PHASE FIELD SIMULATION OF GRAIN GROWTH IN PRESENCE OF SECOND PHASE PARTICLES ASHIS MALLICK (M. Tech, IIT Delhi, India) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Dedicated to my father Shri Suresh Chandra Mallick PREFACE The dissertation entitled ‘Phase field simulation of grain growth in presence of second phase particles’ is submitted in partial fulfillment of the requirements for the award of the degree of Doctor of Philosophy in Mechanical Engineering at the National University of Singapore. The research described herein was conducted under the supervision of Dr. Srikanth Vedantam (Asst. Professor, Mechanical Engineering Department, NUS) and Prof. Lu. Li (Mechanical Engineering Department, NUS). To the best of my knowledge, this work is original, except where acknowledgements and references are made to previous work. In my opinion, the work presented in this dissertation has reached the requisite standard fulfilling the requirement of Doctor of Philosophy. The result contained in this dissertation have not been submitted in part or in full, to any other university or Institute for the award of any degree, diploma or other qualification. Part of this thesis has been published/accepted and under review for publication as listed below: Journal Articles 1. A. Mallick, and S. Vedantam, ‘Phase field study of the effect of grain boundary energy anisotropy on grain growth’, Computational Materials Science, 46, 21–25, 2009 (a part of chapter 5). 2. A. Mallick, S. Vedantam, and L. Lu, ‘Grain size dependent tensile behavior of Mg-3%Al alloy at elevated temperatures, Material Science and Engineering A, 515, 14–18, 2009 (a part of chapter 2). 3. S. Vedantam, and A. Mallick, A Phase model for a bicrystalline grain growth in presence of second phase mobile particle, Acta Materialia (in press) (a part of chapter 4). 4. A. Mallick, Tensile properties of Ultrafine Mg-3%Al alloy at elevated temperatures, Int. J. Mat. Research (accepted) (extended part of chapter 2). i 5. S. Vedantam, and A. Mallick, ‘Theory of grain growth in the presence of mobile second phase particles’, Submitted to Materials Letters (a part of chapter 3). 6. A multiphase field theory and simulation for a polycrystalline grain growth in presence of second phase mobile particle (manuscript in preparation) (a part of chapter 6). Conference Papers (Peer Reviewed) 1. A. Mallick, and S. Vedantam, ‘Phase field simulation of polycrystalline grain growth in presence of mobile second phase particles’, AIP proceedings, 1091, 240 – 242, 2009. 2. S. Vedantam, and A. Mallick, ‘Phase field simulation of grain growth in presence of mobile second phase particles: A bicrystal model’, Appeared in 10th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation of New Materials, 2008, Spain. 3. A. Mallick, ‘Nanocrystalline Mg-3%Al alloy: its synthesis and investigation of its tensile behavior’, Appeared in the proceeding of ICAP 2008: International conference on applied physics, WASET, 33, 10-13, 2008, Germany. 4. A. Mallick, S. Vedantam, and L. Lu, ‘Ultrafine Mg-3%Al alloy: Its synthesis and investigation of its tensile properties at elevated temperature’, Appeared in the proceeding of PMP III, 2008, Bangkok. Conference/workshop presentations 1. ‘Phase field simulation of polycrystalline grain growth in presence of mobile second phase particles’, 10th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation of New Materials, 14-19 September, 2009, Spain. 2. ‘Nanocrystalline Mg-3%Al alloy: its synthesis and investigation of its tensile behavior’, Int. conference on applied physics, 24-26 September, 2009, Germany. 3. ‘The Effect of High Angle Grain Boundary Energy Anisotropy and the Presence of Mobiles Particles on Grain Growth: A Phasefield Approach’, Phase-fieldSimulations:Materials Science meets Biology and Medicine:An International Focus Workshop, 12-14, November, 2008, Max-Planck Institute, Germany. 4. ‘Phase field simulation of grain growth in the presence of mobile second phase particles’, 10th U.S. National Congress for Computational Mechanics, 16-19, July, 2009, USA. ii ACKNOWLEDGEMENT It gives me a great pleasure in expressing my sincere thanks to Dr. Srikanth Vedantam and Prof. Lu Li for suggesting the problem and for their continued guidance. Their keen interest during the work has been a great source of encouragement. I owe a debt of gratitude for the help, critical comments and useful suggestions in preparation of the thesis, given by my supervisors throughout the period of this work. I am also thankful to all staff of the Applied Mechanics Division and Material Science Division for their assistance and valuable advice. I gratefully acknowledge the pleasurable time extended by my friends, specially Mohanraj, Raju, Mustafa, Krishna, Dilip, Swapan, Shivaji, who made my time at Singapore enjoyable. Also, I would like to express my heartiest feelings to all of my friends with whom I spent enjoyable and memorable moments during my B.Tech and M.Tech study and in my job. Worth mentioning are Sandeep (Arizona, USA), Dr. Prabir (Univ. Malaysia Sarawak), Sachin Pawar (PWD, Bombay), Dr. Rajesh (SLIET, India) and Dr. Goutam Barua (IIT, Guwahati). I always discussed my academic, personal and many other problems with them. They also shared my happiness, feelings and success. I acknowledge Prof. Dumir and Prof. Dube, IIT Delhi, India for their support and guidance during my M. Tech. thesis. Dr. Dumir deserves a special place and special mention in this thesis, as it is for him that today I am writing this thesis. If he would not have provided me the moral support in the capacity of Programme Advisor on that dark day when I had lost all hope, I would have withdrawn from the M. Tech. programme owing to the family problems, which were affecting me psychologically to a great extent. iii When I was desperate to withdraw, he came forward and advised me to see my family for one week and come back to continue the programme, assuring me that everything will eventually fall in place. I was not able to refuse the advise of such a great teacher, and later insisted him on becoming my M. Tech. supervisor. At every moment, I feel that without his support on that day and every day after that, it was not possible for me to receive the M. Tech. degree, which is the starting point of my research career and Ph.D. degree. I am sure that Prof. Dumir will be as proud as my father when he reads this thesis. Finally, I will be eternally grateful to my father, mother-in-law, aunty (pisi), sisterin-law (bhabi), wife (Kabita), and daughter (Aditi) for their unconditional love and support. iv ABSTRACT Prediction of nucleation, grain growth, and concomitant microstructure in polycrystalline materials is of great technological importance because the size, shape, and the orientation of grains have a significant effect on the mechanical properties of materials. In this thesis we first examine the effect of grain size - ranging from nano to micron sizes - on the elevated temperature tensile behaviour of a Magnesium alloy. We find a strong dependence of the tensile behaviour on the microstructure. Various characterisation techniques indicate the presence of particles and voids in the alloys which also affects the eventual microstructure formed. The control of microstructure, especially for nanocrystalline materials, has been recognized to be important in high temperature applications. In the remaining work we focus on the effect of second phase particles on the evolution of grains in such polycrystalline materials using a phase field theory. The connection of the microstructure formed to the mechanical properties is the scope of future work in this area. We first construct a theoretical framework for the interaction of mobile second phase particles on a grain boundary. Since the time of Zener, most studies have focused on grain boundary interaction with immobile particles. However, many inclusions, voids and other defects may in fact be mobile and their interaction with grain boundaries is significantly different from immobile particles. Our theoretical study is restricted to the interaction of a single columnar grain with uniformly distributed particles and highlights many of the phenomena of grain boundary interaction with mobile particles. For more realistic polycrystalline grain growth in the presence of mobile particles, we resort to a computational approach. Recently, phase field theories are becoming v popular for numerical simulation of grain growth. The phase field approach has developed for immobile particles and their effect on grain growth. In this thesis we develop this theory for the interaction of mobile particles and grain boundaries and we study the effects of this interaction in detail. The mechanical properties of Mg-3%Al alloys are strongly dependent on the grain size, test temperature and the presence of second phase particles. Theoretical calculation for the interaction of columnar grain boundary and uniformly distributed mobile particles shows that the presence of particles has a strong effect on the grain boundary motion. If the particle mobility is higher than that of the mobility of the grain boundary, the particle will move along with the grain boundary. However, for low particle mobility, the grain boundary will have the tendency to detach from the particle which also depends on the curvature of the grain boundary. We calculate the transition radius for the different mechanisms of grain boundary motion. Our phase field simulation for bicrystal grain growth in presence of particles shows the effect of size and the mobility of particle on the kinetics of grain boundary. The mobile particles are dragged by the grain boundary and create a particle free band very similar to the experimental observation of Ashby and Gentamore (Acta Metallurgica (1968) 16, 1081). Next we perform polycrystalline simulations using the phase field method. In polycrystalline particle-free simulations with grain boundary energy anisotropy, we observe that grain boundary energy anisotropy has a strong effect on the grain growth, grain size distribution and microstructural entropy. In a polycrystalline system with particles, the grain growth is retarded due to the presence of particles. The rate of retardation is higher when the particles are immobile than for mobile particles for the vi same volume fraction and particle size. The average grain size as function of simulation time depends on the size, volume fraction, and mobility of the particles present. vii TABLE OF CONTENTS PREFACE (i) ACKNOWLEDEGEMENTS (iii) ABSTRACT (v) TABLE OF CONTENTS (viii) LIST OF FIGURES (xi) NOMENCLATURE (xvi) Introduction (1) Experimental studies of grain size dependent mechanical properties of Mg-3%Al alloy at elevated temperatures (7) 2.1 Introduction (7) 2.2 Magnesium and its alloy (8) 2.3 Experimental procedures (9) 2.4 Results and discussion 2.4.1 Crystallographic representation and grain size calculation (11) 2.4.2 Tensile properties (13) 2.4.3 Fracture properties (18) 2.5 Conclusions (11) Theory of grain growth in presence of second phase mobile particles (19) (21) 3.1 Introduction (21) 3.2 Theory of grain growth (22) 3.3 Illustrative case study (25) 3.3.1 Case I: Particle mobility(mp) > Grain boundary mobility (mb) (26) viii and 12 µm respectively. It has been observed that the tensile properties are strongly depending on the grain size as well as the test temperature. At room temperature, the reduction of grain size gives better overall tensile properties. However, at elevated temperature the reduction of strength with a non-monotonic failure strain was observed in fine-grained samples. Next our analytical study of the effect of mobile second phase particles described the interaction of a columnar grain boundary and uniformly dispersed mobile particles in an axisymmetric setting. We observed that the presence of particles retards the grain boundary motion. The particles move with the grain boundary when mp>mb while the grain boundary may detach from particles if mb>mp and the driving force of grain boundary motion exceeds the critical attraction force. This situation is consistent with the analytical results of Gottstein [72]. In the numerical model of grain growth in presence of mobile particles, we develop the phase field model for interaction with mobile particles. An additional constitutive kinetic equation (vp = mpκ) for each particle describes the motion of particles. The curvature κ of the grain boundary is expressed explicitly in terms of the phase field variables. Particle interpenetration was prevented by introducing a repulsive force between two particles. A close examination in a bicrystal system using the phase field model shows that the kinetics of the grain boundary is affected by the particle. The growth rate of the circular grain depends on the size and area fraction of particles. The mobility of particle is inversely proportional to the size of particle and thus increase in size of the particles retards the growth kinetics. The grain boundary obtains a dimple-like shape when it 96 approaches or leaves the particle. The first case is due to the local attraction of grain boundary by the particle while the second one is due to three surface tensions (one boundary surface tension (γb) and others boundary/particle surface tension (γAP and γBP)) dictating the equilibrium shape of the boundary near the particle. In a dense particle system it was observed that the grain boundary dragged the particle cloud and creates a clear particle-free band behind the grain boundary. Similar particle dragging by migrating grain boundary in a polycrystalline copper was observed experimentally by Ashby and Gentamore [43]. In polycrystalline simulations we begin with the effect of anisotropic grain boundary enegy on grain growth. Our grain boundary energy anisotropy is not restricted to low misorientations unlike previous work. For the high angle grain boundaries, we used the extended Read-Shockley (ERS) grain boundary energy [50]. We find that the energy anisotropy has a profound affect on the growth rate, grain size distribution and the number of neighboring grains. The average radius of grains is found to be proportional to the simulation time and the relationship is predicted by a power law. However, the energy anisotropy in grain boundary delays the growth rate and it is more pronounced in the ERS grain boundary energy and followed by RS. The normalized grain size distribution in an energy anisotropy shifts toward the smaller grain size indicating that growth is not uniform. The distribution of number of neighbouring grains is broader in the anisotropic case. All the above observations are in very good agreement with the results from literature [9, 32, 35, 101]. In the polycrystalline system, even a small volume fraction of second phase particles delays grain growth and leads to significant change in the grain size distribution. This 97 change is more pronounced in presence of immobile particles than that of mobile particles. It was observed that in the presence of particles, the peak of normalized grain size shifted towards the smaller grain sizes at later simulation times. In particle free grain growth, the above distribution is essentially invariant with time. The immobile particle pinned the grain boundary while the grain boundary dragged mobile particle. In both the cases (either grain boundary pinning or particle dragging) the effect of particles is based on the amount of grain boundary reduced by those particles which are located on the grain boundary. This interaction mechanism is consistent with the theoretical studies on Zener pinning [40, 118] or particle dragging [72, 74]. For immobile particles, we established the relationship of average stagnant radius (RZ) in terms of area fraction. Assuming all particles are on the grain boundary, the average stagnant radius is predicted by RZ = 1.18rp / f a0.48 . This is in close agreement with values in the literature. The main merits of this work to the materials scientist are as follows: • Our experimental result gives an direct evidence that the grain size, structure of microstructure and the presence of second phase particles has a strong effect on the mechanical properties and thus models of grain growth and its control are very important. • The presence of second phase particles plays an important role in controlling the final average grain size. The immobile particles pinned the grain boundary and are more effective to control the final average grain size than mobile particles. Nevertheless, in some cases particularly in sintered material the presence of mobile particles cannot be ignored and thus the consideration of mobile particles in our model of grain growth enhances this understanding. 98 • The evolution of grain growth is a result of the reduction of free energy. The energy anisotropy of grain boundary influenced significantly on grain growth exponent, number of neighboring grains, grain size distribution and more. • The grain growth in a polycrystalline system is affected due to the presence of mobile particles. However, the retardation of growth process is more pronounced in the presence of immobile particles than that of mobile ones. The growth kinetics in presence of particles depends on the particle size, volume fraction of particles and the mobility of particles. 99 7.2 Future work In order to build additional physical realism into the model, theory and experiment, a number of extensions and enhancements immediately suggest themselves, to wit: 1. The particles, as considered in Chapter and Chapter 6, not evolve through any diffusional processes. However, the evolution of particles may also be important in many cases and the particle sizes not remain constant. Particle evolution may be accounted for by using a conserved order parameter field to describe the particles and solving the Cahn-Hilliard equations which describe the evolution in this setting. 2. It should be possible, within the general framework outlined in Chapter 5, to establish a three-dimensional model of high angle grain boundary energy anisotropy without restriction of the number of order parameters taken in simulation. 3. 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King, "Grain rotation and grain boundary selection in thin films," in Materials Research Society Symposium - Proceedings, Boston, MA, USA, 1997, pp. 301-306. 111 [...]... properties In Mg-Al alloys, the presence of second phase particle has been found to be very effective in inhibiting grain growth at higher temperature [47-49] We next focus on the mechanism of grain growth inhibition by second phase particles In chapter 3, we develop a mathematical model of grain growth in presence of second phase mobile particles A single columnar grain with randomly distributed spherical particles. .. constant of grain γb grain boundary energy per unit area γ bp local value of grain boundary energy Ro initial grain radius R(t) grain radius at time t R, A nondimensional radius of grain, nondimensional area of grain RT transition radius of grain R* the critical radius at which particles are shed by the grain boundary vp velocity of mobile particle mp mobility of the particle Fp local driving force... minima value of the free energy density for a given value of ηQ+1 2 2 + η ηQ +1 Figure 4.4: Circular grain containing embedded by an infinite grain Particles are distributed radially inside of the circular grain 47 Figure 4.5: Motion of a particle free grain boundary towards r = 0 (b) Pinning of grain boundary by particle at r = 140 (c) Motion of grain boundary loaded with mobile particle The grain boundary... system in the presence of monodisperse second phase mobile and immobile particles Initially particles are randomly distributed over the volume of the material in two dimensional setting The presence of particles significantly retards the grain growth process The microstructure becomes effectively pinned in presence of immobile particles while the presence of mobile particles allows the motion of grain. .. 4.3 Grain growth mechanism in presence of second phase particles (40) 4.4 Phase field model implementation (43) 4.5 Computational aspects (46) 4.6 Interaction of grain boundary with axisymmetric distribution of particles( 47) 4.7 Interaction of grain boundary with single mobile and immobile particle (54) 4.8 Bicrystal interaction with dense mobile particles 4.9 Conclusions 5 (60) (61) Phase field simulation. .. exerted by the grain boundary V Volume occupied by each particle fv volume fraction of the particles fa area fraction of the particles ′ γb effective grain boundary energy in presence of mobile particles vb p grain boundary velocity in presence of particles υ (= mp/mb) the ratio of particle mobility to grain boundary mobility ζ geometrical constant of the intersecting area of particle on grain boundary... small grain sized samples, the grain growth is expected to be very rapid in single -phase alloys at elevated temperatures due to grain boundary motion without 4 obstacles However in the Mg-Al alloys tested, grain growth even in the smallest grain size samples is found to be minimal [46] Presumably this is because finely dispersed second phase particles present in these alloys control the grain growth. .. energy of grain boundary loaded with particles at radius R(t) Pgg driving force for the particle free grain growth Pdrag pinning force by a single static particle PZ total pinning force per unit area of grain boundary M particle mobility coefficient p total number of particles θ boundary bypass angle r position vector of particle RZ limiting grain size n number of particles per unit volume ns number of particles. .. model for grain growth in the presence of finely dispersed immobile second phase particles Suwa et.al [41] extended the same model to three dimensions However, while there is experimental evidence to indicate that second phase particles may indeed be mobile [42-45], there have been no phase field models for grain growth in presence of dispersed mobile second phase particles The purpose of the research... described in this thesis is to develop a theoretical and computational model for grain growth and its control by the presence of mobile and immobile particles We use the multiphase field approach for simulation of microstructural evolution It is well understood that the growth process is more pronounced in smaller grains than in larger grains In normal grain growth, the driving force for the grain boundary . Phase field simulation of grain growth in presence of second phase particles is submitted in partial fulfillment of the requirements for the award of the degree of Doctor of Philosophy in. mechanisms of grain boundary motion. Our phase field simulation for bicrystal grain growth in presence of particles shows the effect of size and the mobility of particle on the kinetics of grain boundary 1. Phase field simulation of polycrystalline grain growth in presence of mobile second phase particles , 10 th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation