NUMERICAL SIMULATIONS OF CONCENTRATION-DEPTH PROFILES OF CARBON AND NITROGEN IN AUSTENITIC STAINLESS STEEL BASED UPON HIGHLY CONCENTRATION DEPENDENT DIFFUSIVITIES by XIAOTING GU Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Advisor: Gary M. Michal, Ph.D. Department of Materials Science and Engineering CASE WESTERN RESERVE UNIVERSITY May, 2011 Dedicated to Fen and My Parents CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of XIAOTING GU candidate for the (signed) Doctor of Philosophy degree *. GARY M. MICHAL (chair of the committee) ARTHUR H. HEUER FRANK ERNST WALTER R. LAMBRECHT _________________________________________________ _________________________________________________ (date) Nov. 12th, 2010 *We also certify that written approval has been obtained for any proprietary material contained therein. Table of Contents Table of Contents ……………………………………………………………………. i List of Tables ……………………………………………………………………….. vii List of Figures ………………………………………………………………………. ix List of Symbols ……………………...…………………………………………….. xiv Acknowledgements ……………………………………………………….……….. xx Abstract ………………………………………………….…………………………. xxii Chapter 1 Introduction …………………………….…………………………… 1 1.1 Surface Engineering ………………………………………………………….... 1 1.2 Austenitic Stainless Steels …………………………………………………….. 2 1.3 Low Temperature Carburization …………………………………………….… 4 1.4 The Goal ………………………………………………………………..……… 5 Chapter 2 Thermodynamic Modeling ………………………………………. 7 2.1 CALPHAD Model …………………………………………..………………… 7 2.2 Gibbs Free Energy of Compounds ………………………………….…………. 8 2.2.1 Definition of Site Fraction …………………………..………….……….. 8 2.2.2 Gibbs Free Energy Reference State ………………………...………….. 11 2.2.3 Ideal Entropy of Mixing ……………………………………...………… 15 2.2.4 Gibbs Excess Energy of Mixing …………………………………….…. 16 2.2.5 Magnetic Gibbs Free Energy …………………………………...……… 18 2.2.6 Contribution of Pressure to Gibbs Free Energy …………………...…… 20 2.2.7 Total Gibbs Free Energy of Compounds ………………………………. 20 i 2.3 2.4 Chemical Potential …………………………………………………..……….. 23 2.3.1 Derivation of the Chemical Potential of Carbon …………………….…. 24 2.3.2 Expressions of the Chemical Potentials of Carbon and Nitrogen …….... 27 Conversion between Elementary Thermodynamic Model and Compound Energy Model ……………….…………………………………………..……. 29 Chapter 3 3.1 3.2 3.3 3.4 Numerical Simulation ……………………………………………. 31 Concentration Dependent Diffusion ……………………………….………… 31 3.1.1 Fick’s Second Law ………………………….………………………….. 31 3.1.2 Concentration Dependence of Diffusion Coefficient ………….………. 34 Introduction to Finite Difference Method ……………………………….…… 35 3.2.1 Taylor Series Formulation …………………………………...………… 36 3.2.2 Finite Difference Approximation of Derivatives ……………….……… 38 3.2.3 Errors Involved in Numerical Solutions ………………………..……… 39 Simulation Schemes for One-Dimensional Diffusion ……………………….. 40 3.3.1 Simple Explicit Scheme ……………………………………………..…. 41 3.3.2 Fully Implicit Scheme ……………………………………………….…. 43 3.3.3 Crank-Nicolson Scheme ……………………………………………….. 44 Boundary Conditions ……………………………………………..………….. 46 3.4.1 Fixed Boundary Condition ……………………………………..………. 46 3.4.2 Convective Boundary Condition ………………………………….……. 47 3.5 Stability ……………………………………………………………………..... 49 3.6 Fully Implicit Scheme with Concentration-Dependent Diffusion Coefficient ……...………………………….…………………….... 52 3.7 Lagging Properties by One Time Step ……………………………………….. 55 3.8 Computational Environment …………………………………………...…….. 57 ii Chapter 4 4.1 4.2 Experimental Methods ………………..…………………………. 59 Auger Electron Spectroscopy ……………………………………….……….. 59 4.1.1 Introduction …………………………………………………………….. 59 4.1.2 Specimen Preparation …………………………………………..……… 60 4.1.3 Experimental Procedure …………………………………...…………… 61 X-Ray Photoelectron Spectroscopy ………………………………………….. 63 4.2.1 Introduction ………………………………………………………….…. 63 4.2.2 Specimen Preparation ……………………………………………….…. 64 4.2.3 Experimental Procedure ………………………………………...……… 65 Chapter 5 Modeling of the Concentration Dependence of Carbon Diffusivity in Austenite …………………...………... 67 5.1 5.2 Literature Review ……………………………………………………..……… 68 5.1.1 Absolute Reaction Rate Theory / ARRT (1948) ……………………….. 69 5.1.2 Kaufman et al. (1962) ………………………………………………….. 70 5.1.3 Asimow (1964) ………………………………………………………… 70 5.1.4 Siller and McLellan (1970) ……………………………………..……… 72 5.1.5 Tibbett (1980) ………………………………………………………….. 72 5.1.6 Larché and Cahn (1982) ………………………………………………... 73 5.1.7 Ågren (1986) …………………………………………………………… 73 5.1.8 Liu et al. (1991) ………………………………………………….…….. 74 5.1.9 Evaluation of the Models …………………………………….………… 75 Diffusion Coefficient of Carbon at Infinite Dilution ……………………..….. 78 5.3 Application of Asimow’s Model of Carbon Diffusivity to AISI 316L Stainless Steel ………………………………………………………………... 80 5.3.1 Activation Volume ………………………………………………...…… 81 5.3.1.1 Compressibility χ …………………………………………...……. 81 5.3.1.2 Grüneisen Parameter δ …………………………………………… 82 iii 5.3.1.3 5.3.2 5.4 The Value of the Parameter K ……………..………………………….. 84 Physical Origin of Concentration Dependent Diffusion ……………..….…… 85 Chapter 6 6.1 Activation Gibbs Free Energy ΔGA ……………………………… 83 Simulation of the Diffusion of Carbon into Austenite …… 87 Four Types of Simulation ………………………………………………...….. 88 6.1.1 Fixed Boundary Condition and Concentration-Independent Diffusion Coefficient …………...……………………………………… 88 6.1.2 Convective Boundary Condition and Concentration-Independent Diffusion Coefficient …………………...…………………………….... 90 6.1.3 Fixed Boundary Condition and Concentration-Dependent Diffusion Coefficient ……………………………………………………………… 93 6.1.4 Convective Boundary Condition and Concentration-Dependent Diffusion Coefficient ……………………………...…………………… 96 6.2 Simulation of the Swagelok Process …………………………………………. 98 6.2.1 Swagelok Process of Low Temperature Carburization …………...…… 98 6.2.2 Maximum Possible Solubility of Carbon XC,max ……..……………….. 101 6.2.3 Activity of Carbon aC in the Carburization Atmosphere ……….…….. 102 6.2.4 Simulation Parameters ………………………………………….…….. 104 6.2.5 AES Results …………………………………………………...……… 105 6.2.5.1 The Composition of Metal Elements …………………………… 107 6.2.5.2 Concentration-Depth Profile of Carbon ………………………… 108 6.2.5.3 Comparison with Numerical Simulation …………………..…… 109 6.2.6 XPS Results ……………………………………………………...…… 111 6.2.6.1 The Composition of Metal Elements …………………………… 116 6.2.6.2 Concentration-Depth Profile of Carbon ………………………… 117 6.2.6.3 Comparison with Numerical Simulation ……………………….. 119 iv Chapter 7 7.1 7.2 7.3 Simulation of the Diffusion of Carbon and Nitrogen during Plasma Carbonitriding …………......…….. 121 Thermodynamic Analysis ………………………………………………..…. 123 7.1.1 The Gibbs Free Energy of Austenite ………………………….……… 123 7.1.2 The Chemical Potential of Carbon in Austenite ……………..……….. 124 7.1.3 The Chemical Potential of Nitrogen in Austenite …………….………. 126 Numerical Simulation of Diffusion ……………………………………..….. 129 7.2.1 Simulation Conditions ……………………………………….……….. 129 7.2.2 Concentration-Dependent Diffusion …………………………….……. 130 7.2.3 Simulation of Carbon Concentration-Depth Profile for Carburization at 430oC ……………………………..……………...….. 131 7.2.4 Simulation of Nitrogen Concentration-Depth Profile for Nitridation at 430oC …………………………..………………………. 133 7.2.5 Simulation of Carbon and Nitrogen Concentration-Depth Profiles for Plasma Carbonitriding at 430oC ……………….………… 135 7.2.5.1 Simulation Parameters ………………………………….………. 135 7.2.5.2 Computational Logic ………………………………………...…. 136 7.2.5.3 Simulation Results ……………………………………………… 139 Simulation Predictions ……………………………………………………… 146 Chapter 8 Discussion ……………..…….…………………………………. 151 8.1 Mass Transfer Coefficient …………………………………………………... 151 8.2 Selection of the Diffusion Model …………………………………………… 159 8.2 Influence of Carbon Concentration on Young’s Modulus ……..…………… 162 8.2 Influence of Nitrogen on the Drift Velocity of Carbon ………..…………… 163 Chapter 9 Conclusions ……………………………………………………..… 166 Chapter 10 Suggestions for Future Research …………………….…….. 168 v Appendix I CALPHAD Parameters Employed in the Thermodynamic Analysis ……………….………………. 170 Appendix II Evaluation of the Carbon Concentrations Measured by AES and XPS …………...……...…………….. 172 1. Sample Used in the Evaluation …………………………………………...….. 172 2. Experimental Results of AES …………………………………………...…… 173 3. Experimental Results of XPS ………………………………………………… 176 Appendix III The Derivation of the Thermodynamic Factor for FCC Systems ………………………..……………………. 188 References ………………………………………………………………………….. 191 vi List of Tables 1-1 Chemical Composition of AISI 316L Stainless Steel (wt%) ……………….…… 3 3-1 A Range of Values for the Error Function (Jost, 1960) ……………………..…. 33 3-2 The Information of the Computer Employed in This Study ………………...…. 58 5-1 Expressions of DC’ for Austenite and the Corresponding Values at 450oC ….... 78 5-2 The Relevant Parameters of the Grüneisen Parameter δ ……………………….. 82 5-3 The Relevant Parameters of the Activation Gibbs Free Energy ΔGA for AISI 316L Stainless Steel ………..……………………………………..……… 83 6-1 The Activity of Carbon during the Carburization Process with the Maximum and Minimum CO Contents (Volume Percent) ………………...…. 103 6-2 Simulation Parameters for Carbon Concentration-Depth Profile in AISI 316L Stainless Steel ……………..…..………………………………….. 104 6-3 The Energies of the Auger Peaks for the Principal Elements in Carburized AISI 316L Stainless Steel ………………………...……………… 106 6-4 The Average of the Normalized Composition of the Metal Elements Measured by AES Line Scan and the Normalized Nominal Composition of AISI 316L Stainless Steel (at%) ………………..…...……………………... 108 6-5 The Optimized Values for the Mass Transfer Coefficients α Fitting the Experimental Data in Fig. 6-8, Normalized Sum χ2 of Residuals, the Degrees of Freedom (d.f.), and the Probability of the Simulation Being Correct (P) ...………………...…………………………….. 111 6-6 The Binding Energy Values and the Corresponding Baseline Settings for the XPS Energy Peaks Analyzed in Depth Profiling Experiment of AISI 316L Stainless Steel………..…………………………………...……….. 113 vii 6-7 The Average of the Normalized Composition of the Metal Elements Measured by XPS Depth Profiling and the Normalized Nominal Composition of AISI 316L Stainless Steel (at%) …………………………….. 117 7-1 The Optimized Value for the Parameter k in Eq. (3-8) Fitting the Experimental Data in Fig. 7-4, Normalized Sum χ2 of Residuals, the Degree of Freedom, and the Probability of the Simulation Being Correct …………………………………………………………………. 133 7-2 The Optimized Value for the Parameter k in Eq. (3-8) Fitting the Experimental Data in Fig. 7-5, Normalized Sum χ2 of Residuals, the Degree of Freedom, and the Probability of the Simulation Being Correct …………………………………………………………………. 134 8-1 The Interrupted Treatment of the Typical Swagelok Procedure ……………… 153 8-2 The Optimized Value for the Mass Transfer Coefficient α Fitting the Experimental Data in Fig. 8-1, Normalized Sum χ2 of Residuals, the Degree of Freedom, and the Probability of the Simulation Being Correct ………………………………………….…………. 155 A-1 Nominal Composition of Amorphous Alloy SAM 1651-4 (at%) and the Normalized Composition of All the Elements except Carbon (at%) …………………………………………………………………………... 172 A-2 The Auger Peaks of the Elements in the Sample of SAM 1651-4 ………….… 174 A-3 The Statistical Average and the Standard Deviation of the Measured Compositions of the Sample SAM 1651-4 by AES Line Scan ………………. 175 A-4 The Analyzed Energy Peaks of the Obtained X-Ray Photoelectron Spectra and the Corresponding Binding Energy Values of the Peaks for the Sample of SAM 1651-4 ……………………………………………….. 177 viii List of Figures 2-1 (a) Composition space encompassed by the system (A, B)1(C, D)1 and (b) The reference energy surface described by Eq. (2-12) after Hillert and Staffanson (1980). ……………………………….…………………. 12 3-1 Schematic view of one-dimensional diffusion in a semi-infinite system. …..…. 41 3-2 The finite difference mesh squares associated with the simple explicit scheme applied to the diffusion equation. ………………………...……………. 43 3-3 The finite difference mesh squares associated with the fully implicit scheme applied to the diffusion equation. ………………………..…………….. 44 3-4 The finite difference mesh squares associated with Crank-Nicolson scheme applied to the diffusion equation. …………………………...…………. 45 3-5 The influence of the parameter r on the stability of finite difference solution with the simple explicit scheme. ……………..…………….…………. 50 5-1 Concentration dependence of carbon diffusivity predicted by all the introduced models for a fcc Fe-C binary system at 1000oC. The numbers beside the curves represent the models listed previously: (1) ARRT (Eq. (5-2)), (2) ARRT (Eq. (5-3)), (3) Kaufman et al., (4) Asimow, (5) Siller and McLellan, (6) Tibbetts, (7) Larché and Cahn, (8) Ågren, (9) Liu et al. …………………………………………………...…….….……… 75 5-2 Concentration dependence of carbon diffusivity predicted by all the introduced models for a fcc Fe-C binary system at 450oC. The numbers above the curves represent the models listed previously: (1) ARRT (Eq. (5-2)), (2) ARRT (Eq. (5-3)), (3) Kaufman et al., (4) Asimow, (5) Siller and McLellan, (6) Tibbetts, (7) Larché and Cahn, (8) Ågren, (9) Liu et al. ………………………….………………………………....……… 76 5-3 Concentration dependence of carbon diffusivity in AISI 316L stainless steel at 450oC based upon: (1) a Boltzmann-Matano analysis by Ernst et al., and the introduced models (2) Ågren, (3) Asimow, and (4) ARRT (Eq. (5-3)). ….……………………………………………….…. 85 ix 6-1 Concentration-depth profiles based upon the fully implicit scheme with a fixed boundary condition and a concentration-independent diffusion coefficient. .......................................................................................................… 90 6-2 Concentration-depth profiles based upon the fully implicit scheme with a convective boundary condition and a concentration-independent diffusion coefficient. …………………………………………………….……………….. 92 6-3 Simulated and normalized concentration-depth profiles based upon the fully implicit scheme with a fixed boundary condition and concentrationdependent diffusion coefficients. ………………………………...…………….. 95 6-4 Simulated concentration-depth profiles based upon the fully implicit scheme with convective boundary conditions and a concentrationdependent diffusion coefficient. …………………...…………………………… 97 6-5 Schematic drawing of the typical low temperature carburization procedure developed by the Swagelok Company. ………………………………………… 99 6-6 The gas flow during the second carburization step of 39 hours. ……..….…… 100 6-7 The normalized concentration-depth profiles of the metal elements in AISI 316L stainless steel obtained from Auger line scan. ……………….…… 107 6-8 The calibrated concentration-depth profiles of carbon in AISI 316L stainless steel obtained from Auger line scan. …….……….…………………. 109 6-9 The simulated concentration-depth profile of carbon in AISI 316L stainless steel with the best estimates for the mass transfer coefficients α fitting the experimental data in Fig. 6-8. …………………………………… 110 6-10 The obtained X-ray photoelectron spectra for the principal elements in carburized AISI 316L stainless steel: (a) Fe2p1, (b) Cr2p, (c) Ni2p3, (d) Mo3d, (e) C1s …………………………………………………….....…….. 113 6-11 The normalized concentration-depth profiles of the metal elements in AISI 316L stainless steel obtained from XPS depth profiling. ………….……. 116 x 6-12 The concentration-depth profile of carbon in AISI 316L stainless steel obtained from XPS depth profiling and AES line scan. ……………………… 118 6-13 Simulation results: (a) carbon concentration-depth profile obtained from the typical Swagelok procedure before the purging step is performed, (b) carbon concentration-depth profile simulated including the one-hour purging step, based upon the assumption that α = 0 µm·s-1 during that one hour, (c) carbon concentration-depth profile simulated with the inclusion of the one-hour purging step, based upon the assumption that α = 10-4 µm·s-1 during that one hour ………………………………………………………………………….... 119 7-1 Typical concentration-depth profiles of S-phase layers on ASTM F138 (AISI 316L): nitrogen profile for 430oC nitrided S-phase (N430[N]), carbon profile for 430oC carburized S-phase (C430[C]) and nitrogen and carbon profiles for 430oC carbonitrided S-phase NC430[N] and NC430[C]. …………………….. 122 7-2 The chemical potential of carbon relative to graphite in AISI 316L stainless steel at 430oC. The curves correspond to the chemical potential of carbon for a series of carbon contents from 2 to 20 at% (spacing as 2 at% each). ….....… 126 7-3 The chemical potential of nitrogen relative to one half of the pure Gibbs free energy of nitrogen gas in AISI 316L stainless steel at 430oC. The curves correspond to the chemical potential of nitrogen for a series of nitrogen contents from 2 to 24 at% (spacing as 2 at% each). ………………...………... 128 7-4 The simulated profile (solid line) fitting the experimental data (round dots) for the profile C430[C] in Fig. 7-1. ………………………………....………… 132 7-5 The simulated profile (solid line) fitting the experimental data (round dots) for the profile N430[N] in Fig. 7-1. …………………………………………... 134 7-6 The changes of the carbon and nitrogen concentration-depth profiles, C(act.), C(eff.), N(act.), and N(eff.) within the first minute including six computational steps (a) – (f). (g) shows the actual carbon and nitrogen concentration-depth profiles, C(act.) and N(act.), at the end of the first minute. …......................................................................................................…. 137 7-7 The simulated concentration-depth profiles C(act.), C(eff.), and N(act.) for (a) 5 minutes, (b) 10 minutes, (c) 20 minutes, (d) 30 minutes, (e) 1 hour, (f) 3 hours, (g) 5 hours and (h) 10 hours of the plasma nitridation treatment. …………………………………………………………………...… 140 xi 7-8 (a) The simulated final concentration-depth profiles C (act.), C (eff.) and N (act.) for plasma carbonitriding at 430oC. (b) Comparison with the experimental data (round dots) NC430[C] and NC430[N] shown in Fig. 7-1. …………………………………………………………………….…. 143 7-9 The final chemical potential profiles of carbon relative to graphite for 430oC plasma carbonitriding based upon: (a) the experimental data in Fig. 7-1, and (b) the simulation result in Fig. 7-8 (b). …………………….……………. 145 7-10 The simulated profiles of actual carbon and nitrogen concentrations for a 15-hour plasma nitridation followed by a 15-hour plasma carburization at 430oC. ……………………………………………………………………….... 147 7-11 The simulated profiles of actual carbon and nitrogen concentrations for a 15-hour simultaneous plasma carbonitriding process at 430oC. …………..….. 148 7-12 The simulated actual concentration-depth profiles of the total amount of carbon and nitrogen produced by the three different processes at 430oC: (a) 15-hour plasma carburization followed by 15-hour plasma nitridation, (b) 15-hour plasma nitridation followed by 15-hour plasma carburization, and (c) 15-hour simultaneous plasma carbonitriding process. …………...…… 149 8-1 The simulated concentration-depth profile of carbon in AISI 316L stainless steel treated by the procedure in Table 8-1 fitting the experimental data obtained from AES.…………………………………...………………………. 155 8-2 The X-ray photoelectron spectra of the standard XPS survey on the AISI 316L stainless steel specimen carburized by the procedure in Table 8-1. ……. 157 8-3 The analysis of the chemical state of carbon in the X-ray photoelectron spectra in Fig. 8-2. ………………………………………………...………….. 158 8-4 The simulated concentration-depth profiles of carbon in carburized AISI 316L stainless steel for the typical Swagelok procedure in Section 6.2.1 based upon the models of (a) ARRT (Eq. (5-3)), (b) Asimow (Eq. (5-7)), and (c) Ågren (Eq. (5-14)) and the corresponding experimental profile shown in Fig. 6-9. ………………………………………………………..…… 161 xii A-1 The concentration-distance profiles of all the elements in the sample of SAM 1651-4 measured by an AES line scan under a continuous sputtering mode. The carbon concentration-distance profile was adjusted using the calibration as Eq. (4-1). ….…………………………………...……………….. 175 A-2 The X-ray photoelectron spectra obtained from the standard survey after 50 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4. ………………………..……..…………………………. 178 A-3 The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting after 50 nm of sputtering on the sample SAM 1651-4. ………..………………………………..………………...…………… 179 A-4 The X-ray photoelectron spectra obtained from the standard survey after 100 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4. ……………………..………………………………. 180 A-5 The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting after 100 nm of sputtering on the sample SAM 1651-4. …………………..…...……………………………………………….. 181 A-6 The X-ray photoelectron spectra obtained from the standard survey after 150 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4. …………………………..…………………………. 183 A-7 The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting after 150 nm of sputtering on the sample SAM 1651-4. …………………………………………..……………………………. 184 A-8 The X-ray photoelectron spectra obtained from the standard survey under a continuous sputtering mode and the corresponding chemical compositions of the sample SAM 1651-4. …………………………………………..………. 185 A-9 The X-ray photoelectron spectrum of carbon obtained from the survey under a continuous sputtering mode and the relevant parameters of the peak fitting on the sample SAM 1651-4. …..………………………...……….. 186 xiii List of Symbols a General variable, Chapter 3 Activity, Chapter 6 aR Activity based upon a Raoultian standard state, Appendix III aH Activity based upon a Henrian standard state, Appendix III a0 Lattice parameter of austenite matrix (m), Eq. (5-8) A Function of CALPHAD parameter p*, Eq. (2-24) b General variable, Chapter 3 B0 Average magnetic moment per atom, Eq. (2-23) c Concentration variable (mol·m-3) cs Concentration on the surface of materials (mol·m-3) co Original concentration in the materials (mol·m-3) cmax Maximum possible concentration (mol·m-3) c Concentration in the treatment gas atmosphere (mol·m-3) CP Heat capacity at constant pressure (J·kg-1·K-1), Table 5-2 CV Heat capacity at constant volume (J·kg-1·K-1), Eq. (5-21) D Diffusion coefficient (m2·s-1) D Diffusion coefficient at infinite dilution (m2·s-1) D0 Pre-exponential factor (m2·s-1) erf Error function, Chapter 3 erfc Complementary error function, Chapter 6 E Young’s Modulus (Pa), Chapters 5 and 8 xiv f Function, Chapter 5 F Function, Chapter 3 Gm Total Gibbs free energy of molar formula units (J·mol-1) Gm Total Gibbs free energy of molar atoms (J·mol-1) Gmref. Gibbs free energy reference state of molar formula units (J·mol-1) Gmideal Gibbs ideal energy of mixing of molar formula units (J·mol-1) Gmexcess Gibbs excess energy of mixing of molar formula units (J·mol-1) Gmmag. Magnetic contribution to the Gibbs free energy of molar formula units (J·mol-1) Gmpres. Pressure contribution to the Gibbs free energy of molar formula units (J·mol-1) o Gh Gibbs free energy of pure compounds in a hypothetical nonmagnetic state (J·mol-1) o GAC Pure Gibbs free energy of the end member AC (J·mol-1) o Pure Gibbs free energy of the component array I (J·mol-1) GI GI 0 Pure Gibbs free energy of the component array I of zeroth order (J·mol-1) GA Activation Gibbs free energy associated with diffusion process (J·mol-1), o Eq. (5-22) H A Activation enthalpy associated with diffusion process (J·mol-1), Eq. (5-22) i Grid point of distance I Component array I0 Component array of zeroth order J Diffusive flux (mol·m-2·s-1), Eq. (8-1) xv k Enhancement coefficient for the concentration dependence of diffusion coefficient, Eqs. (3-7) and (3-8) K Asimow’s parameter representing a concentration enhancement, Eq. (5-7) L CALPHAD interaction parameters of mixing (J·mol-1) m Lower bound of summation, Chapter 2 Grid point of time, Chapter 3 M Number of discretized distance, Chapter 3 Atomic mobility (mol·m2·J-1·s-1), Eq. (8-2) n qp Number of atoms of component p on the sublattice q q nVa Number of vacancies on the sublattice q nC Number of carbon atoms on the interstitial sublattice nFe Number of iron atoms on the substitutional sublattice nVa Number of interstitial vacancies Nq Total number of sites on sublattice q N int. Total number of interstitial sites N sub. Total number of substitutional sites o Rank of CALPHAD interaction parameters of mixing, Chapter 2 p Component, Chapter 2 p Fraction of the magnetic enthalpy absorbed above the critical temperature, Eq. (2-24) P Pressure (Pa), Eqs. (2-27) and (2-28) Probability of Simulation Being Correct, Chapters 6, 7, and 8 PI Y Product of site fractions for the corresponding components and sublattices from the Y matrix xvi PI 0 Y Product of site fractions of zeroth order for the corresponding components and sublattices from the Y matrix q Sublattice, Chapter 2 Q Activation energy (J·mol-1), Eq. (3-3) r Dimensionless parameter, Eq. (3-28) r0 Radius of an interstitial hole in austenite matrix (m), Eq. (5-8) R Gas constant (8.3145 J·mol-1·K-1) Smideal Ideal entropy of mixing of molar formula units (J·mol-1·K-1), Chapter 2 S A Activation entropy associated with diffusion process (J·mol-1·K-1), Eq. (5- 22) t Time (s) T Thermodynamic temperature (K) Tm Melting temperature (K), Eq. (5-22) TCmag. Critical magnetic ordering temperature (K), Eq. (2-23) u Subscript related to the ratio of sublattice sites, Chapter 2 v Subscript related to the ratio of sublattice sites, Chapter 2 Drift velocity (m·s-1), Eqs. (8-1) and (8-2) VA Activation volume associated with diffusion process (m3·mol-1), Eq. (5-8) Vm Molar volume of austenite matrix (m3·mol-1), Eq. (6-8) x Distance, Depth (m) Xp Atomic fraction of component p Xj Atomic fraction of interstitial component j Xk Atomic fraction of substitutional component k xvii X max Possible maximum atomic fraction y General variable, Eq. (3-5) Y Matrix of site fractions, Chapter 2 Y pq Site fraction of component p on the sublattice q YVaq Site fraction of vacancies on the sublattice q YC Site fraction of carbon on the interstitial sublattice YFe Site fraction of iron on the substitutional sublattice YVa Site fraction of interstitial vacancies Yj Site fraction and of component j on the interstitial sublattice Yk Site fraction and component k on the substitutional sublattice z General variable, Eq. (3-5) z0 Coordination number of the octahedral interstitial site in austenite matrix, Eq. (5-11) Greek Mass transfer coefficient (m·s-1) T Thermal expansion coefficient (K-1), Eq. (5-21) , Simplified parameters, Eq. (3-35) Activity coefficient R Activity coefficient based upon a Raoultian standard state, Appendix III xviii H o Activity coefficient based upon a Henrian standard state, Appendix III Constant value of the activity coefficient of an interstitial, as the site fraction of the interstitial approaches zero, for a Raoultian standard state, Appendix III M Activity coefficient of activated complex, Eqs. (5-2), (5-3), and (5-16) Grüneisen parameter, Eq. (5-9) Elastic strain associated with the difference in the radii between the carbon atom and the interstitial hole in the austenite matrix, Eq. (5-8) Linear expansion per atomic fraction change of carbon in austenite matrix, Eq. (5-13) i , i , i Simplified parameters, Eq. (3-39) Dimensionless thermal coefficient of Young’s modulus, Eq. (5-22) p Chemical potential of component p (J·mol-1) I 0 Chemical potential of component array I0 (J·mol-1) Poisson’s ratio of austenite matrix, Eqs. (5-13) and (5-20) Liu’s parameter representing the blocking effect, Eq. (5-16) Truncation error associated with the finite difference approximation, Chapter 3 Density (kg·m-3), Eq. (5-21) 0 Density of lattice sites in austenite matrix (mol·m-3), Eq. (5-13) The parameter related to the repulsive interaction between neighboring carbon atoms in the model of Siller and McLellan, Eq. (5-11) Ratio between the actual temperature and the critical magnetic ordering temperature, Eq. (2-23) Compressibility (Pa-1), Eqs. (5-9), (5-20), and (5-21) χ2 Normalized sum of chi square residuals, Chapters 6, 7, and 8 xix Acknowledgements First of all, I would like to express my deepest gratitude to my academic advisor, Prof. Gary M. Michal, for his unlimited support, warm encouragement and patient guidance throughout my research. He set an excellent example as a person who actively contributes to the development of the field he pursuits with enthusiasm, courage and determination. I am very thankful to the members of my thesis committee, Prof. Arthur H. Heuer, Prof. Frank Ernst, and Prof. Walter R. Lambrecht. I appreciate all of the inspirations and constructive criticisms from Prof. Arthur H. Heuer and Prof. Frank Ernst throughout my research. I also acknowledge Prof. Walter R. Lambrecht for his time and great interest in my research. I want to thank Prof. J. Iwan D. Alexander for the discussion and help on the finite difference simulation in this study. I would like to express my sincere gratitude to Dr. Wayne D. Jennings for the technical training and data analyses on XPS and AES. I have particularly enjoyed our numerous exciting discussions, which greatly facilitated my understanding on surface chemical analysis. I also want to thank Dr. Reza ShaghiMoshtaghin for training me on metallurgical specimen preparation, and Prof. Harold Kahn for the discussions on my experimental data of XPS and AES and also on the heat treatment processes. Many thanks are due to Prof. John J. Lewandowski and Dr. Xiaojun Gu for providing me with the specimen of amorphous alloy SAM 1651-4, and former Master student Joshua Katz for performing the calibration on AES. I am thankful to the technical representatives from the Swagelok Company, Dr. Sunniva Collins, Mr. Peter C. Williams, and Dr. Steven V. Marx for their valuable xx suggestions on my research. I gratefully thank Dr. Steven V. Marx for providing me with the carburized specimens of AISI 316L stainless steel and the raw data of heat treatments and for helping me understand the actual manufacturing procedures. I also want to thank Mr. Jim Carner at the Swagelok Company for performing all the chemical analysis on the specimen of SAM 1651-4. I deeply appreciate the administrative assistance from Patsy Harris, Meredith Stanton, Cathy Nichols, Jenny Aquino-Pyles, Penny Cooke, Cindy Lepouttre, Gail O’Brien, and Lynne Snyder. Many thanks are also due to the former and current members of low temperature carburization research group. I dedicate my special thanks to my parents for their love and trust all along this journey, without which this dissertation will never be finished. I am indebted with my lovely wife, Fen Qi, who has been incredibly supportive even at the expense of her promising career in China. Thank her for being always with me during this long journey and making this journey much more wonderful than I have ever thought it could be. This dissertation is dedicated to her. Finally, I would like to thank the financial support for this study from Case Prime Fellowship, the Swagelok Company, the U.S. Department of Energy, the National Science Foundation, DARPA, the Naval Research Laboratory and the Office of Naval Research, and the Ohio Department of Development. xxi Numerical Simulations of Concentration-Depth Profiles of Carbon and Nitrogen in Austenitic Stainless Steel Based Upon Highly Concentration Dependent Diffusivities Abstract by XIAOTING GU The typical Swagelok procedure of the low temperature carburization conducted for approximately 40 hours at 450oC can produce an interstitially hardened case in AISI 316L stainless steel with a surface carbon concentration of 10 to 15 at% and a penetration depth of 20 to 25µm. The concentration-depth profile of carbon obtained after such a treatment exhibits a very concave shape that greatly deviates from the profile defined by an error-function solution to Fick’s second law. This result provides a unique opportunity for studying the concentration dependence of the carbon diffusivity in austenite. An extensive literature review focusing on the concentration dependence of the carbon diffusivity in austenite was performed. The model proposed by Asimow provides a very satisfactory description for the concentration dependence of the carbon diffusivity in AISI 316L stainless steel during low temperature carburization. This result indicates that the concentration-enhanced diffusion of carbon in austenite is most likely because the lattice expansion induced by carbon in solution in an austenite matrix greatly decreases the activation energy associated with the diffusion of carbon. xxii A numerical model for a one-dimensional diffusion in a semi-infinite system was designed using CALPHAD-based thermodynamic modeling and a fully-implicit finite difference algorithm. The numerical simulation based upon Asimow’s analytical model generated the concentration-depth profile of carbon in AISI 316L stainless steel carburized by the typical Swagelok procedure and provided an excellent agreement with the experimental results obtained from surface chemical analysis of X-ray Photoelectron Spectroscopy (XPS) and Auger Electron Spectroscopy (AES). The simulation parameters, including the activity of carbon in treatment atmosphere, the maximum possible solubility of carbon, and the mass transfer coefficient at the gas-metal interface, were thoroughly discussed. A similar numerical simulation was performed to reproduce the unusual carbon concentration-depth profile observed during a plasma carbonitriding process. The accumulation of carbon in front of the nitrogen diffusion zone was explained by the classical diffusion theory, recognizing the concentration dependence of both carbon and nitrogen diffusivities in stainless steels. The large nitrogen concentration introduced by plasma nitridation provides a significant driving force for carbon diffusion. Nitrogen greatly increases the chemical potential of carbon that corresponds to a given carbon concentration. The chemical potential gradient of carbon generated during the plasma nitridation process provides the driving force for the diffusion of carbon. xxiii Chapter 1 Introduction 1.1 Surface Engineering Surface engineering is now recognized as one of the fast developing research fields of major importance in the successful and efficient advancement of materials in engineering practices. Surface engineering focuses on three separate and yet interrelated activities: the optimization of surface properties, the characterization of coatings and modified surfaces, and the technologies of coating and surface modification [1]. The optimization of surface properties is particularly concerned with the corrosion and wear performance of surfaces and coatings. The characterization of coatings and modified surfaces is mainly concerned with aspects of their condition, composition, structure and morphology, and mechanical, electrical and optical properties. The technologies of coating and surface modification include some traditional topics such as painting, electroplating, weld surfacing, spraying, thermal and thermochemical treatments, such as carburization and nitridation, and some other emerging technologies, such as laser surfacing, physical and chemical vapor deposition and ion implantation. The technologies of carburization and nitridation produce diffusion based coatings [2], which are quite distinct and different from overlay coatings. Carburization and nitridation have become increasingly important engineering specifications intended to produce hard, corrosion-resistant, wear-resistant and fatigue-resistant surfaces on core materials that are primarily selected for their bulk properties. Diffusion based coatings 1 are produced within the original boundaries of workpieces and are normally characterized by a concentration gradient of the chemical species, such as carbon and nitrogen, generated during the diffusion process with a maximum concentration at the surface of workpieces, which decreases to a minimum value within the core of the original materials. Carburization and nitridation treatments can be performed using a variety of engineering approaches, such as gas-metal reactions, vacuum, plasma and salt bath. 1.2 Austenitic Stainless Steels Stainless steels are iron-base alloys containing at least 10.5 wt% chromium. These steels achieve their stainless characteristics through their ability to form a very thin and adherent chromium-rich oxide film, which greatly reduces the rate of corrosion and therefore effectively provides a resistance to attacks from the atmosphere and many industrial gases and chemicals. The austenitic grades of stainless steels owe their name to their face center cubic crystallographic structure. These steels contain 16 to 30 wt% chromium and 2 to 20 wt% nickel to enhance surface quality, formability and increase corrosion resistance. They are widely used for automotive trim, cookware, food and beverage equipment, processing equipment and a variety of other industrial applications. Many of these steels have high strengths at elevated temperatures, which is the reason for their wide use in such environments. They are also among the primary materials selected for extremely low temperature applications, because they do not exhibit the ductile to brittle transition found in ferritic stainless steels. 2 The austenitic stainless steel type explored in this research is AISI 316L. The chemical composition range specified for AISI 316L stainless steel is listed in Table 1-1. Table 1-1 Chemical Composition of AISI 316L Stainless Steel (wt%) C Mn Si Cr Ni Mo P S Fe 0.03 max 2.00 max 1.00 max 16.0 18.5 10.0 14.0 2.0 3.0 0.045 max 0.03 max Balance This type of alloy has excellent corrosion resistance to a wide range of media, due to its molybdenum content. Molybdenum combined with chromium is very effective in terms of stabilizing the passive film in the presence of chlorides. Molybdenum is especially effective in increasing resistance to the initiation of pitting and crevice corrosion. Therefore, this type of alloy is normally chosen for applications in aggressive environments. Typical applications for AISI 316L include fittings of boats and piers, architectural components, polluted or industrial environments, hot water systems, mineral processing, petrochemical and other industries. Although austenitic stainless steels are widely used because of their excellent ductility, formability and corrosion resistance, these alloys commonly lack hardness and mechanical strength, which greatly restricts their application and makes them often not the first choice for structural components that normally demand high strength or high surface hardness. 3 1.3 Low Temperature Carburization It is well established in surface engineering technology that conventional carburization is a very effective method to improve surface hardness, wear resistance and fatigue resistance. However, it is an inappropriate method to treat stainless steels. In the process of conventional carburization, the alloys are treated at high temperatures where carbide phases can readily form. The chromium in the matrix is reduced due to chromium partitioning to the carbide phases and the corrosion resistance of the stainless steel is thus reduced. Conventional carburization only achieves low solubilities of carbon with the formation of carbides, because the treatment proceeds under equilibrium conditions. Therefore, conventional carburization is not a satisfactory process to treat stainless steels due to these critical disadvantages. However, the low temperature carburization technology developed by the Swagelok Company provides an ideal solution to the problems encountered with conventional carburization of stainless steels. This technology can produce a “colossal” supersaturation of carbon in the near surface region of stainless steels to greatly improve their surface hardness, wear and fatigue resistance without losing corrosion resistance [3] [4]. The treatment proceeds under a paraequilibrium condition in which the temperature is relatively low (about 450oC). Under this condition, essentially no diffusion of substitutional atoms in the matrix occurs due to the extremely low diffusion coefficients of metal elements in the matrix [5]. Carbon can diffuse over considerable distances during the treatment because of its relatively high diffusion coefficient. Another advantage of low temperature carburization is that very high carbon solubilities can be 4 obtained in stainless steels because carbide formation is kinetically suppressed. A layer with a very high concentration of interstitially dissolved carbon extending from the original surface into the core of the materials is produced. A variety of mechanical properties of the treated austenitic stainless steels are remarkably enhanced while their corrosion resistance is maintained or even enhanced. Low temperature carburization technology has been applied to a wide range of stainless steels and some Ni- and Co- based alloys. 1.4 The Goal The success of low temperature carburization technology provides a unique opportunity for studying the concentration dependence of carbon diffusivity in austenite. That topic has been explored by several researchers over the past decades [6-16]. A numerical model is needed to simulate the concentration-depth profiles of carbon during low temperature carburization treatment since the enhanced properties generated are closely correlated to the penetration depth of carbon in the austenite matrix and the precise shape and curvature of concentration-depth profiles for carbon. An extensive literature review has been performed to find a suitable analytical model which can provide a satisfactory quantitative description of the concentration dependence of carbon diffusivity in austenite and a possible explanation of the corresponding physical mechanism responsible for this interesting phenomenon. 5 Numerical simulation for the diffusion of carbon into austenite during a low temperature carburization process has been performed based upon the extraordinarily strong concentration dependent diffusivity of carbon in austenite. The calculated concentration-depth profiles of carbon based upon a finite difference algorithm have been compared with experimental data obtained from near surface chemical analyses conducted using X-ray Photoelectron Spectroscopy (XPS) and Auger Electron Spectroscopy (AES). A similar simulation approach has been applied to the modeling of plasma carbonitriding processes. Such processes currently are a subject of research on a global scale [17-22]. It also is expected that the results and developments of this methodology can be applied to other diffusion processes for different interstitial species in other types of materials. 6 Chapter 2 Thermodynamic Modeling 2.1 CALPHAD Model The thermodynamic analyses performed in this study are based upon the wellknown thermochemical model commonly referred to as CALPHAD [23], which is the acronym derived from CALculation of PHAse Diagrams, but it is also well defined by the sub-title of the CALPHAD journal, The Computer Coupling of Phase Diagrams and Thermochemistry. The CALPHAD method is founded upon the viewpoint that almost all real materials are multi-component in nature. Phase diagrams are generally used to represent only binary and ternary alloys, but it is possible to predict the phase behavior of highly complex, multi-component materials based upon extrapolations to higher-order properties from lower-order binary and ternary systems. Furthermore, the CALPHAD method can also predict the behavior of materials under conditions well away from equilibrium, which considerably enhances its value. The analysis of carburization or nitridation is greatly aided by a multiplesublattice model. The studied alloy, AISI 316L stainless steel, contains two sublattices: one substitutional sublattice and one interstitial sublattice. Metal species occupy the substitutional sublattice; carbon or nitrogen atoms partially occupy the interstitial sublattice. In 1980, Sundman and Ågren proposed a formulism which could be used to describe a regular solution for phases with several components and sublattices [24]. Their 7 paper provides a basis for the CALPHAD or compound energy model that can be applied to interstitial solid solutions. 2.2 Gibbs Free Energy of Compounds 2.2.1 Definition of Site Fraction In the multiple-sublattice model, the concept of a site fraction, Y pq , is introduced [24]. The site fraction is defined as the fractional site occupation of each of the components, p, on the various sublattices identified by the superscript, q, Y pq n qp Nq (2-1) n qp is the number of atoms of component p on the sublattice q and N q is the total number of sites on the sublattice q. The substitutional sublattice in an interstitial solid solution is occupied by a variety of metal atoms. The site fractions of these metal atoms are expressed directly by Eq. (2-1). For example, the site fraction of iron in the alloy could be expressed as: YFe nFe N (2-2) 8 The superscript q is neglected in Eq. (2-2) because all the parameters in this equation are m assumed to refer to the same sublattice, the substitutional sublattice. Obviously, Y p 1 p 1 on the substitutional sublattice. The concept of site fraction is commonly generalized to include vacancies [24], which are important to consider in interstitial phases, so that Eq. (2-1) becomes: Y pq n qp q nVa n qp (2-3) p q is the number of vacancies on sublattice q. The summation is over all nonand nVa vacancy species on the q sublattice. Consider an interstitial solid solution with the interstitial sublattice containing only carbon and vacancies. For example, the site fraction of carbon atoms would be expressed as: YC nC nVa nC (2-4) Meanwhile, the site fraction of vacancies is: YVa nVa nVa nC (2-5) and YC YVa 1 on the interstitial sublattice. Mole fractions are related to site fractions by the following relationship [24]: 9 N Y N 1 Y q Xp q p q q q Va (2-6) q where this summation is over the q sublattices. When this equation is applied to interstitial solid solutions, the ratio of the numbers of interstitial sites, N int. , and substitutional sites, N sub. , needs to be considered for different matrices. The studied alloy, AISI 316L stainless steel, has a face center cubic (fcc) structure and the relevant value of that ratio is: N int. N sub. fcc 1 (2-7) For the interstitial sublattice, the relation between mole fraction and site fraction based upon the value of N int. N sub. 1 is expressed as follows: Yj Xj 1 X j Xj Yj 1 Yj (2-8) (2-9) Y j and X j respectively represent site fraction and mole fraction of interstitial elements, such as carbon. The components j do not include vacancies because X j refers to only the elements with vacancies conventionally excluded. For the substitutional sublattice, the relation between mole fraction and site fraction based upon the value of N int. N sub. 1 is expressed as follows: 10 Yk Xk 1 X j (2-10) Xk Yk 1 Yj (2-11) Yk and X k represent the site fraction and the mole fraction, respectively of substitutional elements, such as iron, chromium or nickel. Eqs. (2-8) to (2-11) apply to the situations when the interstitial sublattice is comprised of only carbon and vacancies or nitrogen and vacancies in a phase with a face center cubic structure. Either N int. or N sub. arbitrarily can be set equal to one. Both protocols are found in the CALPHAD literature [25]. The work of this dissertation will use the convention of N sub. equal to one. 2.2.2 Gibbs Free Energy Reference State The Gibbs free energy reference state [24] is considered to be effectively defined by ‘end members’, which are generated when the sublattices are occupied by pure components. These end members could be simply elements, but more generally are compounds. A model sublattice phase is introduced with the formula (A, B)u(C, D)v, where A and B are on sublattice 1, C and D are on sublattice 2. There are four end members for this two-sublattice solid solution phase. For the most simple example of u = v = 1, the four end members are AC, AD, BC and BD as shown in Fig. 2-1. Possibly, these four end members could completely occupy the four points, which means that pure A exists on sublattice 1 and either pure C or D on sublattice 2, or conversely pure B exists on sublattice 1 with either pure C or D on sublattice 2. These four end members 11 define the composition space of this sublattice phase, which is shown as the square in Fig. 2-1(a) [23]. Then the Gibbs free energy reference state could be represented by the surface in Fig. 2-1(b). This surface is actually [24] given by Eq. (2-12) and represents the Gibbs free energy of a mechanical mixture of the end member compounds corresponding to a particular composition: Gmref. YAYC oGAC YBYC oGBC YAYD oGAD YBYD oGBD (2-12) YD2 1 YB1 (a) (b) Fig. 2-1: (a) Composition space encompassed by the system (A, B)1(C, D)1 and (b) The reference energy surface described by Eq. (2-12) after Hillert and Staffanson (1980). For the sublattice phase with q sublattices and p components, the site fractions, Y’s [24], can be expressed by the following (q + p) matrix, Y: 12 YA1 YB1 YC1 Y p1 YA2 YB2 YC2 Y p2 YA3 YB3 YC3 Y p3 Y pq YAq YBq YCq Rows represent sublattices and columns represent components. For the completely general sublattice phase (A, B, C, D)1(A, B, C, D)1, the four components can reside on either of the two sublattices ( p 4 and q 2 ). So the number of possible compounds in this sublattice phase is 16 ( p q 4 2 16 ), and the matrix of these compounds is defined as a component array I. The Gibbs free energy of the pure compound, oG I , defined by I would be comprised of the following sixteen parameters: o GA1A2 o GA1B2 o o GB1A2 o o GC1A2 o o GD1A2 o GB1B2 o GC1B2 o GD1B2 GA1C2 o GB1C2 o GC1C2 o o GD1C2 GA1D2 GB1D2 GC1D2 o GD1D2 The superscripts associated with the compounds contained in I define the sublattices in which each of the components resides. An additional notation, PI Y , represents the product of site fractions for the corresponding components and sublattices from the Y matrix. Then the general form of the Gibbs free energy reference state is expressed as [24]: Gmref. PI Y oGI (2-13) I 13 For the general sublattice phase (A, B, C, D)1(A, B, C, D)1, this Gibbs free energy is the sum of the 16 terms in Eq. (2-13): Gmref. YA1YA2 oGA1A2 YA1YB2 oGA1B2 YA1YC2 oGA1C2 YA1YD2 oGA1D2 YB1YA2 oGB1A2 YB1YB2 oGB1B2 YB1YC2 oGB1C2 YB1YD2 oGB1D2 (2-14) YC1YA2 oGC1A2 YC1YB2 oGC1B2 YC1YC2 oGC1C2 YC1YD2 oGC1D2 YD1YA2 oGD1A2 YD1YB2 oGD1B2 YD1YC2 oGD1C2 YD1YD2 oGD1D2 For the case when only A and B are on sublattice 1 and C and D are on sublattice 2, YA2 YB2 YC1 YD1 0 . Only the four terms in the dotted rectangle in Eq. (2-14) contribute to Gmref. . The result is the same as Eq. (2-12) if the superscripts shown in Eq. (2-14) are ignored. The expression for the Gibbs free energy reference state can be shown to be equivalent to a mechanical mixture of end members by taking the fcc model steel (Fe, Cr)1(C, X Fe 0.6 Va)1 as an example. Assume the composition of this steel is X Cr 0.3 X C 0.1 . According to Eq. (2-8) and (2-10), the site fractions of all the components are YFe 2 3 YCr 1 3 YC 1 9 YVa 8 9 . The Gibbs free energy reference state of this model sublattice phase could be expressed as a mechanical mixture of the end members – FeC, CrC, pure (FeVa) and pure (CrVa) as follows: Gmref. YFeYVa oGFe:Va YFeYC oGFe:C YCrYVa oGCr:Va YCrYC oGCr:C 16 o 2 o 8 o 1 o GFe:Va GFe:C GCr:Va GCr:C 27 27 27 27 (2-15) The fractional amounts of each end member sum to one indicating a total of one mole of formula units. The numbers of moles of the individual elements are then given by: 14 16 2 18 2 27 27 27 3 8 1 9 1 nCr 27 27 27 3 2 1 3 1 nC 27 27 27 9 nFe which recovers the mole fractions of each of these element, XFe = 0.6, XCr = 0.3, and XC = 0.1. In Eq. (2-15), the elements are separated by a comma – “,”, and sublattices are separated by a colon – “:”. This rule is commonly used in recent papers in the literature, which is the reverse of the way Sundman and Ågren referred to components and sublattices in their original paper [24]. They used colons to separate the elements and commas to separate the sublattices. 2.2.3 Ideal Entropy of Mixing Sundman and Ågren pointed out that the ideal entropy of mixing is made up of the configurational contributions by components mixing on each of the sublattices [24]. The Gibbs ideal mixing energy of molar formula units is expressed as: Gmideal TSmideal RT N q Y pq ln Y pq q (2-16) p where Y pq includes the contribution from vacancies. In interstitial solid solutions, the ideal entropy of mixing comes from the two sublattices. On the substitutional sublattice, the entropy depends on the mixing among the 15 metal atoms. Therefore, the ideal entropy of mixing from the substitutional sublattice is expressed as: Gmideal sub. TS mideal RT Yk ln Yk (2-17) k where Yk is the site fraction of metal atoms on the substitutional sublattice. On the interstitial sublattice, the entropy comes from carbon or nitrogen atoms and vacancies, which is expressed as Eq. (2-18) for a fcc phase: Gmideal int. TS mideal RT YC ln YC YN ln YN YVa ln YVa (2-18) Therefore, according to Eq. (2-16), the ideal entropy of mixing for a fcc interstitial solid solution is expressed as: Gmideal Gmideal sub. Gmideal int. TSmideal (2-19) RT Yk ln Yk RT YC ln YC YN ln YN YVa ln YVa k 2.2.4 Gibbs Excess Energy of Mixing An interaction energy develops when components mix on a sublattice. The twosublattice phase (A, B)1(C, D)1 can also be used to show this Gibbs excess energy of mixing. In this alloy A-C, A-D, B-C, and B-D interactions are reflected in the Gibbs free energy of the compounds AC, BC, AD and BD. Then the Gibbs excess energy of mixing could be represented by making the interactions compositionally dependent on the site occupation in the other sublattice as [24]: 16 Gmexcess YA1YB1YC2 LA,B:C YA1YB1YD2 LA,B:D YA1YC2YD2 LA:C,D YB1YC2YD2 LB:C,D (2-20) L parameters are regular solution parameters for mixing on the sublattices which depend upon the site occupation of the other sublattice. For example, LA,B:D represents the interaction energy between A and B on the first sublattice when the second sublattice is completely filled with D. If D represents a vacancy then LA,B:Va is the interaction coefficient for the elementary regular solution model for an A-B binary system. LA:C,D represents the interaction energy between C and D on the second sublattice when the first sublattice is completely occupied by A. If D represents a vacancy then LA:C,D also represents the interaction of the C components with vacancies on the second sublattice when the first sublattice is completely filled with A. This comes about because YVa 1 YC . Additionally, some L parameters have several different rank terms, which are related to the site fractions of elements. The general forms are [24] as follows: LA,B:C YA1YB1YC2 o LA,B:C YA1 YB1 m o o LA,B:D YA1YB1YD2 o LA,B:D YA1 YB1 m o o LA:C,D Y Y Y 1 2 2 A C D m o LA:C,D Y Y 1 C (2-21) 1 o D o LB:C,D YB1YC2YD2 o LB:C,D YC1 YD1 m o o where the sum is from o = 0 to m. It is noteworthy that the order of the site fractions when the rank of L parameters is higher than zero. The order of the site fractions in the difference terms should be consistent with that in the L parameters expressions. For 17 example with o LA,B:C , its first order term should have the site fraction difference in the form as YA YB , not YB YA . Obeying this protocol is very important for the relevant calculations. Considering interstitial solid solutions, the fcc model sublattice phase (Fe, Cr, Ni)1(C, Va)1 has the Gibbs excess energy of mixing of the form: Gmexcess YFeYCYVa 0 LFe:C,Va YCrYCYVa 0 LCr:C,Va YNiYCYVa 0 LNi:C,Va YCrYFeYVa 0 LCr,Fe:Va 1LCr,Fe:Va YCr YFe YCrYFeYC 0 LCr,Fe:C YCrYNiYVa 0 LCr,Ni:Va 1LCr,Ni:Va YCr YNi YCrYNiYC 0 LCr,Ni:C YFeYNiYVa L 0 Fe,Ni:Va 1LFe,Ni:Va YFe YNi 2 LFe,Ni:Va YFe YNi 2 (2-22) YFeYNiYC 0 LFe,Ni:C 1LFe,Ni:C YFe YNi YCrYFeYNiYVa 0 LCr,Fe,Ni:Va YCrYFeYNiYC 0 LCr,Fe,Ni:C In this representation ternary interaction terms for the metal sublattices are included. 2.2.5 Magnetic Gibbs Free Energy In paramagnetic materials, there is no polarization of electron spins and therefore it is not necessary to consider the magnetic contribution to the Gibbs free energy. However, for ferromagnetic, anti-ferromagnetic and ferrimagnetic materials, there is a competition between different spin arrangements and there is one favorite spin polarization by which the enthalpy of the system can be reduced below a certain critical temperature. These materials are usually transition metals, rare earths and their associated alloys and compounds [23]. This magnetic contribution turns out to be of sufficient 18 magnitude to have a major effect on phase transformations. The general form of the magnetic Gibbs free energy Gmmag. [26] is described by: Gmmag. RT ln B0 1 f (2-23) where T / TCmag. , TCmag. is the critical magnetic ordering temperature, and B0 is the average magnetic moment per atom. For ferromagnetic and ferrimagnetic materials, TCmag. is the Curie temperature, and for anti-ferromagnetic materials, it is the Neel temperature [26]. These two parameters are functions of the composition of solid solutions. The function f depends upon whether the temperature is above or below TCmag. . 79 1 474 1 3 9 15 for 1: f 1 1 6 135 600 A 140 p 497 p 5 15 25 for 1: f A 10 315 1500 (2-24) 518 11692 1 * where A 1 . The value of p , which is thought of as the 1125 15975 p fraction of the magnetic enthalpy absorbed above the critical temperature, depends on the structure (p* = 0.28 for fcc and p* = 0.4 for bcc) [26]. For the current study, the magnetic contribution to Gibbs free energy in AISI 316L stainless steel is very modest because the austenite phase is normally paramagnetic under the conditions of low temperature carburization. Therefore, the magnetic part is ignored in the following process of modeling and computation. 19 2.2.6 Contribution of Pressure to Gibbs Free Energy The changes in pressure also contribute to the Gibbs free energy [26], which is represented by Gmpres. . The values of Gmpres. for pure elements as a function of temperature and pressure are well defined [26] and could be calculated directly. However, the influence of pressure on the Gibbs free energy of the compound phases, such as MC carbides, and solid solutions are not well defined quantitatively. With regard to solid solutions, it is not clear how the pressure can affect the CALPHAD interaction parameters, such as LM:C,Va . The carburization process can yield a compressive stress of 2 to 3 GPa in the near surface region of the alloy matrix [27]. Calculations were performed, which showed that a compressive stress of that magnitude in stainless steels has a negligible influence on the maximum solubilities of carbon or nitrogen based upon the assumptions that the influence of pressure on the compound phases is the same as that on the matrix and the relevant CALPHAD interaction parameters are not changed by compressive stress levels of 2 to 3 GPa [28]. Therefore, the term Gmpres. is not included in the thermodynamic analysis of this dissertation. 2.2.7 Total Gibbs Free Energy of Compounds From the above analysis, the total Gibbs free energy of compounds can be generalized. It is the sum of the following five parts: (1) Mechanical mixture of end members (2) Ideal entropy of mixing (3) Gibbs excess energy of mixing 20 (4) Magnetic Gibbs free energy (5) Contribution of pressure So the general expression of the total Gibbs free energy becomes [23]: Gm PI 0 Y oGI 0 RT N q Y pq ln Y pq Gmexcess Gmmag. Gmpres. I0 q (2-25) p The first term is the Gibbs free energy reference state, which is the Gibbs free energy of a mechanical mixture of the end members. The second and the third terms are respectively the ideal energy of mixing and the excess energy of mixing. The fourth and the fifth terms are respectively the magnetic contribution and the pressure contribution to the Gibbs free energy, which are both ignored in the calculation for AISI 316L stainless steel. The following model phase is selected to represent AISI 316L stainless steel: (Fe, Cr, Ni, Mo)1(C, N, Va)1 In this model phase, the metal elements, Fe, Cr, Ni, and Mo, occupy the substitutional sublattice and the interstitial sublattice is occupied by carbon, nitrogen and interstitial vacancies. Therefore, the molar Gibbs free energy of this model phase can be expressed as Eq. (2-26): 21 hfcc hfcc hfcc Gmfcc YFeYC oGFe:C YFeYN oGFe:N YFeYVa oGFe:Va hfcc hfcc hfcc YCrYC oGCr:C YCrYN oGCr:N YCrYVa oGCr:Va hfcc hfcc hfcc YNiYC oGNi:C YNiYN oGNi:N YNiYVa oGNi:Va hfcc hfcc hfcc YMoYC oGMo:C YMoYN oGMo:N YMoYVa oGMo:V a RT YFe ln YFe YCr ln YCr YNi ln YNi YMo ln YMo YC ln YC YN ln YN YVa ln YVa 0 fcc 0 fcc 0 fcc YCYVa YFe 0 Lfcc Fe:C,Va YCr LCr:C,Va YNi LNi:C,Va YMo LMo:C,Va 0 fcc 0 fcc YNYVa YFe 0 Lfcc Fe:N,Va YCr LCr:N,Va YMo LMo:N,Va 0 fcc YFeYCYN 0 Lfcc Fe:C,N YCrYFeYC LCr,Fe:C 1 fcc YCrYFeYN 0 Lfcc Cr,Fe:N LCr,Fe:N YCr YFe 1 fcc YCrYFeYVa 0 Lfcc Cr,Fe:Va LCr,Fe:Va YCr YFe 2-26 YCrYFeYNYVa 0 Lfcc Cr,Fe:N,Va 1 fcc YFeYNiYC 0 Lfcc Fe,Ni:C LFe,Ni:C YFe YNi 1 fcc YFeYNiYN 0 Lfcc Fe,Ni:N LFe,Ni:N YFe YNi YFeYNiYVa L 0 fcc Fe,Ni:Va 2 fcc 1Lfcc Fe,Ni:Va YFe YNi LFe,Ni:Va YFe YNi 2 0 fcc YCrYFeYNiYC 0 Lfcc Cr,Fe,Ni:C YCrYFeYNiYVa LCr,Fe,Ni:Va 0 fcc 0 fcc YFeYMoYC 0 Lfcc Fe,Mo:C YFeYMoYVa LFe,Mo:Va YCrYNiYC LCr,Ni:C 1 fcc YCrYNiYVa 0 Lfcc Cr,Ni:Va LCr,Ni:Va YCr YNi 0 fcc YCrYNiYNYVa 0 Lfcc Cr,Ni:N,Va YCrYMoYN LCr,Mo:N 1 fcc YCrYMoYVa 0 Lfcc Cr,Mo:Va LCr,Mo:Va YCr YMo 1 fcc YMoYNiYVa 0 Lfcc Mo,Ni:Va LMo,Ni:Va YMo YNi In Eq. (2-26), the terms oG h are the Gibbs free energy of pure compounds in a hfcc hypothetical nonmagnetic state. For example, oGFe:Va represents the Gibbs free energy of pure iron in a hypothetical fcc phase at a nonmagnetic state. In addition, the terms representing the Gibbs free energy contributions from magnetic ordering and pressure are ignored. 22 To perform the relevant computations of the Gibbs free energy, the corresponding CALPHAD parameters have been found in the literature. The CALPHAD parameters are defined by fitting the CALPHAD model to experimental data pertaining to equilibrium, which in transition metal systems is normally attainable above 700oC. For the present work on low temperature carburization, it is necessary to extrapolate these parameters to low temperatures, i.e. 450oC. The parameters used in this modeling are taken from a variety of papers focusing on the simulations of the CALPHAD method for the relevant alloy systems. These papers and related CALPHAD parameters are listed in detail in the reference and Appendix I, respectively. 2.3 Chemical Potential The partial molar Gibb free energy with respect to a component p is defined [24] as: G n p T , P , n j p (2-27) According to the compound energy model, the partial molar Gibbs free energy with respect to a component array of zeroth order, I0, for a phase with several sublattices is defined [24] to be: G n I 0 T , P ,n I 0 (2-28) j0 23 where nI0 is the number of moles of formula units of I0. It is shown that the well known expression for the calculation of the partial molar Gibbs free energy, which is also called the chemical potential, can be generalized as follows [24]: G I 0 Gm q Y m q pq Y jq j Gm Y jq (2-29) where pq is the component in sublattice q of the component array I0. By applying this formula to interstitial solid solutions, the partial molar Gibbs free energy with respect to certain elements would be generalized. 2.3.1 Derivation of the Chemical Potential of Carbon In order to illustrate the application of the chemical potential to interstitial solid solutions, the simple model phase (Fe, Cr)1(C, Va)1 will continue to be used. In the introduction section, the concept of the component array of zeroth order, I0, was introduced. The component array of the model phase (Fe, Cr)1(C, Va)1 is based upon only Fe and Cr occupying the metal sublattice and carbon and vacancies filling the interstitial sublattice: o o GFe:Va o GCr:Va o GFe:C GCr:C The corresponding chemical potential representation of this component array is: Fe:Va Cr:Va Fe:C Cr:C 24 where I 0 represents the chemical potential of each member of the zeroth order component array. By applying Eq. (2-29), expressions for the chemical potentials can be attained as follows: Fe:Va Gm Fe:C Gm Cr:Va Gm G Gm Gm Gm G YFe m YCr YVa YC m YFe YFe YCr YVa YVa YC Gm G Gm Gm Gm G YFe m YCr YVa YC m YFe YFe YCr YC YVa YC G G Gm Gm Gm G Gm m YFe m YCr YVa YC m YCr YFe YCr YVa YVa YC Cr:C Gm (2-30) Gm G Gm Gm Gm G YFe m YCr YVa YC m YCr YFe YCr YC YVa YC where M:Va represents the chemical potential of the compound MVa (actually the chemical potential of the pure element Fe or Cr), in which the metal M completely occupies the substitutional sublattice and the interstitial sublattice is empty. M:C represents the chemical potential of the compound MC in which the metal M also completely occupies the substitutional sublattice and carbon fills the entire interstitial sublattice. The chemical potential of compounds can be expressed as the sum of the chemical potentials of metal and interstitial species [29] for fcc as: Fe:C FeC Fe C Cr:C CrC Cr C Fe:Va FeVa Fe Va Cr:Va CrVa Cr Va (2-31) It may be assumed that the chemical potential of vacancies, Va , is zero at equilibrium. Therefore, the chemical potential of the compound MVa is represented by the chemical potential of the pure metals, such as: 25 Fe:Va Fe Cr:Va Cr (2-32) Base upon the above analysis, the chemical potential of carbon, C , in this model phase (Fe, Cr)1(C, Va)1 can be obtained from Eq. (2-31) and Eq. (2-32) as: or Fe:C Fe:Va Fe C Fe C (2-33) Cr:C Cr:Va Cr C Cr C (2-34) By substituting expression for µFe:C and µFe:Va from Eq. (2-30) into Eq. (2-33), the following expression can be used to define C : G G G G G G Fe:C Fe:Va Gm m YFe m YCr m m YVa m YC m YFe YFe YCr YC YVa YC G G Gm Gm Gm G Gm m YFe m YCr YVa YC m YFe YFe YCr YVa YVa YC Gm Gm YC YVa (2-35) Thus the chemical potential of carbon in the model phase (Fe, Cr)1(C, Va)1 is obtained explicitly from Eq. (2-33) and Eq. (2-35) as follows: C Gm Gm YC YVa (2-36) This is actually the specific form of the chemical potential of carbon for fcc interstitial solid solutions. Similar derivation can be applied to obtain the expression of the chemical potential of nitrogen for fcc interstitial solid solutions as: 26 N Gm Gm YN YVa (2-37) 2.3.2 Expressions of the Chemical Potentials of Carbon and Nitrogen For the model phase (Fe, Cr, Ni, Mo)1(C, N, Va)1 that represents AISI 316L stainless steel, the chemical potential of carbon can be obtained by applying Eq. (2-26) to Eq. (2-36) as: Cfcc Gmfcc Gmfcc YC YVa hfcc hfcc hfcc hfcc YFe oGFe:C oGFe:Va oGCr:Va YCr oGCr:C hfcc hfcc hfcc hfcc YNi oGNi:C oGNi:Va oGMo:Va YMo oGMo:C RT ln YC / 1 YC YN 0 fcc 0 fcc 0 fcc 1 2YC YN YFe 0 Lfcc Fe:C,Va YCr LCr:C,Va YNi LNi:C,Va YMo LMo:C,Va 0 fcc 0 fcc YN YFe 0 Lfcc Fe:N,Va YCr LCr:N,Va YMo LMo:N,Va YFeYN 0 Lfcc Fe:C,N 0 fcc 1 fcc YCrYFe 0 Lfcc Cr,Fe:C LCr,Fe:Va LCr,Fe:Va YCr YFe 2-38 YCrYFeYN 0 Lfcc Cr,Fe:N,Va YFeYNi L 0 fcc Fe,Ni:C 0 fcc 1 fcc 2 fcc 1Lfcc Fe,Ni:C YFe YNi LFe,Ni:Va LFe,Ni:Va YFe YNi LFe,Ni:Va YFe YNi 0 fcc YCrYFeYNi 0 Lfcc Cr,Fe,Ni:C LCr,Fe,Ni:Va 2 0 fcc YFeYMo 0 Lfcc Fe,Mo:C LFe,Mo:Va 0 fcc 1 fcc YCrYNi 0 Lfcc Cr,Ni:C LCr,Ni:Va LCr,Ni:Va YCr YNi YCrYNiYN 0 Lfcc Cr,Ni:N,Va 1 fcc YCrYMo 0 Lfcc Cr,Mo:Va LCr,Mo:Va YCr YMo 1 fcc YMoYNi 0 Lfcc Mo,Ni:Va LMo,Ni:Va YMo YNi Similarly, the chemical potential of nitrogen can be obtained by applying Eq. (2-26) to Eq. (2-37) as: 27 Nfcc Gmfcc Gmfcc YN YVa hfcc hfcc hfcc hfcc YFe oGFe:N oGFe:Va oGCr:Va YCr oGCr:N hfcc hfcc hfcc hfcc YNi oGNi:N oGNi:Va oGMo:Va YMo oGMo:N RT YN / 1 YC YN 0 fcc 0 fcc 0 fcc YC YFe 0 Lfcc Fe:C,Va YCr LCr:C,Va YNi LNi:C,Va YMo LMo:C,Va 0 fcc 0 fcc 1 2YN YC YFe 0 Lfcc Fe:N,Va YCr LCr:N,Va YMo LMo:N,Va YFeYC 0 Lfcc Fe:C,N 1 fcc 0 fcc 1 fcc YCrYFe 0 Lfcc Cr,Fe:N LCr,Fe:N YCr YFe LCr,Fe:Va LCr,Fe:Va YCr YFe 2-39 1 2YN YC YCrYFe 0 Lfcc Cr,Fe:N,Va YFeYNi L 0 fcc Fe,Ni:N 0 fcc 1 fcc 2 fcc 1Lfcc Fe,Ni:N YFe YNi LFe,Ni:Va LFe,Ni:Va YFe YNi LFe,Ni:Va YFe YNi YCrYFeYNi 0 Lfcc Cr,Fe,Ni:Va YFeYMo 0 Lfcc Fe,Mo:Va 1 fcc YCrYNi 0 Lfcc Cr,Ni:Va LCr,Ni:Va YCr YNi 1 2YN YC YCrYNi 0 Lfcc Cr,Ni:N,Va 0 fcc 1 fcc YCrYMo 0 Lfcc Cr,Mo:N LCr,Mo:Va LCr,Mo:Va YCr YMo 1 fcc YMoYNi 0 Lfcc Mo,Ni:Va LMo,Ni:Va YMo YNi Eq. (2-38) and Eq. (2-39) are both very useful in computing the chemical potentials of carbon and nitrogen and the solubilities of carbon and nitrogen in a given austenite phase during low temperature carburization, nitridation and carbonitriding processes. 28 2 2.4 Conversion between Elementary Thermodynamic Model and Compound Energy Model In the elementary thermodynamic model, the Gibbs free energy is usually defined for one mole of atoms in the solution, which herein will be written as Gm . However, in the compound energy model, the Gibbs free energy is defined for one mole of formula units, which is represented by Gm . Based upon the relationship between the number of atoms and the number of formula units, the relationship between these two different bases for the Gibbs free energies for a fcc phase can be shown as: Gm Gm 1 Y j (2-40) j where Y j is the site fraction of interstitial species, such as carbon or nitrogen. The denominator represents the number of moles of atoms with respect to one mole of formula units. For example, in a fcc phase with interstitial carbon and nitrogen atoms, there are 1 YC YN moles of atoms in the system with respect to 1 mole of formula units. By applying Eq. (2-8), the relationship between these two Gibbs free energies could also be expressed as: Gm 1 X j Gm j (2-41) where X j is the mole fraction of interstitial species. 29 Therefore, the relationship between Gm and Gm is obtained, which allows the chemical potential to be expressed in two different ways. The CALPHAD expression of the chemical potential of interstitial species as Eq. (2-36) or Eq. (2-37) can also be successfully converted to the expression of the elementary thermodynamic model. The complete derivation process has been thoroughly introduced in author’s master thesis [28]. In summary, these two thermodynamic models are consistent with each other and can be successfully converted from one to another as is convenient. 30 Chapter 3 Numerical Simulation In the past two decades, with the growing availability of high speed, large capacity computers, the interest in the use and development of numerical methods, such as the finite difference method, for solving problems governed by differential equations has increased significantly. Many complicated engineering problems can now be solved with computers at very little cost in a very short time. Concentration dependent diffusion is also well described by differential equations, so the finite difference method is capable of simulating this kind of problem and certainly becomes the first choice in this study. 3.1 Concentration Dependent Diffusion 3.1.1 Fick’s Second Law The diffusion of carbon in austenitic stainless steels can be described by Fick’s second law [30], which is expressed as Eq. (3-1): cC c DC cC C t x x (3-1) where cC is the carbon concentration, and DC cC is the diffusion coefficient of carbon in austenite, which is a function of temperature and carbon concentration. 31 When the diffusion coefficient is only a function of temperature, i.e., the diffusion coefficient with carbon at infinite carbon dilution applies, Eq. (3-1) simplifies to the following: cC 2cC DC t x 2 (3-2) where DC is the diffusion coefficient of carbon independent of carbon concentration. The temperature dependence of DC can be expressed as Eq. (3-3): Q DC DC,0 exp C RT (3-3) where DC,0 is the pre-exponential factor for carbon, which is a constant, and QC is the activation energy for the diffusion of carbon in austenite, which is equal to the migration enthalpy associated with the diffusion of carbon. In this case, the simplest carburization process, which is conducted at a constant temperature, can be described by the Van-Ostrand-Dewey solution [30] to Fick’s second law (Jost, 1960), which is expressed as Eq. (3-4): x cC x, t cC, s cC, s cC, o erf 2 D t C (3-4) where cC x , t represents the carbon concentration in the material as a function of the carburization time t and the distance x from the outer surface of the material. cC, s is the surface concentration of carbon, cC, o is the original carbon concentration in the material, 32 and “erf ” stands for the error function [30], which is an indefinite integral defined by the following equation: erf z z 2 exp y 2 dy π0 (3-5) A range of values [4] for the error function (Jost, 1960) is shown in Table 3-1: Table 3-1 A Range of Values for the Error Function (Jost, 1960) It should be noted from Table 3-1 that erf 0.5 0.5 , which defines the value of depth x , at which the carbon concentration is midway between cC, s and cC, o . This depth is also commonly defined as the depth of the carburized case [31]. 33 3.1.2 Concentration Dependence of Diffusion Coefficient When the diffusion coefficient of carbon is dependent on carbon concentration, Eq. (3-1) applies and can be expressed as Eq. (3-6): cC c DC cC C t x x DC cC cC 2cC DC cC 2 x x x 2 D c c 2c C C C DC cC 2C c x x (3-6) When the diffusion coefficient of carbon in austenite is strongly dependent on the carbon concentration, the shape of the concentration versus depth profile will deviate from the error-function shape [6] [7]. In order to simulate the corresponding carbon concentration-depth profiles, it is very important to study the concentration dependence of the diffusion coefficient of carbon in austenite. Empirically, the diffusion coefficient of carbon as a function of carbon concentration can be described by a simple exponential function as [32]: or c DC DC exp kC C cC,max (3-7) XC DC DC exp kC X C,max (3-8) where cC,max and X C,max are the maximum concentration and maximum atomic fraction of carbon, which are likely obtained on the surface. The parameter k is the enhancement coefficient, which controls the magnitude of the concentration dependence of the 34 diffusion coefficient. Other results focusing on this subject would be reviewed in Chapter 5. This type of formulism can be incorporated into a finite-difference-based numerical simulation to generate the concentration-depth profiles. 3.2 Introduction to the Finite Difference Method The finite difference method is now a universally used approach for solving linear and nonlinear differential equations governing engineering problems in a very wide range of research fields. It is simple to formulate, can readily be extended to two or three dimensional problems, and is easy to learn and apply. With the advent of numerical grid generation technique, the finite difference method now possesses geometrical flexibility while maintaining the simplicity of the conventional finite-difference technique. The basic principle of this method is briefly introduced as follows. When a differential equation is solved by analytical methods over a given region subject to specified boundary conditions, the solution is able to satisfy the differential equation at every point in that region. However, when the problem could not be solved analytically or the analytic solution is too involved, a numerical technique is generally applied for the solution. When the finite difference approach is used, the problem domain is discretized so that the values of the unknown dependent variables are considered only at a finite number of nodal points instead of every point over that region. If N nodes are selected, N algebraic equations are developed by discretizing the governing differential equations and 35 the boundary conditions for the problem. Therefore, the problem of solving the ordinary or partial differential equations over the problem domain is transformed to the task of development of a set of algebraic equations and their solution by a suitable algorithm. To discretize the derivatives in partial differential equations, a Taylor series expansion is applied. 3.2.1 Taylor Series Formulation The definition of the derivative of the function F a, b at a a0 , b b0 is given as follows: F a0 a, b0 F a0 , b0 F lim 0 a a a (3-9) If the function F a, b is continuous, the right hand side of Eq. (3-9) should be a reasonable approximation to F / a for a finite and sufficiently small a . According to a Taylor series expansion, a function f a about a point a0 in the forward (i.e., positive a) and backward (i.e., negative a) directions can be expanded as shown in Eqs. (3-10) and (3-11), respectively [33]: f a 0 a f a 0 df d2 f a 2 da da df d2 f x 2 f a 0 a f a 0 da da a 2! a 2 d3 f da 3 a d3 f 3 2! da 3! a 2 3 3! (3-10) (3-11) 3 36 These two equations are the basis for developing the finite difference approximations of the first derivative df / da , about a0 . Eqs. (3-10) and (3-11) can be rearranged as Eqs. (312) and (3-13): f a0 a f a0 df + a forward da a (3-12) f a0 f a0 a df + a backward da a (3-13) where the notation a represents the “truncation error” associated with the finite difference approximation, which is the difference between the derivative and its finite difference representation. For instance, the truncation error for Eq. (3-12) is: a f a a f a0 a 0 2 6 2 (3-14) The central difference approximation can be obtained by subtracting Eq. (3-11) from Eq. (3-10) as follows: f a0 a f a0 a df 2 a central da 2 a where a 2 a 6 2 f a0 a 120 5 f a0 (3-15) (3-16) Because the truncation error of the central difference approximation is second order in a , the central difference approximation is a more accurate approximation than the forward and backward differences. 37 3.2.2 Finite Difference Approximation of Derivatives Assume that i is the grid point at a0 . Then the notation i 1 and i 1 will respectively represents the grid point at a0 a and a0 a . The notation i 2 and i 2 will respectively refer to the grid points at a0 2a and a0 2 a , and so on. Based upon these notations, the first derivatives of the function f in a two-point formula can be expressed as [33]: f i 1 f i a a f i forward f i backward f i central (3-17) f i f i 1 a a (3-18) f i 1 f i 1 2 a 2a (3-19) By employing similar process to those described in Section 3.2.1, expressions for d 2 f / da 2 are defined and the results, written with similar abbreviated notations as the first derivative, can be expressed as: f i forward f i 2 f i 1 f i 2 a f i backward f i central 2 a f i 2 2 f i 1 f i a 2 f i 1 2 f i f i 1 a 2 a a 2 (3-20) (3-21) (3-22) 38 where a a f a and a 2 a 2 12 f a for the second derivatives of the function f. The central difference approximations for the first and second derivatives as expressed by Eqs. (3-19) and (3-22) are selected as the simulation basis of the finite difference formulism due to the better accuracy than the forward and backward approximations. 3.2.3 Errors Involved in Numerical Solutions In many situations, some questions would arise regarding the round-off errors and truncation errors involved in the numerical computations, such as the consistency, stability and the convergence of the finite difference method [33]. The first error source is the “round-off error”. Because computations are rarely made in exact arithmetic, the real numbers are represented in “floating point” form and the errors are caused due to the rounding-off of the real numbers. Even though modern computers can represent real numbers to many decimal places, in some cases such errors, so called “round off” errors, can still accumulate and become a main source of error. Another important error source is called “truncation error”. In the finite difference approximation of derivatives, the higher order terms in the Taylor’s series expansion are usually neglected by truncating the series and an error would be caused as a result of such truncation. For instance, in the forward difference approximation of the first derivative to the order a as given by Eq. (3-12), the error term 39 1 2 a a f a0 1 2 a f a0 6 represents this truncation error. In this term, the lowest order term on the right hand side of the equation gives the order of the method, i.e. a . Therefore, the truncation error can be defined as the difference between the exact solution of a differential equation and its finite difference approximation without the round-off error. The terminology “discretization error” is also used to identify the error involved due to the truncation error in the finite difference representation of the governing differential equation and the relevant boundary conditions. 3.3 Simulation Schemes for One-Dimensional Diffusion In this study, a primary objective was is to investigate the precise shape and penetration depth associated with carbon concentration-depth profiles. The diffusion of carbon in stainless steel is considered as a one-dimensional diffusion in a semi-infinite system as shown in Fig. 3-1. In Fig. 3-1, x = 0 represents the gas-metal interface during the carburization treatment. The positive x region represents the treated metal and the negative x region represents the treatment gas atmosphere. Considering that the thickness of an interstitially hardened case is relatively small compared with that of the whole material, the metal can be regarded as a semi-infinite system. The carbon concentration in the treatment gas atmosphere is represented by the term cC x 0 , which is also indicated as cC, . 40 cC (mol·µm-3) cC, Fig. 3-1: Schematic view of one-dimensional diffusion in a semi-infinite system. For this clearly defined problem, a variety of simulation schemes are available for selection. However, it is always challenging to find a well conditioned and highly stable algorithm that can also perform the simulation efficiently and produce the results accurately in a numerical simulation [34]. The numerical schemes considered in this simulation are introduced in what follows. 3.3.1 Simple Explicit Scheme Consider the diffusion problem with a concentration-independent diffusion coefficient, expressed as Eq. (3-2). Neglecting the subscript C for all the parameters, Eq. (3-2) expressed as: 41 c 2c D 2 t x (3-23) where the concentration c is a function of both distance x and time t. The following conditions are added as required by a finite difference discretization: 0 x z, t 0, x z M (3-24) So the region 0 x z is divided into M equal parts establishing a mesh size. By employing Eq. (3-22) for 2 c / x 2 and Eq. (3-17) for c / t , Eq. (3-23) can be discretized as: cim 1 cim c m 2cim cim1 2 D i 1 t , x 2 t x (3-25) where the subscripts i, i – 1 and i + 1 represent the distance steps and the superscripts m and m +1 represent the time steps, which means: c x, t c ix, mt cim (3-26) Eq. (3-25) is rearranged as: cim 1 rcim1 1 2 r cim rcim1 r D where t x 2 (3-27) (3-28) m = 0, 1, 2, …… , and i = 1, 2, ……, M – 1 with a truncation error of the order 2 t , x . 42 In this scheme, the only unknown value is cim1 at the time step m, which can be directly calculated from Eq. (3-27) once the values of cim1 , cim and cim1 at the time step m are all available. Therefore, this kind of scheme is called the simple explicit form for a finite difference approximation. The finite difference mesh squares associated with the simple explicit scheme applied to the diffusion equation are schematically illustrated in Fig. 3-2. m 1 cim 1 m cim1 cim cim1 m 1 i 1 i i 1 Fig. 3-2: The finite difference mesh squares associated with the simple explicit scheme applied to the diffusion equation. 3.3.2 Fully Implicit Scheme For the fully implicit scheme is at each time step, the finite difference equations are solved simultaneously in order to determine all of the nodal concentration values. The general form of the simple implicit method is expressed as Eq. (3-29): cim 1 cim c m 1 2cim 1 cim11 2 D i 1 t , x 2 t x (3-29) 43 2 which is accurate to t , x . Eq. (3-29) can also be rearranged as: rcim11 1 2 r cim 1 rcim11 cim (3-30) where the parameter r is defined by Eq. (3-28). In this scheme, the only known value is cim at the time step m, while the values of cim11 , cim 1 and cim11 at the next time step m + 1 are all unknown. Therefore, this scheme is called the fully implicit form of the finite difference approximation. The finite difference mesh squares associated with the fully implicit scheme applied to the simple diffusion equation are schematically illustrated in Fig. 3-3. cim11 cim 1 m 1 cim11 m cim m 1 i 1 i i 1 Fig. 3-3: The finite difference mesh squares associated with the fully implicit scheme applied to the diffusion equation. 3.3.3 Crank-Nicolson Scheme This finite difference scheme proposed by Crank and Nicolson (1947) can be regarded as a combination of the simple explicit scheme and the fully implicit scheme. 44 By retaining the left hand side of the implicit equation (3-29) and taking the arithmetic average of the right hand sides of the explicit equation (3-25) and the implicit equation (3-29), the implicit finite difference approximation associated with the Crank-Nicolson scheme is expressed as Eq. (3-31): cim 1 cim D cim11 2cim 1 cim11 cim1 2cim cim1 2 2 t , x 2 2 t 2 x x (3-31) It should be noted that this scheme is accurate to second order in both time and space [33], which is better than the other two schemes. The finite difference mesh squares associated with the Crank-Nicolson scheme applied to the simple diffusion equation are schematically illustrated in Fig. 3-4 [33]. m 1 m cim11 cim 1 cim11 cim1 cim cim1 m 1 i 1 i i 1 Fig. 3-4: The finite difference mesh squares associated with Crank-Nicolson scheme applied to the diffusion equation. From Eq. (3-31), the accurate to second order finite difference approximation by the Crank-Nicolson scheme is obtained as Eq. (3-32): 45 rcim11 2 2 r cim 1 rcim11 rcim1 2 2r cim rcim1 (3-32) where the parameter r is again defined as Eq. (3-28). 3.4 Boundary Conditions Taking the fully implicit scheme as an example, in Fig. 3-3, it should be noticed that the system contains M + 1 unknown node values cim1 (i = 0, 1, 2, …, M). However, the M – 1 algebraic relations generated from Eq. (3-30) only provide M – 1 unknown node values. Two additional relations need to be provided by suitable boundary conditions at i = 0 and i = M. Once the boundary node values are prescribed, the number of equations is equal to the number of unknowns, which is a requirement in numerical linear algebra [34]. 3.4.1 Fixed Boundary Condition The simplest situation is that the boundary node values are fixed, such as: c0m cmax cMm co (3-31) where cmax is the maximum concentration value usually achieved at the gas-metal interface (i = 0) and co is the original concentration in the bulk material (i = M). Because carbon transport is regarded as a diffusional process in a semi-infinite system, co is 46 normally assumed to be zero. Both boundary concentrations are fixed during the entire time range (m = 0, 1, 2, …). Once the boundary conditions are defined, all the requirements for a simulation are satisfied. The computational algorithm is performed as follows: (1). Start the calculations with m = 0. All the node values at m = 0 should be available including the two fixed boundary conditions. Compute the values of ci1 (i = 1, 2,…, M –1), based upon Eq. (3-30) and set the values of c01 and c1M as the fixed boundary values at the end of the first time step. (2). Set m = 1 and repeat the calculation process as in step (1) because all the node values are known from the calculation at the previous time step. (3). Repeat this procedure for each subsequent time step and continue until a specified time or a specified value of concentration is reached. 3.4.2 Convective Boundary Condition Simulation becomes more complicated when the boundary condition is “convective” [35]. During low temperature carburization, the surface of the treated alloys is in contact with a carbon-bearing gaseous atmosphere, and there is a mass transport of carbon from the ambient into the alloys through the gas-metal interface. Under a steady state, the mass passing through the surface boundary should be equal to the mass transported from the gaseous atmosphere into the material. This condition is defined as Eq. (3-32): 47 D c c cs x (3-32) where cs now is not a constant and c is normally a constant independent of time. is called the mass transfer coefficient [35] having units of length/time, which represents the efficiency of mass transport based upon a certain concentration gradient. Eq. (3-32) is transformed to a finite difference expression by employing the central difference approximation at the time step m + 1 as Eq. (3-33): D c1m 1 cm11 c c0m 1 2 x (3-33) To eliminate the fictitious concentration value cm11 for a simulation based upon the fully implicit scheme, Eq. (3-30) at i = 0 is applied, which is expressed as: rcm11 1 2r c0m 1 rc1m 1 c0m (3-34) where r is still defined by Eq. (3-28). Therefore, Eq. (3-35) is obtained by eliminating the fictitious concentration value cm11 between Eq. (3-33) and (3-34) as: 1 2r c0m 1 2rc1m 1 c0m 2r where 1 (3-35) x x and c D D Once x , t , D’, and c are known, the parameters r, and are also known. Therefore, the boundary concentration values at i = 0 for all the time steps would be obtained only if the concentration value c0m at the previous time step m is available. 48 Assume that the boundary concentration value at i = M is fixed, which is represented by the same expression as Eq. (3-31): cMm co . Then Eq. (3-30) together with Eq. (3-31) and (3-35) provide M + 1 expressions for the determination of M + 1 unknown concentration values at each time step. 3.5 Stability In finite difference simulations, it is possible that the finite difference equations become divergent under certain conditions. It is always challenging to find a suitable scheme which can give the best combination of accuracy, stability and efficiency. The stability of the three schemes that have been introduced is evaluated as follows. With the simple explicit scheme, for the equations to remain stable, the value of the parameter r defined by Eq. (3-28) should be restricted to: 0r 1 2 (3-36) This stability criterion indicates that, for given values of D and x , the magnitude of the time step t needs to be limited in order to satisfy Eq. (3-36). The influence of the value of the parameter r on the stability of finite difference solution with the simple explicit scheme is shown in Fig. 3-5 [33]. 49 c Fig. 3-5: The influence of the parameter r on the stability of finite difference solution with the simple explicit scheme. Briefly speaking, the physical significance of the restriction on the maximum value of the parameter r is due to a requirement of the finite difference approximation that the concentration value cim1 cannot go below the concentration values of the two neighboring nodes. The violation of this requirement is possible when the coefficient 1 2r in Eq. (3-27) is negative. Therefore, in order to obtain meaningful results, the coefficient 1 2r in Eq. (3-27) should satisfy such a criterion: 1 2r 0 or r 1 2 which is defined by Eq. (3-36). This stability criterion is not only for the internal nodes of the region, but is also for the boundary conditions. When the boundary concentration value is fixed as in Eq. (331), no additional restrictions are needed. However, when the boundary condition is convective, the second order accurate finite difference approximation for the simple 50 explicit scheme will impose another restriction on the parameter r, which is described by Eq. (3-37): 1 2r 0 or 0 r 1 1 2 2 2 x D (3-37) This stability criterion is more restrictive than the one given by Eq. (3-36). Therefore, when the boundary condition is convective, the stability criterion defined by Eq. (3-37) should be applied. This stability issue also implies that reducing the distance corresponding to the mesh size will dramatically increase the computation time needed. It was mentioned that reducing the mesh size by one half could increase the computation time by 8 fold [33]. So this restriction on the maximum time step due to this stability issue is indeed a major disadvantage of the simple explicit scheme. The fully implicit scheme and Crank-Nicolson scheme have been developed with the objective of finding efficient schemes that are more accurate and also have no restriction on the size of the time step. Both of these two schemes have been demonstrated to be “unconditionally stable” [33] and widely used in all kinds of simulation applications. However, in this study, the stability of Crank-Nicolson scheme has been frequently challenged when the simulation involves large latitudes of the mass transfer coefficient and a concentration-dependent diffusion coefficient. It seems that the Crank-Nicolson-based finite difference algorithm would become unstable when the mass transfer coefficient exceeds a critical value for certain forms of the diffusion coefficient. In addition, some other research work in the literature [36] has also discussed that the Crank-Nicolson scheme may only be conditionally stable in some cases and the fully 51 implicit scheme is thus recommended as a better choice for its higher stability and excellent combination of accuracy and computational efficiency. In this study, it is almost impossible to accurately describe the stability of a numerical scheme. Therefore, the fully implicit scheme was empirically determined to be the first choice of the finite difference simulation, when a convective boundary condition with a large latitude of the mass transfer coefficient is involved. Under a fixed boundary condition, the Crank-Nicolson scheme was employed considering its higher accuracy in the dimension of time. 3.6 Fully Implicit Scheme with Concentration-Dependent Diffusion Coefficient The finite difference approximations become more complicated when the diffusion coefficient is dependent upon the concentration, which is expressed as Eq. (3-1). During low temperature carburization, the maximum surface concentration of carbon in treated AISI 316L stainless steel is up to ~ 15 at%. It is expected that the diffusion coefficient of carbon is strongly dependent upon the carbon concentration under this condition. The finite difference approximation of Eq. (3-1) with the fully implicit scheme can be expressed as Eq. (3-38): cim 1 cim c m 1 cim 1 cim11 cim 1 Di 1/2 i 1 D i 1/2 2 2 t x x (3-38) 52 where the truncation error is also second order to t and x . It is noteworthy that the diffusion coefficient D is dependent on the node values in this case. Eq. (3-38) can be compactly expressed as follows: cim 1 cim i cim11 2icim 1 icim11 i Di 1/2 where t x i Di 1/2 i (3-39) 2 t x 2 1 1 t i i Di 1/2 Di 1/2 2 2 2 x By separating the concentration values at m and m + 1 time steps, Eq. (3-39) is rearranged so that all unknown concentration values appear on the left side and all the known concentration values appear on the right side, which is expressed as Eq. (3-40): i cim11 1 2i cim1 icim11 cim (3-40) It should be noted that when the diffusion coefficient is independent of concentration, the following relationships are obtained: i i i D t x 2 r (3-41) and Eq. (3-40) will be simplified to Eq. (3-30). When the relevant values of i , i and i are available, if both of the boundary conditions are fixed, the set of equations, i.e., 53 Eq. (3-40) can provide the complete determination of all the unknown (M – 1) internal node concentration values based upon the fully implicit scheme. Early in this chapter it was shown that the concentration dependence of the diffusion coefficient of carbon in austenite can be empirically expressed as Eq. (3-7): c DC DC exp kC C cC,max In the finite difference approximation, the values of the diffusion coefficient will depend on the concentration values at the nodes because the domain under study in the diffusion direction has already been discretized. For example, the diffusion coefficient at the node i and the time step m can be expressed in the general form as Eq. (3-42): cm Dim D exp k im c0 (3-42) where the subscripts “C” for carbon have been ignored. This expression indicates that the diffusion coefficient at the node i and the time step m, Dim is dependent upon the concentration value at the node i and the time step m, cim . Because the values of D and k are known, the values of the diffusion coefficients can be determined once the boundary condition, essentially the value of c0m , is well defined. Therefore, the diffusion coefficients in Eq. (3-38) can be determined as follows: cm cm cm Dim1/2 D m ci 1/2 D exp k i m1/2 D exp k i 1 m i c0 2c0 (3-43a) 54 cm cm cm Dim1/2 D m ci 1/2 D exp k i m1/2 D exp k i m i 1 c0 2c0 (3-43b) In this case, the diffusion coefficients Dim1/2 would be dependent upon the values of cim1/2 , which are the averages of concentration values at the nodes i and i 1 . It can also be evaluated as the average of the diffusion coefficients based upon the concentration values at the nodes i and i 1 [33]. However, when the concentration-depth profile has a significant gradient as observed in this study, the first protocol as Eq. (3-43) is considered to be a better choice for obtaining more accurate values of the diffusion coefficients. However, for the time step at m + 1, unknown concentration values, such as cim1 , would appear in the exponential function of the diffusion coefficient, which will make the unknown variables very involved and therefore difficult to separate in the finite difference equations as Eq. (3-39). In order to solve this problem, the “lagging property” method is used. 3.7 Lagging Properties by One Time Step The “lagging property” method is to delay the evaluation of the concentrationdependent diffusion coefficients by one time step. That is, during the computation at the time step m + 1, the diffusion coefficients are evaluated at the previous time step m instead of at the time step m + 1. For example, the diffusion coefficient Dim1 at the node i and the time step m + 1 in the similar form as Eq. (3-42) should be expressed as: 55 c m 1 Dim 1 D exp k im 1 c0 (3-44) Using the “lagging properties” method, the diffusion coefficient at the node i is evaluated at the previous time step m as Eq. (3-42). Therefore, the coefficients in Eq. (3-39) are actually all based upon the calculations at time step m: i im , i im , and i im (3-45) This method is the simplest, but least accurate method for computing these coefficients. More complex methods for this computation are widely discussed in the literature [33]. Considering this method can provide sufficient accuracy for the simulation in this study, it was employed in the following simulations. When the boundary condition at i = 0 is convective as defined by Eq. (3-32), the finite difference approximation of this boundary condition with a concentrationdependent diffusion coefficient is expressed as: D0m c1m cm1 c c0m 2 x (3-46) where D0m is the diffusion coefficient at the node i = 0 and the time step m. Correspondingly, the finite difference approximation for this boundary condition at m + 1 time step can be expressed as Eq. (3-47) by lagging the diffusion coefficient by one time step: D0m c1m 1 cm11 c c0m 1 2 x (3-47) 56 where the diffusion coefficient at the node i = 0 and the time step m + 1 is still evaluated at the previous time step m. The fictitious concentration values cm1 and cm11 can be eliminated from Eq. (3-46) and (3-47) utilizing Eq. (3-34) for the time steps m and m + 1. Then an expression of this convective boundary condition with concentration-dependent diffusion coefficients will be obtained by the process similar to the one outlined in Section 3.4.2. Assuming another boundary concentration value fixed at i = M, a complete set of finite difference equations with a convective boundary condition and concentration-dependent diffusion coefficient are provided for the computations of all the M + 1 node concentration values. 3.8 Computational Environment The coding procedure for a finite difference algorithm is based upon the essential concepts of numerical linear algebra and can be realized using fundamental operations of the C language. Generally speaking, the coding in this study is much less complex than the challenging problems in the professional field of numerical linear algebra and applied mathematics, which often explores the solutions of problems with millions of unknown variables [34]. The codes for a finite difference algorithm can be incorporated into any professional software of scientific computation which can perform the operations in numerical linear algebra. In this study, the coding procedure was based upon the MATLAB® software of R2008a version. The information regarding the computer employed in this study is listed in Table 3-2 as follows. 57 Table 3-2 The Information of the Computer Employed in This Study Computer Model Dell Laptop Studio 1558 Processor Intel® CoreTM i3 CPU – 2.13 GHz Physical Memory 4.00 GB Input / Output Cache 128 MB Operation System Window 7 Home Premium 64-bit The simulation processes based upon the introduced computational environment of software and hardware normally only take up to several minutes to generate concentration-depth profiles and perform the fitting procedure. The generated concentration-depth profiles based upon the finite difference simulations are compared with the experimental data obtained from surface chemical analysis and elemental depth profiling, which are introduced in the next chapter. 58 Chapter 4 Experimental Methods In this study, accurate measurements of the concentration-depth profiles of carbon in treated AISI 316L stainless steel is extremely important and very challenging. This is because the greatly improved properties of treated materials are directly related to the penetration depth and precise shape of the concentration-depth profiles and also because the experimental data need to be widely used to evaluate the analytical models based upon concentration-dependent diffusion of carbon in stainless steels. Therefore, two types of surface analysis techniques, Auger Electron Spectroscopy (AES) and X-ray Photoelectron Spectroscopy (XPS), were employed in this study to provide chemical composition surveys and elemental depth profiles. 4.1 Auger Electron Spectroscopy 4.1.1 Introduction A scanning Auger microprobe (Perkin-Elmer PHI-680) was employed to record carbon concentration-depth profiles of carburized specimens of AISI 316L stainless steel. The instrument consists of a scanning electron microscope which operates with a fieldemission gun based on a Schottky emitter, a secondary electron detector, and an axial cylindrical mirror analyzer with a multichannel detector to collect Auger electrons produced during electron imaging. In addition, the Auger microprobe is equipped with a 59 PHI 06-350 ion gun for sputtering the specimen surface with an inert gas. In order to record concentration-depth profiles, the electron probe can be positioned to automatically scan along a line. The system was carefully calibrated with a magnification standard to minimize the error in absolute distances between the scan points. Scanning Auger microprobe analysis essentially has three advantages. Firstly, specimen preparation is not time consuming. Secondly, this technique can record data along the scan line with very high spatial resolution. The typical electron probe diameter is about 50 nm in the line scan mode. Finally, the data can be acquired efficiently with this technique. The regular acquisition time with 32 scan points along a line is up to one hour. A special routine for improving the data quality was developed by Dr. Wayne D. Jennings and was followed very carefully during all of the experiments. The specimen preparation and the experimental procedure of the special routine are introduced in the following. 4.1.2 Specimen Preparation To measure the carbon concentration-depth profiles, metallographic cross sections of the carburized AISI 316L stainless steel were prepared. The carburized specimens were cut with a high speed saw into cross-sectional sheets of 1-2 mm thickness. During the polishing procedure, Buehler P400, P800, P1200 and P2400 grit SiC grinding paper was used, followed by 3µm, 1µm and 0.25µm polycrystalline diamond suspension applied with nap cloths. The specimens were finally polished with 0.05µm alumina suspension, rinsed with ethanol and dried efficiently. 60 Before introduction into the UHV system of the Auger microprobe, each specimen was pre-cleaned ultrasonically in ethanol or isopropanol for five minutes. 4.1.3 Experimental Procedure Auger Electron Spectroscopy line scans were carried out over the cross section of specimens so that a carbon concentration-depth profile could be acquired along a line orthogonal to the carburized surface. The electron beam potential was 10 kV, and the double-pass cylindrical mirror analyzer detector was operated in a Fixed Retard Ratio (FRR) mode. Before the acquisition of each Auger spectrum, the specimen was sputtered with Ar ions at 3 kV in a 2 mm × 2 mm raster with an argon pressure of 15 mPa for 5 minutes, which removed approximately 50 nm of materials at the 10 nm·min-1 sputter rate. This initial sputtering step was intended to remove the oxide layer and any undesired hydrocarbons on the surface of the cross sections. During the acquisition of line scans, Auger electron spectra were recorded while the specimen surface was continuously sputtered with Ar ions at the lowest flux rate available on the instrument, which was under an argon pressure of approximately 0.2 mPa corresponding to a sputter rate of about 0.1 nm·min-1. The purpose of this mode was to eliminate the graphitization of mobile carbon on the surface of the specimens due to the local heating effect generated from the electron beam [37]. However, the sputtering which took place after the graphite was removed did bring about slight changes to the surface chemistry of the specimen, therefore a relevant systematic error may arise from 61 the preferential sputtering or surface roughening that occurs in the course of the continuous sputtering. A calibration procedure was performed by former Master student Joshua Katz [37]. The carbon concentrations for a suite of Fe-C alloys obtained under the continuous sputtering mode were quantitatively evaluated and compared with the known compositions of the alloys. An empirical relationship between the measured carbon concentration under the continuous sputtering mode and the actual carbon concentration in the alloys was obtained enabling a correction to experimental data obtained from Auger Electron Spectroscopy line scans to be made as follows [37]: X Cact. 0.566 X Cmea. 0.986 (4-1) where the carbon concentrations are atomic percentage. This calibration result was further evaluated by a later procedure, which focused on the carbon concentration in a specimen of an amorphous alloy SAM 1651-4. The measured carbon concentration with an AES line scan under the continuous sputtering mode was compared with the result of a LECO analysis performed by the Swagelok Company. Based upon the correction of Eq. (4-1), the measured carbon concentration with the AES line scan was about 1 at% higher than the test result of the LECO analysis at the nominal carbon concentration of ~ 15 at%. The results of this procedure are listed in Appendix II. 62 4.2 X-Ray Photoelectron Spectroscopy 4.2.1 Introduction XPS surveys were acquired with a Perkin-Elmer PHI-5600 Versaprobe XPS system. An aluminum target was used along with a monochrometer producing Kα radiation at 1486.6 eV that irradiated the sample. For standard survey scans or sputtered depth profiles, a 300 µm spot size was used, and at this spot size X-rays were generated with an applied power of around 70 W. The detection of the generated photoelectrons was accomplished using a hemispherical capacitance analyzer with an electron take-off angle of 45o. The Versaprobe’s hemispherical analyzer was employed in the Fixed Analyzer Transmission (FAT) mode, which provides an energy resolution of about 1.5 eV with a pass energy of 93.9 eV. The spectrometer was also equipped with an ion gun that generated sputtering ion flux which was accelerated at 3 kV in a 3 mm × 3 mm raster. The sputter rate was normally about 10 nm·min-1 based upon a Ta2O5 calibration sample of known thickness on a Ta substrate. The most important advantage of XPS is its capability of determining the chemical environment of a specific element. For instance, it is possible to tell whether carbon is present in its adventitious state or in an interstitial solution. The analysis of chemical state is performed by careful examining the obtained spectra and deconvoluting the energy peaks of the elements. This technique is more challenging and complex in terms of the data analysis procedures than data acquisition. It is necessary to have 63 relevant experience in defining the positions and baselines of the elemental peaks in a spectrum in order to obtain reliable and reproducible results. XPS normally has poor spatial resolution, which can be considered as an advantage as well as a disadvantage. It is a disadvantage because the poor resolution restricts the examination of very small regions of a specimen and the investigation of desired specific locations. However, it is also an advantage because the surveyed area is more representative of the average surface chemistry, which increases the data reliability. 4.2.2 Specimen Preparation For this technique, it is very important to work with a surface that is as smooth as possible achieved by polishing or even electropolishing prior to a carburization treatment, because the greater the degree of roughness of the surface, the more adventitious carbon that can be trapped inside troughs on the surface. Such carbon is difficult to remove from the surface. With this adventitious carbon, spectra analysis could become either extremely difficult or even impossible in some cases. For instance, the adventitious carbon peak at 284.8 eV can sometimes completely overlap the carbidic carbon peak only about 2 eV lower in energy. When only carburized specimens with rough surfaces were available for analysis, the specimen was polished prior to XPS analysis with a slurry of 0.05 µm alumina particles on a soft cloth or with a cotton Q-tip for several minutes. The polishing procedure was performed very carefully, in which case the adventitious carbon was removed as much as possible and the surface of the specimen returned to its shiny appearance after such polishing, which was designed to as little materials as was 64 necessary from a surface. Based upon this consideration, a weight measurement was performed before and after polishing, which showed that the weight loss for a specimen of 4 g and with a 1 cm2 polished surface was normally less than 0.1 mg, corresponding to a loss of < 0.12 µm thickness from the surface. Each specimen was also pre-cleaned ultrasonically in ethanol or isopropanol for five minutes before introduction into the PHI-5600 Versaprobe XPS system. 4.2.3 Experimental Procedure The specimens were sputtered prior to data acquisition. The purpose was to remove as much of the oxide layer and the residue adventitious carbon on the surface as possible. For standard surveys to determine chemical compositions, the acquisition was performed after about 50 to 100 nm of sputtering from the surface of a specimen. The survey was then acquired over a binding energy range of 0-1100 eV and was comprised of 8 cycles of data acquisition which regularly took up to 40 minutes. For sputtered depth profiling, the process of sputtering and surveying was performed for a number of times. The number of acquisition cycles and the sputter depth per cycle were determined by the desired depth to be sampled and the depth resolution needed. Due to the sputtering rate, sputtered depth profiles could only feasibly be acquired to a depth of about 2-3 µm. The carbon concentration in a specimen of an amorphous alloy SAM 1651-4 obtained from standard XPS analysis surveys was also evaluated and compared with the 65 result of the LECO analysis performed by the Swagelok Company. The measured carbon concentration with the standard XPS surveys was about 1 at% lower than the test result of the LECO analysis at the nominal carbon concentration of ~ 15 at%. The results of this procedure are listed in Appendix II as well. 66 Chapter 5 Modeling of the Concentration Dependence of Carbon Diffusivity in Austenite As discussed in previous sections, the carbon concentration in AISI 316L stainless steel treated by a low temperature carburization technique can achieve an extraordinarily high level of about 10-15 at%. This provides a unique opportunity to study the concentration dependence of the diffusion coefficient of carbon in austenite over a very broad range of carbon concentrations. The concentration dependence of the diffusion coefficient of carbon in austenite has been widely studied by researchers, but it is still not completely understood. In order to simulate the concentration-depth profiles of carbon in treated AISI 316L stainless steel, an extensive literature review was performed to find the most suitable analytical model of the concentration dependence of the diffusion coefficient for carbon in austenite. The application of the selected model on the concentration dependence of carbon diffusivity in AISI 316L stainless steel was also derived and compared with the published data by Ernst et al. [38] for carbon diffusivity in AISI 316L stainless steel as a function of carbon concentration. 67 5.1 Literature Review A variety of analytical models were considered and applied to the generation of carbon concentration-depth profiles in austenite. The models found in the literature are reviewed in chronological order in the following. To evaluate the concentration dependence of the diffusion coefficient of carbon for all of these analytical models, the normalized ratio of DC / DC’ is commonly employed. To make the expressions as consistent as possible, all the models are presented in the following format: DC YC DC f YC (5-1) where the ratio between the concentration-dependent diffusivity of carbon DC (YC) and the diffusivity of carbon at infinite carbon dilution DC’ equals a function of the site fraction of carbon YC. The relationship between YC and the atomic fraction of carbon XC for austenite has been defined in Eqs. (2-8) and (2-9). The physical meanings of several critical parameters are defined as follows: DC: The diffusivity of carbon in austenite which is dependent upon carbon concentration (µm2·s-1). DC’: The diffusivity of carbon at infinite carbon dilution, which is independent of carbon concentration and only affected by the metal elements in the austenite matrix (µm2·s-1). XC: Concentration unit for carbon as atomic fraction. 68 YC: Concentration unit for carbon as site fraction. T: Thermodynamic temperature (K). R: Gas constant (8.3145 J·mol-1·K-1). γC: The activity coefficient of carbon based upon a Henrian standard state. µC: The chemical potential of carbon (J·mol-1). 5.1.1 Absolute Reaction Rate Theory / ARRT (1948) In 1948, Fisher et al. [8] proposed that the process of interstitial diffusion in metals be treated as a chemical reaction. This model attributes the concentration dependence of diffusivity to the variation of the activity coefficient of the interstitial species with change in its concentration. Two expressions were derived based upon different assumptions associated with a parameter, γM, the activity coefficient of the activated complex: DC YC DC 1 d ln C d ln YC d ln C DC YC DC C 1 d ln YC M C (5-2) M (5-3) constant Eq. (5-2) is obtained if γM is assumed to be proportional to γC, and Eq. (5-3) is obtained if γM is assumed to be a constant, i.e., γM = 1 [8]. The only difference between these two expressions is the extra γC term found in Eq. (5-3). The term in parentheses is called the thermodynamic factor. When γC is replaced with the relevant CALPHAD expression [39], the diffusivity of carbon in a fcc Fe-C binary system can be written as follows: 69 2 8339.9 DC YC DC 1 YC 2 T 1 YC 1 YC 2 8339.9 8339.9 DC YC DC YC 1 YC exp 2 T T 1 YC 1 YC (5-4) (5-5) The derivation of the thermodynamic factor for fcc systems is outlined in Appendix III. 5.1.2 Kaufman et al. (1962) In 1962, Kaufman et al. proposed a model derived from work focusing on the kinetics of the edgewise growth of bainite [9]. The authors expressed the activation energy associated with carbon diffusion as a quadratic function of XC based upon the empirical data of Wells et al. [6] for XC ≤ 0.06 and the kinetic results for bainite growth by Speich and Cohen [40] obtained at XC = 0.16. The model in terms of YC can be expressed as follows: 95621.5 YC 276799.2YC 2 DC YC DC exp 30 2 T 1 YC T 1 YC (5-6) 5.1.3 Asimow (1964) In 1964, Asimow proposed a model [10] based upon the absolute reaction rate theory that also considers the influence of a concentration-dependent pressure induced by carbon in solution on the activation Gibbs free energy associated with carbon diffusion in austenite. The model can be expressed as follows: d ln C DC YC DC 1 exp KYC d ln YC (5-7) 70 where K 16 3 r03 a03 E VA RT (5-8) In Eq. (5-7), the exponential term is related to the influence of a concentration-dependent pressure, which was derived from the elasticity work of Eshelby [41] [42]. The parameters contained in Eq. (5-8) are defined as follows: ε: The elastic strain associated with the difference in radii between a carbon atom and an interstitial hole in the austenite matrix. r0: The radius of an interstitial hole in austenite matrix. a0: The lattice parameter of the austenite matrix. The term (εr03/a03) is correlated with the volume expansion of the austenite matrix and its value is about 1.68 × 10-2 [10]. E: Young’s modulus of the austenite matrix (164.6 GPa at 450oC) [43]. ΔVA: The activation volume associated with carbon diffusion in austenite. Its value is calculated based upon Keyes’s model [44][45]: VA 2 1 / 3 GA (5-9) where δ is the Grüneisen parameter (1.44 at 450oC) [43], χ is the compressibility (8.39 × 10-12 Pa-1 at 450oC) [43] and ΔGA is the activation Gibbs free energy associated with carbon diffusion in austenite (125 kJ·mol-1) [46]. When γC is replaced by the relevant CALPHAD expression [39], the diffusivity of carbon in a fcc Fe-C binary system can be written as the following function of YC: 71 2 8339.9 DC YC DC 1 YC exp KYC 2 T 1 YC (5-10) 5.1.4 Siller and McLellan (1970) In 1970, Siller and McLellan proposed a model [11][12] using a first order quasichemical thermodynamic model for carbon in austenite, together with the absolute reaction rate theory. Their model can be expressed as follows: DC YC 2 z Y 1 Y 8888 Y 1 Y 1 2 Y Y 0 C C C C C C DC 2 z z 1 Y T z0 2 1 YC 0 0 C 1 2 1 YC 2 2 1 1 YC 1 YC 8888YC exp (5-11) T 992.24 T 1 exp where z0 12 5.1.5 Tibbett (1980) In 1980, Tibbetts [13] provided an expression for the diffusivity of carbon in austenite based upon the work of Goldstein and Moren [47], which itself is a linear approximation of the data of Wells et al. [6]. This model is expressed in Eq. (5-12): 3985440 YC DC YC DC exp 1920 T 12YC 55.85 (5-12) 72 5.1.6 Larché and Cahn (1982) In 1982, Larché and Cahn [14] proposed a model that also is based upon the absolute reaction rate theory and considers the influence of a local stress field arising from compositional inhomogeneities on the diffusional flux. The model can be expressed as follows: DC YC DC 1 d ln C 2 2 EYC d ln YC 0 RT 1 1 YC (5-13) The parameters in Eq. (5-13) are defined as follows: η: The linear expansion per atomic fraction change of carbon in the austenite matrix (0.264) [14]. E: Young’s modulus of the austenite matrix (164.6 GPa at 450oC) [43]. ρ0: The density of lattice sites in the austenite matrix (1.44×105 mol·m-3) [14]. ν: Poisson’s ratio of the austenite (0.27) [48]. and γC is still defined by the relevant CALPHAD expression [39]. 5.1.7 Ågren (1986) In 1986, Ågren [15] proposed a model based upon irreversible thermodynamics. The model can be expressed as follows: DC YC DC YC 1 YC C 26436 exp 5.8714 YC RT YC T (5-14) 73 Replacement of µC by the relevant CALPHAD expression [39] enables the diffusivity of carbon in a fcc Fe-C binary system to be expressed as: 8339.9 26436 5.8714 YC exp DC YC DC 1 YC 1 YC T T (5-15) 5.1.8 Liu et al. (1991) In 1991, Liu et al. [16] proposed a model based upon the absolute reaction rate theory and a modification of the activity coefficient of the activated complex, γM, of carbon in austenite based upon consideration of the kinetic energy transfer between carbon atoms during resonance, which is a widely used concept in the field of chemical quantum mechanics. The authors demonstrated that γM is inversely related to carbon concentration, rather than being a constant or proportional to the carbon activity coefficient [8]. Their model can be expressed as follows: DC YC DC 1 YC where C M d ln C / M YC 1 dlnYC 1 YC 8878 13 12 13 12 T M (5-16) 2/35.7 4 34416 exp 17.52 YC T and γC is again defined by the relevant CALPHAD expression [39]. 74 5.1.9 Evaluation of the Models All of the above models were validated through comparison with experimental data acquired at temperatures higher than 750oC and for a carbon concentrations less than 6 at% [6][7]. Fig. 5-1 shows the concentration dependence of carbon diffusivity in a fcc Fe-C binary system at 1000oC predicted by all the models introduced previously. In Fig. 5-1, the normalized ratio of DC / DC’ is plotted as a function of XC. At 1000oC, the predictions of all the models concerning the concentration dependence of carbon diffusivity are close to each other for carbon concentrations less than 6 at%, the concentration range over which each model was originally fitted. (9) (5) (3) (6) (8) (4) (2) (7) (1) Fig. 5-1: Concentration dependence of carbon diffusivity predicted by all the introduced models for a fcc Fe-C binary system at 1000oC. The numbers beside the curves represent the models listed previously: (1) ARRT (Eq. (5-2)), (2) ARRT (Eq. (5-3)), (3) Kaufman et al., (4) Asimow, (5) Siller and McLellan, (6) Tibbetts, (7) Larché and Cahn, (8) Ågren, (9) Liu et al. 75 However, these models need to be evaluated quantitatively when they are extrapolated to the conditions that apply to low temperature carburization processes, i.e., at 450oC and carbon concentrations approaching 0.2 atomic fraction. Fig. 5-2 shows the concentration dependence of carbon diffusivity in a fcc Fe-C binary system at 450oC predicted by all the models. In Fig. 5-2, the normalized ratio of DC / DC’ is again plotted as a function of XC. (9) (3) (8) (5) (4) (6) (2) (7) (1) Fig. 5-2: Concentration dependence of carbon diffusivity predicted by all the introduced models for a fcc Fe-C binary system at 450oC. The numbers above the curves represent the models listed previously: (1) ARRT (Eq. (5-2)), (2) ARRT (Eq. (5-3)), (3) Kaufman et al., (4) Asimow, (5) Siller and McLellan, (6) Tibbetts, (7) Larché and Cahn, (8) Ågren, (9) Liu et al. 76 It is clear that the concentration dependence of carbon diffusivity greatly diverges for the models when they are extrapolated to this much lower temperature, especially for carbon concentrations up to 10-15 at%. An evaluation of some of these models has been published previously [38]. It has been pointed out in [38] and earlier literature [7] that the original models based upon the absolute reaction rate theory [8] underestimate the carbon concentration dependence of the diffusivity of carbon in austenite, which means the concentration dependence of carbon diffusivity cannot be completely attributed to the variation of the activity coefficient of carbon with carbon concentration. The widely used model of Ågren [15] significantly overestimates the concentration dependence when the atomic fraction of carbon is higher than 0.08 [38]. The latest model proposed by Liu et al. [16] predicts too aggressive an enhancement of diffusivity even at carbon concentrations as low as 3 at%. The current study has determined that the model proposed by Asimow [10] provides a reasonable prediction for the concentration dependence of carbon diffusivity in austenite under the conditions relevant to low temperature carburization. Fig. 5-2 demonstrates that based upon Asimow’s model, the diffusion coefficient of carbon in a fcc Fe-C binary system at 450oC within a carbon concentration range of 10-15 at% is about 20-90 times the diffusion coefficient of carbon at infinite carbon dilution. The derivation of Asimow’s parameter K for concentration enhancement is introduced in Section 5.3. Also in Section 5.3 the predicted diffusion coefficient of carbon based upon Asimow’s model is compared with the published result of a Boltzmann-Matano analysis [38] acquired from a carbon concentration-depth profile of treated AISI 316L stainless steel. 77 5.2 Diffusion Coefficient of Carbon at Infinite Dilution The value of DC’ in austenite at the temperature of interest (450oC) is also very important in this analysis. The expressions of DC’ and the corresponding values at 450oC for all of the models discussed are listed in Table 5-1. Table 5-1 Expressions of DC’ for Austenite and the Corresponding Values at 450oC Expression of DC’ Value of DC’ 2 (µm ·s ) at 450oC (µm2·s-1) ARRT [8] Not Specified N/A Kaufman et al. [9] 19275.3 5 107 exp T 1.3×10-4 Asimow [10] Not Specified N/A Siller and McLellan [11][12] 21230 30965.3 T exp T 18619 4.7 107 exp T Model Tibbett [13] Larché and Cahn [14] Ågren [15] Liu et al. [16] -1 Not Specified 1 4.53 105 exp 17767 2.221 104 T 17237.4 1.75 107 exp T 4.0×10-6 3.1×10-4 N/A 5.0×10-4 7.7×10-4 Most of the models defined an expression for DC’ based upon their own formulation procedures and therefore the corresponding values of DC’ at 450oC can be calculated. However, an expression of DC’ is not specified in the original work of ARRT [8], Asimow [10], and Larché and Cahn [14]. 78 According to the data in Table 5-1, the average value of the diffusion coefficient of carbon at infinite carbon dilution in a fcc Fe-C binary system at 450oC is 4.3×10-4 µm2·s-1 based upon the predictions of the analytical models proposed by Kaufman et al. [9], Tibbett [13], Ågren [15], and Liu et al. [16]. The corresponding value of DC’ based upon the model proposed by Siller and McLellan [11][12] is about two orders of magnitude lower than the average. Experimental data for DC’ in a fcc Fe-C binary system also are well represented in the literature [6][7][46]. The latest experimental result in the temperature range 500oC to 900oC has an expression of DC’ as follows [46]: 15050 DC 1.23 106 exp T μm 2 s-1 (5-17) The extrapolated value of DC’ at 450oC based upon Eq. (5-17) is about 1.1×10-3 µm2·s-1, which is 2.5 times the average value predicted by the analytical models. In AISI 316L stainless steel, the diffusion coefficient of carbon at infinite carbon dilution is significantly lower than that in a fcc Fe-C binary system because of the large affinity between carbon and the carbide forming elements, i.e., Cr and Mo [49]. Relevant experimental data also are well recorded in the literature [49] providing an expression for DC’ as follows: 18883 DC 1.9 107 exp T μm 2 s-1 (5-18) 79 The corresponding value of DC’ at 450oC based upon Eq. (5-18) is about 9.4×10-5 µm2·s-1, which is about one twelfth the corresponding value based upon Eq. (5-17) for a fcc Fe-C binary system. 5.3 Application of Asimow’s Model of Carbon Diffusivity to AISI 316L Stainless Steel In order to calculate the diffusion coefficient of carbon in AISI 316L stainless steel based upon Eq. (5-7), the values of DC’, the thermodynamic factor and the parameter K need to be defined. The value of DC’ can be calculated based upon Eq. (518). The derivation of the thermodynamic factor and the corresponding calculation based upon the chemical composition of AISI 316L stainless steel are listed in Appendix III. Therefore, the only unknown parameter is Asimow’s parameter K representing a concentration enhancement. According to Eq. (5-8), the parameter K is expressed as: K 16 3 r03 a03 E VA RT The term r03 a03 is related to the lattice expansion in austenite matrix with a value of 1.68×10-2 [10]. The value of Young’s modulus is calculated using Eq. (5-19) [43]: E 205.91 2.691 10 2 T 4.188 105 T 2 GPa (5-19) 80 and E = 164.6 GPa at 450oC for AISI 316L stainless steel. Therefore, in order to calculate the value of the parameter K, the activation volume associated with carbon diffusion in austenite ΔVA needs to be determined. 5.3.1 Activation Volume Physically, the activation volume associated with carbon diffusion in austenite represents the change in volume of the austenite matrix when one mole of interstitial carbon atoms simultaneously moves from equilibrium sites to “activated sites” [50]. Relevant experimental results do exist in the literature, although the research on this subject is extremely challenging. Based upon the study of Keiser and Wuttig [50], the value of ΔVA associated with carbon diffusion in austenite is essentially within the range of 1.3-3.9 cm3·mol-1. Empirically, the value of ΔVA also can be calculated according to Keye’s model [44][45] as expressed by Eq. (5-9): VA 2 1 / 3 GA where δ is the Grüneisen parameter, χ is the compressibility and ΔGA is the activation Gibbs free energy associated with carbon diffusion. 5.3.1.1 Compressibility χ The compressibility of the austenite matrix is calculated as follows [10]: 3 1 2 E (5-20) 81 where ν is the Poisson’s ratio and ν = 0.27 for austenite [48]. Based upon the value of E, the compressibility of AISI 316L stainless steel at 450oC is 8.39×10-12 Pa-1. 5.3.1.2 Grüneisen Parameter δ The Grüneisen parameter of the austenite matrix is defined as [10]: 3 T CV (5-21) The relevant parameters in the expression of the Grüneisen parameter are listed in Table 5-2 [10][43][48]. The calculated result for the Grüneisen parameter δ at 450oC based upon the relevant parameters in Table 5-2 is 1.44. Table 5-2 The Relevant Parameters of the Grüneisen Parameter δ Parameter Derivation Value at 450oC Density, ρ (g·cm-3) --- 7.96 Compressibility, χ (Pa-1) 3 1 2 E 8.39×10-12 Thermal Expansion Coefficient, αT (106·K-1) T 11.81 1.311 102 T Heat Capacity at Constant Pressure, CP (J·kg-1·K-1) C P 365.43 0.406 T Heat Capacity at Constant Volume, CV (J·kg-1·K-1) 6.137 106 T 2 1.732 10 4 T 2 T 2 CV C P T 18.1 568.8 565.2 82 5.3.1.3 Activation Gibbs Free Energy ΔGA The activation Gibbs free energy associated with carbon diffusion in austenite ΔGA is defined in Zener’s model [51] as follows: T GA H A T S A H A 1 Tm (5-22) where ΔHA is the activation enthalpy, ΔSA is the activation entropy, Tm is the melting temperature of austenite and the parameter κ is the dimensionless thermal coefficient of Young’s modulus. The values of the parameters employed to calculate ΔGA are listed in Table 5-3: Table 5-3 The Relevant Parameters of the Activation Gibbs Free Energy ΔGA for AISI 316L Stainless Steel Parameter Value Reference Activation Enthalpy, ΔHA (kJ·mol-1) 154.8 [49] Melting Temperature, Tm (K) 1643 – 1673 [43] Dimensionless Thermal Coefficient of Young’s Modulus, κ 0.45 [10] Using the average of the melting temperature 1658 K, the calculated value of ΔGA associated with carbon diffusion in AISI 316L stainless steel at 450oC based upon the parameters in Table 5-3 is 124.4 kJ·mol-1. 83 Therefore, the activation volume associated with carbon diffusion in AISI 316L stainless steel at 450oC ΔVA based upon the known parameters is 2.3 cm3·mol-1, which agrees with the experimental data in the literature [50]. 5.3.2 The Value of the Parameter K According to the above analysis, the value of Asimow’s parameter K representing a concentration enhancement for carbon diffusion in AISI 316L stainless steel at 450oC can be calculated using Eq. (5-8). The corresponding value of K is 18. The diffusion coefficient of carbon in AISI 316L stainless steel at 450oC as a function of carbon concentration is now established, considering the value of DC’ (Eq. (5-18)) and the thermodynamic factor (Appendix III). The comparison between the calculated concentration dependence of carbon diffusivity based upon Asimow’s model and the published data by Ernst et al. [38] obtained from a Boltzmann-Matano analysis of low-temperature-carburized AISI 316L stainless steel is shown in Fig. 5-3. The results based upon other analytical models, such as Ågren [15] and ARRT [8], also are included for comparison. It is obvious that the concentration dependence of carbon diffusivity in AISI 316L stainless steel increases first slowly and then more rapidly with increasing carbon concentration. Asimow’s model [10] exhibits an excellent agreement with the Boltzmann-Matano data of carbon diffusivity for a concentration level up to 14 at%, which is much better than the results predicted by other analytical models, such as Ågren and ARRT. Experimentally, it is not clear how the diffusion coefficient of carbon changes with carbon concentration beyond the concentration range in this study. 84 (2) (3) (1) (4) Fig. 5-3: Concentration dependence of carbon diffusivity in AISI 316L stainless steel at 450oC based upon: (1) a Boltzmann-Matano analysis by Ernst et al., and the introduced models (2) Ågren, (3) Asimow, and (4) ARRT (Eq. (5-3)). 5.4 Physical Origin of Concentration Dependent Diffusion The physical origin of the concentration dependent diffusion of carbon in austenite has been thoroughly discussed in the previous publication by Ernst et al. [38]. Several potential mechanisms that might have an impact on diffusion enhancement were discussed: (1) the lattice parameter expansion induced by a high concentration of interstitially dissolved carbon in solution, (2) the elastic repulsion between neighboring carbon atoms, (3) the gradient in biaxial compressive stress that arises from a gradient in interstitial carbon concentration, (4) pipe diffusion along the dislocations generated by plastic yielding of the material, (5) carbon trapping by chromium. 85 After ruling out several potential mechanisms of diffusion enhancement [38], it was pointed out that the significantly faster carbon diffusivity at high carbon concentrations is most likely because the interstitially dissolved carbon greatly decreases the activation energy for diffusion by inducing a lattice parameter expansion in the metal matrix. According to the explanation of Asimow [10], the interstitial carbon generates an internal compressive stress that does induce a lattice parameter expansion decreasing the activation energy associated with carbon diffusion in an austenite matrix. The current study proves that there is an excellent agreement between the quantitative description of the concentration dependence of the diffusion coefficient of carbon based upon Asimow’s model and the published data for the diffusion coefficient of carbon in carburized AISI 316L stainless steel. In addition, results of numerical simulations also show excellent agreement with the experimental data from carburization treatments conducted at 450oC. Comparisons between numerical simulations of carbon diffusion profiles and those obtained from carburization treatments are presented in the next chapter. 86 Chapter 6 Simulation of the Diffusion of Carbon into Austenite During low temperature carburization of AISI 316L stainless steel, the diffusion of carbon into an austenite matrix was simulated by a one-dimensional diffusion into a semi-infinite system as shown by Fig. 3-1. The simulation was based upon a finite difference algorithm using fully implicit scheme that was outlined in Chapter 3. Different situations based upon the boundary conditions and the diffusion coefficients need to be considered during a simulation. Simulated concentration-depth profiles are shown for the following four types of situations: (1) a fixed boundary condition and a concentration-independent diffusion coefficient, (2) a convective boundary condition and a concentration-independent diffusion coefficient, (3) a fixed boundary condition and a concentration-dependent diffusion coefficient, and (4) a convective boundary condition and a concentration-dependent diffusion coefficient. The simulation results based upon Asimow’s analytical model for the concentration dependence of the carbon diffusion coefficient in austenite were compared with the experimental data obtained from AES and XPS analyses in this chapter. 87 6.1 Four Types of Simulation 6.1.1 Fixed Boundary Condition and Concentration-Independent Diffusion Coefficient If the mass transfer rate at the gas-metal interface is extremely high, in which case the value of the mass transfer coefficient α approaches positive infinity, a fixed boundary condition expressed by Eq. (3-31) is used, and the time for the surface concentration to achieve the maximum concentration is close to zero. In addition, if the diffusion coefficient is independent of concentration, i.e., the diffusion coefficient at infinite dilution, the value of D will be a constant at a certain temperature, i.e., D = D’. When the boundary condition is fixed and the diffusion coefficient is independent of concentration, the concentration-depth profile is analytically defined by an errorfunction solution to Fick’s second law [30] given by Eq. (6-1): x c x, t c 1 erf 2 Dt (6-1) Numerically, the concentration-depth profile can be simulated using a finite difference algorithm. In order to simplify the simulation, the following parameters are assumed as dimensionless numbers: D 1 t 1 (6-2) x 1 Therefore, in this simulation, the parameter r D t / x 1 . As mentioned in 2 Section 3.3, in the direction of diffusion, the semi-infinite domain was discretized into 88 100 pieces with 101 grid points. Therefore, the boundary condition in this simulation can be expressed as: c0m c m c100 co (6-3) where the values of c∞ and co can be assumed to be 1 and 0, respectively. The simulated concentration-depth profiles for a series of time steps based upon the fully implicit scheme with a fixed boundary condition and a concentrationindependent diffusion coefficient are shown in Fig. 6-1. In Fig. 6-1, the x ordinate represents the distance of diffusion and the scales on the x ordinate represent the number of the distance steps x . The y ordinate represents the ratio between the concentration at a grid point and the maximum concentration at the gas-metal interface where i = 0. Because the boundary condition is fixed by Eq. (6-3), the concentration at the gas-metal interface c0m equals a constant c , i.e., c0m c 1 , for all the time steps. In addition, the m co 0 , for all the concentration at the grid point i = 100 equals a constant co, i.e., c100 time steps. The simulated concentration-depth profiles after 10, 50, 100, 200, 300, 400 and 500 time steps all have a convex shape defined by an error function solution, which indicates that the diffusion coefficient is independent of the concentration. 89 The Number of Time Steps x Fig. 6-1: Concentration-depth profiles based upon the fully implicit scheme with a fixed boundary condition and a concentration-independent diffusion coefficient. 6.1.2 Convective Boundary Condition and Concentration-Independent Diffusion Coefficient If the mass transfer at the gas-metal interface is less than perfectly efficient, i.e., the passive film on the surface of stainless steels is not completely removed, the value of the mass transfer coefficient α will be finite instead of being infinite. Therefore, a convective boundary condition defined by Eq. (3-32) is applied to the mass transfer at the gas-metal interface. In addition, the diffusion coefficient is again independent of the concentration, i.e., D = D’. 90 When the boundary condition is convective and the diffusion coefficient is independent of concentration, the concentration-depth profile is defined by the following analytical solution expressed by Eq. (6-4) [32]: x 2 Dt c x, t c erfc x t exp D x t erfc D 2 Dt (6-4) where the function “erfc” stands for the complementary error function and is defined as erfc(z) = 1 – erf(z) [32]. Numerical solution can also be obtained based upon the following parameters which are again assumed to be simple and dimensionless numbers: D 1 t 1 x 1 c 1 (6-5) co 1 0.05 It should be noted that a finite value has been assigned for the mass transfer coefficient α. During this simulation, the convective boundary condition at the grid point i = 0 is defined by Eq. (3-35) with the following parameters based upon the values in Eq. (6-5): r 1 1.05 0.05 (6-6) The boundary condition at the grid point i = 100 is again fixed as indicated in Eq. (6-3). 91 The simulated concentration-depth profiles for a series of time steps based upon the fully implicit scheme with a convective boundary condition and a concentrationindependent diffusion coefficient are shown in Fig. 6-2. The Number of Time Steps x Fig. 6-2: Concentration-depth profiles based upon the fully implicit scheme with a convective boundary condition and a concentration-independent diffusion coefficient. In Fig. 6-2, it is obvious that the concentration depth profiles are still convex which is in accord with a diffusion coefficient that is independent of the concentration. However, due to the convection at the gas-metal interface, the concentration on the boundary increases gradually with time and does not achieve the maximum concentration c∞ after 500 time steps based upon the finite value of the mass transfer coefficient α. The larger the value 92 of the mass transfer coefficient α, the less time the concentration at the interface needs to achieve the maximum concentration c∞. When the value of α approaches positive infinity, the convection at the interface is so dramatic that a fixed boundary condition applies. 6.1.3 Fixed Boundary Condition and Concentration-Dependent Diffusion Coefficient When the diffusion coefficient depends upon the concentration, the shape of the concentration-depth profile deviates from that defined by an error function solution. It has been introduced in Chapter 3 that a concentration-dependent diffusion coefficient can be empirically described by a simple exponential function as Eq. (3-7), ignoring the subscript C for all the parameters in Eq. (3-7): c D D exp k cmax where the parameter k controls the magnitude of the concentration dependence of the diffusion coefficient. Eq. (3-7) suggests that at the maximum concentration, i.e., c = cmax, the diffusion coefficient D is exp(k) times the diffusion coefficient at infinite dilution D’. When the boundary condition is fixed as in Section 6.1.1, the concentration-depth profiles can be obtained by either a numerical approach or a classical method, e.g., a Boltzmann-Matano analysis [30][38]. To obtain the concentration-depth profiles numerically, a finite difference algorithm considering the concentration dependence of the diffusion coefficient introduced in Section 3.6 is employed. The simulation is based upon the following parameters with units as listed: 93 D 9.4 105 μm 2 s 1 t 60 s x 0.5 μm c cmax 1.71 10 co 0 mol μm 14 mol μm (6-7) 3 3 where the value of D’ is the diffusion coefficient of carbon at infinite carbon dilution in AISI 316L stainless steel at 450oC based upon Eq. (5-18). The values of c∞ and cmax correspond to the maximum paraequilibrium solubility of carbon in AISI 316L stainless steel at 450oC under unit activity of carbon, which is about 11 at% [52], and is calculated as follows: c cmax XC 1 X C Vm (6-8) where Vm is the molar volume of the austenite matrix (~ 7 cm3·mol-1). Simulated and normalized concentration-depth profiles based upon a fixed boundary condition and concentration-dependent diffusion coefficients are shown in Fig. 6-3. In Fig. 6-3, the numbers above the concentration-depth profiles are equivalent to the values of exp(k) ranging from 1 to 1000. In addition, the x ordinate representing the distance previously is normalized as a dimensionless term x / 4 Dt , which indicates the following important properties for diffusion in semi-infinite systems [32]: 94 x 4D t Fig. 6-3: Simulated and normalized concentration-depth profiles based upon the fully implicit scheme with a fixed boundary condition and concentration-dependent diffusion coefficients. (1) The distance of diffusion of any given concentration is proportional to the square root of time. (2) The time required for any point to reach a given concentration is proportional to the square of its distance from the surface and varies inversely as the diffusion coefficient. (3) The amount of diffusing substance entering the medium through unit area of its surface varies as the square root of time. In Fig. 6-3, it is obvious how the shape of concentration-depth profiles changes with the value of exp(k). When the value of exp(k) is equivalent to 1, the diffusion 95 coefficient at the maximum concentration c∞ equals the diffusion coefficient at infinite dilution, in which case the concentration-depth profile exhibits a convex shape defined by an error function solution. With the increasing value of exp(k), the shape of the concentration-depth profile becomes more and more concave, which indicates the concentration enhancement of the diffusion coefficient increases. When the value of exp(k) increases up to several hundred, the concentration-depth profiles exhibit a shape with a plateau and a very steep leading edge, which is often observed in the nitrogen concentration-depth profiles generated by a low temperature plasma nitridation [17][18]. 6.1.4 Convective Boundary Condition and Concentration-Dependent Diffusion Coefficient Similar to the situation in Section 6.1.2, the mass transfer at the gas-metal interface may be inefficient, in which case the mass transfer coefficient α has a finite value. The mass transfer coefficient can restrict the surface concentration needed to achieve the maximum possible concentration, and therefore limit the concentration enhancement of the diffusion coefficient. When the boundary condition is convective as indicated in Eq. (3-32) and the diffusion coefficient is dependent upon the concentration as in Eq. (3-7), the concentration-depth profile can only be obtained by a numerical approach [32]. A finite difference simulation was performed to generate the concentration-depth profiles for a 20-hour diffusion process based upon convective boundary conditions and a concentration-dependent diffusion coefficient. All of the parameters employed in this simulation are listed as follows: 96 DC 9.4 105 μm 2 s 1 t 60 s x 0.5 μm c cmax 1.71 1014 mol μm-3 (6-9) co 0 mol μm-3 e k 50 where the value of exp(k) was taken to be 50. The simulated concentration-depth profiles based upon the parameters listed above with convective boundary conditions and a concentration-dependent diffusion coefficient are shown in Fig. 6-4. In Fig. 6-4, the numbers besides the concentration- 103 102 105 104 x μm Fig. 6-4: Simulated concentration-depth profiles based upon the fully implicit scheme with convective boundary conditions and a concentration-dependent diffusion coefficient. 97 depth profiles represent the values of the mass transfer coefficient α ranging from 10-5 to 10-2 µm·s-1. When the mass transfer coefficient has a value of 10-5 µm·s-1, the mass transport at the gas-metal interface is so inefficient that the concentration-depth profile does not exhibit the concave shape which should be observed due to the factor of 50 concentration enhancement applied to the diffusion coefficient. The reason is that the limited value of the mass transfer coefficient α greatly restricts the surface concentration and the maximum concentration that can be achieved. When the mass transfer coefficient increases up to 10-2 µm·s-1, the mass transport at the interface becomes very efficient, the profile can exhibit the concave shape at the maximum concentration and is very close to the profile obtained under a fixed boundary condition and a concentration-dependent diffusion coefficient. The role of the mass transfer coefficient will be further discussed in Chapter 8. 6.2 Simulation of the Swagelok Process 6.2.1 Swagelok Process of Low Temperature Carburization The low temperature carburization procedure developed by the Swagelok Company has been studied during the past ten years [3, 4, 27, 28, 37, 38, and 52]. This treatment procedure can produce an interstitially hardened case in AISI 316L stainless steel with a surface carbon concentration ranging from 10 to 15 at% and a penetration depth of about 20 to 25 µm. 98 The typical carburization procedure employed in this study is schematically shown in Fig. 6-5. The treatment process includes a double surface activation of 3 hours each conducted at 250oC. The objective of the surface activation treatment is to remove the Cr2O3-rich passive layer on the surface of an AISI 316L stainless steel so that the surface becomes transparent to the diffusion of carbon. The gas flow during the surface activation treatment is about 90% HCl and 10% N2. Low temperature paraequilibrium carburization is performed in a divided procedure conducted at 450oC. The first carburization step conducted between the double surface activation lasts 4 hours with a gas flow of 46% CO, 46% H2 and 8% N2. The second carburization step conducted after 15 30 Fig. 6-5: Schematic drawing of the typical low temperature carburization procedure developed by the Swagelok Company. 99 the double surface activation lasts 39 hours and is followed by a very rapid fan-cooling step. The variation of the gas flow during the second carburization step is shown in Fig. 6-6. In Fig. 6-6, the gas compositions of CO, H2 and N2 are shown as volume fractions. H2 N2 CO Fig. 6-6: The gas flow during the second carburization step of 39 hours. The gas flow starts with a composition of 46% CO, 46% H2 and 8% N2, which is the same composition employed during the first carburization step, and lasts for 0.5 hour. Then the content of CO gradually decreases to 10% during the following 11.5 hours while the content of H2 is maintained at 50% and the content of N2 increases correspondingly. It should be noted that the second carburization treatment ends with a one-hour purging step with a gas flow of 90% H2 and 10% N2. The objective of this 100 purging step is to effectively eliminate the undesirable soot produced during the carburization process and restore a shiny appearance to the metal surfaces. 6.2.2 Maximum Possible Solubility of Carbon XC, max In order to simulate a concentration-depth profile of carbon, the maximum possible solubility of carbon XC,max in carburized AISI 316L stainless steel was calculated using the CALPHAD model introduced in Chapter 2. It has been addressed in a previous publication [52] that the CALPHAD Cr-carbon interaction parameters are not sufficiently exothermic at the low temperatures used for paraequilibrium carburization, i.e., 450oC. The calculation based upon the CALPHAD Cr-carbon interaction parameters [25] predicts a paraequilibrium carbon solubility of ~ 5.5 at% in AISI 316L stainless steel at 450oC [52]. However, the Swagelok procedure of low temperature carburization can result in a much higher concentration of carbon in AISI 316L stainless steel, ~ 10 to 15 at% [4][38]. Therefore, an adjustment was applied to the CALPHAD Cr-carbon interaction parameters [28] based upon the most comprehensive experimental study of this system [53]. The following CALPHAD parameters were obtained after the adjustment: o hfcc hfcc GCr:C oGCr:Va oGCgra 28722 6.83774T 0 fcc Cr:C,Va 5267.24 2.9067T 0 fcc Cr,Fe:C 246224.5 166.4358T L L (6-10) The paraequilibrium carbon solubility in AISI 316L stainless steel at 450oC for unit activity of carbon is ~ 11at% based upon the adjusted parameters and the following relationship [28]: Csol. Cgra. RT ln aC (6-11) 101 where Csol. is the chemical potential of carbon in AISI-316L-based interstitial solid solution, Cgra. is the chemical potential of carbon in graphite form and aC is the activity of carbon. Therefore, for a unit activity of carbon, i.e., aC = 1, Eq. (6-11) becomes: Csol. Cgra. (6-12) The maximum possible solubility of carbon XC,max, was obtained by substituting Csol. within Eq. (2-38) and then solving Eq. (6-12). The calculated result of ~ 11 at% carbon based upon the adjusted parameters gives a much better agreement with the experimental results than that based upon the parameters in CALPHAD literature [25]. 6.2.3 Activity of Carbon aC in the Carburization Atmosphere From the analysis in Section 6.2.2, it is obvious that under paraequilibrium conditions, the value of XC,max is determined by the activity of carbon aC in the carburization atmosphere. The activity of carbon aC in the carburization atmosphere at 450oC under the standard pressure of one atmosphere is calculated using the professional software, FactSageTM 5.3. The calculated values of aC based upon the gas compositions with the maximum and minimum CO contents during the carburization process are listed in Table 6-1. The calculation is based upon two assumptions: (1) a very modest amount of water vapor is involved in the equilibria of the treatment atmosphere (~ 0.01%), (2) there is no solid phase involved in the calculation of equilibria. The validity of the second assumption needs to be discussed here. 102 Table 6-1 The Activity of Carbon during the Carburization Process with the Maximum and Minimum CO Contents (Volume Percent) CO H2 N2 aC 46% 46% 8% 49.6 10% 50% 40% 0.9 The calculation shows that at the beginning of the carburization treatment, the activity of carbon is about 50 in the carburization atmosphere, which indicates that there is a significant driving force to precipitate a solid phase of carbon on the surface of the metal, such as the soot mentioned in the Section 6.2.1. On one hand, once the soot starts to form on the surface of the metal, the activity of carbon aC will gradually decrease to one within the boundary layer of the gas flow and the paraequilibrium carburization treatment will actually be performed under a unit activity of carbon. On the other hand, under paraequilibrium conditions, a higher solubility of carbon in treated AISI 316L stainless steel can be produced if a higher activity of carbon is achieved in the carburization atmosphere. For instance, if the activity of carbon is about 50, the corresponding paraequilibrium solubility of carbon based upon Eq. (6-11) increases to ~ 28 at%. Empirically, such a high solubility of carbon has never been observed during the experimental procedures in this study or any other study on AISI 316L stainless steel carburized using the Swagelok procedure [4][38]. The maximum carbon concentration in carburized AISI 316L stainless steel has been widely studied using a variety of characterization techniques, such as X-Ray Diffraction (XRD), X-Ray Photoelectron Spectroscopy (XPS), Glow-Discharge Optical Emission Spectroscopy (GDOES), Auger 103 Electron Spectroscopy (AES) and Local-Electrode Atom Probe (LEAP), by which the measured maximum carbon concentrations are always within the range of 10 to 15 at% [4][38]. Considering that the relative systematic error of the characterization techniques is no larger than ~ 5 at% [38], the activity of carbon actually achieved during the paraequilibrium carburization treatment is likely to be about one. Therefore, in the simulation of carbon concentration-depth profiles, the maximum concentration of carbon in AISI 316L stainless steel at 450oC is taken to be 11 at%, which corresponds to a unit activity of carbon. 6.2.4 Simulation Parameters The simulation of the concentration-depth profile of carbon in AISI 316L stainless steel carburized using the typical Swagelok process introduced in Section 6.2.1 was conducted employing the finite difference algorithm with a fully implicit scheme. The relevant parameters used in this simulation are listed in Table 6-2. Table 6-2 Simulation Parameters for Carbon Concentration-Depth Profile in AISI 316L Stainless Steel Temperature 450oC Analytical Model of DC Asimow [10] DC′ 9.4×10-5 µm2·s-1 [49] XC,max 11 at% [28] Δt 60 s Δx 0.4 µm 104 The carburization procedure includes an initial carburization step of 4 hours and a second carburization step of 38 hours. The purging step during the last one hour is temporarily ignored at this time. The mass transfer coefficient α is expected to be different during the two carburization steps. Empirically, the first surface activation of 3 hours conducted at 250oC is insufficient to remove the passive layer on the surface of AISI 316L stainless steel, which indicates that the mass transfer coefficient during the following four-hour carburization step may have a relatively low value. After the second activation step, the surface essentially becomes much more transparent to the diffusion of carbon [4][38], which indicates that the value of the mass transfer coefficient α during the second carburization step may be much higher than that during the first carburization step. In the simulation, the mass transfer coefficients, α1 and α2 for two carburization steps respectively, are evaluated as fitting parameters to generate the carbon concentrationdepth profile which provides the most satisfactory agreement with the experimental profile obtained from AES line scans. 6.2.5 AES Results A standard elemental depth profiling was performed using the Auger electron spectrometer as described in Chapter 4. The experiment was conducted in the line scan mode under the condition of continuous sputtering. The line scan was performed on the cross-sectional surface of a well-polished specimen of AISI 316L stainless steel along a line perpendicular to the carburized surface, across the hardened case, and into the core of the material. 32 sampling points were equally spaced along the scan line with the spacing being about 1.6 µm. The cross-sectional surface of the specimen was pre-sputtered to eliminate any undesirable surface contamination, during that process about 50 nm of the 105 surface was removed. The Auger electron spectra obtained were analyzed using the Multipak software. The energies of the Auger peaks for the principal elements in the carburized specimen are listed in Table 6-3: Table 6-3 The Energies of the Auger Peaks for the Principal Elements in Carburized AISI 316L Stainless Steel Element C Fe Cr Ni Mo Analyzed Auger Transition KLL LMM LMM LMM MNN Energy Peak in Multipak C1 Fe2 Cr2 Ni1 Mo1 Peak Energy (eV) 275 654 531 849 190 The default settings for the baselines for the Auger peaks in the Multipak software are very accurate for most elements. However, it should be noted that the baseline settings need to be adjusted when there are interactions between the Auger peaks of different elements. In this study, the energy peaks of Cr and Mo needed to be adjusted as follows. Firstly, the Auger peaks for the Cr LMM Auger transition, Cr1 at 491 eV and Cr2 at 531 eV, interact with the O KLL Auger transition, O1 at 510 eV. Therefore, the Auger peak of Cr2 was analyzed independently in order to avoid the overlap with the Auger peak of O1. Secondly, the default setting of the peak for the Mo MNN Auger transition is Mo3 at 225 eV, which interacts with Ar LMM Auger transition, Ar1 at 219 eV. During the experiment, the cross-sectional surface of the specimen was continuously sputtered under an Ar ion flow, causing a considerable amount of Ar to be detected 106 during the line scan process. Based upon this consideration, the peak for the Mo MNN Auger transition, Mo1 at 190 eV was analyzed, instead of the Mo3 peak, to provide a better quantification for Mo. 6.2.5.1 The Composition of Metal Elements The quantified concentration-depth profiles of the metal elements based upon the analysis of Auger spectra are shown in Fig. 6-7, in which the compositions have been normalized to exclude carbon. The data points, either sampled off the edge of the specimen at the beginning of the line scan or sampled beyond 40 µm depth from the edge at the end of the line scan, are ignored in Fig. 6-7. Fe Cr Ni Mo Fig. 6-7: The normalized concentration-depth profiles of the metal elements in AISI 316L stainless steel obtained from Auger line scan. 107 It is obvious that the average of the normalized composition of the metal elements measured by AES line scan agrees with the normalized nominal composition of AISI 316L stainless steel, both of which (in at%) are listed in Table 6-4. Table 6-4 The Average of the Normalized Composition of the Metal Elements Measured by AES Line Scan and the Normalized Nominal Composition of AISI 316L Stainless Steel (at%) Element Fe Cr Ni Mo Nominal Composition 66.9 18.2 11.4 1.5 Average of Measured Composition 67.8 19.2 10.8 2.2 Standard Deviation of Measured Composition 1.0 1.3 0.4 0.4 6.2.5.2 Concentration-Depth Profile of Carbon The corresponding concentration-depth profile of carbon, obtained from the analysis of Auger spectra and calibrated using Eq. (4-1) for the continuous sputtering mode, is shown in Fig. 6-8. 108 Fig. 6-8: The calibrated concentration-depth profiles of carbon in AISI 316L stainless steel obtained from Auger line scan. After the calibration, the data points sampled beyond 22 µm depth oscillate at XC = 0 within ± 1.0 at%. These data points are normalized to be zero, considering the low accuracy of the measurements at such a low concentration level of carbon and their modest significance in the evaluation of the numerical simulation. The measured carbon concentration-depth profile obviously has a concave shape as discussed in Chapter 5. The typical Swagelok procedure produced a carbon interstitially hardened case in AISI 316L stainless steel with a surface concentration of ~ 11 at% and a thickness of ~ 22 µm. 6.2.5.3 Comparison with Numerical Simulation The carbon concentration-depth profile corresponding to the typical Swagelok procedure indicated in Section 6.2.1 was simulated using the parameters listed in Section 6.2.4. The simulated concentration-depth profile for carbon with the optimized values for 109 the mass transfer coefficients α fitting the experimental data in Fig. 6-8 is shown in Fig. 6-9. Table 6-5 lists the optimized values for the mass transfer coefficients α fitting the experimental data in Fig. 6-8, the normalized sum χ2 of residuals, the degrees of freedom (d.f.), and the probability of the simulation being correct (P). The fitting procedure was applied to the data points with a carbon concentration XC > 0. Fig. 6-9: The simulated concentration-depth profile of carbon in AISI 316L stainless steel with the optimized values for the mass transfer coefficients α fitting the experimental data in Fig. 6-8. 110 Table 6-5 The Optimized Values for the Mass Transfer Coefficients α Fitting the Experimental Data in Fig. 6-8, Normalized Sum χ2 of Residuals, the Degrees of Freedom (d.f.), and the Probability of the Simulation Being Correct (P) Parameter α1 (µm·s-1) α2 (µm·s-1) χ2 d.f. P Value 0.001 0.1 0.4076 12 0.99999992 Based upon the parameters in Table 6-5, the simulation is in excellent agreement with the experimental data, where the concave shape of the concentration-depth profile for carbon is accurately predicted by Asimow’s model for the concentration-dependent diffusion coefficient of carbon in AISI 316L stainless steel. Secondly, the simulation result is greatly influenced by the mass transfer coefficient α2 for the second carburization step of 38 hours. The relatively large value of α2 indicates that the second activation is very effective in eliminating the passive layer on the surface of AISI 316L stainless steel. Finally, the relatively small value of α1 indicates lack of effectiveness of the first activation step of 3 hours conducted at 250oC, although its influence on the concentration-depth profile is much less than the mass transfer coefficient α2, considering the short period (4 hours) of the first carburization step. Further evaluation on the mass transfer coefficient α1 during a procedure aimed at achieving better surface activation is discussed in Chapter 8. 6.2.6 XPS Results In the standard Swagelok procedure shown by Fig. 6-6, there is a one-hour purging step performed after the second carburization treatment to eliminate the soot accumulated on the surface of treated materials. As introduced in Section 6.2.1, the gas 111 flow during the purging step is comprised of 90% H2 and 10% N2. That step may cause decarburization in the carburized surface layer, during which the interstitially dissolved carbon atoms form CH4 and leave the surface. This process will cause a decrease in the maximum carbon concentration in the surface layer and potentially degrade properties of the surface layer. In order to study the kinetics of decarburization, a carbon concentration-depth profile within the first 2-3 µm of the case was acquired using an X-ray photoelectron spectrometer as discussed in Chapter 4. The experiment was conducted in the depth profiling mode with a series of alternate inert-gas sputtering and chemical composition surveys. The depth profiling process was performed in the direction perpendicular to the carburized surface, based upon an adjusted sputtering rate of 20 nm·min-1 combined with 2 cycles of the chemical composition surveys at each sputtered depth of about 100 nm. Considering the roughness of the carburized surface, a pre-sputtering step was applied on the surface of the specimen to eliminate surface contamination, especially adventitious carbon, during which process about 110 nm of material were removed. The binding energy values and the corresponding baseline settings for the XPS energy peaks analyzed in this experiment are listed in Table 6-6. The X-ray photoelectron spectra for the principal elements in carburized AISI 316L stainless steel are shown in Fig. 6-10, in which the spectra at the bottom always represent the survey results obtained from the least depth of sputtering. 112 Table 6-6 The Binding Energy Values and the Corresponding Baseline Settings for the XPS Energy Peaks Analyzed in Depth Profiling Experiment of AISI 316L Stainless Steel Element Fe Cr Ni Mo C Analyzed Peaks Fe2p1 Cr2p Ni2p3 Mo3d C1s Binding Energy (eV) 723.0 Baseline Settings (eV) 715.6 – 735.0 Cr2p1 Cr2p3 583.0 574.0 569.6 – 597.2 855.0 850.0 – 863.7 Mo3d3 Mo3d5 231.0 228.0 225.4 – 235.6 285.0 280.2 – 291.0 Fe (a) 113 Cr (b) Ni (c) 114 Mo (d) C (e) Fig. 6-10: The obtained X-ray photoelectron spectra for the principal elements in carburized AISI 316L stainless steel: (a) Fe2p1, (b) Cr2p, (c) Ni2p3, (d) Mo3d, (e) C1s. 115 6.2.6.1 The Composition of Metal Elements The quantified concentration-depth profiles of the metal elements based upon the analysis of X-ray photoelectron spectra are shown in Fig. 6-11, in which the compositions have been normalized to exclude carbon. Fe Cr Ni Mo Fig. 6-11: The normalized concentration-depth profiles of the metal elements in AISI 316L stainless steel obtained from XPS depth profiling. Fig. 6-10 shows that no shift in peak energy was detected for the metal elements during the depth profiling procedure, and Fig. 6-11 shows that the metal compositions at different depths are very uniform. These results indicate that the depth profiling procedure was performed in one phase of an interstitial solid solution. It is also obvious that the average of the normalized composition of the metal elements measured by XPS 116 depth profiling is in an excellent agreement with the normalized nominal composition of AISI 316L stainless steel, both of which (in at%) are listed in Table 6-7. Table 6-7 The Average of the Normalized Composition of the Metal Elements Measured by XPS Depth Profiling and the Normalized Nominal Composition of AISI 316L Stainless Steel (at%) Element Fe Cr Ni Mo Nominal Composition 66.9 18.2 11.4 1.5 Average of Measured Composition 68.9 17.9 10.9 2.3 Standard Deviation of Measured Composition 0.7 0.4 0.5 0.1 6.2.6.2 Concentration-Depth Profile of Carbon In X-ray photoelectron spectra, the binding energy of carbon in adventitious form is about 284.8 eV. This value shifts to about 282.0 eV – 283.0 eV for carbon in carbidic form or carbon in solid solution [4]. In Fig. 6-10, it is obvious that the spectra of carbon consistently align at the peak energy of 283 eV and the influence of adventitious carbon is considerable just for the first 2 or 3 spectra, which indicates that the ~ 110 nm presputtering step effectively removed most of the adventitious carbon residue on the surface. Using the Target Factor Analysis (TFA) function in the Multipak software, the carbon energy peaks at 283 eV were extracted from all the carbon spectra and a corresponding concentration-depth profile of carbon, representing the carbon concentration in solid 117 solution, was obtained as shown in Fig. 6-12. The first three data points of the concentration-depth profile measured by AES in Fig. 6-8 are also shown in Fig. 6-12 for comparison. Fig. 6-12: The concentration-depth profile of carbon in AISI 316L stainless steel obtained from XPS depth profiling and AES line scan. The carbon concentration-depth profile within the surveyed depth has a nearly flat shape, which is consistent with the curvature of the carbon concentration-depth profile measure by AES. This result indicates that there is no significant decarburization in the surface layer. The average concentration of carbon within the first 2-3 µm depth measured by XPS is about 9.5 at%, which is about 1.5 at% lower than the concentration value measured by AES. A systematic error may exist between this two characterization techniques as mentioned previously [38], which is further discussed in Appendix II. 118 6.2.6.3 Comparison with Numerical Simulation A numerical simulation based upon a finite difference algorithm was performed to evaluate how a carbon concentration-depth profile changes during the one-hour purging step. The simulation parameters were the same as used in Section 6.2.4 except that the maximum concentration of carbon XC,max is zero in this simulation, because there is no carbon-bearing gas flow during the purging process. In this case, a positive value of the mass transfer coefficient α will generate a mass transport of carbon from the AISI 316L stainless steel to the atmosphere based upon Eq. (3-32). The simulation results are shown in Fig. 6-13. (a) (b) (c) Fig. 6-13: Simulation results: (a) carbon concentration-depth profile obtained from the typical Swagelok procedure before the purging step is performed, (b) carbon concentration-depth profile simulated including the one-hour purging step, based upon the assumption that α = 0 µm·s-1 during that one hour, (c) carbon concentration-depth profile simulated with the inclusion of the one-hour purging step, based upon the assumption that α = 10-4 µm·s-1 during that one hour. 119 In Fig. 6-13, profile (a) is the carbon concentration-depth profile after the typical Swagelok treatment procedure before the purging step is performed, where the carbon concentration decreases monotonously from the surface. Profile (b) is the carbon concentration-depth profile simulated including the one-hour purging step, based upon the assumption that the mass transfer coefficient α equals zero during that one hour, in which case the surface concentration of carbon decreases a little and the concentrationdepth profile becomes nearly flat. Profile (c) is the carbon concentration-depth profile simulated with the inclusion of the one-hour purging step, based upon the assumption that the mass transfer coefficient α has a finite positive value of 10-4 µm·s-1, in which case the surface concentration decreases to a much lower level than the concentration in the deeper case due to the outward mass transport at the interface. Comparing with the experimental data in Fig. 6-12, it is not difficult to see that the outward mass transport at the interface during the purging step has such a modest rate that the mass transfer coefficient α approaches zero. The nearly flat shape of the carbon concentration-depth profile is mostly due to the inward diffusion of carbon during the one hour period, recognizing that the interface becomes almost impermeable when the mass transfer coefficient α approaches zero. Therefore, under this condition, the decarburization is greatly restricted by the kinetics at the gas-metal interface and will not cause a serious degradation of the surface properties. 120 Chapter 7 Simulation of the Diffusion of Carbon and Nitrogen during Plasma Carbonitriding With the development of low temperature carburization and nitridation techniques, plasma carbonitriding has become a focus of research in this field during recent years. Unusual composition profiles have been observed after low temperature carburization and nitridation of austenitic stainless steels [17-22]. In particular, when the nitridation process occurs after carburization, there is an accumulation of carbon in front of the nitrogen diffusion zone, which has been observed with a variety of depth profiling techniques. Some authors describe this phenomenon as a “push” effect [18] of nitrogen on carbon. In this chapter, a quantitative explanation is provided to understand this interesting phenomenon, based upon the thermodynamic analysis contained in Chapter 2 and the overview of numerical simulation found in Chapter 3. According to a CALPHAD-based thermodynamic computation, nitrogen greatly increases the chemical potential of carbon, whose gradient is the actual driving force for carbon diffusion. This means that a given chemical potential of carbon may correspond to a reduced carbon concentration due to the introduction of a large amount of nitrogen during the plasma nitriding process, although the carbon may appear to be depleted within the nitrogen diffusion zone. Therefore, the curious shape of a carbon depth profile during a plasma nitriding treatment can be explained by classical diffusion theory, recognizing the 121 concentration dependence of both carbon and nitrogen diffusivities in stainless steels, instead of the so-called “push” effect of nitrogen on carbon. In this study, a numerical simulation based upon a finite difference method is able to reproduce the depth profiles shown in Fig. 7-1, which has been observed often in low temperature carburization and nitridation treatments [17]. Four concentration depth profiles are shown in Fig. 7-1. The profiles include the typical carbon (C430[C]) and nitrogen (N430[N]) depth profiles formed in ASTM F138 (i.e. AISI 316L) during 430oC / 15h plasma carburization and nitridation treatments, respectively, and the carbon and nitrogen depth profiles (NC430[C] and NC430[N]) formed during a 430oC carbonitriding process (15h plasma nitridation after 15h plasma carburization). Fig. 7-1: Typical concentration-depth profiles of S-phase layers on ASTM F138 (AISI 316L): nitrogen profile for 430oC nitrided S-phase (N430[N]), carbon profile for 430oC carburized S-phase (C430[C]) and nitrogen and carbon profiles for 430oC carbonitrided S-phase NC430[N] and NC430[C]. 122 7.1 Thermodynamic Analysis 7.1.1 The Gibbs Free Energy of Austenite Based upon CALPHAD modeling, the austenite phase of AISI 316L can be treated as two interpenetrating fcc sublattices, one based upon Fe and substitutional solutes (Cr, Ni, Mo) and the other partially occupied by the interstitial solutes (C, N). The ratio between the total number of the substitutional sites and the interstitial sites is 1:1. Therefore, the Gibbs free energy of one mole of formula units in this fcc phase (Fe, Cr, Ni, Mo)1(C, N, Va)1 can be expressed as Eq. (2-26): hfcc hfcc hfcc Gmfcc YFeYC oGFe:C YFeYN oGFe:N YFeYVa oGFe:Va hfcc hfcc hfcc YCrYC oGCr:C YCrYN oGCr:N YCrYVa oGCr:Va hfcc hfcc hfcc YNiYC oGNi:C YNiYN oGNi:N YNiYVa oGNi:Va hfcc hfcc hfcc YMoYC oGMo:C YMoYN oGMo:N YMoYVa oGMo:V a RT YFe ln YFe YCr ln YCr YNi ln YNi YMo ln YMo YC ln YC YN ln YN YVa ln YVa 0 fcc 0 fcc 0 fcc YCYVa YFe 0 Lfcc Fe:C,Va YCr LCr:C,Va YNi LNi:C,Va YMo LMo:C,Va 0 fcc 0 fcc 0 fcc YNYVa YFe 0 Lfcc Fe:N,Va YCr LCr:N,Va YMo LMo:N,Va YFeYCYN LFe:C,N 0 fcc 1 fcc YCrYFeYC 0 Lfcc Cr,Fe:C YCrYFeYN LCr,Fe:N LCr,Fe:N YCr YFe 1 fcc 0 fcc YCrYFeYVa 0 Lfcc Cr,Fe:Va LCr,Fe:Va YCr YFe YCrYFeYNYVa LCr,Fe:N,Va 1 fcc 1 fcc 0 fcc YFeYNiYC 0 Lfcc Fe,Ni:C LFe,Ni:C YFe YNi YFeYNiYN LFe,Ni:N LFe,Ni:N YFe YNi YFeYNiYVa L 0 fcc Fe,Ni:Va Y Y Y Y L 2 fcc 1Lfcc Fe,Ni:Va YFe YNi LFe,Ni:Va YFe YNi 0 fcc Cr Fe Ni C Cr,Fe,Ni:C Y Y Y Y 2 0 fcc Cr Fe Ni Va Cr,Fe,Ni:Va L 0 fcc YFeYMoYC 0 Lfcc Fe,Mo:C YFeYMoYVa LFe,Mo:Va 0 fcc 1 fcc YCrYNiYC 0 Lfcc Cr,Ni:C YCrYNiYVa LCr,Ni:Va LCr,Ni:Va YCr YNi 0 fcc 1 fcc YCrYNiYNYVa 0 Lfcc Cr,Ni:N,Va YMoYNiYVa LMo,Ni:Va LMo,Ni:Va YMo YNi 0 fcc 1 fcc YCrYMoYN 0 Lfcc Cr,Mo:N YCrYMoYVa LCr,Mo:Va LCr,Mo:Va YCr YMo 123 where the magnetic and pressure contributions to the Gibbs free energy are ignored based upon the analysis presented in Chapter 2. 7.1.2 The Chemical Potential of Carbon in Austenite The chemical potential of carbon in austenite is defined by Eq. (2-38): Cfcc Gmfcc Gmfcc YC YVa hfcc hfcc hfcc hfcc YFe oGFe:C oGFe:Va oGCr:Va YCr oGCr:C hfcc hfcc hfcc hfcc YNi oGNi:C oGNi:Va oGMo:Va YMo oGMo:C RT ln YC / 1 YC YN 0 fcc 0 fcc 0 fcc 1 2YC YN YFe 0 Lfcc Fe:C,Va YCr LCr:C,Va YNi LNi:C,Va YMo LMo:C,Va 0 fcc 0 fcc YN YFe 0 Lfcc Fe:N,Va YCr LCr:N,Va YMo LMo:N,Va YFeYN 0 Lfcc Fe:C,N 0 fcc 1 fcc YCrYFe 0 Lfcc Cr,Fe:C LCr,Fe:Va LCr,Fe:Va YCr YFe YCrYFeYN 0 Lfcc Cr,Fe:N,Va YFeYNi L 0 fcc Fe,Ni:C 0 fcc 1 fcc 2 fcc 1Lfcc Fe,Ni:C YFe YNi LFe,Ni:Va LFe,Ni:Va YFe YNi LFe,Ni:Va YFe YNi 2 0 fcc YCrYFeYNi 0 Lfcc Cr,Fe,Ni:C LCr,Fe,Ni:Va 0 fcc YFeYMo 0 Lfcc Fe,Mo:C LFe,Mo:Va 1 fcc 0 fcc YCrYNi 0 Lfcc Cr,Ni:C LCr,Ni:Va LCr,Ni:Va YCr YNi YCrYNiYN 0 Lfcc Cr,Ni:N,Va 1 fcc YCrYMo 0 Lfcc Cr,Mo:Va LCr,Mo:Va YCr YMo 1 fcc YMoYNi 0 Lfcc Mo,Ni:Va LMo,Ni:Va YMo YNi Based upon Eq. (2-38), the chemical potential of carbon in AISI 316L stainless steel with a nominal composition Fe-18Cr-12Ni-2Mo (in wt%) at 430oC can be computed using the relevant thermodynamic parameters recorded in the CALPHAD literatures. All of the parameters employed in this simulation are listed in Appendix I. 124 The chemical potential of carbon relative to graphite in AISI 316L stainless steel, with fixed site fractions on the metal sublattice, at 430oC is calculated using Eq. (7-1): Cfcc oGCgra 52.42YC 55.29YN 5.85ln YC +10.41 1 YC YN kJ mol -1 (7-1) and is plotted in Fig. 7-2 as a function of the atomic fraction of carbon and nitrogen, based upon the conversions of XC = YC / (1+YC+YN) and XN = YN / (1+YN+YC) for a fcc phase. From Fig. 7-2, it is easy to see that nitrogen greatly increases the chemical potential of carbon in austenite. For example, assume there is only 2 at% carbon in the system before the introduction of nitrogen. The chemical potential of 2 at% carbon in AISI 316L relative to graphite is about -11 kJ·mol-1. After introducing about 17 at% nitrogen into the system, the chemical potential of this 2 at% concentration of carbon increases to about 4 kJ·mol-1. The chemical potential level of 4 kJ·mol-1 is equivalent to that of about 10 at% carbon when there is no nitrogen present in the system. The actual carbon content after the introduction of nitrogen can be defined as XC (act.) and the “effective” carbon content with the same chemical potential level when there is no nitrogen present in the system can be defined as XC (eff.). The effect of nitrogen during plasma nitriding is to maintain a driving force for the inward diffusion of carbon, i.e., the chemical potential gradient of carbon, instead of “pushing” the carbon ahead of the nitrogen diffusion zone. In other words, the inward diffusion of nitrogen magnifies the chemical potential gradient of carbon, without an increase in the surface concentration of carbon. 125 Fig. 7-2: The chemical potential of carbon relative to graphite in AISI 316L stainless steel at 430oC. The curves correspond to the chemical potential of carbon for a series of carbon contents from 2 to 20 at% (spacing as 2 at% each). 7.1.3 The Chemical Potential of Nitrogen in Austenite The chemical potential of nitrogen in austenite is defined by Eq. (2-39): Nfcc Gmfcc Gmfcc YN YVa hfcc hfcc hfcc hfcc hfcc hfcc hfcc hfcc YFe oGFe:N oGFe:Va oGCr:Va oGNi:Va oGMo:Va YCr oGCr:N YNi oGNi:N YMo oGMo:N 0 fcc 0 fcc 0 fcc RT YN / 1 YC YN YC YFe 0 Lfcc Fe:C,Va YCr LCr:C,Va YNi LNi:C,Va YMo LMo:C,Va 0 fcc 0 fcc 0 fcc 1 2YN YC YFe 0 Lfcc Fe:N,Va YCr LCr:N,Va YMo LMo:N,Va YFeYC LFe:C,N 1 fcc 0 fcc 1 fcc YCrYFe 0 Lfcc Cr,Fe:N LCr,Fe:N YCr YFe LCr,Fe:Va LCr,Fe:Va YCr YFe 0 fcc 0 fcc 1 2YN YC YCrYFe 0 Lfcc Cr,Fe:N,Va YCrYFeYNi LCr,Fe,Ni:Va YFeYMo LFe,Mo:Va YFeYNi L 0 fcc Fe,Ni:N 0 fcc 1 fcc 2 fcc 1Lfcc Fe,Ni:N YFe YNi LFe,Ni:Va LFe,Ni:Va YFe YNi LFe,Ni:Va YFe YNi 2 1 fcc 0 fcc YCrYNi 0 Lfcc Cr,Ni:Va LCr,Ni:Va YCr YNi 1 2YN YC YCrYNi LCr,Ni:N,Va 0 fcc 1 fcc 0 fcc 1 fcc YCrYMo 0 Lfcc Cr,Mo:N LCr,Mo:Va LCr,Mo:Va YCr YMo YMoYNi LMo,Ni:Va LMo,Ni:Va YMo YNi 126 Based upon Eq. (2-39), the chemical potential of nitrogen in AISI 316L stainless steel with a nominal composition Fe-18Cr-12Ni-2Mo (in wt%) at 430oC can again be computed using the relevant thermodynamic parameters recorded in the CALPHAD literatures. All of the parameters employed in this simulation are also listed in Appendix I. Carbon has a similar influence on the chemical potential of nitrogen as nitrogen has on carbon. By employing the same analysis used in the Section 7.1.2, the chemical potential of nitrogen relative to one half of the pure Gibbs free energy of diatomic nitrogen gas in AISI 316L stainless steel at 430oC is calculated as shown in Eq. (7-2): Nfcc 0.5 oGNgas 87.46YN 55.29YC 5.85ln 2 YN 22.18 1 YN YC kJ mol -1 (7-2) and is plotted in Fig. 7-3 as a function of a carbon content for various nitrogen levels, based upon the same conversions of XN = YN / (1+YN+YC) and XC = YC / (1+YC+YN) for a fcc phase. Fig. 7-3 shows that carbon can also greatly increase the chemical potential of nitrogen. Similarly, the actual nitrogen content with carbon in the system can be defined as XN (act.) and the “effective” nitrogen content with the same chemical potential level when there is no carbon present in the system can be defined as XN (eff.). 127 Fig. 7-3: The chemical potential of nitrogen relative to one half of the pure Gibbs free energy of diatomic nitrogen gas in AISI 316L stainless steel at 430oC. The curves correspond to the chemical potential of nitrogen for a series of nitrogen contents from 2 to 24 at% (spacing as 2 at% each). It is apparent that the magnitudes of this mutual enhancement of the carbon and nitrogen chemical potentials with respect to unit concentration changes are actually the same based upon the results of Eqs. (7-1) and (7-2). However, considering the difference in the actual maximum concentrations in the system, the influence of carbon on the chemical potential of nitrogen is smaller than that of nitrogen on the chemical potential of carbon, because the maximum nitrogen concentration is greater than the maximum carbon concentration. To simplify the numerical program and maintain its stability, the relationships between XC (act.), XC (eff.), XN (act.), and XN (eff.) can be empirically described using linearized forms of Eq. (7-1) and Eq. (7-2) represented by Eq. (7-3) and Eq. (7-4): 128 X C eff. 0.61X N act. X C act. (7-3) X N eff. 0.53 X C act. X N act. (7-4) where the concentration variables X are evaluated as atomic fraction. Eq. (7-3) represents the influence of the actual nitrogen concentration up to 17 at% upon the chemical potential of the actual carbon concentration up to 10 at%. Eq. (7-4) represents the influence of the actual carbon concentration up to 10 at% upon the chemical potential of the actual nitrogen concentration up to 17 at%. Therefore, the chemical potential levels of carbon and nitrogen in the system, indicated by the values of “effective” concentrations, can be computed using Eq. (7-3) and Eq. (7-4) along with the corresponding values of the actual concentrations of carbon and nitrogen in the system during the numerical simulation of diffusion discussed in the next section. 7.2 Numerical Simulation of Diffusion 7.2.1 Simulation Conditions The numerical simulations in this study are again based upon a finite difference algorithm, through which Fick’s second law is discretized as a finite difference approximation in the dimensions of processing time t (s) and penetration depth x (µm) with the spacing between grid points of Δt = 60 s and Δx = 0.5 µm. To increase the accuracy of the numerical simulations, the Crank-Nicholson algorithm introduced in Section 3.3.3 was employed for all of the simulations in this study. Inasmuch as the 129 surface activation during a plasma treatment to remove the Cr2O3-rich passive layer on the surface of stainless steels is very efficient [17][18], a fixed boundary condition introduced in Section 3.4.1 was used in the numerical simulations, which assumes that the mass transfer coefficient α at the gas-metal interface is infinite. Effectively, the surface concentration achieves its maximum solubility at the very beginning of each treatment. 7.2.2 Concentration-Dependent Diffusion The concave shapes of the carbon and nitrogen depth profiles during plasma carburization and nitridation can be successfully modeled recognizing the concentration dependence of the diffusion coefficients of carbon and nitrogen in stainless steels. From Fig. 7-1, the carbon profile for the sample carburized at 430oC (C430[C]) is obviously concave, which is very different from the conventional error-function solution for the diffusion equation [30]. As shown in Chapter 5, the diffusion coefficient of carbon depends strongly on the carbon concentration, particularly when the carbon content is as high as 10 at%. Similar to the carbon profile, the nitrogen profile for the sample nitrided at 430oC (N430[N]) also has a concave shape, indicating that the diffusion coefficient of nitrogen is also highly concentration dependent when the nitrogen content is as high as 17 at%. Such concentration-dependent diffusion coefficients can be empirically described by the simple exponential function as shown by Eq. (3-8), ignoring the subscript C for all of the parameters in Eq. (3-8): 130 X D D exp k X max where D is the concentration-dependent diffusion coefficient, D’ is the diffusion coefficient at infinite dilution, and X and Xmax are the actual concentration and the maximum possible concentration. The parameter k controls the magnitude of the concentration dependence. For instance, the diffusion coefficient at the maximum possible concentration is ek times the diffusion coefficient at infinite dilution: D D exp k at X X max A series of computed concentration-depth profiles based upon Eq. (3-8) were shown in Fig. 6-3. 7.2.3 Simulation of Carbon Concentration-Depth Profile for Carburization at 430oC Based upon the analysis in Section 7.2.1 and Section 7.2.2, a simulated profile was obtained through finite difference modeling of the carbon concentration-depth profile for the 430oC /15h plasma carburization experiment, as shown in Fig. 7-4. The maximum possible concentration at the gas-metal interface XC,max is assumed to be 10 at% [17], which was determined by extrapolating the concentration-depth profile of carbon C430[C] to the gas-metal interface at x = 0. The diffusion coefficient at infinite dilution D’ for carbon in AISI 316L stainless steel, 4.5×10-5 µm2·s-1 at 430oC, is taken from Agarwala et al. [49]. In Fig. 7-4, the dots are the discretized experimental profile C430[C] in Fig. 7-1. The simulated profile is fitted to all of the experimental data points with a carbon 131 concentration higher than 2 at%, taking into consideration the difficulties in chemical analysis of carbon at low concentrations. The optimized value for the parameter k in Eq. (3-8) fitting the experimental data in Fig. 7-4, the normalized sum χ2 of residuals, the degrees of freedom, and the probability of the simulation being correct are listed in Table 7-1. Fig. 7-4: The simulated profile (solid line) fitting the experimental data (round dots) for the profile C430[C] in Fig. 7-1. 132 Table 7-1 The Optimized Value for the Parameter k in Eq. (3-8) Fitting the Experimental Data in Fig. 7-4, Normalized Sum χ2 of Residuals, the Degrees of Freedom, and the Probability of the Simulation Being Correct Parameter k χ2 d.f. P Value 5.0 0.4097 15 0.9999999996 Table 7-1 shows that the parameter k is 5.0, which means that the diffusion coefficient of carbon at a 10 at% carbon content is about e5.0 = 145 times the diffusion coefficient at infinite carbon dilution DC’. 7.2.4 Simulation of Nitrogen Concentration-Depth Profile for Nitridation at 430oC A similar simulation was performed for the nitrogen concentration-depth profile N430[N] after the 430oC / 15h plasma nitridation experiment as shown in Fig. 7-1. The maximum possible concentration at the gas-metal interface XN,max was assumed to be 17 at% [17], again by extrapolating the concentration-depth profile of nitrogen N430[N] to the gas-metal interface at x = 0. The literature review did not provide a good value of the diffusion coefficient at infinite dilution D’ for nitrogen in AISI 316L stainless steel. Therefore, the experimental data on nitrogen diffusivity in gamma iron [54], 2.8×10-5 µm2·s-1 at 430oC, was employed. Fig. 7-5 shows the simulated profile and the discretized experimental profile N430[N] in Fig. 7-1. The simulated profile is again fitted to all of the experimental data points with nitrogen concentration higher than 2 at%. The optimized value for the parameter k in Eq. (3-8) fitting the experimental data in Fig. 7-5, the normalized sum χ2 133 of residuals, the degrees of freedom, and the probability of the simulation being correct are listed in Table 7-2. Fig. 7-5: The simulated profile (solid line) fitting the experimental data (round dots) for the profile N430[N] in Fig. 7-1. Table 7-2 The Optimized Value for the Parameter k in Eq. (3-8) Fitting the Experimental Data in Fig. 7-5, Normalized Sum χ2 of Residuals, the Degrees of Freedom, and the Probability of the Simulation Being Correct Parameter k χ2 d.f. P Value 4.2 0.0901 9 0.99999998 134 Table 7-2 shows that the parameter k is 4.2, which means that the diffusion coefficient of nitrogen at a 17 at% nitrogen content is about e4.2 = 67 times the diffusion coefficient at infinite nitrogen dilution DN’, which is less than that for carbon at a 10 at% concentration level. The diffusion coefficient of nitrogen at 10 at% based upon Eq. (3-8) is 12.2 DN’. Given that DN’ / DC’ is 0.6, carbon diffuses about 20 times faster than nitrogen at the same concentration level in an AISI 316L stainless steel’s austenitic matrix. 7.2.5 Simulation of Carbon and Nitrogen Concentration-Depth Profiles for Plasma Carbonitriding at 430oC 7.2.5.1 Simulation Parameters An unusual carbon concentration-depth profile develops when AISI 316L stainless steel is plasma nitrided after plasma carburization. The simulation parameters that were employed for the carburization and nitridation are maintained in the carbonitriding simulations except for the following two points. Firstly, for the simulation of the carbon concentration-depth profile during plasma nitridation, an impermeable boundary condition was used, which means there was no carbon flux entering or escaping through the gas-metal interface during the plasma nitridation; no carbon enters the metal through the interface because there is no carbonbearing gas in the nitriding atmosphere, and no carbon escapes through the interface because the desorption process for carbon is extremely difficult under these conditions. In this case, the mass transfer coefficient α for carbon atoms is essentially zero. 135 Secondly, the k parameter for carbon during plasma nitridation should be higher than the value used for Fig. 7-4, because the effective carbon concentration increases to a much higher level due to the increased chemical potential of carbon. It is difficult to predict the exact value of k at the higher effective carbon concentration level, which can be up to 18 at% based upon the calculation shown by Eq. (7-3). A variety of k values for carbon were tried in the simulation, and the k value of 6.5 for carbon concentrations up to 18 at% yielded the most satisfactory fit. 7.2.5.2 Computational Logic The simulation starts with the carbon concentration-depth profile shown in Fig. 74. The changes in the carbon and nitrogen concentration-depth profiles within the first minute are demonstrated in order to show the computational logic employed for the simulation. There are a total of six calculations for each time step, as shown in Fig. 7-6. Fig. 7-6 (a) shows the first step. Plasma nitridation starts and the effective nitrogen concentration N (eff.), representing the chemical potential of nitrogen, equilibrates with the maximum possible nitrogen concentration of 17 at% at the gas-metal interface as determined by the nitrogen activity in the nitriding gas atmosphere. Then the effective nitrogen diffuses based upon the highly concentration-dependent diffusion coefficient of nitrogen used to produce Fig. 7-5. The simulated effective nitrogen concentration-depth profile after the diffusion for one time step is shown in Fig. 7-6 (b). A fixed boundary condition was used during this step. Fig. 7-6 (c) shows the third step. Because carbon can increase the chemical potential of nitrogen, the actual nitrogen solubility in AISI 316L stainless steel under a 136 given nitrogen activity decreases, and the actual nitrogen concentration-depth profile was computed based upon the current values of C (act.) and N (eff.), according to Eq. (7-4). Because nitrogen also greatly increases the chemical potential of carbon, the effective carbon concentration-depth profile was computed based upon the actual concentrations of N (act.) and C (act.), according to Eq. (7-3). The computed profile of the effective carbon concentration is shown in Fig. 7-6 (d). (a) (b) (c) (d) 137 (e) (f) (g) Fig. 7-6: The changes of the carbon and nitrogen concentration-depth profiles, C(act.), C(eff.), N(act.), and N(eff.) within the first minute including six computational steps (a) – (f). (g) shows the actual carbon and nitrogen concentration-depth profiles, C(act.) and N(act.), at the end of the first minute. Fig. 7-6 (e) shows the fifth step. The effective carbon diffuses based upon a highly concentration-dependent diffusion coefficient, and the profile was simulated under an impermeable boundary condition as described in the last section. The k parameter used in this simulation is 6.5, which is larger than the value of 5 obtained from Fig. 7-4. Then the actual carbon concentration-depth profile was computed based upon the current 138 values of C (eff.) and N (act.), according to Eq. (7-3) again. The computed profile of actual carbon concentration is shown by Fig. 7-6 (f). The actual carbon and nitrogen concentration-depth profiles at the end of the first minute are shown in Fig. 7-6 (g). Due to the highly concentration-dependent diffusivity of carbon, the actual carbon concentration near the surface decreases to a very low level, while a peak develops along the actual carbon concentration profile. The actual carbon and nitrogen concentration-depth profiles shown in Fig. 7-6 (g) were taken as the initial concentration-depth profiles for the next time step and then repeated until the end of the simulation for the 15-hour plasma nitridation process. 7.2.5.3 Simulation Results Fig. 7-7 (a-h) shows the simulated concentration-depth profiles C (act.), C (eff.), and N (act.) for total elapsed times of 5 minutes, 10 minutes, 20 minutes, 30 minutes, 1 hour, 3 hours, 5 hours and 10 hours of plasma nitridation treatment. Obviously, the actual carbon peak along the concentration-depth profile generated during the beginning of the treatment is gradually homogenized due to carbon diffusion and after about 1hour a very smooth curve develops. 139 (a) (b) (c) (d) 140 (e) (f) (g) (h) Fig. 7-7: The simulated concentration-depth profiles C(act.), C(eff.), and N(act.) for total elapsed times of (a) 5 minutes, (b) 10 minutes, (c) 20 minutes, (d) 30 minutes, (e) 1 hour, (f) 3 hours, (g) 5 hours and (h) 10 hours of the plasma nitridation treatment. 141 Fig. 7-8 (a) shows the simulated final concentration-depth profiles C (act.), C (eff.) and N (act.). Due to the large amount of nitrogen introduced during the plasma nitridation treatment, the effective carbon concentration-depth profile, representing the chemical potential of carbon, develops as a very smooth and monotonously decreasing curve, although the actual carbon concentration seems to be depleted within the nitrogen diffusion zone. In Fig. 7-8 (b), it is obvious that there is a good agreement between the simulated concentration-depth profiles and the experimental results in Fig. 7-1. The general shape of the actual carbon concentration-depth profile agrees with the experimental profile, although the fit is not perfect. Considering the difficulties in obtaining the experimental profiles using the GDOES technique and the assumptions and approximations made during the simulation, the fit is most satisfactory. The smooth effective carbon concentration-depth profile demonstrates that the large nitrogen content introduced by plasma nitridation provides a significant driving force for carbon diffusion, i.e., the chemical potential gradient of carbon during the plasma nitridation process, rather than carbon being “pushed” ahead of the nitrogen diffusion zone. It also should be noted that with the similar influence of carbon on the nitrogen chemical potential gradient, the nitrogen profile becomes a little flatter compared with the original nitrogen concentration-depth profile. In addition, the maximum concentration of nitrogen at the surface decreases a little due to the modest amount of carbon present in the nitrogen diffusion zone. These results also are in excellent agreements with the experimental measured profiles shown in Fig. 7-1. 142 (a) (b) Fig. 7-8: (a) The simulated final concentration-depth profiles C (act.), C (eff.) and N (act.) for plasma carbonitriding at 430oC. (b) Comparison with the experimental data (round dots) NC430[C] and NC430[N] shown in Fig. 7-1. 143 Fig. 7-9 shows the final chemical potential profiles of carbon calculated using Eq. (7-1) based upon the CALPHAD model. Fig. 7-9 (a) is based upon the carbon and nitrogen concentration values taken from the experimental concentration-depth profiles NC430[C] and NC430[N] in Fig. 7-1. Fig. 7-9 (b) is based upon the carbon and nitrogen concentration values taken from the simulated concentration-depth profiles in Fig. 7-8 (b). Both profiles in Fig. 7-9 display features that must be in error. The chemical potential profile of carbon has a valley at around 5 microns of depth in Fig. 7-9 (a) and the profile has a small hump at the equivalent position in Fig. 7-9 (b). Possible uncertainties in the experimental concentration-depth profiles have already been mentioned in the previous sections as a source for these effects. The simulation error is mainly due to the linearization of the relationships between the effective and the actual concentration-depth profiles as Eq. (7-3) and Eq. (7-4). Based upon the above analyses, it is clear that the final chemical potential profile of carbon should be similar to the C (eff.) curve in Fig. 7-8 (a). 144 (a) (b) Fig. 7-9: The final chemical potential profiles of carbon relative to graphite for 430oC plasma carbonitriding based upon: (a) the experimental data in Fig. 7-1, and (b) the simulation result in Fig. 7-8 (b). 145 7.3 Simulation Predictions Similar numerical simulations can be used to predict the other two types of procedures: (1) 15-hour plasma nitridation followed by 15-hour plasma carburization at 430oC, and (2) 15-hour simultaneous plasma carbonitriding process at 430oC. Because the relevant experimental data are not available in the literature, the following simulation results are conservatively considered as predictions, the accuracy of which deserves further discussion. The simulations are essentially based upon the same set of parameters defined in Section 7.1 and Section 7.2, and employed in the previous simulation for a 15hour plasma carburization followed by a 15-hour plasma nitridation at 430oC. The computational logic is adjusted accordingly, based upon the different order or combination of the plasma carburization and plasma nitridation processes. Similarly, to the simulation for a 15-hour plasma nitridation followed by a 15hour plasma carburization at 430oC, an impermeable boundary condition was employed for the diffusion of nitrogen during plasma carburization, and the value of the k parameter for nitrogen was empirically determined to be 5 at nitrogen concentrations up to 19 at%, which is higher than the value of 4.2 at nitrogen concentrations up to 17 at%, used to produce Fig. 7-5. The simulated profiles of the actual carbon and nitrogen concentrations are shown in Fig. 7-10. 146 Fig. 7-10: The simulated profiles of actual carbon and nitrogen concentrations for a 15-hour plasma nitridation followed by a 15-hour plasma carburization at 430oC. Fig. 7-10 shows that the penetration depth of the carbon profile is smaller, and the penetration depth of the nitrogen profile is larger, compared with the carbon and nitrogen profiles in Fig. 7-8. This is reasonable because nitrogen diffuses for 15 more hours and carbon diffuses for 15 less hours for the treatment represented in Fig. 7-10 compared to the one shown in Fig. 7-8. In Fig. 7-10, the surface concentration of nitrogen decreases to about 10 at% during the subsequent plasma carburization, due to the highly concentration-dependent diffusion coefficient of nitrogen. Meanwhile, the carbon concentration within the nitrogen diffusion zone was limited to a very low level of about 3-4 at%. This is because the actual carbon concentration has to decrease in order to maintain the chemical potential of carbon determined by the carburizing gas atmosphere based upon the presence of a large amount of nitrogen introduced before the 147 carburization treatment. Carbon is able to increase to a higher concentration level ahead of the nitrogen diffusion zone and develops a shape similar to that observed in Fig. 7-8, although this tendency is not so obvious in Fig. 7-10 considering the limitation on the length of time for diffusion. In the simulation for a 15-hour simultaneous plasma carbonitriding process at 430oC, the plasma carburization and the plasma nitridation proceed at the same time. Therefore, the chemical potentials of carbon and nitrogen are both determined by the activities of carbon and nitrogen in the carbonitriding gas atmosphere, which would be the same as employed during the individual carburization and nitridation processes used to create Fig. 7-4 and Fig. 7-5 by ignoring the mutual influence on the activities of carbon and nitrogen due to the mixing of the treatment gases. The simulation result is shown in Fig. 7-11. Fig. 7-11: The simulated profiles of actual carbon and nitrogen concentrations for a 15-hour simultaneous plasma carbonitriding process at 430oC. 148 In Fig. 7-11, the penetration depth of carbon is close to that in Fig. 7-10 due to both representing the same length of time for carbon diffusion. The penetration depth of nitrogen in Fig. 7-11 is close to that in Fig. 7-8 due to the same length of time for nitrogen diffusion. In Fig. 7-11, the carbon concentration-depth profile develops a similar shape to that observed in Fig. 7-8. Considering the larger diffusion coefficient of carbon than nitrogen as explained in Section 7.2.4, carbon has the opportunity to develop a peak along its concentration-depth profile ahead of the nitrogen diffusion zone at the very beginning of the simultaneous carbonitriding process. Fig. 7-12: The simulated actual concentration-depth profiles of the total amount of carbon and nitrogen produced by the three different processes at 430oC: (a) 15-hour plasma carburization followed by 15-hour plasma nitridation, (b) 15-hour plasma nitridation followed by 15-hour plasma carburization, and (c) 15-hour simultaneous plasma carbonitriding process. 149 Fig. 7-12 shows the actual concentration-depth profiles of the total amount of carbon and nitrogen, produced by the three different processes: (a) 15-hour plasma carburization followed by 15-hour plasma nitridation, (b) 15-hour plasma nitridation followed by 15-hour plasma carburization, and (c) 15-hour simultaneous plasma carbonitriding. The surface concentrations of the total amount of carbon and nitrogen are very close between process (a) and process (c), which are higher than that produced by process (b). This is mainly because the nitrogen concentration decreases significantly during the plasma carburization in process (b), considering the similar low carbon concentrations at the surface during these three processes. The penetration depth of the total amount of carbon and nitrogen during process (a) is deeper than that during process (b) and process (c), which is mainly because carbon diffuses faster than nitrogen under the condition of these simulations as explained in Section 7.2.4. It is noteworthy that the shapes of the profiles in Fig. 7-12 can only be interpreted qualitatively due to the empirical assumptions and approximations made during the simulation process. The accurate quantification of these simulation deserves further exploration. 150 Chapter 8 Discussion 8.1 Mass Transfer Coefficient The concept of a convective boundary condition was introduced in Section 3.4.2 and the relevant results of finite difference simulations based upon convective boundary conditions were thoroughly presented in Chapter 6. Under a convective boundary condition, the mass transfer at the gas-metal interface is described by Eq. (3-32): D c c cs x where α is the mass transfer coefficient, c∞ is the concentration in the treatment gas atmosphere and cs is the concentration in the metal at the gas-metal interface. (c∞ - cs) represents the concentration difference between the gas atmosphere and the metal, which is also considered to be the driving force for the mass transfer at the interface. The mass transfer coefficient α can be considered as an empirical constant in the mathematics of diffusion [32], in which case the magnitude of the mass transfer coefficient is an approximate description of the general efficiency of mass transfer at the boundary. The larger the magnitude of the mass transfer coefficient, the higher the efficiency of the mass transfer at the boundary. It has been shown in Chapter 3 that the boundary condition can be considered as fixed when the mass transfer coefficient approaches positive infinity, which indicates that the mass transfer efficiency is extremely high. 151 The mass transfer coefficient α in actual situations is influenced by many factors, such as temperature T, the possible maximum concentration in the gas atmosphere c∞, the concentration in the metal at the interface cs and the surface conditions of the metal [55]. A theoretical explanation of the mass transfer coefficient, considering all of these factors, will require a very thorough understanding of the dynamic characteristics of the gas flow, which is beyond the scope of the current study. Possible theoretical models for explaining the convective mass transfer phenomena have been proposed in the literature [35], such as the film model, the penetration model and the boundary layer model. Both of the film and penetration models are related to mass transfer involving an interface between two moving fluids. When one of the phases is a solid, the boundary layer model is commonly used for correlating the data involving a solid subliming into a gas phase or a solid dissolving into a liquid phase. Detailed discussion of this theoretical model is not treated in this thesis. In most cases, the mass transfer coefficient is empirically determined from experimental investigations. For instance, in the current study, the magnitude of the mass transfer coefficient was determined by fitting the simulated concentration-depth profiles to the experimental data obtained from surface chemical analysis. The point was made that the mass transfer efficiency of carbon during a low temperature carburization procedure is greatly influenced by the surface activation treatment, during which the Cr2O3-rich passive layer on the surface of stainless steels is eliminated and the surface becomes transparent to the diffuse of carbon [4]. An empirical study by the Swagelok Company has demonstrated that adjusting the surface activation procedures has a very critical influence on the effectiveness of a low temperature carburization treatment. 152 Therefore, the numerical simulation at the current stage is based upon the assumption that the surface condition for low temperature carburization, determined by surface activation treatments, is the principal factor influencing the magnitude of the mass transfer coefficient α. The other reason for making this assumption is the evaluation of the mass transfer coefficient would become very complex, if the value of the parameter α is also considered to be a function of the driving force during the simulation procedure. However, further evaluation of the mass transfer coefficient, considering the influential factors comprehensively, is valuable in the future. An optimized surface activation also has been applied to the low temperature carburization procedure. The activation process is conducted for 3 hours at 324oC, which is higher than the previous activation temperature of 250oC, and the treatment gas has the same composition as listed in Section 6.2.1 and Table 8-1. In order to evaluate the effectiveness of this surface activation process, an interrupted treatment, as shown in Table 8-1, of the typical Swagelok procedure was designed and applied to specimens of AISI 316L stainless steel. Table 8-1 The Interrupted Treatment of the Typical Swagelok Procedure Step Temperature (oC) Time (Hour) Gas Composition (Volume Percent) Surface Activation 324 3 90% HCl + 10% N2 Carburization 450 4 46% CO + 46% H2 + 8% N2 153 At the end of the treatment as Table 8-1, the carburization step was followed by a rapid fan cooling process. Compared with the typical Swagelok procedure in Fig. 6-5, this interrupted treatment was almost identical to the first surface activation step and the first carburization step of a complete Swagelok procedure, with the only difference being the activation temperature. An AES line scan was performed on the cross-sectional surface of the specimen using the experimental procedure presented in Chapter 4. The Auger spectra were analyzed in the same way as described in Chapter 6. The corresponding carbon concentration-depth profile was adjusted based upon the calibration represented by Eq. (4-1). A simulated profile based upon the parameters in Table 6-2 was fitted to the experimental AES data with the optimized value for the mass transfer coefficient α listed in Table 8-2. Both the experimental and the simulated concentration-depth profiles for carbon are shown in Fig. 8-1. The optimized value for α was obtained by fitting the simulated profile to all of the experimental data with a carbon concentration > 1 at% in Fig. 8-1 except the first data point which was rejected due to the influence of the adventitious carbon residue on the carburized surface. 154 Fig. 8-1: The simulated concentration-depth profile of carbon in AISI 316L stainless steel treated by the procedure in Table 8-1 fitting the experimental data obtained from AES. Table 8-2 The Optimized Value for the Mass Transfer Coefficient α Fitting the Experimental Data in Fig. 8-1, Normalized Sum χ2 of Residuals, the Degrees of Freedom, and the Probability of the Simulation Being Correct Parameter α (µm·s-1) χ2 d.f. P Value 0.005 0.5312 6 0.9977 155 A standard XPS survey was performed using the experimental procedure outlined in Chapter 4. The chemical composition at the surface of the carburized AISI 316L stainless steel specimen was measured after a pre-sputtering process that removed about 50nm from the surface. The X-ray photoelectron spectra from all of the analyzed peaks are shown in Fig. 8-2. The measured chemical composition is also shown in Fig. 8-2 and the measured carbon concentration is 12.5 at%. The carbon peak was deconvoluted into a peak signal from the adventitious carbon at ~ 284.8 eV and a peak signal from carbon in solid solution at 282.0 eV – 283.0 eV employing the Multipak software, which is shown in Fig. 8-3. By factoring out the signal contribution from the adventitious carbon, it is obvious that the carbon concentration in solid solution is about 85.6% of the total amount of 12.5 at% carbon, which is equivalent to ~ 10.5 at%. This result agrees very well with the simulation in Fig. 8-1, which means that the surface concentration is very close to the maximum possible concentration of ~ 11 at%. 156 Fig. 8-2: The X-ray photoelectron spectra of the standard XPS survey on the AISI 316L stainless steel specimen carburized by the procedure in Table 8-1. 157 Fig. 8-3: The analysis of the chemical state of carbon in the X-ray photoelectron spectra in Fig. 8-2. Based upon the above analysis, the optimized surface activation at 324oC can provide a more efficient mass transfer at the gas-metal interface based upon the larger value of the mass transfer coefficient, α = 5×10-3 µm·s-1. This efficient mass transfer at the gas-metal interface enables the surface carbon concentration to increase up to 10 at% within the first 4-hour carburization step, and thus a concave shape of the concentrationdepth profile for carbon is observed. This result is also well predicted by the finite difference simulation based upon Asimow’s analytical model introduced in Chapter 5. Therefore, increasing the value of the mass transfer coefficient is desirable and very critical to maintain the high efficiency of the mass transport at the gas-metal interface and significantly facilitate the potential for concentration-dependent diffusion at high carbon concentration levels. 158 8.2 Selection of the Diffusion Model Based upon the analysis in Chapter 5, Asimow’s analytical model provides the most satisfactory description for the concentration dependence of the diffusion coefficient of carbon in austenite. The diffusion coefficient of carbon in AISI 316L stainless steel predicted by Asimow’s model is in excellent agreement with the published data derived from a Boltzmann-Matano analysis of carburized AISI 316L stainless steel performed by Ernst et al. [38]. It should be noted that a systematic error in the maximum carbon concentration in AISI 316L stainless steel does exist among the measurements obtained from various types of characterization techniques, which was estimated at < 5 at% [38]. Therefore, the derived carbon diffusivities in Fig. 5-3 based upon the Boltzmann-Matano analysis were correspondingly influenced by the systematic error, especially at carbon concentrations higher than 11 at%. Meanwhile, the experimental data for the diffusion coefficient of carbon at infinite carbon dilution in AISI 316L stainless steel DC’ employed to calculate the value of DC in Fig. 5-3 also has some error, which was estimated to be less than a factor of 2 [49]. However, the possible errors mentioned above did not generate a significant influence on the selection of the analytical model during the simulation procedures. The reason for this is discussed as follows. First, Asimow’s model is based upon the point of view that the activation energy associated with carbon diffusion in austenite is reduced due to the typical lattice expansion induced by carbon in solution in an austenite matrix. The excellent agreement in Fig. 5-3 provides quantitative evidence to support the point of view suggested by Ernst et al. [38] of the physical origin for enhanced diffusion, which is 159 consistent with the standpoint proposed by Asimow [10]. Similar perspective was suggested by Christiansen and Somers [56] considering the enhanced diffusion of nitrogen observed during plasma nitridation of stainless steels. Secondly, the selection of the analytical model was also significantly influenced by the effectiveness of the numerical simulation of the concentration-depth profiles, during which several models were tried to fit the experimental data using the simulation procedure introduced in Chapter 3 and Chapter 6. At the current stage, the finite difference simulation based upon Asimow’s model provides the most satisfactory agreement with the experimental data, which is much better than the results that the other models can provide. Fig. 8-4 shows the simulated concentration-depth profiles of carbon in carburized AISI 316L stainless steel for the typical Swagelok procedure in Section 6.2.1 based upon the models of (a) ARRT (Eq. (5-3)), (b) Asimow (Eq. (5-7), and (c) Ågren (Eq. (5-14)) and the corresponding experimental profile shown in Fig. 6-9. The parameters employed in this simulation are the same as those listed in Table 6-4 and the values of the mass transfer coefficients are the same as those in Table 6-5. 160 (a) (b) (c) Fig. 8-4: The simulated concentration-depth profiles of carbon in carburized AISI 316L stainless steel for the typical Swagelok procedure in Section 6.2.1 based upon the models of (a) ARRT (Eq. (5-3)), (b) Asimow (Eq. (5-7)), and (c) Ågren (Eq. (5-14)) and the corresponding experimental profile shown in Fig. 6-9. The point has been made that the original model of the absolute reaction rate theory proposed by Fisher et al. [8], based upon the assumption that γM is a constant, underestimates the concentration dependence of the diffusion coefficient of carbon in austenite [7][38]. Therefore, in Fig. 8-4, the simulated profile (a) based upon Eq. (5-3) of the ARRT model has a smaller penetration depth and a flatter shape for the concentration-depth profile of carbon compared with the experimental data obtained using AES. It was also pointed out that the widely used model of Ågren [15] significantly overestimates the concentration dependence when the atomic fraction of carbon is higher than 0.08 [38]. Therefore, in Fig. 8-4, the simulated profile (c) based upon Eq. (5-14) of 161 Ågren’s model predicts a much larger penetration depth and a much more concave shape for the concentration-depth profile of carbon. Considering all of the above analysis, the simulation based upon Asimow’s model provides the most satisfactory results for the concentration-depth profiles of carbon in austenite and therefore deserves further optimization based upon a more comprehensive and complete understanding of the physical mechanisms of concentration-dependent diffusion of carbon in austenite. 8.3 Influence of Carbon Concentration on Young’s Modulus Section 5.3 contains the derivation of the parameter K in Asimow’s model for the concentration dependence of carbon diffusivity in AISI 316L stainless steel. It is obvious that the calculation of the parameter K involves accurate determinations of several important physical parameters, such as the Young’s modulus of the austenite matrix E and the activation volume associated with carbon diffusion in austenite ΔVA. The typical Swagelok procedure of low temperature carburization can introduce a carbon concentration of ~ 10 to 15 at% into the surface layer of AISI 316L stainless steel. It is possible that such a large carbon concentration has some influence on the relevant physical parameters of AISI 316L stainless steel. As such, the value of the parameter K and the concentration dependence of carbon diffusivity based upon Asimow’s analytical model would be different from the calculated results in Section 5.3 that were based upon the physical parameters for non-carburized AISI 316L stainless steel. However, evaluation of the effect of high carbon concentration on all of the parameters in Asimow’s model is beyond the scope of the current study, and relevant experimental 162 results are lacking in the literature. Therefore, the following discussion is only focused on the possible influence of carbon concentration on the Young’s modulus of the austenite matrix, which is a key parameter in Asimow’s model. This discussion is based upon an extensive literature review of studies concerned with the mechanical properties of bulk cementite (Fe3C), which contains a carbon concentration of 25 at%. According to Glikman et al. [57], Mizubayashi et al. [58], and Umemoto et al. [59], cementite does not have a higher Young’s modulus than pure iron, irrespective of its higher hardness and strength than the pure iron. This behavior is quite different from most of other carbides, which show a much higher Young’s modulus than their pure metal counterparts. The reason to this phenomenon is still not completely clear at present, but the theoretical study of Kimstach [60] on the nature of cementite did confirm that cementite should be considered not as a chemical compound, i.e., a carbide phase, but a solid solution of carbon in iron. Therefore, it is completely reasonable that the ~ 10 to 15 at% interstitially dissolved carbon in an austenite matrix would not have a considerable influence on the Young’s modulus of the austenite matrix. The change of the concentration dependence of carbon diffusivity predicted by Asimow’s analytical model due to a variation in Young’s modulus is expected to be modest. 8.4 Influence of Nitrogen on the Drift Velocity of Carbon In classical diffusion theory, the diffusive flux of atoms is considered as a net drift velocity superimposed on the random jumping motion of each diffusing atom [31]. The 163 drift velocity of carbon is simply related to the diffusive flux of carbon via Eq. (8-1) as follows: J C vCcC (8-1) where J is the diffusive flux, v is the drift velocity and c is the concentration. Because atoms always migrate in order to remove differences in chemical potential, it is reasonable to suppose that the drift velocity has the following relationship with the local chemical potential gradient as indicated in Eq. (8-2): vC M C C x (8-2) where M is known as the atomic mobility. Since the chemical potential gradient, ∂µ/∂x, has units of energy with respect to distance, it is effectively the chemical ‘force’ causing the atoms to migrate [31]. In Chapter 7, it was shown that the interesting curvature of the carbon concentration-depth profile in carburized AISI 316L stainless steel observed during plasma carbonitriding processes can be successfully explained by classical diffusion theory. However, understanding the interdependent diffusion phenomena of interstitial elements needs a much more thorough study on the mutual influence of the interstitial elements on their diffusion behaviors. This task is definitely very challenging. The following discussion is aimed to provide some inspiration from the aspect of the influence of the introduced large amount of nitrogen on the drift velocity of carbon during a plasma carbonitriding process. 164 The influence of nitrogen on the drift velocity of carbon is now discussed in two aspects based upon Eq. (8-2). On one hand, in Chapter 7, it has been discussed that nitrogen can greatly increase the chemical potential of carbon based upon a CALPHAD thermodynamic modeling. Therefore, during the plasma nitridation, the large amount of nitrogen introduced alters the chemical potential gradient of carbon, which is the actual driving force causing the diffusion of carbon. According to Eq. (8-2), nitrogen can increase the drift velocity of carbon by increasing the chemical potential gradient of carbon. On the other hand, carbon diffuses in accord with a highly concentration dependent diffusion coefficient and the actual carbon concentration decreases to a much lower level within the nitrogen diffusion zone. The mobility of an atom and its diffusion coefficient are related [31]. As discussed in Chapter 5, the diffusion coefficient of carbon increases with the concentration of carbon. Therefore, when nitrogen decreases the actual carbon concentration, that causes a decrease in the atomic mobility of carbon within the nitrogen diffusion zone. According to Eq. (8-2), nitrogen can decrease the drift velocity of carbon by decreasing the atomic mobility of carbon. The above discussion indicates that the two influences of nitrogen on the drift velocity of carbon are in opposition, and their overall effect on the diffusive flux of carbon cannot be directly derived from the above analysis. A possible qualitative solution is to compare the penetration depths of the carbon concentration-depth profiles generated by a plasma carbonitriding process with a 15-hour carburization followed by a 15-hour nitridation and another process with a 30-hour plasma carburization. A detailed quantitative solution deserves a more extensive exploration in the future. 165 Chapter 9 Conclusions 1. The typical Swagelok procedure of low temperature carburization conducted for approximately 40 hours at 450oC can produce an interstitially hardened case in AISI 316L stainless steel with a surface carbon concentration of 10 to 15 at% and a penetration depth of 20 to 25µm. The concentration-depth profile of carbon obtained after such a treatment exhibits a very concave shape that greatly deviates from the profile defined by an error-function solution to Fick’s second law. 2. The analytical model proposed by Asimow provides a very satisfactory description for the concentration dependence of the diffusion coefficient of carbon in AISI 316L stainless steel during low temperature carburization. This result indicates that the concentration-enhanced diffusion of carbon in austenite is most likely because the lattice expansion induced by carbon in solution in an austenite matrix greatly decreases the activation energy associated with the diffusion of carbon. 3. Numerical simulation with a finite difference algorithm using Asimow’s analytical model for carbon diffusion can successfully reproduce the concentration-depth profile of carbon in AISI 316L stainless steel carburized by the typical Swagelok procedure. Excellent agreement was achieved between the simulated concentration-depth profiles for carbon and the experimental data obtained from surface chemical analyses conducted with AES and XPS. 166 4. The unusual shape of the carbon concentration-depth profile during a plasma carbonitriding process can be explained by classical diffusion theory, recognizing the concentration dependence of both carbon and nitrogen diffusivities in stainless steels. The large nitrogen concentration introduced by plasma nitridation provides a significant driving force for carbon diffusion. Nitrogen greatly increases the chemical potential of carbon that corresponds to a given carbon concentration. The chemical potential gradient of carbon generated during the plasma nitridation process provides the driving force for the diffusion of carbon. 167 Chapter 10 Suggestions for Future Research 1. The systematic error of the measured carbon concentrations using various types of characterization techniques should be comprehensively investigated, because the corresponding accuracy of the measurements is very critical to determine the precise functional relationship between the diffusion coefficient of carbon in austenite and the carbon concentration. For the surface analysis experiments involving sputtering, the relative sputtering rates of the elements in the analyzed materials should be determined by a systematic study. 2. The specimens of AISI 316L stainless steel employed in the current study were not pretreated by any annealing process and electropolishing procedure. In this case, the influence of the surface damage layer upon the diffusion of interstitial elements is very difficult to estimate. More specimens with a higher initial surface quality should be prepared for studies in the future. 3. The activity of carbon in the carburizing atmosphere should be further studied, because this result significantly influences the maximum possible solubility of carbon in the treated materials. Potential solutions would be using in-situ techniques to investigate the surface adsorption of carbon or the monitoring analysis of the residue gases. 4. A comprehensive optimization of Asimow’s analytical model is highly desirable and the possible application of this model to the diffusion of nitrogen in stainless steels should be discussed, based upon a detailed literature review of the parameters related to 168 the diffusion of nitrogen in austenite. Numerical simulation of the carbonitriding processes should be further studied in order to more comprehensively understand the interdependent diffusion phenomena associated with interstitial elements. 169 Appendix I CALPHAD Parameters Employed in the Thermodynamic Analysis o hfcc hfcc GFe:C [25] oGFe:Va oGCgra 77207 15.877T o hfcc hfcc GCr:C [61] oGCr:Va oGCgra 13748.62 79.953T 9.53T ln T 2701850 / T 2.643 108 T 2 1.2 1010 T 3 0.00142205T 2 1.47721 106 T 3 o hfcc hfcc GNi:C [25] oGNi:Va oGCgra 62000 7.6T o hfcc hfcc GMo:C [62] oGMo:Va oGCgra 22700 8.93T 750000T 1 1 o gas GN 2 35997.6 367.138T 36.45T ln T 6.4 104 T 2 2 1 o hfcc hfcc GCr:N [63] oGCr:Va oGNgas2 131744 141.997T 8.5T ln T 2 1 o hfcc hfcc GNi:N [64] oGNi:Va oGNgas2 38680 143.09T 10.9T ln T 0.00438T 2 2 1 o hfcc hfcc GMo:N [65] oGMo:Va oGNgas2 80544 149.07T 9.78T ln T 2 0 fcc LFe:C,Va [25] 34671 o hfcc hfcc GFe:N [63] oGFe:Va 0 fcc Cr:C,Va [25] 11977 6.8194T 0 fcc Ni:C,Va [25] 14902 7.5T L L 0 fcc Mo:C,Va L [62] 41300 0 fcc Fe:N,Va [63] 26150 0 fcc Cr:N,Va [63] 20000 L L 0 fcc Mo:N,Va L [65] 52565 0 fcc Fe:C,N [66] 21893 0 fcc Cr,Fe:C [25] 74319 3.2353T 0 fcc Cr,Fe:N [63] 128930 86.49T 1 fcc Cr,Fe:N [63] 24330 L L L L 0 fcc Cr,Fe:Va [25] 10833 7.477T 1 fcc Cr,Fe:Va [25] 1410 L L 170 [63] 162516 0 fcc Cr,Fe:N,Va L 0 fcc Fe,Ni:C [67] 49074 7.32T 1 fcc Fe,Ni:C [67] 25800 0 fcc Fe,Ni:N [64] 22710 5.19T 1 fcc Fe,Ni:N [64] 3334 L L L L 0 fcc Fe,Ni:Va [67] 12054.355 3.27413T 1 fcc Fe,Ni:Va [67] 11082.1315 4.45077T 2 fcc Fe,Ni:Va [67] 725.805174 L L L [67] 8215 0 fcc Cr,Fe,Ni:C L 0 fcc Cr,Fe,Ni:Va L [62] 6000 0 fcc Fe,Mo:C L 0 fcc Fe,Mo:Va L 0 fcc Cr,Ni:C L [67] 1618 [62] 28347 17.691T [25] 125935 95T 0 fcc Cr,Ni:Va [25] 8030 12.8801T 1 fcc Cr,Ni:Va [25] 33080 16.0362T L L 0 fcc Cr,Ni:N,Va L 0 fcc Cr,Mo:N L [68] 661270 305T [65] 40000 0 fcc Cr,Mo:Va [69] 28890 7.962T 1 fcc Cr,Mo:Va [69] 5974 2.428T 0 fcc Mo,Ni:Va [70] 4803.7 5.96T 1 fcc Mo,Ni:Va [70] 10880 L L L L Memo: (1) The values of CALPHAD parameters are given in SI units and correspond to one mole of formula units. (2) The significant digits for the CALPHAD parameters were determined during the simulation processes of the relevant phase diagrams, and no universal criteria to date are followed to determine the significant digits that should be associated with CALPHAD parameters. 171 Appendix II Evaluation of the Carbon Concentrations Measured by AES and XPS 1. Sample Used in the Evaluation In order to evaluate the carbon concentrations measured by AES and XPS systems introduced in Chapter 4, a sample of an amorphous alloy SAM 1651-4 was employed with the nominal composition listed in Table A-1. The normalized composition of all the elements except carbon is also shown in Table A-1. Table A-1 Nominal Composition of Amorphous Alloy SAM 1651-4 (at%) and the Normalized Composition of All the Elements except Carbon (at%) Element Fe Cr Mo B C Y Nominal Content 48 15 14 6 15 2 Normalized Content 56.5 17.6 16.5 7 --- 2.4 A two-gram piece of the sample was cut off and submitted to the chemical laboratory at the Swagelok Company for LECO analysis and wet chemical analysis to accurately determine the actual composition of the sample. A one-gram piece of the submitted sample was tested by LECO analysis, which shows that the carbon content in the sample is 3.6 wt%. The other one-gram piece of the 172 submitted sample was tested by wet chemical analysis in order to determine the contents of the other alloying elements in the sample. However, the test was unsuccessful, because there was a considerable amount of insoluble precipitate after dissolution of the sample. No useful information was obtained, although all the elements listed in the nominal composition of the sample were detected during the test. Therefore, assuming the actual content ratios of the other alloy elements except carbon are identical to the normalized content ratios in Table A-1, the carbon content of 3.6 wt% is equivalent to ~ 15.3 at%, which is quite close to the expected nominal carbon content. The remaining part of the sample, as a 15 mm × 15 mm × 3mm thin plate, was polished on both sides by Dr. Xiaojun Gu using P4000 grit SiC grinding paper and was pre-cleaned ultrasonically in isopropanol for 5 minutes before being analyzed by both AES and XPS. 2. Experimental Results of AES An AES line scan was performed on the polished surface of the sample after a pre-sputtering step to eliminate any contamination present on the surface, during which about 100 nm of the material was removed from the surface. The line scan was conducted in the same mode as described in Section 6.2.5 under a continuous sputtering condition. The sampling process was along a scan line 25 µm long with 32 equally spaced data points. The Auger spectra obtained were analyzed using Multipak software. The Auger peaks of the elements in the sample of SAM 1651-4 are listed in Table A-2. The corresponding concentration-distance profiles of all the elements are shown in Fig. A-1. 173 The carbon concentration-distance profile was adjusted using the calibration represented by Eq. (4-1) for carbon concentrations measured by an AES line scan under a continuous sputtering mode. The statistical average and the standard deviation of the measured compositions are listed in Table A-3. Table A-2 The Auger Peaks of the Elements in the Sample of SAM 1651-4 Element Fe Cr Mo Y C B Analyzed Auger Transition LMM LMM MNN LMM KLL KLL Energy Peak in Multipak Fe2 Cr2 Mo1 Y2 C1 B1 Peak Energy (eV) 654 531 190 1748 275 185 174 Fe Mo C Cr Y B Fig. A-1: The concentration-distance profiles of all the elements in the sample of SAM 16514 measured by an AES line scan under a continuous sputtering mode. The carbon concentration-distance profile was adjusted using the calibration as Eq. (4-1). Table A-3 The Statistical Average and the Standard Deviation of the Measured Compositions of the Sample SAM 1651-4 by AES Line Scan Element Fe Cr Mo Y C B Statistical Average (at%) 45.4 12.8 17.4 4.6 16.0 3.8 Standard Deviation (at%) 1.1 0.5 0.7 1.0 0.6 0.6 175 Table A-3 shows that the measured chemical composition is essentially consistent with the nominal composition of the alloy SAM 1651-4. The measured carbon concentration after the calibration associated with Eq. (4-1) is just slightly higher than the result of the LECO analysis, but the difference is no larger than 1 at%, which is actually an excellent agreement. 3. Experimental Results of XPS The same side of the polished sample of alloy SAM 1651-4 was examined by the XPS system. Standard surveys of the chemical composition were performed after sputtering to several different depths below the surface of the sample. The analyzed peaks in the X-ray photoelectron spectra and their corresponding binding energy values are listed in Table A-4. The X-ray photoelectron spectra for carbon were influenced by the peak signals from adventitious carbon and therefore were fit using three peaks with different binding energies as follows: (a) carbon in solid solution at 282.0 eV – 283.0 eV, (b) carbon in graphite at ~ 284.8 eV, and (c) carbon with oxygen at > 286.0 eV. 176 Table A-4 The Analyzed Energy Peaks of the Obtained X-Ray Photoelectron Spectra and the Corresponding Binding Energy Values of the Peaks for the Sample of SAM 1651-4 Element Fe Cr Mo Y C B Analyzed Peaks Fe2p1 Cr2p Mo3d Y3d C1s B1s 158.0 285.0 191.0 Binding Energy (eV) 723.0 Cr2p1 Cr2p3 Mo3d3 Mo3d5 583.0 574.0 231.0 228.0 The X-ray photoelectron spectra obtained from the standard survey after 50 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4 are shown in Fig. A-2. In Fig. A-2, the left column shows the chemical composition of all the elements in the sample and the right column shows the normalized chemical composition of all the elements except carbon. The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting are shown in Fig. A-3. After 50 nm of sputtering, there is still a significant amount of adventitious carbon detected on the surface and the peak fit shows that the signal from carbon in solid solution is only ~ 62% of the total amount of carbon, which is equivalent to a carbon concentration of ~ 14.2 at% in solid solution. 177 Fig. A-2: The X-ray photoelectron spectra obtained from the standard survey after 50 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4. 178 Fig. A-3: The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting after 50 nm of sputtering on the sample SAM 1651-4. The X-ray photoelectron spectra obtained from the standard survey after 100 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4 are shown in Fig. A-4. In Fig. A-4, the chemical composition of all the elements in the sample and the normalized chemical composition of all the elements except carbon are shown in the left and right columns, respectively. The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting are shown in Fig. A-5. After 100 nm of sputtering, it is obvious that the peak signal from the adventitious carbon decreases and the energy peak of carbon in solid solution becomes more apparent in the spectra. The peak fit shows that the signal from carbon in solid solution becomes ~ 70% of the total amount of carbon, which is equivalent to a carbon concentration of ~ 14.6 at% in solid solution. 179 Fig. A-4: The X-ray photoelectron spectra obtained from the standard survey after 100 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4. 180 Fig. A-5: The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting after 100 nm of sputtering on the sample SAM 1651-4. The X-ray photoelectron spectra obtained from the standard survey after 150 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4 are shown in Fig. A-6. In Fig. A-6, the chemical composition of all the elements in the sample and the normalized chemical composition of all the elements except carbon are again shown in the left and right columns, respectively. The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting are shown in Fig. A-7. 181 After 150 nm of sputtering, it seems that the peak signal from carbon in oxidized states with a higher binding energy > 286.0 eV almost disappears, but the energy peak from carbon as a hydrocarbon at 284.8 eV still exists. Therefore, the X-ray photoelectron spectrum of carbon was fitted by two peaks ignoring the modest signal from carbon in oxidized states, and the peak fit shows that the signal contribution from carbon as a hydrocarbon becomes relatively weak and the signal from carbon in solution became ~ 93% of the total amount of carbon, which is equivalent to a carbon concentration of ~ 14.2 at% in solid solution. 182 Fig. A-6: The X-ray photoelectron spectra obtained from the standard survey after 150 nm of sputtering and the corresponding chemical compositions of the sample SAM 1651-4. 183 Fig. A-7: The deconvoluted X-ray photoelectron spectrum of carbon and the relevant parameters of the peak fitting after 150 nm of sputtering on the sample SAM 1651-4. After 150 nm of sputtering, another standard survey was performed under a continuous sputtering mode, in which case the survey process and the sputtering process were conducted simultaneously. The sputtering rate was adjusted to be the minimum value that the instrument could achieve in order to reduce the influence of the sputtering upon the surface chemistry. The X-ray photoelectron spectra obtained from the standard survey under a continuous sputtering mode and the corresponding chemical compositions of the sample SAM 1651-4 are shown in Fig. A-8. In Fig. A-8, the chemical composition of all the elements in the sample and the normalized chemical composition of all the elements except carbon are again shown in the left and right columns, respectively. 184 Fig. A-8: The X-ray photoelectron spectra obtained from the standard survey under a continuous sputtering mode and the corresponding chemical compositions of the sample SAM 1651-4. 185 The X-ray photoelectron spectrum of carbon obtained under a continuous sputtering mode has a very clear peak as shown in Fig. A-9, which is well represented by the peak from carbon in solid solution at ~ 283 eV. The peak signal from adventitious carbon almost completely disappeared, although the setting of the baseline for the carbon peak was a little challenged by the noisy background energy values in the spectra. Therefore, the carbon concentration ~ 14.2 at% shown in Fig. A-8 can be considered as a complete contribution from the carbon in solid solution, which is consistent with the other measurements. Fig. A-9: The X-ray photoelectron spectrum of carbon obtained from the survey under a continuous sputtering mode and the relevant parameters of the peak fitting on the sample SAM 1651-4. 186 In a summary, the carbon concentration measured by the standard surveys conducted with the XPS system is in the range of 14.2 to 14.6 at%, which is slightly lower than the carbon concentration measured by the LECO analysis, but the difference is essentially no larger than 1 at%. Considering the result measured by AES, the systematic error of the measurements of a carbon concentration of 15 at% between these two characterization techniques should be less than 2 at%. Further study along a more systematic path is suggested, i.e., using ideal specimens with a range of carbon contents. 187 Appendix III The Derivation of the Thermodynamic Factor for FCC Systems In the analytical model of the Absolute Reaction Rate Theory (ARRT) proposed by Fisher et al. [8], the expressions of the thermodynamic factor in Eq. (5-2) for fcc systems can be derived using the CALPHAD model. Eq. (5-2) shows the general expression of the thermodynamic factor as follows: Thermodynamic Factor = 1 d ln C d lnYC (A-1) where γC is the activity coefficient of carbon and YC is the site fraction of carbon. The activity coefficient for the interstitial atom, γ, is defined for a Henrian standard state [8]. For a Henrian standard state, γ becomes one as the atomic fraction of the interstitial, Xj , approaches zero. For the Raoultian standard state, γ becomes a constant, oγ, as Xj approaches zero. The change of standard state from Raoultian to Henrian is accomplished by adding the term RT ln o to the expression for the chemical potential of the interstitial, j , based upon a Raoultian standard state. The CALPHAD model for interstitial solid solutions is used to formulate an expression for oγ. The chemical potential for carbon, μC, in austenite for the iron-carbon binary system is given by Eq. (A-2). 188 hfcc hfcc C oGFe:C oGFe:Va RT ln YC 1 2YC 0 Lfcc Fe:C,Va 1 YC (A-2) Eq. (A-3) represents the common concentration that μC equals that of graphite, oGCgra , when the activity of carbon is equal to one. C oGCgra RT ln aCR (A-3) The superscript indicates the activity is based upon a Raoultian standard state. Combining Eqs. (A-2) and (A-3) yields an expression for the term RT ln aCR as follows: hfcc hfcc RT ln aCR oGFe:C oGFe:Va oGCgra RT ln YC 1 2YC 0 Lfcc Fe:C,Va 1 YC (A-4) Inserting X C CR for aCR and YC / 1 YC for X C results in an expression for the term RT ln CR as follows: hfcc hfcc RT ln CR oGFe:C oGFe:Va oGCgra RT ln 1 YC 1 2YC 0 Lfcc Fe:C,Va 1 YC (A-5) As XC approaches zero, YC approaches zero and Eq. (A-5) reduces to Eq. (A-6). hfcc hfcc RT ln CR oGFe:C oGFe:Va oGCgra 0 Lfcc Fe:C,Va (A-6) Adding Eq. (A-6) to Eq. (A-3) results in Eq. (A-7), where μC is now based upon a Henrian standard state. hfcc hfcc H C oGFe:C oGFe:Va 0 Lfcc Fe:C,Va RT ln aC (A-7) Equating Eqs. (A-2) and (A-7) results in the following expression for RT ln aCH . 189 RT ln aCH RT ln YC 2YC 0 Lfcc Fe:C,Va 1 YC (A-8) Inserting X C CH for aCH and YC / 1 YC for X C results in an expression for RT ln CH . 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