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INTRINSIC DIFFUSION SIMULATION FOR SINGLEPHASE MULTICOMPONENT SYSTEMS AND ITS APPLICATION FOR THE ANALYSIS OF THE DARKEN MANNING AND JUMP FREQUENCY FORMALISMS By NAGRAJ SHESHGIRI KULKARNI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Nagraj Sheshgiri Kulkami DEDICATED This dissertation is dedicated to my father, the late Dr. S. N. Kulkarni, himself a scientist and advisor to many aspiring organic chemists. The unwavering support of my mother (Amma) and brothers, Anurag and Nandkishore, was deeply felt during the course of this quest for which I am eternally grateful. ACKNOWLEDGMENTS I would like to express my deepest thanks to my advisor, Prof. Robert T. DeHoff, for his invaluable guidance and support during the course of this long journey. His scientific temperament, intuition and unique approach to our field are hard to put in words. Only those among us, who have had the privilege to have worked and studied under his tutelage, can truly appreciate his contributions. I thank Profs. Rajiv Singh, E. Dow Whitney, Gerhard Fuchs and Timothy Anderson for serving on my supervisory committee. My thanks also to the previous members of my committee, Profs. Michael Kaufman and Robert Scheaffer, who were unable to serve during my dissertation defense, but had previously attended my candidacy exam. The support and assistance of faculty members, staff and colleagues at the Department of Materials Science & Engineering is gratefully acknowledged. My heartfelt appreciation also to the services provided by the University of Florida Libraries. My special thanks to Dr. Theodore Besmann at Oak Ridge National Laboratory and Oak Ridge Associated Universities for their support and understanding during the final stages of my dissertation. Finally, I am forever grateful for the support provided by my friends, both in the U.S. and in India, particularly those with whom I battled on the tennis court or near the blackboard. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................. . . iv LIST OF FIGURES .............................................. viii ABSTRACT .................................................. xiii CHAPTER 1 INTRODUCTION ................................... ...........1 2 PHENOMENOLOGICAL DIFFUSION FORMALISMS .................. 9 2.1 Diffusion Formalisms: An Overview ................. . . ... 9 2.2 Experimental Observables in Diffusion ................... 12 2.2.1 Interdiffusion Experiments ................... ..... 12 2.2.2 Intrinsic Diffusion Experiments ................... 16 2.2.3 Tracer Diffusion Experiments ................... . 20 2.3 Diffusion Formalisms ................... ............ . 21 2.3.1 Interdiffusion Formalism ................... ......... 22 2.3.2 Intrinsic Diffusion Formalism .......... ..... 23 2.4 Problems with the Phenomenological Formalism and its Implementation .................. ................... ...24 2.4.1 Interdiffusion Problems ................... ...... . 24 2.4.2 Intrinsic Diffusion Problems ................... . 28 2.5 Connection Between Tracer, Intrinsic and Interdiffusion Information 30 2.6 Simplifications to the Phenomenological Formalism ........... ... 33 2.6.1 Intrinsic Diffusion Coefficients ................... 34 2.6.2 Interdiffusion Coefficients .................. ......... 34 2.6.3 Problems with the Simplified Version ................... 36 3 INTRINSIC DIFFUSION SIMULATION ................... .... 38 3.1 The Need for an Intrinsic Diffusion Simulation ............... 39 3.2 The Simulation ................... ................... ....... 42 3.2.1 Input for the Simulation .............. ............. 45 3.2.2 The Setup ...........................................46 3.2.3 RunTime Iterations .................. ................ 50 3.3 Test of the Simulation for a Model System ................. 55 3.3.1 Analytical Model System with Constant Molar Volume ....... 55 3.3.2 Analytical Model System with Variable Molar Volume ...... 66 3.4 Test of the Simulation for the FeNi system ................ 76 4 TESTS OF DARKENMANNING THEORIES USING THE INTRINSIC DIFFUSION SIMULATION ................... ... 80 4.1 Importance of DarkenManning Theories ................... 81 4.2 The Darken Theories ................... ......... . 84 4.2.1 Darken A: Intrinsic Diffusion ................... . 84 4.2.2 Darken B: Relation Between Intrinsic and Tracer Diffusion The Mobility Concept ................... ......... 93 4.2.2.1 Relation between intrinsic and interdiffusion coefficient, mobility and activity .............. ... 94 4.2.2.2 Relation between tracer diffusivity and mobility ...... 96 4.3 Analysis of the Darken Theories with the Phenomenological Equations .................. ............... 99 4.4 M anning Theory ................... ............ . 103 4.5 Previous Tests of DarkenManning Theories ................ 106 4.6 Tests of DarkenManning Theories with the Simulation ............ 114 4.6.1 A. Establishing Consistency of Experimental Measurements and Procedures ................. ... ..... ... ....... 115 4.6.2 B. Procedure for the Test of the DarkenManning Theories .. 119 4.6.3 SilverCadmium (AgCd) System ........... . ..121 4.6.3.1 Examining selfconsistency of experimental intrinsic data ......... ................... 122 4.6.3.2 DarkenManning relations for AgCd .......... ... 126 4.6.4 GoldNickel (AuNi) System ............... ........ 143 4.6.4.1 Examining selfconsistency of experimental intrinsic data ............... ............. 144 4.6.4.2 DarkenManning relations for AuNi .......... ... 158 4.6.5 CopperZinc (CuZn) System ................... ...... 177 4.6.5.1 Examining selfconsistency of experimental intrinsic data ............. .............. 178 4.6.5.2 DarkenManning relations for CuZn system ........ 184 4.6.6 CopperNickel (CuNi) System ................... .199 4.6.6.1 Examining selfconsistency of experimental intrinsic data ................ .............. 200 4.6.6.2 DarkenManning relations for CuNi .......... ... 205 5 KINETIC FORMALISM FOR INTRINSIC DIFFUSION BASED ON ATOM JUMP FREQUENCIES ...................... 5.1 Jump Frequency Formalism ......... ........... 5.2 Unbiased Intrinsic Flux ....................... 5.2 .1 C u Z n .............................. 5.2.2 CuN iZn ........................... 5.3 Biased Intrinsic Flux ........... . . . . 5.4 Unbiased Intrinsic Flux Using Effective Jump Frequencies 5.4.1 Effective Jump Frequency Model for FeNiCo 5.4.2 Darken's "Uphill Diffusion" Experiment ...... 5.4.3 Evaluation of Effective Jump Frequencies from Intrinsic Data ......... ............... 6 CONCLUSIONS AND FUTURE WORK ................. . . 219 . . 220 . . 229 ...........229 ...........231 . . 234 . . .. 238 . . .. 239 . . 239 ...........242 . . 244 APPENDIX A FRAMES OF REFERENCE ..... ............. . B THERMODYNAMIC RELATIONS FOR DIFFUSION ANALYSIS C DIFFUSION COEFFICIENTS IN DIFFERENT FRAMES OF REFERENCE ................... ......................... D MOLAR VOLUME EFFECTS ON REFERENCE VELOCITIES ... . . 251 . . 264 .......267 . . 278 E MEASUREMENT OF LATTICE VELOCITY USING AN OBLIQUE INERT MARKER INTERFACE ................. .............. REFERENCES . . . BIOGRAPHICAL SKETCH ................... ................... .... ... 283 . 293 . 302 LIST OF FIGURES Figure Page 21 Composition distribution in a diffusion couple ................... 13 22 Interdiffusion flux in a flowing system. ................... ...... . 14 23 Calculation of the interdiffusion flux directly from the concentration profiles using Dayananda's equation ................... ....... 18 24 Displacement of wires in an interdiffusion experiment for the CuZn system, a phenomenon known as the Kirkendall effect .............. 19 25 Oblique marker plane for the determination of lattice velocities ............ 19 31 Setup for the intrinsic diffusion simulation ................. ..... . 48 32 Flowchart depicting the various steps in the intrinsic diffusion simulation .... 49 33 Variation of the intrinsic diffusion coefficients with composition for the model system ................. ..............................58 34 Composition profile for the model system ................. ......... 62 35 Variation of the lattice velocity and lattice shift as a function of position ..... 63 36 Dependence of the lattice shift and lattice position on the initial lattice position ................ ............................ 64 37 Diffusion fluxes for the model system ................... ...... 65 38 Variation of the molar volume and intrinsic diffusion coefficients with composition for the model system with varying molar volume ......... 68 39 Concentration profile (in mol/cc) for the model system with variable molar volume ................... ................... ......... 72 310 Composition profile (in atom fraction) for the model system .............. 73 viii 311 Mean and lattice velocities, and lattice shift as a function of position ........ 74 312 Diffusion fluxes for the model system with variable molar volume .......... 75 313 Application of the intrinsic diffusion simulation for the FeNi system at 12000C ........ ......................... ............. 77 314 Intrinsic fluxes and lattice velocity for the FeNi system at 12000C.. ........ 78 41 BoltzmanMatano analysis for the determination of the interdiffusion coefficient.......................................................86 42 Procedure for testing the DarkenManning theories using the intrinsic diffusion simulation ................... ................... ..... ... 116 43 Phase diagram and diffusion coefficients in the AgCd system ............ 124 44 Simulated concentration profiles in the AgCd system . . . 127 45 Lattice shift profiles for diffusion couples. ................. . ... 129 46 Simulated output in the AgCd system using the experimental intrinsic diffusion coefficients. ................... .............. . 131 47 Tracer diffusion coefficients and thermodynamic data in the AgCd system at 6000C......................... ................. 133 48 Correction factors obtained from the Manning theory in the AgCd system .. 135 49 Comparison between the diffusion coefficients predicted by the DarkenManning theories and the experimental coefficients .............. 138 410 Test of the DarkenManning theories in the AgCd system .............. 140 411 Composition penetration curves obtained using the theoretical intrinsic diffusion coefficients as the input to the simulation .......... ... .. 141 412 Comparison between experimental and predicted Kirkendall shifts and Kirkendall plane compositions ................... ........... 142 413 Binary AuNi phase diagram ................... ................... 145 414 Lattice parameter and molar volume as a function of composition in the AuNi system ................................... ............. 151 415 Diffusion coefficients in the AuNi system at 900'C ................ 152 416 Output using the intrinsic diffusion coefficients as input ............. 153 417 Procedure for determining the number of iterations in the simulation, that are needed to obtain a stable output ................... ........ 154 418 Output using the intrinsic diffusion coefficients as input ............. 155 419 Diffusion fluxes as a function of the Boltzmann variable ............. 156 420 Mean volume (vv), mean number (vA), and lattice velocity (vK) as a function of the Boltzmann parameter at 900C ............. ...... 157 421 Tracer diffusion of gold as a function of reciprocal temperature for AuNi alloys .............. ................................ 160 422 Tracer diffusion coefficients of Au and Ni in AuNi alloys at 900'C. ..... 161 423 Thermodynamic data for the AuNi system at 900C ......... . 162 424 Intrinsic diffusion coefficients of Au and Ni computed using the Darken theory for the AuNi system at 900C.................. ......... 163 425 Interdiffusion coefficient for the AuNi system computed using the Darken theory ................... ................... ......... 164 426 Vacancy Wind terms for the AuNi system at 900C ......... . 170 427 Output using the intrinsic diffusion coefficients predicted from the Darken theory as input ................... ................... ... 171 428 Concentration profiles using the intrinsic diffusion coefficients predicted from the Darken theory as input. ......... ............... 172 429 Diffusion fluxes obtained using the Darken intrinsic coefficients as input .... 174 430 Mean volume (vv), mean number (vA), and lattice velocity (vK) as a function of the Boltzmann parameter at 900C . ... . 175 431 Binary CuZn phase diagram ......... .. ................. 181 432 Intrinsic diffusion coefficients in the fcc phase of the CuZn system ........ 181 433 Simulated composition profile obtained using the intrinsic diffusion coefficients at 780'C reported by Home and Mehl ................ 182 434 Comparison between experimental and predicted Kirkendall shifts and Kirkendall plane compositions. ................... .......... 183 435 Tracer diffusion coefficients of Cu and Zn as a function of 1/T ............ 185 436 Tracer diffusion coefficients of Cu and Zn as a function of composition ..... 186 437 Thermodynamic data in the afcc CuZn system at 780'C ................ 188 438 Intrinsic diffusion coefficients predicted by the Darken (solid lines) and Manning (dashed lines) theories ......................... .... 189 439 Correction factors obtained from the Manning theory at 780'C ............ 190 440 Ratio of the experimental and predicted intrinsic diffusion coefficients ...... 191 441 Interdiffusion predictions from the Darken and Manning theories for the afcc CuZn system at 780'C ................... ....... 192 442 The "S" correction factor for the interdiffusion coefficient given by the Manning theory ................... ................... ......... 193 443 Comparison between the experimental and predicted composition profiles ... 196 444 Comparison between the experimental and predicted Kirkendall shifts ...... 197 445 Comparison between the experimental and predicted Kirkendall plane compositions. ........................................... 198 446 Binary CuNi phase diagram ............ ................... 200 447 Intrinsic diffusion coefficients in the afcc CuNi system at 800'C ......... 203 448 Simulated concentration (150 h) and lattice shift profiles (55 h) at 1000C ... 204 449 Tracer diffusion coefficients in the CuNi system at 10000C .............. 207 450 Thermodynamic data in the afcc CuNi system at 1000C .............. 209 451 Intrinsic diffusion coefficients predicted by the Darken (solid lines) and Manning (dashed lines) theories at 1000C ................... . 210 452 Correction factors from the Manning theory at 1000C. . . . 211 453 Interdiffusion coefficient predicted from the Darken (solid line) and Manning (dashed line) theories ................... .......... .... 212 454 The "S" correction factor for the interdiffusion coefficient given by the Manning theory ......... .................................. 212 455 Composition profiles (solid lines) using the predicted intrinsic diffusion coefficients ................... .................................213 456 Lattice shift profiles (solid lines) using the predicted intrinsic diffusion coefficients ................... .................................. 214 457 Output for a proposed diffusion couple that is Nirich ................. 217 51 Schematic representation of two adjacent planes in a solid ........... 222 52 Comparison between the unbiased jump frequency and Fick's formalisms in the a fcc CuZn system at 780'C. ........................ . 230 53 Tracer diffusion coefficient of Cu for the ternary CuNiZn system ......... 232 54 Diffusion composition paths in the CuNiZn system at 900'C ............ 233 55 Total, biased and unbiased components of the intrinsic flux for a diffusion couple in the CuZn system ................ ......... 236 56 Composition profiles and bias factors for the diffusion couple in the CuZn system ............................................ 237 57 Composition paths and jump frequency model for the FeCoNi system ..... 241 58 Simulation (solid line) of Darken's experiment demonstrating uphill diffusion of carbon in y iron at 1050'C using ajump frequency model ...... 242 E1 Procedure for preparing an "Oblique Interface" in a diffusion couple ....... 290 E2 Experimental determination of lattice shift as a function of marker position 291 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INTRINSIC DIFFUSION SIMULATION FOR SINGLEPHASE MULTICOMPONENT SYSTEMS AND ITS APPLICATION FOR THE ANALYSIS OF THE DARKEN MANNING AND JUMP FREQUENCY FORMALISMS By Nagraj Sheshgiri Kulkami May 2004 Chairperson: Robert T. DeHoff Major Department: Materials Science and Engineering A multicomponent, singlephase, diffusion simulation based on Darken's treatment of intrinsic diffusion has been developed, which provides all the information available from an intrinsic diffusion experiment, including composition profiles and diffusion paths, lattice shifts and velocities, intrinsic and interdiffusion fluxes, as well as fluxes and mean velocities in different frames of reference. The various steps involved in the simulation are discussed and the selfconsistency of the simulation is tested with the aid of model systems having constant and variable molar volumes. After an examination of the historical development of the DarkenManning theories and a brief discussion of previous tests in the literature, a systematic procedure for the comprehensive assessment of these theories is proposed in which the intrinsic diffusion simulation developed in this work occupies a central role. This procedure is then utilized to perform an assessment of the DarkenManning relations for four binary systems: AgCd, AuNi, CuZn and CuNi. It is shown that the DarkenManning relations that provide the connection between the tracer, intrinsic and interdiffusion coefficients, are unsatisfactory. Hence, it is suggested that the development of multicomponent diffusion databases, which often use the Darken relations for the evaluation of the phenomenological coefficients, may be compromised. As an alternative to the traditional phenomenological formalism of multicomponent diffusion, a kinetic formalism based on atom jump frequencies is proposed. An expression for the intrinsic flux in terms of an unbiased and a biased component is derived. It is demonstrated with the aid of the simulation for the CuZn system, that the biased flux may be evaluated from the experimental intrinsic flux and the unbiased flux (obtained from the tracer jump frequency). An unbiased jump frequency formalism that utilizes effective rather than tracer jump frequencies and avoids the complexities associated with bias effects has shown some success in modeling diffusion in ternary systems such as FeNiCo and the FeCX systems, using simple jump frequency models obtained through an informed trial and error approach. A procedure to compute the effective jump frequency at every composition in a diffusion couple from the experimental intrinsic flux is discussed. It is suggested that the effective jump frequency approach, if validated through intrinsic diffusion measurements in ternary systems, offers the potential for building physically meaningful diffusion databases in multicomponent systems. CHAPTER 1 INTRODUCTION The study of diffusion and diffusion related processes in solidstate materials encompasses a broad spectrum of modern technologies. Semiconductors, precipitation hardened superalloys, metal and ceramic matrix composites, corrosion resistant coatings and alloys are all subject to complex diffusion processes that are dependent on a variety of operational parameters. The ability to analyze and model diffusion processes so as to be able to predict the variation in the relevant properties of such advanced materials assumes even greater significance today given the everincreasing demands that are being placed on them. Global competition and budgetary constraints have reduced the time and finances available for significant experimental investigations due to which modeling efforts in fundamental areas such as diffusion are being increasingly emphasized. The present work is an effort in this direction. Most of the present diffusion modeling efforts owe their origin to the phenomenological formalism first proposed by Adolf Fick [Fic55] in 1855 for the description of diffusion in binary electrolytes, which relates diffusion fluxes to driving forces which in his case were the concentration gradients. Since that time, significant progress has been made especially in the adaptation of the phenomenological formalism to multicomponent systems and the associated developments in the field of irreversible thermodynamics [Ons31, Ons32a, Pri55, Lec53, Gro62, How64, Man68, Haa69, Phi91, A1193]. Typical studies in diffusion examine the diffusion of components with respect to a fixed frame of reference, the socalled laboratory frame. This description of diffusion behavior known as interdiffusion has proven to be rather inadequate for the description of diffusion in solid state crystalline materials (indeed in liquids and gases as well) since variations in the movements of the individual components are not explicitly considered. For example in a singlephase binary system, there exists only a single independent interdiffusion flux and a single interdiffusion coefficient, which effectively means that differences in the actual movements of the two components are not accessible in the description. That these differences are quite important was first suggested by Smigelskas and Kirkendall [Smi47] in their description of the "Kirkendall Effect," a phenomenon in which imbalances in atom flows caused the appearance of porosity in their a CuZn diffusion annealed couples. To explain their observations, Kirkendall proposed an atom vacancy exchange mechanism as the primary diffusion mechanism for their system rather than the direct exchange mechanism [Zen50] that was prevalent at the time. Later, Darken [Dar48] in his seminal paper of 1948 provided a formal treatment that helped explain Kirkendall's observations by describing the intrinsic diffusion of each atom species with respect to a frame of reference that effectively moves with the lattice on which the atoms reside, the socalled lattice frame of reference. In Darken's treatment of binary singlephase diffusion, for example, in case of a CuZn, differences in diffusion behavior are explicitly considered by defining intrinsic diffusion coefficients and intrinsic diffusion fluxes for each component that are related to their respective concentration gradients. Darken's purely macroscopic formalism was followed by the work by Seitz [Sei50, Sei53] who provided an atomic interpretation of the Kirkendall effect and thus firmly established the vacancy diffusion mechanism as the primary diffusion mechanism in metals. Darken was also able to relate the diffusion behavior of atoms in a homogenous system to that in a flowing system in the form of relations commonly referred to as the Darken relations. This led him to suggest a more suitable formalism for diffusion in multicomponent systems in which the driving forces for diffusion are the chemical potential gradients rather than the concentration gradients as embodied in the Fickian formalism. He was able to demonstrate the utility of this formalism by providing an explanation for the phenomenon of "uphill diffusion" (i.e., diffusion of a component in a direction opposite to its concentration gradient) of carbon in the presence of silicon in y iron [Dar49, Dar51]. Darken's analysis of diffusion in solid state systems was based primarily on earlier treatments for aqueous systems by Onsager and Fuoss [Ons32b], and Hartley [Har3 1]. These and other efforts led to the development of the field of irreversible thermodynamics of which solid state diffusion is an important component. Unfortunately, the determination of the kinetic descriptors (e.g., intrinsic diffusion, interdiffusion coefficients or mobilities) of diffusion in the various phenomenological formalisms has turned out to be a rather nontrivial endeavor, especially for multicomponent systems. The difficulty stems primarily from the fact that the kinetic descriptors of diffusion in multicomponent systems are in the form of a matrix, the evaluation of which is extraordinarily difficult. Furthermore this matrix cannot be determined for systems containing more than three components without simplifying assumptions. More worrisome is the fact that the actual values of the terms 4 of the matrix are dependent on the choice of independent concentration variables1 used in the description of the diffusion fluxes, and can even be negative. Indeed phenomenological coefficients by their very definition do not have a direct relation to the actual physics of the diffusion process, i.e., the jumping of individual atoms in a multicomponent system. As a result, efforts geared towards the systematic development of diffusion databases [Cam02, BorOO] in multicomponent systems have not been easy without some simplifying assumptions. These assumptions usually involve ignoring the crossterms in the matrix of phenomenological coefficients (i.e., L, = 0, ij), and relating the ondiagonal terms to the tracer diffusion coefficients using the Darken theory. Thus, the Darken relations that connect tracer diffusion measurements in homogenous systems, with the phenomenological coefficients in gradient systems, still occupy a central role in current diffusion applications. In spite of the fact that the Darken theory [Dark48] has been around for more than fifty years, it was felt that a critical assessment was lacking. It was found that many of the earlier tests of the theory were incomplete in several respects. Particularly lacking was the ability to compare the experimental diffusion data (e.g., concentration profiles and lattice shifts) with those predicted by the Darken theory. Since an intrinsic diffusion simulation is necessary to perform such a comparison, an important goal of the present work was the development of such a simulation, and its utilization for the assessment of 1 In an n component system, there are n concentration variables; however only (n 1) of these are independent since there exists a relation between them. If ck is the concentration and Xk the mole fraction of component k, then since the sum of the mole fractions is one, = X = , where Vis the molar volume. k=1 V k=1 V the Darken relations in a number of binary systems. The Manning corrections [Man68, Man70] to the Darken relations, which take into account correlation and vacancy wind effects during diffusion, are usually too small to be of major consequence; nevertheless, these were also considered in the present endeavor. A kinetic approach that is based on the effective atom jump frequencies was previously proposed by Iswaran [Isw93] and DeHoff [Deh02] as an alternative to the traditional phenomenological formalism. Although kinetic formalisms based on atom jump frequencies had been discussed earlier in the literature by Manning [Man68], LeClaire [Lec58] and Seitz [Sei50], they are usually too complicated or impractical to be utilized for the analysis of diffusion in practical multicomponent systems. Iswaran and and DeHoff were able to devise a rather simple yet efficient kinetic diffusion formalism that could be easily utilized for the analysis of diffusion in practical multicomponent systems with the aid of an intrinsic diffusion simulation that they also developed. Suitable jump frequency models were concocted using an "informed trial and error" approach to describe diffusion in several isomorphous ternary systems with the aid of the simulation [Isw93, Deh02]. In spite of the reasonable success achieved, this approach was incomplete in several respects. Firstly, their kinetic flux expression did not explicitly take into account socalled "bias" effects that are considered in the Darken formalism [Dar48, Dar49] as well as in the previous kinetic formalisms of Manning [Man67, Man70, Man71, Man89] and others [Sei50, Lec58, How64, A1193]. Secondly, although the simulation modeled intrinsic diffusion using the atom jump frequency formalism, Iswaran and DeHoff did not extend its application for the purpose of examining other intrinsic formalisms in the literature, most notably the Darken intrinsic diffusion formalism. Thirdly, the DarkenManning relations that provide a connection between diffusion in homogenous and flowing systems and are an integral part of the prevalent diffusion theory were not critically examined with the aid of the simulation, even though that was not the focus at the time. A fourth point of weakness was in the development of jump frequency models using an "informed trial and error" approach that obviously could not be replicated by an individual not knowledgeable in the field. Lastly, the simulation itself was developed in a now rarely used programming languageTurboPascal, which again requires some expertise and is not very user friendly. These and other issues in diffusion modeling and analysis are considered in the present dissertation. In Chapter 2 of this dissertation, some of the more traditional phenomenological formalisms including those proposed by Fick, Darken and Onsager are considered. The trouble with these formalisms is highlighted. Simplifications employed so as to make the formalisms more tractable are discussed. The development of an intrinsic diffusion simulation in MathCad [MatOl] is the focus of Chapter 3. The details of the simulation are discussed in some length with the aid of a flowchart. Model systems with constant and variable molar volumes, and a practical system (FeNi) are used to illustrate the special features of the simulation. Chapter 4, the main focus of this dissertation, discusses the utilization of the simulation for the analysis of the DarkenManning relations, which are of great importance in present diffusion theory. Previous tests of the DarkenManning relations in the literature are first briefly examined. A systematic procedure for the assessment of these relations with the aid of the simulation is presented. Using this procedure, the DarkenManning relations are critically examined for four wellcharacterized binary systems (AgCd, AuNi, CuZn and CuNi) and important conclusions are reached on these assessments. The jump frequency formalism is the main theme of Chapter 5. An expression for the intrinsic diffusion flux of a component in a multicomponent system is first derived. The significance of the biased and unbiased terms in the flux expression is then considered. Even though the physical content of the bias term in this formalism is unknown at the present time, a novel application of the simulation for computing the bias terms, using the experimental intrinsic diffusion data in the CuZn system, is demonstrated. A simplified version of the jump frequency formalism based on the unbiased version is suggested as a practical way of avoiding the complications associated with bias effects. The construction of effective jump frequency models in this formalism is explored with the aid of a few ternary systems including Darken's "uphill diffusion" system. Rather than using an informed trial and error approach for the development of jump frequency models, an alternate procedure for the direct computation of the jump frequencies from the intrinsic diffusion fluxes is proposed. It is shown that the lattice velocities needed for the determination of the intrinsic fluxes can be obtained using an oblique marker technique, the construction of which is examined in Appendix E. Chapter 7 summarizes the findings from the present work regarding the Darken Manning relations. The utility of the simulation developed in this work is highlighted. Some important advantages of the jump frequency approach for the potential development of diffusion databases are presented. A number of recommendations for further research in the field of solid state diffusion are also suggested. Several rather 8 detailed appendices are provided at the end of the dissertation as an aid to the discussions in various chapters. CHAPTER 2 PHENOMENOLOGICAL DIFFUSION FORMALISMS 2.1 Diffusion Formalisms: An Overview Diffusion must occur in any process that requires a change in local chemistry. In crystalline systems this requires that single atoms change their position on the crystal lattice. The average frequency with which this event occurs in a local volume element is different for each chemical component in the system, and varies with the composition and temperature of the volume element. The pattern of these events ultimately determines the evolution of the spatial distribution of the elements in a single phase crystal. Prediction and control of this chemical evolution require an understanding of the diffusion process. To achieve this understanding it is first necessary to be able to describe the diffusion process, i.e., to develop a formalism that relates how the atoms move to the current condition of the system. The phenomenological equations of irreversible thermodynamics [Haa69, Kir87, She89, Phi91] define fluxes as measures of the atom motions and relate them to forces defined in terms of gradients of the properties of the system calculated from the current condition of the system. These equations have been widely accepted as the basis for the description of the diffusion process. Application of the phenomenological formalism to the description of interdiffusion of the elements in binary systems is straightforward and requires modest experimental effort. However, for three component systems, the formalism becomes 10 cumbersome and the level of experimental effort required to evaluate the matrix of four interdiffusion coefficients defined for the system is significant. As the number of components in the system increases beyond three, the number of diffusion coefficients needed to describe behavior increases rapidly. Further, values for the matrix of diffusivities in quaternary and higher order systems cannot be determined experimentally. Evidently a practical description of diffusion behavior in multicomponent systems would require an ability to understand the physical content of these diffusion coefficients so that their values could be computed or at least estimated from more fundamental information. Tracer diffusion experiments, which report the penetration of a radioactive tracer of an element into a chemically homogeneous alloy, provide fundamental information about the frequency with which atoms of a component make a jump between adjacent sites on a crystal lattice. If the physical content of interdiffusion coefficients is to be understood, their values must be related to this fundamental information about the pattern of atom jump frequencies in the system. Kirkendall's classic observation that the crystal lattice moves during diffusion first complicated, then clarified this situation. Tracer information describes atom motion relative to the crystal lattice, but the Kirkendall observation showed that the crystal lattice moves in interdiffusion. Darken solved this problem by introducing the concept of intrinsic diffusion, describing motion of atoms in the nonuniform flowing system relative to the moving lattice. The stage was set for what became the central theoretical problem in multicomponent diffusion: the determination of the connection between the fundamental tracer diffusion information, through the intrinsic diffusion coefficients to the practical interdiffusion coefficients. Without this theoretical connection the description of diffusion in multicomponent systems would remain impractical. Subsequent sections in this chapter review the experimental observables for each of the three kinds of diffusion experiments: tracer, intrinsic and interdiffusion. The formalisms devised to describe each of these kinds of experimental results are then recalled briefly. Problems with the formalisms themselves are discussed. The established theoretical connection between the three kinds of information and experimental tests is briefly discussed; a detailed presentation of this theory and the experimental results that illuminate the inadequacies of the theory connecting these kinds observations are the focus of Chapter 4. These deliberations support the thesis that the traditional phenomenological description of diffusion does not provide a useful or practical basis for collecting and organizing information about diffusion in a multicomponent system. Alternate approaches to this problem are needed. One strategy for overcoming some of these problems is based on the simplified version of the phenomenological formalism in the lattice frame of reference, which ignores the offdiagonal terms in the matrix of phenomenological coefficients [Kir87, Agr82, And92]. With this assumption, the ondiagonal terms can then be determined either directly from intrinsic diffusion experiments, or more frequently, by invoking the Darken theory that establishes the connection between tracer diffusion coefficients and the ondiagonal phenomenological coefficients or mobilities. Unfortunately, it does not appear that this approach has been examined with the necessary rigor in multicomponent systems. Given the problems with the Darken relations even in binary systems, it appears questionable whether the Darken relations in multicomponent systems can be reliably utilized in this simplified version of the phenomenological formalism. Another, perhaps more preferable, alternative considered in Chapter 5 is a kinetic approach. This is based upon a kinetically derived flux equation which avoids the problems associated with the phenomenological approach. In its rigorous version, it has a potential for providing new insights into influences that operate in diffusion that bias the jumps of components relative to the diffusion direction. A simplified version has been shown to provide relatively simple models for diffusion behavior that successfully describe the experimental observables in a number of ternary systems. The latter has the potential to make the description of diffusion in multicomponent systems practical. The derivation and application of this kinetic formalism is the focus of Chapter 5. 2.2 Experimental Observables in Diffusion 2.2.1 Interdiffusion Experiments Diffusion produces changes in the distribution of chemical elements with time. The primary experiment designed to yield an understanding of diffusion behavior in an alloy system is based on the diffusion couple. Blocks of two alloys, P and Q, are bonded at a temperature that is low enough so that the components in the alloys cannot intermix (Fig. 21). The distribution of each component is initially a step function (Fig. 2la). The couple is then taken to a high temperature and isothermally annealed for many hours or perhaps days. The components interpenetrate; each develops its composition profile after some time, t (Fig. 21b). The couple is cooled, sectioned and chemically analyzed to yield the functions ck(x, t), (k = 1, 2, ..., n) where ck is the molar concentration (moles/cc) of component k. The design of the couple guarantees that the flow of all of the 400 300 200 100 0 100 200 300 400 x (Gm) 400 200 0 200 400 1.0 0.8 0.6 (b) 0.4 0.2 0.0 x(tim) Figure 21. Composition distribution in a diffusion couple: (a) before and (b) after the isothermal anneal. An error function was employed to generate the composition profile where the interdiffusion coefficient, D = 109 cm2/s, and the time for the diffusion anneal, t = 6 hours. < I I XB 0.6 k 0.0 L 50 0 0.4 0.2  0.0 500 1.0 0.8 0.6 U LL E o 0.4 0.2 0.0 , I , Ck dA dV hdxV dnk = C dV = Ck dA dx dx = vk dt dnk Ck dA vk dt dn Jk k C v, k dAdt k Vk Figure 22. Interdiffusion flux in a flowing system. components will be one dimensional (call it the x direction, perpendicular to the original bond interface). Generally the size of the two blocks is large compared to the diffusion zone (the zone of composition change) so that the boundary conditions for the diffusion process is "semiinfinite"; this simply means that for sufficiently large values ofx the composition profiles after the anneal remain at the original concentration values on each side of the couple. The motion of the atoms of a component in such flowing systems is described by the concept of the interdiffusion flux of component k. Focus upon a volume element dV in the couple (Fig. 22). Atoms of component k form a subset of all of the atoms in this volume. Define vk to be the average velocity of k atoms in dV during some small time interval. Focus further upon the area element dA bounding the right side of dV. The number of atoms of k that flow across dA reported per unit area and per unit time is defined to be the flux of component k at dA. The area element is viewed as being at a fixed position in the couple relative to a coordinate system that is external to the couple. The flux of atoms relative to a value of x that is fixed in the laboratory is called the interdiffusion flux of k (see Appendix A), and can be shown to be Jko = Ck vk (k = 1, 2, n) (21) The superscript (0) in this and subsequent equations is used to describe properties associated with the interdiffusion process, i.e., with the motion of atoms in a fixed external (laboratory) reference frame. Note also that Ck represents the concentration (mol/cc) and n the number of components in the systems. Xk is the atom fraction of component k in the volume element dV. It can be shown (Appendix A) that this definition of the interdiffusion flux yields the relationship SJk = 0 (22) k=l if the molar volume of the system is constant. That is, in a n component system there are (n 1) independent interdiffusion fluxes. Dayananda [Day83] showed that these fluxes can be evaluated from the concentration profile measurements without additional information: ckx) Jko(x) = xdc, (k = 1, 2, n) (23) ck where t is the time of the diffusion anneal, ck is the far field concentration of component k on the left side of the couple and ck(x) is the concentration of component k at the 16 position x in the couple. The derivation of this result neglects variations in molar volume in the system, a common assumption in diffusion theory.1 A sample result of this calculation is shown in Fig. 23. The concentration profiles for this demonstration were generated using ternary error function solutions given by Kirkaldy [Kir87]. The typical interdiffusion experiment yields a pattern of composition profiles for all of the n components in the system. The pattern of interdiffusion fluxes in that couple can be computed from this information for each of the components. Because these fluxes sum to zero, there are (n 1) independent interdiffusion fluxes in an n component system. 2.2.2 Intrinsic Diffusion Experiments In 1947 Smigelskas and Kirkendall [Smi47] placed thin molybdenum wires at the original interface between two alloys P (pure Cu) and Q (a brass, Cu 70% Zn 30%) and annealed the resulting couple. The wires moved during the experiment (Fig. 24). Kirkendall concluded that the wires moved with the crystal lattice in the couple. Subsequent investigations established that this behavior is pervasive. The fundamental diffusion process in crystals is the motion of atoms from one lattice site to an adjacent vacant site, i.e., motion relative to the crystal lattice. Kirkendall's experiment established that the crystal lattice is moving in the external fixed reference frame that is used to describe interdiffusion. Thus, in order to follow the fundamental diffusion process, it is necessary to define the diffusion flux relative to the moving crystal lattice: Jk = Ck(vk L) (24) 1 Appendices A to D discuss various issues related to molar volume effects and diffusion fluxes in nonlaboratory reference frames. where v, (or vK)is the local lattice velocity. Darken [Dark48] defined this measure of atom motion relative to the moving lattice to be intrinsic diffusion. (In the notation of the discussion in this chapter, intrinsic diffusion fluxes and other intrinsic properties have no superscript (0) in order to distinguish from the analogous interdiffusion properties.) The intrinsic and interdiffusion fluxes are related by [Dar48] Jk = JkO CkVL (25) Unlike the interdiffusion fluxes (Eq. (23)), all n intrinsic fluxes are independent. They sum, not to zero, but to the local value of the vacancy flux J,: E Jk j, (26) k=l The original Kirkendall experiment and its successors could only provide the lattice velocity at the marker plane inherited from the original bond plane. Thus information on intrinsic diffusion could be obtained from a single composition in a given couple. In later years couples have been constructed with markers distributed through the diffusion zone, see Appendix E. Fig. 25 shows one couple design that incorporates a marker plane that is oblique to the direction of diffusion. Upon sectioning after diffusion, the markers have displaced forming a curve that may be described by a function, g(x). Philibert [Phi91] has devised an analysis that permits the determination of the lattice velocity distribution vL(x) from the marker pattern g(x) without simplifying assumptions. Thus, both terms in Eq. (26) are experimentally accessible and the intrinsic diffusion fluxes of all n components can be computed from experimental composition profiles and marker displacements without simplifying assumptions. 0 E X2 I/ 0 0.4   0.2   0.0 400 200 0 200 400 x (Gm) 400 200 0 200 400 x (Gm) Figure 23. Calculation of the interdiffusion flux directly from the concentration profiles using Dayananda's equation [Day83]. (a) concentration profiles (b) interdiffusion fluxes (a) (b) 0.8 0.6 Original interface 0.0 L 2000 1000 0 1000 2000 3000 Figure 24. Displacement of wires in an interdiffusion experiment for the CuZn system, a phenomenon known as the Kirkendall effect [Smi47]. Oblique marker surface before diffusion Matano (original) Interface Kirkendall Shift Figure 25. Oblique marker plane for the determination of lattice velocities for the entire diffusion zone before and after the interdiffusion anneal. Wire interface Xzn   .1 x (Gm) 4000 2.2.3 Tracer Diffusion Experiments The classic tracer diffusion experiment begins with a block of an alloy of a desired composition. A thin layer containing the desired radioactive element is dispersed on one face of the block. The sample is annealed at a predetermined temperature for a predetermined time. The radioactive isotope permits measurement of the concentration of tracer as a function of depth of penetration even though the tracer element is present in very small quantity. Thus the experiment monitors the motion of one of the components into an alloy that is essentially chemically uniform. A straightforward analysis of the tracer profile yields the tracer diffusion coefficient, Dk', for that element in that alloy. A physical argument that describes the tracer flux in terms of the mean effective jump frequency of component, *, yields the relation [She89, Phi91] Dk; 6 Uk (27) where h is the diffusion jump distance. The jump frequency for each component in a given system is found to depend upon the composition of the alloy and the temperature of the anneal. The number of jumps an atom makes per second between sites is the central physical descriptor of the diffusion process. In principle it would seem that, given the jump frequencies of each of the components as a function of composition one could predict the other experimental observables, i.e., the composition distribution function, ck(x, t), for k = 1, 2, ..., n), and the lattice velocity distribution, v, (x, t), in a chemically evolving system. This connection between tracer, intrinsic and interdiffusion behavior is the focus of the physical theory of diffusion. 2.3 Diffusion Formalisms The information obtained from the diffusion experiments described above is explicit to that experiment. It is desirable to use that information for predicting diffusion behavior in other situations in the same alloy system. This has been achieved by devising a formalism that provides a general description of diffusion phenomena. The experimental results are then used to evaluate parameters introduced in the formal description. With the parameters evaluated for an alloy system, the general, formal equations may be used to predict diffusion behavior in that alloy system for any set of initial and boundary conditions. The time evolution of the distribution of the compositions in a chemically evolving system is formally described by the phenomenological equations adapted from the thermodynamics of irreversible processes [Haa69, Kir87, A1193]. These equations are qualitatively based upon the notion that flow rates increase as a system gets farther from equilibrium. Flow rates are defined in terms of the fluxes of the things that move (quantity per m2 sec) as in Eq. (21). The conditions for equilibrium specify that, in the absence of externally applied fields, gradients of certain local intensive thermodynamic properties are zero at equilibrium, specifically, gradT = 0 gradg k = 0 (k = 1, 2,... n) (28) where R, is the chemical potential of component k. In the absence of external fields in an isothermal system the chemical potential is an algebraic function of composition, so that the condition grad k = 0 implies gradck = 0 (k = 1, 2, ..., n). When the gradients vanish, the corresponding flows are zero. It is natural to write the flow equation based on 22 the assumption that each flux is proportional to all of the independent gradients in a non equilibrium system. 2.3.1 Interdiffusion Formalism For isothermal interdiffusion in a system with n components the phenomenological equations may be written n1 Jk = Lo,n grad (k = 1, 2, ..., n1) (29) The matrix of phenomenological coefficients, Lbo", must be evaluated from diffusion couple experiments. If they are determined as a function of composition and temperature then Eq. (29) permits prediction of the evolution of composition distribution in the system. Recall that, in an ncomponent system there are (n 1) independent interdiffusion fluxes, Eq. (23). An alternate, and more traditional formalism for describing the interdiffusion fluxes in a multicomponent system is a generalization of Fick's law for binary systems: Jko = ED7gradc (k = 1, 2, ..., n1) (210) jl J=1 where D,' is a matrix of interdiffusion coefficients. The suffix superscript "n" for the phenomenological or interdiffusion coefficient is used to denote the dependent variable. Note that in either formalism there are (n 1) independent fluxes in the system (because the interdiffusion fluxes sum to zero, Eq. (23)) and (n 1) terms in each equation. There are (n 1) independent chemical potential gradients in Eq. (29) because the chemical potentials of the n components are related by the GibbsDuhem equation in thermodynamics [Haa69, DeH93]. There are (n 1) independent concentration gradients 23 in Eqs. (210) because the atom fractions sum to 1. There are thus (n 1)2 coefficients in the description of an n component system. In the thermodynamics of irreversible processes the principle or microscopic reversibility devised by Onsager [Ons31, Ons32a] shows that the square matrix of mobility coefficients is symmetrical, i.e., corresponding offdiagonal terms are equal. This reduces the number of independent phenomenological coefficients in a ncomponent system to 12 n (n 1). The two descriptions in Eqs. (29) and (210) are interconvertible since the chemical potentials are functions of composition. Accordingly, the matrix of diffusion coefficients, Dk, or, alternatively, the matrix of phenomenological coefficients, Lko, can be computed from one another [Kir87, Phi91]. The analyses of experimental penetration profiles yields the matrix of diffusivities [Kir87, Tho86]. If desired the matrix of phenomenological coefficients can then be computed by combining the diffusivity results with a thermodynamic solution model that permits calculation of the chemical potentials from the concentrations. 2.3.2 Intrinsic Diffusion Formalism Phenomenological equations used to describe intrinsic diffusion behavior appear to be very similar to Eqs. (29) and (210) with two important differences: a. The fluxes and diffusivities are "intrinsic" properties of the process, in the sense that this word is used in diffusion theory (indicated by dropping the superscript o in the notation. Sometimes the superscript K for the Kirkendall or lattice frame is added for clarification), and b. The fluxes do not sum to zero, see Eq. (26). For intrinsic diffusion: n1 Jk = Ln grad (k = 1, 2, ..., n) (211) nl1 Jk = Djn gradc (k = 1, 2, ..., n) (212) =1 As before, the suffix superscript "n" in these equations is used to denote the dependent variable. There remain (n 1) terms in each flux equation (since there remain (n 1) independent composition variables) but now there are n independent fluxes. The matrix of diffusion coefficients is thus not a square matrix, but contains n (n 1) elements. In principle the matrix of intrinsic diffusivities for a multicomponent system can be evaluated by combining experiments that report concentration evolution data and lattice marker displacement measurements. The corresponding matrix of phenomenological coefficients can be computed from this information and a thermodynamic solution model to compute the chemical potentials. 2.4 Problems with the Phenomenological Formalism and its Implementation Unfortunately, both the interdiffusion and intrinsic diffusion formalisms have their own share of problems besides the obvious difficulties associated with the measurement of the matrix of phenomenological coefficients in multicomponent systems. 2.4.1 Interdiffusion Problems The interdiffusion equations provide a basis for incorporating results of diffusion couple experiments into a framework that permits prediction of the evolution of concentration patterns for that system for any set of initial and boundary conditions. Given c (x, t) for a full range diffusion couple in a binary system, D(c) can be computed from the classic Boltzman [Bol94] Matano [Mat33] analysis, a procedure that has been in introductory textbooks for more than half a century [She89]. Then, given D (c) an appropriate simulation of the governing flux equation can be used to predict the chemical evolution for any other situation involving that binary alloy system. The trouble begins when this analysis is extended to three component systems and beyond. In a three component system there are only two independent compositional variables and two independent interdiffusion fluxes. Thus, a variety of flux equations can be written for the same system, depending upon which component is chosen to be the dependent variable. E.g., Component 3 as dependent variable: J2 = D213 gradc, D223 grad c2 (213) Component 1 as dependent variable: J2 = D2201 gradc2 D2301 gradc3 It becomes necessary to add a superscript to the notation to make explicit the choice of dependent variable made in writing the equation. It is evident that the coefficients D2203 and D2201, both of which report the effect of the gradient of component 2 on the flux of component 2, are not equal. In general, the values of the phenomenological coefficients depend upon the choice of the dependent variable made in setting up the problem. In the light of this observation, it is unlikely that these coefficients will convey much physical understanding of how the atoms move in the system. This approach may be used to describe the composition evolution, but not to explain or understand it. The analysis can be significantly simplified if it is assumed that "offdiagonal terms", Dkjo, which describe the contribution of the gradient of components to the flux of component k, may be neglected as small in comparison with "ondiagonal terms", Dkk, which describes the contribution of gradient of component k upon its own flux, J,. This is a convenient and intuitive assumption because it decouples the flux equations and greatly simplifies the mathematics of the description. This is a dangerous assumption. Analysis of relations between coefficients defined for two different choices of dependent variable, as illustrated in Eq. (213) for example, shows that some offdiagonal terms in one description contain an ondiagonal term in the other description. For example, it can be shown that D323 = D321 D331. In addition one can always find local situations in the patterns that develop in which the concentration gradient for the on diagonal term is small so that the local flux is primarily determined by the offdiagonal term. In order to determine the matrix of four diffusion coefficients in a three component system, it is necessary to prepare and analyze two diffusion couples with composition paths that cross in the Gibbs triangle. At the position of the crossing point both couples will have the same composition (and hence the same set of values for the Dko matrix) but different values of the fluxes and gradients. The two flux equations (2 10) for each couple yield four equations in four unknowns (the Dkjo matrix), and may be solved for their values at that specific composition. To obtain the pattern of composition dependent functions for the diffusivities, Dko(c2, 3) for the whole system, it is necessary to construct and anneal a series of diffusion couples with composition paths that criss cross each other in the Gibbs triangle. Values of the each of the four diffusivities are computed at crossing points. These isolated values may be analyzed statistically to deduce best fit functions, four surfaces over the Gibbs triangle , that complete the database for interdiffusion in the system for the temperature chosen for the diffusion anneals. A complete data base would require that this process be repeated at other temperatures. Because of the level of effort required to obtain these data there have been relatively few determinations of this matrix in the five decades since the procedures were first outlined [Kir87]. In their text, Kirkaldy and Young [Kir87] list 26 references reporting such analyses where "the data is sufficiently accurate and comprehensive that a closure with theory can be illustrated." Only four of these cover the full composition range or even an entire phase field. Evidently the level of effort required to generate interdiffusion data bases for three component alloys is difficult to justify. This situation is not helped by the fact that the interdiffusion coefficients do not convey a physical understanding of how the atoms move in the system as discussed above. The data do not contribute to a physical understanding of the processes going on in diffusion so that patterns might be divined or predictions hypothesized. Determination of the matrix of nine diffusivities for a single composition point in a quaternary system requires an experiment involving a set of three diffusion couples with composition paths that intersect at a point in the three dimensional composition space of the Gibbs tetrahedron. Unfortunately the probability that three space curves intersect in three dimensional space constitutes a set of points of measure zero. Accordingly, this matrix has never been measured for any quaternary system.2 Experimental methods for measuring interdiffusion coefficients in quaternary and higher order systems are evidently impractical. 2 The nine diffusivities have been estimated at a single quaternary composition in the NiCrAlMo system by constructing three incremental couples (i.e., with small composition differences) with crossing diffusion vectors and applying the approximation that the nine diffusion coefficients may be treated as constants in this small composition interval [Sta92]. 2.4.2 Intrinsic Diffusion Problems Because the intrinsic diffusion fluxes are independent, the description of the intrinsic behavior of an n component system requires an additional flux equation requiring addition of (n 1) diffusion coefficients. Darken analyzed the problem of determining intrinsic diffusion coefficients in a binary system [Dark48]. He derived the result D, = D (1 Xa) ax VL (214) D2 = Do + X2 VL Sax2 In a binary system, the interdiffusion coefficient can be obtained by applying the BoltzmanMatano analysis to composition profiles, and the lattice velocity can be obtained from marker displacements during diffusion. D, and D2 may then be evaluated at any composition for which vL is measured. The original Kirkendall experiment placed a marker at a single point, the original interface, and measured its displacement. Thus, vL, and hence D, and D2, could be determined at a single composition in this experiment. That composition was not known apriori; it was determined as the composition at the marker plane after the experiment. To generate a database for a range of compositions requires the preparation of a number of diffusion couples with initial composition pairs that produced marker plane compositions that spanned the range. If the oblique marker plane experiment (Fig. 25) is implemented, vL can be determined at each composition in the couple. Combination with D' values for the couple will permit computation of the two intrinsic diffusivities at all compositions that exist in the couple. The values of D, and D2 are indicative of the relative rates of motions of the two components on the lattice. Darken and others extended his analysis to ternary systems [Dar5 1, Guy65, Kir87, Man70]. Implementation of the analysis requires determination of the matrix of interdiffusion coefficients, Dk o, in the system using a pair of couples with intersecting composition paths with the attendant difficulties described above. In addition, the lattice velocity must be determined in both couples at their common composition. Evidently this cannot be achieved with traditional, single marker couples because the compositions at the marker planes in each couple, where vL may be determined, will not in general coincide with the composition at the intersection point of their composition paths. The oblique marker plane experiment permits determination of vL at all points in the compositions that exist in the couple, including that of the point of intersection of the composition paths. The resulting six equations (three flux equations in each couple) may be solved simultaneously to determine the matrix of six intrinsic diffusion coefficients at the composition of the intersection point. Given the level of effort involved it is not surprising that intrinsic diffusivities have been estimated at but a few points in a few ternary systems [Guy65, Day68, Car72, Car73, Che75]. There has been no systematic study attempting to map this matrix, Dk,, as a function of composition in any ternary system. The extra effort might be justified in the light of the fact that intrinsic fluxes provide a more direct view of how the components move with respect to one another, since they report independent motions of the components relative to the crystal lattice. However this insight is not supplied by the numerical values of the elements of the diffusivity matrix, Dk. These properties suffer from the same problem cited for the interdiffusion coefficients: their values depend upon the choice of independent 30 compositional variable in the analysis. D221 and D223 are different numbers, though both purport to describe the effect of the concentration gradient of component 2 on the intrinsic flux of component 2. Evidently the numerical values of the elements in the diffusivity matrix is not a useful basis for understanding these atom flows in the system. The intrinsic fluxes themselves have the potential to provide a level of physical understanding of the real motion of the components in the system. Eq. (25) provides the relationship between intrinsic fluxes and the interdiffusion fluxes and lattice velocities. On the right side of this equation, Dayananda's analysis permits determination of the interdiffusion flux Jk(x) for each component by applying Eq. (23) to the measured composition distribution. The lattice velocity distribution vL(x) can be determined from a couple with an oblique marker plane, using Philibert's analysis. Thus the pattern of the intrinsic fluxes in a couple can be determined experimentally from the composition profiles and marker shift pattern in that couple. These relationships make no simplifying assumptions. Unfortunately, the oblique marker plane experiment has only been applied to a few binary systems (see Appendix E). It has never been applied to ternary or higher order systems, though the methodology makes such an application relatively straightforward. 2.5 Connection Between Tracer, Intrinsic and Interdiffusion Information The phenomenological formalism was devised so that information extracted from diffusion couple experiments could be used to make more general calculations. In its complete form, it has proven to be a cumbersome basis for developing a diffusion database. Perhaps if a physical understanding of the content of the diffusivities could be developed, patterns could be recognized and correlations evoked that would foster the 31 generation of a more useful diffusion database. Tracer diffusion data provides the most direct link to a physical understanding of atom motions in binary and multicomponent systems. Darken recognized this in his initial paper on the subject [Dar48, Dar51]. He set out to establish the connection between tracer diffusion information and intrinsic and interdiffusion behavior. Since a detailed discussion of Darken's theory is the subject of Chapter 4, only the final Darken relations are presented here. The Darken relations between tracer diffusivities and intrinsic diffusivities are S1 + 1n y, k = D 1 + (k = 1, 2) (215) where yk is the activity coefficient of component k in the alloy system. The Darken relation between interdiffusion and tracer diffusion coefficients is Do = (X2D +1 2 an X (216) In order to test the Darken relations, it is necessary to carry out all three kinds of diffusion experiments (tracer profiles, marker displacement and composition profile) over a composition range in a binary system. There are a few examples of binary systems in the literature for which this information exists. The definitive test combines tracer diffusion coefficients, thermodynamic information and intrinsic diffusivity measurements to test Eqs. (215, 216). Analyses for a number of these systems have been reported in the literature and are discussed in Chapter 4. It has been found in some cases that if the two components have similar tracer behavior, then the tests support Darken's hypothesis within experimental error at least for the interdiffusion coefficient. However if the tracer behavior is significantly different, then the results show that the theory is inadequate to explain the differences between tracer and intrinsic diffusivities. Examples of tests for such systems are given in Chapter 4. Assessments of the data with the aid of an intrinsic diffusion simulation have shown that the disagreement in the intrinsic diffusion coefficients and hence the lattice shifts, are well outside the experimental error in these systems. It must be concluded from these results that Darken's hypothesis yields a theory that is inadequate to explain the connection between tracer and intrinsic diffusion coefficients. Manning [Man68, Man70, Man71] and others [Bar51, How64, Heu79, A1193] have analyzed this connection at a more sophisticated level, introducing concepts such as correlation and vacancy wind effects. Examples of the predictions of Manning's more detailed analysis are also discussed in Chapter 4. In some cases, the additional term produces a small correction to Darken's prediction, within the range of experimental error in most cases. In other cases, Manning's correction is in the wrong direction. Darken's treatment, and Manning's corrections, have been expanded to describe three component systems [Man70, Kir87, Phi91, A1193]. Experimental studies in a few ternary systems would provide a truly definitive test of this explanation of the phenomenological coefficients. The effort involved to make such a series of tests appears daunting. There is but one ternary system for which tracer data exists for all three components over a significant composition range: CuNiZn [Anu72]. Interdiffusion data exists for the same system in the same composition range. There is anecdotal information about marker displacements for couples in this system, but no systematic study exists that would permit determination of the intrinsic diffusion coefficients. It is possible that, with the use of a suitable simulation of intrinsic diffusion, 33 one could arrive at an intrinsic diffusion model, that provides a satisfactory description of the available marker data, but this has not been reported. Thus, the generalization of Darken's theory to ternary systems has not been tested systematically. 2.6 Simplifications to the Phenomenological Formalism Because of the experimental difficulties involved in the determination of the matrix of phenomenological coefficients (even though their meaning is abstruse), an often used simplification is to ignore the crossterms in this matrix. With this assumption, the simplified version of the formalism for the intrinsic flux can be stated: Jk Lk Lkk c C k (L = k i) (k = 1,2,...,n) (217) i=1 8x x ax Mk in this equation is known as the mobility and its evaluation is the focus in this version of the formalism that has been adapted by the DICTRA software [Agr82, And92, BorOO]. Note that in Eq. (217), Mk is related to Lkk by Lkk = ckMk (218) The term "mobility" was first used by Darken [Dar48] for deriving the relations between tracer, interdiffusion and intrinsic diffusion (Chapter 4). From the Darken theory, the connection between Lkk or Mk and the tracer diffusion coefficient of component k is Lkk = k Mk k (219) RT Originally, the relation between the mobility and the tracer diffusion coefficient was proposed by Darken for binary systems; in this formalism it has been extended to multicomponent systems as well. 2.6.1 Intrinsic Diffusion Coefficients In order to relate Lkk or Mk in this formalism to the intrinsic diffusion coefficients, Eq. (212) is rewritten in the following form: n J4= ZD (k = 1, 2, ..., n) (220) j=1 ax The Dk in this equation is related to D'j in Eq. (212) by Dn = Dk Dk (221) Eq. (217) can be rewritten using the chain rule of differentiation and noting that Lk, = 0 when k + i: k n n 9, ac In a9uk 9C S E Lkk' (k =1,2,...,n) (222) ,=l j=i 9cj ax j=1 Cc ax Comparing Eqs. (220) and (222), it is seen that D = Lkk (223) Since Lk is related to Dk' by the Darken relation Eq. (219), relations between Dn or Dk and Dk may be similarly obtained. 2.6.2 Interdiffusion Coefficients In order to compute the interdiffusion coefficients, which are defined in the laboratory frame of reference, from the Lkk' s, which are defined in the lattice frame of reference, a transformation of the intrinsic fluxes to the laboratory frame is necessary. The flux in the laboratory frame is related to the intrinsic flux by [Dark48] ko = X, JkK (224) k=1 35 This follows from Appendix A, Eqs. (A45), (A46), (A51) and (A52), noting that vis zero if the molar volume is constant. The phenomenological flux equation in the laboratory frame in a convenient form is Lk"o ' From Eqs. (217) and (225), and using Eq. (224), it may be shown that (225) Lko = Y (, X) L, (k j Eq. (225) may be rewritten as follows: Jko = a c i=1 '=1 8c ax The interdiffusion flux can also be expressed as n Be Jko = o (k J1 8x Comparing Eqs. (227) and (228), the relation between Dk and Lo is obtained: Dko LI k ci (226) (227) (228) (229) The interdiffusion coefficient Dk in Eq. (229), and D o." in Eq. (210) are related by D'on = D D (230) where n is the dependent species in Eq. (210). Since the L ,'s are related to the Lk 's which are in turn related to the tracer diffusion coefficients, expressions relating the interdiffusion coefficients Do,'" orD to the tracer diffusion coefficients D, can be obtained. The expressions for ternary (k = 1, 2, ..., n 1) = 1, 2,..., n I) (k = 1,2,..., n 1) = 1, 2, ..., n1) systems have been given in the paper by Cserhati et al. [CseOl]. These authors determined the interdiffusion coefficients at crossing points in fcc CuFeNi system and computed the tracer diffusion coefficients at these locations using the above formalism. The thermodynamic data needed for computing the chemical potentials, was also assessed by the research group [Ron96]. The computed tracer diffusion coefficients were found to be significantly different from the experimental values [CseOl]. In some cases, the authors reported that negative values were obtained, which is of course not possible with experimental tracer diffusion coefficients. This suggested to the authors that there were either problems with the experimental data and procedures or with the fundamental formalism. 2.6.3 Problems with the Simplified Version The central question in the simplified version of the phenomenological formalism is whether the L.'s are a unique function of composition in multicomponent systems. This can be tested by measuring the intrinsic fluxes (using for example, the oblique interface technique (Appendix E)) in ternary systems for two or more independent diffusion couples that have crossing diffusion paths. If the Lkk's are unique, the values at the compositions where the diffusion composition paths intersect, should be the same. Unfortunately, it does not appear that this test has been performed at the present time, although indirect tests such as the one by Cserhati et al. [CseOl], which were discussed in the previous section, also cast doubt on this simplified formalism. Nevertheless, the development of diffusion databases using this formalism has progressed [And02, Cam02] using the Darken relations that connect the mobilities or the Lkk's with the tracer diffusion coefficients. In Chapter 4, the Darken relations are examined for several binary 37 systems with the aid of an intrinsic diffusion simulation developed in this work. Since these relations are not satisfactory even in binary systems, it appears unlikely that they will work in multicomponent systems as well. CHAPTER 3 INTRINSIC DIFFUSION SIMULATION The original diffusion experiments in the a CuZn system by Kirkendall and Smigelskas [Smi47] demonstrated the need to consider the "intrinsic" motion of atoms in a crystalline lattice so as to explain the phenomenon of the Kirkendall Effect, a phenomenon in which an imbalance of atom flows on a crystalline lattice results in the displacement of the lattice, thus causing the movement of inert markers residing on the lattice. The intrinsic motion of atoms, i.e., the diffusion of atoms between lattice planes due to an atomvacancy exchange mechanism, first recognized by Kirkendall, was given a firm foundation by Darken in his theory of intrinsic diffusion [Dar48, Dar51]. A discussion of Darken's treatment of intrinsic diffusion is provided in Chapter 4. Although Darken's treatment of intrinsic diffusion has withstood the test of time, there have been relatively few efforts devoted towards the simulation of intrinsic diffusion in the original classical form presented by Darken. The reason may be partly attributed to the lower number of intrinsic diffusion studies in the literature due to the inherent experimental difficulties involved in the measurement of lattice velocities. In Appendix E, a technique for the measurement of lattice velocities known using an "oblique interface technique" [Cor72, Cor74] is presented, which the author hopes should partly mitigate some of these difficulties and encourage more intrinsic diffusion measurements in the future. Indeed in the last decade, there has been an increasing emphasis on intrinsic diffusion in both single and multiphase systems [Dal00a, Dal00b, Loo90a, 38 39 Loo90b]. The development and application of an intrinsic diffusion simulation, which is the focus of this chapter, is expected to provide a valuable predictive tool for the diffusion specialist engaged in intrinsic diffusion studies. 3.1 The Need for an Intrinsic Diffusion Simulation A variety of intrinsic diffusion formalisms have been discussed in the literature, in which the kinetic descriptors of intrinsic diffusion as well as the driving forces are different, even though the meaning and value of the intrinsic flux is still retained. Phenomenological treatments include, for example, Darken's, which considers intrinsic diffusion coefficients as the kinetic descriptors and concentration gradients (in mol/cc) as driving forces while others, such as the diffusion software DICTRA1 [Agr82, And92, BorOO] are coded in terms of mobilities and chemical potential gradients. Kinetic expressions for the intrinsic flux, such as the jump frequency approach (see Chapter 5) [Isw93], involve gradients in jump frequencies and concentrations. The test of any intrinsic formalism is a comparison with experimental observables, for example, concentration profiles and lattice shifts. An intrinsic diffusion simulation, such as the one developed here, has the potential of assessing a variety of formalisms and examining the underlying assumptions inherent in these approaches. For example, in the case of multicomponent systems, the assumption that the cross terms in the matrix of phenomenological coefficients can be ignored (such as in DICTRA) can be tested by utilizing the simulation to examine a number of composition paths and lattice shifts in the 1 Diffusion controlled transformations (DICTRA) for diffusion simulations in multicomponent alloys, Royal Institute of Technology, Stockholm, Sweden. 40 system. Similarly in the jump frequency approach, the assumption that the bias terms can be neglected can be assessed with the aid of the simulation. The measurement of diffusion coefficients using conventional experiments involving semiinfinite diffusion couples is unfortunately not a very accurate process and can result in significant errors [Ior73, Kap90, Loo90a]. The advent and development of the electron microprobe [Zie64, Phi91, Gol02] and other modern techniques for the determination of concentration profiles in diffusion couples has resulted in improved measurement accuracies as compared to older techniques (e.g., wet chemical analysis). Similarly, the availability of the personal computer along with improved algorithms for data analysis has reduced the computational effort and improved the accuracy involved in the determination of diffusion coefficients from experimentally measured concentration profiles and lattice shifts. Nevertheless, errors in diffusion measurements and analysis cannot be eliminated and quite often the practitioner's own expertise or lack of it may be an important contributor. As compared to interdiffusion coefficients, the measurement of intrinsic diffusion coefficients requires, in addition to the concentration distributions, the accurate measurement of lattice shifts. Thus it would be quite desirable for the diffusion researcher to have at his/her disposal a diffusion simulation that can rapidly ascertain the accuracy of the diffusion measurements. The DarkenManning theories have long been used to connect the diffusion of atoms in homogenous solid solutions (i.e., tracer diffusion) with those in gradient systems (i.e., interdiffusion). A discussion of the DarkenManning theories is provided in Chapter 4. The DarkenManning theories are usually tested by comparing the experimentally determined interdiffusion and intrinsic diffusion coefficients with those 41 predicted using these theories. Besides having the capability of assessing the quality of the experimental data and analysis, the simulation developed in this work has the added advantage of reproducing all the experimental information that is available from a diffusion experiment using the intrinsic diffusion coefficients predicted from these theories as an input. Thus the DarkenManning theories that connect the intrinsic and interdiffusion coefficients with the tracer diffusion coefficients, can be examined more critically with the aid of the simulation. This is considered in some depth in Chapter 4. An important contribution of the present effort is the introduction of bias effects in the jump frequency approach, that has been developed for modeling intrinsic diffusion in multicomponent systems. At the time of this writing, a theoretical approach for the analysis of the bias factors in the jump frequency approach, is lacking. However, it has been demonstrated in Chapter 5 with the aid of the simulation, that the bias factors can be directly determined from the total and unbiased intrinsic fluxes. Thus for the first time, it has been shown that the bias contributions of the individual components in the classical a CuZn system are not only unequal but in fact have opposite signs. This unexpected development is expected to provide some critical insights into the origin of bias effects in diffusion. Needless to say, this could not have been discovered without the benefit of the simulation. In principle, the simulation could have been developed using conventional software packages such as C++ or Fortran. In fact, the previous version of the simulation was developed in TurboPascal [Isw93]. Significant improvements, particularly on the application side, have been made in the recent version that has been developed using the math software, MathCad. The author's preference in utilizing MathCad was primarily influenced by its user friendliness, advanced math subroutines, simple programming syntax, and easy display of graphical output. A special advantage of MathCad is the ability to enter expressions and comments in the working document in a manner very similar to that utilized by an instructor teaching traditional topics in Materials Science & Engineering. The development of the simulation in MathCad has been a strong foundation that resulted in valuable educational and research benefits in many diverse fields within the materials sciences. 3.2 The Simulation Given an initial concentration distribution of the components in a system, the goal of the intrinsic simulation is to predict the variation in the concentration of the components and the lattice velocity as a function of position and time. The simulation of multicomponent diffusion is based on a finite difference evaluation of the expression for the intrinsic flux for multicomponent systems. Various approaches for evaluating the intrinsic fluxes are available depending on the choice of the kinetic descriptors and the driving forces. The expression for the intrinsic flux in multicomponent systems based on the mulitcomponent version of the Darken equation for binary systems [Kir87] is JK = nE1 D (i = 1,2,...n) (31) j=1 lx P,T,tt,c. This expression is similar to the multicomponent version of Fick's law (see Chapter 2 or Appendix C), however unlike the fluxes in the other reference frames, the fluxes in the lattice or Kirkendall frame (i.e., the intrinsic fluxes) are all independent. The DK,n are the matrix of intrinsic diffusion coefficients in the n component system with the nth component taken to be the dependent one. Unfortunately, as discussed in Chapter 2, the determination of the matrix of intrinsic diffusion coefficients in ternary or higher order systems is a nontrivial task. Hence the evaluation of intrinsic fluxes using the Darken equation is usually restricted to binary systems. In the case of binary systems, the Darken equation [Dar48] is K = DkK (k = 1,2) (32) ax where DkKis now the binary intrinsic diffusion coefficient of component k in cm2/s and ck is the concentration of component k in cc/mole. The intrinsic flux expression utilized by DICTRA [Agr82, And92, BorOO] uses mobilities as the kinetic descriptors and chemical potential gradients as the driving forces (see Chapter 4). This is essentially obtained from the phenomenological intrinsic flux expression: JK= L (i = 1,2,...) (33) where the L 's are the phenomenological coefficients in the lattice or Kirkendall frame and p, is the chemical potential of components. In the case of substitutional solid solutions, based on Kirkaldy's analysis [Kir87], the crossterms in the matrix of the phenomenological coefficients are neglected (i.e., L, = 0 if isj) and the resultant expression is the one used by DICTRA: K K8 a Kd a J = L,1 cMi a (34) 9x 8x where M ,K is the mobility of component i as defined by DICTRA in the Kirkendall frame. It is evident from the above equation, that neglecting the crossterms in the matrix 44 of phenomenological intrinsic coefficients significantly simplifies the effort involved in the determination of the L,,'s or the mobilities, M' s. Manning and Lidiard have vigorously objected to Kirkaldy's assertion that the crossterms in the matrix of phenomenological coefficients can be neglected (see [Kir85], discussions by Lidiard and Manning, and responses by Kirkaldy therein). In spite of this, the utilization of Eq. (34) has persisted primarily due to the experimental difficulties associated with the measurement of the nondiagonal terms in Eq. (33). The procedure for determining the mobilities in Eq. (34) is discussed by Agren et al. [Agr82, And92, BorOO]. The expression for the intrinsic flux based on intrinsic jump frequencies for multicomponent systems proposed by DeHoff and Iswaran [Isw93, Deh02] is K 1 2 k Fk Jk (35) 6 ax where 1k is the effective intrinsic jump frequency of component k in # jumps/s (Fk not equal to the tracer jump frequency 17) and h is the jump distance between atomic planes. This expression ignores bias effects operating during the diffusion process, i.e., the fraction of the number of jumps in the +x and x directions are assumed to be equal to 1/6 (the isotropic value). A more rigorous expression that takes into account bias effects is discussed in Chapter 5. This expression is K k k ( 36) Jk = 2akkCkk 2 (36) 6 ax where ak is the bias factor for component k, i.e., the difference between the fraction of the total number of jumps in the +x direction and 1/6, and 1F* is the tracer jump frequency of component k. At the present time, the theoretical content for ak is lacking, hence the unbiased version of the jump frequency formalism (Eq. (35)) has been utilized in the simulation for modeling diffusion in multicomponent systems (see Chapter 5). However, in Chapter 5 it is demonstrated, that the Darken flux expression, Eq. (32), and the rigorous jump frequency expression, Eq. (36), can be effectively combined to extract aok with the aid of the simulation. This unique piece of information is expected to provide some critical insights into the physical origin of the bias factor and aid in the development of an analytical expression that can be utilized in the simulation. 3.2.1 Input for the Simulation The current version of the simulation has been set up to model unidirectional, isothermal diffusion in a semiinfinite diffusion couple, a situation in which the initial compositions at the two ends of the diffusion couple remain unchanged during the course of the diffusion experiment. The input for the simulation are: 1. The initial concentration distribution for each component. 2. Depending upon the intrinsic flux formalism, a functional relationship for the kinetic descriptors as a function of composition. For example, the intrinsic diffusion coefficients for the Darken formalism, the mobilities for the DICTRA formalism or the jump frequencies in the jump frequency formalism. 3. The molar volume as a function of composition. The measurement of concentration profiles using the electron microprobe requires knowledge of the molar volume (concentration (mol/cc) = atom fraction (from microprobe) / molar volume (cc/mol) or ck = Xk V). For a single phase, solid state crystalline system, the assumption that the molar volume is constant is commonly employed. This assumption is a reasonable one in single phase systems where molar volumes usually do not vary appreciably (less than 30%), hence the errors involved in neglecting molar volume effects are usually within the experimental errors involved in the determination of the kinetic parameters (20 % or higher reported by Kapoor and Eagar [Kap90], 10 % reported by Dayananda for the AgCd system [Ior73]). If it is still desired to take into account variations in molar volume, a reasonable approximation is to compute the molar volumes from the molar volumes of the pure components assuming ideal mixing [Loo90a], i.e., Vegard's law. For e.g., it is shown in Chapter 4 for the AuNi system [DalOOa], that this approximation works rather well. For the jump frequency formalism, the jump distance, X, is obtained from the molar volume and crystal structure assuming nearest neighbor jumps. 3.2.2 The Setup A schematic of the setup for the simulation is shown in Figure 1. The diffusion zone is divided in the direction of the positional coordinate (the xaxis) into a number of slices or intervals having the same width 6 and crosssectional area A. 240 such intervals (or slices) has been found to give adequate resolution of composition variations in the diffusion zone, however additional intervals can be added if needed. The spacing between the points on the xaxis or the width of the slices 6 is chosen to be commensurate with the order of magnitude of the jump frequencies or the intrinsic diffusion coefficients and the time interval chosen. For values of jump frequencies in the range of 10107 per second or diffusion coefficients in the range of 101_1010 cm2/s, a time interval At of 10 seconds and a spatial interval 6 of 24 microns usually gave useful resolutions of the evolution of the composition distribution in both space and time. In case the differences in the jump frequencies or intrinsic diffusion coefficients of the various components were significant (34 orders in magnitude), the slice width was increased up to 10 microns. It was found that on a Pentium 4 based PC, 128256 iterations, taking less than a minute in real time, were sufficient to obtain sufficient resolution, so as to examine and adjust the choice of the setup parameters. The various slices are numbered from left to right (Fig. 31) along the positive x axis with the position of the left end of the first slice (i = 1) taken to be the origin. From knowledge of the slice widths, which are initially equal for all slices (>li 1e' ithi= 6i), the position of the slices x, are determined. Thus for example, the first slice has the position x, = 0, while the center slice (i = 121) that corresponds to the Kirkendall interface, is located at x, = 240 for 6 = 2 im. The last slice (i = 241) is taken to begin at the end of the 240th interval and is located at x, = 480. Each of the positions x, in the spatial grid, that denotes the initial position of the slice, is assigned an initial value of concentration, c,, (the first suffix "k" denotes the component, the second suffix "i" the slice number), atom fraction, X,,, and a molar volume, V,. The values of these variables at the midpoint of the slice, are also computed for every slice. From knowledge of the mean concentrations, the slice widths and the crosssectional area, A,, the number of atoms of each component in each slice is calculated, hence the total number of atoms in each slice is known at the start of any iteration. If chemical potentials are needed as in the DICTRA software, a thermodynamic solution model for the Gibbs excess free energy as a function of composition is required. Chemical potentials (pu,) and their gradients can then be easily computed from such a model. If the jump frequency formalism is utilized, then the jump frequency of each x, Vi Difusion direction CM <  > A,, t= 0 (a) i\ i \ t=t, (b) Figure 31. Setup for the intrinsic diffusion simulation: (a) The diffusion zone is initially divided into a number of equally spaced slices having a thickness 6. x, is the position of the ith slice with respect to the origin taken to be the position of the first slice (i = 1). c,k, Xk, and D, are the concentration, atom fraction and intrinsic diffusion coefficient respectively of component k in slice i. V, is the molar volume of slice i and AO is the crosssectional area of each slice, here assumed to be constant at any time during the diffusion process. The diffusion direction is normal to AO as indicated in the figure. (b) After performing a number of iterations corresponding to a diffusion time t,, the positions and widths of the slices as well as the concentrations have changed due to the accumulation or depletion of atoms within the individual slices. The schematic illustrates the movement of lattice planes due to the Kirkendall phenomenon. component, k,,, is computed from the composition dependent jump frequency model that is input for the system and is assigned to the position x,. Similarly if the Darken intrinsic K formalism is utilized, the intrinsic diffusion coefficient of each component, Dk is computed from the input model and assigned to x, and likewise for the mobilities, Mk,, in the DICTRA approach. Input: Dk, Vk, Xk, Ck (x, t=O) Initialize: xi, Xk, V, ck, Dk, ti Compute Intrinsic Fluxes: Jk Accumulations: AJAt Mean compositions Atom ifractions Mean molar volume: V, (X2,..Xd) Mean Concentrations Slice widths: 6, Positions: x, Lattice shifts: Ax, Velocity: v, NewX,, V,, ch, Dki C 0 U N T E R Figure 32. Flowchart depicting the various steps in the intrinsic diffusion simulation. Output: Fluxes, concentration profiles, paths, lattice shifts and lattice velocity I !W 3.2.3 RunTime Iterations After the initialization procedure is completed, the main body of the program begins (Fig. 32). The fluxes at the first (i = 1) and last slice (i = 241) are set equal to zero at the beginning of each iteration. The intrinsic flux of component k at x, is computed from a central finite difference formulation of the desired expression for the intrinsic flux, Eqs. (32), (34), (35): K K Ck,i+l k,z1 Jk, D k 1 (37) Xk,i+l k,i1 K K k,z+1 k,z1 Jk Ck, k, + (38) 1 1 xk,i+l k,i1 K 1 2 k,l+1 k,z+l k,z k ,zl 1 Jk,z = (39) 6 xk,+l k,,1 The phenomenological flux expressions given in Eqs. (37) and (38) are applicable to multicomponent systems, if one neglects the cross terms in the matrix of intrinsic diffusion or phenomenological coefficients respectively. Otherwise, both are restricted to binary systems. If in the rare situation where multicomponent intrinsic diffusion coefficients become available, the expression for the intrinsic flux can be modified to include the crossterms, see Eq. (31). In contrast, the kinetic expression represented by Eq. (39) is always valid for multicomponent systems. Since there exists a unique composition dependent jump frequency for each component, the question of neglecting crossterms does not arise in the jump frequency formalism (Eq. (39)). However, as mentioned earlier with reference to Eqs. (35) and (36), bias effects are ignored in this equation but are discussed with some depth in Chapter 5. 51 The flux of each component is thus computed using one of the above equations at each position x,. The accumulation or change in the number of atoms of component k in the ith slice or the interval (x,,, x,) is given by I", = (J^AM (310) dnk, k A( At j1 J 1A,1 At) N (310) where N0 is Avagadro's constant. Hence the total number of atoms that accumulate in the ith interval can be computed. Note that the accumulation depends on the sign and magnitude of the fluxes in the adjacent slices, and hence can be either positive or negative in sign. By adding the accumulation of each component k to the number of atoms of component k existing in the ith slice at the start of the current iteration in (tAt), the total number of atoms of each component k is obtained. nk,,(t =dnk,(t +n(tAt) (311) C Hence the total number of atoms of all components (n, = C nk, ) in the ith slice is k=l computed and thus the mean atom fraction (Xk ) of component k in the ith slice can be determined for the current iteration. If the molar volume is a function of composition, then knowledge of the mean atom fractions permits the computation of the new mean molar volume ( V) in each slice i and hence the mean concentration (Ck,) of slice i is determined. A net change in the total number of atoms in each slice requires that the total volume of the slice be adjusted so as to accommodate this net change. The newly computed molar volume can be used to compute the new slice width 6i: ST Vn, 6 n (312) N A, In making this adjustment for the slice widths, it is assumed that sites are created or destroyed as needed. This is equivalent to the assumption of local equilibrium in the concentration of vacancies. Intervals that have a net gain in atoms must expand to accommodate this accumulation, while those that experience a net loss must contract. In semiinfinite diffusion couples, it has been shown that this adjustment in volume occurs primarily along the diffusion directions, i.e. in the +x or x directions. Balluffi's [Bal52, Bal60] and Ruth's [Rut97] experiments have demonstrated that contractions or expansions in directions normal to the diffusion direction are primarily restricted to regions very close to the surfaces of the specimens. Hence as long as the diffusion specimens are of sufficient dimensions and are not classified as "thin films", volume changes normal to the diffusion direction can be neglected. This is equivalent to assuming that the crosssectional area A, of each slice remains unchanged during the entire diffusion experiment. From knowledge of the new slice widths, 6,, the new position of a slice, x,, can then be determined for the current iteration: x,.1 = x, + 6, (313) Note that the position of the first slice is always zero at all times (x, = 0). Eq. (313) has to be applied from the second slice onwards. Thus, the new position of any slice x, depends upon the cumulative sum of all the slice widths to the left of slice i. The difference between the position of a slice at time t and t + At is the instantaneous lattice shift. The sum over time of all the shifts for the ith plane is the distance that a marker originally placed at this plane would move during the time of the experiment (i.e., the lattice shift): Ax,(t) = x,(t) x(t =0) (314) The instantaneous velocity of the Ith plane or the local lattice velocity (viK) may be calculated from the lattice position using Philibert's (p. 218 in [Phi91] expression (see Appendix E): K 1 ^ a \, 7 17+,1 ~X71 v (x x ) (x x ) (x x ) (x x ) I I1 2 t 0121 dx I2 t 121 121 x x 0, 0+ 0,_, (315) where x = x (t=0) Note that for the Kirkendall interface (i = 121), the lattice velocity simplifies to the well known expression Ax121/2t. Eq. (315) can also be expressed in terms of the lattice shift Ax,, Eq. (314): K 1 dAx, vi = Ax (xoxo2 dx (316) where the differential can be expressed in a finite difference form as in Eq. (315). Alternatively, the lattice velocity may be directly computed from the instantaneous lattice shift: K x(t+At) x,(t) Vi At (317) The vacancy flux and the interdiffusion fluxes can be computed from the intrinsic fluxes and the lattice velocity (Appendix A): J J, K k1 (318) J K k + CVK (319) JO + C'I Vz Note that the lattice velocity cannot be directly obtained from the vacancy flux, unless the reference velocity in the number frame v is zero (Eq. (A23)), which is only true if the molar volume is constant (Appendix D). KN K N Vi Vi V JK =__ Vi V (320) The reference velocity vN as well as the flux J, in the number frame can also be obtained, see Eqs. (A22) and (A20): C Ni=V Jk (321) k=1 JN JO N JkN = NkVi (322) Similarly, mean velocities and fluxes in other reference frames can be computed if needed (see Appendix A). The new molar volume V, and concentration ck,, which are then the starting values at position x, for the next iteration, are obtained by a linear interpolation of the mean values of these quantities in adjacent slices. For example, the updated concentration is given by c k, kz c = ~k,l , Cki = Ck + i (323) The updated values for the atom fractions Xk,, are then obtained from ck,, and V, at x,. Following this, the values for the kinetic descriptors (DK or FK, or Mk, ) are updated. The next iteration then commences. The computation is repeated until the composition profiles stabilize and span a significant fraction of the full range of positions programmed into the simulation. It is found that in most cases, the concentration profiles plotted on normalized coordinates (xA/t) stabilize after about 128256 iterations, and takes less than a minute of computer time on a normal PC. However for lattice velocity profiles, more iterations, usually 512 1024 (or more depending on the diffusion coefficients), are needed for stabilization. 3.3 Test of the Simulation for a Model System An examination of the internal consistency of the simulation can be performed with the aid of model systems. The goal here is to compare the output from the simulation with those obtained using the analytical models. Two such model systems are considered here. The first is for a binary, semiinfinite diffusion couple having constant molar volume; the second is for a similar couple having variable molar volume, but with constant partial molal volumes. 3.3.1 Analytical Model System with Constant Molar Volume The analytical model has to be capable of generating intrinsic diffusion information such as intrinsic fluxes and lattice velocity profiles, along with conventional interdiffusion information such as composition profiles. Such a simple model for a binary system is one in which the ratio R of the intrinsic diffusion coefficients is assumed to be constant over the range of compositions considered: K R K (324) DR where DAK and DB are the intrinsic diffusion coefficients of components A and B respectively. A final simplification of the model assumes that the interdiffusion coefficient D for this system is constant for the range of compositions considered. It should be noted that since the molar volume is assumed to be constant, there exists a single, unique interdiffusion coefficient in this binary, singlephase system (see Eq. (C 15)), which is equal to the diffusion coefficient in the volume frame, 1D Assuming a constant interdiffusion coefficient, the concentration profiles for the two components can be generated using error functions: XB(x) = XB + (XB XB) cerf 2 Dt (325) X (x) = 1 XB(x) XB and XB are the far field compositions of component B in the two binary alloys that comprise the diffusion couple. Cerf(z) is the "complete error function" that is related to the normal error function erf(z) by cerf(z) = (1 + erf(z)) (326) 2 The error function erf(z) is defined as erf(z) = fexp(u2)du (327) The concentration parameters XA and X, are the atom fractions of components A and B respectively Usually, in the error function solution, Eq.(325), the concentrations have to be expressed in terms of ck (moles/cc), however since the molar volume is constant, concentrations in terms of atom fractions are valid (since ck = X/V, the Vterm cancels on both sides of the equation). In Eq. (325), x is the diffusion distance parameter and t, the time for the diffusion experiment. The Darken relation that relates the intrinsic and interdiffusion coefficients [Dar48] is D = X,4D + XB D (328) Using Eq. (328) and Eq. (324), expressions for the intrinsic coefficients in terms of the interdiffusion coefficient and R can be obtained: D (x) = D(x) (329) 1 XB(x) + XB(x)R D (x) = RDBK(x) = R I X(x)+ XB(x)R (330) 1 XB(x) + XB(x)R This variation of the intrinsic diffusion coefficients with composition is shown in Fig. 33 for the specified value of the interdiffusion coefficient and R. The lattice velocity is given by [Dar48] SK K D 1 dx(x Substituting for the intrinsic coefficients given in Eqs. (329) and (330), in Eq. (331), an expression for the lattice velocity in terms of the interdiffusion coefficient and R is obtained: KrX ^D(x) dXB (x) V K(x) = ( R)  (  (332) 1 XB(x) + XB(x)R dx 3 An analytical expression for the differential on the R.H.S. of Eq. (332) can be obtained using Leibnitz' rule: 60 C 50 (_ 20 UD D B S40 0.0 0.2 0.4 0.6 0.8 1.0 XB Figure 33. Variation of the intrinsic diffusion coefficients with composition for the model system. The ratio of the intrinsic diffusion x 30 exp 4 dXB(x) (XB XB) 4Dt (333) o 20 ""' v K x xo0 (334) 0.0 0.2 0.4 0.6 0.8 1.0 The distances x and x, are assumed to be measured with respect to the origin at x = 0. The above equation (334) is a first order linear differential equation that can be solved for xo as a function ofxsiti for any given time t. Rearranging Eq. (334) in a typical form, as a function of x for any given time t. Rearranging Eq. (33 4) in a typical form, vK _) + X = 0 2t 2t dxo (335) dx This equation can be solved numerically using a math software such as MathCad (using the function "rkfixed"). The boundary conditions are known since at the far field compositions of the diffusion couple, the lattice velocity is zero, i.e., the lattice positions are unchanged. Since Eq. (335) has a singularity at the position corresponding to the Kirkendall interface (x = v K 2t), the complete numerical solution (xo(x)) was obtained for two separate intervals, from x = x to x, and then from x = x' to xK. Once x0 was obtained, the lattice shifts (x x0) could be also determined. The intrinsic fluxes are given by D4(x) dXA(x) D D(x) dX(x) J4(x) ; JB(x) B dX () (336) V d dx V V dx and the vacancy flux is J(x)= [J4() + JB(x)] (337) Note that since the sum of the atom fractions is unity, dX (x) dXBx) A B X) (338) dx dx Using Eqs. (330, 332, 336 and 338), the intrinsic fluxes given in Eq. (336) and the vacancy flux, Eq. (337), can be expressed in terms of the interdiffusion coefficient and R. The interdiffusion fluxes are D (x) dX(x) D d(x) dX) J(x) " A Jo(x) X (X) (339) ) \ dx ) V dx Parameters. The following were the parameters for the model system: interdiffusion coefficient D = 1010 cm2/s, ratio of intrinsic coefficients R = 5, molar volume V= 9 cc/mole, diffusion time t = 123600 s for the error function solution. A fullrange diffusion couple, i.e., where the infinite compositions are the pure components, was selected. The initial compositions at t = 0 were: XA = 0, XB = 1 and A = 1, XB = 0. The initial setup conditions for the simulation are: spacing 6 = 0.5 rim, time step for each iteration At = 10 s, # of slices = 240. A total of 512 iterations were performed to obtain a high resolution output. This corresponds to a duration of the diffusion experiment, t,= 5120 s. In Figs. 34 to 37, the circles represent the output obtained using the simulation and the solid lines those from the analytical error function solution. If the output from the simulation were plotted for every slice position x,, the distance between any two points x, and x,,, in the figures would represent the slice width of slice i. However in the present situation, the interval between the data points (every 10th data point was plotted) was varied to permit sufficient resolution between the two outputs. In Fig. 34a, the compositions profiles (in atom fractions, Xk's) for the analytical model and the simulation are plotted as a function of the distance. If x is the distance variable for the model system, the corresponding distance variable for the simulation is x, x1, where the initial position of the 121st slice, xo, corresponds to the position x  0 (the Matano interface) for the analytical model solution. In Fig. 34b, the compositions are plotted as a function of the normalized distance (or Boltzman) SxI x parameter, x for the analytical solution, and x 21 for the simulation. Although the diffusion times in the simulation and the analytical solution are different, the resultant profiles when plotted using the normalized parameter, are identical as predicted from Fick's second law. The output from the simulation can be converted to any desired time by multiplying the normalized distance with the square root of the diffusion time. For example, in Fig. 34a, the composition profiles are plotted as a function of the distance x for time t = 12 hrs. Figs. 35a and 35b depict the variation in the lattice velocity and lattice shift with distance respectively. It is seen that the maximum value of the lattice shift and the lattice velocity both occur at the Kirkendall interface. Note that in general, this is not always true. Cornet and Calais [Cor72, Cor74 ] and van Dal et al. [DalOOb] have discussed models to prove this point. For example, instead of choosing a model where the ratio of the intrinsic diffusion coefficients is constant, a model where the difference between the coefficients is constant, can be shown to have different positions for the maximum in the lattice shift and lattice velocity profiles. Figs. 36a and b show the lattice position x and lattice shift Ax as a function of the initial position xo. An "oblique marker" experiment (Appendix E), that can provide the positions of lattice markers throughout the diffusion zone, results in a lattice position profile similar to that shown in Fig. 36b [Cor72, Cor74, Phi91]. The variation in the intrinsic fluxes and vacancy flux is depicted in Fig. 37a, while Fig. 37b shows the variation in the interdiffusion fluxes with position. Since the molar volume is constant for the present model system, the interdiffusion fluxes are equal and opposite in sign (see Appendix A). It is clear from the figures that the simulation faithfully reproduces the analytical results for the model system considered. 1.0 XB XA 0.8  S0.6 U E (a) o 0.4 0.2 0.0 00 200 100 0 XK 100 200 x (Gm) 1.0 1 . 0.8  (b) 0.6  o 0.4  0.2  0.0  1.0 0.5 O.OXK 0.5 1.0 xA/t (jim s1/2) Figure 34. Composition profile for the model system as a function of (a) distance from the origin (the initial interface) and (b) normalized distance The solid lines denote the error function solution, the circles the output from the simulation. XK is the position of the Kirkendall interface. The parameters for this model system are:D = 1010 cm2/s, R = DA ID = 5, V= 9 cc/mol, t= 123600 s. 100 50 0 XK 50 100 150 x (Gm) 100 50 0 XK 50 100 150 x (Gm) Figure 35. Variation of the lattice velocity and lattice shift as a function of position. (a) Lattice velocity; (b) lattice shift. The maximum for both the graphs is at the Kirkendall interface for the chosen model. 250 (o ) 200 o E 150  100 50 2] 50 0 4 150 150 150 64 20 15 (a) 10  100 50 0 50 100 XO (tm) 100 50 X K (b) j 0 50  100 50 0 50 100 X0 (Gim) Figure 36. Dependence of the lattice shift and lattice position on the initial lattice position, i.e., at t = 0. (a) Lattice shift; (b) lattice position. 20 S10 E J 0 CU (a) C I 10 20 100 50 0 XK 50 100 x (im) 20 E E ste (b) 0 cJ 0 0 4 100 50 0 XK 50 100 x (tim) of A is greater than B, the resultant vacancy flux is positive. Hence the Kirkendall shift is also positive. The interdiffusion fluxes are equal and opposite in sign. equal and opposite in sign. 3.3.2 Analytical Model System with Variable Molar Volume In this case, it is assumed that the molar volume of the system is a function of composition. A simple model for a composition dependent molar volume is one in which the partial molal volumes are equal to the molar volumes of the pure components but different from each other. V4 = V; Vo B V (340) Hence the molar volume is obtained by ideal mixing of the pure components (Vegard's law) and exhibits a linear variation with composition, XB (Fig. 38a). V.x VAX + VBXB = VVAA + VBX(341) As in the previous case, the ratio R of the intrinsic diffusion coefficients is assumed to be constant over the range of compositions considered. In Appendix D, it is shown that if the partial molal volumes are constant and equal to the molar volumes of the pure components, the volume averaged velocity, vv, is zero. Hence, as in the previous case, from Eq. (C15), the interdiffusion coefficient, D for this system is constant for the range of compositions considered and equal to the diffusion coefficient in the volume frame, D D. Hence, the concentration profiles for the two components can be generated using error functions. However, since the molar volume is variable the concentration units have to be expressed in terms of ck (moles/cc). The expressions are CA(x) = CA + (CA CA)cerf 2D (342) CB(X) = CB + (CB CB)cerf 2 t where C4 and cA, and CB and cR are respectively the far field concentrations of components A and B in the two binary alloys that comprise the diffusion couple, and are given by A B A B CA ; C and c ; C (343) V V V+ V The far field molar volumes V and V+ are obtained using Eq. (341): V = VoX4 + VBoXB and V = VAo X + Vo X (344) The variation of the molar volume with position is obtained from the concentrations that are provided by the error function solutions, Eq. (342): 1 1 V (x) c7\(x7) = cC (345) CA(X) + CB(X) C(x) where C is the total concentration in mol/cc. The atom fraction of component k is obtained from the product of the concentration and the molar volume: X,(x) = C,(x) V",(x) (346) The Darken relation that relates the intrinsic and interdiffusion coefficients is now modified (Eq. (C50)). D = cVADf + cBVBD = CAVDR + cBVD (347) As in the previous case, expressions for the intrinsic coefficients in terms of the interdiffusion coefficient and R can be obtained: K D(x) DB (x) = D (348) CA(X) V4 + CB(X)VBR 68 10 9 VA 0 E 3 8 8 w (a) E Vmix ( 3 3 0 > 7 o0 6 5 0.0 0.2 0.4 0.6 0.8 1.0 XB 60 50 x NE 40 3 DA (b) 5 30 0 20 0 S10 0 0  ^ ^ 0.0 0.2 0.4 0.6 0.8 1.0 XB Figure 38. Variation of the molar volume and intrinsic diffusion coefficients with composition for the model system with varying molar volume. (a) Molar volume and (b) intrinsic diffusion coefficients. The parameters are: R = D /D = 5, V1 = 9 and V0 = 5 cc/mol and D = 10 10 cm2/s. Df (x) = RD (x) = R CA(X) +C(x) R (349) c (x) V + cB(x) VBOR The variation of the intrinsic diffusion coefficients with composition is shown in Fig. 3 8b. The lattice velocity measured with respect to the volume frame (Eq. (C48) and (B9)) is given by SKV(x) = [D,(x) D (x)] dcB (350) S'4 B dx However, since v is zero, SKV(x) = vK() vV(x) = K(x) (351) Substituting for the intrinsic coefficients given in Eqs. (348) and (349), in Eq. (351), an expression for the lattice velocity in terms of the interdiffusion coefficient and R is obtained: 1K D\ D(x) dcB(x) v 'K)= (1_ R) B (352) c(x)V + cB(x)VR dx (352) The original position, xo, of a marker or lattice plane at t = 0 and the lattice shift, (x xo), can be obtained from the lattice velocity in a manner similar to that for the previous model system, see Eqs. (334) and (335). The intrinsic fluxes are given by K dc A(x) K K dcB(x) JK(x) = D (x) (= DR (x) (353) dx dx( 70 and the vacancy flux is the negative sum of the intrinsic fluxes, see Eq. (337). As in the earlier case, the intrinsic fluxes and the vacancy flux can be expressed in terms of the interdiffusion coefficient and R. The interdiffusion fluxes are given by 0 dc, (x) dcB(x) J(x) = D(x) ; J (x) = D(x) (354) dx dx The number weighted velocity, vA, is obtained using Eq. (A22): VN(x) = V x)[J(x) + J(x)] (355) and the corresponding flux of component k in the numberfixed frame, Jk is (Eq. (A20)) J =Jko Vk N (356) Parameters. The molar volumes of the two components for this model system were: V = 5 and VBo= 9 cc/mole. The other parameters for this model system were identical to the previous one: interdiffusion coefficient D = 1010 cm2/s, ratio of intrinsic coefficients R = 5, diffusion time t = 123600 s for the error function solution. A full range diffusion couple, i.e., where the infinite compositions are the pure components, was again selected. The initial compositions at t = 0 were: XA = 0, XB = 1 and XA = 1, X = 0. The initial setup conditions for the simulation were identical to the previous case. In Figs. 39 to 313, the circles represent the output obtained using the simulation and the solid lines those from the analytical error function solution. In Fig. 39a, the concentration profiles, Ck (mol/cc), are plotted as a function of distance, x, and in Fig. 39b as a function of the normalized distance (or Boltzman) x parameter, . From knowledge of the concentrations and the molar volume, the V{t composition profiles in atom fractions, X., are plotted as a function of distance and normalized distance in Figs. 310a and b respectively. In Fig. 31 la, the variation in the lattice velocity, v' (Eq. (352)), and the number weighted velocity, v (Eq. (355)), are shown. Since v is nonzero only when the molar volume is variable, it is evident from Fig. 31 la, that the contribution is quite significant. In Fig. 31 Ib, a plot of the lattice shift as a function of distance again reveals that the maximum shift is for the Kirkendall plane. It is to be noted that in spite of choosing a model system with variable molar volume, the lattice velocity and lattice shift are identical to the previous case. This appears to be the case since the particular choice of parameters used for the model system (other than the varying molar volume) were identical to the previous case. A detailed analysis has however not been performed. The variation in the intrinsic fluxes and vacancy flux is depicted in Fig. 312a. In Fig. 3 12b, the number fluxes (Eq. (356)), shown are equal in magnitude and opposite in sign as expected from Eq. (A18). Since the molar volume is not constant, it is seen from Fig. 312c, that the partial molal volume weighted interdiffusion fluxes are equal in magnitude and opposite in sign, see Eq. (A27). It is clear from these figures, that as in the previous case, the simulation again reproduces the analytical results for the model system with variable molar volume. Rather than compare the output of the simulation with analytical results obtained for the model system, the true test of the simulation is in comparison with experimental information for a number of important systems. An application for the FeNi system is discussed in the next section. More examples are provided in Chapter 4. 72 0.20  0cB I J 0.15  0 E \ I cA (a) 0.10 0 o 0.05 0.00 200 100 0 XK 100 200 x (jim) 0.20  CB o 0.15 0 E CA (b) 0.10 o 0.05 0.00 1.0 0.5 0.0 XK 0.5 1.0 xlt (tm s1/2 Figure 39. Concentration profile (in mol/cc) for the model system with variable molar volume as a function of (a) distance from the origin (the initial interface) and (b) normalized distance The solid lines denote the error function solution, the circles the output from the simulation. XK is the position of the Kirkendall interface. The parameters for this model system are:D = 1010 cm2/s, R = D IDB = 5, V = 9 and Vo = 5 cc/mol, t 123600 s. 1.0 ...... XB I XA 0.8 I 0.6 E A (a) o 0.4 0.2 0.0 200 100 0 XK 100 200 x (Gm) 1.0 XB I XA 0.8 0.6 (b) U o 0.4 0.2 0.0 1.0 0.5 O.OXK 0.5 1.0 xMvt (Gm s1/2) Figure 310. Composition profile (in atom fraction) for the model system as a function of (a) distance from the origin (the initial interface) and (b) normalized distance Note that Xk = ckVx,. 74 250 'K V 200 (o 1 150 SL(a) 100 v 50 0 150 100 50 0 xK 50 100 150 x (Jm) 20 15 10 5 averaged (b) 150 100 50 0 XK 50 100 150 x (Jim) Figure 311. Mean and lattice velocities, and lattice shift as a function of position for the model system with variable molar volume. (a) Variation of the lattice velocity (vK) and number averaged velocity (vN), and (b) lattice shift. 20 I CN E oE 0 J I o 10 C 20 100 50 0 XK 50 100 x (jim) 20 C x N S10 JB (C1 n i E z 20 100 50 0 xK 50 100 x (Gm) Figure 312. Diffusion fluxes for the model system with variable molar volume. (a) Intrinsic fluxes of the components and the resultant vacancy flux, and (b) fluxes in the number frame. Note that the number fluxes are equal in magnitude and opposite in sign. 20 x \/OBJOB P 10 10 C 0L 0 10 20 100 50 0 XK 50 100 x (jim) Figure 312. Continued. (c) Partial molal volume weighted interdiffusion flux as a function of distance. Since the molar volume of the system is varying, the partial molal volume weighted interdiffusion fluxes are equal in magnitude and opposite in sign. 3.4 Test of the Simulation for the FeNi system The intrinsic diffusion coefficients in the a FeNi system at 1200'C were measured by Kohn et al. [Koh70] using a multifoil technique [Heu57]. It is seen in Fig. 313a, that the intrinsic coefficient of Fe is much higher than Ni over the entire composition range. It is also noticed that the intrinsic coefficients of both components increase with increasing Ni (the slower diffuser). This is reflected in a deeper penetration depth on the Nirich side of the diffusion couple as shown in Fig. 313b. The Kirkendall shift (XK in Fig. 313b) is significant for the duration of the experiment (48 h) due the large differences in the intrinsic fluxes of Fe and Ni (Fig. 314a). The lattice velocity distribution obtained from the simulation is compared with the experimental data in Fig. 314b. The agreement for all the graphs is reasonable given the fact that the authors 1e9 le10  le11 0.0 0.8 1.0 0.01 I.1 ,I 400 300 200 100 0 XK 100 200 x (inm) Figure 313. Application of the intrinsic diffusion simulation for the FeNi system at 1200'C. (a) Intrinsic diffusion coefficients as a function of composition. (b) Concentration profile for a fullrange FeNi diffusion couple. The solid line represents the output of the simulation based on the intrinsic diffusion data in Table 1 of Kohn et al. [Koh70], that includes data from both full range and incremental couples. Circles indicate the data points obtained from the experimental concentration profile given in Fig. 4 of their paper. XK denotes the Kirkendall interface obtained from the simulation. The location of the Matano interface in their Figure 4 has been corrected by 15 microns to account for the porosity present on the Fe rich side of their diffusion couples. 0.2 0.4 0.6 Atom Fraction Ni FeNi, 12000C, 48 hrs 3 15 s0 S10 x 3 5 ') *I 5 0.2 0.4 0.6 Atom Fraction Ni 0.2 0.4 0.6 Atom Fraction Ni 0.8 1.0 0.8 1.0 Figure 314. Intrinsic fluxes and lattice velocity for the Fe Ni system at 12000C. (a) Intrinsic flux distribution as a function of composition. The intrinsic flux of Fe is much higher than Ni since the intrinsic diffusion coefficient of Fe is higher than Ni. (b) Lattice velocity profile. From Fig. 3 13(b), it is seen that the lattice shift is towards the Ferich side of the diffusion couple, i.e., in the +x direction. Hence, the lattice velocity is positive. 79 had to apply a correction in the measured lattice velocity profile to account for the formation of Kirkendall porosity on the Ferich side of their diffusion couple. Vignes and Badia [Vig69a] have demonstrated that the porosity formation can be attributed to the low purity of the electrolytic Fe utilized in their experiments, the use of highpurity zonerefined Fe prevented this problem. Nevertheless, the simulation demonstrates, that the correction employed by the authors, results in reasonable agreement with the experimental data. In upcoming chapters, the simulation is repeatedly applied to more systems that include both binary and ternary systems. CHAPTER 4 TESTS OF DARKENMANNING THEORIES USING THE INTRINSIC DIFFUSION SIMULATION In the previous chapter, a simulation based on the Darken theory of intrinsic diffusion was introduced [Dar48] and its selfconsistency was demonstrated with the aid of a model system. In this chapter, the DarkenManning (DM) theories [Dar48, Man68, Man70] that have long (about 50 yrs.) been used to connect diffusion behavior in homogenous and gradient systems, are analyzed with the aid of the simulation. Attempts to examine the DM theories are not new, however most of these are based on comparisons between interdiffusion or intrinsic diffusion coefficients and tracer diffusion coefficients in binary systems. A more reliable method of assessing the DM theories, is to compare the experimental observables in a diffusion experiment, e.g., concentration profiles and lattice shifts, with those predicted using these theories. Without the aid of a simulation capable of reproducing the experimental observables using the intrinsic diffusion coefficients predicted from the DM theories as the input, such an endeavor could not be attempted in the past. In this chapter, the relevance of the DM theories to current diffusion theory is first considered. Some of the previous tests of these theories in the literature are critically examined. The Darken theory and the important assumptions involved in its development are then discussed. Since the Manning theory, although theoretically significant, results in relatively minor corrections to the Darken equations, only a brief discussion is presented. A procedure for examining the DM theories using the simulation is discussed. This procedure is then utilized to test the DM theories for four systems: AgCd, AuNi, CuZn and CuNi. 4.1 Importance of DarkenManning Theories There are many situations, where the kinetic parameters, e.g., the intrinsic diffusion coefficients, that are needed as the input to the simulation, are either unreliable or are impossible to determine due to the experimental difficulties involved in their measurement. In thin films, composition gradients are very steep, due to which measurement errors using a technique such as the electron microprobe [Gol02, Zie64], can be rather inaccurate if adequate precautions are not taken [Loo90a]. Stress and gradient energy effects [Ste88, Lar85, Hil69]; electric, magnetic and other external fields [Man62, Man89, Hun75], can cause errors in the measured diffusion coefficients, if these are not explicitly taken into consideration in formulating the diffusion equations in such systems. Impurities can cause notable differences in the measured diffusion coefficients; for e.g., in the FeNi system, Vignes and Badia [Vig69a] have demonstrated, that the utilization of low purity, electrolytic Fe causes the formation of Kirkendall porosity due to which the measurements may be compromised. Similarly, Van Loo [Loo90] has discussed a situation where the presence of oxygen impurities led to the formation of oxide phases during the early stages of diffusion in the TiAl system. The microstructure may also have an important effect on the experimental measurement of diffusion coefficients. For example, in the case of diffusion in fine grained materials, it is necessary to include a grain boundary diffusion contribution in addition to the usual volume diffusion contribution. Furthermore, if there exists anisotropy in the microstructure, the resultant diffusion coefficients may be quite different from those measured for isotropic microstructures. Even if the practitioner is aware of the many factors that may influence the measurement of the diffusion coefficients, quite often, it is too cumbersome and impractical for him/her to engage in deconvolution. As a result, the diffusion coefficients reported by different authors can vary appreciably. Some of the more fundamental problems in the phenomenological theory of diffusion were considered in Chapter 2. For example, the values of the phenomenological diffusion coefficients are not unique, they depend on the choice of the independent composition variables. As the number of components in a system increases, the physical meaning of these coefficients and the connection with the actual jump frequency of atoms in the moving system, becomes increasingly abstruse. A more pressing problem is the wellknown fact that in quaternary or higher order systems, the matrix of diffusion coefficients cannot be determined without simplifying assumptions, for e.g., the crossterms in the matrix of diffusion coefficients are ignored (as in the software DICTRA), or the diffusion coefficients are assumed to be constant within a narrow composition range. While such assumptions may be adequate or justifiable in some cases, they fall short of providing a reliable methodology for developing robust multicomponent databases for diffusion, that can be utilized in a simple and effective manner by the average researcher. In comparison to the phenomenological diffusion coefficients, tracer diffusion coefficients or tracer jump frequencies have unambiguous meaning, i.e., there is a unique value for each component regardless of the number of components. Ignoring the negligible difference in the atomic mass between a component and its isotope (tracer), the tracer diffusion coefficient is practically identical to the diffusion coefficient of that component in a homogenous system (no composition gradients). The measurement of tracer diffusion coefficients is usually carried out on highpurity, homogeneous samples under wellcontrolled laboratory conditions. These measurements involve the use of radioactive tracers and are difficult to perform due to the obvious environmental and health concerns. Nevertheless, they are highly accurate if properly conducted and measurement errors within 3 pct in tracer diffusion measurements from separately annealed specimens within the same laboratory are obtainable [Rot84]. Values of tracer diffusion coefficients from easy duplicable materials (e.g., Ag 99.999 pct) are found to be reproducible within 10 pct between laboratories. This is to be compared to the 20 % or higher error usually reported for diffusion coefficients using diffusion couples [Kap90]. Modem thermodynamic databases based on CALPHAD techniques are being increasingly used in conjunction with calculations based on "firstprinciples" for the purpose of designing and comprehending complex multicomponent, multiphase systems [Kau70, Kat97]. The advantage of a firstprinciples approach is that the fundamental properties of the system, including the basic atom jump frequencies (practically equal to the tracer jump frequencies) of the components can be obtained, although such approaches are primarily restricted to unary and a few binary systems at the present time [Mur84]. Given the inherent difficulties associated with the measurement of diffusion coefficients in multicomponent systems (some of which may contain more than fifteen elements) and the ambiguous meaning of these coefficients, it appears that the best approach for developing modem diffusion databases and simulations for modeling evolutionary processes in multicomponent materials, should be based on the connection between diffusion behavior in homogenous and gradient systems. Thus if the diffusion behavior of a component in a homogenous system can be accessed with the aid of a database, its behavior in a moving system can be predicted based on this connection. Alternatively if the diffusion coefficients have been measured in the moving system, their authenticity may be verified. The DarkenManning theories that provide these connections are therefore of vital importance in such endeavors. 4.2 The Darken Theories 4.2.1 Darken A: Intrinsic Diffusion The diffusion theory of Adolf Fick [Fic55], first proposed in 1855, is still regarded as the foundation behind modern diffusion theory, in spite of the numerous modifications that have occurred since that time. Darken's theory of intrinsic diffusion [Dar48, Dar51], that was initially presented in 1948, was in response to the inherent limitations of Fick's formalism in describing diffusion behavior in more complicated systems. Hence, a discussion of the historical context with which the Darken theories were originally formulated, appears necessary. Fick's first law states that the flux of a diffusing substance along the diffusion direction is proportional to the concentration gradient of the diffusing substance. In the case of unidirectional diffusion in a binary system, this can be written as ack Jk = D k (41) 8x Jk is the flux of component k in mol/cm2s; the implied constant of proportionality in Eq. (41), D, is the diffusion coefficient expressed in units of cm2/s; Ck is the concentration of component k in mol/cc; and x is the distance in cm. Fick's second law is essentially a conservation statement for component k: ack a k a ack 8= k ( D (42) at 8x 8x 8x and ifD is independent of concentration, the second law simplifies to 8c, 82C k D k (43) at 8x2 At the time the laws were proposed, there existed little experimental information for validating whether the constant of proportionality in Fick's first law, i.e., the diffusion coefficient, was indeed a constant as the law implied. Subsequently, experiments conducted (mostly in gaseous or dilute liquid solutions) found that variations in the diffusion coefficient with composition were relatively small (less than 8 %), thus appearing to validate the law [Dar48, Dar51]. It was not until considerably later, that diffusion studies in electrolytes and polymer systems and eventually in metallic systems, conclusively demonstrated that the diffusivity (or the diffusion coefficient) varied significantly with composition. In fact, it was found that the diffusion coefficient varied by several orders in magnitude in some systems (the CuZn system discussed later in this chapter is an example). With these findings, Fick's law evolved to a definition of the diffusion coefficient D as a function of composition. Note that it is also implied in the Fickian formalism, that D is not a function of the concentration gradient, but is only a function of state variables such as the temperature, pressure and composition. This definition of D was facilitated by the works of Boltzmann [Bol94] and Matano [Mat33], who showed that by replacing the distance (x) and time (t) variables in Fick's second law with a single variable, X = , Fick's second law (Eq. 42) in a semiinfinite system {It XM = 0 Figure 41. BoltzmanMatano analysis for the determination of the interdiffusion coefficient. The Matano interface, i.e., the origin, is positioned so that the areas on the two sides of the interface are equal. could be transformed to an ordinary differential equation by virtue of which the composition dependent diffusivity, D, could be easily computed. This expression is 1 dk 1 dx D(ck') I A dck dck (44) 2 dck c 2t dck Ck C where ck is the concentration of component k at a farfield position (x = ), that is unchanged during the course of the diffusion process. In order to determine D in Eq. (4 4,) it is necessary to determine the position of the origin from which x is measured. This choice of origin, first suggested by Matano [Mat33], is known as the "Matano interface," and is obtained by dividing the two areas in Fig. 41, such that they are equal. Mathematically, this can be stated as 