermann and Viskanta [20]. Note also that Eq. (6-18) represents an implicit link between micro- and macrosegregation. The concentration gradient in the liquid phase that contributes to the convective term (first term in Eq. (6-18)) is obtained from the macrosegregation model. The second term in Eq. (6-18) is used to solve the

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

microsegregation (see Eq. (11) in [23]) and is obtained through both the micro- and macro- segregation models. The governing equations for macroscopic transport include the conservation equations for mass, momentum, energies, and species for the geometry of interest. They are solved by the control volume method. Model evaluation

To evaluate the contribution of the “open system” and of coarsening on microsegregation a theoretical sensitivity analysis was performed. The variation of the interface liquid concentration and the fraction of eutectic were calculated for a hypothetical system using different process variables and assumptions with the model. The main data and variables used in calculation, and the calculated fraction of eutectics, are summarized in Tables 6-3 and 6-4. Constant growth and coarsening velocities for each case were assumed so that the solidification time, and the final grain radius, were the same. Other data used for calculation are: A linear variation with time for the local average concentration was assumed to allow for the solute to either enter or leave the domain. For instance, “Open+10%C+10%R” means that the local average concentration is linearly increased (solute enters the domain) from 100% (initial value at the onset of solidification) to maximum 110% (end of solidification), and 10% linear increase in the grain radius (coarsening) is allowed. The Scheil model (closed system, complete diffusion in liquid, no diffusion in solid) was used as a basis for comparison. It is apparent from Tables 6-3 and 6-4 and Fig. 6-6 that the results obtained with the Scheil model and the present model applied to the closed system are very close. Small discrepancies were observed at the end of solidification when the Scheil model predicted larger amount of eutectic than the proposed model. However, large differences in the prediction of both interface liquid concentration and fraction of eutectic were observed, when the “open system” assumption were used (Tables 6-3 and 6-4 and Fig. 6-7). This clearly emphasizes the importance of relaxing the “open system” assumption. When only coarsening and/or coalescence were introduced, small fluctuations in the results were observed. This is because both the size of the final domain and solidification time were maintained constant, such that the overall micro-diffusion was not much affected. Note that considering only coarsening and/or coalescence without allowing the solute to enter in the domain, may be misleading. Both coarsening and “open system” have to

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be considered. Then, the effect of coarsening and/or coalescence on microsegregation can be correctly analyzed. As shown in Tables 6-3 and 6-4, there are significant differences in the prediction of the fraction of eutectic between expandable and non-expandable domains. A possible explanation is that the partition coefficient may vary during solidification [22]. It may be dependent on composition and convection in the liquid phase. Thus, for accurate prediction of microsegregation it is necessary to have correct data for k.

Model validation

Validation of the model was performed using the experimental data of Boeri and Weinberg [21]. The diameter of the measured samples was 15 mm. The physical constants used for calculation are given in Table 6-5. The characteristic distance between equiaxed dendrite center and the last liquid to solidify, that is the final radius in this model is 75 mm, and the local solidification time is 30 s [21]. Since the eutectic temperature range is small, the diffusion coefficients were considered to be independent on

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temperature. The diffusivity values at the equilibrium eutectic temperature were used. The redistribution of Mn and Cu in SG iron within a spherical element containing one graphite spheroid was calculated with the proposed model for the open and closed system cases. The results are compared with the experiments [21] in Fig. 6-8. Manganese produces normal microsegregation (k < 1), while copper produces inverse microsegregation (k > 1) during solidification of SG iron. For copper the model predicts lower concentration than the experimental values.

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As shown in ref. [22], much better agreement was obtained for copper redistribution when a variable k was introduced. Unfortunately, the equation for the variation of k [22] was strictly developed for the closed system assumption and could not be used here. As can also be seen from Fig. 6-8, similar results are observed for redistribution of Cu and Mn for both open and close system assumptions. This is because the convective flow, and therefore macrosegregation developed during solidification of SG iron thin castings (15 mm ID), were small, while the heat extraction rate was high. 6.3.4. Partition Coefficient Evaluation

Calculated microsegregation is very sensitive to the value of the partition coefficient used. Thus, for accurate prediction of microsegregation

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it is necessary to have correct data for Unfortunately, limited experimental data on partition coefficients are available [24]. As emphasized by Kagawa and Okamoto [24], the partition coefficient is not constant during solidification. It is dependent on composition and convection in the liquid phase. Thus, the calculated equilibrium partition coefficient for silicon in the austenite-graphite eutectic (ternary system-Fe-C-Si) is expressed as follows [24]:

where is the silicon liquid concentration at the interface. Also, Burton et al. [25] have derived based on boundary layer theory the dependence between effective partition coefficient and growth rate under the assumption that the solid-liquid interface is planar. This is represented by the following equation:

where is the thickness of the boundary layer and it has to be determined from experiments. Kagawa and Okamoto [24] have experimentally evaluated the thickness of the boundary layer for silicon to be for and growth velocities of up to Thus, using in Eq. (6-21) and this experimental value for can be calculated with Eq. (6-22). It is true that can be directly calculated from experimental data if both the solid and the liquid concentrations are measured during solidification. Nevertheless, these measurements can be unreliable, in particular toward the end of solidification, because in complex alloys compounds precipitation is possible. From the experiments [21] used for validation in this study, only data on the interface solid concentration of silicon were available. The effective partition coefficient of silicon can be obtained through manipulation of Eq. (6-14). Thus, is the solution of the following quadratic equation:

The effective partition coefficient obtained with Eq. (6-22) is plotted in Fig. 6-9 against that obtained from experimental data [21] and Eq. (6-23). It

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can be seen that the evaluated from experimental data on commercial SG iron is smaller than that calculated for ternary alloys. If the interface solid concentration is not available, the liquid interface concentration measured in the quenched region can be used. Then, can be calculated with [22]:

For example, in the case of copper redistribution in SG iron, the following regression was obtained with Eq. (6-24) when using the experimental data obtained by Boeri and Weinberg [21] for the liquid copper concentration at the interface,

Note that Eq. (6-25) is strictly valid only for the given experimental conditions. However, similar equations can be obtained with Eq. (6-24) for different conditions. Figure 6-10 shows a comparison between calculated with Eq. (6-25) and directly obtained from experimental liquid and solid interface concentrations. It is seen that linearly increases during solidification. As can be seen from Fig. 6-1 1a, when a constant obtained from the literature was used, the model predictions were in limited agreement with the experimental data. However, when calculated with Eq. (6-25) was introduced in the microsegregation model for SG iron much better agreement was obtained, as shown in Fig. 6-11b for copper redistribution.

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For systems where microsegregation results in the solidification of a eutectic, can be calculated from Eq. (6-14) as follows:

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95

where eutectic fraction is and is the eutectic composition as given by the phase diagram. Then, if is experimentally evaluated, Eq. (626) can be used to calculate for the particular solidification problem corresponding to the experiment. For systems where no second phase precipitates, can be calculated from Eq. (6-24) for if the maximum interface composition at the end of solidification is known. The effective partition ratio for an Al-4.9 wt.% Cu alloy was calculated with Eq. (6-26) using the diffusivities in Table 6-6 and the experimental data of Sarreal and Abbaschian [26] given in Table 6-7. Plate geometry and constant growth velocity were assumed. is one half of the dendrite arm spacing. The results are given in Fig. 6-12 for various cooling rates. As expected, is not affected significantly in the range of low cooling rates On the contrary, for cooling rates above 1 K/s, increases rapidly.

6.3.5. Predictions of Microsegregation in Commercial Alloys A comparison of predictions of solute redistribution of niobium in Inconel 718 by various models is presented in Fig. 6-13. These models include the Scheil, Brody-Flemings, Clyne-Kurz and Ohnaka models for plate elements, the Ohnaka model for “columnar” dendrite, and the newly proposed model for spherical and plate geometry. Also, the time-dependent and quasi-steady state solutions of the proposed model are compared. The values of the physical parameters used in calculations are given in Table 6-8.

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For this case, the quasi-steady state solution seems to give poor results toward the end of solidification. However, it can be seen that up to 0.9 fraction of solid, the quasi-steady-state solution almost coincides with the time-dependent one. Different models predict different composition profiles and final fraction of Laves phases. Since experimental data on the composition profile were not available for this case, validation can be done only against the final fraction of Laves phase. The proposed model for spherical geometry predicts 1.2% Laves, which is in the experimental range of 0.38 to 1.44% by volume Laves phase, measured by Thompson, et al. [27]. All other models predict higher fractions. Thus, it appears that predictions by models assuming complete diffusion in liquid and plate element, applied to equiaxed geometry, give correct results only accidentally. To assess the validity of the new model not only in terms of final fraction of phase but also as to its accuracy in predicting the composition profile, calculated Nb redistribution for Inconel 625 was plotted in Fig. 6-14 against experimental data from directional solidified samples obtained by Sawai et al. [28] (see Table 6-8). Cylindrical geometry and constant growth velocity were assumed. Comparison was made with samples quenched from 1593 K, which is above the solidus temperature of 1523 K, and with samples quenched 2000 s after solidification, from 1423 K. Reasonably good agreement was obtained. It is considered that the agreement can be improved if an accurate transverse growth velocity can be used. Note that existing

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analytical microsegregation models cannot be used for calculation after the end of solidification, since they rely on solidification time. Additional validation was attempted for microsegregation of Mn and P in a plain carbon (0.13 wt.% C) steel directionally solidified with a columnar structure. Experimental data after Matsumiya et al. [12] were plotted in Fig. 6-15 together with calculated data using the Scheil equation and the proposed model. Cylindrical geometry and constant growth velocity were assumed for the model. The data used in calculation are given in Table 6-8. Experimental data were available for two different cooling rates. At the slower cooling rate of the dendrites in the microstructure did not exhibit clear patterns for secondary arms. Accordingly, calculations were performed for the primary arm spacing When the cooling rate was increased to the dendrites developed clear secondary arms. Thus, calculations with the model were performed for the secondary arm spacing according to the experimental measurements. Unlike the Scheil model, the data obtained with the proposed model fit well the experimental results. 6.3.6. Microsegregation Index (MSI) The microsegregation intensity (defined here as microsegregation index or MSI) will increase with solidification velocity and length scale and will decrease with solidification time liquid and solid diffusion coefficients, and with the partition coefficient (k) as follows [2]:

where is the Fourier number, is the cooling rate, A incorporates material properties, and a, b, c, and n are experimental parameters. Thus, the MSI defined by Eq. (6-27) includes both material and process parameters. The individual effect of grain size and solidification time on MSI number as well as their combined effects is illustrated in Fig. 6-16. As shown by Eq. (6-27), MSI is the product of cooling rates,

and

At low

is very small and the diffusion process is controlled by

the solidification time. Because increases with cooling rate, MSI also increases, which will result in increasing the amount of secondary phases (for example, Laves phase in alloy 718). At high cooling rates,

is very

small, and the diffusion process is mostly controlled by the grain size

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variation. Hence, MSI will decrease with increasing cooling rate, which results in a lower amount of Laves phase. As also shown by Eq. (6-37) and Fig. 6-16, a critical cooling rate exists at which maximum segregation and thus maximum amount of secondary phases form. This has major practical importance because by manipulating process parameters, it should be possible to avoid significant amounts of secondary phases in the microstructure.

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6.4 DETERMINISTIC MODELING OF SECONDARY PHASES In this section, a deterministic approach to simulate the formation of secondary phases is presented. A stochastic modeling approach of secondary phases is described in chapter 9.3.9.

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The importance of modeling of NbC/Laves formation in Inconel 718 is broadly discussed in literature [29-36]. It is also known that the distribution of carbides, Laves phases, and microporosities in alloy 718 are affected by the solidification path. The volumetric solid fraction is a function of the local growth velocity, the solidification time, the solidus temperature, the local temperature gradient at the freezing rate, etc. The redistribution of elements strongly affects the phase evolution of common superalloys with respect to temperature, as well as their mechanical properties and surface stability at elevated temperature. Previous studies on alloy 718 showed that both NbC and Laves produce intergranular liquid films due to intergranular distribution of Nb and C [33, 34]. Also, the ability of Laves to promote intergranular liquation cracking (microfissuring and hot cracking) during heat treatment is much higher than that of NbC. This is because the temperature of Laves phase formation is usually lower than that of NbC, i.e. liquation initiates at the eutectic-Laves temperature. Thompson et al. [33, 34] demonstrated that the carbon content of alloy 718 directly affects the volume fraction of carbides. This is readily explained by a pseudo-ternary phase relationship during solidification [33]. A schematic of an isothermal section through the space diagram of the pseudo-ternary just above the ternary eutectic is represented in Fig. 6-17a. The alloy in as-cast condition could contain a higher volume fraction of NbC and Laves phase than what the phase diagram suggests due to the microsegregation during solidification. The relative volume fractions of NbC and Laves depend on the C/Nb ratio. Alloys with high C/Nb ratio will have a higher volume fraction of carbide than alloys with low C/Nb ratio. The possible solidification paths of the schematically presented in Fig. 6-17a, are shown in Table 6-9:

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The model for predicting the evolution of secondary phases is based on the following assumptions [37, 38]: instantaneous nucleation, carbides grow only in the liquid, negligible interference between growing carbides, carbides are pushed by the solid/liquid interface, volume diffusion limited growth of carbides, binary diffusion couple. The following steps are required to predict the NbC/Laves phase formation in IN718: (1) the nucleation and growth of NbC assuming that the slow step is the volume diffusion of carbon from the liquid to the NbC/L interface, (2) modeling of redistribution of Nb and C, (3) equiaxed dendritic growth, (4) growth of Laves phase, and (5) coupling between macro transport-solidification kinetics models. The growth mechanism of NbC is based on the carbon diffusion from the liquid to the NbC/L interface and the reaction kinetics between Nb and C. Applying the metastable progress variable approach, it can be shown that the chemical reaction rate is very high, i.e. the kinetics of reaction can be neglected (see Appendix I in [38]). Thus, carbon concentration is depleted at the NbC/L interface, and the amount of NbC is based on the volume diffusion of carbon from the liquid to the NbC/L interface.

The schematic representation of the diffusion pattern for NbC is shown in Fig. 6-17b and the main model assumptions are shown in Appendix II in [38]. It is assumed that the carbides instantaneously nucleate at the equilibrium solidification temperature of alloy 718 (T = 1609K). The solution of volume diffusion-limited growth is described by the averaging method:

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where liquid, and

where

is the volume of liquid,

is the number of carbides in the

is the solutal supersaturation defined as:

is the liquid interface concentration,

concentration, and

is the interface NbC

is the average liquid concentration.

The redistribution of both C and Nb are calculated with a modified version of the model described in [3]. The main assumptions are: solute transport is calculated in both solid and liquid phases assuming Fick’s law for binary systems in spherical coordinates, the solid/liquid interface is planar and under local equilibrium, no solute flow into or out of the volume element (closed system), constant initial liquid concentration, and the following overall mass balance:

with

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where is the initial alloy concentration, is the actual volume of carbides, and is the volume of the element over which the mass balance is calculated. The overall mass balance is used to couple the concentration fields in both the solid and liquid phases. Note that the cross interdiffusion coefficients, as required in calculation of pseudo-ternary systems, are usually one order of magnitude lower than normal diffusion coefficients [41, 42] and are neglected in this analysis. The solution of this diffusion couple is a modified version of the model described in [3] and consists of the following equations:

where:

and Q,

and

and

with

Average concentration in the liquid phase is:

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where

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

is the volume of the liquid phase, r is the radial coordinate,

the initial concentration, the eigenvalue, is the

is

is and

root of the equation

The coupling between macro transport–solidification kinetics models is accomplished through Latent Heat Method [46], where the heat source term is described as follows:

Laves phase starts to form when concentration of Nb reaches the eutectic-Laves composition which is 19.1 wt% Nb. Kinetics of Laves is very high due to its morphology (eutectic or globular type-divorced eutectic) and appearance (discontinuous thin film at the grain boundary). It is assumed that the amount of Laves phase is directly related to the Nb concentration [43-45]. The equiaxed dendritic growth is based on the alloy 718 pseudobinary phase diagram [29, 34], where the primary driving force for growth is the liquid Nb concentration. The model for equiaxed dendritic growth of alloy 718 is described elsewhere [43-45]. The physical properties used in calculations are shown in Table 6-10.

The measured grain equiaxed radius and carbide radius as a function of cooling rate are presented in Figs. 6-18 and 6-19, respectively. The experimental procedure is described in [44, 45]. Experimental and calculated amounts of NbC and Laves phase as a function of cooling rate for the initial carbon and niobium contents of 0.06 and 5.25 wt%, respectively, are shown

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in Fig. 6-20. A typical example of calculation is given in Fig. 6-21. Note that as the cooling rate increases, the amount of precipitated NbC decreases, and falls short of the maximum (equilibrium) amount. The influence of cooling rate on the amount of NbC precipitated in cast Inconel 718, when the initial carbon and niobium contents were 0.125 and 5.25 wt.%, respectively, is presented in Fig. 6-21. Note in Fig. 6-21 that and are strongly dependent on the cooling rate. An optimum combination between C and Nb in function of cooling rate has to be predicted to minimize the amount of both NbC and Laves, in order to obtain a Laves free microstructure. This optimum combination may result in desirable mechanical properties, such as good stress-rupture ductilities due to the formation of a higher volume fraction of carbides (spherical shape), improved room temperature tensile strength and ductility due to elimination of Laves phase, etc. Experimental evidence demonstrates that the amount of NbC and Laves in cast alloy 718 is different from that predicted by phase equilibrium. The reason for this difference is that while in equilibrium processes mass diffusion transport is very fast compared with solidification kinetics (V