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 {411} 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 {411} 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 Probability distributions P (w), of a P 9 Tilt boundary (top) and a P 9 Twist boundary (boundary) found through width fluctuations at 880 K. (note: connecting lines are not a fit, just an aid to better see the distribu- tions.)...... 52 5.7 The disjoining potential for a P 9 tilt boundary (left) and a P 9 twist boundary (right). Each part of the curve is symbolized by the associated chemical potential difference. The tilt boundary is fit to a double exponential and the twist is fit to an exponential curve...... 54 5.8 The disjoining potential for a P 9 twist boundary and a P 9 tilt boundary in both alloy (left) and pure (right) cases...... 55

x Chapter 1

Introduction

Premelting is a term referring to the melting at an interface in a material before the melting point has been reached, as approached from below. The liquid-like layer is thermodynam- ically stable, even though the temperature is lower then the bulk melting point. It is a strange concept to except immediately. A Liquid-like layer occurring naturally below the bulk melting temperature doesnt seem thermodynamically sound. But the process of pre- melting does in fact occur in nature. To start off, I would like to first describe the concept of surface melting, which brought attention to premelting, in order to fully understand the process involved. The next chapter of this thesis takes a step back from the topic of premelting and introduces crystallography and grain boundary basics. Crystal structure greatly influences many properties of a material, thus it is important to understand the basics in order to properly describe premelting. Grain boundary basics are also outlined in this section, including a description of the two different boundaries used in this thesis; (i) P 9 h115i 120o twist grain boundary and (ii) P 9 h011i {411} 38.9o tilt grain boundary. Chapter 3 explains the physics involved with grain boundary premelting, and introduces a term “Disjoining potential” which contains interfacial free energy terms. The disjoining potential has a complicated dependence on width of the grain boundary, which varies depending on orientation. The purpose of this thesis is to calculate the disjoining potential for the two different grain boundary orientations introduced in Chapter 2 for an Aluminum- Magnesium alloy. Using the embedded atom method of Aluminum-Magnesium alloy, the disjoining po- tential can be calculated via Monte Carlo simulations for both boundaries. Chapter 4

1 outlines the simulation theory involved with embedded atom method as well as Monte Carlo simulations, and summarizes the Monte Carlo simulation technique. This thesis concludes with the results of the disjoining potential analysis of boundaries (i) and (ii).

1.1 Introduction to Surface Melting

We can begin this thesis by answering a question: why is ice so slippery? It is commonly thought that pressure under the skaters blade melts the ice, accounting for the low coef- ficient of friction on the ice surface. As you will see below there are major flaws in this theory, to the point that it has been disproved altogether. Another theory, which was first brought to light over a hundred and fifty years ago, is the key to fully understanding this problem.

1.2 Regelation

The first person to initiate the investigation into surface melting was Michael Faraday in 1842, when he was simply recording his thoughts on the strange behavior of ice and snow [23]. This began the series of investigations spanning the next 20 years on the topic of surface melting. One of the observations about snow, taken from an excerpt of his diary, was the ability to pack. He noticed that when wet snow is packed together it stays in a lump with water in between, and does not fall apart the way most kinds of matter would; like wetted sand. By preforming simple ice experiments, Faraday hoped to better understand some of these strange properties. One of these simple experiments proved to be significant in the field of surface melting. By wrapping 2 pieces of ice in flannel on a day where the external temperature was near, but under, the melting point of water, Faraday found that the place where the 2 pieces of ice made contact was frozen together; fusing into 1 piece of ice. This effect was known as regelation. [24] Faraday and others [24, 26, 30, 96] preformed similar regelation experiments and con- vinced themselves that a liquid-like film is present on the surface of ice at equilibrium. He also noticed that if this film is sandwiched between two pieces of ice the film is no longer thermodynamically stable. The film changes state and the two pieces solidify together. Faraday’s simple experiment on ice lead him to believe there was a thermodynamically

2 stable liquid-like film over the surface of ice for temperatures below 0oC. He was the first to look into surface melting and started the debate, which would ultimately lead to the topic of premelting.

1.3 Pressure Melting

In response to Faraday’s experimental conclusion on regelation, James Thompson voiced some of his concerns with Faraday’s theories [91, 92, 94]. Both Thompson and his brother William, known later as Lord Kelvin, came up with a rebuttal based on their growing work in pressure melting. Earlier in Thompson’s work he discovered that pressure lowers the freezing point of water, an idea originating from le Chatelier’s principle.

From le Chatelier’s principle, an increase in pressure of the system will result in a change on volume. Water is denser than ice, which results in a 10% decrease in volume per unit mole. As a result, an increase of pressure on ice will decrease the volume of the system by melting the ice surface where the pressure is being applied. So ice under a scale pressure melts, to produce a water film. In 1850, Thompson developed an expression for the linear dependence of freezing point on pressure, confirmed experimentally by his brother. [54]

Thompson stated that the two pieces of ice in Faraday’s experiment could not touch without a pressure being supplied. This pressure would lower the freezing point at the place where the particles touched resulting in a liquid-like layer that would then solidify at the place of contact due to the release of pressure on either side of the ice cubes. He suggested this was the real reason behind Faraday’s regelation results. After receiving Thompson’s response to his experiment, Faraday devised another ex- periment to exclude pressure from factoring into his results [25]. He created a system under water; at constant 0oC. Faraday observed two pieces of ice floating in the water. When the 2 pieces came near each other they repelled one another. If brought into contact they froze together. Regelation occurred and the blocks of ice were adhered to each other with no external factors (ie. Pressure) influencing the results. In Thompson’s note to Faraday in the Proceedings of the Royal Society of London, he commended Faraday on the quality of his more recent experiment, minimizing pressure from the scenario, but Thompson saw the experiments as further proof of his own pressure theories [93]. Since the surface contact of the ice was little more then a geometrical point,

3 the intensity of the force must be large. He thought the pressure on this point was enough for pressure melting to occur. Thompson remained adamant that pressure melting was the cause for regelation in Faraday’s experiments. While Thompson was unconvinced of Faraday’s reasoning behind regelation, another scientist named John Joly started investigating pressure melting in a way that no one had investigated before [42]. John Joly (1886) was the first to relate regelation, and past works of Faraday and Thompson, to ice-skating.

1.4 Why is ice Slippery?

It can be taken as common knowledge that liquid over a surface makes the surface slippery because liquid is more mobile, lowering the coefficient of friction. So the ability to skate on ice is directly related to a liquid or liquid-like film located under a skater’s blade. According to Thompson’s pressure melting, this could occur due to pressure under a blade melting the ice surface into a thin film of water. The pressure under a skate blade is quite large due to the large amount of force dis- tributed over a small surface area. However, this does not account for the actual coefficient of friction of ice except at near melting point temperatures. Testing Thompson’s pressure melting theory, Joly calculated that under a skate blade the amount of pressure corre- sponded to a temperature of −3.5oC. A problem can immediately be seen with this result. The optimal temperature for skating is −5.5oC and for playing hockey −9oC. Also, Skating on ice is possible below −20oC which according to the pressure melting theory is outside the limits for pressure melting to occur. Another thought came to light in 1939 from Bowden and Hughes [8], not about skating, but rather skiing. As reported by Robert Falcon Scott [83] on one of his excursions in Antarctica, the optimal temperature for skiing is −30oC and turns to a sand-like powder below −46oC [83]. Bowden and Hughes tested the pressure applied by skis on the snow for pressure melting and discovered that the pressure is too small for melting at these temperatures. This would suggest another mechanism is at work here.

1.5 Frictional Heating

The idea of frictional heating came from Bowden and Hughes. It is a very intuitive argu- ment. While skiing or skating the blades slide against the snow or ice producing friction, or

4 heat. This heat would melt the ice or snow on contact, thus producing a liquid-like layer to slide over. They did an extensive set of experiments into frictional heating, investigating the static and kinetic coefficients of friction using different materials for skis. Pressure heating depends on the pressure exerted over a surface area, not material. If calibrated according to mass of material, there shouldnt be a difference in coefficient of friction under the blades. Upon completion of the experiments there was a noticeable change in the fric- tion coefficients. They concluded that pressure heating was not the mechanism responsible for the layer of water required on the snow to ski, and it was the friction that was creating the layer of water. Further evidence for frictional heating over pressure melting was found in an experiment done by Colbeck [16]. By attaching a thermocouple to a ski blade, as velocity went up, the temperature rose about the blade; which is consistent with frictional heating. If pressure melting were the mechanism, the temperature would go down as velocity increased due to the endothermic nature of pressure melting. But there exists flaws with the frictional heating argument as well. De Koning and De Groot studied the friction of ice during speed skating [46]. The resultant trends did not support frictional heating because the coefficient of friction measured was exceedingly lower then the expected due to frictional heating alone. The result on the other hand supported the argument of Faraday’s thermodynamically stable film on the surface of ice.

1.6 Premelting

The slipperiness of ice and snow for temperatures near the melting point can be described by pressure melting, but not any lower. Friction heating would explain melting, but it is dependent on a velocity. But ice is still slippery for temperatures far below the melting temperature, and even when standing still. By what mechanism is this possible? To answer this, scientists had to pursue a line of inquiry that was dormant for nearly a hundred years; Faraday’s regelation. With a thermodynamically stable liquid-like film on the surface of ice, it is possible for ice to be slippery in all of the conditions mentioned above. Research on this topic resumed in 1949. Gurney [36] stated that there is an intrinsic liquid-like film over ice. This layer forms due to the instability of the molecules on the sur- face because they dont have a stable layer of molecules above them. The surface molecules migrate inward to the bulk of the solid, which makes the surface unstable. This results in a liquid-like phase. Another scientist agreed with Faraday’s conclusions of regelation. Building on the work

5 of Gurney, Weyl [101] developed a model based on the difference between bulk and surface water molecules. He made qualitative arguments supporting Faraday’s (and Tyndall [96]) hypothesis of a thermodynamically stable liquid-like layer on the surface of ice, using justifications from the laws of thermodynamics. In the mid fifties more research groups were getting involved in surface melting research, starting with recreating Faraday’s first regelation experiment and creating experiments similar in design. [33, 72] In 1963, J. W. Telford and J. S. Turner [90] preformed another type of regelation experiment that Thompson previously used to support his pressure melting argument.

Wire, under pressure, will migrate through a block of ice under appropriate temperature conditions. [91]

To test if pressure melting or frictional heating were the cause of this phenomenon, measurements of the velocity through the block were made. It was shown that the velocity increased linearly with temperature from −3.5oC to −0.5oC, and increased sharply from −0.5oC to the melting point. In this particular experiment under this specific force and size of the wire used, the pressure would decrease the melting point by −0.5oC. This was the first indication that pressure melting was not the cause for regelation in this experiment. Also, there is a quasi-static nature to this experiment due to the exceedingly slow velocities, indicating that frictional melting was not the culprit either. Telford and Turner came to the conclusion that there was a thin layer of viscous fluid on the surface of the ice. This fluid preformed Newtonian shear about the wire, and this was the reason for the migration. The mid sixties brought more experimental approaches into premelting, preformed under a diverse range of conditions. Most were focused on determining the temperature range of premelting and the thickness of the liquid-like layer. Michael Orem and Arthur Adamson preformed one of these experiments in 1969 [75]. He looked at the absorption of hydrocarbon vapors on ice compared to liquid water on the surface of ice. It was concluded in these experiments that the onset of premelting occurs at −35oC. This is consistent with previous reports of ice losing its slipperiness around −30oC.[83] Many different techniques since then have provided strong evidence supporting Fara- day’s idea of a liquid-like film on ice. A convincing experiment was done with X-ray diffraction in 1987 [47], measuring the intermolecular distance on ice surface. The results show direct evidence that there exists a quasi-liquid layer on ice surface near 0oC, which has no long-range order.

6 In 1998, Astrid Doppenschmidt and Hans-Jurgen Butt used atomic force microscopy (AMF) on ice and physically measured the liquid-like layer on the surface [19]. Noting that capillary forces would make the tip jump in softer areas, the onset of premelting was determined to be at −33oC. They also tested premelting when introducing impurities of salt into the system, noting that the width of the liquid-like layer increases when salt is present. Many other experiments were executed that provide evidence of an inherent liquid- like layer on ice, including using neutron magnetic resonance (NMR) [52, 71]], Proton backscattering [34, 59], glancing-angle X-ray scattering [48], and X-ray reflectivity [20]. All these experiments provide evidence supporting the existence of a liquid-like layer on ice surface under 0oC, as well as other specific details about the state of the molecules on the surface. With more knowledge of the trends of surface melting and the increasing capability of computers, computational modeling became an important tool to increase the knowl- edge base of surface melting. Surface melting has been investigated using computational simulation by Molecular dynamics (MD) [89, 43, 27, 40], Phase field methods (PF, PFC) [15, 11, 60, 73, 64, 21] and Monte Carlo (MC) [4,3] simulations.

1.7 Not limited to Ice

Though ice had been used as the prime example of surface melting for over a century, the effect is not limited to this material. In 1985, Joost W. M. Frenken and J. F. van der Veen preformed another type of premelting experiment altogether, completely separated from the idea of surface melting on ice [31]. They sent a beam of ions to a lead crystal and monitored the scattering and noticed that lead has a melting transition far below the bulk meting point. In this phase the surface becomes completely disordered. Through many other studies and experiments, it has been verified that surface melting occurs on semiconductors, metals, molecular solids and rare gases [80]. Melting before the bulk melting point is not limited to the surface of materials either. It also takes place in interfaces and grain boundaries. When the effect is not bound to the surface, surface melting isnt a suitable term to describe what is happening. Premelting encompasses all of these cases as it occurs at interfaces, whether they be solid-vapor or inside a material. Though these experiments complemented Faraday’s theories in the 1800s, the nature of this liquid-like layer is still not clear from experiment alone. Since then, many studies have

7 been preformed under varying conditions, using different substances, through simulations and experiments in a lab. With so many varying conditions it is difficult to compare results to make accurate conclusions about premelting. In order to get a better idea of the underlying mechanisms, a more comprehensive study is needed for a single substance under varied conditions.

8 Chapter 2

Grain Boundary Basics

Coming from a physics background, there are many elementary engineering topics that I did not encounter prior to entering my Masters program. In order to fully understand the terminology involved with premelting, and this thesis, we must first break down the topic of premelting into its most fundamental parts. Contained in this chapter is a brief summary of some of the basic crystallography and material science topics that are relevant to my thesis, most of which were not encountered in physics courses.

2.1 Crystal Structure

Since a crystals structure can greatly influence the properties of the parent material, it is important to understand the basic lattice geometry, as well as how it is analyzed in an engineering setting. Starting with the terminology for crystallographic orientations outlined by miller indices. The next part outlines how to quantify the lattice structure through details of the face-centered cubic crystal system. This section ends with a brief description of dislocations.

2.1.1 Lattice Geometry Identification

There are many properties that depend on the geometry of crystals including the speed of both light and sound, strength of the material, conductivity, etc. Orientation of the crystal itself can vary these properties as well. Because of this, it is beneficial to be able to specify certain directions and planes in a crystal structure.

9 A method to describe orientation of planes and directions in a crystal lattice was de- veloped by William Hallowes Miller [69]. Since a crystal is a periodic structure, the whole structure can be described as a multiple of the smallest repeatable component. This is called a unit cell. Miller indices describe the directions and planes in this unit cell based off an arbitrarily placed x, y, z origin. Miller indices are expressed using the variables h, k, and l for both planes and directions. The difference between plane and direction nomenclature is the use of specific brackets; [hkl] for directions and (hkl) for planes.

Figure 2.1: A simple cubic unit cell with arbitrary placed origin

In order to calculate the miller indices of a direction, a vector must be constructed, which begins at the origin and terminates at a lattice point; see figure 2.2. The vector r can be written as:

r = hx + ky + lz

where x, y and z are the basic vector directions and h, k, and l are the miller indices. Since the length of a side of the unit cell is equal to one, it is possible to run into fractions when calculating the vector coordinates. Multiplying all components by the common denominator will eliminate any fractions in the indices. If there is any negative index, it is indicated by a line over the number. In order to get the miller indices of a plane, an origin must be placed on a lattice site that is not touching the plane; see figure 2.3. The x, y, z intercepts of the plane are calculated in terms of unit cell dimensions. The reciprocal of these coordinates are the miller indices of the plane. As before, a line over the number denotes negatives and

10 Figure 2.2: A simple cubic unit cell with vector r; miller index would be: [212] multiplying by the common denominator eliminates fractions. As a note, if an intercept is at infinity, the reciprocal (hence the miller index) is zero.

Figure 2.3: A simple cubic unit cell containing a plane; miller index of the plane would be: (111)

In a cubic crystal structure, directions or planes with the same miller indices, regardless of direction or order, are equivalent due to symmetry. A group of equivalent directions or

11 planes is called a family. The bracket notation for a family of directions or planes is hhkli and {hkl} respectfully. With Miller indices, referring to one unit cell can reference the entire lattice. It is impor- tant in material science to have this notation for atomic planes because several properties of a material depend on crystallographic orientation; such as optical properties, surface tension and dislocations.

2.1.2 Cubic Crystal Structure

The cubic crystal system is one of the most common and simplest shapes found in nature. This is the case where the unit cell of the crystal system is in the shape if a cube. All cubic systems used in this thesis are of the cubic type, more specifically the Face-Centered Cubic crystal type, abbreviated FCC.

Figure 2.4: A Face-centered cubic (FCC) unit cell, with lattice parameter a.

The FCC unit cell is composed of 4 atoms. Each face of the cube has lattice sites containing half an atom. There are lattice sites on each of the 8 corners, contributing 1/8 atom each. Hence by adding up all contributions of atoms contained within the unit cell the total number of atoms per FCC unit cell is four.

2.1.3 Lattice Parameter

The dimension of a unit cell is defined by lattice parameters, a, b, and c (see figure 2.1). Each one corresponds to the difference to the next unit cell in different directions of 3

12 dimensional space, x, y, and z. For cubic structures the difference in all directions to the next unit cell is the same, so all lattice parameters are equal to a. In cubic systems the miller indices of both direction and plane can be taken as the directions in Cartesian coordinates because the lattice parameter is equal in x, y and z. Because of this we can use the miller indices to calculate the distance between adjacent lattice planes:

a dhkl = √ (2.1) h2 + k2 + l2 The distance between adjacent planes becomes important in simulation to keep strain at a reasonable level when establishing a crystal lattice.

2.1.4 Dislocations

No matter how seemingly perfect a crystal structure may seem, real world materials are filled with defects. Defects in a lattice structure are divided into four groups: point defects (vacancies, interstitial atoms), line defects (dislocations), planar defects (stacking faults) and volume defects (voids). Dislocations are a one-dimensional defect found in almost all crystalline materials. There are two archetypes of dislocations; a pure edge dislocation and pure screw dislocation. Edge dislocations occur when an extra halfplane is inserted into a perfect crystal lattice. The dislocation propagates parallel along the plane. A screw dislocation is of a helical nature and runs perpendicular to the dislocation line, circling like the thread of a screw. Though these are only the extremes for possible dislocations, most are a hybrid of screw and edge. Dislocations have a stress field surrounding them, creating internal strain in the crystal. A large part of this dislocation strain energy can be accounted for by the elastic strain in the lattice having to accommodate the dislocation. The surrounding planes bend around the dislocation so that the crystal structure is perfectly ordered on either side of the dislocation. The strain energy on screw dislocations is typically 50% smaller than edge dislocations. [87] In order to mathematically describe a dislocation within the lattice structure, a disloca- tion core model was invented, based off of previous dislocation models [51]. This introduces the concept of a dislocation core, the center of a dislocation determined by a radius ro such

13 Figure 2.5: Dislocations; edge (left) and screw (right). that the stress at the surface of the core is zero. This core is what breaks up the nor- mal lattice structure and introduces a strain field. By working with the dynamics of the cores, we can describe the stress and strain fields and the elastic energy of an interface. The dynamics of these dislocation cores will become important with the concept of grain boundary orientation later in this thesis.

Burger’s Vector

The Burgers vector [14] is a precise measurement of the magnitude and direction of the shear caused by a dislocation. It is best understood by first picturing a perfect crystal structure, without the dislocation. In this perfect crystal, a closed circuit can be drawn with lengths and widths equal to that of its unit cell. Next, the dislocation can be introduced, encompassed by the circuit. The dislocation will deform the perfect lattice as well as the closed circuit. The circuit will now be disjoined, having sheared perpendicular in some direction, leaving an opening. This opening defines the burgers vector, both in direction and magnitude; see figure 2.6. The burgers vector is written in the same fashion as the miller indices: [hkl]. In edge dislocations, the burgers vector and dislocation line are perpendicular to one another. In screw dislocations, they are parallel [44]. The magnitude of the burgers vector, [hkl], is calculated by:

a√ kbk = h2 + k2 + l2 (2.2) 2 With the lattice parameter as a. The magnitude of the burgers vector is important in the description of a dislocation, and ultimately the dynamics of dislocation cores.

14 Figure 2.6: Edge (top) and screw (bottom) dislocations, including location and direction of the burgers vector [87].

2.2 Grain Boundaries

A materials structure is not limited to a single orientation. In fact virtually every metal we come into contact with is separated into different crystal sections, called grains, with differing orientations with respect to each other. We can think of these polycrystalline solids like a puzzle. The pieces of this puzzle are the grains, and the spaces in between the grains are the grain boundaries. Grain boundaries are the interfaces between crystals of orientation mismatch, and are a type of planar defect in a crystalline material. The width of a grain boundary can range from two molecular spacings to an appreciable number of interplanar spacings. The physical properties of metals are dependent on the physical properties of the inter- faces in the system, so the study of metal interfaces is an important topic. In this section we will outline grain boundary basics, including types of grain boundaries and how grain misorientation about a grain boundary effects a material.

15 2.2.1 Twist and Tilt Boundaries

All grain boundaries or interfaces can be described by dislocation arrangements. One of the simplest types of grain boundaries is called a tilt boundary. It is made up entirely of edge dislocations. Tilt boundaries are created when two grains are brought into contact, but are at differing orientations. There is only one-degree-of-freedom in a tilt boundary, the angle θ where the grains meet. The resultant lattice planes across the grain boundary take a chevron form, adapting around the introduced dislocations.

Figure 2.7: Small-angle tilt boundary (a) Misorientation angle θ. (b) Dislocation core separation. [88]

The simplest type of grain boundary that is made up entirely of screw dislocations is called a twist boundary. The twist grain boundary is also a one-degree-of-freedom boundary and is created when the two grains rotate about an axis normal to the boundary. It can be viewed as holding the two grains, one in each hand, and twisting them opposite ways. The angle θ measures the misorientation perpendicular to the dislocation plane. Since edge and screw are only the extremes to describe dislocations and most are a combination, there are many other types of grain boundaries that are not pure tilt or

16 Figure 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] twist, but a combination of both. Nevertheless, the grain boundaries relevant to this thesis are pure tilt and pure twist boundaries.

2.2.2 Misorientation

Twist and tilt boundaries as well as all of the hybrids are created when the grains have some shift in angle about the grain boundary. The misorientation angle is a measurement of this orientation change. The properties of a grain boundary are greatly influenced by how the grains are oriented with respect to each other, which makes the misorientation angle important to know when studying a material. Typically grain boundaries are split into two distinct types depending on the misori- entation angle; either low angle grain boundary (LAGB) or high angle grain boundary (HAGB). Classically, the transition region between low angle boundaries and high angle boundaries occurs at 15o for cubic materials, which is more of a ballpark figure and was decided upon somewhat arbitrarily [77, 79]. It is commonplace to see the transition angle listed between 10o − 15o [87, 104,9].

17 Grain Boundary Energy

It is generally accepted that the energy of a grain boundary originates from the fact that a grain boundary distorts a perfect lattice, and atoms must shift from ideal positions to make way for this crystalline defect. So the energy is a measurement of the redistribution of atoms around a grain boundary from their positions in a perfect lattice.

Low Angle Grain Boundaries

Read and Shockley (1950) used their well-established dislocation model of grain boundaries to describe how the energy varies with angle, which has been successfully compared with many experimental cases [79]. They showed that the energy of a low angle grain boundary steeply rises with angle, and a high angle grain boundary plateaus with increasing angle (see figure 2.9). When combining the dislocation core energy and the dislocation density, a method for estimating low angle grain boundary energy was devised. They started with a symmetric tilt boundary, where the dislocation density is given using the burger’s vector , b, by:

b b D = ≈ (for small angles) (2.3) θ θ 2 sin 2 The energy of a grain boundary can be found by envisioning a grain boundary as a wall of dislocations. From the spacing of dislocations and the core energy of each dislocation, Read and Shockley predicted the energy of a low angle tilt grain boundary in what is known as the Read-Shockley equation:

γG = γoθ (A − ln θ) (2.4)

where the misorientation angle is b , γ = Gb , A = 1 + ln ( b ), G is the shear D o 4π(1−ν) 2πro modulus, ν is Poissons ratio and ro is the radius of the dislocation core. This figure shows that the misorientation angle has a more significant effect on the grain boundary energy with lower angles, as there is a steep rise in energy with small changes in misorientation. When the grain boundary has transitioned into a high angle grain boundary the energy remains relatively constant over misorientation.

18 Figure 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]

High Angle Grain Boundary

Though many details of low angle grain boundaries, lattice imperfections and structure have been available for a long time, the structure of high angle grain boundaries has proven more illusive. There is no current universal theory for high angle grain boundaries. Though no universal theory exists at the moment, there are abundant experimental results for high angle grain boundaries which show a general pattern. As mentioned above, the grain boundary energy plateaus for high angle boundaries, but there are special orien- tations which decrease the grain boundary energy. These cusps in the grain boundary energy represent the location of special grain bound- aries; specific orientations that lower the grain boundary energy; figure 2.10. Experiments using high purity material and refined experimental methods have made advances, and systematic computer modeling has proven to be an extremely valuable tool in establishing high angle properties. In fact, there have been extensive atomistic modeling to locate the minimum energy configurations; specifically computer modeling of a number of grain boundaries of FCC crystals [87, 49, 32, 37]. The atomistic simulations are essential for better understanding for high angle grain boundary structural formation.

19 Figure 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]

2.2.3 Coincidence-Site Lattice (CSL) Model

In 1949, Kronberg and Wilson [51] proposed a model to classify special grain boundaries, called the coincidence-site lattice model. If we consider a superposition of the two crystals in the grain boundary system, visualized in figure 2.11, there are certain places with atomic sites in common. These are called coincidence sites. Taking perfect alignment of the superposition to be the minimum Gibbs free energy state, if there are a high number of coincidence sites, the grain boundary energy is low because there are minimal broken bonds across the boundary. If there is low number of coincidence sites then the number of bonds broken across the grain boundary is high, and the grain boundary energy is high. The parameter is the reciprocal of the coincidence site density and is used to characterize the coincidence-site lattice. The sigma value is defined as the ratio between area enclosed by a unit cell of the coincidence sites, and the standard unit cell. In cubic lattices, the miller index between misorientation of the grains can be used to calculate the sigma value:

X = δ h2 + k2 + l2
(2.5)

If (h2 + k2 + l2) is odd then δ = 1 and if (h2 + k2 + l2) is even then δ = 1/2. This is because in cubic systems all P values are odd. The parameter is simply a geometric criterion that classifies the misorientation of the

20 Figure 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] two grains. It gives no information on grain boundary atomic structure, or orientation. Despite this, CSL is used as nomenclature to characterize individual grain boundaries in many areas of science and engineering.

2.3 Relevant Grain Boundaries

Now that the terminology is laid out and we have a basic understanding of grain boundaries, the types of grain boundaries used in this thesis can be adequately described in the proper nomenclature. There are two different types of grain boundaries being used in this thesis. One of the grain boundaries is a P 9 h115i 120o twist grain boundary. The other grain boundary is a symmetric tilt grain boundary with orientation P 9 h011i {411} 38.9o.

21 Chapter 3

Premelting Physics

We discussed earlier the concept of surface melting starting from 1854 with Faraday’s ex- periments on ice and snow. Since that time there have been advances in techniques for direct observation of surface melting [25]; for example, the proton scattering experiment of Franken and Van Der Veen [31]. Not only are there experimental studies into premelting, but also there have been a number of atomistic simulations to compliment the experiments [11, 64, 40]. However, premelting on surfaces is not the only location that has been wit- nessed. Premelting at solid-solid interfaces has also been reported, and can take two basic forms: premelting at solid-solid heterophase boundaries and grain boundary premelting. Grain boundary premelting is the formation of a thin, thermodynamically stable, liquid- like film at the interface for temperatures below the bulk melting point [27, 40]. Premelting at grain boundaries has been the subject of numerous atomistic and experimental studies, and even still, the structural forces driving premelting remain moderately understood at best. The purpose of this chapter is to investigate the physics behind premelting within a grain boundary, and the mathematical process to isolate the structural forces responsible.

3.1 Grain Boundary Premelting

Prior to 1984, a large number of atomistic simulations had been performed on ideal grain boundaries in the ground state configurations, using both two-body potentials and molec- ular statics, but not much had been compiled for real grain boundaries. In order to remedy this, studies were expanded to include Leonard-Jones forces and temperature changes ap- proaching the melting point, Tm, from below.

22 In 1984, Broughton and Gilmer [12, 10] preformed a study of melting at grain bound- aries, where atoms interacted under truncated Leonard-Jones forces. As the temperature was raised towards the triple point1, a disordered liquid-like film grew at the interface. Using these results in 1986, Lipowsky [56, 57, 58] related the melting phenomena observed by Broughton and Gilmer to make new predictions for melting at boundaries. Lipowsky investigated premelting within grain boundaries and began his investigation by considering the solid-liquid coexistence curve on a phase diagram. Supposing a droplet of the melt was within the grain boundary, in equilibrium the interfacial tension of the grain boundary would have to be equal to the interfacial tension of the droplet:

γGB = 2γSL cos θ (3.1)

Figure 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.

Where θ is the contact angle of the droplet to the interface (see figure 3.1). Equation 3.1 is known as Antonov’s rule [87]. When theta is zero, the contact angle of the droplet against the grain boundary is non-existent so the droplet is spread along the interface in a thin film, wetting the grain boundary. This film appears as the solid-liquid coexistence curve is approached from T < Tm. Using these conditions Lipowsky assembled a landau free energy of premelting per unit area from the landau free energy of thin films [12, 56, 50]:

V (w) = Btsw + Vo(w) (3.2)

Where ts = (Tm − T )/Tm and B = Tm(SL − SS), where SL and SS are bulk entropies per unit volume, and Vo(w) is a variable dependent on interfacial forces. Also, w denotes the thickness of the premelted region. 1Where the substance has both the temperature and pressure for solid, liquid and gas to be in coexis- tence at thermodynamic equilibrium.

23 Figure 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.

Using equation (3.2) as a base, Hoyt et al [27, 40] constructed an equation for excess Gibbs free energy per unit volume for premelting in a grain boundary. Just like Lipowsky’s 0o contact angle example, the liquid-like region would appear inside a grain boundary sandwiched between two solid grains. The width of the grain boundary, w, is shown in figure 3.2. The new form of the equation is:

∆G(w) = Gvw + Ψ(w) (3.3)

Where the B and ts term from equation (3.2) has been simplified to the bulk free energy difference per unit volume of solid and liquid using the thermodynamic equation of Gibbs free energy; where Gv = ∆SSL∆T . The excess free energy per unit area, equation (3.3), shows the competition between opposing bulk and interfacial thermodynamic factors. The bulk free energy difference between solid and liquid per unit volume, Gv, times the width of the grain boundary, w, is representative of the bulk terms in the system. The disjoining

24 potential, Ψ(w), contains all the interfacial terms and has some intricate dependence on the width of the grain boundary. Note the excess Gibbs free energy for premelting in a three dimensional space has units of J/m3. To express the Gibbs free energy per unit area, the length of the system must be taken into account:

l∆G(J/m3)(w) = Gvw + Ψ(w) (3.4)

Now that the units are balanced, the only real mystery lies within the form of the disjoining potential. Within the disjoining potential holds the driving force of premelting, which depends heavily on the intermolecular forces. The details of the disjoining potential will be discussed below.

3.2 The Disjoining Potential

The disjoining potential is a mathematical function used to quantify the structural forces responsible for driving premelting. There are practical reasons to persue an understanding of these forces, such as modeling “hot tears”; a solidification defect as a result of casting or welding [40,2, 99, 76]. While there is a use in pursuing the disjoining potential, there is still no definite picture of the magnitude and spatial extent of the disjoining potential in metals and alloy systems. In order to understand the disjoining potential we will look back at equation (3.3) showing the total excess grain boundary free energy per unit area for premelting, with a grain boundary width of w. The function of the disjoining potential, Ψ(w), takes the limits of interfacial energies of a dry grain boundary, γGB, and two times the solid-liquid interface free energy, γSL, for widths that are small and infinite respectfully. For the values of width in between these limits the disjoining potential has shown a complicated dependence on width. In order to encompass all of the structural forces involved, both long and short-ranged forces must be taken into consideration. The long-range dispersion forces are dominant at large widths [87] but when taken in the range significant to premelting in the grain boundary, the long-ranged forces are miniscule in comparison to the short-ranged forces. From the general theory of van der Waals forces, if long-range forces were the most signif- icant part of the disjoining potential, then the two solid-liquid interfaces would always be attractive. The interfacial fluctuations would not be able to overcome this attraction in

25 three-dimensional systems, which imply the liquid-like film has a finite horizontal thickness at the melting point. The significance is that the film cannot span the length of the grain boundary, therefore is split into discrete sections of melt. This is known as incomplete wetting of the grain boundary (figure 3.3)

Figure 3.3: Incomplete wetting in the grain boundary. The premelted region does not span the interface but is in contained regions.

Premelting represents a complete liquid-like film in the grain boundary, which would suggest that the short-range forces are most important to the disjoining potential in order for complete wetting to occur. The short-ranged structural forces come about from overlap in the density waves in the diffuse regions of the solid-liquid interfaces. The general math- ematical trends of the short-range forces indicate a decaying exponential form, which was found by both lattice-gas models and mean field theories [40, 58, 102, 107]. From these experiments, the form of the short-ranged disjoining potential is:

 w Ψ (w) = 2γ + ∆γ exp − (3.5) SR SL δ ∆γ = γGB − 2γSL (3.6)

Where the short-range disjoining potential, ΨSR, is equal to the sum of interfacial free energies and a decaying exponential term. The interfacial energy of a solid-liquid boundary is γSL, is multiplied by 2 due to a solid-liquid interface on either side of the grain boundary. The parameter, δ, is called the interface length and is on the order of a few atomic spacings. The free energy of a dry grain boundary is γGB, which is a grain boundary with no premelting. Using equation (3.6), we can see a condition for premelting to occur based solely on the interfacial free energies. If ∆γ < 0, then it takes more energy to form two solid-liquid

26 interfaces then to remain a solid-solid interface (a dry grain boundary, γGB), which implies that there will be no premelting because it is thermodynamically favorable to remain dry in the grain boundary. If ∆γ > 0, the energy to form two solid-liquid interfaces is less then that of a dry grain boundary. Therefore, premelting will be more thermodynamically favorable because the free energy will be reduced.

3.2.1 Calculating Disjoining Potential

Hoyt et al [27, 40] used Molecular Dynamics (MD) simulations to calculate the disjoining potential of premelted grain boundaries. For pure Nickel, the temperature was adjusted ap- proaching the melting point of the material from below in order reach the premelting zone2. While the system was in thermodynamic equilibrium, the width of the grain boundary was highly dynamic. Using the statistical fluctuations in the width of the grain boundary, prob- ability distributions can be made which mathematically lead to the disjoining potential of a premelted grain boundary; se figure 3.4.

 ∆G  P (w) ∼ exp − (3.7) KBT These probability distributions are weighted using the Boltzmann factor, creating a direct link to the excess Gibbs free energy of premelting (equation 3.3). Combining the probability distributions with Gibbs free energy, the equation can be rearranged to solve for disjoining potential.

 G w + Ψ(w) cP (w) = exp − v (3.8) KBT  w Ψ (w) = 2γ + ∆γ exp − = −K T ln (cP (w)) − G w (3.9) SR SL δ B v Where c is a scaling factor attached to the probability distribution function. The P 9 h115i 120o twist boundary and the P 9 h011i {411} 38.9o symmetric tilt boundary represent a repulsive and attractive-repulsive case, respectfully. The P 9 h115i 120o twist boundary shows a typical decaying exponential form to the disjoining potential, as predicted by the short-range disjoining potential in equation (3.5). The same boundary was used in this thesis, but using Aluminum-Magnesium instead of pure Nickel.

2Location of premelted zone is discussed in Chapter 4: Simulation.

27 Figure 3.4: Probability distributions of width (left) were used to calculate a disjoining potential (right) for grain boundary premelting in pure Nickel, for two different grain boundary orientations. [27, 40]

It was briefly stated earlier that an attractive grain boundary will have a ∆γ < 0 indi- cating premelting was not favored in the boundary. So the fact that premelting occurs in the P 9 h011i {411} 38.9o symmetric tilt boundary, even though there is a weak attractive minima, makes this boundary an interesting case to study. Though this type of behavior differs from the predicted short-range potential, the result was not limited to this study. Molecular dynamics studies of three Silicon boundaries, and a symmetric tilt boundary in Copper show similar results [98, 103]. Qualitatively similar results from Phase field crystal model (PFC) were also obtained for a P 9 tilt boundary. [65, 39] Because this boundary is an interesting case, the P 9 h011i{411} 38.9o symmetric tilt boundary was chosen for this thesis. The difference comes from using an alloy model, instead of the pure case shown above.

3.3 Thermodynamic Theory

In Chapter 2, the specific types of grain boundaries pertaining to this thesis were mentioned but not what material was actually being used. This section is a brief summary of the thermodynamics involved with a binary alloy system, specifically of Aluminum-Magnesium,

28 which will have applications in the simulation section of this thesis1. We will begin with the Gibbs free energy for mixing in an ideal case and briefly outlying the definition of a semi-grand canonical ensemble. In this section an important concept is brought up, the chemical potential difference of a binary alloy, in conjunction with the Gibbs free energy. This will be needed later in order to successfully describe the simulation theory.

3.3.1 Gibbs Free Energy

The system we will be looking at is an ideal binary alloy, which is composed of Aluminum- Magnesium. The sum of the mole fractions of aluminum and magnesium is equal to one mole of homogeneous solid solution.

XAl + XMg = 1 (3.10)

Where X is a mole fraction. The mole fraction is defined by the amount of constituent, ni, divided by the total amount of solution, nttl.

Xi = ni/nttl (3.11)

Lets consider an ideal solution where the interactions between all atoms will be identical (Al-Al, Mg-Mg, Al-Mg). Also in the ideal case there is no enthalpy of mixing, ∆Hmix = 0. Therefore the Gibbs free energy of mixing will only be dependent on the entropy. The entropy of mixing is equal to:

∆Smix = Sstep2 − Sstep1

where step1 is equal to the entropy before mixing and step2 is entropy after. Since there is only one configuration of atoms before mixing, Sstep1 = KB ln (1) = 0, the total change in entropy will be equal to the entropy of the mixed solution.

∆Smix = Sstep2

Where the entropy is given by statistical mechanics as:

S = KB ln(Ω) (3.12)

1Chapter 4, section 4.3.1

29 where KB is the Boltzmann constant and Ω is the number of states. In a system made up of only two different types of atoms, the total number of atoms is equal to Nttl = NAl + NMg. Based on this, the number of states the system will have is:

N ! Ω = ttl (3.13) NAl!NMg!   Nttl! where ∆Smix = KB ln (Ω) = KB ln from plugging in the number of states, NAl!NMg! equation (3.13), to equation (3.12). The entropy of mixing can be simplified using Sterlings approximation for large numbers: lnN! ≈ NlnN − N [100]

     NAl NMg ∆Smix = −KB NAl ln + NMg ln NAl + NMg NAl + NMg And can be further simplified by replacing number of atoms with mole fractions:

∆Smix = −R[XAl ln XAl + XMg ln XMg]

∆Gmix = −T ∆Smix = RT [XAl ln XAl + XMg ln XMg] (3.14)

Equation (3.14) is the Gibbs free energy for mixing of an ideal solution. The total Gibbs free energy of the system is given by [107]:

Gttl = XAlGAl + XMgGMg + ∆Gmix

Gttl = XAlGAl + XMgGMg + R[XAl ln XAl + XMg ln XMg]

The general shape of Gibbs free energy for binary alloys is shown in figure ( 3.5).

3.3.2 Chemical Potential for Ideal Solution

The chemical potential of a species is stated in thermodynamics as:

∂G µ = (3.15) i ∂η i P,T,ηi,ηj ...

30 Figure 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 Aluminum is XMg = 0 and XMg = 1 is pure Magnesium.

By rearranging the chemical potential equation (3.15), the difference in Gibbs free energy can be found for a binary alloy.

∂G ∂G dG = ( )dηAl + ( )dηMg ∂ηAl ∂ηMg ∂G ∂G G = ( )ηAl + ( )ηMg ∂ηAl ∂ηMg

G = µAl ηAl + µMg ηMg dη X where : Al = Al dηMg XMg · ·· G = µAlXAl + µMgXMg (molar Gibbs free energy) (3.16)

If a line is drawn tangent to the curve in figure 3.5, plugging in equation (3.11) to (3.16) and rearranging will reveal the intercepts and slope.

31 G = µAl + (µMg − µAl)XMg (3.17)

Figure 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.

Performing a tangent construction of the chemical potentials of Aluminum and Mag- nesium will result in the concentration of the system (figure 3.6). The chemical potential difference is an adjustable parameter in my Monte Carlo simulations, thus it is important to know what concentration is related to a set chemical potential difference, ∆µ.

3.3.3 Semi-Grand Canonical Ensemble

In statistical mechanics, a grand canonical ensemble is a theoretical collection of macro- scopic states created to mirror the statistical fluctuations of the real world system. The system is in equilibrium with respect to an external reservoir for both energy and particle exchange. The control variables for the Grand canonical ensemble are the chemical poten- tial, µ, the temperature, T and the volume, V. This allows the total number of particles to fluctuate.

32 A semi-grand canonical ensemble is similar to a grand canonical ensemble (µ, V, T ), but has the total number of atoms in the system fixed and varying concentration. While the total number of particles stays the same, the particles of different species are inserted and removed according to the configuration energy and the set chemical potential difference. The common use for semi-grand canonical ensembles is in Monte Carlo simulations of mixtures.

33 Chapter 4

Simulation Methodology

Before the mid 1980s, atomistic simulations were performed using pair-potentials to de- scribe atomistic interactions [1]. However pair potentials do not encompass all the energy of a material, especially the nearly free-electron metals such as Aluminum and the alkalis. Only about 10% of the cohesion energy of a metal is represented in pair-potentials. Pair potentials have another folly. The local density variations are completely ignored even though pair potentials depend a great deal on the density. Consequently, large changes of density, such as free-surfaces or vacancies in high electron density metal, were prob- lematic. Nevertheless, the density dependence of pair potentials was ignored due to the difficultly defining local density. These facts highlight some of the inadequacies of the pair potential method, yet until 1984 most atomistic simulation was done using this scheme. In the early 1980’s, a new method, which involved many-body potentials, rectified the local density problem. The embedded atom method [17] describes complex metal systems by splitting the binding en- ergy into two terms. The first being the repulsive interaction described by a pair potential, and the second introduces an embedding energy; the attractive portion which is a result of embedding an atom in the local charge-density of the lattice. Later, the Finnis-Sinclair method [28] was introduced which builds on the embedded atom method, altering the embedding energy for a better description of local density. The EAM and Finnis-Sinclair potentials have similar computational timescales to pair- potentials, which results in a much better description of atomistic interactions with a similar computational cost. The Embedded atom method and the Finnis-Sinclair potentials are used in simulation

34 to properly describe a metal system. Both are briefly outlined below, as they are relevent to the Monte Carlo simulations for this thesis.

4.1 Embedded Atom Method

In 1980, Norskov and Lang [74], as well as Scott and Zaremba [82], invented a scheme called effective medium theory, where they use density functional theory to relate energy of an atom in a solid to the energy of embedding an atom into that solid. Building on this work, Baskes and Daw generalized the theory creating what is commonly known as Embedded Atom Method (EAM) [17]. The embedded atom method is a technique to approximate the energy between two atoms using two different interatomic potential schemes. The total energy of a metal is viewed as the energy associated with pair-wise interactions of the atoms, a two-body potential, and the energy associated with embedding an atom into the charge density of the other atoms in the system, an N-body embedding function. The analytical expression for the total energy of an atom i is:

! 1 X X E = φ (r )) + F ρ (r ) (4.1) 2 αβ ij α β ij i6=j i6=j

where rij is the distance between atoms i and j, φαβ is the pair-wise potential function, ρβ is the contribution to the electron charge density from atom j of type β at the location of atom i, and F is the embedding function that represents the energy required to place atom i of type α into the electron cloud. [86] The pair potential and the embedding function are both found by fitting bulk parame- ters such as lattice parameter, elastic constants, cohesive and vacancy-formation energies.

For a binary alloy of Nttl atoms, the EAM potential requires seven functions: three pair-wise interactions (A-A, A-B, B-B), two embedding functions, and two electron cloud contribution functions. The total energy is a summation of all of these contributions.

N 1 X X E = φ (r )) + Ei (4.2) 2 αβ ij N i6=j 1

Nttl i X where EN = F [ ρ(rij)] (4.3) i6=j,j=1

35 Where this multi-body potential is limited by a cut-off radius due to the weakening interaction energy from increasing distance away from electron cloud.

The two embedding functions contained in EN come from the two different elements involved. Let FA be the embedding function for component A, and FB be the embedding function for component B. Similarly, let ρA be for component A and ρB for component B α in the embedding part of equation (4.3). Now define δA so that when i is an atom is of i i type A, δA = 1 , and δA = 0 if it is of type B. Define similarly for type B atoms. Then for a binary system we have [22]:

i i i i i EN = FA(f )δA + FB(f )δB (4.4)

Nttl i X j j f = ρA(rij))δA + ρB(rij)δB (4.5) i6=j,j=1

This adapts the embedded atom model from a single atom type to a binary alloy case. The EAM has been widely used in FCC metals to model interfaces. With adaptions made for binary cases, alloy interfaces have also been widely explored using the EAM [17, 86, 22, 87, 41]. There are many advantages to using EAM over simply the pair-potentials. The most significant advantage is the ability to accurately describe the tendency in metallic bonding for an increase in the bond strength as the coordination number decreases. In a pair potential model, the energy of a particular atom is proportional to the coordination number in a linear fashion. In a real metal, there is a non-linear dependence on coordination number that is rectified in the embedding function; that is often approximated by a square root1. Also, if all cohesion is represented only by the pair potentials, then for cubic materials the Cauchy relation for the elastic constants is obeyed: c12 = c44. This is not true for any real material, which shows a deficiency for solely using pair potentials. The embedded atom describes real material more accurately with the insertion of the embedded function. Computation time is an important factor when considering simulation methods for a number of reasons. With a comparable size, the computational cost of an EAM simulation is no more then twice a simple pair potential [87]. It is advantageous to use this model for complex metal systems with only a minor increase in computation time as the penalty.

1Known as Finnis-Sinclair potentials

36 4.2 Finnis-Sinclair Potentials

In the 1980’s a variety of n-body potential were employed to study metals in order to properly describe the local density dependence of atomic interactions in metals; the em- bedded atom theory included. In 1984, Finnis and Sinclair outlined their scheme to solve this local density dependence [28]. The Finnis-Sinclair potential (FS) is an alteration to the bonding energy of the EAM, that sets the embedding function to a square root in order to capture the dependence of atomic interactions on the local density. It has been shown that the Finnis-Sinclair Potential has good results in simulation of grain bound- aries, as well as surfaces, point defects, and amorphization transitions for metals and alloys [28, 63, 105, 45, 53, 106].

4.3 Monte Carlo Computing

In 1950, a group of researchers led by Metropolis developed the Monte Carlo method [68]. Monte Carlo methods are a class of computer algorithms that depend on random sampling to compute the results. It is important in thermodynamics to be able to calculate the equilibrium of an observable, which would insinuate evaluation of an ensemble average. By random sampling, the probability distribution of the statistical ensemble is weighted such that the important contributions will be sampled most frequently [87,5, 62]. This is a good starting point to begin the discussion of simulating the interaction of an Aluminum- Magnesium alloy using a Monte Carlo technique. The goal is to find equilibrium states in the alloy and attempt to gather information pertaining to the width of the grain boundary. Assuming there is a minimum free energy of the system, by taking variations of position and concentration of each atom in the lattice, attempts to reach the minimum energy state will create probability distributions of states close to equilibrium. With the embedded atom method giving the ability to calculate the energy of the state, a Monte Carlo simulation will give the lattice the ability to move to new states to test the energy again, fluctuating about a minimum free energy. In this section, a description of the Monte Carlo method is presented for a binary alloy of Aluminum-Magnesium.

4.3.1 Monte Carlo Technique

The Monte Carlo (MC) method was named according to the use of random numbers to make statistical fluctuations in order to numerically generate probability distributions [5].

37 As noted by Hoyt et al. [40], the width of a premelted grain boundary is highly dynamic under constant temperature for a pure material. Exploiting the statistical fluctuations of the width gives rise to probability distributions, which have applications towards to disjoining potential. In this work the skeletal structure of molecular dynamics (MD) method used by Hoyt was used in order to compute width fluctuations of the grain boundary for a binary alloy of Aluminum-Magnesium. For a pure material, temperature was increased in order to premelt, but the process is different for an alloy. By changing concentrations, while keeping temperature constant, the premelting zone can be reached, much in the same way as raising the temperature to the melting point. Exploiting the width fluctuations of the grain boundary for each chemical potential difference in the simulation will lead to a disjoining potential; much the same way as temperature in the Hoyt et al case. There are a couple reasons why MC is a good choice for the alloy simulations. Monte Carlo methods are prefered when there is no dynamical model to describe phenomena, and are controlled just by instructing energy. Also, MC is best for grand and semi-grand canonical ensembles due to atoms being removed or added into the system, which is diffi- cult to do in MD. For the simulations in this thesis, we want to change chemical potential difference of the system, hence the number of atoms of each type, all while keeping total atoms the same. This is defined by a semi-grand canonical ensemble, which is best simu- lated by Monte Carlo methods. Outlined below is the semi-grand canonical Monte Carlo method used to manipulate the Aluminum-Magnesium lattice by setting parameters such as chemical potential difference, lattice parameter, temperature, etc. The Monte Carlo simulations monitor the configuration of the lattice as a function of computer timesteps. With the ability to calculate the energy of a binary alloy, outlined in the embedded atom method section, now what is needed is the ability to generate a new state in order to use a Monte Carlo algorithm. Once the location of each atom is established in a lattice, then the energy of the state is computed. For each step of the Monte Carlo simulation, each atom of the lattice is given a chance to move or switch atom type. Once the Monte Carlo step is complete, the energy of the new state of the lattice is computed. The whole purpose of this is to check whether these changes have moved the system closer to a minimum free energy. However, there are rules that influence the choice to move or switch atom type.

38 Accepting or Rejecting Position Change

The rule to accept or reject a movement in the system is weighted so that after many Monte Carlo steps the probability of finding any given atom is specified by the Boltzmann factor:

∆E exp(− ) (4.6) kBT Where ∆E is the difference between the new and previous energy state of the system. Using a random number generator to choose z a random number between 0 and 1, an atoms movement is accepted or rejected depending on if z is less than or greater than the Boltzmann factor, respectfully.

Preferred Concentration

The rule to accept a change in atom type depends on the chemical potential difference of the binary alloy. This can be visualized using the Gibbs free energy versus concentration for a binary alloy (figure 3.5). In the Monte Carlo Simulation, one of the controlled parameters is the chemical poten- tial difference. By controlling the chemical potential difference of the alloy, it essentially defines a preferred concentration for the system at which to equilibrate. As demonstrated in figure 4.1, the dark line marks a chosen chemical potential difference, as shown on the side of the graph. Where this line touches the Gibbs free energy curve sets the equilibrium concentration. By changing the slope of this line, which is equivalent to the chemical po- tential difference, a new preferred concentration is set, as shown by the light line on figure 4.1. Monte Carlo techniques are statistical by nature, thus the system will fluctuate about this preferred concentration. These fluctuations rule whether the system will accept or reject the atom type change. If the system is lacking in Aluminum, as dictated by the preferred concentration of the system, then Magnesium atoms will be more likely to switch to Aluminum to appease the preferred concentration. If there is too much Aluminum in the system, it will not be favorable for Magnesium to change to Aluminum. In summary, for a set chemical potential difference there is a preferred concentration that the system will fluctuate around. Atom types will change according to deficiencies or abundances in reference to the preferred concentration.

39 Figure 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.

4.3.2 Equilibration

Monte Carlo simulations take time to reach equilibrium states. Equilibration is essential in Monte Carlo simulations because statistical averages are only believable after this condition has been met. Equilibration can be checked by monitoring a physical property of the material, such as the internal energy or pressure. Once averages of the parameter are constant, then equilibration has been met. Seki et al. [84, 85] found that in order for a material to equilibrate 400-2000 time steps were needed per atom for kinetic Monte Carlo Simulations. This range was generated for a pure material, where all atoms in the simulation are of the same size. If the atom sizes differ, the physical properties of the material take much longer to converge to equilibrium [87]. This comes about from larger atoms in the system trying to travel into smaller lattice sites. It takes much more attempts for a larger atom to move into a smaller space due to the compression required. Monte Carlo simulations are computationally expensive to begin with, but are increasingly so if they are a binary alloy.

40 Figure 4.2: Equilibration of pressure and energy for a P 9 h011i {411} symmetric twist boundary in Al-Mg, with a chemical potential difference of 1.715 eV/atom.

The internal energy and pressure of the lattice was plotted versus time in order to see where the simulation lay in terms of equilibration; figure 4.2. Since the material is a binary alloy, it is expected to take much longer to equilibrate. After 12 billion Monte Carlo steps the material was equilibrated; which is equivalent to 300000-500000 steps per atom. The amount of MC steps per atom for the binary alloy case is increased by a factor of 100 from Seki’s time steps in kinetic Monte Carlo for the pure material.

4.4 Simulation Methodology

Phase diagrams are tools to visualize the phase that a material will have given a specific temperature and composition. Using the phase diagram of Aluminum-Magnesium, an esti- mate of a materials phase can be made given the simulation input parameters. Premelting occurs below the solidus line on a phase diagram, as approached from below by increasing temperature or concentration. Figure 4.3 shows the phase diagram of the Finnis-Sinclair model of Aluminum-Magnesium alloy [28]. In order to reach the premelting region, the concentration was increased from

41 Figure 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. pure Aluminum, all while keeping the temperature constant at 880K. Premelting occurs before the melting point of the material, but the exact location of the onset of premelting is unknown. In order to obtain an encompassing view, simulations were run beginning with essentially pure Aluminum, and ending just under the solidus line, as shown by the dashed line in figure 4.3. The step size is controlled by the chemical potential difference, with the approximate increase of 1% Magnesium per step.

Rescaling

The original Nickel grain boundaries were retrieved from Hoyt et al [27, 40] and reworked in the MC code in order to create a binary alloy. Using pure Nickel as the base, an Aluminum lattice was formed. Since both are an FCC type, the orientation of the crystallographic

42 planes would be correct, the only difference was the physical size of the lattice. By scaling the lattice up using the lattice parameters of both Nickel and Aluminum, Aluminum could be created from this pure nickel lattice. Starting with a pure Aluminum lattice, Magnesium solute was added via the Monte Carlo simulations in accordance with set chemical potential differences.

43 Chapter 5

Results and Discussion

We will begin this chapter by visualizing the system via Atomeye, an atomistic viewing program.

Figure 5.1: Aluminum-Magnesium lattice, including grain boundary. Atoms are shaded by the coordination number.

The MC snapshot in figure 5.1 shows the system of Aluminum-Magnesium which consists of a grain boundary sandwiched between two solid grains. The atoms in this snapshot have been shaded according to the coordination number. The lighter shades indicate atoms located in one of the two grains. The darker shades represent regions of disorder, which denote most notably a grain boundary in the center. The grain boundary can be recognized by the distinct disordered environment as compared to bulk solid. The snapshot was taken in the P 9 tilt boundary with ∆µ = 1.69 eV/atom, corresponding to a concentration of approximately 3.6% Magnesium, estimated via figure 5.2, and the stable phase solid. The x, y, z coordinate axis is also shown in figure 5.1.

44 Figure 5.2: This figure shows the relationship between chemical potential difference, ∆µ, and concentration of Magnesium. The dotted line represents an exponential fit.

Comparing different snapshots of the same boundary while in thermodynamic equi- librium, fluctuations occur in the width of the grain boundary as can be seen in figure 5.3 . The snapshots were obtained from MC simulation at 880K, with chemical potential difference ∆µ = 1.69 eV/atom. The fluctuations in grain boundary width are the basis for the histogram analysis underlying calculation of the disjoining potential.

5.1 Grain Boundary Widths

The goal of this project is to model the disjoining potential of premelting in an Aluminum- Magnesium system. This was done using Monte Carlo simulations, outlined in chapter 4. Using the knowledge that Monte Carlo simulations are highly dynamic by nature, statistical analysis of fluctuations in the grain boundary width can be done, which can be used to find the disjoining potential; equation (3.10). In turn, finding the disjoining potential requires

45 Figure 5.3: The width of the grain boundary is highly dynamic in thermodynamic equilib- rium. Atoms are shaded by the coordination number and the solid line in the inner section represents the approximate width of the grain boundary. a method to collect the widths of many grain boundaries. This section outlines the two different methods employed to find the width of the grain boundary; concentration profiles and the centro-symmetry parameter.

5.1.1 Concentration Profiles

When a system is being quenched or under an application of stress, non-equilibrium seg- regation occurs. This means that vacancies move towards the grain boundary or sinks in the system. If there is solute in the system, it will couple with the vacancies and also move towards the grain boundary [18]. A method of finding the width of a premelted grain boundary was developed from the knowledge that solute will segregate at an interface. After the system has been equilibrated, a concentration profile along the x-axis can be made. Due to non-equilibrium segregation, there should be a peaks of solute at the interfaces. Since a premelted grain boundary has 2 solid-liquid interfaces, measuring the distance between these peaked regions results in a grain boundary width. This method was executed using a histogram of the concentration of magnesium atoms; as shown in figure 5.4 . Binning the number of Magnesium atoms across the x-axis of the

46 Figure 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. system 1, the width of the elevated region of magnesium atoms was calculated. The width of the grain boundary was found by multiplying this value by the width of the bins in the histogram. Unfortunately, there is a distinct disadvantage with this method due to the dependence of the clarity of the interfaces. Though the majority of solute has congregated to the grain boundary, there is still some located in the bulk. On concentration profiles, solute in the bulk appears as a peak in the bulk region. In these situations, though a general location of

1See figure 5.1 for orientaion of axis

47 the grain boundary can be seen, the interfaces are not clear-cut. Thus the width of these grain boundaries may be wildly over estimated. Another disadvantage lies in the binning method for the histograms. There is a delicate balance required in spacing of the bins. If the bins are too large, the interfaces become much more clear, but there is a loss of accuracy to the width. If the binned width is too small, the overall shape of the histogram is too noisy to pinpoint the interface locations. A proper balance must be made in order to see clean peaks and still have a decently accurate width. A way to remedy this issue is to have many snapshots of the width in order to identify and remove outliers in the data. While it would have been preferable to have at least 500 snapshots for reasonable statistics, only 100 snapshots could be taken. While this little data will not lead to an accurate disjoining potential, especially with massively fluctuating concentration profiles, a general trend can be found.

5.1.2 Nearest Neighbor Analysis

Since concentration profiles could not supply distinct solid-liquid interfaces in which to measure the width of the grain boundary, the next idea was to use the coordination number for each atom. The coordination number is a measurement of how many nearest neighbors an atom has. At the premelted grain boundary there is a liquid-like region that is more amorphous then the rest of the system. Theoretically by examining a 1-dimensional slice perpendicular to the grain boundary for the number of nearest neighbors of each atom, there will be a peak in the amorphous region. Unfortunately, the change in coordination number between solid and liquid is very small, so the chances of resolving a distinct peak are minimal. But there is a method that will result in a larger difference between solid and liquid states that is based on nearest neighbor locations. It is called the centro-symmetric parameter.

The Centro-Symmetry Parameter

A centro-symmetric lattice is a structure where any given atom has pairs of equal and opposite bonds to its nearest neighbors. These bonds remain approximately equal and opposite under elastic deformation2 of the material. It is defects in the lattice, or plastic

2 Elastic deformation: a temporary shape change that is reversible.

48 deformation 3 that cause the equal and opposite nearest neighbor pairs to deform and break symmetry. The centro-symmetry parameter is used to characterize the degree in which a material holds this equal and opposite nearest neighbor pair relationship. Since an FCC material has 12 nearest neighbors, there are 6 nearest neighbor pairs. The centro-symmetry parameter for an FCC lattice is defined as follows:

X 2 P = |Ri + Ri+6| (5.1) i−1,6

Where Ri and Ri+6 are the vectors associated with the 6 nearest neighbor bonds of the atom in the FCC lattice. The 12 vectors of the undistorted bulk FCC lattice are calculated in tandem with the orientation of the lattice. The vectors for the distorted lattice are calculated by finding the neighbors in the distorted lattice with the closest distance to the undistorted vectors. There may be repeated uses of the same atom in this case as well as the use of non-nearest neighbors if the center atom has fewer then 12 nearest neighbors, or if the local surroundings are exceedingly distorted. Each of these pairs is added together, resulting in 6 vectors. The sum of the squares of the 6 vectors is found, resulting in a final number, known as the centro-symmetry parameter.

Using CSP to Find Grain Boundary Width

The centro-symmetry parameter is a measure of the variance from centro-symmetry, where each bond about a given atom has equal and opposite pairs. Since the variance from centro- symmetry indicates a deformation in the local vicinity of the atom, it indicates the location of a defect or identifies an atom in the liquid. It is this property that is of great use for finding the width of a premelted grain boundary. Amorphous material presents more of a disordered lattice then solid crystal, so the centro symmetry parameter in the grain boundary region is expected to be significantly higher then in the surrounding bulk FCC lattice. Also, Magnesium atoms, which are smaller then the Aluminum solvent, segregate to the interface. The presence of Magnesium in the solid-liquid interfaces increases the disorder, creating an even more distinguishable difference of the centro-symmetry parameter from the bulk state. This is the idea behind using the centro-symmetry parameter to find the grain boundary width.

3 Plastic deformation: irreversible change

49 Using a histogram of the centro-symmetry parameter across the x-axis (refer to figure 5.1 ), a curve can be constructed that extends from one crystal, through the grain boundary and into the other crystal; figure 5.5. Since the centro-symmetry parameter will be increased in the grain boundary region, the horizontal length of this spike across the x-axis is indicative of the grain boundary width in the premelted region.

Figure 5.5: Superposition of many histograms of centro-symmetry parameter across binned x-position in the system. The grain boundary is located in the elevated region of CSP. The approximate average width of the grain boundary is w.

Using this scheme, a series of grain boundary widths were calculated for the premelted regions taken from snapshots of the grain boundary over a number of steps of the Monte Carlo simulation; figure 5.5. There is Magnesium present in the bulk FCC lattice, which creates regions of increased centro-symmetry, which can be seen in the histogram as spikes in the bulk part of the

50 lattice. By comparing the location of these spikes with an image of the lattice, through the atomistic visualization program Atomeye, the large peaks in the bulk correspond with areas where a couple of Magnesium atoms happen to be nearest neighbors in the bulk. Nevertheless, the fluctuations in the data are significantly less then the concentration profiles such that a reasonable distribution of grain boundary widths can be obtained. These widths are vital in finding the disjoining potential for the premelted region in a grain boundary.

5.2 Probability Distributions

There is expected to be fluctuations in the value of the width under constant chemical potential difference, due to the fluctuation about the equilibrium concentration. Monte Carlo simulations capture these dynamics, which allows a statistical approach to be utilized. Using many simulations, a number of widths can be gathered for each chemical potential difference. With many widths, a probability distribution of the widths can be made. A different probability function was formed for each chemical potential difference sim- ulated; see figure 5.6 . As the chemical potential difference was raised, the system moved closer to the melting point, and the width of the grain boundary on average increased. When compared to the pure case of Hoyt et al (figure 3.4 ), this was true as well. The fluctuations in width for each chemical potential difference was expected to increase as the chemical potential difference increased, due to observations made in the pure case. The distributions in the alloy case show a similar trend. This can be seen in the width of the probability function, which increases in general for a higher chemical potential difference (which also represents approaching the melting point). There is significantly more scatter in the probability distributions of the alloy in (figure 5.6 )then in the pure case (figure 3.4 ). The reason for the high fluctuations in the alloy probability distribution, P (w), as compared with the pure material may be due to the shear amount of statistics used in the pure case versus the alloy. In the pure case, over 2000 snapshots of the width were taken and analyzed. Unfortunately, the number of snapshots in the alloy case is significantly less; about 100. Since the accuracy of statistical methods depends on the amount of available data, the distributions for the alloy are expected to be drastically less accurate. Though there is clearly not enough data to obtain an accurate probability distribution, there is enough to capture the general trend and the location of the most probable width can be estimated.

51 Figure 5.6: Probability distributions P (w), of a P 9 Tilt boundary (top) and a P 9 Twist boundary (boundary) found through width fluctuations at 880 K. (note: connecting lines are not a fit, just an aid to better see the distributions.)

5.3 The Disjoining Potential

As discussed in previous sections, the probability of observing a certain width at a given chemical potential difference (hence concentration) is related to the total Gibbs free energy of the system as follows:

52 ∆G(w) = Gvw + Ψ(w)

Which is related to the probability distributions by:

 ∆G  P (w) ∼ exp − KBT and solving for the disjoining potential leads to:

 w Ψ (w) = 2γ + ∆γ exp − = −K T ln (cP (w)) − G w SR SL δ B v Where c is a scaling factor attached to the probability distribution function. There are seven different chemical potential differences used for the P 9 tilt boundary and nine P for the 9 twist boundary: µ1 = 1.61, µ2 = 1.64, µ3 = 1.67, µ4 = 1.69, µ5 = 1.695, µ6 = 1.70, µ7 = 1.705, µ8 = 1.71(twist only) and µ9 = 1.715 (twist only) eV/atom. This leads to seven different probability distribution functions for the tilt boundary and nine for the twist boundary. Once the disjoining potential has been computed for each different probability func- tion, the unknown scaling constant c is used to fix any offset between different disjoining potential sections. To construct a continuous disjoining potential function, the offset is estimated to give optimal overlap between the data. Each disjoining potential section is then shifted by this offset. The resultant disjoining potential curves are shown in figure 5.7. While it is immediately clear there is insufficient data to obtain an accurate disjoining potential, the general trend of the data can be discerned. The P 9 twist boundary, which from the pure Nickel simulations obeyed a decaying exponential form, appears to follow the same trend in the alloy case. This is indicitive of a purely repulsive grain boundary, where the disjoining potential is completely contained in the positive region of the disjoining potential curve. As the temperature approaches the melting point of the material from below, the grain boundary width logarithmically diverges. The disjoining potential for theP 9 symmetric tilt boundary on the other hand is the interesting case that exhibited a minimum in simulations of pure Nickel; modeled using a double exponential function. It appears that the alloy follows this trend as well. There is a clear difference in the shape of the P 9 tilt curve in the alloy as compared to the P 9 twist boundary of the alloy; denoted by the disjoining potential curving back towards the positive

53 Figure 5.7: The disjoining potential for a P 9 tilt boundary (left) and a P 9 twist boundary (right). Each part of the curve is symbolized by the associated chemical potential difference. The tilt boundary is fit to a double exponential and the twist is fit to an exponential curve. axis for the highest width values. Though there is large error due to the small number of data points, it is likely the disjoining potential is actually curving back towards the positive axis. A reason to support the double exponential fit is higher chemical potential differences have the most data points and when pieced together have the most overlap of data points. More than one disjoining potential segment4 is curving back up for the same range of widths, which indicates a trend rather than random scatter. Secondly, the data points in the disjoining potential curve are the ones associated with high probabilities in the probability distribution curves. By eliminating the data points with very low probability, a relatively smooth curve was constructed for the disjoining potential. The disjoining potential of the tilt boundary shows attractive-repulsive behaviour. The attractive-repulsive behavior emerges because at small widths (less than 4 nm, figure 5.7) the disjoining potential is positive but if the widths get any higher and the curve dips into the negative region. Premelting is favored in the replusive part of the curve becuase of the condition in equation (3.6), where it is more thermodynamically favorable to have two solid-liquid interfaces then a solid-solid interface. In the attractive portion of the

4Piece of the curve constructed from a specific chemical potential difference

54 curve, it is more thermodynamically favorable to have no premelting. Thus as the system approaches the melting point, the grain boundary will grow until it has reached 4 nm where the attractive region is reached. From here the width will stay finite, in the minimum well at approximitely 11 nm, until the melting point is reached.

Figure 5.8: The disjoining potential for a P 9 twist boundary and a P 9 tilt boundary in both alloy (left) and pure (right) cases.

By comparing both disjoining potentials to the pure case, there is a definite trend between both data sets; figure 5.8. The alloy case shows an increase in grain boundary width as compared to the pure case. This is an expected result since impurities promote premelting. While I would not trust the exact value until more data is collected, the disjoining potential has the same order of magnitude and general shape as the pure case; which is in agreement with a number of atomistic simulations spanning across MD, PFC and MC methods. In order to be sure of the result, many more MC snapshots are needed to obtain reasonable statistical results.

55 Chapter 6

Conclusions

Premelting is the formation of a thin thermodynamically stable, liquid-like region in an interface below the bulk melting point. While premelting was first studied in the 1850’s, the topic was left dormant for over a century. Research began again in the 1950’s, analyzing surface melting on ice. While surface melting was the first type of premelting discovered, there are other locations for premelting, including grain boundaries. Grain boundary structure has shown liquid-like behavior near the melting point, which was shown to be temperature dependent in pure materials; using MD, PFC and MC simulation. This behavior has been quantified using thermodynamics, which involves a width dependent interfacial free energy, termed the disjoining potential. Using a Monte Carlo technique, the disjoining potential of a binary alloy of Aluminum- Magnesium can be directly calculated. In order to simulate premelting in an alloy, the concentration of Magnesium solute was increased from esentially pure Aluminum ending just below the bulk melting point of Al-Mg binary alloy. Seven to nine different chemical potential differences were utilized, and probability distributions of the width of the grain boundary were made for each chemical potential difference (see figure 5.6 ). Using proba- bility distributions, a disjoining potential can be directly calculated using equation (3.10). This technique was applied to two different types of grain boundaries; P 9 h011i {411} 38.9o tilt boundary and a P 9 h115i 120o twist boundary. The disjoining potential of the twist boundary displayed a decaying exponential form. This behavior is consistent with a logarithmically diverging premelting layer thickness as the melting point as approached from below. The disjoining potential for the symmetric tilt boundary is the interesting case that exhibited an attractive minimum, which was modeled using a double exponential function. The behavior of the tilt boundary demonstrated a

56 finite grain boundary width at the melting point. The disjoining potential by Hoyt et al [27, 40] had similar results using the same grain boundaries in pure Nickel, though shifted due to smaller grain boundary widths (figure 5.8). This is due to solute aiding the effects of premelting, creating a larger thickness of the liquid-like layer in the alloy. In the future it would be advantageous to compile more data. With more data points better statistical analysis can be made for the probability distributions, resulting in more accurate disjoining potentials. The result of which are useful to the theoretical study of premelting, and have applications to late-stage solidification studies in alloys.

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