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Uwaterloo Latex Thesis Template Structural Disjoining Potential of Grain Boundary Premelting in Aluminum-Magnesium via Monte Carlo Simulations by Tara Power A thesis presented to McMaster University in fulfillment of the thesis requirement for the degree of Master of Science in Physics Hamilton, Ontario, Canada, 2013 c Tara Power 2013 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Premelting is the formation of a thin, thermodynamically stable, liquid-like film at the interface for temperatures below the equilibrium melting temperature. Using a Monte Carlo technique, the underlying short range structural forces for premelting at the grain boundary can be directly calculated. This technique is applied to a (i) P 9 h115i 120o twist boundary and a (ii) P 9 h011i f411g symmetric tilt boundary in an embedded atom model of Aluminum-Magnesium alloy. Both grain boundaries exhibit disordered structures near the melting point that depend on the concentration of Magnesium. The behavior is described quantitatively with sharp interface thermodynamics, involving an interfacial free energy that depends on width of the grain boundary, referred to as the disjoining poten- tial. The disjoining potential calculated for boundary (i) displays a decreasing exponential dependence on width of the grain boundary, while the disjoining potential of (ii) features a weak attractive minimum. This work is discussed in relation to a previous study using pure Nickel, results of which can be useful to the theoretical study of thermodynamic forces underlying grain boundary premelting in an alloy. iii Acknowledgements I would like to especially thank my immediate supervisor, Dr. Jeff Hoyt, for his great interest and assistance in the pursuit of these studies and preparation of this thesis. I would also like to thank him for his patience and understanding. No matter how many photo-shopping study breaks were taken, or lab pranks were pulled, he took it with very good humor. Thank you. I would also like to thank Liz and Mary, who were my partners in crime. We came up with some pretty awesome shenanigans. Besides my advisor, a special thanks to my thesis committee, Dr. Nik Provatas and Dr. Bruce Gaulin for their guidance, helpful comments and encouragement. To Wilson, Harith and Rameez, thank you for all your help. You guys are a great technical support team. Whenever I needed help programming in a new language, you guys were there and I am definitely a better programmer for it. To David, Nana, Jonathan, Joel, M.J., and everyone else in the ACSR, you guys have been awesome! Lastly, I would like to thank Nino for being there no matter what. The past two years have not been the smoothest in terms of health, but whenever I needed help you were there. Thank you for carrying my backpack to school when it was too heavy with crutches, for all the coffee breaks and baked goods. Thanks for everything! iv Dedication This thesis is dedicated to Nino, Lee-Anne and Cleary. v Table of Contents List of Figures viii 1 Introduction1 1.1 Introduction to Surface Melting........................2 1.2 Regelation....................................2 1.3 Pressure Melting................................3 1.4 Why is ice Slippery?..............................4 1.5 Frictional Heating................................4 1.6 Premelting....................................5 1.7 Not limited to Ice................................7 2 Grain Boundary Basics9 2.1 Crystal Structure................................9 2.1.1 Lattice Geometry Identification....................9 2.1.2 Cubic Crystal Structure........................ 12 2.1.3 Lattice Parameter............................ 12 2.1.4 Dislocations............................... 13 2.2 Grain Boundaries................................ 15 2.2.1 Twist and Tilt Boundaries....................... 16 2.2.2 Misorientation.............................. 17 2.2.3 Coincidence-Site Lattice (CSL) Model................ 20 2.3 Relevant Grain Boundaries........................... 21 vi 3 Premelting Physics 22 3.1 Grain Boundary Premelting.......................... 22 3.2 The Disjoining Potential............................ 25 3.2.1 Calculating Disjoining Potential.................... 27 3.3 Thermodynamic Theory............................ 28 3.3.1 Gibbs Free Energy........................... 29 3.3.2 Chemical Potential for Ideal Solution................. 30 3.3.3 Semi-Grand Canonical Ensemble................... 32 4 Simulation Methodology 34 4.1 Embedded Atom Method............................ 35 4.2 Finnis-Sinclair Potentials............................ 37 4.3 Monte Carlo Computing............................ 37 4.3.1 Monte Carlo Technique......................... 37 4.3.2 Equilibration.............................. 40 4.4 Simulation Methodology............................ 41 5 Results and Discussion 44 5.1 Grain Boundary Widths............................ 45 5.1.1 Concentration Profiles......................... 46 5.1.2 Nearest Neighbor Analysis....................... 48 5.2 Probability Distributions............................ 51 5.3 The Disjoining Potential............................ 52 6 Conclusions 56 References 58 vii List of Figures 2.1 A simple cubic unit cell with arbitrary placed origin............. 10 2.2 A simple cubic unit cell with vector r; miller index would be: [212].... 11 2.3 A simple cubic unit cell containing a plane; miller index of the plane would be: (111)..................................... 11 2.4 A Face-centered cubic (FCC) unit cell, with lattice parameter a....... 12 2.5 Dislocations; edge (left) and screw (right)................... 14 2.6 Edge (top) and screw (bottom) dislocations, including location and direction of the burgers vector [87]............................ 15 2.7 Small-angle tilt boundary (a) Misorientation angle θ. (b) Dislocation core separation. [88]................................. 16 2.8 Small-angle twist boundary (a) Top view of an unrelaxed lattice with twisted grains. The misorientation angle is θ. (b) Dislocation core separation in a relaxed lattice. [88]............................... 17 2.9 Calculated (Left) and measured (Right) grain boundary energies of [110] tilt boundaries in Aluminum as a function of misorientation (between [001] directions) [37]................................. 19 2.10 Calculated (Left) and measured (Right) grain boundary energies of [100] tilt boundaries in Aluminum as a function of misorientation (between [001] directions) [101]................................ 20 2.11 Schematic of a P 7 coincident site lattice (CSL) model. 1 in 7 lattice sites are coincident between the two lattice regions. [95]............. 21 3.1 (a) A droplet is on the interface. (b) If the contact angle θ goes to zero, the droplet spreads out to a thin layer....................... 23 viii 3.2 Premelting inside a grain boundary; with width of the premelted layer, w, the cross sectional area, A, and length of the system, l............ 24 3.3 Incomplete wetting in the grain boundary. The premelted region does not span the interface but is in contained regions................. 26 3.4 Probability distributions of width (left) were used to calculate a disjoin- ing potential (right) for grain boundary premelting in pure Nickel, for two different grain boundary orientations. [27, 40]................ 28 3.5 Example of the general shape for a Gibbs free energy vs concentration curve. Concentration is listed as mole fractions of Magnesium; where pure Alu- minum is XMg = 0 and XMg = 1 is pure Magnesium............. 31 3.6 Location of chemical potential of aluminum and magnesium on the example Gibbs free energy curve. The slope is the chemical potential difference, ∆µ = µMg − µAl................................. 32 4.1 Example Gibbs free energy of a binary alloy curve. Each chemical potential difference has a equilibrium concentration of the system. Slopes of Dark and light lines tangent to the curve equal different chemical potential differences and the location of the equilibrium concentration............... 40 4.2 Equilibration of pressure and energy for a P 9 h011i f411g symmetric twist boundary in Al-Mg, with a chemical potential difference of 1:715 eV=atom. 41 4.3 Simulated Phase diagram of the Al-Mg system calculated from the Finnis- Sinclair potential [66].The expected region for premelting is highlighted above. 42 5.1 Aluminum-Magnesium lattice, including grain boundary. Atoms are shaded by the coordination number........................... 44 5.2 This figure shows the relationship between chemical potential difference, ∆µ, and concentration of Magnesium. The dotted line represents an exponential fit......................................... 45 5.3 The width of the grain boundary is highly dynamic in thermodynamic equi- librium. Atoms are shaded by the coordination number and the solid line in the inner section represents the approximate width of the grain boundary. 46 5.4 Superposition of many histograms of the concentration of Magnesium across the lattice. The grain boundary is denoted by the elevated concentration of Magnesium. The width of the grain boundary is approximated at w..... 47 ix 5.5 Superposition of many histograms of centro-symmetry parameter across binned x-position in the system. The grain boundary is located in the ele- vated region of CSP. The approximate average width of the grain boundary is w........................................ 50 5.6
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