Molecular Dynamics Investigation of Crystallization in the Hard Sphere System

Transcription

1 Molecular Dynamics Investigation of Crystallization in the Hard Sphere System Brendan O Malley B. App. Sci. (Hons.) RMIT Department of Applied Physics Faculty of Applied Science Royal Melbourne Institute of Technology Melbourne, Australia June 2001 A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy

2 Declaration Except where due acknowledgement has been made, this work is mine. This work has not been submitted previously, in whole or in part, to qualify for any other academic award. Brendan O Malley

3 Acknowledgements I would like to thank for my supervisor, Ian Snook, for his enthusiasm, support and encouragement throughout my thesis. I would also like to thank my second supervisor, Peter Daivis, for his many suggestions and insightful comments on my thesis. I would also like to thank Steve Williams for use of his hard sphere simulation code and Bill van Megen for many interesting discussions. Thanks also go to all my fellow students in the Computational Physics Group for their camaraderie and good humour, which has made working on this thesis a fun and pleasurable experience. Thanks also go to all of the staff and students of the Applied Physics Department who have contributed to an extraordinarily friendly and helpful working environment. And lastly I would also like to thank my family and friends who have been a constant source of support throughout my thesis.

4 Table of Contents Summary...26 Chapter Background Nucleation Theory Experimental Studies of Crystallisation Computer Simulation of Crystallisation Bibliography Chapter Introduction Voronoi Tessellation Definitions and Terminology Applications of Voronoi Description Voronoi definition of nearest neighbours Modified Voronoi Tessellation Description Application to Hard Sphere Crystal Planar Graph Description of Local Order Description Application to Hard Sphere Crystal Shortest Path Rings Definition and Terminology Geometric Analysis of Ring Structures Application to Hard Sphere Crystal Spherical Harmonics Summary Bibliography Chapter Introduction Theoretical and Semi-Empirical Relations Topology Average Geometry

5 3.2.3 Lewis and Desch Laws Aboav-Weaire Law Voronoi face area distributions and distance distributions Results and Discussion: Ideal gas Average Geometry Lewis and Desch Law Aboav-Weaire Law Voronoi face area and diagonal distributions Summary of Ideal Gas (RVP) Results Results and Discussion: Hard Sphere System Average Voronoi Cell Geometry Voronoi Polyhedra Types Lewis and Desch Law Aboav-Weaire Law Voronoi face areas Voronoi diagonal distributions Summary Bibliography Chapter Introduction Simulation Details Results and Discussion Topology of Hard Sphere Fluid Decomposition of Pair Correlation Function, g(r) Decomposition of Bond Angle Distribution High Density Fluid: Shoulder and Split Peak in g(r) Summary Bibliography Chapter Introduction Simulation Details Thermodynamic Changes Compressibility Factor Crystal fraction versus time: Planar Ring Method

6 5.3.3 Radial Distribution Function Ring Statistics Topology Six-Membered Rings Spherical Harmonics Density of Crystal and Volume Fluctuations Crystal density from lattice parameter Density fluctuations from Voronoi Volume Distribution Structure and Growth of Crystal Nuclei Nucleation at Coexistence: Size of Nucleus versus time Form of Nuclei at Coexistence Growth of Nuclei Stacking Probability Close Packed Planes Nucleation near the melting density Summary Bibliography Conclusion...271

7 List of Figures Figure 2.1: The Voronoi decomposition and the dual Delaunay tessellation in 2D. The black circles represent particles, while the black lines are the edges of the Voronoi cells and the light coloured lines are the bonds between Voronoi neighbours (particles sharing a Voronoi face) Figure 2.2: The Voronoi construction in 2D. The Voronoi cell of a particle is the smallest polygon (or polyhedron in 3D) formed by the intersecting bisectors (a line in 2D or a plane in 3D) drawn from the particle to all other particle centres. Note that the perpendicular bisector of the bond between the central particle and a candidate neighbour j does not intersect the Voronoi cell of the central particle. Hence j is not a Voronoi neighbour of the central particle Figure 2.3: A Voronoi polyhedral cell in 3D. Three faces of the Voronoi cell meet at each vertex and two faces share each edge Figure 2.4: (a) The circumcircle (point 1) defined by three particles a, b and c. The radius of the circumcircle is indicated by ccircle. By definition no other particle lies within this circle. The three perpendicular bisectors of the bond between the spheres define the circumcircle. (b) The circumsphere of a tetrahedron in the Delaunay decomposition. The radius of this circumsphere is denoted by cr (circumradius). No other sphere lies within this sphere. The circumcentre is defined as the intersection of the lines drawn from the circumcircle of each face (1-4) and perpendicular to that face. The circumcircle of face abd is denoted by point 2, acd by point 3 and that of bcd by point Figure 2.5: (a) An octahedron decomposed into four tetrahedra along the long diagonal ij. The individual tetrahedra are adij, dcij, cbij and baij. The other two possible long diagonals that the octahedron could be decomposed into are ac and bd. (b) A view along the diagonal ij Figure 2.6: (a) An octahedron decomposed into five tetrahedra, the long diagonal ij intersects the face abd. The tetrahedra are ijab, ijad, bdjc, bdci and ijbd. The last tetrahedron is a sliver or Kije simplex of near zero volume and is composed of the four nearly coplanar particles i, j, b and d. (b) The Kije simplex ijbd

8 Figure 2.7: The Voronoi polyhedron of a particle in a body-centred cubic lattice. Two 6-sided faces and one 4-sided face meet at each vertex Figure 2.8: The Voronoi polyhedron of a particle in a face-centred cubic lattice. All 12 faces are 4-sided. Note that four faces meet at six of the fourteen vertices.1-58 Figure 2.9: The Voronoi polyhedron of a particle in a perturbed face-centred cubic lattice. (a) The formation of a small face is a result of the intersection of a plane perpendicular to the bond between the central particle and a second nearest neighbour with the Voronoi cell, which in an ideal lattice is tangent to the vertex. (b) The formation of a small edge is due to 4 faces of the Voronoi cell (or more correctly the plane bisectors of the bonds defining them) not meeting at exactly the same point in space Figure 2.10: A view along a long diagonal of an octahedron in a face-centred cubic lattice. The two dashed bonds are also associated with an arrangement of 4 tetrahedra about each dashed bond Figure 2.11: The distribution of face areas of the Voronoi polyhedra at a reduced density of ρ = 1.05 near the melting density (ρ = 1.044) Figure 2.12: Decomposition of the distribution of face areas of the Voronoi polyhedra into contributions from different sided faces polyhedra at a reduced density of ρ = Figure 2.13: Distribution of face areas for real neighbours and extra neighbours due to the arrangement of four tetrahedra about a common edge, or equivalently, the common sharing of 4 particles by a pair of neighbours Figure 2.14: Different ways in which a packing of two tetrahedra and two octahedra about a common edge can be decomposed in a Delaunay tessellation. The dashed lines indicate the long diagonals along which the octahedra have been decomposed into tetrahedra. This is the arrangement of octahedra and tetrahedra in a face-centred crystal Figure 2.15: Comparison of initial distribution of Voronoi face areas and the final distribution after the deletion of bonds between certain Voronoi nearest neighbours Figure 2.16: Comparison of initial distribution of distances between Voronoi neighbours and the final distribution after the deletion of bonds between certain Voronoi nearest neighbours

9 Figure 2.17: The local environment about a particle in a face-centred cubic lattice and the planar graph indicating the network of bonds between the twelve neighbours of the central particle (which is not shown) Figure 2.18: (a) The packing of alternating tetrahedra and half octahedra about a nearest neighbour bond in a face-centred cubic lattice. (b) The packing of tetrahedra and half octahedra about half of the nearest neighbour bonds in a hexagonal close packed lattice Figure 2.19: (a) The planar graph of particles in a face-centred cubic arrangement and ring sequence (86). (b) The planar graph of particles in a hexagonal close packed arrangement and ring sequence (86). (c) Planar graphs of particles in perturbed face-centred cubic arrangements: face-centred + extra nearest neighbour and minus a related bond (106), (d) face-centred with a single extra bond between neighbours (10 5) and (e) face-centred less one neighbour (8 5) and plus two related additional bonds Figure 2.20: Different views of the fluctuations in the bond network that give rise to the defect structures shown in Fig The blue (extreme right and left pairs) particles have the planar graph ring sequence (10 5), the orange particles (middle left and right pairs) are (10 6) and the grey particles (middle pair) are (8 5) Figure 2.21: (a) and (b) are 4 membered shortest path rings while (c) is not a shortest path ring due to the path a-b Figure 2.22: (a)-(b) are 5-membered shortest path rings while (c) is not a shortest path 5 membered ring as the path bd is a shorter way around the network than the path bcd. Note that by describing the path as shorter we do not mean that the physical distance is less but the number of bonds traversed along the path bd is smaller (=1) than along the path bcd (=2) Figure 2.23: (a) Flat six-membered SP rings passing through each carbon particle of graphite; (b) Three of the twelve buckled six-membered SP rings through each carbon particle of diamond Figure 2.24: An SP six-membered ring containing three antipodal pairs. Particles A1 and A2 are antipodal pairs, as are B1 and B2 and C1 and C2. The shortest path distance between A1 and A2 is through the particles B1 and C1, involving three bonds Figure 2.25: An SP five-membered ring containing six antipodal pairs. The antipodal pairs are (1,3), (1,4), (2,4), (2,5), (3,5) and (3,1). The shortest path distance

10 between 1 and 3 is through the particle 2 and involves two bonds 1-2 and Figure 2.26: A modified six-membered ring. C is the central particle Figure 2.27: One of the four close packed planes, made up of overlapping sixmembered rings, which pass through every particle of a face centred cubic crystal Figure 2.28: (a) One of the four identical 6-membered rings passing through a locally fcc ordered particle (see note at end). (b) The single flat 6-membered ring passing through a locally hcp particle - this defines the stacking direction in this lattice. (c) A side view of one of the particles making up one of the three bent 6- membered ring passing through a locally hcp ordered particle (see note at end). Three of the particles about the ring are hidden from view by the other particles. (d) One of the three bent six-membered rings of the hcp ordered particle. (Note: The lighter coloured bigger spheres are the members of the modified sixmembered ring.) Figure 2.29: (a) One of the four identical flat 6-membered rings passing through a particle in a bcc lattice. (b) A view of one of the six-membered rings in the bcc lattice. (c) A six-membered ring chair structure in a bcc lattice. The labelled particles correspond to those in Fig 2.28(a) and the particles of this ring all belong to the closer of the two sets of nearest neighbours in a bcc lattice (the 14 neighbours of a particle in a bcc lattice occur at two distances, if d is the distance between the 8 closest neighbours then the distance between to the other 6 neighbours is (4/3)d Figure 2.30: (a) One of the ten identical 6-membered rings passing through an icosahedron; (b) An individual twisted six-membered ring. (c) One of the five identical bent six-membered rings passing through a twisted icosahedron. The ring has an identical from to the three bent six-membered rings associated with a hcp particle Figure 2.31: Bond Angle formed by the triplet of particles (i,j,k) Figure 2.32: (a) Bond Angle for Type 2 six-membered ring bond angle distribution. (b) Bond Angle for Type 3 six-membered ring bond angle distribution Figure 2.33: Distances r ij, r ik that contribute to the five-membered ring component of g(r)

11 Figure 2.34: Pairs of particles that contribute to the two different ring g(r) components for a six-membered ring (a) the Type A component and (b) the Type B component Figure 2.35: (a) (111) plane and 3 membered rings; (b) (100) planes and 4 membered rings Figure 2.36: The distribution of bond angles for rings of various sizes for a hard sphere crystal at a reduced density of Figure 2.37: Comparison of total bond angle distribution and sum of ring contributions for a hard sphere crystal at a reduced density of Figure 2.38: A close packed plane in a fcc crystal. A close packed plane can be considered as a 2-dimensional tiling of overlapping hexagons, with particles decorating the vertices of the polygon Figure 2.39: Overlap of two six-membered rings. The angle BCD (Type 1), associated with the ring centred on A, is counted before the same angle BCD, but associated with particle C and of Type 2 is counted. Thus no Type 2 contributions will be counted in an ideal crystal Figure 2.40: Decomposition of the pair correlation function into contributions from various sized rings for a hard sphere crystal at a reduced density of Figure 2.41: Comparison of g(r) with sum of ring contributions for a hard sphere crystal at a reduced density of Figure 3.1: A T1 elementary topological transformation. In two dimensions cell B loses cell D as a neighbour, while cell A gains cell C as a neighbour. In 3D, these transformations are applied to the Voronoi polyhedron of a single atom, which has A-D as neighbours. However, the same neighbour switching occurs as in two dimensions Figure 3.2: A T1 elementary topological transformation. In two dimensions cell B loses cell D as a neighbour, while cell A gains cell C as a neighbour. In 3D these transformations are applied to the Voronoi polyhedron of a single atom, which has A-D as neighbours. However the same neighbour switching occurs as in two dimensions Figure 3.3: Elementary topological transformation in 3D. In this case cells do not disappear, but the face of a Voronoi cell can disappear. The Voronoi polyhedron of atom I loses the face it shares with atom J by a T2 transformation

12 Figure 3.4: The T2 transformation above (Figure 3.3) also leads to the concurrent T1 transformation applied to the Voronoi cell of atom A. Atoms I and J are no longer neighbours (that is, share an edge) while the Voronoi faces associated with atoms B and C now share an edge of A s Voronoi cell Figure 3.5: Test of Lewis Law for V(f) of ideal gas Figure 3.6: Test of Lewis Law for A(f) of ideal gas Figure 3.7: Quadratic fit of V(f) versus f for ideal gas Figure 3.8: Quadratic fit of A(f) versus f for ideal gas Figure 3.9: fm(f) versus f for ideal gas Figure 3.10: m(f) versus f for ideal gas Figure 3.11: Comparison of fm(f) for the ideal gas with quadratic form of fit to fm(f) referred to in the text Figure 3.12: Distribution of Voronoi face areas for 3-sided faces Figure 3.13: Distribution of Voronoi face areas for different n-sided faces 4-sided faces Figure 3.14: Distribution of Voronoi face areas for 5-sided, 6-sided and 7-sided faces Figure 3.15: Normalised distribution of diagonal distances, d(n), between points sharing a common n-sided face Figure 3.16: Average Voronoi cell area A o for fluid and crystal Figure 3.17: Average Voronoi cell perimeter L o for fluid and crystal Figure 3.18: Density dependence of linear coefficient, c 1, of quadratic form of Lewis law for hard sphere fluid and crystal Figure 3.19: Density dependence of quadratic coefficient, c 2, of quadratic form of Lewis law for hard sphere fluid and crystal Figure 3.20: Slope of V(f), A(f) and L(f) at f = <f> calculated from quadratic form of Lewis law as a function of density Figure 3.21: Value of constant a appearing in Aboav-Weaire law for both fluid and crystal densities Figure 3.22: m(f) versus f for a number of fluid and crystal densities Figure 3.23: fm(f) at a density of ρ = 0.6 and quadratic fit to data Figure 3.24: Density dependence of linear coefficient of quadratic form of Aboav- Weaire law for hard sphere fluid and crystal

13 Figure 3.25: Density dependence of quadratic coefficient of quadratic form of Aboav- Weaire law for hard sphere fluid and crystal Figure 3.26: Slope of fm(f)/<fm(f)> for fluid and crystal using both linear and quadratic from of fm(f) Figure 3.27: Voronoi face and packing of atoms that define the face. A pentagonal Voronoi face (a) and the packing of atoms that define the face (b)-(c). The face is defined by the two light coloured atoms, which are shown in two slightly different viewpoints in (b) and (c). The dark coloured atoms are the five neighbours that they share Figure 3.28: Decomposition of distribution of Voronoi face areas into contributions from different sided faces for fluid at ρ = Figure 3.29: Decomposition of distribution of Voronoi face areas into contributions from different sided faces for crystal at ρ = Figure 3.30: Topological transformations and small-sided faces. The formation of a new five-sided face of the Voronoi cell of an atom requires the overlap of the plane bisector of the atom and a new neighbour with three vertices of the Voronoi cell of the atom Figure 3.31: Normalised distribution of Voronoi face areas for 3-sided faces at a number of fluid densities Figure 3.32: Normalised distribution of Voronoi face areas for 4-sided faces at a number of fluid densities Figure 3.33: Distribution of Voronoi face areas for 5-sided faces normalised by average 5-sided face area A o (5) Figure 3.34: Normalised distribution of diagonal distances P(r) for the hard sphere fluid (ρ = 0.7) and decomposition into contributions from different sided Voronoi faces Figure 3.35: Normalised distribution of diagonal distances P(r) for the hard sphere crystal near the melting density (ρ = 1.05) and decomposition into contributions from different sided Voronoi faces Figure 3.36: Normalised distribution of diagonal distances at different fluid densities for 3-sided faces Figure 3.37: Normalised distribution of diagonal distances at different fluid densities for 4-sided faces

14 Figure 3.38: Normalised distribution of diagonal distances at different fluid densities for 5-sided faces Figure 3.39: Normalised distribution of diagonal distances at different fluid densities for 6-sided faces Figure 3.40: Radial distribution function g(r) of the stable fluid, metastable fluid and stable face-centred cubic crystal of the hard sphere system Figure 3.41: Comparison of pair correlation function g(r) and appropriately normalised distribution of diagonal distances at a fluid density of ρ = Figure 3.42: Comparison of pair correlation function g(r) and appropriately normalised distribution of diagonal distances at a fluid density of ρ = Figure 3.43: Comparison of pair correlation function g(r) and appropriately normalised distribution of diagonal distances at a crystal density of ρ = 1.05, near melting Figure 4.1: (a) Average number of shortest path rings per atom versus density for a hard sphere fluid and crystal. The data points at densities of 1.00 are for the metastable fluid and crystal within the coexistence region Figure 4.2: (a) Three-sided Voronoi face and (b) arrangement of atoms associated with this face Figure 4.3: (a) Radial distribution function, g(r), of the hard sphere fluid at selected densities.(b) Close up view of second peak of radial distribution function, g(r), of the hard sphere fluid at selected densities Figure 4.4: (a) Decomposition of the radial distribution function, g(r), of the hard sphere fluid at a number density of (b) Close up of second peak of decomposed radial distribution function, g(r), of the hard sphere fluid at a number density of Figure 4.5: Close up of second peak of decomposed radial distribution function, g(r), of the hard sphere fluid at a number density of Figure 4.6: (a) Three membered ring contributions to the total g(r). (b) Four membered ring contribution to the total g(r) Figure 4.7: (a) Five membered ring contribution to the total g(r). (b) Six membered ring Type A distance contribution to the total g(r) Figure 4.8: Six membered ring Type B distance contribution to the total g(r)

15 Figure 4.9: Variance, Fisher skewness and Fisher kurtosis of normalized 4-membered ring g(r) contribution as a function of density. The dashed lines are only a guide to the eye Figure 4.10: Gaussian fit to 4-membered ring contribution to g(r) at a reduced density of Figure Comparison of different packings of 4 tetrahedra about a common edge ( octahedral packings as described in Chapter 2) in (a) a face-centred cubic lattice and a body-centred cubic lattice Figure 4.12: 4-membered ring component of g(r) after inclusion of a maximum Voronoi face area (0.05 in reduced units) for deletion of Voronoi nearest neighbour bonds Figure 4.13: Configuration of four hard spheres that give rise to the second nearest neighbour distances (one example is the distance between the spheres i and j in the figure) across a five-membered ring and a six-membered ring Figure 4.14: The mean of the normalised ring g(r) contributions at a number of fluid densities Figure 4.15: The variance of the normalised ring g(r) contributions at fluid densities Figure 4.16: The Fisher kurtosis of the normalised ring g(r) contributions at fluid densities Figure 4.17: The Fisher skewness of the normalised ring g(r) contributions at fluid densities Figure 4.18: Value of mean ring distance versus density Figure 4.19: Peak position of each ring g(r) distribution for various ring sizes versus the density. The lines are only a guide to the eye Figure 4.20: Scaled Peak position of each ring g(r) distribution for various ring sizes versus the density Figure 4.21: Scaled average distance between antipodal pairs of each ring size versus density Figure 4.22: Total bond angle distribution at different fluid densities Figure 4.23: Decomposition of bond angle distribution into different ring contributions at a fluid density of Figure 4.24: Decomposition of bond angle distribution into different ring contributions at a fluid density of

16 Figure 4.25: Average bond angle versus density for various sized rings. The average values are for bond angles that do not overlap with a bond angle associated with a smaller ring Figure 4.26: The average absolute torsional angle formed by adjacent members of a ring as a function of density Figure 4.27: The fraction of overlaps between different six-membered rings, measured using the overlap between the internal and external angles of the rings. This fraction is equal to 1.0 in the crystal Figure 4.28: Fraction of ring bond angles relative to the total number of bond angles observed Figure 4.29: Distribution of bond angle contributions from 3-membered rings Figure 4.30: Distribution of bond angle contributions from 4-membered rings Figure 4.31: Distribution of bond angle contributions from 5-membered rings Figure 4.32: Distribution of bond angle contributions from Type 1 bond angles of 6- membered rings Figure 4.33: Distribution of bond angle contributions from Type 2 bond angles of 6- membered rings Figure 4.34: Distribution of bond angle contributions from Type 3 bond angles of 6- membered rings Figure 4.35: Configurations of atoms leading to the split second peak of a dense fluid. All distances between bonded atoms are the same and equal to d, the average interatomic spacing. (a) Distance between two non-bonded spheres sharing a pair of bonded spheres as neighbours; (b) Distance between non-nearest neighbours of a collinear arrangement of atoms; (c) Distance between non-bonded spheres of two face-sharing regular tetrahedra; (d) Distance between spheres in a square arrangement of spheres, found in the (100) planes of the fcc lattice Figure 4.36: Radial distribution function of hard sphere fluid at densities near and above the freezing density Figure 4.37: Decomposition of g(r) of metastable fluid (ρ = 1.00) into ring contributions Figure 5.1: Compressibility factor (Z) versus time after density quench for a number of different simulation runs near the melting density (ρ = 1.05)

17 Figure 5.2: Compressibility factor (Z) versus time after density quench for selected simulation runs near the melting density (ρ = 1.05) and the mean over all runs Figure 5.3: Compressibility versus time for two different crystallization runs. The arrows denote the times (listed in Table 5.1) at which each different stage in the crystallization process starts. For run 6 the time listed in brackets for stage 2 in Table 5.1 is also shown Figure 5.4: Crystal fraction as a function of time for selected simulation runs at a density of The data has been separated for clarity Figure 5.5: Relative change in the compressibility factor (Z) and crystal fraction (CF) relative to a time 15τ after the start of the simulation (Run 4). -δz is the difference between the compressibility factor at time t and at a time t = 30τ after the beginning of the simulation Figure 5.6: Fraction of hard spheres in an icosahedral environment as function of time Figure 5.7: Fraction of hard spheres in a twisted icosahedral environment as function of time Figure 5.8: The fraction of hard spheres in an fcc environment as a function of time for different crystallization runs Figure 5.9: Fraction of hard spheres in an hcp environment as a function of time for different crystallization runs Figure 5.10: The stacking probability (ratio of fcc to total number of fcc and hcp particles) as a function of time for different crystallization runs Figure 5.11: Crystal fraction predicted from local orientational order parameter Figure 5.12: Fraction of crystal predicted from the planar graph method (left axis) and that from the spherical harmonic method for Run 1. The right axis has been shifted and scaled to highlight the similarities in the qualitative behavior of the crystal fraction predicted from the two methods Figure 5.13: Close up of radial distribution function at various times for simulation Run Figure 5.14: Number of SP rings per particle as a function of time for crystallization Run 1. The equilibrium values in the crystal for four, five and six-membered rings are three, zero and four respectively

18 Figure 5.15: Number of 6-membered rings per particle for a number of selected simulation runs. The equilibrium face-centered cubic crystal value is four Figure 5.16: Number of 4-membered SP rings per particle for a number of selected simulation runs. The equilibrium face-centered cubic crystal value is three Figure 5.17: Number of 5-membered SP rings per particle for a number of selected simulation runs. The equilibrium face-centered cubic crystal value is zero Figure 5.18: The fraction of overlaps between the Type 1 and Type 2 bond angles of six-membered rings as a function of time during crystallization Run 1. This fraction is unity in a perfect crystal Figure 5.19: The average bond angle across a six-membered ring (Type 3 bond angle) versus time for crystallization Run 1. The equilibrium value for the face-centered cubic crystal at ρ = 1.05 is 170 while the ideal crystal value is Figure 5.20: Average distance between antipodal pairs of particles that are part of the same six-membered ring versus time for crystallization Run Figure 5.21: Distribution of distances across a six-membered ring (Type 2 distances) at different times (initial fluid, growth stage and final crystal) during Run Figure 5.22: The average torsional angle, τ 6, of a six-membered ring as a function of time for crystallization Run Figure 5.23: Variation of Q 6 spherical harmonic with time for different crystallization runs Figure 5.24: The distribution of the angle between the local orientational order parameters of neighboring (bonded) particles at various times during crystallization run 7 (listed in the legend in units of τ) Figure 5.25: Decomposition of the distribution of angles between the local orientational order parameters of neighboring (bonded) particles at the end of simulation run 7 (t = 300τ) into contributions from fcc and hcp particles Figure 5.26: The distribution of bond angles formed by the central particle of a sixmembered ring and two antipodal particles of the ring (Type 3 bond angles) Figure 5.27: Crystal density for selected runs as measured from those four-membered rings composed of fcc and hcp ordered hard spheres. The results for Run 4 are

19 virtually identical to those of Run 1and the results of Run 7 are also virtually identical to those of Run Figure 5.28: The crystal density versus time for the coexistence run. The density is derived from the average distance across a 4-membered ring composed of ordered fcc or hcp particles. The equilibrium number density of the stable crystal is Figure 5.29: Variance of distribution of Voronoi volumes, σ 2 V, during different crystallization runs. Note that the results for Run 6 are virtually indistinguishable from those of Run 4 while the results for Run 7 and Run 5 are also very similar to each other. The equilibrium value for the variance of the Voronoi volume of the stable crystal at 1.05 is , the minimum value of the plotted variance scale Figure 5.30: The total fraction of ordered particles (using the spherical harmonic method) compared with the contributions from the two nuclei observed in the simulation at the coexistence number density of Nucleus 1 is the larger of the two observed nuclei Figure 5.31: Total number of particles versus time for two observed nuclei. A power law fit to the data for the larger nuclei is also shown Figure 5.32: Core of larger observed nucleus at a time of 170 τ for the coexistence run. The darker colored (brown) particles are fcc ordered while the lighter colored (yellow) particles are hcp ordered. The blue (very dark coloured) particle in the center has a twisted icosahedral local environment. The main five-fold symmetry axis of the crystal is orientated into the page. (a) All particles within a radius of 3.3σ from the central particle along the fivefold symmetry axis. (b) All particles within a radius of 6.0σ from the central particle along the fivefold symmetry axis Figure 5.33: Core of larger observed nucleus at a time of 170 τ for the coexistence run. (a) All particles within a radius of 7.1σ from the central particle along the fivefold symmetry axis. (b) All particles within a radius of 7.9σ from the central particle along the fivefold symmetry axis Figure 5.34: A side view of the same cluster of particles shown in Figure 5.33(b) displaying the (100) faces of the decahedral core

20 Figure 5.35: A view of all twisted icosahedral particles and the five fcc tetrahedron within the central core of the nucleus at a time 170τ after the start of the simulation. The two lightly shaded particles (yellow) are located at the center of their respective five five-fold axes. The views in previous figures are along the main symmetry axis in the middle of the diagram. The secondary five-fold axis is located to the right Figure 5.36: (a) Packing of fcc tetrahedral about final dominant symmetry axis of larger nucleus, looking down and to the side of the main symmetry axis. (b) Packing of tetrahedral about secondary symmetry axis Figure 5.37: Nucleus of smaller observed nucleus at a time of 170 τ for the coexistence run. (a) All particles within a distance of 6.0 σ from the particle midpoint along the main symmetry axis. (b) All particles in a twisted icosahedral configuration as well as all fcc particles contained within the main decahedral unit Figure 5.38: Form of the nucleus at times (a) 100τ and (b) 50τ. The nucleus is orientated as in Figure Note that the presence in the upper left fcc tetrahedral sub-unit of hcp ordered particles is only the outer bounding surface. The interior particles are fcc ordered. The coloring of particles is the same as in Figure Figure 5.39: Form of the nucleus at times (a) 100τ and (b) 50τ. The nucleus is orientated along the initial five-fold symmetry axis. Growth about this symmetry axis is inhibited by the presence of more than one twinning plane, actually three Figure 5.40: View along the initial symmetry axis at a time of 170τ of a cut through the center of the nucleus. Only ordered particles are drawn, hence the holes that appear in the top part of the nucleus. Note the complex nature of the orientation of twinning planes. The colors are as in Figure 5.32 except that the very light coloured (grey) particles are solid-like (as defined by the spherical harmonic method) but not categorized as fcc or hcp like Figure 5.41: The relation between the initial and dominant final five-fold symmetry axis. The nucleus is shown at the nucleation time of 75τ. (a) Nucleus view along the main final symmetry axis. (b) View of slice through nucleus showing twinning plane containing two five-fold symmetry axes, in the direction

21 indicated by the arrow in (a). The dark colored particles particles arranged in a line are twisted icosahedra ordered and define the five-fold symmetry axes Figure 5.42: Stacking probability, α, versus time for all ordered particles and for the two observed nuclei Figure 5.43: Five hexagonally close packed twining planes defining the main symmetry axis of larger nucleus found at coexistence (t = 100τ). The colors are as in previous figures except that the darker colored particles (brown) in (b) are defective hcp. The light colored particles are hcp ordered. The black particles have a twisted icosahedron local environment. (a) The close packed planes found passing through the main symmetry axis are shown and the particles drawn are ordered hcp or twisted icosahedra. (b) Same as (a) except that particles in a defect hcp environment have been shown as well Figure 5.44: Schematic diagram of fcc tetrahedral block of nucleus at t = 25 τ. Each face of the tetrahedron is a (111) face of the crystal nucleus. The shaded face A (ijk) contains both the initial and dominant final five-fold symmetry axes. The back face is denoted B (ikl), the top face by C (ijl), while the front face is denoted D (jkl) Figure 5.45: The form of the larger nucleus (coexistence) at t = 25τ. The orientation is the same as in Figure 5.46: Coherent close-paced planes parallel to face B of the fcc tetrahedral nucleus Figure 5.47: The bond network of the three hcp planes bounding the fcc tetrahedral unit of the larger nucleus at t = 25τ. In the left figure those particles that are solid-like (as defined by the spherical harmonic method) have been drawn Figure 5.48: A connected cluster containing the dominant partial decahedral nucleus and several smaller fluctuations for Run 1 at the induction time (75τ). Only the fcc and hcp particles described by the planar graph method are shown Figure 5.49: (a) The nucleus at 50τ for Run 1, displaying a central tetrahedral block of fcc ordered particles bounded by twinning planes. The other block of fcc to the top right is displaced behind the central nucleus but is still within a few diameters of the edge of the nucleus. (b) The main five fold symmetry axes at the midpoint of the growth regime (t = 120τ) for Run

22 Figure 5.50: Network of 5-fold axis of decahedron and the particles within a radius 5.5σ of the central axis of one of the MTP s for Run 6 at the crossover time Figure 5.51: The dominant nucleus for Run 5 at the induction time (t = 105τ) displaying a set of three parallel twining planes. Only the ordered fcc and hcp particles have been drawn for clarity

23 List of Tables Table 2.1: Average number of Voronoi nearest neighbours and average number of nearest neighbours after deletion of 3-sided faces (modified Voronoi) and also after deletion of 4-sided faces (final) for a hard sphere crystal Table 2.2: Most frequently occurring polyhedra before and after the deletion of 3- sided faces for a hard sphere face-centred cubic crystal at the melting density (ρ=1.05) Table 2.3: Most frequently occurring planar graph ring sequences (RS) of a hard sphere crystal near melting (ρ=1.05). The indices (I J) indicate the number of triangular (I) and square or 4-sided rings (J) Table 2.4: Number of Shortest Path rings per particle for an equilibrium hard sphere fcc crystal and an ideal fcc crystal Table 2.5: Average Bond angles for a hard sphere crystal at melting Table 3.1: Empirical and theoretical 18 values for <f> and µ Table 3.2: Theoretical 18 and empirical values of average geometrical properties of RVP froth Table 3.3: Values of k L, k A and k V obtained from a linear regression fit to the data Table 3.4: The coefficients of the quadratic fit to the three geometric functions V(f), A(f) and L(f). The values are for fits using the range f = The slope of each function at f = <f> is also included to aid comparison with the linear fit. In the linear case ( Table 3.5: Value of the Aboav-Weaire constant a from a linear regression fit to fm(f). The range of f values used was Table 3.6: Parameters of quadratic fit to fm(f) using above equation Table 3.7: Coefficients of linear relation between density scaled interfacial area and perimeter and density Table 3.8: Number of distinct Voronoi polyhedra (as defined by Bernal s indices) before and after the deletion of 3-sided faces for the hard sphere crystal

24 Table 3.9: Most frequently occurring polyhedra before and after the deletion of 3- sided faces for a hard sphere face-centred cubic crystal at the melting density (ρ=1.05) Table 4.1: The percentage change in the number of SP rings per atom as the density of the hard sphere fluid increases (relative to a number density of 1.00). * The change is measured for the 3-membered rings from the reference density of 0.60; the rings have a minimum value with respect to density for this value Table 4.2: Parameters for power law fits to peak and average ring distances versus density as well as average value of the peak position when scaled by the density. All distances are in units of σ Table 4.3: Mean scaled value of ring distances and ideal values for regular arrangements of hard spheres at contact Table 4.4: Average ring bond angle and percentage increase from a density of 0.20 to a density near freezing of The bond angles are those defined in Chapter 2. Note that two of the bond angles associated with the six-membered rings - Type 2 and Type 3 are angles formed between the central atom of the ring and the atoms about the ring Table 5.1: Times after density quench at which different stages in crystallization process can be discerned based on qualitative behavior of compressibility factor with time. All runs are at a density of 1.05 except the coexistence run which is at a density of Table 5.2: The final compressibility factor Z at the end of the simulation run (t = 300 τ) and fraction of crystal (as measured by fraction of fcc + hcp ordered and simple defects) Table 5.3: Times indicating different stages in crystallization process derived from fraction of crystal at these stages (as measured by fraction of fcc + hcp ordered and simple defects). The midpoint time of growth is simply the time halfway between the induction and crossover time. ** This is the mean value for the simulations performed at a density of 1.05, near the melting density Table 5.4: Total number of particles for each nucleus at different times during the simulation run within the coexistence region (ρ = 1.03) Table 5.5: Dominant Nucleus or Nuclei at Midpoint of Growth Regime. The stacking probability (α) at this time as well as at the end of the simulation is also shown.

25 The crossover time of the growth stage for each simulation is listed in Table 5.3. MTP means a multiply twinned particle, an ideal version of which is a perfect decahedron. SF refers to stacking faults

26 Summary The crystallisation behaviour of the hard sphere system has been examined using molecular dynamics methods. A main focus of this thesis has been to investigate the existence of structural precursors to freezing in the dense equilibrium or metastable fluid that indicate the onset of the fluid-solid transition. The role these precursor structures play, if any, in initiating the crystalline phase has also been examined. Another main aim has been to determine the form in which the stable phase first appears in the melt i.e. the form of the critical nucleus or nuclei. The approach taken to investigate the existence of structural precursors to freezing has been to characterise the equilibrium fluid from the ideal gas limit to the dense metastable fluid using a variety of structural measures. These methods are used to describe both the local and medium range structure. What we are looking for is distinct qualitative and quantitative changes in the microscopic structure of the fluid that appear around the freezing density. In order to accomplish this it was felt necessary to develop a deeper understanding of the statistical geometry of the hard sphere fluid. Two main approaches have been taken in describing the structure of the hard sphere fluid. The first approach has been to treat the hard sphere fluid as an example of a random cellular network. The cells for the hard sphere system are defined as the Voronoi cell of each particle in the system. The applicability of a number of semiempirical laws, obeyed by many random cellular networks, to the hard sphere system has then been tested. The hard sphere system is completely determined by its statistical geometry and hence it is an ideal system in which to look for precursor structures to freezing as no energetic factors come into play. A main result of the Voronoi analysis is that a modified version of both the Lewis and Aboav-Weaire law has been found to be applicable to the hard sphere system. We have also discovered an empirical equation that very accurately (<0.06 %) predicts the average geometry of the Voronoi cells of the hard sphere fluid purely in terms of the

27 geometry of the ideal gas limit. This link is provided by the Lewis law, a simple relation between the average topology and geometry of the cells of a random cellular network. The investigation of the geometry of the hard sphere system using the Voronoi approach has identified a density at which many different geometric properties of the hard sphere fluid show qualitative changes. This density is not the freezing density but a lower number density in the vicinity of 0.70 (φ = 0.37). This is the density around which qualitative changes are also seen in the dynamical behaviour of the hard sphere fluid; in particular the emergence of the cage effect. The second method of characterising the geometry of the hard sphere system focuses on its medium range structure. The geometric bonds between particles in a hard sphere system have been treated as a network and the topological and geometric changes of the system with density have been investigated using this outlook. A main aim has been to describe geometric changes in terms of a small number of topologically defined structures. One utility of this approach has been the ability to decompose the radial distribution function, g(r), (or more specifically the first two peaks) and the distribution of bond angles between neighbours into contributions from a small number of ring structures - four, five and six membered rings. It is found that the split peak in the g(r), which emerges as a shoulder in the dense fluid below freezing, is the most obvious qualitative change in the structure of the fluid about the freezing density. This finding is based on a comprehensive investigation of the changes with density of a wide range of structural properties of the hard sphere fluid. It is found that the emergence of the split second peak of the radial distribution function, g(r), is due to the competition between local pentagonal and hexagonal order - local five-fold pentagonal packings versus an increasingly long range six-fold hexagonal packing. This split peak form of the g(r) is typically described as an indicator of either a glass or undercooled fluid phase but the results of this work indicate that this is not so. It indicates the suppression or frustration of crystallisation, in particular the suppression of the evolution of precursor structures that if allowed to, can form the templates for nucleation of close-packed order. In the fluid these six-

28 membered rings are non-planar but their average bond angles approach crystalline values as the density increases towards freezing. It is found that as the density increase towards the freezing value qualitative changes occur in the geometry of this ring structure, in particular the distribution of distances between particles across a ring. These changes are reflected in the emergence of the split peak above freezing. The first sub-peak of the split second peak of the g(r) can be identified with distances between particles across either a five or six-membered ring while the second peak can be uniquely identified with distances between particles that are part of a sixmembered ring. The close packed planes of the face-centred and hexagonal close packed crystal can be described as an overlapping tiling of the six-membered rings defined in this work. It is found that as the density increases the degree of overlap between the sixmembered rings increases. Nevertheless the rings are not planar and hence it is not expected that a growing correlation length can be identified with these overlap regions. As the density increases a combined increase in the degree of coherence between six-membered rings as well as an increased geometric ordering takes place. The precursor structures in the dense fluid that indicate the onset of freezing are identified as overlapping six-membered rings of the type defined in this work. The description of the structure of the hard sphere system in terms of the network of geometric bonds between particles has compelled us to take considerable care in the definition of such a bond. A modified Voronoi definition of the neighbours of a particle is developed in this work to accomplish this. It has also overcome some of the peculiar problems that have plagued investigators when describing the local environment of a particle in a close packed crystal in terms of the topology of its Voronoi cell. The end result of this approach is the description of the structure of a fluid or crystal in terms of a packing of tetrahedral and octahedral units, with particles located at the vertices of these two units. Any computer simulation investigation of the formation of crystalline order at a microscopic level necessarily entails some method of identifying particles as being in an ordered environment. This has always been a challenging area in computer simulation studies of crystallisation as one is attempting to use a local order parameter

29 to describe the emergence of long-range translational and orientational order. Although the appropriate order parameter for the liquid-solid fluid is a set of Fourier component of the periodically varying density in the crystal this is of little use when a local description is required. The planar ring method described in chapter 2 is such an attempt at describing the local order about a particle. It has been combined with the spherical harmonic method of ten Wolde and has allowed us to categorically identify and describe in detail the form, and in some cases the growth, of crystal nuclei in the metastable hard sphere fluid. The planar graph description of the environment around a particle is topological in nature, as is the Voronoi approach, and in a sense is the dual of the latter method. Both focus on the arrangement of neighbours about a particle, but with different emphasises. The crystallisation of a hard sphere fluid density quenched into both the coexistence region of the phase diagram (ρ =1.03, φ 0.539) and to a density slightly above melting (ρ =1.05, φ 0.549) has been examined in detail. It is found that within the coexistence region, compact well-separated nuclei are observed. The form of both observed nuclei is that of a multiply twinned particle with an overall decahedral symmetry. The growth of these nuclei has been followed in detail. It is found that the initial fluctuation is not a decahedral core, but a tetrahedral shaped block of facecentred cubic arranged particles bounded by twinning planes, five of which when packed about a common edge form the decahedral crystallite. The induction period is found to end when two or more of these sub-units have formed and share a face - growth proceeds rapidly after this occurs. It is also found that the nuclei merge well before the crossover time (the time at which a substantial slowing down of crystallisation is observed). The density of the growing crystal has also been estimated and at coexistence a value of 1.11 ± 0.01 (φ = 0.58) was found at early times, during the induction period. This density drops towards the equilibrium value as the crystal grows in size with qualitative changes following the pressure. The density of the initially formed crystallites at the density beyond melting was found to be only slightly higher at 1.12, ± 0.01 (φ = 0.59). Again changes with time followed the drop in pressure of the system.

30 It was found that nucleation at a density slightly above melting, ρ = 1.05 (φ 0.549) could not be described in terms of the appearance of multiple but separated nuclei. It was found that not only where multiple nucleation events observed but also that nuclei appeared close to one another interacting in a highly complicated fashion. Nevertheless one nucleus tended to dominate the growth process with other nuclei often appearing during the growth stage of this nucleus. It was also found that the fluid region between nuclei was partially ordered leading to occasional bridges between nuclei. It is concluded that it cannot a priori be assumed that nuclei forming around this density grow independently, although are results at least indicate the existence of dominant nuclei that starts to grow at a time predicted by the induction time. This is consistent with experimental results on hard sphere colloidal suspensions in which a burst of nucleation and a maximum in the nucleation rate is found at the melting density. A variety of different nuclei forms were observed. Yet again multiply twinned particles of decahedral morphology were observed. The growth pattern followed that observed at coexistence. Other nuclei formed consisted of planar lamella containing parallel twin planes. The number and distance between twin planes differed, but as would be expected their formation lead to a preference for particles arranged in a face-centred environment compared to a hexagonal environment. All simulations showed a slight preference for particles in a face-centred cubic environment compared to those in a hexagonal close packed environment. This behaviour can easily be rationalised in terms of the growth behaviour of the main two types of nuclei observed here. Small decahedral nuclei, of the type observed in this work at both densities studied, can produce a stacking probability of around 0.50, the random stacking value. Given this fact the observation of a crystallised sample with such an estimated packing probability does not imply that the system is randomly stacked, it is just as possible that the crystal has formed from many small decahedral nuclei. The ultimate origin of the nuclei was followed back in time and it was found that precursor structures of the form described above formed the template for growth. Whether the fluctuations consisted of parallel or crossing close packed planes determined the eventual form of the nuclei.

31 The appearance, structure and growth of nuclei in a hard sphere system quenched to both a density in the coexistence region and to a density above melting has been studied. The precursor structures to freezing have been identified and it has been shown that these fluctuations in the dense fluid, observed at early times before the growth stage, provide the templates for the nuclei observed to grow.

32 Chapter 1 Introduction 1.1 Background Crystallisation is a first order phase transition in which a homogeneous, isotropic liquid transforms into a crystalline structure characterised by long-range translational and orientational order 1. The casting process in metal production is one example of an industrially relevant crystallisation process 2. Metallurgists would like to control many aspects of this process, from processing conditions to the final average grain size of the metal 2. Understanding crystallisation is also of interest to the food industry, in both food processing 3, 4 and storage. In the latter case the goal is to inhibit crystallisation, specifically the freezing of water 5. The first step in identifying the structure of a protein is to produce a crystalline sample for use in scattering measurements 6. Crystallisation is an as yet poorly understood phenomenon due to the numerous experimental difficulties in studying crystallisation processes. Nucleation rates are observed to be a very sensitive function of the degree of undercooling of a material 7. For small undercoolings nucleation is extremely slow, but within a very small temperature the nucleation rate can increase by many orders of magnitude 8. Crystal growth rates can often approach the speed of sound in the material and hence in many cases the nucleation processes is over before the experiments can be begun 7. Recent development of millisecond resolution X-ray diffraction techniques has allowed the study of crystallisation in metallic glasses, but this is still a comparatively slow process compared to the liquid-solid transition 9. In this regard studies of crystallisation in silicate glasses 10 and metallic glasses 2, 11 have also aided in an understanding of crystallisation. But how similar crystallisation from a solid glass phase to a crystal phase is to nucleation from the liquid melt is an open question. 1-32

33 The spontaneous formation of a crystal nucleus is a rare event and it is often difficult to observe compared to the much more common and competing process of heterogeneous nucleation - the formation of nuclei initiated by impurities or by the container in which the fluid is kept 2. Controlling these effects is very difficult and great care often has to be made in sample preparation and subsequent analysis of crystallisation data to ensure it is avoided 2. Minute levels of impurities in a material can become nucleation sites, as can the surface of the container holding the liquid or the liquid meniscus. Heterogeneous nucleation of a crystal by a surface or impurities is by far the more common and probable process in nature but from a theoretical perspective, this process is likely to be very specific to the nucleating template and hence less amenable to a general theory 12. For this reason the process of homogenous crystallisation, the spontaneous formation and growth of crystallites in the bulk of a liquid, is the main theoretical focus. Another experimental difficulty is the time and length scales associated with crystallisation. It is typically a very fast process, growth rates of the order of metres per nanosecond are common in many metals, and so obtaining information on the dynamics of crystallisation on such short time scales is difficult. Also how crystalline order first appears in an undercooled liquid requires information on very small length scales. In the early stages of nucleation, scattering from nucleated crystallites is easily washed out by scattering from the surrounding liquid. 1.2 Nucleation Theory It is an old observation that while liquids can be easily undercooled it is extremely difficult to superheat a solid. Water can be easily supercooled to -30ºC while gallium can be cooled to 150º C below its freezing point 13. This asymmetry between freezing and melting has given rise to the idea of a kinetic barrier to the formation of a crystalline phase. Although the solid is the thermodynamically stable phase in an undercooled liquid, a large but finite free energy fluctuation is required in order for it to appear

34 The difference between two phases is most conveniently characterised in terms of an order parameter. The most obvious difference between the liquid and solid is the existence of long-range order in the latter. Thus an appropriate order parameter is a Fourier component of the density evaluated at a reciprocal lattice vector of the crystal or a set of such components evaluated at lattice vectors of the crystal. Nucleation describes the process by which the stable phase first appears in the metastable state. In the classical picture of crystal nucleation the new phase is assumed to appear as a local, large amplitude fluctuation. This order parameter fluctuation is called the critical nucleus and acts as a seed for further growth of the crystalline phase 15. The probability of a critical nucleus forming can be related to the free energy required for the fluctuation using standard fluctuation theory 16. The nucleation of a crystal in an undercooled liquid is commonly described using Classical Nucleation Theory (CNT), originally developed to describe gas-liquid nucleation 14. Turnbull and Fisher 17 first applied this theory to the liquid-solid transition. The primary goal of CNT is to predict the nucleation rate and growth rate of a crystallising substance as a function of the degree of undercooling. The nucleation rate is the number of nuclei that form per unit volume and unit time, while the growth rate measures the rate at which these nuclei subsequently grow. The interplay between these two rates is the dominant mechanism determining the grain size in crystals. The basic approach of CNT is to assume that the new phase spontaneously appears as a spherical crystalline droplet with the same density as the equilibrium bulk phase. In the capillarity approximation the interface between the solid and liquid is assumed to be sharp, the density and order parameters change discontinuously at the interface. It is a phenomenological theory as macroscopic equilibrium arguments and quantities are used to describe the process. The nuclei are assumed to be non-interacting and to appear randomly in the bulk of the fluid. 1-34

35 The probability W of a thermally activated free energy fluctuation G associated with the formation of a crystalline nucleus is W exp( G /kt) where T is temperature and k is Boltzmann s constant. The inverse of this is proportional to the lifetime of the metastable state. The fluctuation in free energy required for a nucleus of radius R to form is dependent on two terms: a negative bulk term and positive surface term, 4π G = R 3 3 ρ µ + 4πR s 2 σ ls The density of the solid is denoted by ρ s while the surface free energy density is σ ls.. The surface term equals the work required to form an interface between the solid and liquid and hence is always positive. The bulk term is negative when the difference in chemical potential between the liquid and solid is positive, µ = µ l µ s > 0 It can be seen that there will be a maximum in the free energy of formation of the nucleus at a certain radius. This is called the critical radius, R*, and can be found by setting the derivative with respect to R of G to zero giving, R * 2σ = ρ µ s The height of the free energy barrier, G*, is given by 16π G* = 3 ( ρ σ s 3 ls µ ) 2 which decreases as the system moves further from coexistence. Thus as the system moves further into the metastable state, the driving force for nucleation increases. 1-35

36 A nucleus that spontaneously forms at the critical radius has an equal chance of growing or decaying and is thus called a critical nucleus. Smaller nuclei will decay away as this reduces their free energy and larger nuclei will have their free energy reduced by growing. The liquid-solid surface tension is typically taken as the value at coexistence and bulk values are also used for the chemical potential difference. No account is taken of the finite size of the nucleus or the effect of curvature on the surface tension. This assumes that the properties of very small nuclei can be adequately described by bulk values. Also it in principle assumes that a very small nucleus has a well-defined interface with the liquid and so there is no partial ordering of the liquid surrounding the nucleus. In order to calculate the actual rate at which nuclei form it is assumed that the free energy of formation of a critical nucleus can be used to calculate the nucleation rate. The nucleation rate I is equal to the product of two terms: the rate at which new particles attach to the nuclei, I o, and the number density of nuclei 18. The second term is assumed equal to the probability of formation of a critical nucleus. I = Io exp( G* / kt) The exponential term is typically dominant and changes in the prefactor, I o, by a few orders of magnitude do not have a significant affect on the predicted nucleation rates. The attachment rate of new particles can be proportional to a number of factors. If it is thermally limited then the phonon frequency is a reasonable measure of the time scale for growth. In the case of interface limited growth the diffusion constant is used to estimate the time over which a particle moves a distance equal to its size. However, attachment of atoms to the crystal-liquid interface is often more a case of rearrangement rather than large-scale motion. The above equations are commonly used to analyse experimental results although the kinetic prefactor and the surface free energy density are often treated as free parameters. The coalescence of nuclei is assumed not to occur and different growing 1-36

37 nuclei are assumed not to affect each other. Kinetic equations governing the growth and decay of nuclei of different sizes can be taken from the gas-liquid case but these assume growth and decay is by the addition or loss of single particles. As the liquid is in physical contact with the growing crystalline nuclei this condition is not met in the liquid-solid case. CNT contains no microscopic information about the state of the system and is essentially an absolute chemical rate state theory, in which the transformation from liquid to solid is regarded as a chemical reaction. A modern approach that includes microscopic information about the structure of the fluid is based on density functional theories of the liquid state Density functional theory starts from the well-known result that the free energy of a system can be described as a functional of the density ρ(r) of the system. The approach is used for both the liquid and crystal and hence it amounts to describing the crystal as an inhomogeneous liquid 1. Density functional theory (DFT) can be used to predict the location of the phase transition, the fractional change in density on freezing, the lattice spacing of the equilibrium crystal and in most cases can decide between different input crystal structures 22. Density functional theories have been applied extensively to the hard sphere system in order to predict the phase diagram 23, investigate the crystal melt interface 24 and the kinetics of crystallisation 25, 26. Classical models of growth have also been investigated and evaluated in the light of experimental findings on hard sphere colloidal suspensions Experimental Studies of Crystallisation Colloidal systems have played a large role in experimental studies of crystallisation. An advantage of colloids is that the slow dynamics of these systems allows crystallisation to be observed over an accessible time window. The size of colloidal particles is comparable to the wavelength of visible light and hence a range of optical 1-37

38 techniques from dynamic light scattering to confocal laser microscopy can be used to investigate the crystallisation process 28. One particularly well-characterised system is nearly monodisperse hard sphere colloidal suspensions. This system mimics the behaviour of the ideal hard sphere system, allowing a comparison between the results of computer simulations and experiment 29. As with other colloidal systems the suspension medium provides a large thermal reservoir and hence any latent heat evolved during crystallisation is rapidly dissipated. Hard spheres are a system of impenetrable spheres interacting purely via a hard-core repulsion. As the structure of most simple liquids is dominated by sphere packing considerations the hard sphere system is to real liquids what the ideal gas is to a real gas. It models the dominant influence of the repulsive interactions between molecules on the structure of liquids, i.e. entropic effects 30. The main difference between a hard sphere system and a colloidal approximation of this system is the degree of polydispersity, or range of particle diameters that exist in the latter. The degree of polydispersity has been found to have an influence on the crystallisation behaviour through both experiment 31 and computer simulation 32. The main difference found is a simple increase in the induction time before nucleation and growth commences. The nucleation and growth of hard sphere colloids has been studied extensively. Small angle light scattering has been used to investigate the growth and coarsening behaviour of these systems The length scales probed are much larger than the radius of the particles studied and so each crystal acts as a single diffracting object. Information can then be obtained on the average crystal size as crystallisation proceeds. It is found that at both early and late times the peak intensity and half width peak position display power law scaling behaviour with time. At intermediate times there is a crossover from nucleation and growth of crystals to ripening during which no scaling behaviour is observed 33, 36. The exponents derived from these power law fits predict diffusion limited growth within the coexistence region and interface limited growth beyond the melting density if a constant nucleation rate is assumed. 1-38

39 Growth of the radius of the crystal is proportional to the square root of time within the coexistence region and linear in time for densities beyond melting. By monitoring the peak position of the (111) Bragg reflection, Harland et al 37 observed that the initial crystal nuclei have a higher density than the equilibrium solid 37. Hard sphere colloids are maintained at constant volume rather than constant pressure and hence this behaviour is rationalised in terms of the higher pressure of the metastable fluid acting to compress the growing crystallites. The decrease in the crystal density with time suggests that as the crystal grows the (osmotic) pressure drops and hence the crystal expands in response. Ackerson 38 found a constant nucleation rate is consistent with their scattering data for crystallisation within the coexistence region but at densities near melting there appears to be a burst of nucleation. Harland et al 39 have also found that the nucleation rate is a maximum at the melting density from an analysis of the evolution of the main (111) Bragg peak using dynamic light scattering. They find that the fraction of crystal is proportional to the time cubed (the radius of the crystal increases linearly with time) while at densities near melting they observe accelerated nucleation, obtaining an exponent of 4 for the power law time dependence of the crystal fraction. Henderson 31 has also found that the induction time is a minimum at the melting density. The induction time is a transient time during which the initial critical nuclei spontaneously form. A recent debate on hard sphere systems concerns the equilibrium structure of the hard sphere crystal. The thermodynamically stable form is known to be a face-centred cubic crystal but the entropic difference between the hcp and fcc structure is extremely small 40. The stacking of close packed planes can be characterised in terms of a stacking probability α. This is the probability that a close packed layer at n will have a different registry than a layer at n+2. It is zero for a hcp crystal and unity for a fcc crystal while a random stacked crystal has a stacking probability of 0.5. Pusey et al 41 found that near the melting density the results of light scattering measurements of powder diffraction patterns suggested a random close packing 1-39

40 structure with a tendency towards fcc on increasing or decreasing the density. Later experiments indicated that an fcc preference evolved over time 42. However, microgravity experiments by Zhu 43 found no evolution from random stacking over several days. In order to resolve some of these discrepancies, Kegel and Dhont 44 studied systems with a small degree of polydispersity under simulated microgravity conditions. Their experiments are consistent with a fraction of what they label as faulted-twinned fcc structure which grows at the expense of random packing. Estimates of the timescales involved suggest that the microgravity experiments of Zhu did not have sufficient time to evolve into a faulted fcc structure, although they conceded that gravitational stress would speed the process. A recent direct estimate using confocal laser microscopy 45 suggests a value of 0.4 ±0.2, the large error due to the extremely limited number of layers analysed. Overall estimates of the stacking probability range from 0.4 to 0.75, although recent free energy calculations tend to support the fcc structure having the highest entropy relative to all possible stackings 46. Another avenue that has been used to investigate the possible form of a crystal nucleus is through computational and experimental work on small clusters 47, 48. As the critical nucleus is simply a small cluster of particles and a balance between bulk and surface contributions determines its equilibrium form, it is thought that these clusters may be in some sense representative of the critical nucleus in a fluid. Experimental information on small clusters has come via a number of sources. Transmission electron microscopy (TEM) investigations of small metallic clusters grown on surfaces have revealed a pronounced preference for decahedral and icosahedral structures 47, These five-fold structures have also been observed in electron diffraction studies of clusters formed by free expansion of noble gases such as argon In some materials e.g. silver, tabular forms with single or multiple parallel twinning planes have also been observed 55, 56. The latter observations have given rise to a number of models of nuclei based on structures that provide favourable growth sites 57, 58 and selectively prefer one type of packing over another e.g. a facecentred cubic structure over a hexagonally close packed structure

41 A number of computational studies have focused on the structure and binding energies of small clusters of atoms interacting via a Lennard Jones potential or Morse potential. Again it is found that small particles with decahedral or icosahedral form are dominant for cluster sizes of less than 1000 to atoms depending on the precise range of the potential This preference for what are often called multiply twinned particles is due to the relatively low surface to volume ratio of these particles compared to the equilibrium shape of the crystal. 1.4 Computer Simulation of Crystallisation Computer simulation methods are of use in modelling crystallisation for two main reasons: they provide insight into processes occurring on timescales and length scales that are so short as to be experimentally inaccessible and they enable a wealth of structural and dynamical information to be obtained that at best can only be inferred from experimental measurements 8. The early stages of crystallisation are a particularly pertinent topic to treat by computer simulation methods. A number of studies have been conducted on homogeneous nucleation and heterogeneous nucleation modelling predominantly noble elements, metals 75 and silicon 76. Computer simulations have been used to study several simple model fluids including hard spheres 28, soft repulsive spheres 64, 65 and Lennard-Jones fluids 63, 66, 68. Another useful feature of computer simulations is that significant problems exist in experimentally examining homogeneous nucleation while avoiding heterogenous nucleation. As we have complete control in a computer simulation and typically use periodic boundary conditions to avoid surface effects these problems do not manifest in a computer simulation study. Having said this a main theme of much early work in crystallisation studies was ensuring that the small number of atoms simulated and hence the small length scales involved did not influence the results of the computer experiments on crystallisation 68, 70,

42 There are however several limitations to simulation studies - growth studies are limited as the small length scales mean that when the crystal dimension is of the order of the simulation box side periodic boundary effects will come into play, i.e. a nucleus may start interacting with an image of itself; this also limits the time interval over which an individual growing crystallite can be studied. A more significant problem is due to the small size of the systems - as homogeneous nucleation is a rare event a liquid must be deeply undercooled before a nucleation event is likely to occur. This raises the question as to whether the crystallisation processes observed is representative of what occurs at more moderate undercoolings or at coexistence - the more typical experimental and industrial situation 78. A main result of many of these studies is that the critical nucleus is small, a few tens of atoms. Swope and Anderson 68, in their million-atom study of the Lennard Jones system, estimated the critical nucleus size as between 10 and 20 atoms. However the actual structure of the critical nucleus has not been investigated in detail. What information exists suggests that the nucleus has a highly ramified structure, and shows a preference for face-centred cubic order. In some cases body-centred nuclei have also been observed 63, 70. The difficulty of drawing conclusions about the structure of the critical nucleus from these studies is two-fold. The diffuse nature of the critical nucleus may be due to the extreme undercoolings used in the simulation studies or equally may be due to the choice of order parameter used to identify a particle as part of the growing crystal. A distinct advantage of the hard sphere system over other model systems is that, contrary to previous assertions 79, 80, hard spheres are very easy to crystallise and the spontaneous appearance of multiple well-defined nuclei can be observed even within the coexistence region. Thus crystallisation can be investigated at both moderate and extreme undercoolings; where the degree of undercooling in a hard sphere system relates to the density of the system relative to the freezing density. Qualitative differences in the crystallisation behaviour of the hard sphere system (if any) can then aid in the interpretation of the results of simulations of other systems where only extreme undercoolings are possible. A novel approach to the nucleation problem is the direct calculation of the free energy barrier to nucleation using a biased Monte Carlo method The basic approach is to 1-42

43 bias the sampling of configuration space so that configurations with a small probability, e.g. highly ordered configurations are more commonly sampled. When applied to the study of nucleation it necessarily requires some order parameter to define the degree of crystallinity of a configuration. van Duijneveldt and Frenkel 85 use the global bond orientational order parameter Q 6, originally introduced by Steinhardt 86, 87 to study orientational ordering in liquids and glasses. This technique has been applied recently to calculate the free energy barrier to nucleation at a number of densities within the coexistence region of the hard sphere phase diagram. Auer and Frenkel 88 found that the critical nucleus varies in size from approximately 250 to 70 particles at densities within the coexistence region (φ = to φ = ). At higher densities multiple nucleation events were observed. They find large discrepancies between their predicted nucleation rates and that deduced from experiment, the latter are several orders of magnitude larger than predicted. The structure of the critical nucleus has also been investigated using this approach. They find that the dominant structural signature can be identified with random hexagonal close packing and find that body-centred cubic and icosahedral structures play no role in the nucleation process. This is given as an example of the Ostwald 89 step rule: the phase that first nucleates need not be the most thermodynamically stable but simply the one closest in free energy to the fluid. In the case of the hard sphere system this is suggested as being the random hexagonally close-packed lattice while for the Lennard-Jones system the metastable phase is the body-centred cubic lattice 90, 91. Note that in a hard sphere system the body-centred cubic phase is mechanically unstable and hence is not a metastable phase 92. The main subject of this thesis is the investigation of the crystallisation behaviour of the hard sphere system by the use of computer simulation methods using novel structural analysis techniques. In particular we have investigated how a crystalline phase first appears in a melt, i.e. the structure of the critical crystal nucleus. We have 1-43

44 also investigated a related question: are there any precursor structures in a melt that indicate the onset of the ordered state? In seeking to shed light on these questions a significant component of this thesis has been an analysis of the significant structural features that distinguish the fluid from the crystalline state at a detailed microscopic level. At a macroscopic level the distinguishing difference between a fluid and crystal is obvious: the translational and orientational order of the latter, but at an atomic level differences are more subtle and this is particularly pronounced in the transition from one phase to another. In combining these two topics we hope to provide information on not only the details of the crystallisation process but also on the reasons why the fluid crystallises at the density it does. A practical problem in computer simulations of crystallisation has been the choice of an appropriate local order parameter that can clearly distinguish between fluid and crystalline environments. Therefore a concurrent goal of this thesis has been to develop a robust method of describing the local environment about a particle in order to identify those that are part of a growing crystal nucleus. 1.5 Bibliography 1 A. D. J. Haymet, Science 236, 1076 (1987). 2 D. M. Herlach, Mater. Sci. Eng. R-Rep. 12, 177 (1994). 3 S. Adapa, K. A. Schmidt, I. J. Jeon, T. J. Herald, and R. A. Flores, Food Rev. Int. 16, 259 (2000). 4 P. Fryer and K. Pinschower, MRS Bull. 25, 25 (2000). 5 M. Griffith and K. V. Ewart, Biotechnol. Adv. 13, 375 (1995). 6 J. L. Martin, Curr. Med. Chem. 3, 419 (1996). 7 K. F. Kelton, in Solid State Physics, edited by H. Ehrenreich and D. Turnbull (Academic, Boston, 1991), Vol. 45, p D. Frenkel and J. McTague, Ann. Re. Phys. Chem. 31, 491 (1980). 1-44

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49 Chapter 2 Structural Analysis Methods 2.1 Introduction In this Chapter a number of different approaches that have been developed to analyse the microscopic structure of the hard sphere system are described. A particular focus is on structural measures that are of use in characterizing the onset of crystalline order in a metastable fluid. One aim of this thesis is to identify the form in which the crystalline phase first appears in a hard sphere fluid when density quenched into the two-phase coexistence region or beyond. This requires a method to identify particles as being in a local crystalline environment. A second aim is to investigate the existence of precursor structures in the dense equilibrium or metastable fluid. These structures can either indicate the approach of a first order transition and/or play a role in the early stages of nucleation as templates for the nucleation of the solid phase. A classification scheme based on the Delaunay tessellation of space, the dual of the Voronoi decomposition, has been developed to identify particles in a local crystalline environment. In order to accomplish this a modified version of the Voronoi definition of the nearest neighbours of a particle has also been developed, which is also unique to this work. These structural measures have also been used to investigate the equilibrium fluid and crystalline states as well as the crystallizing fluid. An analysis of the equilibrium system is an important component of our studies, in order to identify the qualitative and quantitative differences between the two phases at the microscopic level. This has proven to be of use in understanding the reasons why at a given density a first order 1-49

50 liquid -solid phase transition takes place in the hard sphere system. The measures are not specific to the hard sphere system and can be applied to other systems as well. A novel structural analysis approach has also been developed that describes the short and medium range order of the hard sphere system in terms of ring statistics. A benefit of this particular methodology is that it allows the decomposition of the local structure of the system into structural motifs that are of particular relevance to dense fluids and crystals. In particular it allows the decomposition of the radial distribution function g(r) of an atomic system into contributions from different ring structures, and hence provides a link with experimentally measurable quantities. 2.2 Voronoi Tessellation A common method of describing the structure of a hard sphere system and indeed any ordered or disordered packing of particles is in terms of its Voronoi cells 1. The Voronoi cell of a particle is a convex polyhedron surrounding it that contains all points closer to it than to any other particle. The Voronoi tessellation of a system of particles partitions the volume they occupy into a set of non-overlapping convex polyhedral cells in three dimensions (3D) or polygonal cells in two dimensions (2D), Figure 2.1. Bernal 2-4 and Finney 1 first applied this particular structural analysis method to the problem of describing the microscopic structure of the dense hard sphere fluid. It provides information on the range of local environments about a particle in a fluid in terms of the different types of polyhedra surrounding each particle. It is also a particularly useful scheme for classifying the local environment around a particle due to the uniqueness of the definition of a Voronoi cell, except for certain degenerate cases discussed below. It is also one of the more common methods used to identify locally ordered environments in computer simulations of crystallisation processes

51 A number of papers describe the basic terminology and properties of Voronoi tessellation s as well as practical methods of computing them 4 6 ; for completeness we will give a brief description here Definitions and Terminology The Voronoi construction divides the volume occupied by an atomic system into a number of non-overlapping polyhedral cells centred on particle centres. It is a unique decomposition of space except for certain locally degenerate arrangements of particles 6. In the solid-state physics of crystals the Voronoi cell is known as the Wigner-Seitz cell. For a single Voronoi polyhedron each edge of the polyhedron is shared by two of its faces, while three faces meet at each vertex. Two Voronoi cells share each face of a Voronoi polyhedron of the tessellation. Three Voronoi cells share each edge while four cells share each vertex 6. This is true except for certain degenerate cases where more than four Voronoi cells may meet at a vertex. Such is the case for certain vertices of the Voronoi tessellation of the face-centred and hexagonally close packed lattices. In practice this degeneracy has been removed by infinitesimal displacements of the particles from their lattice sites. The Voronoi cell enclosing a particle is produced by first drawing bonds between the particle and all particles within a certain distance from the particle. A plane is then drawn through the midpoint of each bond and perpendicular to it. The intersections of the planes define the vertices of the Voronoi cell, Figure 2.2. This is a polygon in 2D space and convex polyhedron in 3D space, Figure 2.3. Except for certain degenerate configurations, 3 planes of a Voronoi polyhedral cell meet at each vertex. Likewise four Voronoi cells meet at each vertex of the Voronoi tessellation. The dual to the Voronoi decomposition is the Delaunay tessellation of space. This can be constructed from the Voronoi decomposition by drawing bonds between particles that share a common face of a Voronoi cell. This is shown in Figure 2.1 for a 2D case. 1-51

52 Figure 2.1: The Voronoi decomposition and the dual Delaunay tessellation in 2D. The black circles represent partic les, while the black lines are the edges of the Voronoi cells and the light coloured lines are the bonds between Voronoi neighbours (particles sharing a Voronoi face). Figure 2.2: The Voronoi construction in 2D. The Voronoi cell of a particle is the smallest polygon (or polyhedron in 3D) formed by the intersecting bisectors (a line in 2D or a plane in 3D) drawn from the particle to all other particle centres. Note that the perpendicular bisector of the bond between the central particle and a candidate neighbour j does not intersect the Voronoi cell of the central particle. Hence j is not a Voronoi neighbour of the central particle. 1-52

53 In 2D the Delaunay tessellation results in a tiling of space with triangles having particles at their corners. In 3D it results in a packing of tetrahedra with particles at the corners of the tetrahedra. The circumcentre of a tetrahedron of the Delaunay tessellation is a point equidistant from the four particles defining the tetrahedron, while the circumcircle of each of its triangular faces is equidistant from three of the particles and within the plane formed by them 7. The circumcentre of each tetrahedron is a vertex of the Voronoi tessellation and a line drawn from this point to any of the circumcircles of the triangular faces of the tetrahedron is an edge of the Voronoi tessellation, Figure 2.4. This is because the circumcentre of a tetrahedron is the point of intersection of four lines, each line perpendicular to one of the faces of the tetrahedron and passing through the circumcentre of its face. The distance from the circumcentre of a tetrahedron to any of the particles defining it is called the circumradius and by construction, no other particle in the packing is closer to it than those four defining it. This empty circumsphere property has important consequences and is also exploited in calculating the Voronoi tessellation 6. If more than 4 particles are equidistant from the same point in space then degeneracy is said to have occurred. One common example is the case of a regular octahedron. The reason for this is that an octahedron can be decomposed along any one of its three long diagonals into 4 tetrahedra each sharing the same long diagonal, Figure 2.5. The circumradius of each tetrahedron is at the same location in space, namely midpoint along the diagonal. Thus there are 6 particles equidistant from the circumradius rather than the usual four. The degeneracy in the Delaunay construction is due to the three equivalent long diagonals that the octahedron can be decomposed into. The degeneracy in the Voronoi case is the meeting of four faces of a Voronoi cell at the same point in space. In the case of a perfect lattice this degeneracy is usually overcome by perturbing perfect octahedron slightly so that one diagonal is preferentially selected. The resulting Voronoi face created has a near zero measure. 1-53

54 Figure 2.3: A Voronoi polyhedral cell in 3D. Three faces of the Voronoi cell meet at each vertex and two faces share each edge. (a) (b) Figure 2.4: (a) The circumcircle (point 1) defined by three particles a, b and c. The radius of the circumcircle is indicated by ccircle. By definition no other particle lies within this circle. The three perpendicular bisectors of the bond between the spheres define the circumcircle. (b) The circumsphere of a tetrahedron in the Delaunay decomposition. The radius of this circumsphere is denoted by cr (circumradius). No other sphere lies within this sphere. The circumcentre is defined as the intersection of the lines drawn from the circumcircle of each face (1-4) and perpendicular to that face. The circumcircle of face abd is denoted by point 2, acd by point 3 and that of bcd by point

55 Another degeneracy occurs with a perfect octahedral arrangement of particles because it can be decomposed into either 4 or 5 tetrahedra. In the latter case one of the tetrahedron has a near zero volume and this type of tetrahedron is called a sliver or Kije simplex 8, Figure 2.6. The Kije simplex consists of 4 particles that are nearly coplanar. This also results in a 3-sided Voronoi face. Two of the sides of the sliver tetrahedron are the long diagonals of the octahedron Applications of Voronoi Description The Voronoi approach has been used to analyse the structure of a wide range of simulated materials such as sodium glasses 9, lipid membranes 10 and Frank-Kasper phases 11. It has also been used as a general method of distinguishing between crystalline structures 12. Bernal first suggested the use of a set of indices to describe different polyhedra 2. These indices are a set of numbers (n 3 n 4 n 5 n 6.. n k ) describing the number of k-sided faces of a given polyhedron. The number of different Voronoi polyhedra observed in a liquid is then a useful measure of the range of local environments. In a perfect crystal there is only one type of polyhedron eg. (0608) for a body-centred cubic (bcc) crystal, where the indices mean that the Voronoi cell for a bcc lattice has 14 faces of which 6 have four sides and the other 8 have 6-sides, Figure 2.7. However Bernal s indices are not sufficient to uniquely define a polyhedron. As an example the Voronoi polyhedron for the face-centred (fcc) and hexagonally close packed lattices (hcp) have the same indices (01200) but differ in the relative arrangement of 4-sided faces about each vertex of the polyhedron. Thus in practice the arrangement of faces about each vertex is used to supplement Bernal s description in order to unambiguously identify the local environment as fcc or hcp. 1-55

56 (a) (b) Figure 2.5: (a) An octahedron decomposed into four tetrahedra along the long diagonal ij. The individual tetrahedra are adij, dcij, cbij and baij. The other two possible long diagonals that the octahedron could be decomposed into are ac and bd. (b) A view along the diagonal ij. (a) (b) Figure 2.6: (a) An octahedron decomposed into five tetrahedra, the long diagonal ij intersects the face abd. The tetrahedra are ijab, ijad, bdjc, bdci and ijbd. The last tetrahedron is a sliver or Kije simplex of near zero volume and is composed of the four nearly coplanar particles i, j, b and d. (b) The Kije simplex ijbd. 1-56

57 The Bernal indices have been exploited in molecular dynamics simulations of crystallisation to identify particles in a local crystalline environment 5. Ideally a single Voronoi polyhedral cell is identified with a given lattice structure, in accordance with the idea that all particles in a crystal have the same local environment. Although this approach is found to work for a stable body-centred cubic crystal it fails in practice for the face-centred and hexagonally close packed lattices due to the degeneracy problems discussed above. These problems occur as the Voronoi polyhedron of a particle or sphere in an fcc or hcp lattice is sensitive to thermal fluctuations. This is because more than three faces of the Voronoi polyhedron meet at the same vertex, Figure 2.8. Thermal fluctuations result in the formation of small faces or edges at these vertices that remove degeneracy problems but result in a wide range of Voronoi polyhedra being associated with the same local crystalline environment, Figure 2.9. In order to reduce the topological range of Voronoi environments a number of methods have been developed to remove small faces 13, 14 and our approach is in the same spirit, as it too is focused on the removal of these faces. Due to the correspondence between shared Voronoi faces and nearest neighbours the deletion of a Voronoi face is equivalent to the deletion of a geometric bond between spheres Voronoi definition of nearest neighbours The Voronoi tessellation also provides a topological definition of the nearest neighbours of a particle and hence it is independent of any metric such as a maximum bond length 2. Two particles are defined as nearest neighbours if their Voronoi cells share a common face. Thus the number of neighbours of a particle, that is, its coordination number is equal to the number of faces of its associated Voronoi cell. These neighbours are often termed geometric neighbours as there is no actual chemical bond between the particles and the term neighbour does not necessarily imply that the particles are physical neighbours. 1-57

58 Figure 2.7: The Voronoi polyhedron of a particle in a body-centred cubic lattice. Two 6-sided faces and one 4-sided face meet at each vertex. Figure 2.8: The Voronoi polyhedron of a particle in a face-centred cubic lattice. All 12 faces are 4-sided. Note that four faces meet at six of the fourteen vertices. (a) (b) Figure 2.9: The Voronoi polyhedron of a particle in a perturbed face-centred cubic lattice. (a) The formation of a small face is a result of the intersection of a plane perpendicular to the bond between the central particle and a second nearest neighbour with the Voronoi cell, which in an ideal lattice is tangent to the vertex. (b) The formation of a small edge is due to 4 faces of the Voronoi cell (or more correctly the plane bisectors of the bonds defining them) not meeting at exactly the same point in space. 1-58

59 The number of neighbours of a given particle has a well-defined meaning in a Voronoi approach but it is often an overestimate of the true number of physical neighbours of a particle 2. By this we loosely mean those neighbours that directly affect the motion of the particle, i.e. interact strongly with it, or are responsible for observed structural correlations in the system. It is these interactions between neighbours that will have a dominant effect on the physical properties of the system. In a dense fluid it typically implies those particles that inhibit the ballistic motion of the particle and form a shell around the particle. The structural evidence for this shell or cage of neighbours is the existence of a well-defined first peak in the pair correlation function of a liquid. The influence of this cage of neighbours on the motions of each particle is also indicated by the qualitative difference in the velocity autocorrelation function of a liquid or dense fluid compared to a gas. This function becomes negative at small times due to the backscattering of the particle from the cage. The simplest definition of a physical neighbour is due to Bernal 2 who defined a physical neighbour as a Voronoi neighbour that has a maximum bond length of 1.25 σ, where σ is the ratio of the position of the first peak in the g(r) to the hard core diameter. Other definitions may use the position of the first minimum in the g(r) of the liquid, which is slightly larger, at a distance of about 1.4 σ in a hard sphere system near freezing. This of course introduces a degree of arbitrariness into the definition of physical neighbour. One problem is that it assumes a global maximum for the largest distance between neighbours, which does not take into account the large range of local environments in a liquid. In the next section we describe a modified Voronoi definition of the neighbour of a particle that attempts to remove those Voronoi neighbours of a particle associated with small faces in a self-consistent manner and so overcome the problems caused by degenerate configurations in the close packed crystal lattices. No minimum distance of Voronoi face size is assumed, hence avoiding the arbitrary nature of associated with many definitions of the neighbours of a particle. 1-59

60 2.3 Modified Voronoi Tessellation Description In defining the nearest neighbours of a sphere our main motivation in this work is to produce a definition of use in analysing local crystalline environments. A common method is to define the nearest neighbours of a particle as those Voronoi neighbours sharing a Voronoi face with an area larger than some lower limit 13. This is often taken as the minimum value in the distribution of Voronoi face areas, assuming it exists. The motivation for this approach is the observation that for a face-centred or hexagonally close packed crystal the distribution of Voronoi face areas is bimodal with one peak centred on small Voronoi faces and associated predominately with what would be classed as second nearest neighbours in a close packed lattice. These neighbours are those associated with the decomposition of the octahedral arrangements of particles that occur in face-centred or hexagonally close packed lattices. There are however problems with the straightforward application of this method. In particular defining a minimum face size can lead to a certain degree of arbitrariness in the definition of a neighbour, particularly if the two peaks in the area distribution are separated by a non-zero minimum. One method of alleviating this problem is to somehow mitigate the effects of thermal fluctuations. A simple approach is simply to average positions over a short time compared to the average time for diffusion of particles from the cages formed by their neighbours 15. Another common method is to use the inherent structure approach 16 in which a configuration of particles is mapped onto its local potential energy minimum by a steepest descent minimisation in order to obtain its temperature independent inherent structure 13. These approaches go some way towards reducing the range of observed Voronoi polyhedra observed in a high temperature crystal but do not completely mitigate the effects of thermal fluctuations. The latter approach is also not applicable to a hard sphere system. In this work we wish to produce a method of deleting extraneous geometric bonds between spheres so that the Delaunay tessellation of the spheres into non-overlapping 1-60

61 tetrahedra are modified into one consisting of tetrahedra and octahedra. Thus we focus on deleting bonds between Voronoi nearest neighbours associated with octahedral arrangements of spheres. This is motivated by our scheme for classifying the local environment about a sphere. In this context, octahedral implies four tetrahedra are arranged around a common edge. The meaning of octahedral here is topological rather than geometrical and does not imply that the geometric properties of this particular atomic arrangement are close to that of a perfect octahedron, although this may turn out to be the case. The following methodology is employed in order to define the appropriate modified Voronoi neighbours of a particle. First all 3-sided faces are deleted. This is equivalent, in the case of a close packed crystal to deleting all sliver tetrahedra. Hence all octahedra that are decomposed into five tetrahedra by the Voronoi method are converted into four tetrahedra arranged about a long diagonal of the octahedral arrangement of particles. If we were not careful this would immediately create a problem as two of the sides of the sliver tetrahedra, the two long diagonals are both associated with a 3-sided face. Which 3-sided face should be deleted is decided on the basis of which has the smallest face associated with it. Alternatively we could choose the shorter diagonal associated with the faces. A similar problem can arise with deleting neighbour bonds associated with small four sided faces. In Figure 2.10 the two dashed diagonals are themselves associated with a four-sided face. An iterative process in which the bond with the smallest face is eliminated if conflicts arise is used in order to overcome this problem. In order to avoid eliminating large four-sided faces e.g. those associated with octahedral arrangements of particles in a bcc crystal a maximum face size for deletion can be incorporated. The impacts of both approaches on the resulting topology will be discussed latter; at this point it is sufficient to say that the effect is minimal. Large faces tend not to be deleted as a result of conflicts, although if we were analysing a bcc crystal we would need to include an upper limit. The degeneracy problems associated with fcc and hcp crystal do not arise in the case of bcc crystals. A bcc lattice is said to be stable with respect to small perturbations. Only three faces meet at each vertex of the Voronoi polyhedron of the bcc lattice and hence it is not sensitive to small perturbations of the particles from their lattice sites. 1-61

62 2.3.2 Application to Hard Sphere Crystal In order to investigate in more detail the utility of this modified Voronoi definition of the nearest neighbours of a hard sphere a number of hard sphere simulations of a fcccentred cubic crystal were performed in the number density range (ρ) of 0.97 to The reduced number density ρ is equal to ρ n σ 3 where ρ n is the actual number density and σ is the hard sphere diameter. This density range spans both metastable and stable densities of the solid phase. The simulations started with 4000 particles in a facecentred cubic lattice and were simulated for a time of 400 τ. Configurations were stored every 1 τ and a Voronoi analysis performed on the last 350 configurations. The time constant?τ is the average time scale for short time diffusion in the dense fluid and is equal to σ (m/kt). At high fluid densities it represents the mean time taken for a hard sphere to break out of the cage formed by its nearest neighbours. The distribution of face areas of the Voronoi polyhedra at the melting density of 1.05 is shown in Figure It can be seen that there is a large peak at zero while there is a broad peak at larger Voronoi face area values. In order to examine the Voronoi decomposition of an equilibrium hard sphere crystal system in more detail, the distribution of face areas of the Voronoi polyhedra is decomposed into contributions from different sided faces, Figure It can be seen that the small peak at nearly zero is due to the 3-sided Voronoi faces, associated in a close packed crystal almost exclusively with the decomposition of octahedra into five tetrahedra. The distribution of four sided faces has a double peaked structure, where the peak at smaller areas is again due to the octahedra. In our simulations we start from an ideal lattice and hence if we assume that at the melting density the particles are still vibrating about their ideal lattice positions then we know what the real neighbours of each particle are. By calculating the distribution of face areas for these real neighbours one can identify the extra neighbours due to the octahedra. This is done in Figure 2.13, and it can be clearly seen that the face areas associated with the 3-sided Voronoi faces and that due to the first peak of the 4-sided Voronoi face distribution is due to the extra neighbours. 1-62

63 Figure 2.10: A view along a long diagonal of an octahedron in a face-centred cubic lattice. The two dashed bonds are also associated with an arrangement of 4 tetrahedra about each dashed bond P(A) Area (A) Figure 2.11: The distribution of face areas of the Voronoi polyhedra at a reduced density of ρ = 1.05 near the melting density (ρ = 1.044). 1-63

64 0.05 P(A) Area (A) Figure 2.12: Decomposition of the distribution of face areas of the Voronoi polyhedra into contributions from different sided faces polyhedra at a reduced density of ρ = P(A) real 4-sided face extra 4-sided face total 4-sided faces Area (A) Figure 2.13: Distribution of face areas for real neighbours and extra neighbours due to the arrangement of four tetrahedra about a common edge, or equivalently, the common sharing of 4 particles by a pair of neighbours. 1-64

65 The distribution of different sided Voronoi faces is a result of the 3 different ways in which the octahedra can be decomposed along one of the 3 long diagonals of the octahedra. In an fcc crystal each neighbour bond is part of 2 octahedra and two tetrahedra. There are a number of possible ways in which the octahedra can be decomposed in order to produce between 4 and 6 tetrahedra packed around this diagonal, and hence between four and six sided faces of the Voronoi polyhedron, Figure The first case results in a four-sided Voronoi face. In general the octahedra will decompose along their shortest diagonal. For this reason we expect the mean area to increase with the number of sides of the Voronoi face. The origin of the second peak of the distribution of areas of four-sided Voronoi faces can know be seen. One can see from Figure 2.13 that there is a problem in deleting all Voronoi neighbours bonds possessing a face size less than some minimum value. This approach would delete a number of real bonds due to the overlap of the two distributions composing the distribution of areas for 4-sided Voronoi faces. There is no such problem with 3-sided faces and hence these can be deleted without problem. The final distribution of Voronoi face areas is shown in Figure There is another motivation for deleting 3-sided faces that will be discussed later in reference to our method for defining the local environment about a particle in terms of a planar graph. One of the interesting consequences of deleting sliver tetrahedra is that the average number of Voronoi nearest neighbours is very close to 14, the value expected for the average number of Voronoi neighbours for a perfect close packed lattice 17. Deviations from this value can only be due to the presence of 3-sided faces, and hence this gives us confidence that the deletion of 3-sided faces is removing the degeneracies associated with the decomposition of octahedra into 5 rather than 4 tetrahedra. In Table 2.1 the number of Voronoi nearest neighbours before and after the removal of slivers is shown as well as the average number of nearest neighbours after the deletion of 4-sided faces as well. 1-65

66 Figure 2.14: Different ways in which a packing of two tetrahedra and two octahedra about a common edge can be decomposed in a Delaunay tessellation. The dashed lines indicate the long diagonals along which the octahedra have been decomposed into tetrahedra. This is the arrangement of octahedra and tetrahedra in a face-centred crystal P(A) initial final Area (A) Figure 2.15: Comparison of initial distribution of Voronoi face areas and the final distribution after the deletion of bonds between certain Voronoi nearest neighbours. 1-66

67 Table 2.1: Average number of Voronoi nearest neighbours and average number of nearest neighbours after deletion of 3-sided faces (modified Voronoi) and also after deletion of 4-sided faces (final) for a hard sphere crystal. Density Voronoi nn Modified Voronoi nn Final nn P(r) initial final r Figure 2.16: Comparison of initial distribution of distances between Voronoi neighbours and the final distribution after the deletion of bonds between certain Voronoi nearest neighbours. 1-67

68 The effect of deleting small faces in this way on the resulting distribution of bond lengths is shown in Figure 2.16 and compared with the results after the initial Voronoi decomposition. It can be see that the distribution goes smoothly to zero and is also nearly identical with that which would be obtained if we had used the list of neighbours obtained from the ideal lattice. The peak at large distances due to second nearest neighbours has disappeared. 2.4 Planar Graph Description of Local Order Description In computer simulation studies of crystallisation one key challenge is to develop a measure of the local order around a particle that can distinguish between particles in a liquid-like environment and those in a crystalline environment, and perform this in a robust manner. The approach taken here is topological in nature and is a development of the reduced simplical graph method of Bernal. The local structure around each particle is defined in terms of the bonds between the nearest neighbours of the central particle. Specifically the planar graph formed by the nearest neighbours and the bonds between them is used to characterise this connectivity. A planar graph is a set of vertices (particles) joined by lines (bond between neighbours) such that no two lines cross one another. At the simplest level the number of primitive rings of various sizes making up the planar graph can be used to characterise it. By primitive rings we mean those closed paths through the network of bonds that are not part of any larger ring. The possible combinations of rings of various sizes for a particle with a given coordination number can be easily calculated using Euler s equation for planar graphs. A more complete description also characterises the arrangement of rings about each vertex (nearest neighbour). This is done is this work in order to categorically identify crystalline or well-ordered local environments. 1-68

69 As an example consider the local environment about a particle in a face-centred cubic lattice. Each particle has 12 nearest neighbours and each neighbour is coordinated with 4 other nearest neighbours, Figure The local structure can be described as a packing of 8 regular tetrahedra and 6 regular half-octahedra about the central particle Figure 2.18(a). A regular tetrahedron has all 6 sides equal in length while a regular octahedron has all 8 sides equal in length and the distance between non-bonded particles is equal to 2 times the distance between bonded particles. Figure 2.17: The local environment about a particle in a face-centred cubic lattice and the planar graph indicating the network of bonds between the twelve neighbours of the central particle (which is not shown). The packing of tetrahedra and octahedra about each neighbour is identical. In a hexagonal close packed lattice the local environment is similarly described, the one difference is that for half of the neighbours, the arrangement of the 2 tetrahedra and 2 octahedra about the vertex differs from the fcc case Figure 2.18(b). These particles define the single stacking direction in an hcp lattice. Quantifying the face centred cubic planar graph in terms of rings, there are 8 threemembered rings and 6 four membered rings. Around each vertex of the graph there are four rings, alternating between 3 and 4 membered rings, i.e. an alternating packing of 2 tetrahedra and 2 half octahedra. It can be easily shown that for a particle with 12 nearest neighbours for which each neighbour is bonded to exactly four nearest neighbours the only two arrangements are fcc and hcp. 1-69

70 (a) (b) Figure 2.18: (a) The packing of alternating tetrahedra and half octahedra about a nearest neighbour bond in a face-centred cubic lattice. (b) The packing of tetrahedra and half octahedra about half of the nearest neighbour bonds in a hexagonal close packed lattice. One advantage of this approach is that it is simple to treat the problem of defining crystalline order in a system sensitive to thermal fluctuations. Consider a planar graph of a particle in an fcc crystal at finite temperature. One can easily characterise perturbations of this fcc planar graph e.g. caused by a single extra bond between neighbours or an extra nearest neighbour contained within a four-membered ring. Those perturbations of the underlying fcc planar graph, which arise from thermal fluctuations, can be identified easily and hence a particle in such an environment would be treated as locally fcc. 1-70

71 When this classification method is combined with the previously described method for defining nearest neighbours this planar graph approach can also be interpreted as a description of the local environment about a particle in terms of the packing of tetrahedra and half-octahedral units that contain the particle. Alternatively the only rings that occur are either primitive three or four-membered rings. Each packing unit contains spheres at its vertices and is either a Delaunay tetrahedron of the Voronoi tessellation or part of an octahedral unit formed by the merger of four Delaunay tetrahedra into an octahedral unit. An advantage of the approach developed here is the overcoming of the black box approach used in many structural measures of crystalline order. A particle is typically defined to be in a crystalline environment if an order parameter used to define it as crystal-like has a value above a certain threshold. At finite temperature there will be a range of possible values and hence a calibration approach is typically used. A simulation is run of a hot crystal and the results are used to calibrate the structural measure 18, 19. But it is often unclear as to the meaning of this minimum value. Thus this approach can be considered as complementary to these approaches but which still maintains a link between the order parameter and a detailed description of the local environment. The disadvantage is that the method may not be as robust, in terms of its sensitivity to large fluctuations from local crystalline order, as say the approach of van Duijneveldt et al 20 at identifying solid-like particles Application to Hard Sphere Crystal In order to display the utility of the particular approach developed here the number and type of different Voronoi polyhedra for a hard sphere face-centred crystal near the melting density was calculated as well as the number and type of different planar graphs. The simulations started with 4000 particles and were performed for a time of 400 τ. Configurations were stored every 1 τ and a Voronoi analysis performed on the last 350. The number of different polyhedra was calculated before and after the deletion of 3-sided faces and it was found that this alone reduced the number of polyhedra from 295 to 37. The most commonly occurring polyhedra and their 1-71

72 frequencies are shown in Table 2.2. These 8 different polyhedral types represent 71% and 85% of all observed polyhedra, respectively. The planar graph analysis reveals that 97.9% of the particles are identified as facecentred cubic while a further 2% can be identified with simple fluctuations (Table 2.3) giving a total of 99.9% identified as crystalline. The fluctuations are related to either a missing or extra nearest neighbour or a single extra bond between nearest neighbours of otherwise perfect local fcc environment. Note that these latter two will be related which is why the percentages are nearly equal. Table 2.2: Most frequently occurring polyhedra before and after the deletion of 3-sided faces for a hard sphere face-centred cubic crystal at the melting density (ρ=1.05). Voronoi Polyhedra Frequency (Before) Frequency (After) (3 6 4) (4 4 5) (5 2 6) (4 4 6) (3 6 5) (2 8 4) (4 4 7) (3 6 6) An analysis of a metastable crystal at a density of 1.00 in the middle of the coexistence region reveals that even at this density these four planar graphs represent 97.6 % of all particles with the decrease in the number of particles in a perfect facecentred neighbourhood (86%) compensated by an increase in the number of these simple defect local environments. 1-72

73 (a) (b) (c) (d) (e) Figure 2.19: (a) The planar graph of particles in a face-centred cubic arrangement and ring sequence (86). (b) The planar graph of particles in a hexagonal close packed arrangement and ring sequence (86). (c) Planar graphs of particles in perturbed facecentred cubic arrangements: face-centred + extra nearest neighbour and minus a related bond (106), (d) face-centred with a single extra bond between neighbours (10 5) and (e) face-centred less one neighbour (8 5) and plus two related additional bonds. 1-73

74 Table 2.3: Most frequently occurring planar graph ring sequences (RS) of a hard sphere crystal near melting (ρ=1.05). The indices (I J) indicate the number of triangular (I) and square or 4-sided rings (J). Planar Graph RS Coordination Frequency (%) (8 6) (10 5) (10 6) (8 5) The actual arrangement of particles that give rise to the above ring sequences is shown in Figure 2.19 (a)-(c). In Figure 2.20 the local fluctuations that give rise to these defect structures are shown. It can be seen that the defect structures are associated with one another as well as with 5-membered rings. The defect structures are the result of fluctuations in the (100) planes of the fcc crystal. 2.5 Shortest Path Rings A useful method for describing the structure of an atomic system is in terms of the network of bonds formed by the constituent particles of the system. Using this approach both topological and geometric information can be extracted about the structure of the system in a unified manner. It is common to analyse the connectivity of such a bond network in terms of the ring structures formed by the bonds, that is, closed paths through the network of bonds 21. In this work we use a particular definition of a ring called the shortest path rings (SP) to investigate the short and medium range order of the hard sphere system. 1-74

75 2.5.1 Definition and Terminology There are a number of different definitions of a ring, ranging from the broadest definition: any path through the network of bonds starting and finishing on the same particle, to the most restrictive: the smallest ring structures that are not part of another larger ring, often called primitive rings. The ring types most commonly used in this work, unless otherwise stated are shortest path rings. Franzblau 21 defines an SP ring in the following way: a ring is a shortest path (SP) ring if the number of bonds passed through in moving from one particle of the ring to another particle is equal to the shortest possible path, considering all possible paths through the network of bonds from one particle to the other. The distance between two particles is defined as the minimum number of bonds that one would have to transverse in order to move from one particle to the other. The distance is one for nearest neighbours, two for second nearest neighbours and so on. In Figure 2.21 and Figure 2.22 a number of examples and counter examples of 4-membered and 5- membered shortest path rings are shown in order to illustrate the above definition. Figure 2.20: Different views of the fluctuations in the bond network that give rise to the defect structures shown in Fig The blue (extreme right and left pairs) particles have the planar graph ring sequence (10 5), the orange particles (middle left and right pairs) are (10 6) and the grey particles (middle pair) are (8 5). 1-75

76 (a) (b) (c) Figure 2.21: (a) and (b) are 4 membered shortest path rings while (c) is not a shortest path ring due to the path a-b. (a) (b) (c) Figure 2.22: (a)-(b) are 5-membered shortest path rings while (c) is not a shortest path 5 membered ring as the path bd is a shorter way around the network than the path bcd. Note that by describing the path as shorter we do not mean that the physical distance is less but the number of bonds traversed along the path bd is smaller (=1) than along the path bcd (=2). The simplest information that can be obtained from a ring analysis of an atomic structure is the number and type of rings formed. As an example in an open network such as graphite or diamond there are only 6 membered rings present in the structure. Three six-membered rings pass through each carbon particle in sp 2 bonded graphite (Fig 2.23) while 12 rings pass through each particle in sp 3 bonded diamond. The rings in graphite are planar while in diamond they are buckled. The incorporation of 5 membered rings in an sp 2 network can lead to geometric curvature and is the main topological difference between graphitic structures and fullerenes, such as buckyballs and buckytubes. 1-76

77 In an SP ring antipodal pairs are pairs of particles of a ring that are opposite each other, Figure For odd numbered rings there are two such antipodal pairs for each particle of the ring, Figure Thus there are three antipodal pairs in a shortest path six-membered ring and the distance between these antipodal pairs is three. One different type of ring structure is also used in this work, due to its importance in close packed crystalline structures. The definition of six-membered rings, which in the case of shortest path rings occurs in open low-density networks such as in graphite, is modified for dense systems. The criteria for a modified 6-membered ring, shown in Figure 2.26, are as follows: Every particle of the six-membered ring is bonded to the same one and only one central particle. The distance between antipodal pairs is 2, along a path through the central particle. For antipodal pairs there are no other paths between the two particles that have a distance of 2. In other words there is only one nearest neighbour shared between the two particles and this is the central particle. The six outer particles of the ring have the same shortest path relations as for a normal shortest path ring, if the central particle is ignored. (a) (b) Figure 2.23: (a) Flat six-membered SP rings passing through each carbon particle of graphite; (b) Three of the twelve buckled six-membered SP rings through each carbon particle of diamond. 1-77

78 Figure 2.24: An SP six-membered ring containing three antipodal pairs. Particles A1 and A2 are antipodal pairs, as are B1 and B2 and C1 and C2. The shortest path distance between A1 and A2 is through the particles B1 and C1, involving three bonds. Figure 2.25: An SP five-membered ring containing six antipodal pairs. The antipodal pairs are (1,3), (1,4), (2,4), (2,5), (3,5) and (3,1). The shortest path distance between 1 and 3 is through the particle 2 and involves two bonds 1-2 and 2-3. A close packed plane can be viewed as an overlapping set of these modified sixmembered rings. The third condition excludes six-membered rings formed by particles in different close packed (111) planes of a face centred cubic (fcc) lattice. Thus in a face centred cubic lattice each particle is a centre of four six-membered rings, Figure The last condition is included so that so-called chair ring structures are ignored. An example of a chair structure in a body-centred cubic lattice is shown in Figure 2.29(c). One of the useful attributes of these six-membered rings is that they appear in all close packed structures as well as in a number of icosahedral structures Figure 2.30(a)-(c). 1-78

79 Figure 2.26: A modified six-membered ring. C is the central particle. Figure 2.27: One of the four close packed planes, made up of overlapping six-membered rings, which pass through every particle of a face centred cubic crystal. 1-79

80 (a) (c) (b) (d) Figure 2.28: (a) One of the four identical 6-membered rings passing through a locally fcc ordered particle (see note at end). (b) The single flat 6-membered ring passing through a locally hcp particle - this defines the stacking direction in this lattice. (c) A side view of one of the particles making up one of the three bent 6-membered ring passing through a locally hcp ordered particle (see note at end). Three of the particles about the ring are hidden from view by the other particles. (d) One of the three bent six-membered rings of the hcp ordered particle. (Note: The lighter coloured bigger spheres are the members of the modified six-membered ring.) 1-80

81 (a) (b) (c) Figure 2.29: (a) One of the four identical flat 6-membered rings passing through a particle in a bcc lattice. (b) A view of one of the six-membered rings in the bcc lattice. (c) A six-membered ring chair structure in a bcc lattice. The labelled particles correspond to those in Fig 2.28(a) and the particles of this ring all belong to the closer of the two sets of nearest neighbours in a bcc lattice (the 14 neighbours of a particle in a bcc lattice occur at two distances, if d is the distance between the 8 closest neighbours then the distance between to the other 6 neighbours is (4/3)d. (a) (b) (c) Figure 2.30: (a) One of the ten identical 6-membered rings passing through an icosahedron; (b) An individual twisted six-membered ring. (c) One of the five identical bent six-membered rings passing through a twisted icosahedron. The ring has an identical from to the three bent six-membered rings associated with a hcp particle. 1-81

82 2.5.2 Geometric Analysis of Ring Structures One of the most common descriptions of the local order in a molecular system is the radial distribution function, g(r). This is the conditional probability of finding two particles separated by distance r and measures local density fluctuations in an atomic system. It is the Fourier transform of the static structure factor, S(q), which can be measured in scattering experiments. In experimental studies of crystallisation the behaviour of one or more peaks in the S(q) is used to characterize the nucleation and growth processes. A main advantage of computer simulations is that the positions of all particles in the system are known and hence measures can be used that provide more detailed information on the evolution of the microscopic structure of the system. This is a main advantage of simulations for crystallisation studies: they can provide information on the form of the crystal nucleus at times too early to be observed experimentally and measure properties not accessible to experiment. In computer simulations the information contained in the g(r) is too averaged to provide the more detailed information that we would like to know and also does not sufficiently exploit the available data. Obviously it is of particular benefit if the structural measures that are developed can be related back to the pair correlation function and this is what is done here. Another commonly used measure of the structure of a hard sphere system is the distribution of bond angles formed by triplets of neighbours. This can be viewed as a reduced three-body distribution function and hence provides qualitatively different information than the g(r). But yet again it is fairly featureless at low density and its evolution with density is quite subtle. In order to extract more useful and detailed information from these two distribution functions they are decomposed into contributions from different shortest path rings that are formed by the bonds between nearest neighbours in the system. In this way it is hoped that more detail about the changes in the geometry of the hard sphere system 1-82

83 with density can be obtained as well as geometric changes that occur as the system crystallises. Geometric information can also be extracted as a function of the ring size. The distribution of bond angles formed by adjacent triplets of particles in a ring can be calculated, and hence the total bond angle distribution for an atomic structure can be decomposed into contributions from rings of various sizes, Figure It is possible that triplets of particles may be part of more than one ring. In cases where it is desired that the ring contributions be properly normalised a simple rule can be applied to avoid double counting. Bond angles are counted in an ascending hierarchy of ring size, and any repeating triplets in larger rings are ignored. By the definition of shortest path rings there is no overlap between 3 and 4 membered rings. The bond angle distribution for the six-membered rings is split into three components due to its modified definition. The first component uses the same definition of triplets of particles as for the other SP rings, ignoring the central particle, and is labelled Type 1 in this work. The second distribution records bond angles formed by the central particle and the 6 separate pairs of particles of the ring separated by a shortest path of 2, Figure 2.32(a), and is labelled Type 2. The third distribution records bond angles formed by the central particle and the 3 antipodal pairs of the ring, Figure 2.32(b), and is labelled Type 3. We avoid double counting by considering the contributions from Type 1 before Type 2 bond angles. The first and second peak of the radial distribution function of an atomic system can also be decomposed into contributions from rings of various sizes by separately summing over each ring size and every antipodal pair of particles in the same ring, Figure Note that for 3-membered rings each of the three possible pairs are antipodal. In a typical calculation of the radial distribution function of an atomic system one loops over every pair of particles in the system, summing the number of pairs of particles separated by distances between r and r+dr. In our ring analysis program this procedure is modified. During the ring analysis for every ring found a list is stored of all antipodal pairs of particles that are part of the same ring. In calculating the radial 1-83

84 Figure 2.31: Bond Angle formed by the triplet of particles (i,j,k). (a) (b) Figure 2.32: (a) Bond Angle for Type 2 six-membered ring bond angle distribution. (b) Bond Angle for Type 3 six-membered ring bond angle distribution. Figure 2.33: Distances r ij, r ik that contribute to the five-membered ring component of g(r). 1-84

85 distribution function for the atomic system one first performs a loop over all pairs of particles found in this way starting from the smallest rings. Individual distributions are stored for each ring size. The remaining contributions to the g(r) of pairs of particles, which are not both part of the same ring, are then calculated. The present decomposition is given in Equation 2.1, where the first sum extends over all distinct antipodal pairs of particles i and j that are part of the same 3-membered ring, and so on for larger rings. The radial distribution function is then given by, g(r) = g (r) + g (r) + g (r) + g (r) + g (r) g (r) Eqn a b + rem where each ring distribution function is normalised as for the total g(r). In order to ensure there is no double counting, pair contributions are counted only once in an ascending hierarchy of ring sizes. As an example, if the pair of particles, i and j, are part of a five-membered ring but are also part of a six-membered ring, they will only contribute to the ring g(r) distribution of the former. The contribution to the radial distribution function of each pair of particles that are part of the same six-membered ring is divided into two parts. Note that in the analysis of six-membered rings the central particle is ignored in calculating pair contributions. The first contribution, Type A, is defined as the contribution to the total g(r) of particles that are separated by a shortest path of 2 in traversing the ring. The second contribution, Type B, comes from the antipodal pairs, Figure Application to Hard Sphere Crystal In order to illustrate this approach a ring analysis of a hard sphere face-centred cubic crystal at a density near melting (ρ = 1.05) will be presented. A hard sphere simulation of 4000 particles was performed for a total time of 400 τ. Configurations were stored every 1 τ and a ring analysis was performed on the last 350 of these. Nearest neighbours of a particle were defined using the modified Voronoi method described earlier. 1-85

86 (a) (b) Figure 2.34: Pairs of particles that contribute to the two different ring g(r) components for a six-membered ring (a) the Type A component and (b) the Type B component (a) (b) Figure 2.35: (a) (111) plane and 3 membered rings; (b) (100) planes and 4 membered rings. The number of rings per particle for different ring sizes is shown in Table 2.4. In an ideal face-centred cubic crystal there are three different types of rings. In an fcc crystal there are 4 (111) planes through each point. For each (111) plane a particle is part of six 3-membered rings, Figure 2.35(a). Thus it is part of twenty-four 3- membered rings in total. As each ring is counted 3 times there are 8 (= 24/3) 3- membered rings per particle in the crystal. 1-86

87 Another way of looking at this is to consider the local environment about each particle in an fcc crystal. Each particle has 12 nearest neighbours. Each of these particles is bonded to four other nearest neighbours of the central particle, that is, it has a local coordination of 4. Each triplet of central particle and two bonded neighbours is a 3 membered ring. The total number of 3-membered rings so formed is thus given by (4*12)/2 = 24 where the factor 2 is included to avoid double counting. Each particle is also part of 3 (100) planes, which are 2D square lattices, Fig. 33(b). Hence for each (100) planes a particle is part of four 4-membered rings, and in total it is part of twelve 4-membered rings. Thus there are 3 (=12/4) 4-membered rings per particle. As stated before there are 4 (111) planes in an fcc crystal and hence 4 6-membered rings per particle. Table 2.4: Number of Shortest Path rings per particle for an equilibrium hard sphere fcc crystal and an ideal fcc crystal. Ring Size Density = 1.05 Ideal The distribution of bond angles for rings of various sizes is shown in Figure Nearly all bond angles formed between triplets of bonded particles are resolved into contributions from different rings. This is shown in Figure 2.37 where the sum of the ring contributions is compared to the total bond angle distribution. The distributions are symmetric with the exception of the bond angle distribution for the triplets of nearly collinear particles of the six-membered rings (Type 3). This is due to the limited range of the bond angle distribution from 0º to 180º. The bond angle in this case will fluctuate about 180º. 1-87

88 total 3 rings 4 rings 5 rings 6 rings (Type 1) 6 Rings (Type 3) P(q ) Bond Angle (q ) Figure 2.36: The distribution of bond angles for rings of various sizes for a hard sphere crystal at a reduced density of total sum of rings 0.02 P(q) Bond Angle (q) Figure 2.37: Comparison of total bond angle distribution and sum of ring contributions for a hard sphere crystal at a reduced density of

89 Also there is a slight skew to larger bond angles for three membered rings. The average bond angle for each ring is given in Table 2.5 and it can be seen that it is equal to the ideal case when all bond lengths are equal and all particles are in the same plane. This is even the case for the small number of five-membered rings. The fraction of bond angles due to various rings is also recorded and this should be compared with Table 2.1. Table 2.5: Average Bond angles for a hard sphere crystal at melting. Ring Size Average Bond Angle Ideal Angle º 60º º Type A 120.4º 120º 6 Type B º 6 Type C 165.2º 180º The contribution of the second bond angle distribution associated with the sixmembered rings (Type 2) can be seen to be negligible. This is due to the nature of close packed planes. In Figure 2.38 the close packed planes are shown as a tiling of overlapping six-membered rings, of the type defined here. In calculating the ring contributions to the bond angle distribution we avoided double counting by calculating all Type 1 bond angles formed by triplets of particles before Type 2 bond angles. As each triplet of particles of Type 1 is also part of a triplet of particles for a Type 2 bond angle, and since the former are processed for all particles before any of the latter, no triplets will remain that have not already been counted in an ideal fcc crystal. This is the main motivation for including this particular contribution as it gives us some idea of the overlap or extent of close-packed like layering in a dense or metastable fluid. If no two rings are connected in the way shown in Figure 2.39 then we expect the two contributions from Type 1 and Type 2 bond angles to be equal. 1-89

90 Figure 2.38: A close packed plane in a fcc crystal. A close packed plane can be considered as a 2-dimensional tiling of overlapping hexagons, with particles decorating the vertices of the polygon. Figure 2.39: Overlap of two six-membered rings. The angle BCD (Type 1), associated with the ring centred on A, is counted before the same angle BCD, but associated with particle C and of Type 2 is counted. Thus no Type 2 contributions will be counted in an ideal crystal. 1-90

91 At this point it is worthwhile to note one consequence of the ring analysis. A commonly used method of distinguishing liquid- like local environments from solidlike, ordered environments is by the identification of nearly collinear triplets of particles. These triplets of particles are associated with close packed planes and are the Type C triples discussed above. Honeycutt and Anderson 19 have used this criterion in their seminal work on crystal nucleation. They defined three particles, as being collinear if the angle formed by the triplet was greater than a preset value of about 165º. However they found that this criterion was a very sensitive function of the cut-off angle used and from an examination of Figure 2.36 we can see why. It can be seen that the distribution of angles is fairly broad showing no distinct signature at the cut-off angle. The situation in a nucleating system would probably be worse. 5 g(r) g(r) 3 rings 4 rings 5 rings 6 rings (Type A) 6 rings (Type B) r Figure 2.40: Decomposition of the pair correlation function into contributions from various sized rings for a hard sphere crystal at a reduced density of

92 The decomposition of the pair correlation function into contributions from various sized rings is shown in Figure All distributions, with the exception of the 3- membered rings, have a gaussian shape. The 4-membered ring peak is associated with the (100) planes of the fcc crystal. It can be seen from Figure 2.41, which compares the total g(r) with the sum of the ring contributions, that all first and second neighbours contributions to the pair correlation function, g(r) have been resolved. Note that in the case of a crystal there is no overlap between the various distributions, as each antipodal pair of particles can be uniquely identified with a given ring size g(r) sum of ring contributions g(r) r Figure 2.41: Comparison of g(r) with sum of ring contributions for a hard sphere crystal at a reduced density of

93 2.6 Spherical Harmonics A geometric measure of the orientational order of an atomic system is the spherical harmonic bond order parameters. A set of spherical harmonics is associated with each geometric bond, r, joining a particle to one of its neighbours, Q lm ( r) = Y ( θ( r), φ( r)) Eqn 2. 2 lm where ( θ( r), φ( r)) are spherical harmonics and θ(r) and φ(r) are the polar angles Y lm of the bond measured relative to a reference coordinate system, typically that defined by the simulation cell. The overall orientational order is determined by taking appropriate averages over all bonds in the system, Q lm 1 = Q lm ( r) Eqn 2. 3 N b As both θ and φ depend on the reference coordinate system used rotationally invariant combinations of the Q lm are often defined, such as Q 1 4π = l 2l + 1 Q lm m= l 1/ 2 Eqn 2. 4 If both orientations of a bond between two particles are counted in the above average () then the odd harmonics will vanish and hence attention is usually focused on the even l spherical harmonics, as these are invariant under inversion. Baranyai et al 22 has shown that the Q l s are a measure of three body angular correlations between particles and higher order functions can also be defined using the Q l s, such as the third-order invariants, 1-93

94 l l l w l = Q lm Q 1 lm Q 2 lm Eqn m m m m,m,m ŵ l = m Q w lm l 2 3/ 2 Eqn 2. 6 The coefficients in the third-order invariants are Wigner 3j symbols and the w l s are a measure of four body angular correlations. Steinhardt et al 16 originally introduced the even l spherical harmonics in order to study orientational order in liquids and glasses. They have also been used by a number of authors to provide an order parameter for a crystallizing system. In such studies l = 6 spherical harmonic Q l is often used to monitor the evolution of crystalline order. This order parameter is large for a number of different crystalline and icosahedral structures, and hence gives an overall measure of the crystallinity of the system that is insensitive to the actual underlying crystal lattice. Although the above spherical harmonics are in principle zero for an isotropic system they can locally be large and hence if the above average is taken over the bonds between a particle i and its N b (i) neighbours they can be used to characterize local orientational order by defining a set of invariants for each particle, N (i) b 1 q lm (i) = Ylm (rˆ ij ) Eqn 2. 7 N (i) b j= 1 q l 4π 2 l (i) = q lm (i) 2l + 1 m= l 3 / 2 Eqn

95 In a liquid the even l spherical harmonics are zero when averaged over all bonds in the system, as the spherical harmonics of neighbouring particles are not in phase and hence do not add up coherently, as they do in a crystal. This has led van Duijneveldt et al 20 to suggest a local measure of the crystallinity or solid-like nature of the environment of a particle in terms of the coherence of the spherical harmonics of a particle with that of its neighbours. They define for each particle a normalized 2l + 1 vector (l = 6), ~ q 6m (i) = 6 m= 6 q q 6m 6m (i) (i) 2 1/ 2 Eqn 2. 9 q m m= 6 (i) q (j) = q ~ (i) q ~ (j)* Eqn m with q 6 (i).q 6(i) = 1 Eqn The dot product is then taken between this particle and its neighbours. If this dot product is greater then some preset value (0.5 in their work) the two particles are regarded as being connected (their spherical harmonics are in phase). A particle is regarded as being solid-like if it has at least 7 such connections with its neighbours. The number of connections is empirically based - by comparing the distribution of connections for each particle in both a liquid and a crystal at the same pressures (used for their simulations) van Duijneveldt et al 20 find that there is little overlap between the liquid and crystal distributions for seven or greater connections. In this work we have also used van Duijneveldt s definition of a solid-like particle in order to compare our topological approach with this geometric measure of order. The order parameters Q 4 and Q 6 were also calculated for both the equilibrium hard sphere system and during the crystallisation runs. The distribution of connections for both a hard sphere fluid and crystal at the same density of 1.05 reveals that if van 1-95

96 Duijneveldt s solid-like criteria is used then 100% of hard spheres in the crystal are regarded as solid while in the fluid less than 4% of spheres are regarded as solid-like. A comparison of the average number of connections in both a fluid and a crystal at a lower density of 1.00, within the coexistence region, suggests that a more appropriate minimum number of connections may be 10 for the hard sphere system. Using this stronger criterion, 99.9% of the spheres in the crystal are classed as solid-like, while only 1.1% are classed as solid-like in the fluid. The fraction of solid-like particles in the hard sphere fluid at a density of 1.05 (before crystallisation occurs) using this more stringent criterion is equal to 2.7%. 2.7 Summary The different structural analysis methods described in this Chapter can provide a range of information about the microscopic structure of a fluid or crystal. In particular the ring analysis method developed here is able to decompose the radial distribution function and the bond angle distribution function into different contributions based on the topologically defined ring structures. In the case of the hard sphere crystal this method provides a complete decomposition of these functions. The planar graph method is a development of the topological description of the local environment of a particle in terms of its Voronoi cell. When it is combined with the modified definition of a Voronoi neighbour it provides a significantly more robust method of describing the local environment about a particle without any loss of information regarding the local arrangement of neighbours about a particle. When applied to the case of a crystal it can identify all particles as being in a crystalline environment while when applied to a liquid at the same density (ρ = 1.0) it identifies less than 1%of the particles as being crystalline. Hence it is of great value in monitoring and describing the growth of a crystal nucleus in computer simulations of crystallisation. The reduction in the range of topologically distinct environments compared to the Voronoi description provides a description of the local environment of particles in a growing crystallite that is both detailed but still manageable. 1-96

97 2.8 Bibliography 1 J. L. Finney, Proceedings of the Royal Society London A 319, 479 (1970). 2 J. D. Bernal, Nature 183, 141 (1959). 3 J. D. Bernal, Proceedings of the Royal Society London A 280, 299 (1964). 4 J. D. Bernal, in Physics of Simple Liquids, edited by H. N. V. Temperley, J. S. Rowlinson and G. S. Rushbrooke (North-Holland Pub. Co., Amsterdam, 1968), p J. N. Cape, J. L. Finney, and L. V. Woodcock, Journal of Chemical Physics 75, 2366 (1981). 6 J. L. Finney, Journal of Computational Physics 32, 137 (1979). 7 M. Tanemura, Journal of Computational Physics 51, 191 (1983). 8 V. P. Voloshin, Y. I. Naberukhin, and N. N. Medvedev, Molecular Simulation 4, 209 (1989). 9 M. I. Aoki and K. Tsumuraya, Journal of Chemical Physics 104, 6719 (1996). 10 W. Shinoda and S. Okazaki, Journal of Chemical Physics 109, 1517 (1998). 11 P. Jund, D. Caprion, J. F. Sadoc, and R. Jullien, Journal of Physics: Condensed Matter 9, 4051 (1997). 12 N. Thomas, Acta Crystallographica B52, 939 (1996). 13 W. C. Swope and H. C. Anderson, Physical Review B 41, 7042 (1990). 14 N. N. Medvedev and Y. I. Naberukhin, J. struc. Chem. 26, 369 (1985). 15 M. Tanemura, Y. Hiwatari, H. Matsuda, T. Ogawa, N. Ogita, and A. Ueda, Pro. Theor. Phys. 58, 1079 (1977). 16 F. H. Stillinger and T. A. Weber, Physical Review A 25, 978 (1982). 17 J. P. Troadec, A. Gervois, and L. Oger, Europhysics Letters 42, 167 (1998). 18 P. ten Wolde, M. J. Ruiz-Montero, and D. Frenkel, Physical Review Letters 74, 2714 (1995). 19 J. D. Honeycutt and H. C. Anderson, Journal of Physical Chemistry 90, 1585 (1986). 20 J. S. van Duijneveldt and D. Frenkel, Journal of Chemical Physics 96, 4655 (1992). 21 D. S. Franzblau, Physical Review B 44, 4925 (1991). 22 A. Baranyai, A. Geiger, P. R. Gartrellmills, K. Heinzinger, R. McGreevy, G. Palinkas, and I. Ruff, Journal of the Chemical Society-Faraday Transactions Ii 83, 1335 (1987). 1-97

98 Chapter 3 Voronoi Analysis of the Hard Sphere System 3.1 Introduction In this Chapter, the statistical geometry of the hard sphere fluid and crystal has been characterised in terms of various topological and geometric measures associated with the Voronoi cells of the system. This is motivated by the fact that the structure and thermodynamics of the hard sphere system is completely determined by entropic factors, that is, by its statistical geometry. The Voronoi tessellation of the space occupied by a system of hard spheres or atoms can be regarded as one example of a random cellular network. In parallel to applications within the materials field (see Chapter 2, section 2.2.2) there has been a great deal of empirical and theoretical work on random networks that appear in other fields, such as within biology, geology and metallurgy. The partitioning of an area or volume into non-overlapping cells is often called a random froth in three dimensions and a mosaic or tiling in two dimensions 1 within this area. Rivier 2 and Weaire 1 have given two excellent reviews of the field, focusing mainly on 2D cellular structures 3. A surprising feature of many of these random mosaics or froths is that they obey several of the same empirical relations. These relations include geometrical ones such as the Lewis 4 and Desch laws 5 and topological ones such as the Weaire identity 6, the Aboav-Weaire law 7 and those due to Peshkin 8. These will be described in detail later. One of the purposes of this current work is to attempt to gain a better understanding of the geometry and topology of the hard sphere fluid and crystal by using the Voronoi approach. In particular the applicability of the above mentioned semi-empirical laws 1-98

99 to the hard sphere system will be tested. One aim is to be able to describe the average geometry and topology of the hard sphere fluid and crystal in terms of a small number of parameters. The Voronoi cell is the random equivalent of the Wigner-Seitz cell in the solid-state physics of crystals. Rivier 2 has described the Voronoi tessellation of space as statistical crystallography applied to random space-filling structures. This echoes comments made by Bernal 9 in his early studies of the structure of liquids. He envisaged a statistical geometry for random systems that could do for fluids what crystallography had accomplished for the solid-state field. In particular he imagined that there might be limits on the range of possible polyhedra in a fluid as well as relations between the topology and geometry of the Voronoi tessellation, due to sphere packing constraints, that could be uncovered. This could help in putting limits on the form of distribution functions describing the average structure of a fluid. It may also aid in uncovering the minimum number of parameters needed to describe the average structure of an atomic system. A variety of random space-filling cellular networks exist in nature. Examples in three dimensions include soap bubble froths, metallurgical grains, foams and columnar arrangements such as biological tissues and the Giant s Causeway 2. In two dimensions examples include plane sections of polycrystals, biological tissues, random Poisson point processes and ecological territories. Rivier 2 has made the point that to a first approximation many of these cellular patterns are indistinguishable, apart from a specific length scale. This has motivated a great deal of theoretical work to explain why cellular structures produced and maintained by vastly different physical and chemical forces appear so similar in form. One proposed reason 2, 10 is that all random assemblies of a vast range of atoms, tissue cells or grains take up the most probable configuration subject to a finite number of constraints. These cellular structures are in statistical equilibrium in the sense that any topological rearrangement of the cells leaves its arbitrariness invariant, the arbitrariness being measured precisely by the entropy or information contained in the structure

100 The maximum entropy inference method (MAXENT), which draws its inspiration in part from statistical mechanics, has been used to give theoretical justification for many of the above laws. The actual physical processes associated with a given cellular structure only determine the constants appearing in the linear laws. A detailed description of maximum entropy methods and their application to random cellular networks is given in two reviews by Rivier 10, 11. Hard sphere systems are a particularly pertinent system in which to test the applicability of the above semi-empirical relations. This is because the equilibrium structure of a hard sphere fluid or crystal is that which maximizes the statisticalmechanical entropy of the system. This is different from the entropy defined in the formalism of maximum entropy inference. However there are similarities between the two entropies for a hard sphere system. The information entropy appearing in maximum entropy formalism is a measure of the information content associated with a given statistical distribution. The mathematical form of the entropy measure is identical with that employed in statistical mechanics. However, the distribution functions are different. In applications of maximum entropy theory to random networks the distribution function used is typically p(n), the probability of an n-sided cell (in 2D networks) whereas in statistical mechanics the relevant distribution function is the phase space distribution. In a canonical ensemble, each configuration is weighted by a factor proportional to its energy. This Boltzmann factor is either 0 or 1 for a hard sphere system, corresponding to whether there are sphere overlaps or not, respectively. Thus all physically realizable configurations are weighted equally for a hard sphere system. The ensemble of equilibrium configurations is then the most uniform one (all equal probability) and it is here that the link with information theory lies 12. The structure of a hard sphere fluid can be loosely described as an irregular packing of identical spheres. The actual equilibrium structure is that which maximizes the entropy of the system, and hence colloquially maximizes the disorder or randomness, within the constraint that no two spheres overlap. This sphere packing constraint has a 1-100

101 particularly dominant influence at high densities on both the equilibrium structure and the dynamics of a hard sphere system. Bernal 9 was one of the first to suggest that there must be a limited number of parameters defining the statistical structure of a homogeneous irregular assembly of atoms. He also expressed the view that there must be some theorems limiting the possible forms of distribution functions of any array of points or spheres. Since then there has been a wide range of both empirical and theoretical work on random spacefilling structures or froths. The Voronoi tessellation of a random assembly of atoms is only one example of a random froth. The geometrical and topological properties of many of these froths can be described by a small number of parameters. These in turn can often be linked to the physical processes that produce the froth 2. A limited number of studies have been done on 3D cellular networks compared to 2D mosaics Many of the empirical relations have been generalized to threedimensional systems simply by taking the analogue of their 2D counterpart. A 3D version of the Lewis law, relating the volume of a cell to the number of its faces has been found to apply to the volume of columnar froths 15. The 3D version of the Aboav-Weaire law has been found to apply to random hard sphere packings by Oger 16. Richard et al. 17 has calculated a number of topological features of the hard sphere system at a wide range of densities including the metastable and crystalline branches of the phase diagram. They have also analysed topological changes that occur on crystallisation of a metastable fluid. Our work is complementary to theirs and includes a more detailed analysis of the applicability of various semi-empirical laws to the hard sphere system

102 3.2 Theoretical and Semi-Empirical Relations Topology In 2D it is found that for Voronoi tessellation s in which three cells always meet at a vertex the average number of sides of a cell <n> is equal to 6. This is a direct consequence of Euler s theorem that relates the number of cells in a mosaic to the number of its edges and vertices. An interesting consequence of this theorem is that it has often led to the erroneous suggestion that certain patterns in nature are hexagonally ordered arrangements. Examples include the Giant s Causeway 1, which is in fact a random cellular structure with a small variance in the number of sides about its hexagonal mean. In 3D the Euler theorem (Appendix 1) gives a relation between the average number of sides of a Voronoi face <n> and the mean number of faces of the Voronoi cells <f>, <n> = 6 12/<f> Eqn 3-1 The average number of faces of a Voronoi cell, or alternatively the average number of Voronoi neighbours, varies from for a Poisson point distribution, also called a Random Voronoi-Poisson point distribution (RVP), 18 to exactly 14 for a perfect close packed lattice 19. This range is similar for most non-covalent materials. This is a consequence of the dominance of excluded volume effects, due to repulsive interactions, on the structure of liquids. It can be seen from the above relation that since the second term in the above equation is always of order unity the average number of sides is approximately 5. This is a topological consequence that is independent of the detailed form of interaction between the atoms, provided that it is non-directional. Bernal 9 observed a preponderance of pentagonal or five-sided faces in the Voronoi polyhedra formed by his dense sphere packings. A pentagonal face is a packing of seven atoms arranged into five tetrahedra about a common edge. Frank 20 has shown 1-102

103 that this particular arrangement of atoms leads to a locally optimum volume fraction or minimal Voronoi cell. In an icosahedral arrangement of 12 atoms about a central one every atom has a surface coordination of 5 and hence its Voronoi cell has only pentagonal faces. Thus the local packing fraction is quite high (i.e. small Voronoi volume) and is actually larger than the maximum packing fraction in a close packed crystal. However a crystal does not have icosahedral symmetry, as icosahedra cannot be packed together without gaps appearing between them, just as in 2D a plane cannot be tiled by identical regular pentagons. If the atoms have an attractive interaction, this arrangement also leads to a high binding energy 21. All this has led to concurrent geometric and energetic arguments that in dense systems there should be a large number of five-sided Voronoi faces. But as can be seen from the above topological relation, even for a system of point particles pentagonal faces should dominate. Although Oger 16 has pointed this out before it should be re-emphasised that the high frequency of pentagonal faces are a general consequence of topology not geometry. The fact that pentagonal arrangements are both geometrically and energetically favourable in dense systems is a fortuitous coincidence. It is undoubtable that the actual frequency of occurrence of pentagonal faces in dense systems is in part due to geometric reasons but it must be recalled that this is on top of an already established topological trend. It would be more accurate to say that the decrease in the coordination number with density observed in many atomic systems is due to the increase in the frequency of occurrence of five-sided Voronoi faces caused by a combination of geometric and energetic factors. The same coincidence applies in 2D where the average number of Voronoi nearest neighbours of a disc is equal to six, irrespective of the density. The densest local packing of discs is also achieved if each of the six discs touches the central one. The densest packing in 2D consists of all discs in the packing having exactly six touching nearest neighbours, i.e., a triangular lattice

104 3.2.2 Average Geometry Another useful measure of the average geometry of a 3D froth is the average surface area (A 0 ) and perimeter (L 0 ) of the Voronoi cells of the froth. Oger 16 has found that both these quantities vary linearly with packing fraction (φ) if they are appropriately scaled, that is, A 0 and L 0 are linear in φ -2/3 and φ -1/3, respectively. Meijering 18 has derived a number of exact results for his cell model of a polycrystal, which is equivalent to a three-dimensional Random Voronoi Poisson point distribution (RVP). This particular froth is also a model of an ideal gas 22. The average volume V 0 is equal to the inverse of the number density. The constants K 1 and K 2 are the sphericity coefficient 23 and the average length per unit volume 18, K 1 36π = A 2 V0 3 0 K 2 2 / 3 1 4π L = Eqn V 0 1/ 3 0 The former measures the degree to which the froth deviates on average from spherical symmetry. A sphere has a sphericity coefficient of 1.0 while regular space-filling polyhedra such as the cuboctahedra or rhombododecahedra (face centred cubic lattice) have coefficients of ~ Lewis and Desch Laws The earliest semi-empirical law observed in 2D systems is the Lewis law 4. Lewis observed that there is a linear relation between the area of a biological tissue cell and its number of sides. This law has been found to apply to a number of other systems including 2D Voronoi tessellation s based on random Poisson point processes 24, plane sections of polycrystals 25 and convective fluid flow patterns 26. It has the general form, A(n) = 1 + ka(n n ) Eqn 3-3 A

105 where A 0 is the average cell area and k A is a constant, 4. Another geometrical relation is the Desch Law 5, which applies to plane sections of metallurgical grains, and relates the perimeter of a cell to its number of sides, L(n) L 0 = 1+ k (n n ) Eqn 3-4 L Again L 0 is the average perimeter and k L is a constant. Oger 16 has suggested the simple generalization to 3D for the average volume of a f-faceted Voronoi cell, V(f ) V 0 = 1+ k (f f ) Eqn 3-5 V The Lewis and Desch laws can be derived using maximum entropy arguments 3, 11. In the former, the law is called an equation of state within the MAXENT formalism, in analogy with statistical mechanics, as it interrelates macroscopic properties of the froth. The form of the Lewis law is such that the effective number of constraints (topological and space-filling) on the system is reduced. This leads to an increase in information entropy or arbitrariness of the froth 3. In practice, the Lewis law is obtained by a Lagrange multiplier approach in which two constraints are linearly combined in such a way that a third is made redundant. The parameter K A is related to the Lagrange multiplier and in the case of biological tissues, the multiplier is proportional to the age of the cells i.e. time. to time 2. In this way, the physical mechanisms responsible for producing and maintaining the froth are related to its macroscopic properties. The Desch law is derived if an additional constraint is imposed on a mosaic 11. In the case of metallurgical grains, this constraint is related to the grain boundary energy associated with the perimeter of the cells of the mosaic

106 3.2.4 Aboav-Weaire Law In contrast to the Lewis law, the semi-empirical Aboav-Weaire law is topological relation that predicts correlation s between neighboring cells in a 2D mosaic, n a + µ m(n) = n a + 2 Eqn 3-6 n where m(n) is the average number of sides of the neighbours of an n sided cell 7. It predicts that small-sided cells will have on average larger sided neighbours and vice versa. The parameter a depends on the particular type of packing and µ 2 is the variance in the distribution of the number of sides of the cells. This relation is consistent with the rigorous Weaire identity 6, nm(n) = n > =µ + n = µ + 36 Eqn 3-7 where < > is the average over cell sides for an arbitrary function B(n), B p(n)b(n) Eqn 3-8 n and p(n) is the probability of an n-sided cell. It has been found to be applicable to a variety of natural and simulated mosaics including plane sections of soap foams 7, 2D ferromagnetic Ising models 27 and convective fluid flows 26. Other relations exist for m(n); in one case, a term of order 1/n 2 is added to the above equation for a particular mosaic that is related to 2D planar Feynman diagrams 28. The generalizations to 3D are straightforward fm (f ) = f = f + µ Eqn

107 m(f ) f a + µ = f a + 2 Eqn 3-10 f where f is the number of faces of a cell and m(f) is the average number of faces of cells adjacent to an f-faceted cell. This law predicts a linear relation between fm(f) and f. The Weaire identity and Aboav-Weaire law have been justified by arguments based on maximum entropy theory as well as those based on the dynamics that maintains the froth in statistical equilibrium 2, that is, keeps it s average properties stationary, while allowing it to explore different configurational arrangements. Two types of elementary topological transformations (ETT) keep a mosaic or froth in statistical equilibrium 30. The first type, called a T1 transformation, is known as cell neighbour switching, Figure 3.1, and the second type, a T2 transformation, is the creation or disappearance of a cell, Figure 3.2. Typically, T2 transformations only apply to 3-sided faces. In some systems only type T1 transformations may occur. As an example in a 2D hard disc system, the number of atoms is conserved and hence cells neither appear nor disappear. Figure 3.1: A T1 elementary topological transformation. In two dimensions cell B loses cell D as a neighbour, while cell A gains cell C as a neighbour. In 3D, these transformations are applied to the Voronoi polyhedron of a single atom, which has A-D as neighbours. However, the same neighbour switching occurs as in two dimensions

108 Figure 3.2: A T1 elementary topological transformation. In two dimensions cell B loses cell D as a neighbour, while cell A gains cell C as a neighbour. In 3D these transformations are applied to the Voronoi polyhedron of a single atom, which has A-D as neighbours. However the same neighbour switching occurs as in two dimensions. In a 3D Voronoi froth based on hard spheres both transformations may occur and are consequences of the changing neighbours of an atom (Voronoi faces), due to thermal motion and collisions. A T2 transformation results when an atom loses a neighbour and T1 transformations are an indirect effect of a T2 transformation. They occur when the edges of the Voronoi cell of a given atom rearrange due to neighbours of the central atom switching faces between themselves, Figure 3.3. In this particular system the two types of transformations are interrelated. Figure 3.3: Elementary topological transformation in 3D. In this case cells do not disappear, but the face of a Voronoi cell can disappear. The Voronoi polyhedron of atom I loses the face it shares with atom J by a T2 transformation

109 Figure 3.4: The T2 transformation above (Figure 3.3) also leads to the concurrent T1 transformation applied to the Voronoi cell of atom A. Atoms I and J are no longer neighbours (that is, share an edge) while the Voronoi faces associated with atoms B and C now share an edge of A s Voronoi cell. The meaning of the parameter a in the Aboav-Weaire law is still unclear in many systems. In two-dimensional random networks is has been related to the kurtosis of the distribution function of n-sided cells, p(n), assuming the skewness in the distribution can be ignored Voronoi face area distributions and distance distributions The dependence of the average area of a Voronoi face on the number of its sides has also investigated in this work. This is particularly useful at high densities where preferences for certain dense local arrangements of atoms may occur, e.g.. pentagonal face arrangements. The distribution of areas for each face size, p(a,n) was also studied, where A is the area of an n-sided Voronoi face. One of the most useful quantities describing the structure of a fluid is the radial distribution function, g(r). This is related to the conditional probability of a finding an atom at a distance r from another atom. It is a measure of microscopic density fluctuations and is one of the few experimentally accessible measures of the average structure of a fluid

110 The sphere packing constraints in a hard sphere system are due to a minimum possible distance of approach between spheres, a distance σ. Thus g(r) = 0 for r < σ. This is an obvious constraint on the Voronoi tessellation of a hard sphere system that does not occur in many other 2D or 3D cellular structures. Thus deviations from the case of an RVP should occur as sphere packing constraints restrict the range of possible local environments and hence dominate over the simple space filling and topological constraints that determine the equilibrium structure in many other cellular networks. The first peak of the g(r) indicates the degree of local ordering about an atom, that is, the average distribution of distances to nearest neighbours and their number. The neighbours of a given atom have a well-defined meaning in a Voronoi approach but they are often an overestimate of the true physical neighbours of an atom 9. By this is loosely meant those neighbours that directly affect the motion of the atom, i.e. interact strongly with it, or are responsible for observed structural correlation s in the system. It is these interactions between neighbours that will have a dominant effect on the physical properties of the system. As an example the number of Voronoi neighbours in an ideal gas, or RVP, is ~ 15.5 but the number of physical neighbours is zero. No correlation s exist between the positions of the atoms in an ideal gas, and g(r) = 1 for all distances. The Voronoi neighbours of an atom are still a reasonable indication of the geometric neighbours of an atom and hence the first peak in the g(r) will have dominant contributions from atoms that are Voronoi neighbours. Thus the distribution of distances between Voronoi neighbours as a function of face size has been investigated. This is also of particular value for studying crystalline systems as it is often the case that second nearest neighbours of an atom in a crystal are defined in the Voronoi approach as nearest neighbours. This is due to degeneracy problems caused by octahedral arrangements of atoms that are present in close packed lattices

111 3.3 Results and Discussion: Ideal gas The structure of a Random Voronoi-Poisson (RVP) point distribution is that of a hard sphere fluid when steric exclusion effects are absent. It describes the random cellular network that a system of point particles possesses when subject only to simple spacefilling and topological constraints. In this way it can be regarded as a structural reference system for hard spheres of finite radii. The diversity of local environments in the RVP compared to a hard sphere fluid at a finite density implies that this system will be the most sensitive to any system size effects. The accuracy of the results of this section can hence be used as a guide for the following section on the hard sphere fluid and crystal. A set of 200 three-dimensional Random Voronoi-Poisson point configurations was generated by randomly placing 4000 points within a cubic box of nominal density 1.0. This point distribution is equivalent to a configuration of atoms or spheres in an ideal gas 22. A Voronoi analysis was performed on each configuration and the results averaged. The Voronoi tessellation was checked to ensure the froth always exactly obeyed the Euler relation, which was found to be the case. Table 3.1: Empirical and theoretical 18 values for <f> and µ 2 Empirical Standard Error Theoretical <f> (48/35π 2 + 2) µ Average Geometry The average volume (V 0 ), surface area (A 0 ) and perimeter (L 0 ) of the Voronoi cells were computed and again found to be in excellent agreement with the cell model predictions of Meijering (1953), Table 3.2. Note that the average volume is equal to 1-111

112 the inverse of the density and hence this is simply a measure of the accuracy of the geometric calculations. Also shown is the sphericity (K1) and average volume per unit length (K 2 ). Table 3.2: Theoretical 18 and empirical values of average geometrical properties of RVP froth. Empirical Standard Error Theoretical V < 1.0e (V 0 A L K K Lewis and Desch Law The average volume V(f), surface area A(f) and perimeter L(f) of an f-faceted Voronoi cell is expected to be linearly proportional to f, if the generalizations of the Lewis and Desch laws to 3D are valid. It can be seen from Figure 3.5 to Figure 3.6 that, to a first approximation, this is the case. A linear regression analysis was performed for these three quantities and the values of k V, k A and k L are given in Table 5, for the limited range f = Table 3.3: Values of k L, k A and k V obtained from a linear regression fit to the data. k V, k A, k L Volume Area Perimeter

113 V(f)/Vo RVP 1 Linear Fit f Figure 3.5: Test of Lewis Law for V(f) of ideal gas A(f)/Ao RVP Linear Fit f Figure 3.6: Test of Lewis Law for A(f) of ideal gas 1-113

114 However, a closer inspection of the data, specifically the residuals of the linear fit, reveals a systematic deviation from linearity that cannot be accounted for by the uncertainties in the data. This is most pronounced for V(f) while A(f) and L(f) show a much subtler deviation form linearity. This discrepancy is observed regardless of the range chosen over which to fit the data. A quadratic fit to the three quantities does give an excellent fit to the data, within the uncertainties in the data points, Figure 3.7 to Figure 3.8. The functions are all scaled relative to their average values. The coefficients of this fit are given in Table 3.4, where the form of the quadratic equation used is, L(f ) / > Lo = (c + c (f < f > ) + c (f < f ) Eqn 3-11 Table 3.4: The coefficients of the quadratic fit to the three geometric functions V(f), A(f) and L(f). The values are for fits using the range f = The slope of each function at f = <f> is also included to aid comparison with the linear fit. In the linear case ( Table 3.3) the values of k i (i = L,A,V) are equal to the slope at f = <f>. c 0 c 1 c 2 Slope at f = <f> Volume e Area e Perimeter e

115 V(f)/Vo RVP Quadratic Fit f Figure 3.7: Quadratic fit of V(f) versus f for ideal gas L(f)/Lo RVP Quadratic Fit f Figure 3.8: Quadratic fit of A(f) versus f for ideal gas

116 A number of different f-ranges were used with very similar results. The smallness of the quadratic coefficient gives the appearance of a linear fit and earlier work may not have detected this subtle non-linearity. The ratio of L(f)/A(f) does not differ much from a value of 3. In the range for which p(f) > 1% this ratio varies from 3.0 by at most 1.5%. The theoretical value of L o /A o, from the work of Meijering 18, is (see Table 3.2). As can be seen by comparing Table 3.3 and Table 3.4 the value of the slope of each function at f = <f> does not differ significantly from those obtained by a linear fit. The value of c 0 should equal unity and it can be seen that this is the case Aboav-Weaire Law The Weaire identity is an exact relation based on sum rules 6 and excellent agreement between calculated and predicted values is found here, the difference being less than The 3D version of the Aboav-Weaire law predicts a linear relation between fm(f) and f, and this is plotted in Figure 3.9. Again performing a linear regression analysis of the data an average value of a of 0.46 is obtained. The values of a obtained from the slope and intercept is shown in Table 3.5. As Fortes 14 found the value of a is negative. Although good agreement was obtained between the values of a obtained from the slope and that from the intercept, it was observed that fitting over different ranges of f produced a variation of around 5% in the value of a, from 0.41 to If the average of these is taken, obtained from fits over different f ranges, is used a value of 0.44 is obtained

117 fm(f) RVP Linear Fit f Figure 3.9: fm(f) versus f for ideal gas m(f) /f Figure 3.10: m(f) versus f for ideal gas

118 Table 3.5: Value of the Aboav-Weaire constant a from a linear regression fit to fm(f). The range of f values used was Slope Error Intercept Error A a The linear fit, Figure 3.9, appears to be excellent, but this is misleading. A plot of m(f) versus 1/f, which should be a straight line, indicates more clearly the poor representation of the function fm(f) by a linear relation with f, Figure If fm(f), normalized by its average value, is fitted to a quadratic function of the form, fm(f) 2 = c c (f - f ) c (f f ) < fm(f) > Eqn 3-12 then excellent agreement is obtained with the data, Figure This form is chosen so that fm(f) is correctly normalised and c 0 equals unity m(f) m(f) data m(f) fit f Figure 3.11: Comparison of fm(f) for the ideal gas with quadratic form of fit to fm(f) referred to in the text 1-118

119 The coefficients are shown in Table 3.6 and it can be seen that the appearance of a linear relation for fm(f) is due to the relative smallness of the coefficient c 2 of the quadratic term. The value of c 0 ideally equals 1.0 and hence this is a measure of the overall goodness of the fit. This was the case independent of the range of f-values chosen over which to fit the data. The values of the coefficients were particularly insensitive to the range over which the data was fitted as demonstrated in Table 3.6. Table 3.6: Parameters of quadratic fit to fm(f) using above equation. f range c 0 c 1 c e e e-4 In the Aboav-Weaire law the slope of fm(f)/<fm(f)> is related to the constant a through, Slope = ( f a) / fm (f ) Eqn 3-13 If the slope of the quadratic form of fm(f) at f = <f> is used instead to estimate the constant a, a value of is obtained a standard deviation of 0.01 (estimated by using a variety of f-ranges) Voronoi face area and diagonal distributions The distribution of Voronoi face areas as a function of the number of sides of a Voronoi face was calculated, p(a,n), Figure 3.12-Figure 3.14 in order to investigate qualitative differences between the distributions

120 7 6 Probability p(a/ao, 3) Scaled Area (A/A o ) Figure 3.12: Distribution of Voronoi face areas for 3-sided faces Probability p(a/ao, 4) Scaled Area (A/Ao) Figure 3.13: Distribution of Voronoi face areas for different n-sided faces 4- sided faces

121 Probability p(a/ao, n) Scaled Area (A/A o ) Figure 3.14: Distribution of Voronoi face areas for 5-sided, 6-sided and 7- sided faces. The distribution of 3-sided faces displays a large peak at around zero area. A near zero face area is typically indicative of more than four points, in this case five, being nearly equidistant from the same point in space. However, a three-sided face of near zero measure is also indicative of four points being nearly collinear. This type of degeneracy is also common in dense systems with a high proportion of degenerate octahedral arrangements of atoms e.g. a close packed cubic crystal. It is surprising to find such a large peak in a random distribution of points; it indicates that this may be a common feature in three-dimensional systems or froths and hence in general have little to do with sphere packing. Both three-sided and four sided faces have a peak value at a zero area, while the peak values for n greater than five are non-zero and increase with the number of sides. The variance of the distribution of face areas broadens with n while the skewness towards small areas decreases with n

122 The normalised distribution of distances between Voronoi neighbours, that is, points sharing a common Voronoi face was also calculated and is shown in Figure The distribution is gaussian in shape with a slight skewness towards low r. To investigate this further the normalised distribution of distances between points for each Voronoi face size, i.e. number of sides (n), was calculated. The function p(r,n) is the normalised probability that two atoms sharing an n-sided Voronoi face are separated by a distance r p(r/ro,n) r/r o Figure 3.15: Normalised distribution of diagonal distances, d(n), between points sharing a common n-sided face. In order to investigate qualitative differences between the distributions, the distance distributions are normalised such that their integrated areas are the same. They are also plotted versus r/r o (n), where r o (n) is the average diagonal distance of an n-sided face. It can be seen that all distributions are peaked at their average value, Figure It can also be seen that as n increases so does the variance of the associated distribution. The distributions are gaussian in shape but limited by the fact that points still have to be separated by a distance of at least zero. The probability of two points being a distance zero apart for the Poisson distribution of points is zero. Thus the form of the distribution is a compromise between a gaussian form and this constraint, as the value of the distribution must be zero at r =

123 3.3.5 Summary of Ideal Gas (RVP) Results The predictions of Meijering 18 concerning the average topological and geometric properties of a three dimensional Random Voronoi-Poisson froth have been confirmed to a very high accuracy. The disagreement is less than 0.01 % for the average number of faces of Voronoi polyhedra in a RVP (<f>) while the average surface area and perimeter values are within < % of the theoretical predictions. Oger 16 have calculated a number of similar properties to us for a few realizations of a 3D RVP but they obtain a value of <f> ~ 15.3 rather than the expected value found here of (Table 3.1). The high frequency and small size of 3-sided faces may be responsible for this estimate as many may be missed in the Voronoi decomposition. Given that the number of 3-sided faces is ~ 14% it is important that Voronoi calculations are executed with the highest possible accuracy. A comparison between the few exact theoretical results for the RVP and our data reveals excellent agreement. This gives us confidence that the system sizes used are sufficiently large to obtain accurate results. 3.4 Results and Discussion: Hard Sphere System A hard sphere system has been simulated using molecular dynamics methods at a range of reduced number densities (ρ) between 0.10 and This density range spans both the stable fluid and crystal phases. Two metastable states were also simulated, one fluid state and one crystal state, both with a density of This density is approximately midpoint between the freezing (0.945) and melting (1.054) densities. The reduced number density ρ is equal to ρ n σ 3 where ρ n is the actual number density and σ is the hard sphere diameter. The number of atoms used in the simulations was 4000 and the simulation time was 400 τ. To obtain the equilibrium fluid states an 1-123

124 initial face-centred crystal was melted by simulating for a time of 50 τ. This time was found sufficient to produce an equilibrated fluid by monitoring the pressure of the system. Configurations were stored every 1τ with the first 50 configurations discarded and the Voronoi analysis performed on the remaining 350. The time constant?τ is an average time scale for short time diffusion, and is defined in Chapter 2. A hard sphere undergoing ballistic dynamics will, on root mean square average, travel its own diameter in a dimensionless time of 1/ Average Voronoi Cell Geometry The average perimeter (L o ) and surface area (A o ) was calculated for a hard sphere fluid at a number of densities. It was found that both A o and L o scaled with the density as roughly ρ 1/3 and ρ 2/3, the actual values being 0.32 and 0.69, respectively. These quantities, L o and A o, were also first scaled by ρ +1/3 and ρ +2/3 respectively so that they would approach the RVP values at low density (see Table 3.2). In this the RVP is being taken as the limiting case when the radius of the spheres becomes zero, at fixed density, rather than when the density is zero. Remarkably a plot of these scaled quantities versus the density displays a linear relation, Figure 3.16 and Figure The scaled average perimeter (L o ) and interfacial area (A o ) were fitted to equations of the form, L o 1/ 3 ρ = L (1 a ρ) Eqn 3-14 o(rvp ) L A o 2 / 3 ρ = A (1 a ρ) Eqn 3-15 o (RVP ) A The constants L o(rvp ) and A o(rvp ) are the theoretical values for the RVP point distribution, while the constants a L and a A are the fitting parameters. The coefficients resulting from a linear regression fit of these scaled quantities to the above equations are given in Table

125 Fluid Linear Fit Metastable Fluid Crystal Ideal Close Packed Crystal Aoρ 2/ density, ρ Figure 3.16: Average Voronoi cell area A o for fluid and crystal Loρ 1/ Fluid Linear Fit Metastable Fluid Crystal Ideal Close Packed Crystal density, ρ Figure 3.17: Average Voronoi cell perimeter L o for fluid and crystal

126 The linear fit is excellent, with the residuals of the fit of the order or less than the standard errors (~ 10-4 ). The percentage difference between the calculated values and those predicted from this equation is less than 0.06%. Table 3.7: Coefficients of linear relation between density scaled interfacial area and perimeter and density. a A, a L Error Area e-4 Perimeter e-4 A similar approach was used to analyse the results of a Voronoi analysis of the hard sphere crystal. In this case the obvious reference system is the perfect crystal at close packing ( ρ = 2 ). The perimeter and interfacial area of a Voronoi cell of an atom in a perfect face centred cubic lattice is, L o 3/ 2 3/ 2 1/ 6 1/ 3 (cp) = 6 = 6 (2 ρ ) Eqn 3-16 A o 6 6 1/ 3 2 / 3 (cp) = = (2 ρ ) Eqn / 2 1/ The values of L 0 and A 0 at close packing are and , respectively. The average geometry of the Voronoi cells of atoms arranged in a face-centred cubic lattice is only subtly altered by small perturbations of the atom positions. Analysis of the average perimeter and interfacial area of the Voronoi cells of such a perturbed lattice reveals that both these quantities deviate by less than % and % from their ideal values. The perturbation was achieved by randomly displacing each atom from its lattice site by a maximum distance of 0.001σ. This is distinct from the topology, which is fundamentally altered by any perturbations

127 In a hard sphere crystal the scaled average interfacial area of the Voronoi cells decreases with density while the perimeter increases. That is, the average perimeter of a hard sphere crystal is smaller than the ideal value at each density, while the average interfacial area is larger. Both quantities approach the ideal values as the density increases towards the close packed limit. Again both L o and A o were found to scale roughly with the density, the actual values were 0.31 and However the crystal data could not be fitted to an equation of the form used for the fluid data. Note that the close packed crystal is not necessarily an ideal reference system for the crystal state as the system is deeply modified by small perturbations. A more appropriate reference system may be a crystal at zero density but the question then is what the reference constants should be. If the average geometry of a hard sphere fluid were identical to that of an ideal gas, i.e., an RVP point distribution, then the coefficients a L and a A would equal zero. The change in density would only rescale the actual area and perimeter values. The negative signs in the above forms indicate that the average surface area and perimeter of the Voronoi cells surrounding the hard spheres is reduced compared to a random point distribution. The only reason for this reduction is the finite size of the spheres, or alternatively, the existence of a minimum distance of approach between spheres. The influence of these steric exclusion effects on any of these averaged geometric quantities, including the average shape of the Voronoi cells (measured by the sphericity), can hence be characterised by the two parameters a L and a A. The smallness of the coefficients indicates that the change in the average geometry of the Voronoi partition due to sphere packing constraints is small. The percentage difference between the average perimeter and area of a RVP and a hard sphere fluid at the freezing density (0.95) is only 6.6% and 5.9%, respectively. On the other hand the same comparison for the sphericity gives a percentage difference of ~ 24%. Another point to note is that the value of L o and A o for the metastable fluid falls on the predicted curve and the percentage differences between the values from the linear fit and the calculated one are of the order of those for the stable fluid. Thus the average Voronoi geometry of the metastable fluid within the coexistence region can 1-127

128 be found by extrapolating from the stable fluid state. In Figure 3.16 and Figure 3.17 the differences between the average geometry of the metastable fluid and that of the crystal can be clearly observed. Hence these geometric measures may be useful in identifying metastable states that have partially crystallized due to the significant differences between the fluid and crystal and the accuracy of the above empirical laws for A o and L o of the fluid. A significant observation is the remarkable correspondence between the constants determining the density dependence of the average interfacial area A o and perimeter L o of a hard sphere fluid, a L and a A (Table 3.7) and the linear coefficients appearing in the quadratic form of the Lewis law for the ideal gas or RVP (Table 3.4). The two sets of constants are virtually identical. This is a surprising correspondence and strongly suggests that the average interfacial area and perimeter of the Voronoi cells of the hard sphere fluid can be predicted simply from a knowledge of the topology and geometry of the ideal gas! One way to understand this correspondence is to recall that the average topology of the hard sphere fluid at any given density is different from that of the ideal gas. In particular the mean number of faces of a Voronoi cell decreases with density. Thus the decrease in the average interfacial area and perimeter with density is due to the change in the topology of the hard sphere fluid relative to the ideal gas. The link with the Lewis law should then be clear as this is simply a relation between the topology of a Voronoi cell, specifically its number of faces, and its interfacial area or perimeter. Thus the average interfacial area or perimeter of the Voronoi cells of a hard sphere fluid is different from that of the ideal gas because its topology is different. However, the exact geometric change can be discerned from the Lewis law for the ideal gas. Another equally important point to note is that the above correspondence is only found for the quadratic form of the Lewis law and not the linear form. This gives us great confidence that this particular form of the Lewis law is both appropriate and the correct form to use

129 3.4.2 Voronoi Polyhedra Types The Voronoi description of the local environment about an atom has been used by a number of authors to identify local crystalline environments in a number of different nucleating systems 34, 35. One problem with this method of characterizing local structure is a degeneracy associated with the Voronoi cells of the face-centred cubic and hexagonally close packed lattices. In a face-centred cubic lattice a 3-sided Voronoi face is related to a particular degenerate decomposition of octahedral arrangements of atoms. In general an octahedron can decompose into 4 or 5 tetrahedra. In the latter case, one of the tetrahedra is a sliver of near zero volume, and produces a 3-sided Voronoi face (actually two 3-sided faces). This 3-sided face also has near zero Voronoi face area. If it is assumed that all of these 3-sided faces are associated with degenerate octahedra and apply the methods described in Chapter 2 to eliminate them, a particular Voronoi decomposition comprised of a minimum number of tetrahedra is obtained. In essence, this method is attempting to remove one of the degeneracies that occur in close packed lattices, due to octahedral arrangements of atoms. Note that this elimination process can only be applied to 3-sided Voronoi faces. The Bernal indices for each Voronoi polyhedra were recalculated after the deletion of these 3-sided faces. The number of polyhedra is given in Table 3.8 as a function of density. It can be seen that there is a dramatic drop in the number of distinct polyhedra. Table 3.8: Number of distinct Voronoi polyhedra (as defined by Bernal s indices) before and after the deletion of 3-sided faces for the hard sphere crystal. Density Number of VP Types Number of VP Types (Before) (After)

130 In Table 3.9, for example, the most commonly observed polyhedra in the face-centred cubic crystal at the melting density are shown, and the percentage of each type before at after the deletion. If these indices were used to identify atoms in a locally facecentred cubic environment, only 70% of the atom would be identified as crystalline while approximately 85% are identified after the modification. In addition, the number of Voronoi faces with greater than six sides is essentially zero, in the modified Voronoi case. Note that the percentage of each of the listed polyhedra does not change significantly with density, for the modified Voronoi case. Table 3.9: Most frequently occurring polyhedra before and after the deletion of 3-sided faces for a hard sphere face-centred cubic crystal at the melting density (ρ=1.05). Voronoi Polyhedra Frequency (Before) Frequency (After) (3 6 4) (4 4 5) (5 2 6) (4 4 6) (3 6 5) (2 8 4) (4 4 7) (3 6 6) Lewis and Desch Law A similar analysis was performed on the average volume V(f), surface area A(f) and perimeter L(f) of an f-faceted Voronoi cell for the hard sphere fluid and crystal as was done for the RVP. Firstly, a linear regression analysis was performed at a number of densities to assess the validity of the Lewis Law. Except for the crystal, deviations from linearity were observed that could not be accounted for by the standard errors

131 A quadratic form for L(f), A(f) and V(f) of the same type used for the ideal gas case (RVP) was then used to fit the data, L(f ) / > Lo = (c + c (f < f > ) + c (f < f ) Eqn 3-18 A systematic study of the dependence of the above coefficients on the range of f- values chosen to fit the data was conducted at four densities (0.3, 0.6, 0.9, 1.10 crystal). It was found that the values of the coefficients were only weakly dependent on the data range chosen, with a typical deviation of less than 0.04%. The constant coefficient c o is equal to unity as expected to within at most 0.01%. The density dependence of the linear and quadratic coefficients as a function of density is shown in Figure 3.18 and Figure The density dependence of the slope of each function at f = <f> was also calculated and is shown in Figure c fluid V(f) crystal V(f) fluid A(f) crystal A(f) fluid L(f) crystal L(f) density Figure 3.18: Density dependence of linear coefficient, c 1, of quadratic form of Lewis law for hard sphere fluid and crystal

132 Overall the linear coefficient c 1 for the three functions decreases with density in the fluid with A(f) and V(f) showing similar behaviour compared to L(f). In the case of the former both are zero at a density of approximately 0.70 and become negative at higher densities. At this density the value of c 1 for L(f) plateaus and increases slightly as the freezing density is approached. In the crystal the coefficients approach zero as the close packed density is approached, although this trend is less clear with the volume function fluid V(f) crystal V(f) fluid A(f) crystal A(f) fluid L(f) crystal L(f) c density Figure 3.19: Density dependence of quadratic coefficient, c 2, of quadratic form of Lewis law for hard sphere fluid and crystal. In general the quadratic coefficients for V(f), A(f) and L(f) are small and decrease with density. This suggests that at high densities the Lewis law is better obeyed, but as the range of non-zero values also decreases with density systematic deviations are harder to discern. This is particularly the case for the crystal where the data range is typically from f = However as is observed for the linear coefficient it is observed that as the close packed density is approached c 2 goes smoothly to zero. The density dependence of c 2 is different for each geometric measure. Both V(f) and L(f) appear to have maximum values in the fluid at low densities while the non-linearity 1-132

133 increases with density for A(f). The quadratic coefficient for A(f) and L(f) also changes sign at a density of As can be seen in Figure 3.20 the slope of the three functions V(f), A(f) and L(f) decreases with density in the fluid. There is a discrete drop to smaller values in crossing over to the crystal as well as a more rapid decrease in the slope with density. For all three quantities the value of the slope smoothly approaches zero as the crystal close packed density is approached. The latter indicates that at high densities there is little dependence of the volume of a Voronoi cell on its number of faces. Since the different Voronoi cells of a crystal are really perturbations of a single type of cell, this should not be too surprising. The smaller slope for the metastable crystal than for the fluid is also indicative of this fact. 0.1 Slope of V(f), A(f) and L(f) at f = <f> fluid V(f) crystal V(f) fluid A(f) crystal A(f) fluid L(f) crystal L(f) density Figure 3.20: Slope of V(f), A(f) and L(f) at f = <f> calculated from quadratic form of Lewis law as a function of density. One would expect that the slope of V(f) decreases with density simply because the range of volume fluctuations will decrease with density, and hence the possibility of large changes in V(f) with f will be suppressed. This is also likely for the other two 1-133

134 functions L(f) and A(f). This suggests that there may be a simple relation between the variance of the volume distribution and the slope of V(f) with f, as well as for A(f) and L(f). However no simple relation between the relevant variances and the calculated slopes could be found Aboav-Weaire Law The mean number of facets of an f-faceted cell, m(f), has been calculated at a number of densities spanning the stable fluid and crystal phases. The Aboav-Weaire relation predicts a linear relation between fm(f) and f of the form, fm (f) µ = ( f a)f + f a + 2 Eqn 3-19 To a first approximation this law is obeyed and the value of a appearing in the above equation is shown in Figure 3.21 for both the fluid and crystal phases. It can be seen that a is negative at low densities becoming positive for densities above It then slowly increases, saturating as the freezing density is approached. There is also a significant change in the value of a between a metastable fluid and a crystal at the same density. The change in a with density is roughly linear for the crystal. Note that the value of a appears to approach 1.0 as the density of the crystal increases towards the close packed limit. A more detailed analysis of the data was conducted in order to investigate any systematic deviations from linearity. A plot of m(f) versus 1/f, which should be a straight line, is shown in Figure 3.22 for a number of densities. The RVP case suggests that a quadratic relation between fm(f) and f may be a more accurate representation of the data. In particular the residuals of the linear fit were checked for any systematic patterns, taking into account the standard errors in the data

135 a density Figure 3.21: Value of constant a appearing in Aboav-Weaire law for both fluid and crystal densities m(f) RVP Figure 3.22: m(f) versus f for a number of fluid and crystal densities. f 1-135

136 In order to assess the sensitivity of the results to the range of f values used in the fitting a more detailed analysis of the data was conducted at a number of selected fluid densities (0.2,0.6 and 0.9). The variation in the value of the above coefficients with the range used was in general less than 0.05%. The function fm(f) was fitted to the same quadratic formula used for the RVP, fm(f) 2 = c + c1(f-< f > ) + c 2 (f - < f 2 < fm(f) > 0 > ) Eqn 3-20 This particular form is chosen so that when normalised c 0 should equal unity. Excellent agreement is obtained with the data, The variation of the linear and quadratic coefficients with density for both the fluid and crystal are shown in. The value of c 0 varied only slightly from the ideal value of 1.0 (< 0.2 %) in the fluid and even less so in the crystal (<0.01 %) fm(f) f Figure 3.23: fm(f) at a density of ρ = 0.6 and quadratic fit to data 1-136

137 c density Figure 3.24: Density dependence of linear coefficient of quadratic form of Aboav-Weaire law for hard sphere fluid and crystal. 2.5E E E E E-05 c2 0.0E E E E E E-04 density Figure 3.25: Density dependence of quadratic coefficient of quadratic form of Aboav-Weaire law for hard sphere fluid and crystal

138 The linear coefficient decreases with density in the fluid but does not do so in a systematic manner, Figure Note that c 1 increases with density for the crystal. This different behaviour between the fluid and crystal is also reflected in the density dependence of the variance in the distribution of Voronoi neighbours (µ 2 ). In general the linear term is larger in the crystal than the fluid. It can be seen that the quadratic coefficient, Figure 3.25, changes sign at a density between 0.7 and 0.8. The quadratic term is also small indicating the subtlety of the non-linear relation between fm(f) and f. It should be recalled that although this term is small, it s inclusion is necessary to predict the correct qualitative form of m(f), particularly at low densities. In the crystal the quadratic term is always negative. The density region of around is associated with the emergence of backscattering dynamics in the motion of hard spheres, as their velocities are reversed by interactions with the cage of neighbours surrounding it. The slope of fm(f)/<fm(f)> at f = <f> was also calculated in order to compare with the results of the linear fit, Figure 3.26.It can be seen that the most significant differences are at low densities in the fluid while the results are very similar for the crystal. Note that the normalised form used here probably scales out most of the influence of the change in both the mean and variance of the distribution of Voronoi neighbours on the function fm(f). The constant a essentially characterises overall topological changes of the hard sphere system with density. This parameter smoothly changes with density for both phases but with a discrete change between them. The slope of fm(f) /<fm(f)> increases with density for the fluid indicating an decreasing topological inhomogeneity as the freezing density is approached. Recall that the slope of fm(f) indicates the degree to which small f-faceted cells are surrounded by large faceted cells and vice-versa. Note that the slope of fm(f) for the metastable fluid and crystal are similar. In addition, the opposite trend is observed for the crystal in which the slope of fm(f)/<fm(f)> decreases as the density increases. This is consistent with the behaviour of the variance of the number of Voronoi neighbours in the crystal, which increases with density

139 Slope of fm(f) at <f> density Figure 3.26: Slope of fm(f)/<fm(f)> for fluid and crystal using both linear and quadratic from of fm(f) Voronoi face areas Two atoms that share a Voronoi face are defined as neighbours and hence the frequency of a given face side and its area provides us with information on the local packing of atoms about a nearest neighbour bond, see Figure The distribution of face areas is also a decomposition of the total interfacial area into contributions from individual Voronoi faces. This is shown in for a typical fluid density in Figure 3.28 and in Figure 3.29for a typical crystal density. There are a number of important features to note in. First the distribution is bimodal with one large peak at approximately zero. The second peak at a finite face area is separated at high densities by a distinct minimum in the distribution that grows in depth with density. In the case of a crystal the two peaks are clearly separated by a zero minima at a sufficiently high density. These features are representative of the results at all fluid and crystal densities

140 (a) (b) (c) Figure 3.27: Voronoi face and packing of atoms that define the face. A pentagonal Voronoi face (a) and the packing of atoms that define the face (b)- (c). The face is defined by the two light coloured atoms, which are shown in two slightly different viewpoints in (b) and (c). The dark coloured atoms are the five neighbours that they share P(A,n) Total Area (A) Figure 3.28: Decomposition of distribution of Voronoi face areas into contributions from different sided faces for fluid at ρ =

141 0.06 P(A,n) Total Area (A) Figure 3.29: Decomposition of distribution of Voronoi face areas into contributions from different sided faces for crystal at ρ = In order to understand these features in more detail the total distribution of face areas was decomposed into the contributions from different sided faces, Figure 3.28 and Figure At all densities the near zero peak are due to small area 3 and 4-sided faces. The simplest explanation of this peak is in terms of topological transformations that take place in the hard sphere system. The Voronoi cell of an atom changes by either the gain or loss of a face or by the indirect consequences of this happening to its neighbours. The gaining of an n-sided face depends on the overlap of a plane bisector with n-2 vertices of the Voronoi polyhedra of an atom, Figure Obviously as the distance between atoms is a smoothly varying function, there will be a finite number of near zero face areas indicating that a new face is just appearing or about to be lost. Thus the zero face areas indicate the fraction of Voronoi faces in transition, that is, the fraction of neighbour bonds forming or about to break. This then gives use some measure of the rate of topological transformations

142 . Figure 3.30: Topological transformations and small-sided faces. The formation of a new five-sided face of the Voronoi cell of an atom requires the overlap of the plane bisector of the atom and a new neighbour with three vertices of the Voronoi cell of the atom. Note that although this picture is appropriate for a hard sphere fluid account must be taken of the fact that small-sided faces with zero area also exist in the RVP. This is simply a geometric fact a near zero face indicates that more than four atoms are equidistant from the same point. Note that an n-sided face is defined by n + 2 atoms. It should be obvious that for probabilistic reasons alone this will be more likely for small-sided faces and for the hard sphere systems packing constraints will restrict the possibilities particularly at high densities. This can be considered as the probability of n atoms being packed with the centres on the same sphere. The center of the sphere is the common circumcentre of the Delaunay simplexes associated with the sphere centres. As the average radius of this sphere is reduced, that is as the atoms are packed closer together, the probability of n-spheres being packed about the sphere without overlaps will decrease. The crystal results are similar to the fluid except that the face area distribution is more sharply peaked at zero and is almost entirely a consequence of 3 and 4-sided faces. The average size of both 3-sided and 4-sided faces drops radically in the crystal compared to the fluid. In a close packed crystal both 3-sided and 4-sided faces are predominately associated with the prevalence of ideal octahedral arrangements of atoms. It can be seen that in Figure 3.32, the distribution of 4-sided faces is bimodal. This can be easily explained in terms of the different types of decomposition of the perturbed Voronoi cell of a perfect face-centred cubic lattice. The 3-sided faces are nearly entirely due to the degenerate decomposition of octahedra into 5 tetrahedra and are discussed in more detail in Chapter

143 The distribution of areas for 3 and 4-sided Voronoi faces is shown in Figure 3.31 and Figure 3.32, for a number of fluid densities. It can be seen that at all densities the distributions are peaked at a zero density. A small shoulder can be discerned at high densities for the 4-sided Voronoi faces indicating a small qualitative change in the packing geometry with density. In Figure 3.33 the distribution of face areas of a 5-sided Voronoi faces at a number of densities in the fluid range is shown, normalised by the average area of a 5-sided face at that density. The peak value increases with density and all distributions intersect at the same normalised face area. At a density around 0.70 the peak switches from the low area side of the common intersection point to the high side. Inspection of Figure 3.33 indicates that for densities above 0.70 this peak grows and hence local arrangements of dense 5-sided packings become important only at densities above this. In general the dominant contributions to the main non-zero peak in the Voronoi face area distribution are due to 5 and 6-sided faces. The peak position of the latter is always greater than the former p(a/ao,3) A/A o Figure 3.31: Normalised distribution of Voronoi face areas for 3-sided faces at a number of fluid densities

144 2.5 p(a/ao,4) A/A o Figure 3.32: Normalised distribution of Voronoi face areas for 4-sided faces at a number of fluid densities p(a/ao(5)) Scaled Area (A/A o (5)) Figure 3.33: Distribution of Voronoi face areas for 5-sided faces normalised by average 5-sided face area A o (5)

145 3.4.6 Voronoi diagonal distributions A complementary measure of the local order around an atom is the distribution of distances to its Voronoi neighbours. This is simply related to the first peak of the radial distribution function of a fluid, and often implies an upper bound to a nearest neighbour bond distance. The distribution of diagonal distances is shown at a number of densities in Figure Included as well are the individual contributions from Voronoi faces of different numbers of sides. The average diagonal distance of a n- sided face was also calculated. The general trend is that the larger the number of sides of a Voronoi faces the closer the two atoms forming the face. The distribution of diagonal distances as a function of the number of sides of the associated Voronoi face is shown in Figure 3.36-Figure It can be seen that the 3- sided faces, Figure 3.36, are always peaked at a value of r greater than 1.0. The distribution has the shape of a skewed gaussian, which narrows and shifts towards lower values of r as the density is increased P(r) Total r Figure 3.34: Normalised distribution of diagonal distances P(r) for the hard sphere fluid (ρ = 0.7) and decomposition into contributions from different sided Voronoi faces

146 The value of p(r,3) at r = 1.0 is essentially zero and the distributions have a minimum value at approximately r = 1.2 irrespective of density. The distribution of diagonal distances of 4-sided faces, Figure 3.37, is also peaked at a value of r greater than 1.0 but is non-zero at r = 1.0. The peak position is approximately given by the position of the first minima in the g(r) of the fluid. The zero value of p(r=1,4) increases with density. There is a qualitative change in the form of this distribution as the density is increased due to the existence of a minimum diagonal distance. The distribution of diagonal distances for 5-sided and 6-sided faces is qualitatively similar, Figure 3.38 and Figure At low densities they are peaked at values of r > 1 while at higher densities their peak value shifts to r = 1.0. This occurs at a density of approximately 0.70 for 5-sided faces and 0.50 for 6-sided faces. 0.1 P(r) Total r Figure 3.35: Normalised distribution of diagonal distances P(r) for the hard sphere crystal near the melting density (ρ = 1.05) and decomposition into contributions from different sided Voronoi faces

147 P(r,3) r Figure 3.36: Normalised distribution of diagonal distances at different fluid densities for 3-sided faces P(r,4) r Figure 3.37: Normalised distribution of diagonal distances at different fluid densities for 4-sided faces

148 P(r,5) r Figure 3.38: Normalised distribution of diagonal distances at different fluid densities for 5-sided faces P(r,6) r Figure 3.39: Normalised distribution of diagonal distances at different fluid densities for 6-sided faces

149 The pair correlation function at a number of densities is shown in Figure A common and simple definition of the neighbours of an atom are all those within a certain distance of the atom. This distance is usually taken as the minima separating the first and second peaks of the g(r). For the densities 0.70, 1.00 and 1.05 in Figure 3.40 these maximum bond lengths would be 1.60, 1.50 and 1.40 respectively. A comparison of an appropriately normalised Voronoi diagonal distribution and the g(r) is shown in Figure 3.41 to Figure 3.43 for a number of densities. The associated distribution of Voronoi diagonal distances for the fluid has an inflection point at the distance corresponding to the first minimum in the g(r). In the case of the crystal the minimum in the distribution of diagonal distances coincides with the first minimum in the g(r). 7 6 g(r) fluid 1.05 crystal r Figure 3.40: Radial distribution function g(r) of the stable fluid, metastable fluid and stable face-centred cubic crystal of the hard sphere system. It can be seen that the Voronoi neighbours, within this definition, include a number of what would be classed as second nearest neighbours. A comparison of the diagonal distributions for different sided Voronoi faces shows that the major contribution of 1-149

150 these distances is from 3 and 4-sided faces. These are both peaked at a value of r > 1. Again this indicates that many of the small faces are again due to the loss or gain of a neighbour, which by definition is a second nearest neighbour before or after the topological transformation. Hence they are second nearest neighbour distances. In the case of the crystal the particular nature of the Voronoi cell for close packed structures means that many second nearest neighbours can be classed as Voronoi neighbours as a result of small fluctuations. Thus at all densities the bonds associated with 3-sided faces are also overwhelmingly associated with what would be classed as second nearest neighbours. This is also partially true for 4-sided faces as well, but only in the fluid. Thus by deleting the bonds associated with these faces in an appropriate manner Voronoi neighbours are deleted that are actually second nearest neighbours. g(r) g(r) p(r) p(r,3) p(r,4) p(r,5) p(r,6) r Figure 3.41: Comparison of pair correlation function g(r) and appropriately normalised distribution of diagonal distances at a fluid density of ρ =

151 g(r) g(r) p(r) p(r,3) p(r,4) p(r,5) p(r,6) r Figure 3.42: Comparison of pair correlation function g(r) and appropriately normalised distribution of diagonal distances at a fluid density of ρ = g(r) g(r) p(r) p(r,3) p(r,4) p(r,5) p(r,6) r Figure 3.43: Comparison of pair correlation function g(r) and appropriately normalised distribution of diagonal distances at a crystal density of ρ = 1.05, near melting

152 3.5 Summary The generalization of the Lewis Law to a 3D froth of the Voronoi cells of hard spheres is not strictly valid. However, the observed non-linearity is subtle and so to a first approximation a linear relation exists between the average volume, area or perimeter of a f-faceted cell and its number of faces (f). The deviations from linearity appear to increase as the dimension of the geometric object under study increases. Nevertheless within the small uncertainties of the data, a quadratic form of the Lewis law, does give excellent agreement with the data. In a similar spirit it has been found that the Aboav-Weaire Law is only approximately linear, closer inspection again reveals systematic deviations. Again the quadratic dependence is not strong but it is obvious if the function m(f) is studied directly rather than fm(f). The form for m(f) that is proposed here is similar to one suggested before by Fortes 36, who reanalysed data produced by Kumar 37 ; the difference is that a linear term in the functional form of m(f) is retained here. This work is consistent with results on 2D Random Voronoi Poisson point distributions, which do not obey the Aboav-Weaire law. The semi-empirical laws of Lewis and Aboav-Weaire are only approximately obeyed, careful analysis reveals that a quadratic form of the laws produces a good fit to our data. This quadratic term is small and hence could be easily missed. A new highly accurate empirical equation for the average interfacial area and perimeter of the Voronoi cells of a hard sphere fluid has also been discovered. One implication is that an accurate equation for the average sphericity of the Voronoi cells of the hard sphere fluid can also be obtained. This latter quantity characterises the average anisotropy of the local environment about the hard spheres. A new connection between the quadratic form of the Lewis law and the density dependence of the average interfacial area and perimeter of the Voronoi cells of the hard sphere fluid has been found. In particular it has been found that the coefficients describing the decrease in the latter quantities with density are numerically equal to the linear term in the corresponding quadratic form of the Lewis law

153 The average interfacial area and perimeter of the Voronoi cells of a metastable hard sphere fluid is also predicted by the above-mentioned empirical equation. As there is a distinct difference between the fluid and crystal values for Ao and Lo at the same density, this implies that these quantities may be useful in characterizing whether or not a metastable fluid has partially crystallized. At low densities the distribution of diagonal distances for different sided Voronoi faces is well described by a gaussian. At higher densities and for larger sided faces the form of each distribution is modified by the existence of a minimum distance of approach between spheres. For both the high-density fluid as well as the hard sphere crystal, Voronoi neighbours with bond lengths equal to or greater than the first minima in the g(r) can be identified with particular sided Voronoi faces i.e. 3 and 4- sided Voronoi faces. This is particularly useful if a modified definition of the Voronoi neighbours of an atom that does not include what are properly classed as second nearest neighbours of the atom is desired. Two fluid densities are observed to appear repeatedly when qualitative changes in many geometrical and topological measures as a function of density are observed. The first density is around 0.10 and indicates that in some cases the RVP may be distinctly different from the hard sphere fluid. Note that a density of corresponds to a percolation threshold for the hard sphere fluid. This is the density above which the void space or available volume becomes disconnected. By void space is meant the volume of space in a hard sphere packing available for the addition of another sphere. The other density is at around 0.70 and this is approximately equal to the density at which the velocity autocorrelation function of the hard sphere fluid becomes negative. This is indicative of the appearance of cage effects or liquid-like dynamics in the hard sphere fluid. These cage effects are important when the free volume of the hard sphere fluid becomes localized. The free volume of a sphere is the volume over which the center of the sphere can translate when all other spheres in the packing are kept fixed. At low densities it is approximately equal to the available volume but as the density is increased each atom only has access to a free volume defined by its nearest neighbours; a fuller discussion of these concepts can be found in Chapter 4. Hence 1-153

154 qualitative changes in both the topology and geometry of the hard sphere fluid at these two densities may be indicative of similar changes in the statistical geometry of the void and free volume of the hard sphere fluid. 3.6 Bibliography 1 D. Weaire and N. Rivier, Contemp. Phys. 25, 59 (1984). 2 N. Rivier, Phil. Mag. B 52, 795 (1985). 3 N. Rivier and A. Lissowski, J. Phys. A: Math. Gen. 15, L143 (1982). 4 F. T. Lewis, Anat. Record. 38, 341 (1928). 5 C. D. Desch, J. Inst. Metals 22, 241 (1919). 6 D. Weaire, Metallography 7, 157 (1974). 7 D. A. Aboav, Metallography 13, 43 (1980). 8 M. A. Peshkin, K. J. Strandburg, and N. Rivier, Phy. Rev. Lett. 67, 1803 (1991). 9 J. D. Bernal, Nature 183, 141 (1959). 10 N. Rivier, in From Statistical Physics to Statistical Inference and Back, edited by P. Grassberger and J.-P. Nadal (Kluwer, Deventer, 1994), p N. Rivier, in Maximum Entropy and Bayesian Methods, edited by P. F. Fougere (Kluwer Academic Publishers, Deventer, 1990), p E. T. Jaynes, in Papers on Probability Statistics and statistical Physics, edited by R. D. Rosenkrantz (Reidel, Dordrecht, 1957), p P. N. Andrade and M. A. Fortes, Phil. Mag. B 58, 671 (1988). 14 M. A. Fortes, Phil. Mag. Lett. 68, 69 (1993). 15 M. A. Fortes, J. Phys. A: Math. Gen. 28, 1055 (1995). 16 L. Oger, A. Gervois, J. P. Troadec, and N. Rivier, Phil. Mag. B 74, 177 (1996). 17 P. Richard, L. Oger, J. P. Troadec, and A. Gervois, Phys. Rev. E 60, 4551 (1999). 18 J. L. Meijering, Philips Res. Rep. 8, 270 (1953). 19 J. P. Troadec, A. Gervois, and L. Oger, Europhys. Lett. 42, 167 (1998). 20 F. C. Frank, Proc. R. Soc. London A 214, 43 (1952). 21 M. R. Hoare, Adv. Chem. Phys. 40, 49 (1979). 22 A. Rahman, J. Chem. Phys. 45, 2584 (1966)

155 23 V. P. Voloshin and N. N. Medvedev, J. struc. Chem. 26, 376 (1985). 24 H. Hermann, H. Wendrock, and D. Stoyan, Metallography 23, 189 (1989). 25 I. K. Crain, Comput. and Geosci. 4, 131 (1978). 26 P. Cerisier, P. Rahal, and N. Rivier, Phys. Rev. E 54, 5086 (1996). 27 R. Delannay, G. L. Caer, and M. Khatun, J. Phys. A: Math. Gen. 25, 6193 (1992). 28 C. Godreche, I. Kostov, and Yekutieli, Phy. Rev. Lett. 69, 2674 (1992). 29 M. A. Fortes, J. Phys. France 50, 725 (1989). 30 M. A. Fortes, Acta metall. 33, 1697 (1985). 31 C. J. Lambert and D. Weaire, Phil. Mag. B 47, 445 (1983). 32 W. Brostow, Phys. Rev. B 57, (1998). 33 J. Lemaitre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, Phil. Mag. B 67, 347 (1993). 34 J. N. Cape, J. L. Finney, and L. V. Woodcock, J. Chem. Phys. 75, 2366 (1981). 35 W. C. Swope and H. C. Anderson, Phys. Rev. B 41, 7042 (1990). 36 M. A. Fortes and P. Pina, Phil. Mag. B 67, 263 (1993). 37 S. Kumar, S. K. Kurtz, J. R. Banavas, and M. G. Sharma, J. Statist. Phys. 67, 523 (1992). 38 S. B. Lee and S. Torquato, J. Chem. Phys. 89, 3258 (1988). Chapter

156 Ring Analysis of the Hard Sphere System 4.1 Introduction The structure of a hard sphere fluid has been analysed using the ring analysis methods described in Chapter 2. The analysis is focused on two common distribution functions used to characterise the structure of a fluid. These are the radial distribution function g(r) and the bond angle distribution function. The first describes two body correlations between atoms in the fluid while the second describes three-body correlations, albeit in a limited fashion. In this Chapter these two distribution functions are decomposed into contributions from ring structures of different sizes, and the changes in the geometry of these topologically defined structures are analysed as a function of increasing density. By taking this approach it is hoped that more detailed information about the structure of a fluid can be extracted that may be washed out by the wide range of possible local geometric environments in a fluid, particularly at low densities. The particular decomposition chosen is motivated by our main focus on dense hard sphere systems and their crystallisation behaviour. It has long been noted that at sufficiently high fluid densities local pentagonal order plays an important role in packing 1-2 and this is investigated using our particular 5-membered ring structures. Our other interest lies in the emergence of crystalline order and to this end the 6- membered rings defined in this work are used to monitor the evolution of local hexagonal order. Note that these rings are found in all close packed structures as well as two important local icosahedral structures: the icosahedron and the related twisted icosahedron or bi-centred prismatic icosahedron. The six-membered rings provide a connection between the fluid and crystal phases, as their number is large in both states. As will be shown it is differences in their geometry and coherence that distinguish the two states

157 There are a number of questions that we wish to address using these methods. The first question is a rather broad one: is there a qualitative change in the structure of a hard sphere fluid as it densifies or are the observed changes in the radial and bond angle distribution function simply the result of an increasingly more restricted geometry, due to the influence of sphere packing constraints at higher densities? Particular attention was paid to changes in the structure that occurred as the density approached the freezing density. This was partly motivated by a second question relating to the crystallisation of a fluid or liquid, specifically whether there is any sign of precursor structures in the dense fluid near freezing that suggest the onset of an ordered crystal. Another motivation for studying a metastable fluid is to see whether we can relate the observed splitting of the second peak of the g(r) at high densities 2 to any particular ring structures. This splitting of the second peak is most pronounced in the glassy phase and has been observed in a wide range of materials such as metallic glasses, random hard sphere packings and model systems such as the Lennard-Jones and soft sphere glasses Simulation Details A hard sphere fluid was simulated using molecular dynamics methods at a range of number densities between 0.2 and The number of atoms used in the simulations was 4000 and the simulation time was 400 τ. A face-centred cubic lattice was used as a starting configuration. The first 50 configurations were discarded and the ring analysis was performed on the remaining 350. The neighbours of each atom were defined using the modified Voronoi method described in Chapter 2. Two other definitions of bonds between nearest neighbours were also used to analyse the structure of the hard sphere fluid at the densities of 0.70 and This was done in order to examine the dependence of the results on the method used to define bonds. In the first method two atoms are defined as neighbours if their separation is less than some maximum bond distance, taken as the position of the first minimum in the g(r). In the second method the Voronoi neighbours of an atom are used. Although there are 1-157

158 some quantitative changes in the topology, which is not surprising as the number of rings found depends on the bond network, the qualitative behaviour is very similar in both cases. A metastable fluid at a density of 1.00, within the coexistence region, was also prepared, as described in Chapter 2. The number of atoms and simulation time were the same as for the equilibrium fluid. 4.3 Results and Discussion Topology of Hard Sphere Fluid The number of shortest path (SP) rings of various sizes as a function of density is shown in Figure 4.1(a), and the main features are summarised in Table 4.1. Of particular interest is the decline in the number of four and five membered rings with increasing density. This is surprising, as intuitively the opposite would be expected. The reason is that as the freezing density is approached one would think that the number of octahedral arrangements of atoms, which are related to the number of 4-membered rings, would increase. This is because they have a relatively high packing fraction and are one of the two common structural units in a close packed crystal. The decrease in the number of five membered rings is also unusual as one of the characteristic features of a dense fluid is meant to be a high proportion of pentagonal arrangements of atoms. However the average coordination number is a decreasing function of the density and hence one would expect that this would have an effect on the average number of shortest path rings found. Overall the change in the number of 4 and 5 membered rings is roughly linear with density while the 6-membered rings increase at a greater than linear rate. Also the number (or fraction) of 5 and 6 membered rings appear to be approaching a common 1-158

159 Number of SP Rings per atom crystal 4 crystal 5 crystal 6 crystal density Figure 4.1: (a) Average number of shortest path rings per atom versus density for a hard sphere fluid and crystal. The data points at densities of 1.00 are for the metastable fluid and crystal within the coexistence region Fraction crystal 4 crystal 5 crystal 6 crystal density Figure 4.1: (b) The fraction of SP Rings of various sizes versus density for a hard sphere fluid and crystal

160 value at a density slightly greater than 1.0. The fraction of rings, relative to the total number of rings, was also calculated and is shown in Figure 4.1(b), revealing the same qualitative trend with density as the number of SP rings per atom. The number of shortest path rings for the metastable and stable face-centred cubic crystal is also shown in Figure 4.1(a). There is a smaller number of 3- membered rings than in the fluid, which is simply a reflection of the lower coordination number in the crystal. The increase in the number of 4-membered rings in the crystal is to be expected, as the face-centred cubic lattice is a packing of regular tetrahedra and octahedra, with each octahedron contributing three 4-membered rings. Note that this increase relative to the fluid occurs even though the average coordination number, and hence the total number of bonds in the system, is smaller in the crystal. A substantial drop in the number of 5-membered rings in the crystal is also observed for densities greater than these rings indicate topological defects in the crystal and are negligible in the stable crystal. Table 4.1: The percentage change in the number of SP rings per atom as the density of the hard sphere fluid increases (relative to a number density of 1.00). * The change is measured for the 3-membered rings from the reference density of 0.60; the rings have a minimum value with respect to density for this value. Ring Size % Change from * In general as the density increases the number of three membered rings increases. For densities less than or equal to 0.50 there is a very small increase in the number of three membered rings. This is attributed to the modified voronoi method rather than something intrinsic about the structure of the low-density fluid. The reason is that one source of a three membered ring is a three-sided Voronoi face, Figure 4.2. In our 1-160

161 modified voronoi method these are all but eliminated at higher densities but we find that at densities equal to or below 0.50 due to the preponderance of their numbers, only a smaller fraction can be eliminated. The percentage change in the number of 3- membered rings, from 0.60 to 0.90 is only ~ 2% and hence quite subtle. (a) (b) Figure 4.2: (a) Three-sided Voronoi face and (b) arrangement of atoms associated with this face. The number of six-membered rings is slightly greater in the metastable fluid than in the stable crystal and is also equal to approximately 4 at a density near freezing (ρ = 0.95). If these rings are seen as indicators of precursor structures to crystallisation then in the dense fluid their frequency is comparable to that in the crystal at freezing but, as will be shown, it is their average geometry that differs. It is particularly heartening that at least locally there is a substantial increase in the number of sixmembered rings, associated with close packed planes in the crystal, as the freezing density is approached

162 4.3.2 Decomposition of Pair Correlation Function, g(r) In Figure 4.3 the pair correlation function is show at a number of densities within the fluid region. The main qualitative features that change with density can be summarised as follows. An increase in the value of g(r) at r/σ = 1. The contact value of g(σ) is trivially related to the compressibility for a hard sphere system and reflects the increasing number of contacts, or touching spheres, as the volume fraction increases. The development of a second peak, is also obvious for densities > 0.50 as well as the subsequent appearance of further peaks at even higher densities. These peaks shift inwards with increasing density. The appearance of a shoulder in the second peak of the g(r) at a density of approximately 0.90, which develops into a split peak as the density increases into the metastable region of the phase diagram. A typical decomposition of the g(r) into ring contributions is shown in Figure 4.4 at a density of In Figure 4.5 a close up of the decomposed second peak of the g(r) at a density of 0.95 is shown for comparison. It can be seen that the first and second peak of the g(r) distribution can be successfully resolved into distance contributions from a small set of shortest path ring structures. As expected the first peak is associated with three membered rings. The Voronoi tessellation of a system of particles can be described as a decomposition of space into a packing of tetrahedra, with atoms arranged at the vertices of the tetrahedra. The edges of the tetrahedra represent bonds between nearest neighbours. As each face of a tetrahedron is a three membered ring, every bond between Voronoi nearest neighbours will be part of a three membered ring

163 g(r) r (a) 2 g(r) r (b) Figure 4.3: (a) Radial distribution function, g(r), of the hard sphere fluid at selected densities.(b) Close up view of second peak of radial distribution function, g(r), of the hard sphere fluid at selected densities

164 g(r) Type A 6 Type B rgt g(r) r (a) g(r) Type A 6 Type B rgt g(r) r (b) Figure 4.4: (a) Decomposition of the radial distribution function, g(r), of the hard sphere fluid at a number density of (b) Close up of second peak of decomposed radial distribution function, g(r), of the hard sphere fluid at a number density of

165 g(r) Type A 6 Type B rgt g(r) r Figure 4.5: Close up of second peak of decomposed radial distribution function, g(r), of the hard sphere fluid at a number density of In the modified Voronoi definition of a nearest neighbour we attempt to merge four tetrahedra that are arranged about a common edge and hence form an octahedral arrangement of atoms. The long diagonal of the octahedron is the edge common to all four tetrahedra. This merger is equivalent to breaking the nearest neighbour bond between the atoms forming the common edge that the tetrahedra share. In general it is found that this nearest neighbour distance is larger than the mean neighbour distance. This is reflected in the contribution to the g(r) of four-membered rings. Distances about the first minima are mainly associated with four membered rings, while the second peak of the g(r) is essentially due to different distances across two larger ring structures. It can be seen that the small peak due to four-membered rings, which cannot be observed in the full g(r) is clearly resolved using the ring analysis methods. This is particularly useful for identifying whether a particular system has partially crystallised as this peak is often very small in crystallising systems, and so it can be hard to detect its presence when examining the total pair correlation function. The second peak of the radial distribution function g(r) is composed of two main contributions. The lower r contribution comes from both the 5-membered rings and 1-165

166 one of the two distances associated with the 6 membered rings (Type A distances). The higher r contribution is due to distances across the 6 membered rings (Type B distances) i.e. the triplets of atoms consisting of the central atom of the 6-membered ring and an antipodal pair of atoms of the ring. In a crystal these three atoms are collinear. The contribution of each ring structure to the radial distribution function is shown as a function of density in Figure 4.6-Figure 4.8. The four-membered contribution is roughly gaussian in character with a slight skewness towards larger distances. The variance µ 2 and fisher skewness γ 1 and kurtosis γ 2 of the (normalised) four membered ring contribution to the g(r) is shown as a function of density in Figure 4.9. The fisher skewness and kurtosis are defined as, γ 1 = µ 3 /(µ 2 ) 3/2 γ 2 = µ 4 /(µ 2 ) 2-3 The symbols µ 3 and µ 4 represent the central third and fourth moments of the distribution. A positive kurtosis implies a distribution with a high peak while a negative value indicates a flat-topped curve. A gaussian distribution has a kurtosis of zero and hence the kurtosis is often called the non-gaussian parameter

167 g(r) r (a) 0.3 g(r) r (b) Figure 4.6: (a) Three membered ring contributions to the total g(r). (b) Four membered ring contribution to the total g(r)

168 g(r) r (a) 0.14 g(r) r (b) Figure 4.7: (a) Five membered ring contribution to the total g(r). (b) Six membered ring Type A distance contribution to the total g(r)

169 g(r) r Figure 4.8: Six membered ring Type B distance contribution to the total g(r). The variance of the four-membered ring g(r) contribution shows a relatively rapid decrease at low density with a slower decrease at high densities. This is a common feature of the variances of all ring g(r) contributions. A gaussian fit is also shown in Figure 4.10 at the low density of It can be seen that due to the positive skewness of the distribution the mean value of the gaussian fit is shifted to higher r. The normalised 4-membered ring g(r) also has a flatter shape than gaussian, which is reflected in the positive value of the fisher kurtosis. At high densities near freezing a small shoulder appears on the low r side of the fourmembered ring g(r) distribution. This we attribute to the modified Voronoi method of defining neighbours. Within the Voronoi definition all octahedral arrangements of atoms, as defined by 4-sided Voronoi faces are deleted, if possible. In some cases we will have octahedral arrangements that are closer in form to those say in a bcc crystal, Figure

170 Moments variance skewness kurtosis density Figure 4.9: Variance, Fisher skewness and Fisher kurtosis of normalized 4- membered ring g(r) contribution as a function of density. The dashed lines are only a guide to the eye data g(r) fit r Figure 4.10: Gaussian fit to 4-membered ring contribution to g(r) at a reduced density of

171 (a) (b) Figure Comparison of different packings of 4 tetrahedra about a common edge ( octahedral packings as described in Chapter 2) in (a) a facecentred cubic lattice and a body-centred cubic lattice. Thus they will have relatively short diagonal distances and their deletion will result in a four-membered ring. Assuming the spheres are touching, then the distance between the non-bonded atoms (i and j) appearing in Figure 4.11 is 2 σ for the fcc octahedron, where σ is the hard sphere diameter, and 2/3 σ for the bcc octahedron. In a bcc lattice there are two first nearest distances with the larger distance being that between the non-bonded atoms shown here. If a maximum area of face deletion is set then this problem will be avoided and indeed when this is done, the small shoulder disappears, Fig 12. The low r shoulder in the distribution gives an indication of the small number of bcc-like octahedral arrangements of atoms near freezing and hence indicates that atoms in a locally bcc environment are rare fluctuations in the fluid near freezing

172 g(r) original modified r Figure 4.12: 4-membered ring component of g(r) after inclusion of a maximum Voronoi face area (0.05 in reduced units) for deletion of Voronoi nearest neighbour bonds. The five-membered ring contributions to the g(r) are shown in Figure 4.7(a) at a number of fluid densities. This distribution has a gaussian shape with a skewness that increases with density. Note that the position of the minimum distance will correspond to approximately the maximum first neighbour distance. This is due to the definition of the five-membered rings the antipodal pairs of atoms whose separation contributes to this distribution are by definition second nearest neighbours. Thus as the density decreases and the range of nearest neighbour distances with it we expect the effect of a minimum distance to become more pronounced. The contribution to the g(r) of Type A pairs that are part of a six-membered ring is shown in Figure 4.7(b). The contribution of these distances is much smaller than for the six membered ring but it must be recalled that we ignore any pairs that are also part of a five-membered ring, to avoid double counting. In any case we can see that in both cases the contributions are due to the same arrangement of atoms, shown in Figure The shape of these two distributions is remarkably similar and can be seen by comparing Figure 4.7(a) and Figure 4.7(b)

173 Figure 4.13: Configuration of four hard spheres that give rise to the second nearest neighbour distances (one example is the distance between the spheres i and j in the figure) across a five-membered ring and a six-membered ring. The contribution to the g(r) of antipodal pairs of atoms that are part of a sixmembered ring (Type B pairs) is shown in Figure 4.8. The peak height is shifted to larger distances compared to the two other distributions reflecting the different topological definition of these pairs. The minimum distance is roughly constant and near freezing is equal to about 3σ. This is equal to the distance between second nearest neighbours of a six-membered ring of spheres when all are in contact. Unlike the other distance distributions there is a qualitative change in the shape of this distribution with density. At low densities the shape is distinctly gaussian, while at densities above 0.80 the kurtosis markedly increases from zero, displaying a very rapid increase from its minimum value. In particular note the significant skewness towards distance around 2σ which becomes particularly pronounced as the system approaches the freezing density. The mean, variance, skewness and kurtosis of the normalised ring g(r) contributions are shown in Figure 4.14 to Figure The distributions are normalised so that the area under each curve is equal to unity. This is done so that qualitative changes in the shape of different ring distributions can be compared. We do not wish to attach to much importance to the actual values, particularly the higher moments, but will more focus on qualitative trends with density. The variance of each distribution decreases with density with a particular rapid decrease at low densities. Interestingly the 1-173

174 variances seem to approach a common value at the freezing density. The skewness of the 5 and 6 membered ring distributions is always positive, indicating a preference for larger distances than the mean, a result consistent with a minimum possible distance imposed by the hard sphere interactions. The skewness of the 4-membered rings decreases with density, and at high densities becomes negative. This has already been remarked upon above and we take this to indicate a very small increase in the number of bcc-like octahedral arrangements of atoms. The kurtosis of each distribution is a roughly increasing and generally positive function of density. The relatively small value for the 6-membered ring distributions at densities less than 0.70 indicates a qualitative change as the system approaches the freezing point cf. Figure 4.7(a)-(b) Type A 6 Type B Mean density Figure 4.14: The mean of the normalised ring g(r) contributions at a number of fluid densities

175 Variance Type A 6 Type B density Figure 4.15: The variance of the normalised ring g(r) contributions at fluid densities. Skewness Type A 6 Type B density Figure 4.16: The Fisher kurtosis of the normalised ring g(r) contributions at fluid densities

176 0.6 Kurtosis Type A 6 Type B density Figure 4.17: The Fisher kurtosis of the normalised ring g(r) contributions at fluid densities. The variation of the average ring distance with density is shown in Figure The mean value displays power law behaviour as a function of density with an average coefficient of Taking into account the skewness of the distributions we assert that this reflects on overall behaviour of distances scaling as the cube of the density. A power law fit to the peak position of each ring g(r) distribution versus density was also performed and the results again show reasonable power law behaviour with an average exponent of -0.33, Figure The results are summarised in Table 4.2. In Figure 4.20 the peak position scaled by the cube of the density is plotted as a function of density and it can be seen that the coefficients vary by less than 5% with density. The average values are summarised in Table 4.2. If the peak position of the Type B six-membered ring distance is calculated from this data at a density equal to the random close-packed limit (ρ 1.22) then it is found to be equal to 1.99σ

177 average ring distance A 6B ρ 4 Figure 4.18: Value of mean ring distance versus density Peak Position Type A 6 Type B density Figure 4.19: Peak position of each ring g(r) distribution for various ring sizes versus the density. The lines are only a guide to the eye

178 Peak Position/ -1/ Type A 6 Type B density Figure 4.20: Scaled Peak position of each ring g(r) distribution for various ring sizes versus the density ri/r Type A 6 Type B density Figure 4.21: Scaled average distance between antipodal pairs of each ring size versus density

179 Table 4.2: Parameters for power law fits to peak and average ring distances versus density as well as average value of the peak position when scaled by the density. All distances are in units of σ. Distance 4 5 6A 6B Peak Position Coefficient Power Average Ring Distance Coefficient Power Scaled Peak Position Coefficient Power That each of these distance scales in such a simple manner with the density is a startling result given that particularly at low densities the influence of sphere packing constraints will be minimal. These results indicate that as far as the average ring distances are concerned these simply scale with the density, and it is the change in the moments of these distributions, that is, the reduction in their variance and increasing sharpness (or kurtosis) that characterises the observed changes in the pair correlation function with density. In order to investigate this scaling behaviour further the average ring distances were scaled versus the mean nearest neighbour distance (r 1 ), Figure It can be seen that there appears to be a simple relation between the average ring distance and the density. The mean values of the scaled ring distances are shown in Table 4.3 and are compared with a number of ideal values that assumes all hard spheres are in contact and r 1 = σ. The configuration of hard spheres for which these ideal values apply are described later in the this chapter

180 Table 4.3: Mean scaled value of ring distances and ideal values for regular arrangements of hard spheres at contact. Ring Type Mean value (r/r 1 ) Ideal Values ( 2) ( (8/3)) 6 A ( 3) 6 B ( 4) Decomposition of Bond Angle Distribution The bond angle distribution at various densities is shown in Figure In this work a bond angle is defined as the angle formed between a hard sphere and two of its nearest neighbours. It can be seen that at low densities the distribution of bond angles is very flat. At higher densities two clear peaks emerge, one at around 60º and another at around 109º. At densities near freezing a hump also appears at higher bond angles, at around 140º. The decomposition of the bond angle distribution into bond angle contributions from different size rings is shown in Figure 4.23 and Figure 4.24, for two different fluid densities, 0.6 and It can be seen that the qualitative form of each individual distribution is similar in both cases. The difference between the two distributions is mainly due to the more restricted range of bond angles for each ring size, as the density increases. The average bond angle as a function of density for each of the defined ring bond angles is shown in Figure Remarkably the average bond angle for each distribution is practically independent of density, with the exception of the bond angle associated with triplets of atoms across a six-membered ring, Table 4.4. The average value is also very close to the values they would have for a planar n-sided polygon. The value of exactly 60 for three membered rings is to be expected as three atoms define a plane. Note that for the Type 3 6-membered ring bond angle the average 1-180

181 value will be underestimated. This is because for any triplet of atoms there are two possible bond angles formed, the second one is equal to 360 minus the smaller one. As there is no way of distinguishing between these two angles the smaller one is always chosen. Thus if the bond angle fluctuates about 180 the distribution of bond angles will not be distributed about 180 but will be skewed towards smaller angles. Hence the mean bond angle will be shifted to smaller angles. Overall the data suggest that deviations of the ring structures from planarity are small at all densities and decrease as the freezing density is approached. The angles and distances between members of a ring are calculated such that if a particular distance or bond angle is associated with a smaller ring then it is excluded from the sum. If the averages are calculated over all rings then a slightly different picture emerges. In general the average bond angles is less than for the first case, as the smaller angles associated with the smaller rings are also included in the average. Nevertheless the differences are still small e.g. the average bond angle of a fivemembered ring varies from 98 to 103 over the studied fluid range. This more clearly suggests the non-planarity of the rings in the fluid state and in Figure 4.26 we show the average absolute torsional angle of the rings as a function of density. The significant planarity of the six-membered rings in the crystal compared to the fluid is the most significant geometric difference between these states. A comparison of the two ways of calculating the mean bond angles indicates that the rings for which there are no overlaps with smaller rings are significantly more planar than average. The increasing overlap between the six-membered rings with density is shown in Figure In the metastable fluid the data indicates that on average each sixmembered ring overlaps with two others. This does not necessarily indicate that there is a growing correlation length in the fluid as it approaches the freezing density as the lack of planarity will militate against this. But the data does indicate a degree of decurving i.e. increasing planarity that may at first lead to an increasing degree of local orientational ordering in the fluid state

182 P(q) Bond Angle (q) Figure 4.22: Total bond angle distribution at different fluid densities total sum of rings Type 1 6 Type 2 6 Type 3 P(q) q Figure 4.23: Decomposition of bond angle distribution into different ring contributions at a fluid density of

183 0.018 P(q) total sum of rings Type 1 6 Type 2 6 Type Figure 4.24: Decomposition of bond angle distribution into different ring contributions at a fluid density of q Type 1 6 Type 2 6 Type <qi> density Figure 4.25: Average bond angle versus density for various sized rings. The average values are for bond angles that do not overlap with a bond angle associated with a smaller ring

184 Table 4.4: Average ring bond angle and percentage increase from a density of 0.20 to a density near freezing of The bond angles are those defined in Chapter 2. Note that two of the bond angles associated with the six-membered rings - Type 2 and Type 3 are angles formed between the central atom of the ring and the atoms about the ring. Ring Size Average Bond Angle % Increase Ideal Planar Value Type Type Type <ti> density Figure 4.26: The average absolute torsional angle formed by adjacent members of a ring as a function of density

185 fraction of overlaps density Figure 4.27: The fraction of overlaps between different six-membered rings, measured using the overlap between the internal and external angles of the rings. This fraction is equal to 1.0 in the crystal. Figure 4.34 displays the variation of the distribution of six-membered ring bond angles (Type 3) as a function of density. It can be seen that there is a significant increase in both the area of the distribution and a strong shift towards 180º, that is, towards a collinear arrangements of triplets of atoms. Included for comparison is the bond angle distribution for both a metastable fluid and crystal at a density of Bernal 1 observed a preponderance of triplets of nearly collinear atoms in his random packings of hard spheres. These triplets of atoms could occasionally be seen to link up to form chains containing up to six or seven roughly collinear atoms. We propose here that these triplets are associated with those observed by Bernal but they themselves are associated, although not exclusively, with 6-membered rings. Their observation at high densities, where the crystal is the stable phase, is indicative of local environments that are topologically similar to that in a crystal but geometrically disordered

186 The fraction of bond angles decomposed into ring contributions is shown in Figure The fraction increases steadily with density and hence the ring analysis can successfully resolve most of the bond angles into contributions from rings of various sizes. This is important as it shows that triplet configurations of bonded atoms can be associated with a small number of structural units. The ring structures are a general way of describing and quantifying many body correlations between atoms (Type 1) 6 (Type 2) 6 (Type 3) fraction density Figure 4.28: Fraction of ring bond angles relative to the total number of bond angles observed. The near completeness of the description for local three body correlations as described by the bond angle distribution - is a good example of its utility. The distribution of bond angles for all rings is shown in Figure 4.29 to Figure A significant skewness in the 3-ring distribution can be observed, particularly at high densities. This is not surprising, as there is a minimum bond angle associated with a triplet of bonded atoms that increases with density due to sphere packing constraints. The 4 and 5 ring distributions on the other hand are well approximated by a gaussian, particularly the former. It can be seen that the main qualitative change in the total bond angle distribution is due to a decrease in the variance of all distributions and an increase in the contribution from six-membered rings

187 0.014 P(q) q Figure 4.29: Distribution of bond angle contributions from 3-membered rings P(q) q Figure 4.30: Distribution of bond angle contributions from 4-membered rings

188 P(q) q Figure 4.31: Distribution of bond angle contributions from 5-membered rings P(q) q Figure 4.32: Distribution of bond angle contributions from Type 1 bond angles of 6-membered rings

189 P(θ) θ Figure 4.33: Distribution of bond angle contributions from Type 2 bond angles of 6-membered rings P(θ) θ Figure 4.34: Distribution of bond angle contributions from Type 3 bond angles of 6-membered rings

190 4.3.4 High Density Fluid: Shoulder and Split Peak in g(r) It has long been observed that the radial distribution function for many model glasses and dense metastable fluids exhibits a split second peak 2 as can be seen in Fig 21 for the dense hard sphere fluid. This has been observed in hard-sphere 2, soft sphere 3 and Lennard Jones 4 systems as well as in nearly hard-sphere colloidal systems 5. Many binary metallic glasses also show this behaviour 6 which has led to the use of binary hard sphere systems as a simple model of the structure of these glasses 7, 8. The splitting of the second peak becomes more pronounced as the fluid enters the glass phase and has been used in the past as a structural indicator of the formation of a glass phase 4. The relative heights of the two peaks vary depending on the particular system and the method of preparation 5. In dense random packings of hard spheres (DRHS) it is observed that the radial distribution function has a double peak structure at approximately r/σ 3 and r/σ 2 2, where σ is the hard-core diameter. An early explanation of the structural origin of this split second peak, due to Bennet 9, is that the first distance is that between the nonbonded pair of spheres in a planar configuration of two equilateral hard bonded triangles and the second distance is due to a collinear arrangement of three hard spheres in contact, Figure In some work a peak is also observed at a distance of r/σ (8/3) 9 which is equal to the distance between the non-bonded hard spheres in face-sharing regular tetrahedra, Figure 4.35(c). In a regular tetrahedron all sides have equal length. The absence of a peak at a distance of r/σ 2, Figure 4.35(d) indicative of close packed crystalline order, is also often used as an indicator that the fluid has no crystalline regions. The four hard sphere configurations described above can be related to the distance across the different ring structures defined in this work, as was done in Table 4.3. In particular 5-membered rings will be commonly associated with a packing of 5 tetrahedra about a common edge or bond, hence their identification with Bennet s tetrahedral arrangement 9. As the dominant local coordination is 5, as reflected in the high fraction of Voronoi pentagonal faces, many of the 5-membered rings will be due to this local packing arrangement

191 (a) (b) (c) (d) Figure 4.35: Configurations of atoms leading to the split second peak of a dense fluid. All distances between bonded atoms are the same and equal to d, the average interatomic spacing. (a) Distance between two non-bonded spheres sharing a pair of bonded spheres as neighbours; (b) Distance between non-nearest neighbours of a collinear arrangement of atoms; (c) Distance between non-bonded spheres of two face-sharing regular tetrahedra; (d) Distance between spheres in a square arrangement of spheres, found in the (100) planes of the fcc lattice

192 It can be seen from Table 4.3 that there is quite reasonable agreement between the scaled average values and the ideal values expect for the Type B distance across a sixmembered ring. It is the peak position rather than the average that should be used for comparison in this case, as discussed later. The reduction in the average distance from 2σ is due to the fact that the six-membered rings are not flat but buckled. Note that if we assume the average bond angle was 144 and all distances are equal then we recover the distance This angle is that formed between the central atom and ring atoms of the buckled six-membered rings associated with an atom in a locally hexagonal or twisted icosahedral arrangement (see Chapter 2 for more detail). Tsumuraya and Watanabe 10 have also investigated the origin of the double peak structure of the g(r) of a metastable fluid and suggested that atomic distances associated with atoms in an icosahedral environment where responsible for producing the second split peak. This was investigated by calculating the local g(r) of an atom as a function of its Voronoi polyhedral type. They found that the local g(r) of atoms in an icosahedral environment showed the sharpest splitting of the g(r) compared to any other polyhedral type. An icosahedron can be regarded as a packing of twenty tetrahedra about a common point, with atoms decorating the corners of the tetrahedra. Each tetrahedron edge is shared by five tetrahedra. If each of the neighbours of the icosahedra is assumed to be touching the central atom then a distance of 1.05 separates them. Tsumuraya and Watanabe 10 identified the first peak with the distance from a central atom to an atom located in the pocket formed by one of the tetrahedra of the icosahedra and the second peak with the distance to an atom located on top of the first nearest neighbour and in line with the central atom. Truskett 11 has suggested that the appearance of a shoulder in the dense liquid near freezing is a structural indicator of the approaching phase transition. In their analysis of the 2D hard disc system (as well as a 3D hard sphere system) they associate the shoulder, spanning the range r/d = 3 to r/d 1.95, with the appearance of a distinct next-nearest neighbour shell, where the distances correspond to those in the triangular lattice at close packing. It is suggested that this is an indicator of the appearance of ordered hexagonal domains in the fluid, which grow in size as the freezing transition is approached. By direct inspection of 2D configurations they come to the conclusion 1-192

193 that the shoulder in the g(r) corresponds to the appearance of a distinct structural motif associated with the crystalline triangular lattice, which they describe as a fourparticle hexagonally close packed configuration Figure 4.35 (a). This qualitative interpretation of the structural origin of the shoulder in the g(r), and by implication the split second peak that evolves as the density is increased, is at odds with the usual explanation, in which the split peak is an indicator of the formation of a glassy state. In Figure 4.36 the pair correlation function at five different densities is shown, three within the equilibrium fluid region, another near the freezing density of and the last within the coexistence region, a metastable fluid at a reduced density of 1.0. It can be seen that for densities equal to or greater than 0.90 a distinct shoulder appears on the low r side of the second peak of the g(r) and as the densities is increase this evolves into a split peak. This has also been noted by recently by Truskett et al 11 in both 2D and 3D hard sphere systems. A decomposition of the g(r) of the metastable fluid is shown in Figure It can be seen that the first split peak is associated with both five-membered rings and Type A distances in six-membered rings. The second split peak is due to Type B distances. In this context we can now interpret our results. The first peak is due to a combination of two different distances related to local structural arrangements that are important at high densities: dense local pentagonal arrangements and the appearance of local compact hexagonal domains. The second peak is due to the distance across these hexagonal domains. The work of Bennet is consistent with our results, but their configurations of atoms can be seen to be a decomposition of two different ring structures i.e. 5 and 6 membered rings, cf. Figure 8. The appearance of a distance of (8/3) is due to pentagonal arrangements of atoms while the distance of 3 is due to the six-membered rings. The work of Tsumuraya and Watanabe can also be interpreted in the light of our results. As noted in Chapter 2 an icosahedron is at the centre of 12 six-membered rings of the type defined in this work. It is the preponderance of 5-membered and

194 g(r) r Figure 4.36: Radial distribution function of hard sphere fluid at densities near and above the freezing density g(r) Type A 6 Type B sum of ring g(r)'s g(r) r Figure 4.37: Decomposition of g(r) of metastable fluid (ρ = 1.00) into ring contributions

195 membered rings associated with local icosahedral arrangements that produce the welldefined split peak observed by Tsumuraya. The emergence of the split peak as the density increases towards freezing is hence due to two contributing factors. First the variance of each of the distributions associated with the second peak decreases due to the increasing severity of sphere packing constraints. This causes the split peak to emerge first as a shoulder in the dense equilibrium fluid and as the variance of the distributions decreases the split peak can be clearly discerned. The second contributing factor is the qualitative change in the distribution of distances associated with the 5 and 6 rings. This is particularly marked for the second peak (Type B distances) where at higher densities the distribution rises rapidly from zero to its maximum value at a distance of around 2σ. The kurtosis of these distributions oscillates around zero until the density reaches approximately and then increases markedly. The change in the relative peak heights and position of the first peak of the split peak observed in different systems or under different preparation conditions 5 can also be interpreted as due to the differing contribution of 5 and 6 membered rings to the split peak. 4.4 Summary The structure of a hard sphere fluid has been investigated using the ring analysis methods described in Chapter 2. This has shown that the short and medium range structure can be successfully decomposed into contributions from a small number of ring structures. The geometric changes that occur as a function of density can then be investigated using these particular structural motifs. Qualitative changes in the average geometry of the system that may occur with density can be discerned above the noise generated by the high degree of randomness in the fluid state. We find that the average bond angle of the ring structures is practically independent of density with the exception of one bond angle. This bond angle is the angle across the hexagonal or six-membered rings (Type 3) and corresponds to a triplet of atoms 1-195

196 that is collinear in the crystal. The change in the bond angle distribution with density for the hard sphere fluid is predominately determined by a decrease in the range of bond angles as the sphere packing constraints becomes increasingly restrictive. With the exception of the angle across a six-membered ring (and the three-membered ring bond angle distribution) the distribution of each defined bond angle displays a reasonably gaussian profile. The skewness to large angles and the increase in the mean with density of the bond angles associated with this angle across a sixmembered ring strongly suggest that this particular ring structure is quantifying the increasing predominance of collinear triplets of atoms. Due to the preponderance of these rings at high density it would not be surprising to find chains of several atoms arranged roughly in a line, as originally noted by Bernal in his ball and stick models 1. A further interesting point to note is that the average bond angle of a 4-membered ring is very close to 90 and the scaled average distance across the ring is very close to 2. Recall that the modified Voronoi method we use to define the bonds between spheres describes the structure of the fluid as a packing of tetrahedra and octahedra. Due to the definition of the rings employed here these octahedra will be exclusively associated with the 4-membered rings. The finding that the average geometry of these rings conforms to the octahedral ideal is unexpected but welcome support for our particular decomposition method - the octahedra we define as simply four tetrahedra sharing a common edge have at least some of the same average properties as a perfect octahedra. We find that the emergence of a split peak in the radial distribution function at high density is a combination of both a simple decrease in the range of packing environments that at lower densities buries it in the width of the distributions its is associated with and another feature that occurs only at high densities: an increasing preference for triplets of atoms (at the least) to occur in approximate collinear chains in contact. The observation that the peak height scales with density in such a way that it is very close to the random close packing value of 2 σ is a very satisfying result but may be fortuitous. Further work at higher densities would help to confirm this result. As was noted before the distribution of distances across a 4-membered shows a small low r bump at densities near freezing. This may indicate a very small degree of body

197 centred cubic ordering as the freezing density is approached. But the small contribution of these arrangements such they are not important in the hard sphere fluid e.g. as possible precursor structures. The six-membered rings defined here show the most significant geometrical changes with density in terms of their angular behaviour. In particular the results indicates that these rings flatten out as the freezing density is approached. As these six-membered rings are also the structural motif for close-packed planes, in which they are on average planar it appears natural to regard the change in their shape as indicators of the onset of crystalline order. We suggest that the observed split peak in the hard sphere system is due to a combination of two structural signatures: that of the dense locally pentagonal liquid state and that due to precursor structures associated with local hexagonal domains that become more crystal-like i.e. flatten out as the system approaches the freezing density. These six-membered rings are the 3D structural analogue of the precursor structures (which at the same time are no identified with a particular motif) identified by Truskett et al 11 in the 2D hard disc fluid

198 4.5 Bibliography 1 J. D. Bernal, Proc. R. Soc. London A 280, 299 (1964). 2 J. L. Finney, Proc. R. Soc. London A 319, 479 (1970). 3 Y. Hiwatari and T. Saito, J. Chem. Phys. 81, 6044 (1984). 4 A. Rahman, M. J. Mandell, and J. P. McTague, J. Chem. Phys. 64 (1976). 5 I. K. Snook, W. vanmegen, and P. Pusey, Phys. Rev. A 1, 1 (1991). 6 G. S. Cargill, J. Appl. Phys. 41, 2248 (1970). 7 P. Chaudhari, B. C. Giessen, and D. Turnbull, Sci.Am. 242, 98 (1980). 8 A. S. Clarke and J. D. Wiley, Phys. Rev. B 35, 7350 (1987). 9 C. H. Bennet, J. Appl. Phys. 43, 2727 (1972). 10 K. Tsumuraya and M. S. Watanabe, J. Chem. Phys. 92, 4983 (1990). 11 T. Truskett, S. Torquato, S. Sastry, P. G. Debenedetti, and F. H. Stillinger, Phys. Rev. E 58, 3083 (1998)

199 Chapter 5 Hard Sphere Crystallisation 5.1 Introduction The simplest system in which a fluid-solid transition is observed to occur is the hard sphere system. The only interaction between hard spheres is a hard-core repulsion and hence its equilibrium properties are purely determined by entropy or equivalently by its statistical geometry 1. Likewise, the crystallization of the hard sphere system is completely determined by entropy, unlike other systems in which energetic effects also play a role, e.g. the evolution of latent heat during crystallization. Hence by studying the crystallization of a hard sphere system the role of packing effects on the crystallization process can be singled out. Hard sphere systems can also be experimentally realized in the form of colloidal suspensions 2 and hence the results of computer simulation and experiment can be compared. From a simulation standpoint the hard sphere system is unusual in that the simulation results are in principle an exact description of the system while the experimental models are an approximation to this ideal (typically due to the presence of very short range attractive forces, a small degree of polydispersity and the presence of a background solvent). This is a reversal of the usual situation in which the computer simulation of a real system is always fraught with the problem that it may not realistically model the material. A main reason for the importance of the hard sphere system is its use as a reference system in perturbation theories of the liquid state. Its use is motivated by the observation that the dominant influence on the structure and dynamics of a liquid is the repulsive interactions between particles (or molecules) of the liquid i.e. packing 1-199

200 effects. The crystallization of a liquid is a process driven by packing considerations as this transition is characterized by the development of ordered structures, which in many non-covalent elements are the most efficient possible packing of spheres (particles) - the face-centered or hexagonal cubic lattice. An altogether different but equally remarkable reason to study the hard sphere system is that it is possible to observe crystal nucleation within the coexistence region of the phase diagram. This observation is counter to previous work that suggests that hard sphere systems are hard to crystallize and can form a glass phase readily 3, 4. The opposite is true; the only condition is that sufficiently large systems of the order of 10 4 particles must be studied. Williams 5 has also found that it is only for reduced densities of 1.03 or greater that nucleation can be observed and in this case at least particles are required. We know of no other common model system such as the Morse, soft sphere or Lennard-Jones systems in which nucleation has been observed at coexistence. In fact ten Wolde and Frenkel have estimated that for the LJ system nucleation is impossible to study using conventional molecular dynamics methods with present computing power 6. The main aim of this chapter is to investigate how the stable solid phase forms in a hard sphere fluid compressed to densities above the freezing density. In a hard sphere system the density of the system is the only thermodynamically relevant control parameter. The form of the nuclei that appear during the crystallization process is investigated using the structural analysis methods described in previous chapters that were ultimately developed for this purpose. Our main focus is on the very early stages of nucleation before significant growth has occurred, which is currently experimentally inaccessible. A related aim is to investigate the structural environment around the nuclei after their formation in order to ascertain the role, if any, of precursor structures in the dense fluid on the formation of the nuclei. The crystallization behavior of a hard sphere system compressed to a number of densities on either side of the melting density was analyzed using the methods described in previous chapters. We focused particularly on nucleation near the melting density with a more limited analysis of crystallization within the coexistence region

201 Ideally we would like to have one nucleus per simulation but as the nucleation rate is a sensitive function of the degree of undercooling it was found to be impractical to accomplish this 5. The simulations either crystallized with more than one nucleus or remained metastable depending on the density. 5.2 Simulation Details A number of simulations were performed at a reduced number density of 1.05 (φ 0.549) near the melting density (ρ m = 1.044, φ = 0.546) using particles. This system size was found to be sufficient to clearly observe the appearance of compact nuclei in the fluid as well as follow their subsequent growth. The evolution of the system from the initial density quenched fluid to a defective close packed crystal was followed for at least 300τ. Standard hard sphere simulation techniques were used and the initial configurations were generated by the compression scheme of Williams 5. A single simulation, using particles, was also performed at a number density of 1.03 within the coexistence region in order to compare with the melting density results. The reduced temperature, kt, is equal to unity. 5.3 Thermodynamic Changes The change in the compressibility factor, Z, was calculated as a function of time for each simulation. The radial distribution function g(r) was also calculated from individual stored configurations. The fraction of crystal identified by the planar graph method, as well as the fraction of solid-like particles found using the spherical harmonic method, were also calculated from stored particle configurations. Unless otherwise stated the results are for the six different simulations at a number density of

202 5.3.1 Compressibility Factor The change in the compressibility factor (Z) of the system as the metastable fluid crystallizes is shown as a function of time in Figure 5.1 and Figure 5.2 for all seven runs. As these systems are simulated at constant temperature (T) and density, the pressure (P) is trivially related to the compressibility factor, P = Z(ρkT), The second simulation (Run 2) was produced by restarting a hard sphere simulation using the coordinates and velocities of simulation Run 1 after 100 t. This was done in order to assess the effect of small perturbations (introduced due to rounding errors in storing the positions and momentum) on the subsequent evolution of the system. The final pressure for this simulation is nearly identical to that of the first simulation (Run 1) but there are still small deviations between them. The compressibility data from different runs has also been averaged (excluding Run 2) and this is shown in Figure 5.2. In a crystallization experiment data is gathered over multiple nucleation events and hence we expect that the experimental data will be qualitatively similar to averaging over our different runs. Four main stages can be clearly identified in each simulation. The first stage is a rapid decrease in the compressibility factor for the first 15τ. This is followed by a second stage in which there is a slow decrease in the compressibility factor or pressure that lasts between τ (relative to the start of the simulation). It is during this stage that the first nuclei can be clearly discerned (see later, section 5.7.2). Differences in the behavior of the compressibility factor with time between simulations cannot be clearly discerned within the first 50 τ. A transient time, in which the compressibility remains constant, cannot be clearly identified in any of the simulations. Rather there is a slow drop in the compressibility compared to later stages. This is in contrast to the assumptions of classical nucleation theory 7 but similar to what has been observed in other computer simulations of crystallization 8. A nucleation time can be identified by noting the time at which the compressibility differs from a straight line fit to the second stage compressibility factor data

203 A third growth stage follows this nucleation stage in which there is a much faster drop in the compressibility with time compared to the second stage. The time at which this transition from the initial appearance of nuclei to their subsequent growth occurs can be characterized by an induction time that varies between 75 τ and 125 τ (relative to the beginning of the simulation t = 0) and depending on the simulation, Table 5.1. This time was determined by the intersection of two linear fits to the early (stage 2) and later (stage 3) time compressibility data. The method employed is the same as that used to monitor crystallization in hard-sphere colloidal systems 9. The time at which the compressibility differs noticeably from a linear fit to the early time behaviour is noted in brackets in the Stage 2 column of Table 5.1. During this stage other smaller nuclei sometimes appear. A fourth annealing or coarsening stage is also observed during which the compressibility factor changes very slowly with time. This occurs at a time, after the beginning of the simulation, which varies between 170 τ and 200 τ. In this last stage the system has predominantly crystallized and changes in the compressibility factor with time are the smallest for the entire simulation Z time (τ) Figure 5.1: Compressibility factor (Z) versus time after density quench for a number of different simulation runs near the melting density (ρ = 1.05)

204 Run 3 Run 5 Run 6 mean Z time (τ) Figure 5.2: Compressibility factor (Z) versus time after density quench for selected simulation runs near the melting density (ρ = 1.05) and the mean over all runs. Table 5.1: Times after density quench at which different stages in crystallization process can be discerned based on qualitative behavior of compressibility factor with time. All runs are at a density of 1.05 except the coexistence run which is at a density of Run Stage 1 (Initial Z Stage 2 (Induction Time Stage 3 (Slower Z Stage 4 (Annealing) drop) -Nuclei Formation) change) 1 15 (50) (50) (60) (--) (60) (--) (90) Coexistence 15 (75) Mean 15 (60)

205 In Figure 5.3 the different stages for two different runs have been highlighted. These are the runs with the smallest and largest compressibility factor at the end of the simulation, respectively. The simulations can be grouped into three sets depending on their final compressibility at the end of the simulation as well as from their qualitative behavior of Z with time. The members of the sets are detailed in Table 5.2. Note that for the two simulation runs 5 and 7 the change in the compressibility factor with time is nearly identical, with a much faster drop in the compressibility factor with time during the growth stage than the other simulation runs. Likewise the change in the compressibility factor with time for Runs 1-3 is similar up to 150 τ, while Runs 4 and 6 show similar behavior up to the same time (~140τ). The qualitative behavior of Run 4 is similar to Runs 1-3 except that the induction time (end of stage 2) is delayed Run 6 Run 7 17 Z time (τ) Figure 5.3: Compressibility versus time for two different crystallization runs. The arrows denote the times (listed in Table 5.1) at which each different stage in the crystallization process starts. For run 6 the time listed in brackets for stage 2 in Table 5.1 is also shown

206 Table 5.2: The final compressibility factor Z at the end of the simulation run (t = 300 τ) and fraction of crystal (as measured by fraction of fcc + hcp ordered and simple defects). Run Final Z Final % Crystal Set A A A B C B Mean Stable Crystal Crystal fraction versus time: Planar Ring Method The variation of the fraction of crystalline particles with time is shown in Figure 5.4 for each simulation. The fraction of crystal is defined as the fraction of particles in a perfect face-centered or hexagonally close packed environment plus those particles in a defect fcc or hcp environment. The defect local environments are those described in Chapter 2 and are observed in the equilibrium crystal. In Figure 5.5 the change in the compressibility relative to the time after the initial equilibration period (15 τ) and the change in the crystal fraction relative to the same reference time have been plotted for Run 4. It can be seen that by appropriate scaling of the axes these two curves fall on top of each other. Hence we obtain an excellent correspondence between the behavior of the compressibility factor and the change in the amount of crystal, which is the cause of the decrease in the former with time. This is a feature of all the simulations. Note that even small changes are faithfully reproduced, as for example the small dip at t = 260. This very close correspondence gives us confidence that our particular measure of crystallinity can accurately monitor 1-206

207 the appearance and growth of the new crystalline phase in our simulations. The correspondence between the behavior of the compressibility factor with time and the fraction of perfectly ordered fcc and hcp particles has also been assessed and it is found that the inclusion of those hard spheres in a defect fcc/hcp environment (those observed in the stable crystal, see Chapter 2 for more details) leads to better results. The sigmoidal shape of the plots of the crystal fraction versus time from these simulation runs is similar to those observed in experimental studies of crystallization e.g. hard sphere colloidal systems 10. The initial slow increase in the crystal fraction is typically identified with an induction time in which nucleation dominates (stage 2). The sharp increase in the fraction of crystal at longer times (stage 3, in this work) is associated with the growth of these nuclei as well as the formation of additional nuclei. The slow change at long times is associated with ripening effects in which large crystallites grow at the expense of smaller ones 9. In order to aid comparison with experiment two times characteristic of the crystallization process are extracted from our data in the same manner as has been done for colloidal hard spheres 9. An induction time, τ i, is obtained by extrapolating a linear fit to the steepest increase in the crystal fraction versus time to zero crystal fraction, in contrast to those obtained from the compressibility factor data. The crystal fraction is assumed to be zero at t = 15τ. A crossover time, characterizing the transition to coarsening behavior, is determined by the intersection of linear fits to the growth (stage 3 in this work) and late time (stage 4 in this work) sections of the crystal fraction data. The results are summarized in Table 5.3, which also shows the times extracted from the average crystal fraction data i.e. averaging over each simulation run

208 Run 1 Run 4 Run 7 crystal fraction time (τ) 1 crystal fraction Run 3 Run 5 Run 6 mean time (τ) Figure 5.4: Crystal fraction as a function of time for selected simulation runs at a density of The data has been separated for clarity

209 δz Z crystal fraction crystal fraction time (τ) Figure 5.5: Relative change in the compressibility factor (Z) and crystal fraction (CF) relative to a time 15τ after the start of the simulation (Run 4). - δz is the difference between the compressibility factor at time t and at a time t = 30τ after the beginning of the simulation. Table 5.3: Times indicating different stages in crystallization process derived from fraction of crystal at these stages (as measured by fraction of fcc + hcp ordered and simple defects). The midpoint time of growth is simply the time halfway between the induction and crossover time. ** This is the mean value for the simulations performed at a density of 1.05, near the melting density. Run Induction % Crystal at Midpoint Crossover % Crystal at Time (τ) Induction of Growth Time (τ) Crossover Coexistence Mean**

210 A comparison of the times in Table 5.1 and Table 5.3 shows a close correspondence between the two different methods of calculating the induction time and the crossover time (start of stage 4 in this work). In general the crossover time is approximately twice the induction time. In all cases the percentage of crystal was approximately 50% or greater at the beginning of the third stage. If it is assumed that the nucleus was spherical then this fraction of crystal corresponds to the case when the nucleus would come into contact with an image of itself due to the periodic boundary conditions used in the simulation. Hence this time could indicate the stage at which a nucleus is either interacting with copies of itself or multiple nuclei have come in contact. This situation is the same as that occurring in coarsening processes except of course in our simulation results there are only a few individual crystals. In fact we find that the merger of nuclei happens at earlier times for simulations at both densities studied. The fraction of crystal at the crossover time suggests that at this time the fluid-solid interfaces of the different nuclei begin to overlap. Although the time and spatial scales in these simulation runs are much smaller than in a nucleation experiment the qualitative features of the change in the fraction of crystal observed in the experimental work are reproduced here. The fraction of crystal of the averaged data set has also been used to extract a time power law for the growth process. We find that the exponent of the power law fit is equal to 2.5 from the crystal fraction data. This compares with an exponent of 3.2 found in hard-sphere colloidal suspensions at a similar volume fraction of φ = The variation in time of the fraction of each type of ordered particle (close-packed or icosahedral-like) was also monitored for the simulations. Particles in a body centered cubic local environment were not observed in any significant fraction. These were monitored using a Voronoi polyhedron approach, and we find that this fraction is at most 2% for a typical run. As a check on the planar graph method, the fraction of icosahedra measured from the Voronoi method was compared with that calculated using the planar graph method. The fraction of icosahedra found using the Voronoi approach is larger than that from the planar graph method (an expected result as the merging of tetrahedra used in our 1-210

211 modified Voronoi approach will change some icosahedral polyhedra) but the qualitative behavior with time is virtually identical for each simulation. It was found that the fraction of particles in an icosahedral environment was at most 1.5% and this maximum fraction was observed at the same time for each simulation, after the first 15τ, see Figure 5.6. Hence a small number of icosahedra are initially observed as a response to the large pressure at the beginning of the simulation (due to the compression scheme used to create the starting configuration) but their number linearly decreases with time. At the end of the simulation the fraction of icosahedral particles is less than 0.1 % and the change in the fraction of these particles with time during the simulation is very similar for all simulations icosahedron fraction Run 3 Run 6 mean time (τ) Figure 5.6: Fraction of hard spheres in an icosahedral environment as function of time. The fraction of particles in a pentagonal bipyramidal local environment (i.e. twisted icosahedron) was also very small, generally between 0.5 and 1.2% but unlike the fraction of icosahedral particles, the quantitative changes with time varied between the simulations Figure 5.7. Unlike the icosahedral particles the fraction of these particles tends to peak at later times and the changes in time depend on the simulation

212 0.014 twisted icosahedron fraction Run 1 Run 3 Run 5 Run time (τ) twisted icosahedron fraction Run 3 Run 4 Run 6 Run time (τ) Figure 5.7: Fraction of hard spheres in a twisted icosahedral environment as function of time

213 The fraction of hard spheres in an fcc environment and the corresponding fraction of those in an hcp environment are shown in Figure 5.8 and Figure 5.9. These environments include the defect ones listed in Chapter 2. For all simulations it is found that at early times the fraction of hard spheres in a hcp environment dominates over the fraction in an fcc environment. The actual difference is small (~2%) but is independent of the simulation, with differences only appearing for t > 100 τ, see Figure For times greater than either 110 τ (Runs 1,3,5) or 160 τ (Runs 4,6,7) the fraction of fcc dominates and the ratio of fcc to the total fraction of crystal increases above fcc fraction Run 1 Run 3 Run 4 Run 5 Run 6 Run time (τ) Figure 5.8: The fraction of hard spheres in an fcc environment as a function of time for different crystallization runs

214 0.35 hcp fraction Run 1 Run 3 Run 4 Run 5 Run 6 Run time (τ) Figure 5.9: Fraction of hard spheres in an hcp environment as a function of time for different crystallization runs stacking probability, α Run 1 Run 3 Run 4 Run 5 Run 6 Run time (τ) Figure 5.10: The stacking probability (ratio of fcc to total number of fcc and hcp particles) as a function of time for different crystallization runs

215 Thus as the stable crystal phase starts to grow rapidly (as judged by the pressure) both the fraction of fcc and hcp particles increases but the former increases at a much faster rate. The ratio of fcc to hcp particles and the relative difference between them also increases with time until the end of the simulation, for each run. There are also significant differences between the simulations after the induction time. A noticeable feature is the predominance of hard spheres arranged in a face-centered cubic environment. In a hard sphere system particles only interact at contact and hence one intuitively expects no overall preference for face-centered particles over hexagonal close packed layers. However in entropy calculations of the relative stability of the face-centered and hexagonal close packed crystal a very small preference for fcc is observed but is of the order of 0.005RT 11 (units). Experiments on colloidal hard sphere systems have also lead to a great deal of controversy over whether the stable hard sphere system has a preference for fcc 12. Comparing the compressibility factor of the system with the fraction of fcc hard spheres at the end of the simulation it can be seen that the runs that are dominantly fcc, also have the lowest pressure. These two runs 5 and 7 also showed the fastest growth of all simulations, Table 5.2. After the growth stage (stage 3) the percentage of hexagonally close packed particles for most simulations (excepting runs 4 and 7) either stays the same or decreases while the fraction of face-centered cubic particles increases. Thus the decrease in the compressibility factor during the last annealing stage is in general a consequence of the increase in the number of face-centered cubic particles, which is either due to the annealing of defect fcc particles or the conversion of hexagonally close packed particles to fcc particles, i.e. the annealing out of stacking faults. This conversion has also been observed during the growth stage in a number of the simulation runs as well. This shows that the spontaneous annealing out of stacking fault defects is possible in the hard sphere crystal. The simulation results can also be grouped into two sets depending on behaviour of the ratio of fcc particles to total crystal. In all cases this fraction is greater than 0.50 at the end of the simulation run. Differences cannot be clearly discerned for time less than approximately 70τ. The simulations can be grouped into two branches at the time 1-215

216 where fcc particles begin to dominate over hcp particles i.e. where the stacking probability shows a preference for fcc over random close packing; for most runs this is at 100τ and for three others this is at 160τ (Runs 4,6 and 7). The variation with time of the fraction of ordered particles as defined by the spherical harmonics method of van Duijneveldt et al 13 (see Chapter 2) has also been calculated and is shown in Figure This is a more local measure of the orientational order in a crystal and is hence less sensitive to the overall crystallinity i.e. the relative orientation of different nuclei. The fraction of crystal predicted by this spherical harmonic method and that using the planar graph method has been compared for each simulation and it is found that both methods give very similar quantitative behaviour. 1 fraction Run 1 Run 3 Run 4 Run 5 Run 6 Run time (τ) Figure 5.11: Crystal fraction predicted from local orientational order parameter

217 0.7 fraction of crystal fcc+hcp all fraction from sph. harmonics time (τ) Figure 5.12: Fraction of crystal predicted from the planar graph method (left axis) and that from the spherical harmonic method for Run 1. The right axis has been shifted and scaled to highlight the similarities in the qualitative behavior of the crystal fraction predicted from the two methods. The fraction of crystal predicted by the latter method is typically only 1-2 % higher than from the spherical harmonic method at early times (< 110 τ) while deviations plateau to about 5% at late times but in the opposite direction. In a computer simulation study of nucleation the measured crystal fraction can easily be sensitive to the order parameter used and hence it seems wise to compare different methods in order to assess the sensitivity of our chosen approach. The fraction of ordered particles that are also regarded as solid-like by the spherical harmonic method of van Duijneveldt et al 13 was monitored during each simulation run. In this way we hoped to obtain some idea of the correspondence between our methods and those of other workers in the field. It was found that practically all (> 97%) of the fcc ordered particles where found to be solid-like, for times greater than 50 τ. The fraction of hcp ordered particles that were also solid-like increased from approximately 70-80% at early times (t > 50 τ) to approximately 100% at a time equal to the induction time

218 However it was observed that only a few percent of the twisted icosahedral ordered particles were identified as solid-like by the spherical harmonic method before the induction time and this only increased to at most 70% by the crossover time. We will show later that this lack of correspondence is because the latter two types of ordered particles, particularly the twisted icosahedral order, appears predominately on the surface of the growing nucleus. Hence many of their neighbours are fluid particles and thus they are only partially orientationally ordered with their neighbours. As the criterion for a solid-like particle, using the spherical harmonic method, is the number of neighbours it is orientationally ordered with, particles on the surface of the crystal will not be included. This we feel emphasizes the importance of dual methods of quantifying the degree of ordering in a system, as what one method may be insensitive to, another may pick up Radial Distribution Function In an experimental investigation of crystallisation the evolution of one or more Bragg peaks of the static structure factor, S(q), is typically used to monitor the formation and growth of the crystal phase 14. An analysis of the peak position, width and area of the main (111) peak can provide information on the lattice spacing, average nucleus dimension and number density of nuclei. No direct comparison with experiment is attempted here due to the poor statistics of this work compared to the experimental situation in which data is collected as an average over thousands of growing crystallites. But nevertheless qualitative changes in the S(q) and the related radial distribution function g(r) that are observed as crystallisation proceeds in the experimental work that may be reflected in the simulation results. The focus on the observation of a single Bragg peak means that the radial distribution function for the crystallising system cannot be obtained experimentally. This is one advantage of computer simulations over experiment. The radial distribution function at various times during the crystallisation simulation for Run 1 is shown in Figure 1-218

219 5.13. The initial dense fluid has a split second peak at distances of r/σ = 1.86 and It can be seen that the double peaked structure of the second peak of the initial fluid evolves into a single peak at r/σ 1.86 with the second peak of the split peak disappearing at the same time. Another peak also appears about the minima in the g(r) at a distance of r/σ 1.54 for times greater than 100τ. This distance corresponds to the lattice spacing and hence the appearance of crystalline cubic order. In the equilibrium crystal this distances would occur at r/σ 1.91 ( 1/ 3 3( 2 / ρ ) ) and r/σ 1.56 ( 1/ 3 2( 2 / ρ ) ), respectively, at a reduced density (ρ) of g(r) r (σ) Figure 5.13: Close up of radial distribution function at various times for simulation Run 1. This suggests that the crystal is formed is at a slightly higher density than the equilibrium value. The effective density of the crystal can be estimated by finding the mean value of the density suggested from the first four peaks of the g(r) at a time of 170τ after the density quench. This gives an estimate of ρ c = 1.14 for the density of the crystal. This is probably an overestimate as a similar analysis of the peaks of the g(r) for the stable crystal (ρ = 1.05) predicts a density of

220 The dominant changes in the peak heights occur at times between 75 τ and 170 τ after the density quench. At this earlier time the height of both sub-peaks of the second split peak are equal in height (for Run 1). As crystallisation proceeds the height of the first peak increases while its position varies little while the height of the second peak decreases and shifts outwards to larger r. Between the times 75 τ and 120 τ other peaks associated with the crystal state appear, in particular the peak associated with (100) planes. 5.4 Ring Statistics Topology The change in the network topology of the crystallising hard sphere system, as defined using the shortest path rings defined in Chapter 2, is shown for a representative simulation run in Figure The results are similar for all simulations near the melting density as well as for the coexistence density simulation. The number of 5-membered rings decreases sharply as the system crystallizes while the number of four membered rings shows a substantial increase as cubic order evolves. These topological changes with time follow the drop in the pressure of the system. In particular, differences between the number per particle of the five and six-membered rings can be correlated with the initial nucleation time listed in brackets in Table

221 5 Number of SP Rings / Atom time (τ) Figure 5.14: Number of SP rings per particle as a function of time for crystallization Run 1. The equilibrium values in the crystal for four, five and six-membered rings are three, zero and four respectively. The change in the number of six-membered rings with time is less pronounced and surprisingly is practically independent of the simulation. Differences between simulations are distinguished by the topological changes of the smaller rings. The number of four-membered rings at the end of the simulation is greatest for Runs 5 and 7 reflecting the higher fraction of particles in a cubic crystalline environment. For all simulations the number of six-membered rings is equal to the number of fivemembered rings after the fast initial equilibration period t ~ 15τ. Differences between simulations runs are also correlated with the initial appearance of the first nucleus (listed in brackets in the stage 1 column of Table 5.1)

222 number of SP rings per atom Run 1 Run 5 Run 6 mean time (τ) Figure 5.15: Number of 6-membered rings per particle for a number of selected simulation runs. The equilibrium face-centered cubic crystal value is four. number of SP rings per atom Run 1 Run 5 Run 6 mean time (τ) Figure 5.16: Number of 4-membered SP rings per particle for a number of selected simulation runs. The equilibrium face-centered cubic crystal value is three

223 4.5 number of SP rings per atom Run 1 Run 5 Run 6 mean time (τ) Figure 5.17: Number of 5-membered SP rings per particle for a number of selected simulation runs. The equilibrium face-centered cubic crystal value is zero. The change from a liquid state, often characterized by local pentagonal or five-fold order, to a crystal is neatly summarized in the above changes in the network topology. Note that the frequency of six-membered rings does not change significantly on crystallization, see Figure 5.15; it is the change in the frequency of four and five membered rings that dominates i.e. the replacement of cubic ordering (here measured as the increase in the number of four membered rings) at the expense of pentagonal ordering, Figure 5.16 and Figure As will be seen it is the change in the geometry, rather than the topology, of the six-membered rings that is most significant, in terms of the evolution of crystalline order Six-Membered Rings The change in the average geometry of the six-membered rings was followed through the crystallisation process. These six-membered rings are the basic structural components of the close packed planes of the fcc or hcp crystal. A close packed plane 1-223

224 can be considered as being an overlapping tiling of the six-membered rings described in this work ( see Chapter 2 for more details). A pivotal issue in crystallisation studies is the question of whether precursor structures in the dense metastable fluid initiate the formation of critical nuclei or whether the formation of a nucleus is due to purely random fluctuations. A high proportion of individual six-membered rings does not by itself imply any ordering in the system. It is the overlap or tiling of these six-membered rings, i.e. their coherence, which suggests the existence of ordered domains in the early stages of crystallisation. These domains may then act as templates for pre-critical nuclei. The benefit of this particular outlook is that a lack of registry between close-packed planes, which will effect any particle centred description of ordering, does not affect our results. In other words in searching for the growth of order in the early stages of crystallisation we are breaking the problem down into two stages. The first stage is to look for close packed planes in the system and then through the use of the planar ring description of local order, investigate the formation of proper registry between these close packed planes. Whether this approach is valid or not of course depends on the existence of close packed planes in the early stages of nucleation. A measure of the coherence of six-membered rings i.e. whether six-membered rings overlap with one another, is provided by the fraction of outer Type 2 bond angles of a given six-membered ring that overlap with the inner Type 1 bond angles of a neighbouring ring, (see Chapter 2 for a fuller discussion). In a crystal this fraction is equal to unity. The fraction of such bond angle overlaps is shown as a function of time for crystallisation run 1 in Figure It can be seen that even at early times the fraction of overlaps is quite large and increases at a linear rate from 30% at the start of the simulation to approximately 78% as crystallisation proceeds. The qualitative changes with time also follow the pressure. As there are 6 such bond angles per six-membered ring, this implies that even at early times each six-membered ring overlaps with at least two others. Obviously this is an average and many ordered domains will be larger (or smaller) than this. Hence it can be seen that at least instantaneously there 1-224

225 exists a profusion of locally close packed ordered domains in the density quenched fluid that can serve as templates for further growth fraction of overlaps time (τ) Figure 5.18: The fraction of overlaps between the Type 1 and Type 2 bond angles of six-membered rings as a function of time during crystallization Run 1. This fraction is unity in a perfect crystal. The average bond angle across a six-membered ring (Type 3 bond angle, defined in Chapter 2, section 2.5.2) as well as the average for rings composed exclusively of fcc or hcp ordered particles is shown in Figure The averages in this case are taken over all observed rings. In an ideal crystal this angle is 180 while for the stable fcc crystal at a density of 1.05 the mean angle is equal to 170. It can be clearly seen that as the system crystallises this bond angle increases substantially indicating the increasing geometric ordering of these rings. Of particular interest is the fact the average bond angle for those rings that contain only ordered particles (168 ) remains relatively constant through the simulation and is very close to the equilibrium value (170 )

226 The average distance across a six-membered ring (Type B distance, defined in Chapter 2, section 2.5.2) is shown in Figure 5.20 as a function of time for the same crystallisation run. The average distance for those rings composed exclusively of ordered fcc and hcp particles is also shown. In the stable fcc crystal (ρ = 1.05) this distance is equal to 2.21 σ where all six-membered rings are planar (no hcp particles). It can be seen that the average distance is substantially smaller at early times. This will in part be due to the presence of bent six-membered rings due to hcp particles. As the fraction of fcc particles relative to hcp particles increases with time one would expect this distance to increase. But it may also suggest that the initial crystal forms at a higher density than that of the final phase, and that as the crystal grows it expands as the overall pressure drops. This assertion is supported by the change in the mean distance across a six-membered ring (Type B distance) that is composed of only fcc and hcp ordered particles, shown in Figure The average distance for these rings, associated with the growing crystal, is significantly smaller than the overall average all rings fcc/hcp ordered <θ6(3)> time (τ) Figure 5.19: The average bond angle across a six-membered ring (Type 3 bond angle) versus time for crystallization Run 1. The equilibrium value for the face-centered cubic crystal at ρ = 1.05 is 170 while the ideal crystal value is

227 <r 6B > all rings fcc/hcp ordered time (τ) Figure 5.20: Average distance between antipodal pairs of particles that are part of the same six-membered ring versus time for crystallization Run 1. The contribution of these distances to the total radial distribution function is shown at different stages during Run 1 in Figure The distribution evolves from one sharply peaked at a distance slightly larger than 2σ to a gaussian shaped distribution centred on the expected crystal lattice value (tacking into account the compressed density). Note that this peak is visible only as a shoulder in the total g(r) at middle to late times, cf. Figure The torsional angle formed by four consecutive particles about a six-membered ring was calculated and the average absolute value of this angle, averaged over each element of the ring, was used to assess the planarity of the six-membered ring. A value of zero corresponds to a planar or flat ring. The mean torsional angle of the sixmembered rings during the crystallisation process is shown for a representative simulation in Figure Also shown is the torsional angle for those rings composed exclusively of ordered fcc particles. The average torsional angle drops substantially as crystallisation proceeds and approaches a value of approximately 22 by the end of the simulation, still quite far from the equilibrium crystal value of 9.9 at ρ = 1.05, but 1-227

228 still a factor of 2 drop from the fluid phase. The torsional angle of the fcc and hcp ordered rings changes little through the simulation with a final value of approximately 12 reasonably close to the equilibrium value. The geometric changes in the six-membered rings as crystallisations proceeds indicates that there is an increasing coherence between rings, and that the rings flatten out and approach crystal values as the system orders. The fact that the two angular terms for the six-membered rings composed of ordered particles changes little during the simulation while the distances across the ring change substantially again suggest that the crystal is initially formed at a higher density. Note that for all three quantities their behaviour with time mirrors that of the compressibility factor with a qualitative change in the time dependence of the average bond angles and distances at the crossover time of 170 τ (for Run 1) g(r*) r* (r/σ) Figure 5.21: Distribution of distances across a six-membered ring (Type 2 distances) at different times (initial fluid, growth stage and final crystal) during Run

229 all rings fcc/hcp ordered <τ6> time (t) Figure 5.22: The average torsional angle, τ 6, of a six-membered ring as a function of time for crystallization Run Spherical Harmonics A common method of assessing the overall degree of crystallinity in a system is by monitoring the spherical harmonic Q 6 (defined in Chapter 2), Figure The equilibrium crystalline value for a perfect fcc crystal is equal to With the exception of Runs 5 and 7 there is not a close correspondence between changes in the value of Q 6 and the total crystal fraction. If the ratio of the value of Q 6 to the equilibrium crystal value is used as an estimate of the crystal fraction then it is found that the value of Q 6 underestimates the latter by between 10 to 40% depending on the simulation, with the best being with the runs 5 and 7. These two runs also have the highest fraction of hard spheres in an fcc environment as well as the largest difference between the fraction of fcc to hcp particles

230 0.45 Q Run 1 Run 3 Run 4 Run 5 Run 6 Run time (τ) Figure 5.23: Variation of Q 6 spherical harmonic with time for different crystallization runs. By comparing the results of different runs at the end of the simulation it is also found that the value of Q 6 does not correlate well with the total fraction of crystal. If an attempt is made to correlate the final Q 6 value of a given simulation run with the various fraction s of ordered particles then it is found that the closest correspondence is, surprisingly, with the fraction of twisted icosahedrons. The reason for this will be discussed latter. The final value of Q 6 is also sensitive to both the fraction of particles in an fcc environment and that fraction relative to the fraction of hcp spheres. As the spherical harmonic Q 6 measures the global degree of orientational order it is sensitive to the existence of stacking faults and multiple crystal nuclei in the system. A local measure of crystallinity is the local orientational bond order parameter described in Chapter 2. In particular attention is focused on the l = 6 or hexagonal symmetry parameter as the value of this parameter is non-zero for a number of closepacked and icosahedral structures

231 In the spherical harmonic method of van Duijneveldt et al 13 a particle is regarded as ordered if it is connected to a certain minimum number of its neighbours. By connected is meant that the spherical harmonics of the particle is in phase with its neighbour i.e. in some sense the two particles possess a certain degree of coherence in the orientation of their respective bonds i.e. there is a certain degree of orientational order between the two particles. The degree of phase coherence is measured by the angle between the spherical harmonic vectors of two particles where the components of the vector are the m = (2l+1) values of the l = 6 local orientational order parameter q 6m (i) of the particle. If this angle is less than 60 (their dot product is greater than 0.5) then the two particles are regarded as connected or in phase. The geometric origin of these phase relations is somewhat obscure. In order to explore the reasons why this method appears to work so well the distribution of these angles has been examined in detail for a number of the crystallised systems. The distribution of the spherical harmonic angles (θ q6 ) between neighbouring particles was calculated at the end of each simulation and the results for Run 7, the most fully crystallised simulation run is shown in Figure The actual angle plotted is the difference from 180 of this angle, for reasons that will become apparent latter. It can be seen that the increase in orientational order is most pronounced in the time region between τ, the start of the growth stage for this simulation. The distribution is double peaked and the cause of this is the existence of both hcp and fcc ordered particles as can be seen in Figure 5.25, where the contributions from these two ordered environments has been calculated. At early times the height of each peak is similar, with the fcc peak dominating for times greater than 140τ where the fraction of fcc particles dominates over hcp particles. Note that in the stable crystal only the larger peak at is 165 is observed

232 frequency θ q6 Figure 5.24: The distribution of the angle between the local orientational order parameters of neighboring (bonded) particles at various times during crystallization run 7 (listed in the legend in units of τ) hcp fcc frequency θ q6 Figure 5.25: Decomposition of the distribution of angles between the local orientational order parameters of neighboring (bonded) particles at the end of simulation run 7 (t = 300τ) into contributions from fcc and hcp particles

233 0.006 frequency θ Figure 5.26: The distribution of bond angles formed by the central particle of a six-membered ring and two antipodal particles of the ring (Type 3 bond angles). The simplest reason why the fcc particles display a small peak at the lower angle of 144 is that a small fraction of their neighbours are hexagonally ordered and hence have a different orientational order. They are both ordered but their bonds will not be completely in phase. Note that the larger of the two peaks for the hcp particle is peaked at about 160 and for the fcc particles at 166 while the smaller peak is at 144. An intriguing aspect of this distribution is that the two peaks are located at the approximate angles of 144 and 165. The distribution of bond angles across a sixmembered ring (Type 3 angles) is also peaked close to these two values, and this is shown in Figure 5.26 for the same times and simulation run. In particular for the sixmembered rings described in this work, planar rings such as those associated with fcc particles, are peaked at 167 for the stable crystal at a density of 1.05, while the bent six-membered rings associated with hcp particles have an ideal bond angle of 144. This distribution of bond angles is peaked at 145 and 170 suggestively close to the values obtained above from the spherical harmonic angle

234 The above correspondence suggests a simple and sensible origin for the apparent utility of the local order orientational order parameters of ten-wolde. The order parameter has a strong empirical character and is in practice calibrated with respect to the stable crystal. It describes a particle as being solid-like dependent on the number of neighbours it is orientationally ordered with. The degree to which a particle is orientationally ordered or in phase with a neighbour is quantified by taking the dot product between the complete local l = 6 spherical harmonics of a particle and its neighbour. Two particles are regarded as connected if the value of the dot product exceeds The physical meaning of a 13 dimensional dot product is, to put it mildly, somewhat obscure. Nevertheless we find a strong correlation between the distribution of angles so found for this dot product and the distribution of bond angles formed across a six-membered ring i.e. the bond angle formed by the central particle of the six-membered rings defined in this work and pairs of particles on opposite sides of the ring. Hence we suggest that the local bond orientational order parameters measure both the size and geometric ordering of two-dimensional hexagonally ordered domains - precursors to close packed planes and when appropriately registered a close packed crystal. That there should be a correspondence is not altogether surprising. The l = 6 spherical harmonics measure hexagonal order while the six-membered rings describe a six-fold arrangement of particles. The difference is that our definition is topological in nature rather than geometric. It is this broader approach that has enabled us to suggest a structural origin of these bond-order parameters. Our results also suggest a reason why the global orientational spherical order parameter (Q 6 ) has been found to be large for all close packed structures as well as the icosahedral structure. In all these structures we observe planar, bent or buckled sixmembered rings. An icosahedron is the central particle of ten buckled six-membered rings, while a particle in face-centred cubic environment is the central particle of four planar six-membered rings. We conjecture that the different non-zero values of the spherical harmonics for these structures are related to the actual number and geometric shape of their associated six-membered rings

235 5.6 Density of Crystal and Volume Fluctuations Crystal density from lattice parameter The density of the crystal was calculated from the mean value of the distance across those four-membered rings whose members were either fcc of hcp ordered. In an fcc lattice this distance a is equal to the lattice spacing and is related to the crystal density by ρ = 4/a 3. The results indicate that the initial crystal forms at a density of 1.12 ±0.01 (φ =0.59 ± 0.005) at early times (50-100τ) and then as the system crystallizes this density decreases towards the equilibrium value (ρ = 1.05). Also shown is the mean value of the density versus time obtained as the average all simulation runs. This is consistent with the common value of the pressure during the induction period (start of stage 2) and indeed suggests that the reason for the compressed density is the large pressure of the surrounding fluid. In the case of the most fully crystallized simulation runs (5 and 7) the final density is 1.054, slightly above the equilibrium value of The small number of ordered particles at the beginning of the simulation is the reason for the large fluctuations at early times. Nevertheless by averaging the results from each simulation a reliable density can be obtained. That the crystal forms at a higher density than the equilibrium value is also emphasized by the similarity between the pressure and density changes with time. By comparing Figure 5.27 and Figure 5.1 it can be seen that the lower the final pressure the lower the crystal density is. Also at a time of ~ 170 τ both the pressure and crystal density are identical for the same simulation runs (1,3,5,7). The same analysis was conducted on the coexistence simulation and it was found that the crystal density was yet again higher than the equilibrium value of by 1%, Figure At early times the initial crystal density is estimated as 1.11 (φ =0.58). The time dependence of the density follows that of the overall pressure

236 1.15 crystal density Run 1 Run 3 Run 5 Run 6 mean time (τ) Figure 5.27: Crystal density for selected runs as measured from those fourmembered rings composed of fcc and hcp ordered hard spheres. The results for Run 4 are virtually identical to those of Run 1and the results of Run 7 are also virtually identical to those of Run crystal density time (τ) Figure 5.28: The crystal density versus time for the coexistence run. The density is derived from the average distance across a 4-membered ring composed of ordered fcc or hcp particles. The equilibrium number density of the stable crystal is

237 5.6.2 Density fluctuations from Voronoi Volume Distribution The volume of the Voronoi cell surrounding an particle can be used to give an estimate of the local density in the system. As each Voronoi cell contains one particle and the Voronoi volumes add to the volume of the system the range of Voronoi volumes the hard spheres possess during crystallisation is a measure of local density fluctuations. The distribution of Voronoi volumes was calculated for each simulation as a function of time and the variance of the volume distribution for a number of simulation runs is shown in Figure 5.29 as a function of time Run Run 3 Run 4 Run 5 σ 2 V time (τ) Figure 5.29: Variance of distribution of Voronoi volumes, σ 2 V, during different crystallization runs. Note that the results for Run 6 are virtually indistinguishable from those of Run 4 while the results for Run 7 and Run 5 are also very similar to each other. The equilibrium value for the variance of the Voronoi volume of the stable crystal at 1.05 is , the minimum value of the plotted variance scale

238 It can be seen that for times less than approximately 75 τ the volume variance for each simulation is identical and that as crystallisation proceeds the variance increases. At a time that is commensurate with the crossover times given in Table 5.3 the volume variance reaches a maximum value and then decreases. Note that at this crossover time the fraction of crystal is approximately 50%. The equilibrium value of the volume variance for the stable crystal at 1.05 is equal to and corresponds to the minimum value of the variance scale in Figure The interpretation of these results is straightforward. As has already been shown the density of the initial crystal is higher than that of the surrounding fluid. As the crystal is more compact than the fluid this will lead to an increase in the volume available for the remaining fluid particles. The combination of these effects will lead to an increase in the variance of the distribution of local i.e. Voronoi volumes. At the same time the crystal itself has a smaller range of local volume fluctuations than the fluid so as crystallisation proceeds and the fraction of crystal increases the variance should at some point stop increasing. Note that as the crystal density decreases with time it also occupies a larger relative volume and this too will reduce the possibility of volume fluctuations in the remaining fluid. The above results show that when the amount of crystal is equal to approximately 50% this maximum is reached. These results for the volume variance have a very simple interpretation if it is assumed that the crystal forms at a higher density than the surrounding fluid. The behaviour of the variance of the local Voronoi volumes lends weight to this conclusion. The variance of the distribution of surface areas of the Voronoi cells of the hard spheres also shows the same qualitative behaviour with time as for the volume variances

239 5.7 Structure and Growth of Crystal Nuclei The following analysis was performed in order to extract the structure of each nucleus and follow its growth. The modified Voronoi method was first used to define the neighbors of each particle. This information was then used to calculate the local environment about each particle in terms of the planar graphs described in Chapter 2. Apart from identifying particles in a well-ordered cubic or icosahedral environment, a certain number of perturbations from these structures were also recorded. A particle was regarded as ordered if either (i) the number of connections it had to neighboring particles, using the spherical harmonic approach, was equal to 10 or more or (ii) the particle was fcc or hcp ordered, as defined in Chapter 2, using the planar graph method. The combination of these two structural measures means that we can make fewer assumptions about what local structures exist in the crystal nucleus. Although, as we shall see, the use of just those ordered and simple defect does easily identify the nuclei, the broader approach of using the spherical harmonic approach does mean that we are not totally dependent on a priori assumptions about the local structure. Having said this the planar graph method is used to identify the actual local environment of each solid-like particle identified by the spherical harmonic method. The inclusion of ordered fcc, hcp and twisted icosahedral particles is motivated by the observation that these environments are occasionally not identified as solid-like by the spherical harmonic method. The problem is that a particle on the surface of a growing crystal may not be regarded as solid-like due to the spherical harmonic definition. This assumes that the particle is coherent with its neighbours and hence depends somewhat on its neighbours being ordered. These ordered structures are included since we observed that they are often located on the surface of the growing nucleus. Each crystal nucleus was identified by performing a cluster analysis on the ordered particles in each configuration. An particle was regarded as part of a cluster if it was bonded to it, as defined by the modified Voronoi definition of a bond. The cluster analysis was performed backwards in time; the number and membership of clusters 1-239

240 was first analyzed at the time at which two or more clusters i.e. nuclei merged together. This was typically found to occur at a time midpoint between the induction and crossover time. Each cluster was linked with that cluster in the previously stored configuration with which it shared most members. In this way the crystal nuclei could be followed back in time allowing an investigation of the structure of the nuclei at early times Nucleation at Coexistence: Size of Nucleus versus time Two compact and well-separated nuclei were observed in the coexistence simulation run and changes in their size and structure were followed in detail. The total fraction of ordered particles (using the spherical harmonic method) as a function of time for the coexistence simulation is compared in Figure 5.30 as well as the contribution to the total from the two observed nuclei. The nucleation time is estimated from the early time behaviour and is found in the same manner as described in Section The induction time was estimated from the change in the number of nuclei versus time using the method described in Section Note that only the crystal fraction data for times less than 175τ after the quench are used as the two nuclei merge at this point. This time is in the middle of the growth regime. The results are summarised in Table 5.4. The induction times are the same for both nuclei and are slightly less than predicted by using the compressibility or crystal fraction data. On the other hand if the data used is for the same time scale then good agreement is obtained. This suggests that a better method of estimating the induction time may be to use the initial growth data not the fastest growing region. Both nuclei appear 20τ after the start of the simulation and have a size of approximately 30 particles. Note that this is after the initial fast drop in the pressure of the system, which lasts for the first 15τ. As for the simulations near the melting density, it was found that the fraction of crystal, identified using either the spherical harmonic of planar ring method, faithfully followed changes in the overall pressure of the system

241 nucleus 1 nucleus 2 total for nuclei All fraction time (τ) Figure 5.30: The total fraction of ordered particles (using the spherical harmonic method) compared with the contributions from the two nuclei observed in the simulation at the coexistence number density of Nucleus 1 is the larger of the two observed nuclei. Table 5.4: Total number of particles for each nucleus at different times during the simulation run within the coexistence region (ρ = 1.03). Nucleus 1 Nucleus 2 Time Size Time Size Nucleation Induction Merger

242 The qualitative change in the number of particles of each nucleus is similar if the data are appropriately scaled, Figure Hence the large difference between the sizes of the nuclei is predominately due to the difference in their initial sizes. The data for both nuclei was fitted to a simple time power law of the form, b N = at, where N is the number of particles in the nucleus, t is time and a and b are constants. Exponents (b) of 3.04 and 2.46 were found for the larger and smaller nucleus, respectively. As we are dealing with an individual nucleus this suggest that the growth is interface rather than diffusion limited i.e. the radius (R) of the crystal increases linearly with time (R~ N 1/3 ) size (nucleus 1) nucleus 1 power law fit nucleus size (nucleus 2) time (τ) 0 Figure 5.31: Total number of particles versus time for two observed nuclei. A power law fit to the data for the larger nuclei is also shown Form of Nuclei at Coexistence The two nuclei observed in the coexistence simulation are shown at a time of 170 τ (corresponding to the time before merger) in Figure 5.32 and Figure Each nucleus was identified using the cluster analysis approach described above. It can be 1-242

243 seen that both nuclei have an overall decahedral (five-fold) morphology. The structure of both nuclei is that of a pentagonal multiply twinned crystal. A decahedral shaped crystal can be considered as being composed of five tetrahedral subunits of face-centered cubic ordered particles joined together along a common edge. This edge is the single five-fold axis of the decahedral crystal. The shared faces are first order twin planes i.e. a single layer of particles arranged in a locally hexagonally close packed environment. The core of both observed nuclei are of this form as can be seen readily in Figure A perfect decahedron possesses 10 (111) faces, 15 edges and 7 vertices. Further twinning can occur on any of these outer faces 15. This face can then become the site for growth of secondary tetrahedral blocks, where the edges of the decahedron become a new five-fold symmetry axis. (a) (b) Figure 5.32: Core of larger observed nucleus at a time of 170 τ for the coexistence run. The darker colored (brown) particles are fcc ordered while the lighter colored (yellow) particles are hcp ordered. The blue (very dark coloured) particle in the center has a twisted icosahedral local environment. The main five-fold symmetry axis of the crystal is orientated into the page. (a) All particles within a radius of 3.3σ from the central particle along the fivefold symmetry axis. (b) All particles within a radius of 6.0σ from the central particle along the fivefold symmetry axis

244 This growth behaviour is observed in both nuclei and is shown in Figure 5.33(a) for the larger nuclei where the hcp layer is also associated with a set of further five-fold symmetry axes - shown as the blue (very dark colored) particles arranged in a pentagonal pattern. In general this twinning plane is found on most, but not all, tetrahedral blocks. One example of the latter is shown in Figure 5.33(b) where the bottom left face is covered with an additional hcp layer rather than an fcc layer. In this case a set of two adjacent twinning planes has formed. Note that further growth about the central region shown in Figure 5.33(b) would produce a Mackay icosahedron except that the extra twinning plane prohibits such a perfect form. A Mackay icosahedron consists of twenty fcc tetrahedral blocks sharing the same vertex (rather than edge as is the case for decahedral particles). (a) (b) Figure 5.33: Core of larger observed nucleus at a time of 170 τ for the coexistence run. (a) All particles within a radius of 7.1σ from the central particle along the fivefold symmetry axis. (b) All particles within a radius of 7.9σ from the central particle along the fivefold symmetry axis

245 The equilibrium shape of a decahedral crystal is often not of the simple type described above, but may be also be bounded by (100) faces as well. This is motivated by both experimental observations and theoretical arguments These faces are parallel to the five-fold axis. A view of the core of the larger nucleus perpendicular to a [100] direction is shown in Figure In this work we find that the outer faces of the larger of the two observed nucleus are bounded by both (111) and (100) faces. The overall form of the nucleus is shown in Figure 5.35 and Figure 5.36 where the two main five-fold axes and the fcc tetrahedral sub-units are also shown. For clarity not all particles of the sub-units have been shown. The decahedral shape of these nuclei is quite remarkable. Figure 5.34: A side view of the same cluster of particles shown in Figure 5.33(b) displaying the (100) faces of the decahedral core

246 Figure 5.35: A view of all twisted icosahedral particles and the five fcc tetrahedron within the central core of the nucleus at a time 170τ after the start of the simulation. The two lightly shaded particles (yellow) are located at the center of their respective five five-fold axes. The views in previous figures are along the main symmetry axis in the middle of the diagram. The secondary five-fold axis is located to the right. Figure 5.36: (a) Packing of fcc tetrahedral about final dominant symmetry axis of larger nucleus, looking down and to the side of the main symmetry axis. (b) Packing of tetrahedral about secondary symmetry axis

247 The smaller nucleus is shown in Figure 5.37 and again it has the same character as the larger nucleus. The main difference between the two observed nuclei is simply the size of the face-centered cubic tetrahedral blocks. For the larger nucleus the triangular base of the block is 11 particles long while for the smaller nucleus it is 4 particles long. Each face of these blocks is covered with a hcp layer. This is observed for both nuclei and defines the size of the five central fcc tetrahedral blocks. We also observe the formation of secondary tetrahedral sub-units about five-fold axes aligned predominately along the edges of these central twinning planes. (a) (b) Figure 5.37: Nucleus of smaller observed nucleus at a time of 170 τ for the coexistence run. (a) All particles within a distance of 6.0 σ from the particle midpoint along the main symmetry axis. (b) All particles in a twisted icosahedral configuration as well as all fcc particles contained within the main decahedral unit

248 5.7.3 Growth of Nuclei The growth of each nucleus was studied using the cluster methods described previously at the start of section 5.7. The growth of the larger of the two decahedral nuclei observed in the simulation at coexistence will be examined in detail. The form of the nucleus at the induction time of 100τ and the earlier time of 50τ is shown in Figure The size of the nucleus at this earlier time is 300 particles. The orientation is the same as in Figure It can be clearly seen that there is a strong asymmetry in the size of different tetrahedral blocks. The best way to interpret the growth of this type of nucleus is in terms of the growth of tetrahedral blocks of particles in a face-centred cubic local environment. An examination of a sequence of images of the growth of this nucleus supports this contention. Overall the growth pattern is not a simple matter of particle by particle addition to a single preferential growth site but is a much more complicated process. We find in general that growth is not from layer by layer addition to a decahedral core. There are multiple growth sites and we find overall preferences for growth about particular axes that change with time. Stacking faults that inhibit the growth of the tetrahedral blocks are also common in the early stages of growth, as is their occasional annealing out. The form of the crystal at late times suggests that the tetrahedral blocks about the main symmetry axis formed first. In fact growth about this symmetry axis only started after the inhibition of growth about another five-fold axis. A view of the nucleus at the same times as above, but orientated with this initial symmetry axis into the page is shown in Figure 5.39, for the same times as above

249 (a) (b) Figure 5.38: Form of the nucleus at times (a) 100τ and (b) 50τ. The nucleus is orientated as in Figure Note that the presence in the upper left fcc tetrahedral sub-unit of hcp ordered particles is only the outer bounding surface. The interior particles are fcc ordered. The coloring of particles is the same as in Figure

250 (a) (b) Figure 5.39: Form of the nucleus at times (a) 100τ and (b) 50τ. The nucleus is orientated along the initial five-fold symmetry axis. Growth about this symmetry axis is inhibited by the presence of more than one twinning plane, actually three

251 Again the asymmetry of the growth of the nucleus can be clearly seen. The completion of the tetrahedral blocks about this symmetry axis is inhibited by the existence of two sets of triple rather than single twinning planes covering the top half of the two main fcc tetrahedral blocks. The persistence of these defects is highlighted in Figure 5.40 where the nucleus is shown at the merger time orientated with the initial growth axis into the page. Figure 5.40: View along the initial symmetry axis at a time of 170τ of a cut through the center of the nucleus. Only ordered particles are drawn, hence the holes that appear in the top part of the nucleus. Note the complex nature of the orientation of twinning planes. The colors are as in Figure 5.32 except that the very light coloured (grey) particles are solid-like (as defined by the spherical harmonic method) but not categorized as fcc or hcp like

252 The relation between the two symmetry axes is shown in Figure 5.41, where the nucleus is shown at the initial nucleation time of 75τ. The main twinning plane has a triangular form and is bounded by the final dominant five-fold axis (top of Figure 5.41(b)) and the initial five-fold axis (to the left of Figure 5.41(b)). The rapid growth of the crystal nucleus proceeds only after this time, when the main symmetry axis, with correctly located twinning planes has evolved. Therefore the initial stage can be seen as due to a waiting time for the formation of appropriately orientated secondary twinning planes after the formation of a single set of twinning planes about one of the tetrahedral blocks. Growth is then about the five-fold axis that appears later. (a) (b) Figure 5.41: The relation between the initial and dominant final five-fold symmetry axis. The nucleus is shown at the nucleation time of 75τ. (a) Nucleus view along the main final symmetry axis. (b) View of slice through nucleus showing twinning plane containing two five-fold symmetry axes, in the direction indicated by the arrow in (a). The dark colored particles particles arranged in a line are twisted icosahedra ordered and define the five-fold symmetry axes

253 5.7.4 Stacking Probability The fraction of particles in a face-centred environment relative to the total fraction of crystal (i.e. fraction of particles in either an fcc and hcp environment) was calculated for both nuclei as a function of time, Figure By the induction time this stacking probability approaches a constant value for both nuclei with an average value of 0.43 ± 0.02 for the smaller nucleus (labelled 2 ) and 0.56 ± 0.02 for the larger nucleus (labelled 1 ). If only the fraction of ordered fcc and hcp particles are used (excluding those in a defect fcc or hcp environment) the stacking probability behaviour with time is very similar, as are the mean values at late times, 0.59 and 0.43, respectively. Note that due to the larger size of nucleus 1 the stacking probability calculated for the total number of fcc and hcp particles increases with time. At the end of the simulation its value is α Nucleus 1 Nucleus 2 Total time (τ) Figure 5.42: Stacking probability, α, versus time for all ordered particles and for the two observed nuclei

254 For a single stacking direction the stacking probability can be defined as the ratio of particles in an fcc environment to those in either an fcc or hcp environment. In the case of a decahedral particle bounded by twinning planes we have derived the stacking probability as function of the number of sides of a face of the fcc tetrahedral units (n). This is given by, 1 α = Eqn (n 3) /(n)(n 1) For the two nuclei found this gives a stacking probability of 0.57 for the smaller nucleus (n = 4) and (n = 11) for the larger nucleus. A decahedron with n = 3 will be composed completely of particles arranged in a hexagonal close packed environment and hence have a stacking probability of zero. The fact that our nuclei are not perfect means that this is an upper bound to the actual stacking probability due to the formation of additional stacking faults, which is observed in both nuclei. This formula implies that for small nuclei random packing is preferred. The size of a decahedron formed by n-sided fcc tetrahedrons bounded by twinning planes is equal to, 5 2 N = n(n + 12n 7) Eqn which for n = 4 (the size for our smaller nucleus) gives 190 particles. Note that this size is in the range of critical nucleus sizes predicted by direct free energy calculations of the nucleation barrier using classical nucleation theory Close Packed Planes In this section we describe a more general method of describing the ordering in a crystallising system which entails using the six-membered rings to search for ordered close packed planes. This approach is taken in order to better understand the initial fluctuation that produces the nuclei that we have observed

255 As described in Chapter 2 a close packed plane can be considered as being composed of a set of overlapping six-membered rings, of the type described in this section. What we wish to do is extract close packed planes from the early crystallization stage by looking for appropriate overlapping six-membered rings. The rings defined in this work include some that are not flat e.g. the three bent rings associated with particles in an hcp environment (see Chapter 2). A set of criteria has been developed so that only those rings that are planar are used to search for close packed planes. The following approach was found to be useful. Only rings that are associated with particles that had at least seven connections to neighboring particles, as defined by the spherical harmonic method, were considered. This we feel is broad enough to investigate partial ordering around the growing nucleus but not so broad that fluctuations in the fluid will also be included. The second criterion relates to the registry between neighboring planes, albeit in an indirect manner. A six-membered ring generally divides the neighbours of the central particle of the ring (those not part of the ring) into two disconnected groups. We only consider rings that divide the neighbours into two groups such that there are between 3 and 4 neighbours on each side of the ring and the number of bonds between these neighbours is equal to or greater than the number of neighbours. Although we wish to look for close packed planes we are still interested in those with some possible registry with neighboring planes. In Figure 5.43 is shown the twinning planes about the main symmetry axis of the larger nucleus at coexistence, found using the above method, at the induction time (100τ). A number of features can be clearly seen. It can be seen that in general the majority of the close packed layered particles are identified as hcp ordered and also that the majority of the particles that are classed as defect ordered (either fcc or hcp) reside on what would be the surface of the crystal. It can also be seen that beyond the ordered regime there is still some partial hexagonal ordering in to the fluid regime. This is to be expected as the interface between the crystal and fluid is of a finite size. What we have been able to do is extract those particles that display local hexagonal 1-255

256 ordering but reside in the interface between the fluid and solid. This picture is consistent with a degree of decreasing partial ordering as the crystal-fluid interface is transferred form the interior i.e. ordered, defect ordered, locally hexagonally ordered and then fluid. Note also that the layers are not completely triangular but truncated. This reflects the partial (100) faceting of the nuclei. (a) (b) Figure 5.43: Five hexagonally close packed twining planes defining the main symmetry axis of larger nucleus found at coexistence (t = 100τ). The colors are as in previous figures except that the darker colored particles (brown) in (b) are defective hcp. The light colored particles are hcp ordered. The black particles have a twisted icosahedron local environment. (a) The close packed planes found passing through the main symmetry axis are shown and the particles drawn are ordered hcp or twisted icosahedra. (b) Same as (a) except that particles in a defect hcp environment have been shown as well. The form of the larger nucleus just after formation at t = 25τ is found to be a partial tetrahedral subunit of face-centered cubic particles bounded by three hexagonal closepacked planes. Each face of the tetrahedron is a (111) close packed plane and the location of the initial and dominant final five-fold symmetry axis is shown in the schematic diagram, Figure

257 Initial Growth Axis Dominant Final Growth Axis Figure 5.44: Schematic diagram of fcc tetrahedral block of nucleus at t = 25 τ. Each face of the tetrahedron is a (111) face of the crystal nucleus. The shaded face A (ijk) contains both the initial and dominant final five-fold symmetry axes. The back face is denoted B (ikl), the top face by C (ijl), while the front face is denoted D (jkl). The actual form of the nucleus is shown in Figure 5.45 parallel to two different set of (111) planes indicated in Figure The majority of particles can be classified as being in either an ordered fcc or hcp environment or one of the simple defect fcc or hcp environments found in the stable crystal (and listed in Chapter 2). No fourth bounding hcp plane is observed (which would be parallel to plane D in Figure 5.44) rather these hcp planes intersect a common fcc plane. In order to shed more light on the local environment about the nucleus the close packed planes observed at this time (t = 25τ) were extracted from the configuration using the aforementioned approach and are shown in Figure All close packed planes that either intersected one another or were parallel to one other were grouped together. The largest cluster of planes contains the particles making up the largest nuclei observed in the simulation. Note the evident curvature in each of the planes

258 (a) (b) Figure 5.45: The form of the larger nucleus (coexistence) at t = 25τ. The orientation is the same as in Figure 5.39, along the initial five-fold axis (The view is parallel to the plane A in Figure 45); (b) A second view orientated parallel to one of the other set of fcc planes (Plane B in Figure 45). The hcp plane to the left is parallel to plane B in Figure 45 while the top hcp plane is parallel to plane C in Figure 45. (a) (b) Figure 5.46: Coherent close-paced planes parallel to face B of the fcc tetrahedral nucleus

259 Only planes containing more than 30 particles were recorded - hence the missing top two small planes. The close packed planes associated with the hcp bounding planes are shown in Figure 5.47 and it can be seen that close packed ordering extends beyond that defined by the particles, but yet only to their neighbours. This suggests a fairly sharp interface between the ordered nucleus and the surrounding fluid. Figure 5.47: The bond network of the three hcp planes bounding the fcc tetrahedral unit of the larger nucleus at t = 25τ. In the left figure those particles that are solid-like (as defined by the spherical harmonic method) have been drawn

260 The smaller nucleus could not be identified at this time (t = 25τ) and indeed the form of the nucleus at this stage from the cluster analysis is a ramified structure. This does not mean to imply that the nucleus actually is ramified - this is simply a limitation of the cluster analysis approach. We have observed the spontaneous appearance of compact crystalline nuclei in a hard sphere system within the coexistence region. The observed nuclei have a decahedral shape similar to many crystallites observed in vapor condensation and metal clusters 15. The formation of these nuclei is not from a decahedral core; rather each tetrahedral block appears separately, one after the other. The initial nucleus is a partial octahedral crystal bounded by twinning (111) faces. Growth is about triangular twinning planes or more specifically the axis defined by the sides of these planes. The difficulty of analyzing the growth of such a nucleus, and concurrently the necessity of reliable structural measures of the local environment, is exemplified by the three-step growth pattern of the larger nucleus. Growth is first about one axis which then switches to a second when it is inhibited by stacking faults. After the completion of a single decahedra growth continues on the twinned surface of a number of the outer faces of the main decahedral crystal Nucleation near the melting density Nucleation near the melting density was found to be qualitatively different from the coexistence case. In particular we observed multiple nuclei appearing at early times that were undoubtedly interacting with one another due to their close proximity. In fact applying the cluster analysis approach to the data revealed that even at early times (around the induction time) a single cluster was found to be composed of several distinct nuclei, Figure Nevertheless the cluster approach could still be applied to find the dominant nucleus and the initial fluctuation, although even at quite early times (50τ) the nuclei are not independent. In order to see if a more restrictive definition of ordering could disconnect the individual clusters a stronger definition of solid-like particle was used; a particle was 1-260

261 regarded as solid-like only if it had 12 connections to neighboring particles. As the mean number of neighbours is also 12 we are focusing only on these particles that are completely in phase with their neighbours i.e. orientationally correlated with all of its neighbours. It was found that fluctuations in the fluid still conspired to produce on average a single linking cluster. Nucleation cannot be regarded at this density, near melting, as due to independent non-interacting fluctuations, even at very early times. Figure 5.48: A connected cluster containing the dominant partial decahedral nucleus and several smaller fluctuations for Run 1 at the induction time (75τ). Only the fcc and hcp particles described by the planar graph method are shown

262 (a) (b). Figure 5.49: (a) The nucleus at 50τ for Run 1, displaying a central tetrahedral block of fcc ordered particles bounded by twinning planes. The other block of fcc to the top right is displaced behind the central nucleus but is still within a few diameters of the edge of the nucleus. (b) The main five fold symmetry axes at the midpoint of the growth regime (t = 120τ) for Run 1 The form of the dominant nucleus or nuclei observed in each simulation at the midpoint of the growth regime (times given in Table 5.3) is summarized in Table 5.5, including the simulation at coexistence. A wide range of qualitatively different nuclei has been observed in the crystallization runs near the melting density. A particularly common form of nucleus is multiply twinned particles, as was observed for the coexistence case. The simulations with a significant number of multiply twinned particles (MTP) or a single large MTP were also those that possessed a small value of the bond orientational order parameter,q 6, Runs 1 and 6. The reason for the poor correlation between the fraction of crystal and the magnitude of Q 6 can now be seen to be due to the lack of global orientational order, which this order parameter is sensitive to, due to the different orientation of the individual tetrahedral crystallites that make up the MTP s. It was also found that the simulation runs 3 and 4, with a Q 6 value at the end of the simulation between that of the MTP runs and that of the similar runs 5 and 7, contained a mixture of MTP s and other nuclei, while the latter runs contained no MTP s. This is also the reason for the correlation between the magnitude of the Q 6 spherical harmonic and the fraction of 1-262

263 particles in a twisted icosahedral environment, as the latter structure is that of particles along the five fold axes of the MTP s. Table 5.5: Dominant Nucleus or Nuclei at Midpoint of Growth Regime. The stacking probability (α) at this time as well as at the end of the simulation is also shown. The crossover time of the growth stage for each simulation is listed in Table 5.3. MTP means a multiply twinned particle, an ideal version of which is a perfect decahedron. SF refers to stacking faults. Stacking Probability (α) Simulation Run Dominant Nuclei Crossover Time Final 1 MTP MTP and Planar SF MTP and Planar SF Twin Lamellae MTP Twin Lamellae Mean (ρ = 1.05) Coexistence MTP 0.56 The average stacking probability found at the mean crossover time for the near melting density simulations is The overall stacking probability depends on a number of factors. For ideal decahedral nuclei the packing fraction is greater than 0.50 for small nuclei and increases slowly with the radius of the nuclei. In practice the decahedral nuclei we observed are not perfect and show an enhanced preference for hcp ordered particles. Hence the stacking probability can be less than random close packing. If only decahedral nuclei are present the actual stacking probability will depend on both the radii of the crystal nuclei present and their relative frequency. The stacking probability for the other main type of nuclei, the twinned lamellae form, is greater than 0.5 and hence these nuclei also show a preference for fcc over hcp ordering

264 Unlike the coexistence case we find a diffuse interface separating the crystal from the fluid with significant ordering in the latter. The small stacking probability for Run 6 and Run 4, relative to the other runs is due to the preponderance of small decahedral nuclei, one of which is shown for the former simulation along with the five-fold axis of other decahedra in Figure As growth proceeds in this particular situation a complex network of stacking faults will evolve due to the larger number of small decahedral nuclei that initially form. Therefore it is expected that the annealing out of these defects would be a slow process and hence lead to the observation of a suppression of the growth rate. Figure 5.50: Network of 5-fold axis of decahedron and the particles within a radius 5.5σ of the central axis of one of the MTP s for Run 6 at the crossover time. For systems with a single preferred stacking direction (parallel stacking faults) due to the formation of lamellae structures, it would be expected that the annealing out of defects would be less frustrated and the growth of fcc at the expense of hcp would be easier. In practice one would expect that a mix of the two types of nuclei would appear at the melting density - intersecting stacking faults e.g. MTP and planar stacking faults such as twin lamellae

265 On the other hand in hard sphere colloidal suspensions polydispersity may favor one type of nuclei over another (as may gravity) and hence the stacking probability may depend sensitively on the actual colloid suspension which at least would explain the wide range of stacking probabilities observed experimentally. Figure 5.51: The dominant nucleus for Run 5 at the induction time (t = 105τ) displaying a set of three parallel twining planes. Only the ordered fcc and hcp particles have been drawn for clarity. 5.8 Summary A number of simulations of the crystallization of a hard sphere system at the number densities 1.03 (φ 0.539) and 1.05 (φ 0.549) have been analyzed to investigate both the nucleation process in general and the form of individual nuclei. It is found that the fraction of crystal at both densities studied scales with the pressure of the system, a simple yet important result. As has been found in studies of the crystallization of hard sphere colloids we find that the initial crystal forms at a higher density than the equilibrium value and as it grows it s density decreases. We also find 1-265