MODELING OF DIFFUSION IN MECHANICAL ALLOYING YANG CHENG NATIONAL UNIVERSITY OF SINGAPORE 2003 MODELING OF DIFFUSION IN MECHANICAL ALLOYING YANG CHENG, M ENG A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2002 Acknowledgement ACKNOWLEDGEMENTS I would like to thank my project supervisors, Associate Professors L Lu and M O Lai for their guidance and encouragement during the course of this research I would also like to acknowledge the support from technicians in Materials Science Laboratory at the Department of Mechanical Engineering, without which this project would not be successful The time we spent together will always be remembered This work is accomplished under a Research Scholarship offered by National University of Singapore Finally, I would like to dedicate this thesis to my family, in the appreciation of their support and inspirations Yang Cheng July 2003 I Contents CONTENTS Acknowledgements I Contents II Summary V List of Figures VII List of Tables X Chapter Introduction Chapter Literature Review 2.1 Introduction 2.2 Diffusion Mechanism 2.3 Thin Film Solution 14 2.4 Grain Boundary Diffusion 16 2.4.1 Introduction 16 2.4.2 Basic Equations 17 2.4.3 Classification 20 Diffusion Through Dislocation 25 Modeling and Computer Simulation 32 Diffusion Model 32 2.5 Chapter 3.1 II Contents 3.2 3.3 Chapter 3.1.1 Introductions 32 3.1.2 Diffusion Model 33 3.1.3 Diffusion Flux 38 Diffusivity 41 3.2.1 Activation Energy 41 3.2.2 Diffusivity 43 Computer Simulation 48 3.3.1 Simulation Program 48 3.3.2 49 Simulation Results Experimentation 51 4.1 Materials and Methods 51 4.2 Procedures and Equipments 52 4.2.1 Sintering and Quenching 52 4.2.2 SPEX Ball Milling 55 4.2.3 XRD Analysis 56 Results and Discussions 58 Lattice Parameter 58 5.1.1 Sintering and quenching of Ni, Al powders 58 5.1.2 Ball Milling of Ni3Al 60 Diffusion Analysis 65 5.2.1 Grain Boundary Diffusion 65 5.2.2 Dislocation Diffusion 66 Chapter 5.1 5.2 III Contents 5.2.3 Stress-assisted Diffusion 69 5.2.4 Diffusion Reaction 70 Diffusion Modeling 71 Conclusions and Recommendations 74 6.1 Conclusions 74 6.2 Recommendations 75 5.3 Chapter References 76 Appendix 83 IV Summary SUMMARY Mechanical Alloying (MA) is a ball milling process where a powder mixture placed in a ball mill is subjected to high-energy collision from the balls The process leads to repetitive plastic deformation, fracturing and cold welding of the powders Diffusion is a fundamental process during mechanical alloying In this thesis, a mathematical model, taking into consideration the critical factors which influence the diffusion process during MA, has been developed to predict the kinetics of diffusive intermixing in a binary miscible system during MA This model divides the MA process into three stages: 1) At the initial stage, the powder particles are cold-welded together to form a laminated structure The chemical composition of the composite materials varies significantly 2) At the second stage, the laminated structure is further refined as fracture takes place The thickness of the lamellae is decreased Although dissolution may have taken place, the chemical composition of the powders may still not be homogeneous A very fine crystalline size can be obtained 3) At the final stage, the lamellae become finer and eventually disappear A homogeneous composition is achieved for all the powder particles, resulting in a new alloy with the composition corresponding to the initial powder mixture In view of the symmetrical and repetitive configuration of these layered structures, a sandwich structure consisting of two adjacent halves of the A and B layers V Summary may constitute a representative element for numerical analysis With periodically repeated deformation in MA, the process of diffusion and homogenization continues till the final stage where the average A element concentration is roughly the same in both A and B composite layers The lamellar structure gradually disappears till A-B alloy is formed Taking into the relative factors of the SPEX ball milling process, a mathematical model could be setup to predict the diffusion process and result during MA During the ball milling process, a change in the lattice parameter is an indication of the diffusion process of the two different elements For ball milling of Ni3Al, the status of the diffusion process is experimentally studied through the calculation of the change in lattice parameter by using the XRD method Comparison between the kinetics predicted by the present model with the relevant experimental data from Ni-Al shows that changes in composition and completion time of diffusion are in good agreement VI LIST OF FIGURES LIST OF FIGURES Chapter Introduction Figure 1-1 A Schematic of mechanical alloying modeling Chapter Literature Review Figure 2-1 Schematic geometry in the Fisher model of GB diffusion [44] 18 Figure 2-2 Schematic illustration of type A, B and C diffusion kinetics according to Harrison's classification [49] 21 Figure 2-3 A typical penetration profile of GB self-diffusion in polycrystalline Ag measured in the C regime (α=17) [53] 25 Figure 2-4 The geometry of the model of diffusion along a curved dislocation: (a) Both the acceleration of diffusion and the difference in geometry are taken into account in calculating the solute concentration at the depth corresponding to the point A (b) The total amount of the solute diffused into the curved and the straight dislocation lines with the same length and the same boundary conditions is calculated 29 VII LIST OF FIGURES Chapter Modeling and Computer Simulation Figure 3-1 The typical lamellar microstructure developed in ball milling of Ni3Al after120 minutes Figure 3-2 The illustration of modeling of A-B alloy diffusion process 34 37 Figure 3-3 Activation energy for diffusion and other processes in Ni-Al system as a function of Ni mole fraction Figure 3-4 Change in effective crystalline size in Ni3Al with MA time 41 45 Figure 3-5 Computer simulation result: A contents change in B-solid solution versus milling time (t) for A-B alloy milled at SPEX ball milling Chapter 49 Experimentation Figure 4-1 Ni-Al phase diagram [90] Changing in lattice parameter of Ni is measured through the different Al contents in Ni, illustrated as broken line 51 Figure 4-2 Sintering process for Ni-Al powder samples 53 Figure 4-3 Quenching process for Ni-Al powder samples 53 Figure 4-4 (a) SPEX 8000 mixer/mill in the assembled condition (b) Stainless steel vial set consisting of vial, lid, gasket, and tungsten carbide balls 54 Figure 4-5 The LAB MERAUN glove box 56 Figure 4-6 Shimadzu XRD-6000 diffractometer and control/analysis system 57 VIII Chapter Conclusions and recommendations Chapter Conclusions and Recommendations 6.1 Conclusions (1) Diffusion is a fundamental process during mechanical alloying, and grain boundary diffusion plays the key role in this process (2) A mathematical model to predict the process of diffusion during mechanical alloying has been established The model takes into three critical factors that influence diffusion in mechanical alloying (3) The kinetics of diffusion involved in the mechanical alloying of Ni3Al binary miscible system has been predicted (4) New findings such as the different stages of diffusion consideration and changing diffusivities predicted by this diffusion model will help to develop greater understanding in the thermodynamic and kinetic considerations during ball milling 74 Chapter Conclusions and recommendations 6.2 Recommendations (1) Mechanical alloying is a very complicated process To conduct thoroughly investigation of the diffusion mechanism, microstructure analysis especially during the MA process must be carried out using SEM/TEM and electron energy spectrum etc (2) The diffusion model proposed in this thesis has been applied only to a two composition alloying system A-B, and only diffusion process has been considered during MA For further study, a multi phase diffusion model which including phase transformation and other phenomena could be set up, using similar analyses methods as provided in present studies 75 References References J S Benjamin, Science America 234-5(1976), p.40 J S Benjamin, New materials by mechanical alloying techniques Oberursel, Germany: DGM Informationgesellschaft, 1989, p 3 J S Benjamin, Metal Powder Rep 45 (1990), p.122 L Lu and M O Lai, Mechanical Alloying, Kluwer Academic Publishers,1998 P Shewmon, Diffusion in Solids, A Publication of The Minerals, Metals & Materials Society, 1989 E M Gutman, Mechanochemistry of materials Cambridge : Cambridge International Science, 1998 E M Gutman, Mechanochemistry of solid surfaces Singapore : World Scientific , c1994 T H Courtney, Reviews in Particular Materials, 2(1994), p.63 D R Maurice and T H Courtney, Metallurgical Transactions, 21A(1990), p.289 10 D R Maurice and T H Courtney, Metallurgical Transactions, 25A(1994), p.147 11 D R Maurice and T H Courtney, Metallurgical Transactions, 26A(1995), p.2431 76 References 12 H Hashimoto and R Watanabe, Materials Science Forum, 88-90 (1992), p 2437 13 N N Thadhani, Progress of Materials Science, 37 (1993), p.117 14 N N Thadhani, Journal of Applied Physics, 76 (1994), p 2129 15 M Li, W J Johnson, and W A Goddard, Materials Science Forum, 179-181 (1995), p 855 16 M Li and W J Johnson, Physics Review Letter, 70 (1993) p 1120 17 V Rosato and C Massobrio, Journal of Alloys and Compounds, 194 (1993), p.439 18 M Li, W J Johnson, and W A Goddard, Materials Science Forum, 179-181 (1995), p 855 19 M Li and W J Johnson, Physics Review Letter, 70 (1993) p 1120 20 V Rosato and C Massobrio, Journal of Alloys and Compounds, 194 (1993), p.439 21 G V Kidson, Journal of Nucleaaar Materials, (1961), p 21 22 A M Gusak and K P Gurov, Fiz Met Metalloved, 63 (1982) P 842 23 D S Williams, R A Rapp And J P Hirth, Thin Solid Films, 142 (1986), p.47 24 Ya Ye Geguzin et al Physics Metal Metallurgy, 47 (1980), p 127 25 U Gosele and K N Tu, Journal of Applied Physics, 53 (1982), p 3252 26 U Gosele and K N Tu, Journal of Applied Physics, 66 (1989), p 2619 27 B M Khina, Physical Review B 44 (1991) P 10778 77 References 28 P J Desre and A R Yavari, Physics Review Letter, 64 (1990), p 1533; 65 (1990), p 2571 29 P J Desre and A R Yavari, Materials Science Forum, 88-90 (1992), p 43 30 A M Gusak and K P Gurov, Solid State Phenomena, 23/24 (1992), p 117 31 R J Highmore and A L Greer, Materials Letter, (1999), p 401 32 R J Highmore, Philosophy Magazine B, 62 (1990), p 455 33 A L Greer, Philosophy Magazine B, 61 (1990), p 525 34 L Lu, M O Lai and S Zhang, Evolution and characterization of a Ni3Al intermetallic compound during mechanical alloying, Materials and Design, Volume 15, Number 2, 1994 35 T H Courtney Process Modeling of mechanical alloying (overview) Materials Transactions, JIM, vol 36, Issue 2, Feb 1995, p110 36 L Lu, M O Lai and S Zhang, Modeling of the mechanical-alloying process, J of Materials Processing Technology, Vol 52, Issue 2-4, June-July 1995, p539.A L Greer, Journal of Magnetism and Magnet Materials, 126 (1993), p 89 37 A K Bhattacharya and E Arzt Scripta Metall Mater 27 (1992), p 749 38 Butyagin, P Journal of Materials Synthesis and Processing, Volume 8, Issue 34, 2000, p 205 39 Kaur, Y Mishin and W GustFundamentals of Grain and Interphase Boundary Diffusion Wiley, Chichester, UK (1995) 40 Y MishinPhil Mag A72 (1995), p 1589 41 Y MishinDefect Diff Forum 143¯147 (1997), p 1357 78 References 42 M N Barber and B W NinhamRandom and Restricted Walks Gordon and Breach, New York (1970) 43 R S BarnesNature 166 (1950), p 1032 44 J C FisherJ Appl Phys 22 (1951), p 74 45 I Kaur, Y Mishin and W GustFundamentals of Grain and Interphase Boundary Diffusion Wiley, Chichester, UK (1995) 46 I Kaur, W Gust and L Kozma Handbook of Grain and Interphase Boundary Diffusion Data, Ziegler, Stuttgart (1989) 47 T Suzuoka Trans Jpn Inst Metal (1961), p 25 48 T Suzuoka J Phys Soc Jpn 19 (1964), p 839 49 L.G HarrisonTrans Faraday Soc 57 (1961), p 1191 50 I Kaur, Y Mishin and W GustFundamentals of Grain and Interphase Boundary Diffusion Wiley, Chichester, UK (1995) 51 E W HartActa Metall (1957), p 597 52 A D Le ClaireBr J Appl Phys 14 (1963), p 351 53 J Sommer and Chr HerzigJ Appl Phys 72 (1992), p 2758 54 R B SchwarzMater Sci Forum 269¯272 (1998), p 665 55 J Friedel Dislocations Pergamon Press, Oxford (1964) 56 D Udler and D.N Seidman Scripta Metall Mater 28 (1992), p 449 57 A A Predvoditelev, N A Tjapunina, G M Zinenkova and G V Bushueva Physics of Crystals with Defects Moscow University Press, Moscow (1986) 58 N F Fiore and C.L BauerProgr Mater Sci 13 (1968), p 85 79 References 59 R W Smith, R Najafabadi and D J SrolovitzActa Metall Mater 43 (1995), p 3621 60 J P Hirth and J LotheTheory of Dislocations McGraw-Hill, New York (1968) 61 S Benhaddad, S Bhan, A Rahmat Effect of ball milling time on Ti-Al and Ni-Al powder mixtures J of Materials Science Letters, Vol 16, Issue 10, 15 May 1997, p855 62 J Borg Richard and G J Dienes, An Introduction to Solid State Diffusion, Publ Academic Press, INC., Harcourt Brace Jovanovich, PUBLISHERS, 1998 63 W D Kingery, H K Bowen and D R Uhlmann, Introduction to Ceramics, 2nd ed., p217-257, John Wiley & Sons, New York 1976 64 D B Miracle Acta metall mater 41 (1993), p 649 65 T Ikeda, A Almazouzi, H Nakamura, M Koiwa, W Sprengel and H Nakajima Acta mater 46 (1998), p 5369 66 C Michaelsen and K Barmak J Alloys Compounds 257 (1997), p 211 67 H J Frost, and M F Ashby, Deformation-Mechanism Maps-The Plasticity and creep of metals and ceramics, Pergamon Press, New York, 1982 68 S K Pabi Computer Simulation of the two-phase diffusion-controlled dissolution in the planar finite multiplayer couples Phys Status Solidi (A) Appl Res Vol 51, Issue 1, Jan 1979, p 281 69 M Sherif El-Eskandarany, A A.Bahgat, N S Gomma, and N A Eissa, Kinetics and Formation Mechanism of Amorphous Fe52Nd48 alloy powder fabricated by mechanical alloying Journal of Alloys and compounds 290 (1999) p 181 80 References 70 A K Bhattacharya and E Arzt, Plastic deformation and it’s influence on diffusion process during mechanical alloying Scripta Metallurgica et Materialia Vol 28, Issue 15 Feb 1993 p 395 71 F Cardellini, G Mazzone, A Montone, and M Vitton Antisari Solid-state reaction between Ni and Al powders induced by plastic deformation Acta Metallurgica et Materialia, Vol 42, Issue 7, July 1994, p2445 72 K Krivoroutchko, T Kulik, H Matyja, V.K Portnoy, V I Fadeeva, Journal of Alloys and Compounds 308 (2000), p 230 73 S K Pabi, B S Murty, Mechanism of mechanical alloying in Ni-Al and CuZn systems, Materials Science and Engineering A214 (1996), p 146 74 J Zbiral, G Jangg and G Korb, Materials Science Forum, 88-90 (1992), p 19 75 C C Koch and Y S Cho, Nanostructured Materials, (1992), p 207 76 L Lutterotti, P Scardi, J Appl Cryst 23 (1990), p 246 77 C N J Wagner, Local Atomic Arrangement Studied by X-ray Diffraction, Vol 36, Gordon and Breach, New York, 1966 (Chapter 7) 78 H Pal, S K Pradhan, M De, Jpn J Appl Phys 32 (1993), p 1164 79 H Pal, S K Pradhan, M De, Mater Trans JIM 36 (1995), p 490 80 S K Shee, S K Pradhan, M De, J Alloys Comp 265 (1998), p 249 81 G K Williamson, R E Smallman, Philos Mag (1956), p 34 100 A Chanda, H Pal, M De, S Kajiwara, T Kikuchi, Mater Sci Eng A265 (1999), p 110 101 A Guinier, X-ray Diffraction, Freeman, San Francisco, 1963, p 124 102 H Pal, A Chanda, M De, J Alloys Comp 278 (1998), p 209 81 References 103 S K Shee, S.K Pradhan, M De, Mater Chem Phys 52 (1998), p 228 104 B E Warren, X-ray Diffraction, Addison-Wesley, Reading, MA, 1969 (Chapter 13) 105 C N J Wagner, Local Atomic Arrangement Studied by X-ray Diffraction, Vol 36, Gordon and Breach, New York, 1966 (Chapter 7) 106 J A Brusso, D E Mikkola, J Mater Res (1994), p 1742 107 L Lutterotti, S Gialanella, R Caudran, Mater Sci Forum 551 (1996), p 228 108 P Shewmon, Diffusion in solids, A Publication of The Minerals, Metals & Materials Society 109 M E Glicksman, Diffusion In Solids, John Willy and Sons, INC., 1999 82 Appendix A Computer simulation program Appendix A Computer Simulation Program /*This program computes the diffusion completion time during Mechanical Alloying According to the diffusion theory, The diffusion concentration C (x,t) = C/2 * ( - erf( x / 2sqrt(DT))) Diffusion flux N=Integrate (J*dt)=Integrate (D (dC/dx)dt)=C * sqrt(D*t/3.14159) Input: The original crystal diffusion coefficient Dl, Db; The collision time interval of ball milling InterTime; The original and final powder particle size d1 and d2; Output: The collision times I during ball milling, The diffusion completion time CompTime -*/ #include #include #include 83 Appendix A Computer simulation program int main(void) { const double Pi = 3.14159; //arithematic constant; cout > Dl; cout > Db; cout > InterTime; cout > d1 >> d2; double AreaIncrease = pow((d1/d2),(2/500)), //area increase for each collision; DiffIncrease = pow((d1/d2),(1/500)), //diffusivity increase for each collision; DiffQuant = sqrt ( Dl * InterTime / Pi ),//diffusion quantity for the first time; Sum = DiffQuant, I = 1; 85 Appendix A Computer simulation program { DiffQuant = (1-DiffQuant) * (sqrt(Dl*InterTime / Pi)), I++, //to calculate the collision times; Sum += DiffQuant; //to calculate the total quantity of diffusions; cout [...]... the increased concentration of vacancies and dislocations Like most reactions in solids, diffusion is a fundamental process during mechanical alloying [34-36] By analysis of the special characteristics of the ball milling process, three critical factors which influence diffusion in mechanical alloying have been found: 1) decrease in crystalline size which increases the diffusion area, and changes in diffusion. .. gap and bring together the most interesting and promising results This is schematically shown in Figure 1-1 Further progress in this area necessitates the development of novel macro kinetic models linking the plastic strain and generation of defects to the mechanisms and kinetics of metastable phase transformation In particular, the intermixing of atoms at the interfaces of metals A and B in lamellar... mechanical alloying of ductile-ductile A-B alloy, since A and B form layered structure and the diffusion time is very short, about 100 seconds [37], the thin film solution can be used 2.4 Grain Boundary Diffusion 2.4.1 Introduction Grain boundary (GB) diffusion plays a key role in many processes occurring in engineering materials at elevated temperatures, such as Coble creep, sintering, diffusion- induced... Modeling (Tad, ∆G) Plastic Strain Strain Rate Lamellar Thickness Solute Distribution Defect Density Kinetic Modeling Concentration Changes Solubility Metastable Extension Phase Lattice Modeling Phase Formation Kinetics Nano grains Amorphous Phase Figure 1-1 A schematic of mechanical alloying modeling 5 Chapter 1 Introduction (5) MA modeling as present study The analyses discussed indicate that there is an... volume diffusion to grain boundary diffusion; 2) increase in density of defects in alloy which can decrease the activation energy, resulting in an increase in diffusivity; and 6 Chapter 1 Introduction 3) repeated fracturing and cold-welding of the powder particles which enable the powder particles to be always in contact with each other with atomically clean surfaces and minimized diffusion distance In. .. the rearrangement of atoms in the nucleus of dislocation Dislocation movement and formation of dislocations are obvious manifestation of heavy plastic deformation The rearrangement of atoms during milling can be analyzed using classical chemical kinetics During mechanical alloying, mechanical energy can partially be stored by the creation of dislocations and grain boundaries in the mechanically alloyed... phase-formation kinetics The development of kinetic models is important for a deeper understanding of the intricate mechanisms involved in MA (4) Kinetic Models Most of the phenomenological kinetic models used for describing metastable phase formation were originally developed for a crystalline–phase layer growth and solid-state amorphization during thermal annealing of planar-binary multiplayer diffusion. .. very rapidly to result in a decrease in diffusion coefficient Defects probably contribute little to the increase in homogenization kinetics in such diffusion process However, the density of defect during mechanical alloying increases with mechanical alloying duration and therefore it significantly contributes to homogenization kinetics, which help to complete the diffusion process 12 Chapter 2 Literature... of the second (intermetallic) phase in thin films However, all these models do not 4 Chapter 1 Introduction consider the generation of defects (i.e., vacancies and dislocations) in metal particles during MA because of repeated plastic deformation Direct observation of defects during the formation of phase in MA can be found in some works [21-33] Disorder in intermetallic compounds during ball milling... based on the consideration of the three critical factors above has been developed to predict the kinetics of diffusive intermixing in a binary miscible system in the course of mechanical alloying Comparison between the kinetics predicted by the present model with the relevant experimental data from Ni-Al shows that changes in composition and completion time of diffusion are in good agreement 7 Chapter .. .MODELING OF DIFFUSION IN MECHANICAL ALLOYING YANG CHENG, M ENG A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE... contribute little to the increase in homogenization kinetics in such diffusion process However, the density of defect during mechanical alloying increases with mechanical alloying duration and therefore... influence diffusion in MA: (a) Decrease in crystalline size increases the diffusion area, and changes the diffusion mechanism from volume diffusion to grain boundary diffusion; (b) Increase in